Course Materials - Asian G-WADI

G-WADI ASIA
Global Network for Water and Development Information in
Arid and Semi-Arid Regions of Asia
Groundwater Modeling for Arid and Semi-Arid Areas
G-WADI Workshop, Lanzhou, China
11-15 June 2007
COURSE MATERIALS
UNESCO Beijing, New Delhi, Almaty and Tehran Offices
IHP-UNESCO Paris
Asian G-WADI Network
Cold and Arid Regions Environmental and Engineering Research Institute,
Chinese Academy of Sciences
Chapter 1
Workshop Introduction:
Hydrological processes, groundwater recharge and surface
water-groundwater interactions in arid and semi arid areas
Howard S. Wheater
Department of Civil and Environmental Engineering
Imperial College of Science, Technology and Medicine,
London SW7 2BU, UK
1.1 Introduction to the workshop
The management of water resources in arid and semi-arid areas has been
identified by UNESCO as a global priority. In arid and semi-arid areas, water
resources are, by definition, limited, and these scarce resources are under increasing
pressure. Population expansion and economic growth are generating increased
demand for water - for domestic consumption, agriculture and for industry. At the
same time, water resources are under threat. Point and diffuse pollution is increasing,
due to domestic, industrial and agricultural activities, and over-abstraction of
groundwater has associated risks of deterioration of water quality, particularly in
coastal areas where saline water can be drawn into the aquifer. Ecosystems are fragile,
and under threat from groundwater abstractions and the management of surface flows.
Added to these pressures is the uncertain threat of climate change.
The hydrology of arid and semi-arid lands is distinct from that of humid areas,
and raises particular challenges. Rainfall is highly variable, both in time and in space.
Surface flows are generally ephemeral, and runoff is focused on stream (wadi)
channels that provide a source of groundwater recharge through channel bed
infiltration. Vegetation is dynamic, responding to rainfall, and affects runoff and
recharge processes. The available resource, i.e. surface flows and/or groundwater
recharge, is a small component (typically less than 5%) of the water balance.
Scientific understanding of these processes is improving, but still limited, and, due to
sparse populations, extreme climate and limited resources, available data are scarce.
UNESCO has recognized that there is a need to share knowledge and
understanding of the hydrology of arid and semi-arid areas to provide access to
state-of-the art information, and to share experience of water management, with
respect to traditional methods and new technology. A global network, G-WADI, has
been established to achieve this. G-WADI is placing emphasis on the sharing of
information, including new data sources, and providing access to state-of-the-art
modelling tools and case studies, to support integrated water resources management.
In March, 2005, a workshop was held in Roorkee, India, to focus on the
problems of modelling surface water systems. The world’s leading experts were
brought together to provide lectures, case studies and tutorial material to an invited
1
audience from the world’s arid regions. That material has been made available
through the G-WADI web-site, and will shortly be published, with some additional
material, as a book by Cambridge University Press.
This Lanzhou workshop aims to provide the same support for issues of
groundwater modelling in arid and semi-arid areas. In many developed countries in
humid climates, groundwater modelling is well established as a management tool, and
recent research developments, building on the increase in available computing power
to incorporate models within a framework of risk assessment, are rapidly finding their
way into practice. For arid areas, there are particular challenges, associated with
climate and data, and little information that addresses the specific problems of arid
regions. The motivation for this workshop is to provide a guide for researchers and
practitioners to the state of the art of groundwater modelling, as applied to the specific
needs of arid and semi-arid areas. Once again, UNESCO has been able to bring
together a set of the world’s leading experts, and key representatives of the major arid
regions of the world, hosted on the edge of the Gobi desert by our friend and
colleague Prof. Xin Li from Lanzhou, and with the support and assistance of
UNESCO’s Asian Region. We are aiming, with the support and advice from the
workshop participants, to produce a set of workshop outputs that will be of assistance
to practitioners and managers world-wide.
1.2 Groundwater resources, groundwater modelling and the quantification of
recharge.
The traditional development of water resources in arid areas has relied heavily on
the use of groundwater. Groundwater uses natural storage, is spatially-distributed, and
in climates where potential evaporation rates can be of the order of metres per year,
provides protection from the high evaporation losses experienced by surface water
systems. Traditional methods for the exploitation of groundwater have been varied,
including the use of very shallow groundwater in seasonally-replenished river bed
aquifers (as in the sand rivers of Botswana), the channelling of unconfined alluvial
groundwater in afalaj (or qanats) in Oman and Iran, and the use of hand dug wells.
Historically, abstraction rates were limited by the available technology, and rates of
development were low, so that exploitation was generally sustainable.
However, in recent decades, pump capacities have dramatically increased, and
hence agricultural use of water has grown rapidly, while the increasing concentration
of populations in urban areas has meant that large-scale well fields have been
developed for urban water supply. A common picture in arid areas is that groundwater
levels are in rapid decline, and in many instances this is accompanied by decreasing
water quality, particularly in coastal aquifers where saline intrusion is a threat. And
associated with population growth, economic development and increased agricultural
intensification, pollution has also become an increasing problem. The integrated
assessment and management of groundwater resources is essential, so that aquifer
systems can be protected from pollution and over-exploitation. This requires the use
of groundwater models as a decision support tool for groundwater management.
Some of the most difficult aspects of groundwater modelling concern the
2
interaction between surface water and groundwater systems. This is most obviously
the case for the quantification of long-term recharge, which ultimately defines
sustainable yields. Quantification of recharge remains the major challenge for
groundwater development world-wide, but is a particular difficulty in arid areas,
where recharge rates are small – both as a proportion of the water balance, and in
absolute terms. However, more generally, the interactions between surface water and
groundwater systems are important. In arid areas, infiltration from surface water
channels, as a ‘transmission loss’ for surface flows, may be a major component of
groundwater recharge, and there is increasing interest in the active management of
this process to focus recharge (for example in an extensive programme of construction
of ‘recharge dams’ in Northern Oman). Conversely, the discharge of groundwater to
surface water systems can be important in terms of valuable ecosystems. Hence the
main thrust of this Chapter is to review hydrological processes in arid and semi-arid
areas, to provide the context for surface-groundwater interactions, and their analysis
and modelling.
1.3 Hydrological processes in arid areas
Despite the critical importance of water in arid and semi-arid areas, hydrological
data have historically been severely limited. It has been widely stated that the major
limitation of the development of arid zone hydrology is the lack of high quality
observations (McMahon, 1979; Nemec and Rodier, 1979; Pilgrim et al., 1988). There
are many good reasons for this. Populations are usually sparse and economic
resources limited; in addition the climate is harsh and hydrological events infrequent,
but damaging. However, in the general absence of reliable long-term data and
experimental research, there has been a tendency to rely on humid zone experience
and modelling tools, and data from other regions. At best, such results will be highly
inaccurate. At worst, there is a real danger of adopting inappropriate management
solutions which ignore the specific features of dryland response.
Despite the general data limitations, there has been some substantial and
significant progress in development of national data networks and experimental
research. This has given new insights and we can now see with greater clarity the
unique features of arid zone hydrological systems and the nature of the dominant
hydrological processes. This provides an important opportunity to develop
methodologies for flood and water resource management which are appropriate to the
specific hydrological characteristics of arid areas and the associated management
needs, and hence to define priorities for research and hydrological data. The aim here
is to review this progress and the resulting insights, and to consider some of the
implications.
1.3.1 Rainfall
Rainfall is the primary hydrological input, but rainfall in arid and semi-arid areas
is commonly characterized by extremely high spatial and temporal variability. The
temporal variability of point rainfall is well-known. Although most records are of
relatively short length, a few are available from the 19th century. For example, Table
1 presents illustrative data from Muscat (Sultanate of Oman) (Wheater and Bell,
3
1983), which shows that a wet month is one with one or two rain days. Annual
variability is marked and observed daily maxima can exceed annual rainfall totals.
For spatial characteristics, information is much more limited. Until recently,
the major source of detailed data has been from the South West U.S.A., most notably
the two, relatively small, densely instrumented basins of Walnut Gulch, Arizona
(150km2) and Alamogordo Creek, New Mexico (174km2), established in the 1950s
(Osborn et al., 1979). The dominant rainfall for these basins is convective; at Walnut
Gulch 70% of annual rainfall occurs from purely convective cells, or from convective
cells developing along weak, fast-moving cold fronts, and falls in the period July to
September (Osborn and Reynolds, 1963). Raingauge densities were increased at
Walnut Gulch to give improved definition of detailed storm structure and are currently
better than 1 per 2km2. This has shown highly localised rainfall occurrence, with
spatial correlations of storm rainfall of the order of 0.8 at 2km separation, but close to
zero at 15-20km spacing. Osborn et al. (1972) estimated that to observe a correlation
of r2 = 0.9, raingauge spacings of 300-500m would be required.
Recent work has considered some of the implications of the Walnut Gulch data
for hydrological modelling. Michaud and Sorooshian (1994) evaluated problems of
spatial averaging for rainfall-runoff modelling in the context of flood prediction.
Spatial averaging on a 4kmx4km pixel basis (consistent with typical weather radar
resolution) gave an underestimation of intensity and led to a reduction in simulated
runoff of on average 50% of observed peak flows. A sparse network of raingauges (1
per 20km2), representing a typical density of flash flood warning system, gave errors
in simulated peak runoff of 58%. Evidently there are major implications for
hydrological practice, and we will return to this issue, below.
The extent to which this extreme spatial variability is characteristic of other arid
areas has been uncertain. Anecdotal evidence from the Middle East underlay
comments that spatial and temporal variability was extreme (FAO, 1981), but data
from South West Saudi Arabia obtained as part of a five-year intensive study of five
basins (Saudi Arabian Dames and Moore, 1988), undertaken on behalf of the Ministry
of Agriculture and Water, Riyadh, have provided a quantitative basis for assessment.
The five study basins range in area from 456 to 4930 km2 and are located along the
Asir escarpment (Figure 1), three draining to the Red Sea, two to the interior, towards
the Rub al Khali. The mountains have elevations of up to 3000m a.s.l., hence the
basins encompass a wide range of altitude, which is matched by a marked gradient in
annual rainfall, from 30-100mm on the Red Sea coastal plain to up to 450mm at
elevations in excess of 2000m a.s.l.
4
5
38.9
143.0
2.03
78.7
Standard deviation
Max.
Mean number of raindays
Max. daily fall (mm)
63.0
31.2
Mean
Number of years record
Jan.
Monthly rainfall (mm)
64
57.0
1.39
98.6
25.1
19.1
Feb.
62
57.2
1.15
70.4
18.9
13.1
Mar.
63
51.3
0.73
98.3
20.3
8.0
Apr.
61
8.9
0.05
8.89
1.42
0.38
May
61
61.5
0.08
64.0
8.28
1.31
June
60
30.0
0.10
37.1
4.93
0.96
July
61
10.4
0.07
14.7
2.09
0.45
Aug.
61
0.0
0.0
0.0
0.0
0.0
Sept.
Table 1. Summary of Muscat rainfall data (1893 -1959) (after Wheater and Bell, 1983)
60
36.8
0.13
44.5
7.62
2.32
Oct.
61
53.3
0.51
77.2
15.1
7.15
Nov.
60
57.2
1.6
171.2
35.1
22.0
Dec.
Figure 1. Location of Saudi Arabian study basins
The spatial rainfall distributions are described by Wheater et al.(1991a). The
extreme spottiness of the rainfall is illustrated for the 2869km2 Wadi Yiba by the
frequency distributions of the number of gauges at which rainfall was observed given the
occurrence of a catchment rainday (Table 2). Typical inter-gauge spacings were 8-10km,
and on 51% of raindays only one or two raingauges out of 20 experienced rainfall. For
the more widespread events, sub-daily rainfall showed an even more spotty picture than
the daily distribution. An analysis of relative probabilities of rainfall occurrence, defined
as the probability of rainfall occurrence for a given hour at Station B given rainfall at
Station A, gave a mean value of 0.12 for Wadi Yiba, with only 5% of values greater that
0.3. The frequency distribution of rainstorm durations shows a typical occurrence of one
or two-hour duration point rainfalls, and these tend to occur in mid-late afternoon. Thus
rainfall will occur at a few gauges and die out, to be succeeded by rainfall in other
locations. This is illustrated for Wadi Lith in Figure 2, which shows the daily rainfall
totals for the storm of 16th May 1984 (Figure 2a), and the individual hourly depths
(Figure 2b-2e). In general, the storm patterns appear to be consistent with the results from
6
the South West USA and area reduction factors were also generally consistent with results
from that region (Wheater et al., 1989).
Figure 2a Wadi Al-Lith daily rainfall, 16th May 1994
7
G
Figure 2b Wadi Al-Lith hourly rainfall, 16th May 1994
G
8
G
th
Figure 2c Wadi Al-Lith hourly rainfall, 16 May 1994
G
9
G
th
Figure 2d Wadi Al-Lith hourly rainfall, 16 May 1994
G
10
G
th
Figure 2e Wadi Al-Lith hourly rainfall, 16 May 1994
G
11
Table 2 Wadi Yiba raingauge frequencies and associated conditional probabilities for
catchment rainday occurrence
Number of Gauges
Occurrence
Probability
1
88
0.372
2
33
0.141
3
25
0.106
4
18
0.076
5
10
0.042
6
11
0.046
7
13
0.055
8
6
0.026
9
7
0.030
10
5
0.021
11
7
0.030
12
5
0.021
13
3
0.013
14
1
0.004
15
1
0.004
16
1
0.005
17
1
0.004
18
1
0.004
19
0
0.0
20
0
0.0
TOTAL
235
1.000
The effects of elevation were investigated, but no clear relationship could be
identified for intensity or duration. However, a strong relationship was noted between the
frequency of raindays and elevation. It was thus inferred that once rainfall occurred, its
point properties were similar over the catchment, but occurrence was more likely at the
higher elevations. It is interesting to note that a similar result has emerged from an
analysis of rainfall in Yemen (UNDP, 1992), in which it was concluded that daily rainfalls
observed at any location are effectively samples from a population that is independent of
12
position or altitude.
It is dangerous to generalise from samples of limited record length, but it is clear
that most events observed by those networks are characterized by extremely spotty
rainfall, so much so that in the Saudi Arabian basins there were examples of wadi flows
generated from zero observed rainfall. However, there were also some indications of a
small population of more wide-spread rainfalls, which would obviously be of
considerable importance in terms of surface flows and recharge. This reinforces the need
for long-term monitoring of experimental networks to characterise spatial variability.
For some other arid or semi-arid areas, rainfall patterns may be very different. For
example, data from arid New South Wales, Australia have indicated spatially extensive,
low intensity rainfalls (Cordery et al., 1983), and recent research in the Sahelian zone of
Africa has also indicated a predominance of widespread rainfall. This was motivated by
concern to develop improved understanding of land-surface processes for climate studies
and modelling, which led to a detailed (but relatively short-term) international
experimental programme, the HAPEX-Sahel project based on Niamey, Niger (Goutorbe
et al., 1997). Although designed to study land surface/atmosphere interactions, rather than
as an integrated hydrological study, it has given important information. For example,
Lebel et al. (1997) and Lebel and Le Barbe (1997) note that a 100 raingauge network was
installed and report information on the classification of storm types, spatial and temporal
variability of seasonal and event rainfall, and storm movement. 80% of total seasonal
rainfall was found to fall as widespread events which covered at least 70% of the network.
The number of gauges allowed the authors to analyse the uncertainty of estimated areal
rainfall as a function of gauge spacing and rainfall depth.
Recent work in southern Africa (Andersen et al., 1998, Mocke, 1998) has been
concerned with rainfall inputs to hydrological models to investigate the resource potential
of the sand rivers of N.E.Botswana. Here, annual rainfall is of the order of 600mm, and
available rainfall data is spatially sparse, and apparently highly variable, but of poor data
quality. Investigation of the representation of spatial rainfall for distributed water
resource modelling showed that use of convential methods of spatial weighting of
raingauge data, such as Theissen polygons, could give large errors. Large sub-areas had
rainfall defined by a single, possibly inaccurate gauge. A more robust representation
resulted from assuming catchment-average rainfall to fall uniformly, but the resulting
accuracy of simulation was still poor.
1.3.2 Rainfall-runoff processes
The lack of vegetation cover in arid and semi-arid areas removes protection of the
soil from raindrop impact, and soil crusting has been shown to lead to a large reduction in
infiltration capacity for bare soil conditions (Morin and Benyamini, 1977). Hence
infiltration of catchment soils can be limited. In combination with the high intensity, short
duration convective rainfall discussed above, extensive overland flow can be generated.
This overland flow, concentrated by the topography, converges on the wadi channel
network, with the result that a flood flow is generated. However, the runoff generation
13
process due to convective rainfall is likely to be highly localised in space, reflecting the
spottiness of the spatial rainfall fields, and to occur on only part of a catchment, as
illustrated above.
Linkage between inter-annual variability of rainfall, vegetation growth and runoff
production may occur. Our modelling in Botswana suggests that runoff production is
lower in a year which follows a wet year, due to enhanced vegetation cover, which
supports observations reported by Hughes (1995).
Commonly, flood flows move down the channel network as a flood wave, moving
over a bed that is either initially dry or has a small initial flow. Hydrographs are typically
characterised by extremely rapid rise times, of as little as 15-30 minutes (Figure 3).
However, losses from the flood hydrograph through bed infiltration are an important
factor in reducing the flood volume as the flood moves downstream. These transmission
losses dissipate the flood, and obscure the interpretation of observed hydrographs. It is
not uncommon for no flood to be observed at a gauging station, when further upstream a
flood has been generated and lost to bed infiltration.
14
G
Figure 3 Surface water hydrographs, Wadi Ghat 12 May 1984 – observed hydrograph
and unit hydrograph simulation
As noted above, the spotty spatial rainfall patterns observed in Arizona and Saudi
Arabia are extremely difficult, if not impossible, to quantify using conventional densities
of raingauge network. This, taken in conjunction with the flood transmission losses,
means that conventional analysis of rainfall-runoff relationships is problematic, to say the
least. Wheater and Brown (1989) present an analysis of Wadi Ghat, a 597 km2
sub-catchment of wadi Yiba, one of the Saudi Arabian basins discussed above. Areal
rainfall was estimated from 5 raingauges and a classical unit hydrograph analysis was
undertaken. A striking illustration of the ambiguity in observed relationships is the
relationship between observed rainfall depth and runoff volume (Figure 4). Runoff
coefficients ranged from 5.9 to 79.8%, and the greatest runoff volume was apparently
generated by the smallest observed rainfall! Goodrich et al. (1997) show that the
combined effects of limited storm areal coverage and transmission loss give important
15
differences from more humid regions. Whereas generally basins in more humid climates
show increasing linearity with increasing scale, the response of Walnut Gulch becomes
more non-linear with increasing scale. It is argued that this will give significant errors in
application of rainfall depth-area-frequency relationships beyond the typical area of storm
coverage, and that channel routing and transmission loss must be explicitly represented in
watershed modelling.
Figure 4 Storm runoff as a function of rainfall, Wadi Ghat
The transmission losses from the surface water system are a major source of
potential groundwater recharge. The characteristics of the resulting groundwater resource
will depend on the underlying geology, but bed infiltration may generate shallow water
tables, within a few metres of the surface, which can sustain supplies to nomadic people
for a few months (as in the Hesse of the North of South Yemen), or recharge substantial
alluvial aquifers with potential for continuous supply of major towns (as in Northern
Oman and S.W. Saudi Arabia).
The balance between localised recharge from bed infiltration and diffuse recharge
from rainfall infiltration of catchment soils will vary greatly depending on local
circumstances. However, soil moisture data from Saudi Arabia (Macmillan, 1987) and
Arizona (Liu et al., 1995), for example, show that most of the rainfall falling on soils in
arid areas is subsequently lost by evaporation. Methods such as the chloride profile
method (e.g. Bromley et al., 1997) and isotopic analyses (Allison and Hughes, 1978)
16
have been used to quantify the residual percolation to groundwater in arid and semi-arid
areas.
In some circumstances runoff occurs within an internal drainage basin, and fine
deposits can support widespread surface ponding. A well known large-scale example is
the Azraq oasis in N.E. Jordan, but small-scale features (Qaa’s) are widespread in that
area. Small scale examples were found in the HAPEX-Sahel study (Desconnets et al.,
1997). Infiltration from these areas is in general not well understood, but may be
extremely important for aquifer recharge. Desconnets et al. report aquifer recharge of
between 5 and 20% of basin precipitation for valley bottom pools, depending on the
distribution of annual rainfall.
The characteristics of the channel bed infiltration process are discussed in the
following section. However, it is clear that the surface hydrology generating this recharge
is complex and extremely difficult to quantify using conventional methods of analysis.
1.3.3 Wadi bed transmission losses
Wadi bed infiltration has an important effect on flood propagation, but also provides
recharge to alluvial aquifers. The balance between distributed infiltration from rainfall
and wadi bed infiltration is obviously dependant on local conditions, but soil moisture
observations from S.W. Saudi Arabia imply that, at least for frequent events, distributed
infiltration of catchment soils is limited, and that increased near surface soil moisture
levels are subsequently depleted by evaporation. Hence wadi bed infiltration may be the
dominant process of groundwater recharge. As noted above, depending on the local
hydrogeology, alluvial groundwater may be a readily accessible water resource.
Quantification of transmission loss is thus important, but raises a number of difficulties.
One method of determining the hydraulic properties of the wadi alluvium is to
undertake infiltration tests. Infiltrometer experiments give an indication of the saturated
hydraulic conductivity of the surface. However, if an infiltration experiment is combined
with measurement of the vertical distribution of moisture content, for example using a
neutron probe, inverse solution of a numerical model of unsaturated flow can be used to
identify the unsaturated hydraulic conductivity relationships and moisture characteristic
curves. This is illustrated for the Saudi Arabian Five Basins Study by Parissopoulos and
Wheater (1992a).
In practice, spatial heterogeneity will introduce major difficulties to the up-scaling
of point profile measurements. The presence of silt lenses within the alluvium was shown
to have important effects on surface infiltration as well as sub-surface redistribution
(Parissopoulos and Wheater, 1990), and sub-surface heterogeneity is difficult and
expensive to characterise. In a series of two-dimensional numerical experiments it was
shown that “infiltration opportunity time”, i.e. the duration and spatial extent of surface
wetting, was more important than high flow stage in influencing infiltration, that
significant reductions in infiltration occur once hydraulic connection is made with a water
table, and that hysteresis effects were generally small (Parissopoulos and Wheater,
17
1992b). Also sands and gravels appeared effective in restricting evaporation losses from
groundwater (Parissopoulos and Wheater, 1991).
Additional process complexity arises, however. General experience from the Five
Basins Study was that wadi alluvium was highly transmissive, yet observed flood
propagation indicated significantly lower losses than could be inferred from in situ
hydraulic properties, even allowing for sub-surface heterogeneity. Possible causes are air
entrapment, which could restrict infiltration rates, and the unknown effects of bed
mobilisation and possible pore blockage by the heavy sediment loads transmitted under
flood flow conditions.
A commonly observed effect is that in the recession phase of the flow, deposition of
a thin (1-2mm) skin of fine sediment on the wadi bed occurs, which is sufficient to
sustain flow over an unsaturated and transmissive wadi bed. Once the flow has ceased,
this skin dries and breaks up so that the underlying alluvium is exposed for subsequent
flow events. Crerar et al., (1988) observed from laboratory experiments that a thin
continous silt layer was formed at low velocities. At higher velocities no such layer
occurred, as the bed surface was mobilised, but infiltration to the bed was still apparently
inhibited. It was suggested that this could be due to clogging of the top layer of sand due
to silt in the infiltrating water, or formation of a silt layer below the mobile upper part of
the bed.
Further evidence for the heterogeneity of observed response comes from the
observations of Hughes and Sami (1992) from a 39.6 km2 semi-arid catchment in
S.Africa. Soil moisture was monitored by neutron probe following two flow events. At
some locations immediate response (monitored 1day later) occurred throughout the
profile, at others, an immediate response near surface was followed by a delayed response
at depth. Away from the inundated area, delayed response, assumed due to lateral
subsurface transmission, occurred after 21 days.
The overall implication of the above observations is that it is not possible at present
to extrapolate from in-situ point profile hydraulic properties to infer transmission losses
from wadi channels. However, analysis of observed flood flows at different locations can
allow quantification of losses, and studies by Walters (1990) and Jordan (1977), for
example, provide evidence that the rate of loss is linearly related to the volume of surface
discharge.
For S.W. Saudi Arabia, the following relationships were defined:
LOSSL =
4.56 + 0.02216 UPSQ - 2034 SLOPE + 7.34 ANTEC
(s.e. 4.15)
LOSSL =
3.75 × 10-5 UPSQ0.821 SLOPE -0.865 ACWW0.497
(s.e. 0.146 log units (±34%))
LOSSL =
5.7 × 10-5 UPSQ0.968 SLOPE -1.049
(s.e. 0.184 loge units (±44%))
18
Where:
LOSSL
=
Transmission loss rate (1000m3/km)
(O.R. 1.08-87.9)
3)
UPSQ
=
Upstream hydrograph volume (1000m
(O.R. 69-3744)
SLOPE
=
Slope of reach (m/m)
(O.R. 0.001-0.011)
ANTEC
=
Antecedent moisture index
(O.R. 0.10-1.00)
ACWW
=
Active channel width (m)
(O.R. 25-231)
and O.R.
=
Observed range
However, generalisation from limited experience can be misleading. Wheater et al.
(1997) analysed transmission losses between 2 pairs of flow gauges on the Walnut Gulch
catchment for a ten year sequence and found that the simple linear model of
transmission loss as proportional to upstream flow was inadequate. Considering the
relationship:
Vx
V0 (1 D ) x
where Vx is flow volume (m3) at distance x downstream of flow volume V0 and D
rҏ epresents the proportion of flow lost per unit distance, then D was found to decrease
with discharge volume:
D 118.8(V0 ) 0.71
The events examined had a maximum value of average transmission loss of 4076 m3
km-1 in comparison with the estimate of Lane et al. (1971) of 4800-6700 m3 km-1 as an
upper limit of available alluvium storage.
The role of available storage was also discussed by Telvari et al. (1998), with
reference to the Fowler’s Gap catchment in Australia. Runoff plots were used to estimate
runoff production as overland flow for a 4km2 basin. It was inferred that 7000 m3 of
overland flow becomes transmission loss and that once this alluvial storage is satisfied,
approximately two-thirds of overland flow is transmitted downstream.
A similar concept was developed by Andersen et al. (1998) at larger scale for the
sand rivers of Botswana, which have alluvial beds of 20-200m width and 2-20m depth.
Detailed observations of water table response showed that a single major event after a
seven weeks dry period was sufficient to fully satisfy available alluvial storage (the river
bed reached full saturation within 10 hours). No significant drawdown occurred between
subsequent events and significant resource potential remained throughout the dry season.
It was suggested that two sources of transmission loss could be occurring, direct losses to
the bed, limited by available storage, and losses through the banks during flood events.
It can be concluded that transmission loss is complex, that where deep unsaturated
alluvial deposits exist the simple linear model as developed by Jordan (1977) and implicit
in the results of Walters (1990) may be applicable, but that where alluvial storage is
limited, this must be taken into account.
19
1.3.4 Groundwater recharge
While the estimation of groundwater recharge is essential to determine the
sustainable yield of a groundwater resource, it is one of the most difficult hydrological
fluxes to quantify. This is particularly so in arid and semi-arid areas, where values are low,
and in any water balance calculation groundwater recharge is likely to be lost in the
uncertainty of the dominant inputs and outputs (i.e. precipitation and evaporation,
together with another minor component of surface runoff).
It can be seen from the preceding discussion that groundwater recharge in arid and
semi-arid areas is dependent on a complex set of spatial-temporal hydrological
interactions that will be dependent on the characteristics of the local climate, the land
surface properties that determine the balance between infiltration and overland flow, and
the subsurface characteristics. Precipitation events in arid areas are generally infrequent,
but can be extremely intense. Where convective rainfall systems dominate, precipitation
is highly localised in space and time (e.g. the monsoon rainfall of Walnut Gulch, Arizona),
but in other climates more widespread winter rainfall may be important, and may include
snow. Precipitation may infiltrate directly into the ground surface or, if overland flow is
generated, be focussed as surface runoff in a flowing channel. We have seen that for arid
climates and convective rainfall, extensive overland flow can occur, enhanced by reduced
surface soil permeabilities as a result of raindrop impact. The relationship between
surface infiltration and groundwater recharge will depend on the subsurface properties,
antecedent conditions, and the duration and intensity of the precipitation.
1.3.4.1 Groundwater recharge from ephemeral channel flows
The relationship between wadi flow transmission losses and groundwater recharge
will depend on the underlying geology. The effect of lenses of reduced permeability on
the infiltration process has been discussed and illustrated above, but once infiltration has
taken place, the alluvium underlying the wadi bed is effective in minimising evaporation
loss through capillary rise (the coarse structure of alluvial deposits minimises capillary
effects). Thus Hellwig (1973), for example, found that dropping the water table below
60cm in sand with a mean diameter of 0.53mm effectively prevented evaporation losses,
and Sorey and Matlock (1969) reported that measured evaporation rates from streambed
sand were lower than those reported for irrigated soils.
Parrisopoulos and Wheater (1991) combined two-dimensional simulation of
unsaturated wadi-bed response with Deardorff’s (1977) empirical model of bare soil
evaporation to show that evaporation losses were not in general significant for the water
balance or water table response in short-term simulation (i.e. for periods up to 10 days).
However, the influence of vapour diffusion was not explicitly represented, and long term
losses are not well understood. Andersen et al. (1998) show that losses are high when the
alluvial aquifer is fully saturated, but are small once the water table drops below the
surface.
Sorman and Abdulrazzak (1993) provide an analysis of groundwater rise due to
20
transmission loss for an experimental reach in Wadi Tabalah, S.W. Saudi Arabia and
estimate that on average 75% of bed infiltration reaches the water table. There is in
general little information available to relate flood transmission loss to groundwater
recharge, however. The differences between the two are expected to be small, but will
depend on residual moisture stored in the unsaturated zone and its subsequent drying
characteristics. But if water tables approach the surface, relatively large evaporation
losses may occur.
Again, it is tempting to draw over-general conclusions from limited data. In the
study of the sand-rivers of Botswana, referred to above, it was expected that recharge of
the alluvial river beds would involve complex unsaturated zone response. In fact,
observations showed that the first flood of the wet season was sufficient to fully recharge
the alluvial river bed aquifer. This storage was topped up in subsequent floods, and
depleted by evaporation when the water table was near-surface, but in many sections
sufficient water remained throughout the dry season to provide adequate sustainable
water supplies for rural villages. And as noted above, Wheater et al. (1997) showed for
Walnut Gulch and Telvari et al. (1998) for Fraser’s Gap that limited river bed storage
affected transmission loss. It is evident that surface water/groundwater interactions
depend strongly on the local characteristics of the underlying alluvium and the extent of
their connection to, or isolation from, other aquifer systems.
Very recent work at Walnut Gulch (Goodrich et al., 2004) has investigated
ephemeral channel recharge using a range of experimental methods, combined with
modelling. These included a reach water balance method, including estimates of near
channel evapotranspiration losses, geochemical methods, analysis of changes in
groundwater levels and microgravity measurements, and unsaturated zone flow and
temperature analyses. The conclusions were that ephemeral channel losses were
significant as an input to the underlying regional aquifer, and that the range of methods
for recharge estimation agreed within a factor of three (reach water balance methods
giving the higher estimates).
1.3.4.2 Spatial variability of groundwater recharge
The above discussion of groundwater recharge from ephemeral channels highlighted
the role of subsurface properties in determining the relationship between surface
infiltration and groundwater recharge. The same issues apply to the relationships between
infiltration and recharge away from the active channel system, but data to quantify the
process response associated with distributed recharge are commonly lacking.
An important requirement for recharge estimation has arisen in connection with the
proposal for a repository for high level nuclear waste at Yucca Mountain, Nevada, and a
major effort has been made to investigate the hydrological response of the deep
unsaturated zone which would house the repository. Flint et al. (2002) propose that soil
depth is an important control on infiltration. They argue that groundwater recharge (or net
infiltration) will occur when the storage capacity of the soil or rock is exceeded. Hence
where thin soils are underlain by fractured bedrock, a relatively small amount of
21
infiltration is required to saturate the soil and generate ‘net infiltration’, which will occur
at a rate dependent on the bedrock (fracture) permeability. Deeper soils have greater
storage, and provide greater potential to store infiltration that will subsequently be lost as
evaporation. This conceptual understanding was encapsulated in a distributed
hydrological model, incorporating soil depth variability, and the potential for
redistribution through overland flow. The results indicate extreme variability in space and
time, with watershed modelling giving a range from zero to several hundred mm/year,
depending on spatial location. The high values arise due to flow focussing in ephemeral
channels, and subsequent channel bed infiltration. The need to characterise in detail the
response of a deep unsaturated zone, led to the application at Yucca Mountain of a wide
range of alternative methods for recharge estimation. Hence Flint et al. also review results
from methods including analysis of physical data from unsaturated zone profiles of
moisture and heat, and the use of environmental tracers. These methods operate at a range
of spatial scales, and the results support the modelling conclusions that great spatial
variability of recharge can be expected at the site scale, at least for comparable systems of
fractured bedrock.
The Yucca Mountain work illustrates that high spatial variability occurs at the
relatively small spatial scale of an individual site. More commonly, a broader scale of
assessment is needed, and broader scale features become important. For example, Wilson
and Guan (2004) introduce the term ‘mountain-front recharge’ (MFR) to describe the
important role for many aquifer systems in arid or semi-arid areas of the contribution
from mountains to the recharge of aquifers in adjacent basins. They argue that mountains
have higher precipitation, cooler temperatures (and hence less evaporation) and thin soils,
all contributing to greater runoff and recharge, and that in many arid and semi-arid areas
this will dominate over the relatively small contribution to recharge from direct
infiltration of precipitation from the adjacent arid areas.
It is clear from the above discussion that appropriate representation of spatial
variability of precipitation and the subsequent hydrological response is required for the
estimation of recharge at catchment or aquifer scale, and that even at local scale, these
effects are important.
1.4 Hydrological modelling and the representation of rainfall
The preceding discussion illustrates some of the particular characteristics of arid
areas which place special requirements on hydrological modelling, for example for flood,
water resources or groundwater recharge estimation. One evident area of difficulty is
rainfall, especially where convective storms are an important influence. The work of
Michaud and Sorooshian (1994) demonstrated the sensitivity of flood peak simulation to
the spatial resolution of rainfall input. This obviously has disturbing implications for
flood modelling, particularly where data availability is limited to conventional raingauge
densities. Indeed, it appears highly unlikely that suitable raingauge densities will ever be
practicable for routine monitoring. However, the availability of 2km resolution radar data
22
in the USA can provide adequate information and radar could be installed elsewhere for
particular applications. Morin et al. (1995) report results from a radar located at
Ben-Gurion airport in Israel, for example. Where convective rainfall predominates,
rainfall variability is extreme, and raises specific issues of data and modelling. However,
in general, as seen from the discussion of Mountain Front Recharge, the spatial
distribution of precipitation is important.
One way forward for the problems of convective rainfall is to develop an
understanding of the properties of spatial rainfall based on high density experimental
networks and/or radar data, and represent those properties within a spatial rainfall model
for more general application. It is likely that this would have to be done within a
stochastic modelling framework in which equally-likely realisations of spatial rainfall are
produced, possibly conditioned by sparse observations.
Some simple empirical first steps in this direction were taken by Wheater et al.
(1991a,b) for S.W.Saudi Arabia and Wheater et al. (1995) for Oman. In the Saudi Arabian
studies, as noted earlier, raingauge data was available at approximately 10km spacing and
spatial correlation was low. Hence a multi-variate model was developed, assuming
independence of raingauge rainfall. Based on observed distributions,
seasonally-dependent catchment rainday occurrence was simulated, dependent on
whether the preceding day was wet or dry. The number of gauges experiencing rainfall
was then sampled, and the locations selected based on observed occurrences (this allowed
for increased frequency of raindays with increased elevation). Finally, start-times,
durations and hourly intensities were generated. Model performance was compared with
observations. Rainfall from random selections of raingauges was well reproduced, but
when clusters of adjacent gauges were evaluated, a degree of spatial organisation of
occurrence was observed, but not simulated. It was evident that a weak degree of
correlation was present, which should not be neglected. Hence in extension of this
approach to Oman (Wheater et al., 1995), observed spatial distributions were sampled,
with satisfactory results.
However, this multi-variate approach suffers from limitations of raingauge density,
and in general a model in continuous space (and continuous time) is desirable. A family
of stochastic rainfall models of point rainfall was proposed by Rodriguez-Iturbe, Cox and
Isham (1987, 1988) and applied to UK rainfall by Onof and Wheater (1993,1994). The
basic concept is that a Poisson process is used to generate the arrival of storms.
Associated with a storm is the arrival of raincells, of uniform intensity for a given
duration (sampled from specified distributions). The overlapping of these rectangular
pulse cells generates the storm intensity profile in time. These models were shown to
have generally good performance for the UK in reproducing rainfall properties at
different time-scales (from hourly upwards), and extreme values.
Cox and Isham (1988) extended this concept to a model in space and time, whereby
the raincells are circular and arrive in space within a storm region. As before, the
overlapping of cells produces a complex rainfall intensity profile, now in space as well as
23
time. This model has been developed further by Northrop (1998) to include elliptical cells
and storms and is being applied to UK rainfall (Northrop et al., 1999).
Work by Samuel (1999) explored the capability of these models to reproduce the
convective rainfall of Walnut Gulch. In modelling point rainfall, the Bartlett-Lewis
Rectangular Pulse Model was generally slightly superior to other model variants tested.
Table 3 shows representative performance of the model in comparing the hourly statistics
from 500 realisations of July rainfall in comparison with 35 years from one of the Walnut
Gulch gauges (gauge 44), where Mean is the mean hourly rainfall (mm), Var its variance,
ACF1,2,3 the autocorrelations for lags 1,2,3, Pwet the proportion of wet intervals, Mint
the mean storm inter-arrival time (h), Mno the mean number of storms per month, Mdur
the mean storm duration (h).This performance is generally encouraging (although the
mean storm duration is underestimated), and extreme value performance is excellent.
Table 3 Performance of the Bartlett-Lewis Rectangular Pulse Model in representing July
rainfall at gauge 44, Walnut Gulch
Mean
Var
ACF1
ACF2
ACF3
Pwet
Mint
Mno
Mdur
Mode
0.103
1.082
0.193
0.048
0.026
0.032
51.17
14.34
1.68
Data
0.100
0.968
0.174
0.040
0.036
0.042
53.71
13.23
2.38
Work with the spatial-temporal model was only taken to a preliminary stage, but
Figure 5 shows a comparison of observed spatial coverage of rainfall for 25 years of July
data from 81 gauges (for different values of the standard deviation of cell radius) and
Figure 6 the corresponding fit for temporal lag-0 spatial correlation. Again, the results are
encouraging, and there is promise with this approach to address the significant problems
of spatial representation of convective rainfall for hydrological modelling.
24
1
0.9
0.8
0.7
Coverage
0.6
Data
1
2
5
10
0.5
0.4
0.3
0.2
0.1
0
0.75
0.8
0.85
0.9
0.95
1
Frequency
G
Figure 5 Frequency distribution of spatial coverage of Walnut Gulch rainfall.
Observed vs. alternative simulations
1
0.9
0.8
0.7
0.6
Data
1
2
5
10
0.5
0.4
0.3
0.2
0.1
0
0
5000
10000
15000
20000
Distance (m)
Figure 6 Spatial correlation of Walnut Gulch rainfall.
Observed vs. alternative simulations
25
25000
Where spatial rainfall variability is less extreme, and coarser time-scale modelling is
appropriate (e.g. daily rainfall), recent work has developed a set of stochastic rainfall
modelling tools that can be readily applied to represent rainfall, including effects of
location (e.g. topographic effects or rainshadow) and climate variability and change (e.g.
Chandler and Wheater, 2002, Yang et al., 2005). These use Generalised Linear Models
(GLMs) to simulate the occurrence of a rainday, and then the conditional distribution of
daily rainfall depths at selected locations over an area. Although initially developed and
evaluated for humid temperate climates, work is currently underway to evaluate their use
for drier climates, with application to Iran.
1.5 Integrated modelling for groundwater resource evaluation
Appropriate strategies for water resource development must recognise the essential
physical characteristics of the hydrological processes. As noted above, groundwater is a
resource particularly well suited to arid regions. Subsurface storage minimises
evaporation loss and can provide long-term yields from infrequent recharge events. The
recharge of alluvial groundwater systems by ephemeral flows can provide an appropriate
resource, and this has been widely recognised by traditional development, such as the
“afalaj” of Oman and elsewhere. There may, however, be opportunities for augmenting
recharge and more effectively managing these groundwater systems. In any case, it is
essential to quantify the sustainable yield of such systems, for appropriate resource
development.
It has been seen that observations of surface flow are ambiguous, due to upstream
tansmission loss, and do not define the available resource. Similarly, observed
groundwater response does not necessarily indicate upstream recharge. Figure 7 presents
a series of groundwater responses from 1985/86 for Wadi Tabalah which shows a
downstream sequence of wells 3-B-96, -97, -98, -99 and -100 and associated surface
water discharges. It can be seen that there is little evidence of the upstream recharge at
the downstream monitoring point.
In addition, records of surface flows and groundwater levels, coupled with
ill-defined histories of abstraction, are generally insufficient to define long term
variability of the available resource.
26
Figure 7 Longitudinal sequence of wadi alluvium well hydrographs and associated
surface flows, Wadi Tabalah, 1985/6
To capture the variability of rainfall and the effects of transmission loss on surface
flows, a distributed approach is necessary. If groundwater is to be included, integrated
modelling of surface water and groundwater is needed. Examples of distributed surface
water models include KINEROS (Wheater and Bell, 1983, Michaud and Sorooshian,
1994), the model of Sharma (1997, 1998) and a distributed model, INFIL2.0, developed
by the USGS to simulate net infiltration at Yucca Mountain (Flint et al., 2000).
A distributed approach to the integrated modelling of surface and groundwater
response following Wheater et al.(1995) is illustrated in Figure 8. This schematic
framework requires the characterisation of the spatial and temporal variability of rainfall,
distributed infiltration, runoff generation and flow transmission losses, the ensuing
groundwater recharge and groundwater response. This presents some technical
difficulties, although the integration of surface and groundwater modelling allows
maximum use to be made of available information, so that, for example, groundwater
response can feed back information to constrain surface hydrological parameterization. A
distributed approach provides the only feasible method of exploring the internal response
of a catchment to management options.
27
DATA
MODELS
RAINFALL DATA
ANALYSIS
STOCHASTIC SPATIAL
RAINFALL SIMULATION
RAINFALL RUNOFF
CALIBRATION
DISTRIBUTED RAINFALL-RUNOFF
MODEL
WABI BED TRANSMISSION LOSS
GROUNDWATER RECHARGE
GROUNDWATER
CALIBRATION
DISTRIBUTED GROUNDWATER
MODEL
Figure 8 Integrated modelling strategy for water resource evaluation
This integrated modelling approach was developed for Wadi Ghulaji, Sultanate of
Oman, to evaluate options for groundwater recharge management (Wheater et al., 1995).
The catchment, of area 758 km2, drains the southern slopes of Jebal Hajar in the
Sharqiyah region of Northern Oman. Proposals to be evaluated included recharge dams to
attenuate surface flows and provide managed groundwater recharge in key locations. The
modelling framework involved the coupling of a distributed rainfall model, a distributed
water balance model (incorporating rainfall-runoff processes, soil infiltration and wadi
flow transmission losses), and a distributed groundwater model (Figure 9).
28
yhpumhssGnlulyh{vyG
JEBEL PLANE
Water Balance Model
ALLUVIAL PLANE
Water Balance Model
WADI BED
Transmission Lose
Model
nyv|uk~h{lyGtvklsG
Figure 9 Schematic of the distributed water resource model
The representation of rainfall spatial variability presents technical difficulties, since
data are limited. Detailed analysis was undertaken of 19 rain gauges in the Sharqiyah
region, and of six raingauges in the catchment itself. A stochastic multi-variate temporalspatial model was devised for daily rainfall, a modified version of a scheme orgininally
developed by Wheater et al., 1991a, b. The occurrence of catchment rainfall was
determined according to a seasonally-variable first order markov process, conditioned on
rainfall occurrence from the previous day. The number and locations of active raingauges
and the gauge depths were derived by random sampling from observed distributions.
The distributed water balance model represents the catchment as a network of
two-dimensional plane and linear channel elements. Runoff and infiltration from the
planes was simulated using the SCS approach. Wadi flows incorporate a linear
transmission loss algorithm based on work by Jordan (1977) and Walters (1990).
Distributed calibration parameters are shown in Figure 10.
29
Figure 10 Distributed calibration parameters, water balance model
Finally, a groundwater model was developed based on a detailed hydrogeological
investigation which led to a multi-layer representation of uncemented gravels,
weakly/strongly cemented gravels and strongly cemented/fissured gravel/bedrock, using
MODFLOW.
The model was calibrated to the limited flow data available (a single event) (Table
4), and was able to reproduce the distribution of runoff and groundwater recharge within
the catchment through a rational association on loss parameters with topography, geology
and wadi characteristics. Extended synthetic data sequences were then run to investigate
catchment water balances under scenarios of different runoff exceedance probabilities
(20%, 50%, 80%), as in Table 4, and to investigate management options.
Table 4 Annual catchment water balance, simulated scenarios
Rainfall
Evaporation
Groundwater
recharge
Runoff
%Runoff
Wet
88
0.372
12.8
4.0
4.6
Average
33
0.141
11.2
3.5
4.0
Dry
25
0.106
5.5
1.7
3.2
Scenario
30
This example, although based on limited data, is not untypical of the requirements to
evaluate water resource management options in practice, and the methodology can be
seen as providing a generic basis for the assessment of management options. An
integrated modelling approach provides a powerful basis for assimilating available hard
and soft surface and groundwater data in the assessment of recharge, as well as providing
a management tool to explore effects of climate variability and management options.
1.5 Conclusions
This Chapter has attempted to illustrate the hydrological characteristics of arid and
semi-arid areas, which are complex, and generally poorly understood. For many
hydrological applications, including the estimation of groundwater recharge and
surface/groundwater interactions, these characteristics present severe problems for
conventional methods of analysis. Much high quality experimental research is needed to
develop knowledge of spatial rainfall, runoff processes, infiltration and groundwater
recharge, and to understand the role of vegetation and of climate variability on runoff and
recharge processes. Recent data have provided new insights, and there is a need to build
on these to develop appropriate methods for water resource evaluation and management,
and in turn, to define data needs and research priorities.
Groundwater recharge is one of the most difficult fluxes to define, particularly in
arid and semi-arid areas. However, reliable quantification is essential to maximise the
potential of groundwater resources, define long-term sustainable yields and protect
traditional sources. This requires an appropriate conceptual understanding of the
important processes and their spatial variability, and the assimilation of all relevant data,
including surface water and groundwater information. It is argued that distributed
modelling of the integrated surface water/groundwater system is a valuable, if not an
essential tool, and a generic framework to achieve this has been defined, and a simple
application example presented. However, such modelling requires that the dominant
processes are characterised, including rainfall, rainfall-runoff processes, infiltration and
groundwater recharge, and the detailed hydrogeological response of what are often
complex groundwater systems.
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37
by Wolfgang Kinzelbach, Peter Bauer, Tobias Siegfried, and Philip Brunner
Sustainable groundwater management —
problems and scientific tools
Institute for Hydromechanics and Water Resources Management, ETH Zürich, Switzerland
Groundwater is a strategic resource due to its usually
high quality and perennial availability. However,
groundwater management all over the world often lacks
sustainability as evidenced by falling water tables, drying wetlands, increasing sea-water intrusion and general deterioration of water quality, As groundwater cannot be renewed artificially on a large scale, sustainable
management of this resource is vital. A number of scientific tools are available to assist in his task. Three items
are discussed here. They include methods for the determination of groundwater recharge, groundwater modeling including the estimation of its uncertainty, and the
interfacing to the socio-economic field. Generally the
quality of water management work can be largely
enhanced with new tools available, including remote
sensing, digital terrain models, differential GPS, environmental tracers, automatic data collection, modeling
and the coupling of models from different disciplines
water cannot be renewed artificially on a large scale its non-sustainable use is a serious danger. This will be illustrated in the following
with various examples.
Sustainable water management
For an assessment of the ground water situation in a catchment, a
definition of sustainable resource utilization is necessary as a starting
point. In the following we will define it as a set of management practices, which avoids an irreversible or quasi-irreversible damage to
the resource water and the natural resources depending on it such as
soil and ecosystems. Such management allows the resource water to
extend its service, including ecological service, over very long periods of time.
The abstraction from a ground water reservoir should in the
long term not be larger than the long-term average recharge. The
storage property of course allows temporary overpumping. As the
quantities abstracted may be used consumptively (e.g. by evapotranspiration in agriculture) and reduce the downstream flows, sustainable management with respect to quantity requires that abstraction is
limited to a fraction of recharge in order to guarantee a minimum
availability of water in the downstream. These principles are violated
in many aquifers all over the world.
Introduction
Overpumping of an aquifer
Steadily falling ground water levels are an indicator of overpumping. A famous example is the Ogallala aquifer in the United
States. Up to 1990, 162 km3 of water above the natural recharge
were abstracted from this aquifer. Finally, the pumping had to be
reduced drastically as the pumping costs rose to a level which made
irrigation with ground water unfeasible (HPUWC, 1998). In Northern China, Eastern India and North- and South Africa ground water
levels drop at a speed of 1 to 3 m/yr. The abstractions in the Sahara
and in Saudi Arabia empty aquifers, which since the last ice age have
not had any recharge worth mentioning. It is estimated that by 2010
the water reserves of the deeper aquifers of Saudi Arabia will contain
less than one half of what there was in 1998.
The cause of overpumping is in all cases the large-scale irrigation with ground water. The quantities pumped for drinking water
are small in comparison. Globally, the water use in households is
only 8% of the total consumption, while irrigation accounts for 70%.
In arid and semi-arid regions, the proportion of irrigation in total
water use may be as large as 90%.
Long before an aquifer is dried up by overpumping other phenomena occur, which indicate a non-sustainable situation.
Ground water contributes worldwide about 20% of people’s fresh
water. Despite this relatively small proportion its role is important
for two reasons: On the one hand, ground water is well suited for the
supply of drinking water due to its usually high quality. On the other
hand, ground water basins are important long-term storage reservoirs, which in semi-arid and arid countries often constitute the only
perennial water resource. The storage capacity is evident if one compares the volumes of surface and ground water resources. Globally
the volume of fresh water resources in rivers and lakes is about
100,000 km3. With about 10,000,000 km3, the volume of ground
water is two orders of magnitude larger (e.g. Gleick, 1993, Postel et
al., 1996). For sustainable water management, however, the renewal
rate is more relevant, and for this quantity the situation is reversed.
The renewal rate of surface water resources is 30,000 km3/a, that of
ground water only about 3,000 km3/a. Worldwide, about 800 km3 of
ground water are utilized by mankind annually. This number still
looks considerably smaller than the yearly renewal rate. However,
the global comparison does not do justice to the real situation. Average figures hide the fact that of the yearly withdrawal rate about one
quarter is supplied by non-renewable fossil ground water reserves
(Sahagian et al., 1994).
Similarly, in a globally averaged analysis the average pollutant
concentrations of ground water become negligible. The natural balance unit for water resources is the catchment of a river, a spring or
a well. And on that scale things look very different on a case-by-case
basis. Ground water resources are degraded in the long term by overpumping and pollution, and the local and regional utilization of
ground water is impaired or even has to be abandoned. As ground
Consequences of ground water table decline
Large drawdowns of the ground water table or pressure head
lead to increased pumping costs, which will limit pumping rates for
economic reasons. But increased cost is not the only consequence of
drawdowns. Pressure reduction may lead to land subsidence in soft
strata, as happens in Bangkok and Mexico City for example. Long
39
280
before an issue of water supply develops, the stability of buildings
requires the reduction or even halt of ground water abstraction.
If a ground water table is declining too quickly or to a too-deep
level the roots of trees relying on ground water (Phreatophytes) may
not be able to follow, which leads to their dying off. This is critical
in dry areas such as the Sahel, where the loss of the capability of trees
to withstand wind leads to an increase in aeolic soil erosion. In wetlands and swamps, in which the visible water table is often a manifestation of a ground water table above ground, a drawdown will
lead to their drying up.
not be solved by artificial drainage. Of the irrigated 230 Mio. ha agricultural land worldwide around 80 Mio. ha are in one way or the
other afflicted by soil salination.
Soil salination is not irreversible, but the time spans for rehabilitation of the soil can be very long. Soil salination is a problem in all
irrigated areas. Strongly affected countries are Iraq, Egypt, Australia,
the USA, Pakistan and China, to mention only a few.
Scientific methods
Sea water intrusion and upconing of salt water
The implementation of water resources management strategies is in
the end a political and economic question. Still, in the analysis of
sustainability and the search for solutions, modern methods of water
research can make a contribution. Three areas are discussed further:
1. The quantification of ground water recharge
2. The use of ground water models including the estimates of
uncertainty; and
3. The incorporation of economical aspects of sustainability
A special consequence of ground water level decline is shown
in coastal areas. Due to the density difference between fresh water on
the land side and salt water on the sea side a salt water wedge develops, which progresses inland until the pressure equilibrium at the
salt-fresh water interface is reached. Every perturbation of this equilibrium by reducing the fresh water flow will lead to a further progression of the salt water wedge inland, until it eventually reaches
and destroys the pumping wells. Sea water intrusion is notorious
along the coasts of India, Israel, China, Spain and Portugal, to mention just a few. In the Egyptian Nile Delta, the zone where ground
water quality is impaired by sea water, reaches up to 130 km inland.
On islands in the sea the fresh water lens formed by recharge from
precipitation is often used for drinking water supply. In this situation
the drawdown due to pumping can cause a rise of the salt-fresh water
interface (upconing). If the interface reaches the well, the well has to
be abandoned.
Similarly, saline water from salt lakes (Chotts) in Tunisia and
Algeria, in the vicinity of which ground water is pumped, can be
attracted to the wells. A ground water level decline inland can lead to
the rise of saline water from underlying aquifers. This tendency is
seen in Brandenburg, Germany. It can also not be excluded in the
Upper Rhine Valley.
Determination of ground water recharge
rates
Ground water recharge is the most important parameter for sustainability in arid and semi-arid regions. Despite that fact, it is not yet
possible to measure this important hydrological quantity with sufficient accuracy. Recharge is not directly measurable, and indirect
methods introduce various uncertainties. To make things worse, rain
and recharge events in dry climates are of an extremely erratic
nature. Integral methods such as the Wundt method, based on low
flow analysis of rivers are very useful in humid climates, but do not
work in arid zones, where the low flow is usually zero. The method
of hydrological water balance is notoriously inaccurate: In the longterm, average and under neglect of surface runoff recharge is the difference between precipitation and evapotranspiration. As these
quantities are both of the same magnitude and inaccurately known,
their difference is even more inaccurate. Only if long time series of
balance components are available is the balance method feasible, as
in this case systematic errors accumulate, and can be identified as
such.
The methods based on Darcy's law, in which the flow Q through
an aquifer cross-section is computed from the cross-sectional area A,
the hydraulic conductivity K and the hydraulic gradient I as
Degradation of ground water quality
The discussion of seawater intrusion shows, that the definition
of sustainability must include both quantity and quality aspects. The
available resources are diminished by pollution. Sea water intrusion
is certainly the most widely spread pollution of ground water worldwide. It is followed by pollution by nitrates and pesticides from agriculture. In industrial areas, chlorinated hydrocarbons and mineral oil
hydrocarbons are still the main problem.
Ground water pollution is in principle not irreversible. In the
case of sea water intrusion, a diminishing or abandoning of abstractions will in the long term let the system return to the original state.
In other pollution cases eradication of the source will lead to cleaning up within the typical renewal time of the resource. But this time
span may be so long that the aquifer cannot be used for drinking
water purposes for several generations, or alternatively money has to
be spent on the processing of this water to make it suitable for drinking.
The experience of industrialized nations shows that aquifer pollution is a lengthy affair even if active remediation measures are
taken. The remaining option of treating the water to bring it up to
drinking water quality is not available for small and decentralized
ground water users in the third world.
Q = A*K*I
(1)
are very inaccurate, as it is extremely difficult to find a representative effective value on the large scale for the usually abnormally distributed hydraulic conductivity K.
Both methods can easily be a factor of 10 to 50 off. Better estimates can be obtained at least in sand-gravel aquifers using environmental tracers. The classical tracer for recharge estimation is Tritium
from the nuclear bomb experiments in the 1960s. The atmospheric
peak of those years reaches the springs and pumping wells with a
delay. The interpretation of the concentration observed in the well
requires a model for the residence time distribution in the aquifer.
Often, simple black box model concepts are used. In the simplest
case a plug flow is assumed. If more information on the geometry of
the aquifer is available, more realistic residence time distributions
can be derived by means of numerical ground water flow models,
which are able to take into account a more realistic aquifer structure.
It has to be remembered however that the interpretation leads to pore
velocities and not specific fluxes. Both are connected by the effective porosity, a quantity, which in sand gravel aquifers is reasonably
well known and much less variable than the hydraulic conductivity.
In principle, every non-reactive trace substance in the atmosphere with a well-known concentration history and distribution can
Soil salination
A further non-sustainable practice, which must be mentioned
here due to its connection with ground water, is soil salination. It is
caused by too high ground water tables. If in arid climates the ground
water table rises (e.g. due to irrigation) to a distance of 1 to 2 m
below the ground surface evaporation of ground water through capillary rise starts. The dissolved salts precipitate and accumulate in the
topsoil, finally leading to infertility of the land. In regions with very
small gradients and little permeable soil, this problem can usually
December 2003
40
be used in a similar fashion as Tritium. This is of great interest, as
the Tritium peak has lost much of its usefulness by decay. A new
group of tracers are the freons and SF6, the atmospheric concentrations of which have been growing exponentially since the middle of
the last century. The freon concentrations have been stagnating or
declining for the last few years due to the ban on their production.
These molecules are however not a complete replacement for Tritium, as Tritium moves as part of the water molecule, while freons as
gases diffuse much faster through the unsaturated zone and only in
the ground water are they transported with the velocity of the water
in which they are dissolved.
In Southern Botswana we used freon tracers for a crude test. In
this desert-like region one can occasionally find fresh water in the
middle of a very saline environment. The question is now whether
this water has been formed recently or whether it is a just a nest of
fossil fresh water, which would not survive a prolonged pumping
test. Freons were found only in wells in which another method — the
chloride method — which is described below, also indicated relatively large recharge. But while the chloride method cannot say anything about the time of recharge, the presence of freons confirmed
that these aquifers have had recent recharge, and pumping them
makes sense.
In combination with its decay product Helium-3, Tritium can be
employed as a stop watch. The ratio of the concentration of both isotopes allows a direct age determination over a time of up to 40 years.
In the beginning, there is no Helium-3 as long as it can escape as gas
to the atmosphere. As soon as a water drop reaches the saturated
zone, Helium-3 can no longer escape, and is enriched relative to Tritium. This method is independent of source strength. Corrections are
necessary due to oversaturation of ground water with air and possible inputs of Helium from mantle and crust (Schlosser et al., 1989).
Stable isotopes such as oxygen 18 and Deuterium can also yield
information on recharge and relevant processes. Their concentration
in water is determined by phase transitions. In the project in
Botswana we could have, for example, very reliably determined the
proportion of water in a well coming from the nearby river or from
the native ground water as the river water has a clear evaporation signature. Stable isotopes can be employed as mixing tracers as well as
transient tracers. For very old water, carbon 14 can be used as an age
tracer. The progress in mass spectroscopy will in future make many
more tracer groups available for hydrology e.g. isotope ratios of rare
earths or metals.
A method often employed successfully in arid regions is the
chloride method. This method uses the deposition of wind-blown sea
salt aerosol. The resulting chloride concentration in soil water
increases by evaporation and transpiration. If the surface runoff is
negligible, the concentration increase is proportional to the evapotranspiration rate and the recharge rate can be derived from
R = (cP*P+D)/(cB)
Figure 1 10-yearly mean of the difference between precipitation
and evapotranspiration in (mm/a) from remotely sensed data.
year in similar fashion. This can be proved by principal component
analysis. The spatial information from satellite images therefore
already allows a more realistic spatial distribution of recharge than
would be arrived at with the assumption of homogeneity over the
whole region. But the absolute value must be adjusted, and it can be
obtained from an average of all chloride values of the region. That
this procedure makes sense can also be seen from a comparison of
recharge rates obtained from remote sensing data and the recharge
rates obtained from well clusters in the same pixels. These show a
correlation coefficient of about 70%. The correlation can be used to
calibrate the recharge map obtained from remote sensing. It is
encouraging to see that the sub-region with the highest computed
recharge rates is also the region with the highest observed ground
water levels.
Generally, remote sensing, be it from an airplane or a satellite
platform, opens up new avenues for water resources management
especially in large countries with weak infrastructure. Local hydrological research can be interpolated and completed into more areal
data sets. Generally, remote sensing data are always of interest if
they allow reduction of the degrees of freedom of a model by an
objective zonation of the whole region considered. We use satellite
images not only in the determination of precipitation and evapotranspiration; from the vegetation distribution and the distribution of
open water surfaces after the rainy season hints of the spatial distribution of recharge can be abstracted. Multi-spectral satellite images
can show the distribution of salts at the soil surface. For large areas,
the GRACE mission will allow the contribution of soil water balances calculated from variations in the gravitational field of the
earth. New developments in geomatics allow us to obtain accurate
digital terrain models on small scales from laser scanning or
stereophotometric methods and on a large scale from radar satellite
images. In both cases, checkpoints on the surface are required which
can be obtained with differential GPS. Digital terrain models are
important in all cases where the distance to ground water table or a
flooded area at varying water surface elevation play a role. The combination of remote sensing, automatic sensors with data loggers,
GPS, and environmental tracers in combination with the classical
methods of hydrology and hydrogeology allows us to introduce into
water resource studies a new quality, especially in those cases in
which in earlier times lack of infrastructure and poor accessibility
were limiting.
(2)
where cP is the chloride concentration in rain water, P the precipitation and D the dry deposition of chloride. cB is the chloride
concentration below the zero upward flux plane. It can be approximated by the concentration in shallow ground water. The method
requires the long term observation of wet and dry deposition of chloride, and of course there must not be any other source of chloride
such as evaporates in the subsoil.
Contrary to the inaccurate large-scale water balance methods,
the chloride method yields more accurate but very local estimates of
recharge rates. Because of this complementarity a combination of the
two methods is promising. In northern Botswana the difference
between precipitation and evapotranspiration was determined using
satellite images over 10 years (Brunner et al., 2002). This difference
indicated a recharge potential. The precipitation can be obtained
from METEOSAT data, and the evapotranspiration can be estimated
from NOAA-AVHRR data using for example the SEBAL algorithm
(Bastiaanssen et al., 1998). The result (on the basis of 97 utilizable
NOAA images) its shown in Figure 1. While the numerical values
themselves are unreliable the pattern is stable, and shows up every
Ground water modeling
Modeling is indispensable in the field of ground water, as in the geosciences in general, as the accessibility of the objects of study is very
limited. Because of the rather slow reactions of a ground water system to changes in external conditions, a tool for prediction and planning is necessary. Ground water models allow us to bring all available data together into a logical holistic picture on a quantitative
Episodes, Vol. 26, no. 4
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282
Figure 3 Development of chloride concentration in the pumping
wells for different configurations of wells with the same total
abstraction rate.
Despite the widespread use of models, care is advisable. Models are notoriously uncertain, and their prognostic ability is often
overestimated. The necessary input data are usually known only partially or only at certain points, instead of in their spatial distribution.
Lacking input parameters have to be identified during calibration,
which, as a rule, does not yield a unique solution. Ground water
models are usually overparameterized, and good model fits are feasible with different sets of parameters, which in a prognostic application of the model however would yield different results. A responsible use of models acknowledges their uncertainty and takes it into
account when interpreting results. For the estimation of model
uncertainty, several methods are available today, which go far
beyond the classical engineering philosophy of best case-worst case
analysis. The spectrum of stochastic modeling methods contains
error propagation, as do Monte Carlo simulation and many others.
Let us look for example at the often-used formula (1) which
estimates the flow through an aquifer cross-section with Darcy's law.
None of the three factors is known exactly, and would therefore be
considered a stochastic variable with uncertainty. The hydraulic conductivity for example is usually lognormally distributed and even if
we know the value of this variable at several points in the aquifer the
spatial average value estimated from those still allows for a considerable estimation variance. The measured hydraulic gradients are
also not exact numbers, not so much due to the measurement accuracy of the dipper, but due to inaccuracies in the leveling of the well
and temporal variations of heads. Finally there is some leeway in the
cross-sectional area, which might be obtained from a geophysical
method. The product of three stochastic variables is again a stochastic variable into which the uncertainties of the factors propagate. In a
numerical model this process takes place in every cell of the discretized model domain, as every flux across a finite difference cell
for example is also calculated by (1). It is therefore advisable to show
modesty when heading towards prognostic use of models. How a
Figure 2 Computed horizontal and vertical distributions of the
freshwater lens under Weizhou Island (In the vertical section the
upconing caused by the central well is visible).
basis. As the first step, the system with all its dominant processes
must be understood. In this calibration phase observation data are
integrated and parameters identified. Only a calibrated model can be
utilized for predictions. In the prognostic application the strength of
models lies in the ease with which scenarios can be compared to each
other. Finally, the integrative effect a model can develop in a project
should not be underestimated. It forces the participants to undertake
disciplined and goal-oriented cooperation.
An example of a model application done in cooperation with
Chinese colleagues is given in Figure 2 (Li, 1994). A ground water
model is used to study the sustainability of ground water abstraction.
On the island of Weizhou, off the South China coast, an increase of
ground water pumping for the extension of tourism is planned.
About a third of the available recharge is required for the additional
water supply. In principle, shallow or even horizontal wells would be
advisable in order to avoid saltwater upconing. But due to the unfavorable conductivity of the upper strata, the fresh
water lens on Weizhou is best pumped in a deeper
aquifer layer. If the pumping is concentrated in one
well, within a few years, the salinity of the pumped
water becomes unacceptable due to saltwater upconing. If the pumping is distributed over two wells at a
4 km distance, an acceptable final salinity is
reached. It the pumping rate is distributed over four
wells sitting on a square with side length 1 km, the
chloride concentration stays below 200 mg/l all of
the time (Figure 3). The model in this case allows us
to find a technical solution to the problem, as the
total pumping rate stays well below the total
Figure 4 Catchment area of a well: deterministic and stochastic approaches, a)
recharge. If the pumping rate is higher than total
Result of deterministic modeling, b) Unconditional stochastic modeling, c) Stochastic
recharge the best model in the world can not find a
modeling with conditioning by 11 long term observed heads.
pumping strategy that avoids salt water intrusion.
December 2003
42
283
As long as the pumping rate of the n users is smaller than QN,
the difference between recharge and consumptive use will flow out
of the basin to a receiving stream. It is clear that in the long term the
total abstraction cannot surpass the recharge rate. Sooner or later, the
abstraction has to adapt to the availability of water. It is however
wise to get to that equilibrium at an early stage, i.e. at a relatively
high water table. First, the higher water table allows larger shortterm overpumping to overcome a drought, for example. Secondly,
the higher water table reduces the specific pumping cost for all users.
If every user goes for a large drawdown and a deep well, all users
will in the end have to shoulder higher specific pumping costs.
Ground water is a resource which rewards partnership and punishes
egotism. Negri showed (Negri, 1989) how a collective decision in
the common pool problem leads to a better solution than the sum of
all independent, individual decisions. In the soil salination problem,
the situation is similar, only with the opposite sign in the water table
change. All irrigators infiltrate water and cause the ground water
table rise. At a depth to water table of less than 2 m, the already
saline ground water starts to evaporate, leading to rapid salinization.
Collective self-restraint will again lead to a better solution. The soil
salination problem in the Murrumbidgee Irrigation District in Australia is of this type. The irrigation of rice leads to considerable percolation, and results in ground water table rise. If the present practices are continued, up to another 35% of the arable land could be
salinized before an equilibrium between the input of irrigation water
from the river, the outflow from drains and the output by evaporation
is reached. With model calculations it has been shown (Zoellmann,
2001) that reducing the rice growing area by one half will basically
halt the salination process at present levels. Figure 5 shows a comparison between the ground water table rise after 30 years for the
present rice growing area and for a 50% reduced rice growing area.
In a project of our institute in Xinjiang, China, it was investigated how far the pumping of ground water for irrigation could keep
the water table at a suitable depth in order to prevent salination.
Again, this is primarily an economic problem, as the pumped ground
water costs about 10 times as much as river water, which has been
used exclusively up to now and which has led to considerable ground
water table rise. The choice of the system boundary for optimization
is crucial. If downstream ecological benefits are taken into account,
the higher cost of ground water may be bearable.
The soil salination problem illustrates another basic economic
aspect. If one applies the classic economic principle of optimizing
discounted net benefits, the discount rate plays an important role. At
a discount rate of 5%, costs and benefits in 40 years' time are already
so small that they have no impact on the optimization. The shortterm gain has much more weight. For slow processes such as salination, the time scale of which can be easily 40 years or more, the usual
economic optimization is unable to take those future penalties into
more appropriate stochastic procedure could look like is discussed
with the example of the determination of catchment areas of wells
(Vassolo et al., 1998, Stauffer, et al., 2002).
It is reckless to draw the catchment of a well by a precise line,
as shown in Figure 4a. It will never be possible to determine a catchment with this precision. A stochastic approach calculates all catchments which result from parameter combinations within the uncertainty limits of the parameters (e.g. transmissivities of n zones and
recharge rates of m zones). In the simplest procedure, this is possible
by sampling the parameter combination from the individual distributions. Then, for each set of parameters a catchment is computed,
yielding a set of catchments which can be analyzed for their statistical properties. By superimposing all catchments, every point on the
surface can be assigned a probability of belonging to the real catchment of the well. This distribution is usually rather broad (Figure
4b). It does not yet take into account measured data for state variables such as heads, spring flows or tracer concentrations. Through
this information the distribution can be conditioned. Only the subset
of catchments is selected which is compatible with the observations.
Incorporating 11 long-term observation data of piezometric heads
the distribution of Figure 4b is narrowed considerably (Figure 4c).
This is the typical way we gain knowledge in the earth sciences. In
the beginning, almost everything is conceivable. Measurements
allow us to filter the realistic cases which are compatible with the
data from the set of imaginable cases. This procedure corresponds to
the Bayes principle. From an a priori distribution, an a posteriori distribution is obtained by conditioning with data. Data are valuable if
they lead to a reduction of the width of the stochastic distribution. To
be fair, it has to be mentioned that stochastic modeling needs data to
yield reasonably narrow distributions. These data are however of a
slightly different nature than usual. What is required besides the
usual scarce observation data of state variables are distribution characteristics of parameters.
The usefulness of models need not suffer from the admission
that they are uncertain. The resource engineer is not interested in
every detail of nature; he wants to propose a robust solution which
will function also if the underground does not exactly look as anticipated. For this type of decision the stochastic approach is adequate.
It allows even the quantification of the risk to take a wrong decision.
Socio-economic aspects
Decisions on sustainability take place in the economic-political
sphere. Models are therefore more useful the more they manage to
interface with socio-economic aspects. The aquifer exploitation
problem is a typical common pool problem. Consider the situation of
n users of an aquifer, which receives a total recharge QN.
Figure 5 Modeled groundwater table rise over 30 years in Murrumbidgee Irrigation District, Australia, a) at present rice growing area, b)
at 50% reduction of rice growing area.
Episodes, Vol. 26, no. 4
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284
account. Alternatives to the classical optimization are required
which give a greater value to the future, e.g., by constraints on the
final value of the agricultural land and its salinity. Of course the state
can repair this deficiency by regulations, which enter the optimization problem as constraints.
Brunner, P., M. Eugster, P. Bauer, W. Kinzelbach, Using Remote Sensing to
regionalize local precipitation recharge rates obtained from the Chloride
Method, Journal of Hydrology, accepted, 2003.
Gleick P.H. (editor), 1993. Water in Crisis, A Guide to the World's Fresh
Water Resources, Oxford University Press, New York.
HPUWC, 1998. High Plains Underground Water Conservation District No.1,
The Ogallala Aquifer, http://www.hub.of the.net/hpwd/ogallala.html.
Li, G.-M., 1994. Saltwater Intrusion in Island Aquifers, Dissertation (in Chinese), China University of Geosciences, Wuhan.
Negri, D.H., 1989. The common property aquifer as a differential game.
Water Resources Research 25, 9-15.
Postel S. L., Daily G. C., Ehrlich P. R., 1996. Human Appropriation of
Renewable Fresh Water, SCIENCE, Vol. 271.
Sahagian D.L., Schwartz F. W., Jacobs D.K., 1994. Direct Anthropogenic
Contributions to Sea Level Rise in the Twentieth Century, NATURE,
Vol. 367.
Schlosser, P. Stute M., Sonntag, C. und Münnich, K. O., 1989. Tritiogenic
3He in shallow ground water. Earth Planet. Sci. Lett. 94, p. 245-256.
Stauffer, F., S. Attinger, S. Zimmermann, W. Kinzelbach, 2002. Uncertainty
estimation of well catchments in heterogeneous aquifers, WRR, 38(11),
p. 20.1-20.8.
Vassolo, S., W. Kinzelbach, W. Schäfer, 1998. Determination of a well head
protection zone by stochastic inverse modelling. Journal of Hydrology
206, p. 268-280.
Zoelmann, K., 2001. Nachhaltige Bewässerungslandwirtschaft in semi-ariden Gebieten, Diss. ETH Zürich, Nr. 1399
Conclusions
The sustainable management of aquifers is a burning problem in
many countries. New scientific tools can assist in its solution, both in
the analysis of whether the present management is sustainable and in
the definition of strategies and measures to achieve sustainability.
Models will always play a role in this task. A new model generation
can use new data sources such as environmental tracers, remote sensing data and geophysical data. It must however not only simulate an
average deterministic situation but also analyze the uncertainty of its
predictions by a stochastic approach in order to remain credible.
Finally, the natural and engineering sciences have to interface much
more with economics and politics to be of real practical use.
References
Bastiaanssen, W.G.M., Menenti, M. Feddes, R. A., Holtslag, A. A. M.
(1998). A remote sensing surface energy balance algorithm for land
(SEBAL). 1. Formulation. Journal of Hydrology, 212-213:213-229.
Wolfgang Kinzelbach was born in
Germersheim, Germany, in 1949.
Educated as a physicist he turned to
environmental engineering and
devote his research to flow and
transport phenomena in the aquatic
environment. He received his doctoral degree from Karlsruhe University in 1978. After professorships at
Kassel University and Heidelberg
University he is presently the head
of the Institute of Hydromechanics
and Water Resources Management
at the Swiss Federal Institute of
Technology (ETH) in Zurich. His
research interests are in environmental hydraulics, nuclear waste
disposal and sustainable water
resources management.
Tobias Siegfried studied environmental sciences at Swiss Federal
Institute of Technology (ETH). He
received his master degree for the
development of a groundwater flow
and transport model of a small basin
in Botswana. He deepened his studies at the London School of Economics in the field of International Relations as a postgraduate. In his current PhD work at the Institute of
Hydromechanics
and
Water
Resources Management at ETH, he
is concerned with optimal utilisation
of a large-scale non-renewable
groundwater resource in Northern
Africa.
Philip Brunner studied environmental engineering at the Swiss
Federal Institute of Technology
(ETH). He currently works at the
Institute of Hydromechanics and
Water Resources Management as a
PhD student. His research focuses
on the modelling of the water and
salt fluxes through an agriculturally
used basin in Xinjiang, China. The
final goal of the PhD- project is to
understand and reproduce the
process of salination and to develop
sustainable strategies of water- and
salt management.
Peter Bauer is developing model
concepts for the sustainable water
management in the Okavango Delta,
Botswana. He received his Diploma
at the Swiss Federal Institute of
Technology thesis in 1999.
December 2003
44
Coinceptual Models For Recharge Sequences In Arid And Semi-Arid
Regions Using Isotopic And Geochemical Methods
W.M. Edmunds 1
ABSTRACT
The application of hydrogeochemical techniques in understanding the water
quality problems of semi-arid regions is described and follows the chemical pathway
of water from rainfall through the hydrological cycle. In this way it becomes possible
to focus on the question of modern recharge, how much is occurring and how to
recognise it as well as the main processes controlling water composition. It is vital to
be able to recognise the interface between modern water and palaeowaters as a basis
for sustainable management of water resources in such regions. The relative
advantages of chemical. isotopic and related methods are introduced and it is
emphasized that a conjunctive multitracer approach is usually required. Chemical and
especially isotopic methods help to distinguish groundwater of different generations.
In fact, the development of water quality may be viewed as an evolution series over
time: i) undisturbed evolution under natural conditions; ii) the development phase
with some disturbance of natural conditions, especially stratification; iii) development
with contamination, and iv) artificially managed systems. A number of case studies
from Sudan, Nigeria, Senegal as well as North African counties and China are used to
illustrate the main principles.
INTRODUCTION
Groundwater quality issues now assume major importance in semi-arid and arid
zones, since scarce reserves of groundwater (both renewable and non-renewable) are
under threat due to accelerated development as well as a range of direct human
impacts. In earlier times, settlement and human migration in Africa and the Middle
East region were strictly controlled by the locations and access to fresh water mainly
close to the few major perennial rivers such as the Nile, or to springs and oases,
representing key discharge points of large aquifers that had been replenished during
wetter periods of the Holocene and the Pleistocene. This is a recurring theme in
writings of all the early civilisations where water is valued and revered (Issar 1990).
The memory of the Holocene sea level rise and pluvial periods is indicated in the
story of the Flood (Genesis 7:10) and groundwater quality is referred to in the book of
Exodus (15:20). More recent climate change may also be referred to in the Koran
(Sura 34; 16).
In present times groundwater forms the primary drinking water source in arid
and semi-arid regions, since river flows are unreliable and large freshwater lakes are
1
Oxford Centre for Water Research, School of Geography and the Environment,
South Parks Road,,Oxon. OX1 3TB, UK. E-mail: [email protected].
45
either ephemeral (e.g. Lake Chad) or no longer exist. The rules of use and exploitation
have changed dramatically over the past few decades by the introduction of advanced
drilling technology for groundwater (often available alongside oil exploitation), as
well as the introduction of mechanised pumps. This has raised expectations in a
generation or so of the availability of plentiful groundwater, yet in practice falling
water levels testifies to probable over-development and inadequate scientific
understanding of the resource, or failure of management to act on the scientific
evidence available. In addition to the issues related to quantity, quality issues
exacerbate the situation. This is because the natural groundwater regime established
over long time scales has developed chemical (and age) stratification in response to
recharge over a range of climatic regimes and geological controls. Borehole drilling
cuts through the natural quality layering and abstraction may lead to deterioration in
water quality with time as water is drawn either from lower transmissivity strata, or is
drawn down from the near-surface where saline waters are commonplace.
Investigations of water quality are therefore needed alongside, and even in
advance of widespread exploitation. The application of hydrogeochemical techniques
are needed to fully understand the origins of groundwater, the timing of its recharge
and whether or not modern recharge is occurring at all. The flow regime at the present
day may differ from that at the time of recharge due to major changes in climate from
the original wetter recharge periods; this may affect modelling of groundwater in
semi-arid and arid regions and geochemical investigation may help to create and to
validate suitable models.
Study is needed of the extent of natural layering and zonation as well as chemical
changes taking place along flow lines due to water-rock interaction (predicting for
example if harmful concentrations of certain elements may be present naturally). The
superimposed impacts of human activity and the ability of the aquifer system to act as
a buffer against surface pollution can then be determined against natural baselines.
Typical landscape elements of (semi-)arid regions are shown in Figure 1, using
the example of North Africa, where a contrast is drawn between the infrequent and
small amounts of modern recharge as compared with the huge reserves of
palaeowaters recharged during wetter climates of the Pleistocene and Holocene.
Broad wadi systems and palaeo-lakes testify to former less arid climates. Against this
background the main water quality issues of arid and semi-arid regions can be
considered both for North Africa and other similar regions. Geochemical techniques
can be applied for defining and mitigating several of the key water issues in (semi)arid areas. These issues include salinisation, recharge assessment, residence time
estimation and the definition of natural (baseline) water quality, as a basis for
studying pollution and human-induced changes more generally. In this context it is
noted that the mainly continental sandstones indicated in the diagram are highly
oxidised and that dissolved O2 can persist to considerable depths in many basins.
Under oxidising conditions several elements (Mo, Cr, As) or species such as NO3
remain stable and may persist or build up along flow pathways.
46
Figure 1. Landscape hydrogeology and hydrogeochemical processes in arid and semiarid regions. The generalised cross-section illustrates the geological environment
found in North Africa, where sedimentary basins of different ages overstep each other
unconformably and may be in hydraulic continuity with each other.
This paper describes the application of hydrogeochemical techniques in
conceptualising groundwater quality in semi-arid regions, especially in Africa, and
follows the chemical pathway of water from rainfall through the hydrological cycle.
In this way it becomes possible to focus on the question of modern recharge, how
much is occurring and how to recognise it. It is vital to be able to recognise the
interface between modern water and palaeowaters as a basis for sustainable
management of water resources in such regions. Chemical and especially isotopic
methods can help to distinguish groundwater of different generations and to
understand water quality evolution.
METHODS OF INVESTIGATION
Hydrogeochemical investigations consist of two distinct steps – sampling and
analysis. Special care is needed with sampling because of the need for
representativeness of sample, the question of mixtures of water (especially
groundwaters which may be stratified), the need to filter or not to filter, as well as the
stability of chemical species. For groundwater investigations a large range of tools is
available, both chemical and isotopic, for investigating chemical processes and overall
water quality in (semi-)arid regions. However, it is essential that unstable variables
such as pH, temperature, Eh and DO be measured in the field. Some details on field
approaches are given by Appelo & Postma (1993) and also by Clark & Fritz (1997).
47
Table 1. Principal geochemical tools for studies of water quality in (semi-)arid zones.
Examples and references to the literature are given in the text
Geochemical/
Role in evaluating water quality and salinity
isotopic tool
Cl
Br/Cl
Master variable: Inert tracer in nearly all geochemical processes, use in
recharge estimation and to provide a record of recharge history.
Use to determine geochemical source of Cl.
Cl
Half-life 3.01u105 years. Thermonuclear production - use as tracer of Cl
cycling in shallow groundwater and recharge estimation. Potential value
for dating over long time spans and also for study of long-term recharge
processes. However, in situ production must be known.
Cl/35Cl
Fractionation in some parts of the hydrological cycle, mainly in
saline/hypersaline environments, may allow finger printing.
36
37
Inorganic tracers:
Mg/Ca
Sr, I, etc.
Mo,Cr,As,U,NO3,
Fe2+
Nutrients: (NO3,K,
PO4 – also DOC)
į18O, į 2H
Diagnostic ratio for (modern) sea water.
Diagenetic reactions release incompatible trace elements and may provide
diagnostic indicators of palaeomarine and other palaeowaters.
Metals indicative of oxidising groundwater. Nitrate stable under aerobic
conditions.
Indicative of reducing environments.
Nutrient elements characteristic of irrigation return flows and pollution.
Essential indicators, with Cl, of evaporative enrichment and to quantify
evaporation rates in shallow groundwater environments. Diagnostic
indicators of marine and palaeomarine waters.
Sr,11B
Secondary indicators of groundwater salinity source, especially in
carbonate environments.
į 34S
Indicator for evolution of sea-water sulphate undergoing diagenesis.
Characterisation of evaporite and other SO4 sources of saline waters.
CFC’s, SF6
Recognition of modern recharge: past 50 years. Mainly unreactive. CFC-12, CFC11 and CFC-113 used conjunctively.
87
3
H
14
C, į 13C
Recognition of modern recharge (half-life 12.3 years).
Main tool for dating groundwater. Half-life 5730 years. An understanding
of carbon geochemistry (including use of 13C) is essential to interpretation.
A summary of potential techniques for the hydrogeochemical study of
groundwaters in arid and semi-arid areas is given in Table 1. Chloride can be regarded
as a master variable. It is chemically inert and is therefore conserved in the
groundwater system - in contrast to the water molecule, which is lost or fractionated
during the physical processes of evapotranspiration. The combined use of chloride
and the stable isotopes of water (į18O, į2H) therefore provide a powerful technique
48
for studying the evolution of groundwater salinity as well as recharge/discharge
relationships (Fontes 1980; Clark & Fritz 1997; Coplen et al. 1999). A major
challenge in arid zone investigations is to be able to distinguish saline water of
different origins including saline build-up from rainfall sources, formation waters of
different origins, as well as relict sea water. The Br/Cl ratio is an important tool for
narrowing down different sources of salinity (Rittenhouse 1969; Edmunds 1996;
Davis et al. 1998), discriminating specifically between evaporite, atmospheric and
marine Cl sources. The relative concentrations of reactive tracers, notably the major
inorganic ions, must be well understood, as they provide clues to the water-rock
interactions which give rise to overall groundwater mineralisation. Trace elements
also provide an opportunity to fingerprint water masses; several key elements such as
Li and Sr are useful tools for residence time determinations. Some elements such as
Cr, U, Mo and Fe are indicative of the oxidation status of groundwater. The isotopes
of Cl may also be used: 36Cl to determine the infiltration extent of saline water of
modern origin (Phillips 1999) and į37Cl to determine the origins of chlorine in saline
formation waters. The measurement precision (better than ±0.09‰) makes the use of
chlorine isotopes a potential new tool for studies of environmental salinity (Kaufmann
et al. 1993). In addition, several other isotope ratios: į15N (Heaton 1986), į87Sr
(Yechieli et al. 1992) and į11B (Bassett 1990; Vengosh & Spivak 1999) may be used
to help determine the origins and evolution of salinity. Accumulations of 4He may
also be closely related to crustal salinity distributions (Lehmann et al. 1995).
For most investigations it is likely that conjunctive measurement by a range of
methods is desirable (e.g. chemical and isotopic; inert and reactive tracers). However,
several of the above tools are only available to specialist laboratories and cannot be
widely applied, although it is often found that the results of research using detailed
and multiple tools can often be interpreted and applied so that simple measurements
then become attractive. In many countries advanced techniques are not accessible and
so it is important to stress that basic chemical approaches can be adopted quite
successfully. Thus major ion analysis, if carried out with a high degree of accuracy
and precision, can prove highly effective; the use of Cl mass balance and major
element ratios (especially where normalised to Cl) are powerful investigative
techniques.
TIME SCALES AND PALAEOHYDROLOGY
Isotopic and chemical techniques are diagnostic for time scales of water
movement and recharge, as well as in the reconstruction of past climates when
recharge occurred. In this way the application of hydrogeochemistry can help solve
essentially physical problems and can assist in validating numerical models of
groundwater movement.
A palaeohydrological record for Africa has been built up through a large number
of palaeolimnological and other archives which demonstrate episodic wetter and drier
interludes throughout the late Pleistocene and Holocene (Servant & Servant-Vildary
1980; Gasse, 2000). The late Pleistocene was generally cool and wet although at the
time of the Last Glacial Maximum much of North Africa was arid, related to the much
49
lower sea surface temperatures. The records show, however, that the Holocene was
characterised by a series of abrupt and dramatic hydrological events, mainly related to
monsoon activity, although these events were not always synchronous over the
continent. Evidence (isotopic, chemical and dissolved gas signatures) contained in
dated groundwater forms important and direct proof of the actual occurrence of the
wet periods inferred from the stratigraphic record as well as the possible source,
temperature and mode of recharge of the groundwater. Numerous studies carried out
in North Africa contain pieces of a complex hydrological history.
There is widespread evidence for long-term continuous recharge of the
sedimentary basins of North Africa based on sequential changes in radiocarbon
activities (Gonfiantini et al. 1974; Edmunds & Wright 1979; Sonntag et al. 1980).
These groundwaters are also distinguishable by their stable isotopic composition;
most waters lie on or close to the global meteoric water line (GMWL) but with lighter
compositions than at the present day. This signifies that evaporative enrichment was
relatively unimportant and that recharge temperatures were lower than at present. This
is also supported by noble gas data, which show recharge temperatures at this time
typically some 5-7 oC lower than at present (Fontes et al. 1991; 1993; Edmunds et al.
1999). Following the arid period at the end of the Pleistocene, groundwater evidence
records episodes of significant recharge coincident with the formation of lakes and
large rivers, with significant recharge from around 11 kyr BP. Pluvial episodes with a
duration of 1-2000 years are recorded up until around 5.5 kyr BP, followed by onset
of the aridity of the present day from around 4.5 kyr BP.
In Libya, during exploration studies in the mainly unconfined aquifer of the Surt
Basin, a distinct body of very fresh groundwater (<50 mg L-1) was found to a depth of
100 m which cross-cuts the general NW-SE trend of salinity increase in water dated to
the late Pleistocene (Edmunds & Wright 1979). This feature (Figure 2) is around 10
km in width and may be traced in a roughly NE-SW direction for some 130 km. The
depth to the water table is currently around 30-50 m. Because of the good coverage of
water supply wells for hydrocarbon exploration a three-dimensional impression could
be gained of the water quality. It is clear that this feature is a channel that must have
been formed by recharge from a former ancient river system. No obvious traces of this
river channel were found in the area, which had undergone significant erosion,
although Neolithic artefacts and other remains testify that the region had been settled
in the Holocene. Whereas the regional, more mineralised groundwater gave values of
0.7-5.4% modern carbon, indicating a late Pleistocene age, the fresh water gave
values from 37.6-51.2% modern carbon (corresponding ages ranging from 5000-7800
years) and were also distinctive in their hydrogeochemistry. Evidence from shallow
wells in the vicinity, however, proved that fresh water of probable Holocene age was
also present more widely at the water table, indicating that direct recharge was
simultaneously occurring at a regional scale. Similar channels, preserving the
Holocene/ Pleistocene interface, have been described by Pearson & Swarzenski (1974)
from Kenya. In Mali the records of Holocene floods from northward migration of the
Niger River inland delta are also well recorded (Fontes et al. 1991).
50
Figure 2 Chloride concentration contours in groundwater from the Surt Basin, Libya,
define a freshwater feature cross-cutting the general water quality pattern. This feature
is younger water from a former wadi channel, superimposed on the main body of
palaeowater.
In comparison with the distinctive light isotopic composition of late Pleistocene
recharge which lies close to the GMWL, Holocene groundwater is characterised by
heavier, more evaporated compositions (Edmunds & Wright 1979; Darling et al. 1987;
Dodo & Zuppi 1997). Extrapolation of these light isotopic compositions along lines of
evaporation on the delta diagrams to the GMWL allow identification of the
composition of the parent rains. It is proposed (Fontes et al. 1993) that these
systematically light isotopic compositions were caused by an intensification of the
monsoon coincident with northward movement of the ITCZ, resulting in convective
heavy rains with low condensation temperatures. From the distribution of such
groundwaters over the Sahara it is implied that the ITCZ moved some 500 km to the
north during the Holocene.
51
RAINFALL CHEMISTRY
Information on the conservative chemical components of rainfall is required by
hydrologists to perform chemical mass balance studies of river flow and recharge.
Rainfall may also be considered as the 'titrant' in hydrogeochemical processes, since it
represents the initial solvent in the study of water-rock interaction. A knowledge of
rainfall chemistry can also contribute to our fundamental understanding of air mass
circulation, both from the present day synoptic viewpoint, as well as for past climates. In
fact, very little information is available on rainfall composition. This is particularly true
for Africa, where only limited rainfall chemical data are available. However, a
reasonable understanding of the isotopic evolution of rainfall in (semi-) arid areas is
emerging, for example within the African monsoon using isotopes (Taupin et al. 1997;
2000). The stable isotope (į18O and į2H) composition of precipitation can aid
identification of the origin of the precipitated water vapour and its condensation history,
and hence give an indication of the sources of air masses and the atmospheric circulation
(Rozanski et al. 1993). Several chemical elements in rainfall (notably Cl) behave inertly
on entering the soil and unsaturated zone and may be used as tracers (Herczeg &
Edmunds 1999). The combined geochemical signal may then provide a useful initial
tracer for hydrologists, as well as providing a signal of past climates from information
stored in the saturated and unsaturated zones (Cook et al. 1992; Tyler et al. 1996; Tyler
& Edmunds 2001). In fact, over much of the Sahara-Sahel region rainfall-derived solutes,
following concentration by evapotranspiration, form a significant component of
groundwater mineralisation (Andrews et al. 1994).
Rainfall chemistry can vary considerably in both time and space, especially in
relation to distance from the ocean. This is well demonstrated in temperate latitudes
for North America (Junge & Werby 1957); the basic relationship between decreasing
salinity and inland distance has also been demonstrated for Australia (Hingston &
Gailitis 1976). The chemical data for Africa are very limited and it is not yet possible
to generalise on rainfall chemistry across the continent. The primary source of solutes
is marine aerosols dissolved in precipitation, but compositions may be strongly
modified by inputs from terrestrial dry deposition. Because precipitation originates in
the ocean, its chemical composition near the coast is similar to that of the ocean.
Aerosol solutes are dissolved in the atmospheric moisture through release of marine
aerosols near the sea surface (Winchester & Duce 1967). This initial concentration is
distinctive in retaining most of the chemical signature of sea water, for example the
high Mg/Ca ratio as well as a distinctive ratio of Na/Cl. As rainfall moves inland
towards the interior of continents, sulphate and other ions may increase relative to the
Na and Cl ions.
The more detailed studies of African rainfall have used stable isotopes in relation
to the passage of the monsoon from its origins in the Gulf of Guinea (Taupin et al.
2000) towards the Sahel, where the air masses track east to west in a zone related to
the position and intensity of the Inter-tropical convergence zone (ITCZ). While the
inter-annual variations and weighted mean compositions of rainfall are rather similar,
consistent variations are found at the monthly scale which are related to temperature
52
and relative humidity. This is found from detailed studies in southern Niger, where
the isotopic compositions reflect a mixture of recycled vapour (Taupin et al. 1995).
Chloride may be regarded as an inert ion in rainfall, with distribution and
circulation in the hydrological cycle taking place solely through physical processes. A
study of rainfall chemistry in northern Nigeria has been made by Goni et al. (2001).
Data collected on an event basis throughout the rainy seasons from 1992 to 1997
showed weighted mean Cl values ranging from 0.6 to 3.4 mg L-1. In 1992, Cl was
measured at five nearby stations in Niger with amounts ranging from 0.3 to 1.4 mg L-1.
The spatial variability at these locations is notable. However, it seems that apart from
occasional localized convective events, the accumulation pattern for Cl is temporally
and spatially uniform during the monsoon; chloride accumulation is generally
proportional to rainfall amount over wide areas. This is an important conclusion for
the use of chloride in mass balance studies.
Bromide, like chloride, also remains relatively inert in atmospheric processes,
though there is some evidence of both physical and chemical fractionation in the
atmosphere relative to Cl (Winchester & Duce 1967). The Br/Cl ratio may thus be used
as a possible tracer for air mass circulation, especially to help define the origin of the Cl.
The ratio Br/Cl is also a useful palaeoclimatic indicator (Edmunds et al. 1992), since
rainfall ratio values may be preserved in groundwater in continental areas. Initial Br
enrichment occurs near the sea surface with further modifications taking place over
land. The relative amounts of chloride and bromide in atmospheric deposition result
initially from physical processes that entrain atmospheric aerosols and control their
size. A significant enrichment of Br over Cl, up to an order of magnitude, is found in
the Sahelian rains (Goni et al. 2001), when compared to the marine ratio. Some Br/Cl
enrichment results from preferential concentration of Br in smaller sized aerosol
particles (Winchester & Duce 1967. However, the significant enrichment in the
Nigerian ratio is mainly attributed to incorporation of aerosols from the biomass as
the monsoon rains move northwards, either from biodegradation or from forest fires.
What is clear is that atmospheric dust derived from halite is unimportant in this region,
since this would give a much lower Br/Cl ratio.
THE UNSATURATED ZONE
Diffuse recharge through the unsaturated zone over time scales ranging from
decades to millennia is an important process in controlling the chemical composition
of groundwater in (semi-)arid regions. In this zone fluctuations in temperature,
humidity and CO2 create a highly reactive environment. Below a certain depth (often
termed the zero-flux plane) the chemical composition will stabilise, and in
homogeneous porous sediments near steady-state movement (piston flow) takes place
towards the water table. It is important that measurements of diffuse groundwater
recharge only consider data below the zero-flux plane. Some vapour transport may
still be detectable below these depths at low moisture fluxes, however, as shown by
the presence of tritium at the water table in some studies (Beekman et al. 1997). The
likelihood of preferential recharge via surface runoff (see below) is also important in
many (semi-)arid regions.
53
In indurated or heterogeneous sediments in (semi-)arid systems in some terrains,
by-pass (macropore or preferential) flow may also be an important process. In older
sedimentary formations joints and fractures are naturally present. In some otherwise
sandy terrain where carbonate material is present, wetting and drying episodes may
lead to mineralisation in and beneath the soil zone, as mineral saturation (especially
calcite) is repeatedly exceeded. This is strictly a feature of the zone of fluctuation
above the zero-flux plane, however, where calcretes and other near-surface deposits
may give rise to hard-grounds with dual porosities. Below a certain depth the
pathways of soil macropore movement commonly converge and a more or less
homogeneous percolation is re-established. In some areas, such as the southern USA,
by-pass flow via macropores is found to be significant (Wood & Sandford 1995;
Wood 1999). In areas of Botswana it is found that preferential flow may account for
at least 50% of fluxes through the unsaturated zone (Beekman et al. 1999; De Vries et
al. 2000).
Four main processes influence soil water composition within the upper
unsaturated zone:
i) The input rainfall chemistry is modified by evaporation or evapotranspiration.
Several elements such as chloride (also to a large extent Br, F and NO3) remain
conservative during passage through this zone and the atmospheric signal is retained.
A build-up of salinity takes place, although saline accumulations may be displaced
annually or inter-annually to the groundwater system.
ii) The isotopic signal (į2H, į18O) is modified by evaporation, with loss of the
lighter isotope and enrichment in heavy isotopes with a slope of between 3 to 5
relative to the meteoric line (with a slope of 8). Transpiration by itself, however, will
not lead to fractionation (Fontes 1982).
iii) Before passage over land, rainfall may be weakly acidic and neutralisation by
water-rock interaction will lead rapidly to an increase in solute concentrations.
iv) Biogeochemical reactions are important for the production of CO2, thus assisting
mineral breakdown. Nitrogen transformations are also important (see below),
leading frequently to a net increase in nitrate input to the groundwater.
These processes may be considered further in terms of the conservative solutes
that may be used to determine recharge rates and recharge history. The reactive
solutes provide evidence of the controls during reaction, tracers of water origin and
pathways of movement, as well as an understanding of the potability of water supplies.
Tritium and 36Cl
Tritium has been widely used in the late 20th century to advance our knowledge
of hydrological processes, especially in temperate regions (Zimmerman et al. 1967). It
has also been used in a few key studies in (semi-)arid zones where it has been applied,
in particular, to the study of natural movement of water through unsaturated zones. In
several parts of the world including the Middle East (Edmunds & Walton 1980;
Edmunds et al. 1988), North Africa (Aranyossy & Gaye 1992; Gaye & Edmunds
54
1996) and Australia (Allison & Hughes, 1978), classical profiles from the unsaturated
zone show well-defined 1960s tritium peaks some metres below surface, indicating
homogeneous movement (piston flow) of water through profiles at relatively low
moisture contents (2-4 wt%). These demonstrate that low, but continuous rates of
recharge occur in many porous sediments. In some areas dominated by indurated
surface layers, deep vegetation or very low rates of recharge, the tritium peak is less
well defined (Phillips 1994), indicating some moisture recycling to greater depths (up
to 10 m), although overall penetration of modern water can still be estimated. Some
problems have been created with the application of tritium (and other tracers) to
estimate recharge, through sampling above the zero-flux plane, where recycling by
vegetation or temperature gradients may occur (Allison et al. 1994).
The usefulness of tritium as a tracer has now largely expired due to weakness of
the signal following cessation of atmospheric thermonuclear testing and radioactive
decay (half-life 12.3 years). It may still be possible to find the peak in unsaturated
zones, but this is likely to be at depths of 10-30 m based on those areas where it has
been successfully applied. Other radioisotope tracers, especially 36Cl (half-life
301,000 years), which also was produced during weapons testing, still offer ways of
investigating unsaturated zone processes and recharge at a non-routine level. However,
in studies where both 3H and 36Cl have been applied, there is sometimes a discrepancy
between recharge indications from the two tracers due to the non-conservative
behaviour of tritium (Cook et al. 1994; Phillips 1999). Nevertheless, the position and
shape of the tritium peak in unsaturated zone moisture profiles provides convincing
evidence of the extent to which 'piston displacement' occurs during recharge, as well
as providing reliable estimates of the recharge rate.
An example of tritium profiles (with accompanying Cl profiles) sampled in 1977
from adjacent sites in Cyprus is shown in Figure 3, which shows peaks at between 9
and 13 m depth. These peaks correspond with rainfall tritium (uncorrected) for the
period from the 1950s to 1970s. The profiles approximate piston displacement and the
peaks align well. A vertical moisture movement of 0.9 m yr-1 is calculated and the
recharge rates are 52 and 53 mm yr-1; these figures compare well with estimates using
Cl (Edmunds et al. 1988). A further example of tritium as a timescale indicator is
shown in Figure 4.
55
C l (mg l-1)
100
-1
0
Cl (mg l )
100
200
0
0
0
AK 2
AK3
5
5
10
15
20
1963
10
1963
Depth (m)
Depth (m)
200
C s =120
15
20
C s = 119
25
25
Water Table
30
30
0
100
200
0
100
Tritium (TU)
Tritium (TU)
200
Figure 3. Tritium and chloride profiles from Akrotiri Cyprus with corresponding
rainfall tritium.
0
MC
5
G18O
GH
Cl
3
H
NO3
Depth (m)
10
SAHEL
DROUGHT
15
1963
20
25
30
35
40
0
2
4
6
8 10
Moisture Content
-5
18
0
5
G O (per mil)
-60 -40 -20 0 20 400 20 40 60 80 100 0 20 40 60 80 100
0 10
2
G H (per mil)
Chloride (mg l-1)
20
30
40
3
H (Tritium Units) NO3-N (mg l-1)
Figure 4. Profiles of tritium, stable isotopes, chloride and nitrate in the unsaturated
zone from the same location – profile L18, Louga, Senegal. This profile records the
impact of the Sahel drought from 1969 to 1989.
Stable isotopes
Stable isotopes have been used in the study of recharge but in general only semiquantitative recharge estimates can be obtained. In (semi-)arid regions, however, the
56
fractionation of isotopes between water and vapour in the upper sections of
unsaturated zone profiles has been used successfully to calculate rates of discharge
(Barnes & Allison 1983; Fontes et al. 1986) and to indicate negligible recharge
(Zouari et al. 1985).
At high rainfall, the recharging groundwater undergoes seasonal fractionation
within the zone of fluctuation (Darling & Bath, 1988), but any seasonal signal is
generally smoothed out with the next pulse of recharge and little variation remains
below the top few metres. In (semi-) arid zones, however, where low recharge rates
occur, the record of a sequence of drier years may be recorded as a pulse of 18Oenriched water, as recorded for example from Senegal (Gaye & Edmunds 1996).
Isotopic depletion in deep unsaturated zones in North America has been used to infer
extended wet and dry intervals during the late Pleistocene (Tyler et al 1996). Extreme
isotopic enrichment in the unsaturated zone accompanies chloride accumulation over
intervals when recharge rates are zero (Darling et al. 1987).
Chloride
Numerous studies using Cl as a conservative tracer in recharge calculations have
been reported, and Cl mass-balance methods probably offer the most reliable
approach to recharge estimation for semi-arid and arid regions (Allison et al. 1994). In
addition, Cl analysis is inexpensive and is widely applicable, bringing it within the
budgets of most water resource organisations, although the capacity for accurate
measurements at low Cl concentrations is required. The methods of investigation are
straightforward and involve the recovery of dry samples by augur, percussion drilling,
or dug wells followed by the measurement of moisture content and the elution of Cl.
A number of criteria must be satisfied or taken into account for successful application
of the technique: That no surface runoff occurs, that Cl is solely derived from rainfall,
that Cl is conservative with no additions from within the aquifer, and that steady-state
conditions operate across the unsaturated interval where the method is applied
(Edmunds et al. 1988; Herczeg & Edmunds 1999; Wood 1999). As with tritium, it is
important that sampling is made over a depth interval which passes through the zone
of fluctuation. For the example shown in Figure 3, from Cyprus, the mean (steadystate) concentrations of Cl in the pore waters (119 and 122 mg L-1 respectively)
provide recharge estimates of 56 and 55 mm yr-1. These estimates are derived using
mean Cl data over three years at the site for rainfall chemistry and the measured
moisture contents. Macropore flow is often present in the soil zone, but in the Cyprus
and northern African profiles, in the absence of large trees this is restricted to the top
2-4 m of the profile.
Nitrate
Groundwater beneath the arid and semi-arid areas of the Sahara is almost
exclusively oxidising. The continental aquifer systems contain little or no organic
matter and dissolved oxygen concentrations of several mg L-1 may persist in
palaeowater dated in excess of 20,000 years old (Winograd & Robertson 1982,
57
Heaton et al. 1983). Under these conditions nitrate also is stable and acts as an inert
tracer recording environmental conditions. Nitrate concentrations in Africa are often
significantly enriched and frequently exceed 10 mg L-1 NO3-N (Edmunds & Gaye
1997; Edmunds 1999). High nitrate concentrations may be traced in interstitial waters
through the unsaturated zone to the water table and are clearly the result of
biogeochemical enrichment. This enrichment is well above concentrations from the
atmosphere, allowing for evapotranspiration, and the source is most likely to be
naturally occurring N-fixing plants such as acacia, though in the modern era some
cultivated species may also contribute. In Senegal (Edmunds & Gaye 1997), the NO3N/Cl ratio can even exceed 1.0. A record of nitrate in the unsaturated zone can
therefore be used as an indicator of vegetation changes, including land clearance and
agriculture.
Reactive tracers and water-rock reactions in the unsaturated zone
The main process of groundwater mineralisation takes place by acid-base
reactions in the top few metres of the earth's crust. Natural rainfall acidity is between
pH 5 and 5.5. Atmospheric CO2 concentrations are low, but may increase significantly
due to microbiological activity in the soil zone. In warmer tropical latitudes the
solubility of CO2 is lower than in cooler mid-high latitude areas and weathering rates
will be somewhat lower (Tardy, 1970; Berner & Berner, 1996). The concentration of
soil CO2 (pCO2) is nevertheless of fundamental importance in determining the extent
of a reaction; an open system with respect to the soil/atmosphere reservoir may be
maintained by diffusion to a depth of several metres.
Investigation of the geochemistry of pore solutions in the unsaturated zone is
usually hampered by low moisture contents. Elutriation using distilled water is
possible for inert components but not for reactive solutes, since artefacts may be
created in the process. Where moisture contents exceed about 5 wt% it may be
possible to extract small volumes by immiscible liquid displacement (Kinniburgh &
Miles 1983). Results for one profile from Nigeria (see Edmunds et al. 1999) illustrate
the trends for some elements in relation to Cl (Figure 5). An increase in Na/Cl above
that in rainfall indicates that some reaction with the rock is taking place (probably
feldspar dissolution). Concentrations of Fe and Al show strong gradients, probably
related to low pH in the top 10 m of the profile.
58
0
rain
rain
Cl
2
Depth (m)
4
6
Al
8
10
NO3-N
12
Fe (total)
14
16
mNa/Cl
water table
18
0
20 40 60 80 100 0
4
mgl-1
8
12
mgl-1
16 0.0
0.5
1.0
1.5
mNa/Cl
mMg/Cl
2.00.0 0.2 0.4 0.6 0.8 1.0 0
mMg/Cl
Sr/Ca.103
5
150.0
10
mSr/Ca.10
3
0.2
0.4
mgl-1
0.6 0
5
10
15
20
mg l-1
Figure 5. Geochemistry of interstitial waters in the unsaturated zone – profile MD1
from northern Nigeria with concentrations and ratios for a number of elements. The
molar N/Cl (u10) is also plotted on the nitrate diagram.
Recharge history
In deep unsaturated zones where piston flow can be demonstrated and where
recharge rates are very low it may be possible to reconstruct climate and recharge
history. Such profiles have been used to examine the recharge history over the decadal
to century scale (or longer) in several parts of the world, as well as to reconstruct the
related climate change (Edmunds and Tyler, 2002). More recently global estimates of
recharge have been made based mainly on Cl mass balance and these are examined in
relation to climate variability (Scanlon et al. 2006). Records up to several hundreds of
years have been recorded from China (Ma and Edmunds 2005) and for several
millennia from USA (Tyler et al. 1996)
An example of an integrated study from Senegal using tritium, Cl and stable
isotopes is given in Figure 4, based on Gaye & Edmunds (1996) and Aranyossy &
Gaye (1992). Samples were obtained from Quaternary dune sands where the water
table was at 35 m and where the long-term (100 year) average rainfall is 356 mm yr-1
(falling by 36% to 223 mm since 1969 during the Sahel drought). The tritium peak
clearly defines the 1963 rainfall as well as demonstrating that piston displacement is
occurring. A recharge rate of 26 mm yr-1 is indicated and little if any by-pass flow is
taking place.
The profile has a mean Cl concentration (23.6 mg L-1) that corresponds to a
recharge rate of 34.4 mm yr-1, calculated using a value for rainfall Cl of 2.8 mg L-1
and an average rainfall of 290 mm yr-1. This rate is in close agreement with the
tritium-derived value, indicating homogeneous movement with little dispersion of
both water and solute. Oscillations in Cl with depth are due to variable climatic
conditions, and the prolonged Sahel drought is indicated by the higher Cl
concentrations. These profiles have been used to create a chronology of recharge
events (Edmunds et al. 1992). The stable isotope values are consistent with the trends
in recharge rates indicated by Cl, the most enriched values corresponding to high Cl
59
and periods of drought, but the stable isotopes are unable to quantify the rate of
recharge, unlike Cl and 3H.
Spatial Variability
The use of the chloride mass balance in the unsaturated zone is now being widely
adopted for diffuse recharge estimation in semi-arid regions (Allison et al., 1994; Gaye
and Edmunds, 1996; Wood and Sandford, 1995). Conservative rainfall solutes,
especially chloride, are concentrated by evapotranspiration and the consequent buildup in salinity is inversely proportion to the recharge rate. This method is therefore
ideally suited to the measurement of low recharge rates; it is also attractive since, in
sharp contrast to instrumental approaches, decadal or longer average rates of recharge,
preserved in thick unsaturated zone profiles may be obtained. The results of recent
applications of the chloride mass balance approach using profiles to derive recharge
assessments in different parts of the northern hemisphere (Africa and Asia) are shown
in Figure 6. Some of these are single data points but for some areas multiple sites are
represented for the same area; only those profiles are shown which are •8m in length,
where steady state moisture and chloride are recorded, indicating net downward flux of
moisture. The sources of data are given in the caption. Several interesting features
emerge:
i) There is a cut off point around 200 mm mean annual rainfall which signifies the
lower limit for effective recharge.
ii) Above this threshold there is the possibility for recharge as in Cyprus where 100
mm recharge is measured from 400 mm rainfall. Nevertheless, the technique
records small but finite recharge rates in some areas with rainfall as high as 550
mm.
iii) At any one site a large range of recharge may occur in a small area implying that
spatial variability is likely to be considerable, due to soil, geology or vegetation
controls, since seasonal rainfall variability is likely to be evened out over the
decadal timescales of most profiles.
iv) Finite, if low, recharge rates still occur even in those areas with mean annual
rainfall below 200mm (eg. Tunisia, Saudi Arabia). However these low rates also
signify moderate salinity in the profiles; with the absence of recharge, salinity
accumulation will be continuous as found in southern Jordan.
60
Figure 6. Recharge estimates from different countries using the Cl mass balance
Most of the profiles are from sandy terrain (sandstones or unconsolidated dune
sands). Few examples are shown of clay-rich sediments. An exception is the Damascus
Basin profile which is predominantly clay-loam (Abou Zakhem and Hafez, 2001) and
which shows very low recharge. Two profiles from China are from loess sediments and
these show modest rates of recharge. The recharge phenomenon is examined further
with reference to the data from Senegal where a wide range of recharge rates has been
measured across the Sahel region with rainfall ranging from 290-720 mm/yr.
One way to determine the spatial variability of recharge using geochemical
techniques is to use data from shallow dug wells. Water at the immediate water table in
aquifers composed of relatively homogeneous porous sediments represents that which
is derived directly from the overlying unsaturated zone. Recharge estimates may then
be compared with those from unsaturated-zone profiles to provide robust long term
estimates of renewable groundwater, taking into account the three dimensional
moisture distributions and based on timescales of rainfall from preceding decades. A
series of 120 shallow wells in a 1600 km2 area of NW Senegal (Figure 7) were used to
calculate the areal recharge from the same area from which the point source recharge
estimates were obtained (Edmunds and Gaye, 1994). Average chloride concentration
for seven separate profiles at this site is 82 mg L-1, giving a spatially averaged recharge
of 13 mm yr-1.
61
Figure 7. Spatial distribution of Cl measured in dug wells in NW Senegal and regional
recharge estimates. Unsaturated zone profiles were measured at the research sites
indicated and gave good correlation for diffuse recharge rates
The topography of this area has a maximum elevation of 45m and decreases
gently towards the coast and to the north; the natural groundwater flow is to the northwest. The dug wells are generally outside the villages and human impacts on the water
quality are regarded as minimal; the area is either uncultivated savannah or has been
62
cleared and subjected to rainfed agriculture. The Cl concentrations in the shallow wells
have been contoured, and range from <50 to >1000 mg/l Cl (Figure 3a). These are
proportional to recharge amounts which vary from >20 to <1mm a-1 (Figure 3b) The
spatial variability may be explained mainly by changes in sediment grain size between
the central part of the dune field and its margins where the sediments thin over older,
less permeable strata. Over the whole area the renewable resource may be calculated at
between 13 000 and 1100 m3/ km2/yr.
Focused recharge
The paper has focused so far on diffuse recharge where regional resource
renewal takes place through mainly homogeneous sandy terrain, typical of much of
North Africa. In areas of hard rock or more clay-rich terrain or where there is a well
developed drainage system, it is likely that recharge will be focused along wadi lines
and some recharge will occur via storms and flash floods, even where the mean
annual average rainfall is below 200mm.
One example of this has been described from Sudan (Darling et al. 1987;
Edmunds et al. 1992). The wadi system (Wadi Hawad) flows as a result of a few
heavy storms each year in a region with rainfall around 200mm/yr. The regional
terrain (the Butana Plain) is mainly clay sediments and it can be shown that there is no
net diffuse recharge, demonstrated by fact that Cl is accumulating in the unsaturated
zone. The deep groundwater in the region is clearly a palaowater with recharge having
occurred in the Holocene. Shallow wells at a distance of up to 1 km from the wadi
line contain water with tritium, all of which indicated a large component of recent
recharge (24-76 TU). Samples of deep wells from north of the area, away from the
main Wadi Hawad, all gave values d 9 T.U. Thus the wadi acts as a linear recharge
source and sustains the population of small villages and one small town through a
freshwater lens that extends laterally from the wadi.
HYDROCHEMISTRY OF GROUNDWATER SYSTEMS IN (SEMI-)ARID
REGIONS
From the foregoing discussion it is evident from geochemical and isotopic
evidence that significant recharge to aquifer systems does not occur at present in most
arid and semi-arid regions. In the Senegal example illustrated above, recharge rates to
sandy areas with long-term average rainfall of around 350 mm are between <1 and 20
mm. It is likely that these results could be extrapolated to other regions with similar
landscape, geology and climate over much of northern Africa, though preferential
recharge along drainage systems is also important, providing areas with small but
sustainable supplies. These present-day low recharge conditions will have an impact
on the overall chemistry due to lower water-rock ratios and larger residence times,
leading to somewhat higher salinities and evolved hydrogeochemistry.
Africa is characterised by several large overstepping sedimentary basins (Figure
1) which contain water of often excellent quality. Much of this water is shown to be
63
palaeowater recharged during the late Pleistocene or Holocene pluvial periods.
Hydrogeochemical processes observed in these aquifers have principally been taking
place along flow-lines under declining heads towards discharge points in salt lakes or
at the coast. Present rates of groundwater movement in the large sedimentary basins
are likely to be less than one metre per year; in the Great Artesian Basin of Australia
rates as low as 0.25 mm yr-1 are recorded (Love et al. 2000). Piezometric decline has
generally been occurring, since fully recharged conditions and transient conditions are
likely to be still operating at the present day as adjustment continues towards modern
recharge conditions. Groundwater in large basins is never ‘stagnant’ and water-rock
interaction will continue to occur. The evolution of inorganic groundwater quality is
considered here in the light of the isotopic evidence, which provides information on
the age and different recharge episodes in the sedimentary sequences.
Input conditions – inert elements and isotopic tracers
The integrated use of inert chemical and isotopic techniques provides a powerful
means of determining the origin(s) of groundwater in semi-arid regions. To a large
extent these tracers (Cl, į18O, į2H) are the same as used in studies of the unsaturated
zone and may be adopted to clearly fingerprint water from past recharge regimes.
However, the longer time scales of groundwater recharge require the assistance of
radiocarbon dating and, if possible, other radioisotopic tools such as 36Cl, 81Kr and
39
Ar (Loosli et al. 1999). Noble gas isotopic ratios also provide evidence of recharge
temperatures which are different from those of the present day (Stute & Schlosser
2000).
Several elements, especially Br and Br/Cl ratios, remain effectively inert in the
flow system and may be used to follow various input sources and the evolution of
salinity in aquifers (Edmunds 1996a; Davis et al. 1998). An example is given from the
Continental Intercalaire in Algeria (Edmunds et al. 2003) along a section from the
Saharan Atlas to the discharge area in the Tunisian Chotts (Figure 6). The overall
evolution is indicated by Cl, which increases from around 200 to 800 mg L-1. The
sources of salinity are shown by the Br/Cl ratio; the increase along the flow-lines for
some 600 m is due to the dissolution of halite from one or more sources, whilst in the
area of the salt lakes (chotts) some influence of marine formation waters is indicated
(possibly from water flowing to the chotts from a second flow-line). Fluorine remains
buffered at around 1 mg L-1 controlled by saturation with respect to fluorite and iodine
follows fluorine behaviour, being released from the same source as F possibly from
organic rich horizons. The Cl and Br concentrations and ratio are related to inputs and
differ from F and I, which are principally controlled by water-rock interaction.
Reactions and evolution along flow lines
The degree of groundwater mineralisation in the major water bodies in semi-arid
and arid zones, recharged during the Holocene and the late Pleistocene, is likely to be
a reflection of atmospheric/soil chemical inputs, the relatively high recharge rates
during the pluvial episodes, the aquifer sedimentary facies and mineralogy. Most of
64
North Africa south of latitude 28oN comprises variable thicknesses of continental
sediments overlying crystalline basement, whereas to the north, marine facies may
occur containing residual formation waters and intra-formational evaporites which
lead to high salinities. Similar relationships are found in the coastal aquifers of West
Africa, as well as on the Gulf of Guinea. In the Surt Basin of Libya the change of
facies to marine sediments is marked by a progressive change in chemistry (especially
Sr increase) as groundwater moves north of latitude 28o30c (Edmunds 1980). For
many of the large basins of the Sahara and Sahel, away from coastal areas, the
groundwater chemistry for inert solutes reflects, to a significant extent, inputs from
the atmosphere (Fontes et al 1993). Superimposed on these inputs, the effects of
water-rock interactions in mainly silicate dominated rock assemblages are recorded.
Important changes in reactive tracers (major and trace elements) also occur,
which can be followed by (time-dependent) water-rock reactions. In addition to the
chemical signature derived from atmospheric inputs, the chemistry of groundwater is
determined to a significant extent by reactions taking place in the first few metres of
the unsaturated or saturated zone and reflecting the predominant rock type. Any
incoming acidity will be neutralised by carbonate minerals or, if absent, by silicate
minerals. In the early stages of flow the groundwater will approach saturation with
carbonates (especially calcite and dolomite) and thereafter will react relatively slowly
with the matrix in reactions where impurities (e.g. Fe, Mn and Sr) are removed from
these minerals, and purer minerals are precipitated under conditions of dynamic
equilibrium towards saturation limits with secondary minerals (e.g. fluorite and
gypsum).
During this stage, other elements may accumulate with time in the groundwater
and indicate if the flow process is homogeneous or not; discontinuities in the
chemistry are likely to indicate discontinuities in the aquifer hydraulic connections.
An example from the Continental Intercalaire aquifer in Algeria/Tunisia (Edmunds et
al. 2003) shows how the major elements may vary along the flow line. The ratios of
reactive versus an inert tracer are used to indicate the evolution. The very constant
Na/Cl ratio (weight ratio of 0.65) throughout the flow path as salinity increases
indicates the dissolution of halite with very little reaction; at depth a different source
is indicated. The Mg/Ca ratio indicates that saturation with respect to calcite (0.60) is
approached after a short distance but that this is disturbed at depth by water depleted
in Mg, probably from the dissolution of gypsum. The K/Na ratio increases along the
flow-line from 0.05 to 0.18, suggesting a time-dependent release of K from feldspars
or other silicate sources.
Redox reactions
Under natural conditions, groundwater undergoes oxidation-reduction (redox)
changes moving along flow-lines (Champ et al. 1979; Edmunds et al. 1984). The
solubility of oxygen in groundwater at the point of recharge (around 10-12 mg L-1)
reflects the ambient air temperature and pressure. The concentration of dissolved
oxygen (DO) in newly recharged groundwater may remain high (8-10 mg L-1),
indicating relatively little loss of DO during residence in the soil or unsaturated zone
65
(Edmunds et al. 1984). Oxygen slowly reacts with organic matter and/or with Fe2+
released from dissolution of impure carbonates, ferromagnesian silicates or with
sulphides. Oxygen may persist, however, for many thousands of years in unreactive
sediments (Winograd & Robertson 1982). Complete reaction of oxygen is marked by
a fall in the redox potential (Eh) by up to 300 mV. This provides a sensitive index of
aquifer redox status. A decrease in the concentration of oxygen in pumped
groundwater therefore may herald changes in the input conditions. An increase in
dissolved iron (Fe2+) concentration as well the disappearance of NO3 may be useful as
secondary indicators.
The Continental Intercalaire aquifer illustrates redox conditions for a typical
continental aquifer system in an arid zone. The redox boundary corresponds with a
position some 300 km along the flow path in the confined aquifer and in waters of late
Pleistocene age (Edmunds et al. 2003). Neither Eh nor O2 measurements were
possible in the study, but the position of the redox change is clearly marked by the
disappearance of NO3, the increase in Fe2+, as well as in the sharp reduction in
concentrations of several redox-sensitive metals (Cr, U for example), which are
stabilised as oxy-anions under oxidising conditions.
Salinity generation
Salinity build up in groundwater in (semi-)arid regions has several origins, some
of which are referred to in the preceding sections. Edmunds & Droubi (1998) review
the topic and discuss the main issues and techniques for studies in such regions. Three
main sources are important – atmospheric aerosols, sea water of various generations
and evaporite sequences, all of which may be distinguished using a cocktail of
chemical or isotopic techniques (Table 1).
Atmospheric inputs that slowly accumulate over geological time scales are of
great importance as a source of salinity. The impact on groundwater composition will
be proportional to the inputs (including proximity to marine or playa source areas),
climate change sequences and the turnover times of the groundwater bodies. The
accumulation of salinity from atmospheric sources is most clearly demonstrated from
the Australian continent (Hingston & Galitis 1976), where the landscape and
groundwaters reflect closely the deposition of aerosols over past millennia.
Formation waters from marine sediments are important as a salinity source in
aquifers near to modern coastlines. Different generations of salinity may be
recognised, either from formation waters laid down with young sediments or from
marine incursions arising from eustatic or tectonic changes. Evaporites containing
halite and/or gypsum are an important cause of quality deterioration in many aquifers
in present day arid and semi-arid regions. Formation evaporites are usually associated
with inland basins of marine or non-marine origin. The different origins and
generations of salinity in formation waters may be characterised by a range of isotopic
and chemical tracers, such as į87Sr and į11B (Table 1).
66
CONCLUSIONS
This paper has focused on water quality from a number of different viewpoints: a)
as a diagnostic tool for understanding the occurrence evolution of groundwater; b) as
a means for determining the recharge amounts; and c) as a means for studying the
controls on groundwater quality dependent on geology and time scales. Emphasis has
been placed on natural processes rather than direct human impacts such as pollution.
Nevertheless, in arid regions the deterioration of groundwater quality due to human
activity, such as excessive pumping, may lead to loss of resources through salinisation
and the approach adopted here will assist in a better understanding of what may be
going on. Salinity increase is also a consequence of point source (single well/village)
or regional (town and city) pollution, and any overall quality deterioration may be
monitored using simple parameters such as specific electrical conductance (SEC) or
chloride.
The approach adopted here has been to recognise that water contains numerous
items of chemical information that may be interpreted to aid water management.
Measurement in the field and in the laboratory of the basic chemistry can
inexpensively provide additional information during exploration, development and
monitoring of groundwater resources. Too often only physical parameters are seen as
important. For example, it is important to understand not only that water levels are
falling, but why they are falling. Some key recommendations that emerge from this
chapter may be summarised as:
Data requirements
Useful information may be obtained through the measurement and careful
interpretation of a few key parameters. In the field, the measurement of T, pH and
SEC are thus recommended. Temperature is essential, since it provides a proxy for the
depth of sample origin in cases where details of production are unknown. Field
measurement of pH is essential if meaningful interpretation of the results, especially
for groundwater modelling, is to be made, since loss of CO2 will take place between
the well head and the laboratory, leading to a pH rise; laboratory pH is therefore
meaningless. SEC is a key parameter in both surveys and monitoring, enabling the
hydrogeologist to quickly detect spatial or temporal changes in the groundwater. In
the field, two filtered (0.45ȝm) samples (one acidified to and the other not acidified)
can then be taken for laboratory analysis (Edmunds 1996b).
In hydrochemical studies the measurement of chloride is particularly important,
since Cl behaves as an inert tracer which allows it to be used as a reference element to
follow reactions in the aquifer and to interpret physical processes such as groundwater
recharge, mixing and the development of salinity. Chloride should always be
measured, and it is recommended that rainfall Cl also be measured in collaboration
with meteorologists.
Most central laboratories are able to cope with at least major ion analyses and
these should be measured as the basic data set for water quality investigations, after
first establishing that there is an ionic balance (Appelo & Postma 1993). Minor and
67
trace elements described above are helpful in the diagnosis of processes taking place
in the aquifer as well as in determining potability criteria. The more detailed
measurements described in this chapter are desirable, but are not essential for routine
investigations. Nevertheless, the implications of case studies where combined isotopic
and chemical data have been used should be digested and extrapotated for local
studies. The results of expensive and specialised case studies are often of considerable
value, since at practical and local level these then allow simple tools such as nitrate or
chloride to be subsequently used for monitoring.
Groundwater resources assessment
In (semi-)arid regions the main resources issue is the amount of modern recharge.
In this paper stress has been placed on how to determine whether modern recharge is
taking place and, if so, how much. In sandy terrain, or other areas with unconsolidated
sediments, it is recommended that attention be paid to information contained in the
unsaturated zone. The use of Cl to measure the direct recharge component is
recommended. However, the distribution of Cl on a regional scale is also important,
since salinity variations will be proportional to the amount of recharge. The use of
isotopes and additional chemical parameters will then be needed to help confirm
whether this recharge belongs to the modern cycle or is palaeowater.
Groundwater exploration and development
Important information can be obtained during the siting and completion of new
wells or boreholes. Chemical information obtained at the end of pumping tests
provides the 'initial' baseline chemistry against which subsequent changes may be
monitored. It is recommended that analyses of all major ions, minor ions and also
stable isotopes of oxygen and hydrogen if possible, be carried out during this initial
commissioning phase.
In more extensive groundwater development it is recommended that exploration
of the quality variation with depth be made. One or more boreholes should be
regarded as exploration wells for this purpose. Depth information can be obtained
during the drilling using packer testing, air-lift tests or similar, which can then help
determine the optimum screen design or well depth, for example to minimise salinity
problems. Alternatively, single wells can be drilled to different depths in a well field
for purposes of studying the stratification in quality or in age (e.g. whether modern
groundwater overlies palaeowater). It should be remembered that subsequent pumped
samples are inevitably going to be mixtures in terms of age and quality, and this initial
testing will enable assessment of changes to be made as the wells are used.
Groundwater quality and use
Most water sampling programmes are heavily oriented towards suitability for
drinking use, having regard to health-related problems. In this chapter emphasis has
been placed on understanding the overall controls on water quality evolution. Natural
68
geochemical reactions taking place over hundreds or thousands of years will thus give
rise to distinctive natural properties. It is important to recognise this natural baseline,
without which it will be impossible to identify if pollution from human activity is
taking place. It has also been shown how salinity distribution is related to natural
geological and climatic factors. In addition, the oxidising conditions prevalent in
many (semi-)arid regions may give rise to enhanced concentrations of metals such as
Cr, As, Se and Mo. Prolonged residence times may also lead to high fluoride and
manganese concentrations. Under reducing conditions, high Fe concentrations may
occur.
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1000
Cl
Cl (mg l-1)
800
600
400
30
200
0
8
7
6
5
4
3
2
1
0
0.006
0.005
0.004
0.003
0.002
0.001
0.000
0.0005
I/Cl (wt)
Br/Cl (wt)
Br (mg l-1)
Br
Br/Cl
seawater
halite
I/Cl
0.0004
0.0003
0.0002
0.0001
-6
(seawater - 3.1x10 )
F (mg l-1)
0.0000
2.0
F
1.5
1.0
0.5
0.0
0
100
ATLAS MTS
(ALGERIA)
200
300
400
500
Distance (km)
600
700
800
CHOTTS
(TUNISIA)
Figure 8 Down-gradient hydrogeochemical profile (west-to-east) in the Continental
Intercalaire aquifer from the Atlas mountains of Algeria to the discharge area in the
Chotts of southern Tunisia. The chloride profile indicates a progressive increase in
salinity: The other halogen elements Br, I, F help to explain the origins of salinity and
the general chemical evolution down-gradient.
74
Numerical Modeling of Groundwater Flow in Heihe River Basin
at Middle Reaches: A Review
Xu-Sheng Wang
School of Water Resources & Environment, China University of Geosciences, Beijing
PR China
1. Introduction
Heihe River Basin is located in the arid west-north part of China with area of land
surface about 13u104 km2. Heihe River, the primary stream in the basin with a length
about 821 km, runs down from Qilian Mountain (upper reaches)to the sedimentary
basin on Hexi Corridor(middle reaches), travels to the Ejina Basin in inner Mongolia
(lower reaches) and then disappears when it reaches Juyan Lakes. The middle
segment of Heihe River is controlled by two outlets from upper reaches to lower
reaches: Yingluo Gorge and Zhengyi Gorge.
In the middle reaches of Heihe River Basin, oasis is strongly developed around
Zhangye City and Jiuquan City by introducing stream flow into channels to irrigate
farm land in this arid area near by Badain Jaran Desert. Since average annual
precipitation is less than 200 mm locally on the contrast, annual potential evaporation
is greater than 2000 mmˈthe oasis is strictly dependent on the surface water resources
came from Qilan Mountain. Together with Heihe River, there are more than twenty
streams developed in Qilan Mountain which provide about 30u108 m3 of water
resources annually to the north area. 2/3 or more of the water resources is expended at
middle reaches. With increasing of population and strength of economic activities,
demand of water resources in the middle reaches area increases rapidly and reduces
the residual stream flow. It is a disadvantage to the oasis at lower reaches. In 1961,
West Juyan Lake was dried up. Thirty years later, the same fate fell on East Juyan
Lake. To prevent further deteriorating of ecological environment at Ejina oasis,
Chinese government started a project from 2001 to control export of surface water at
middle reaches of Heihe River. It leaded to a good result: water level in East Juyan
Lake was rising and has covered 35 km2 until 2004. However, West Juyan Lake is
still drying.
Although surface water is the initial water resources in the middle reaches area,
groundwater also plays an important role in terrestrial water cycle and utilization of
water resources. Firstly, more than half of water flow out from Qilian Mountain leaks
down as recharge of groundwater through beds of rivers and channels. Secondly,
groundwater flows out through springs or directly discharges into Heihe River at the
lower north part of the middle reaches area, provides water resources to irrigation.
75
The amount of discharge flow is about 10u108 m3 annually. In addition, groundwater
is directly abstracted, by over 6000 wells, from aquifers to irrigated farm lands. The
amount of annual groundwater pumping flow is about 3u108 m3 recently.
An aquifer system is composed of Quatemary sediments with sandy gravel or
sand with calyey interbeds under the middle reaches area. Underlying Tertiary
sediments of sand-stones and shale-stones is relatively weak permeable and is
normally regarded as the impermeable bed of the groundwater basin. Thickness of the
aquifer system varies from 20 m to 1800 m. Groundwater table can be deeper than
200 m under the slopy plain (pluvial fans) near Qilian Mountain and comes to be
shallow or overflow in the flat fluvial plain on the north area.
To research the behavior of groundwater and its contribution to water resources at
middle reaches of Heihe River Basin, numerical modeling has been developed in the
last two decades.
West Juyan Lake
Scale
km
East Juyan Lake
Heihe River
Zhengyi Gorge
Jiuquan
Middle
Reaches
Area
Heihe River
Yingluo Gorge
Zhangye
Qilian
Mountain
Heihe River
Figure 1 Plan view of Heihe River Basin and the area of middle reaches
76
2. Browsing on the numerical models developed
Groundwater flow in the aquifer system in the middle reaches area of Heihe
River Basin has been studied through numerical modeling since 1990. Zhou X.Z., et.
al. (1990) treated the aquifer system as a single-layer unconfined aquifer and built a
two-dimensional numerical model with finite-difference method on a polygonal grid.
It is called Model-1990 in this presentation. Control equation of groundwater flow in
the aquifer is written as follow
w §
wH
¨ K xb
wx ©
wx
wH
· w §
¸ ¨¨ K y b
wy
¹ wy ©
·
¸¸ W
¹
Sy
wH
wy
(1)
Where: H is the groundwater level; Kx and Ky are the hydraulic conductivity along
X-axis and Y-axis respectively (in fact, the aquifer is treated as homogenous porous
medium in which Kx=Ky); b is the thickness of saturated zone from groundwater table
to the bottom of the aquifer; Sy is the specific yield; W is the recharge rate of
groundwater. In this numerical model, transmissivity T=Kxb=Kyb is applied in fact and
treated as constant parameter identified by statistic analysis of pumping testes. The
transmissivity varies from 200 m2/d to 6000 m2/d in five parameter-zones and
increases with increasing of the thickness of Quatemary deposit. The value of
transmissivity and specific yield were adjusted to make the transient numerical
modeling of groundwater flow satisfied to phreatic water table changes in 65
observation wells from 1987 to 1989. The result shown that total discharge of
groundwater is 11.7~12.5u108 m3/a through springs and 4.7~5.1u108 m3/a through
evaporation in 1987~1989.
Two numerical model of groundwater flow in the middle reaches area were
developed by Zhang G.X., et. al. (2004) with finite-element method: a single-layer
model (Model-2004-1) and a double-layers model (Model-2004-2). Model-2004-1 is
similar to Model-1990. However, 60 parameter zones are arranged for distribution of
transmissivity and specific yield. Purpose of the model is to evaluate the regional
groundwater balance under changes of hydrological and agricultural conditions since
1990. In Model-2004-2, the aquifer system is treated as an unconfined aquifer (with
shallow groundwater) overlies on a confined aquifer (with deep groundwater). Leaky
condition is simulated to configure interaction of shallow groundwater and deep
groundwater. This double-layers model can show a three-dimensional pattern of
groundwater flow: in the recharge area near Qilian Mountain a downward flow is
developed; in the discharge area far away Qilian Mountain a upward flow is
developed. However, it is a quasi-three-dimensional model.
The numerical model developed by Hu L.T. and Chen C.X. (2006, 2007), which
is called Hu-Chen Model in this presentation, is a three-dimensional model with
8-layers. Finite difference method with random polygon grid is applied in the model.
77
The governing equation for groundwater flow is given in three-dimensional pattern as
follows
w § wH · w § wH · w § wH ·
¸ ¨ Kv
¨ Kh
¸ ¨ Kh
¸w
wx ©
wx ¹ wy ¨©
wy ¸¹ wz ©
wz ¹
Ss
wH
wt
(2)
Where Kh and Kv are horizontal and vertical hydraulic conductivity respectively, w is
the source-sink term, Ss is the specific storage of the aquifer medium. Kh is
0.01~43.75 m/d and Kv is 0.002~11.0 m/d as assigned in 20 parameter zones of 8
model-layers. Relationships between groundwater and surface water such as rivers
and springs are treated. Also multi-layer wells are simulated with coupling
seepage-pipe flow model. Both the trial-and-error method and the Fibonacci
optimization method (Chen and Tang, 1994) are applied to calibrate the model
parameters against observed groundwater level (1995-2000), flow rate of springs and
baseflow of Heihe River along the lower-part in the middle reaches area. The results
of Hu-Chen Model show that, in the main middle reaches area (without the West part
of Jiuquan sub-basin) of Hiehe River Basin, 9.65u108 m3/a of surface water leaks
from river-channel beds and irrigated farm lands as recharge of groundwater, however,
9.93u108 m3/a of groundwater flow out through springs and discharges as baseflow of
lower-part Heihe River. Taking into account the exploitation of groundwater from
wells, 2.25 m3/a, the total discharge rate of groundwater is significantly greater than
the recharge rate. Consequently, a continued withdrawal of groundwater storage was
arising before 2000 and leaded to decreasing of flow rate of springs and residual flow
of Heihe River across Zhengyi gorge.
A double-layers model was also developed by Jia Y.W., et al, (2006) as a part of
the hydrological model WEP-Heihe. It is called WEP-GW Model in this presentation.
The structure of WEP-GW Model is similar to Model-2004-2. Exchange of
groundwater and river water is treated. However, arrangement of aquifer-parameters
and processing of groundwater-surface water interactions are not reported in detail.
3. Issues
While numerical models are developed for groundwater flow in Heihe River Basin
at middle reaches, the modelers face the complexity of terrestrial water cycle in the
studied area. Surface water run down from Qilian Moutain to the sedimentary basin,
recharges to groundwater through river-channel leakage and irrigated farm land or
losses to atmosphere by evaportranspiration. A thick vadose zone always exists over
groundwater table in the recharge area. Almost all of natural river-beds are dried up
early before arriving the lower reaches of Heihe River. At the other part of the same
sedimentary basin, groundwater flows out the aquifer system from springs or directly
discharge to the Heihe River as baseflow across Zhengyi gorge for the lower reaches
area. A large number of wells are spread in agriculture districts abstracting
78
groundwater. Also lumped pumping wells are developed at several locations to supply
water for industrial purpose.
These characteristics of terrestrial water cycle make difficulties in numerical
modeling of groundwater flow with conventional approaches. Modelers try to handle
the main features of this groundwater system with different methods.
3.1 Vadose Zone
In traditional groundwater models, recharge of groundwater is input to the
phreatic surface directly (as done in Model-1990, Model-2004-1 and Model-2004-2).
However, it is well known that when infiltrated water reaches to groundwater table, it
should travel through vadose zone. Recharge patterns would be quite different from
the patterns of infiltration. In the middle reaches area of Heihe River Basin,
groundwater table can be deeper than 100 m under slopy pluvial fans in the vicinity of
Qilian Moutain. Then thick vadose zone exists. It do significant effect to the recharge
of groundwater.
In considering of the effect of vadose zone to groundwater recharge in numerical
modeling of groundwater flow, Chen (1998) present a method called Delayed
Recharge Function (DRF) Method. It assumed that infiltration rate P at the time i
(month) is distributed in recharge rate at the time i, i+1, i+2, and so on. The
contribution in recharge at the time j (month, j >i), Rij, of infiltration at the time i
(month), Pi, is given as
Rij
Z ( j i ) Pi
(3)
Where, Z(j-i) is the distribution power. Chen (1998) gave an empirical function of
Z(j-i) as follows
Z ( j i)
( D / B ) j i
exp( D / B )
( j i )!
(4)
Where D is the depth of groundwater table, B is a parameter dependent on soil
properties in the vadose zone. This method provides a simple way to treat the effect of
vadose zone to groundwater recharge with only an additional parameter.
In Hu-Chen Model, DRF Method is applied. The factor B is identified in several
parameter-zones to make the numerical results match the observed response of
groundwater level to recharge from surface water.
In WEP-GW Model, a transition layer is hypothetically arranged between
top-soil layer and the underlying unconfined aquifer to consider the thick vadose zone.
An experiential function is introduced to configure the flow and storage of this
transition layer. However, it is not described in detail in publications
79
3.2 Interactions between groundwater and surface water
3.2.1 Rivers
River is usually treated as boundary condition in groundwater system. The type
of boundary is dependent on the relation between groundwater and river. In the
middle reaches area of Heihe River Basin, Heihe River can not be regarded as a single
type of boundary for groundwater flow.
When Heihe River and the other rivers run out of Qilian Moutain, they lie on a
thick vadose zone of the aquifer system. Surface water within them can not touch
groundwater directly but leaks as groundwater recharge. In 1967 and 1985,
investigation of leakage of Heihe River was conducted. It was found that 0.32~1.52 %
of stream flow leaks along per kilometer of the sharp river beds just leave Qilian
Moutain, 4~10 % of stream flow leaks along per kilometer of the wide river beds
before arriving the flat fluvial plain. In Model-1990, Model-2004-1 and
Model-2004-2, this kind of leakage is calculated previously as well as leakage from
channels, and prepared as linear sources for groundwater flow model. However, in
Hu-Chen Model, a semi-permeable riverbed is considered. When groundwater level is
less than the bottom of a riverbed, the hydraulic connection of groundwater and the
river is called “infiltration type” and leakage from surface water is determined as
follow
Qr
K zc
Ar ( H s Z s )
Mc
(5)
Where: Qr is the leakage flux to a node-zone where the river goes through in the
discrete grid; Kcz and Mc are the hydraulic conductivity and thickness of the riverbed
respectively; Ar is the area controlled by river node; Hs and Zs are the river stage and
bottom elevation of the riverbed respectively. In addition, Qr is not treated as direct
recharge of groundwater but DRF Method is applied to consider the effect of vadose
zone. In WEP-GW Model, a similar treatment of river leakage is selected and delay of
recharge is handled by a transition layer.
In the flat fluvial plain far away Qilian Moutain where groundwater level is
shallow buried and river beds cuts down the aquifer in directly touching with
groundwater, general head boundary can be arranged for Heihe River as done in
Model-1990, Model-2004-1 and Model-2004-2. In Hu-Chen Model, exchange of
water between river and the aquifer in this condition is determined as follow
Qr
K zc
Ar ( H s H s1 ) Qh
Mc
(6)
Where Qr is the exchange rate of water (it is positive when river recharges to the
aquifer), Hs is the river stage and Hs+1 is the hydraulic head of aquifer-layer lower
than the riverbed-within layer, Qh is the discharge of groundwater at the node duce to
80
horizontal flow among the riverbed-within layer. However, the river stage, Hs, is
given previously according to observed water level within wells in the vicinity of
Heihe River. In WEP-GW Model, Darcy law is also applied to calculate the exchange
of water between river and the aquifer through riverbed, a stream flow model is
coupled with the groundwater flow model in the regional hydrological model.
3.2.2 Springs
In numerical modeling of groundwater flow, a spring could be regarded as flux
boundary with constant discharge rate, or head boundary with constant head equal to
its elevation, or drain boundary similar to a segment of river discharge groundwater.
In the middle reaches area of Heihe River Basin, springs are widely developed in the
fluvial plain where local low-lying land release the phreatic surface. They can not bee
treated as constant flow rate boundary because the flow rate is varying with time
associated to local or regional groundwater level.
In Model-1990, a two steps method is applied for a spring: at the first step, it is
treated as constant head boundary and a flow rate is calculated through local water
balance; at the second step, if the calculated flow rate is zero or less than zero, the
spring-node switches to a normal node. However, this model is a single-layer
unconfined aquifer model with a large thickness. How groundwater flows to a
point-sink (a spring) at the top of the aquifer can not been configured successfully in
this type of two-dimensional model.
In Model-2004-1 and Model-2004-2, no single spring is identified but the springs
are hypothetically distributed in average in a overflow region. The overflow region is
defined as the location where depth of groundwater table under normal surface ground
is less than 3.5 m. Elevation of all the nodes in overflow region is artificially lowered
(abstract 3.5 m) to produce spring-nodes (lowered elevation is less than groundwater
level). A spring-node is treated as drain boundary in which discharge of groundwater
is proportional to the difference of groundwater level and elevation of drain bottom.
In Hu-Chen Model, the elevation of a spring, Zs, is a constraint to spring flow
occurrence. A semi-permeable layer is considered under the spring and the discharge
rate, Qsp, is determined as follow
Qsp
K zc
Asp ( H s1 Z s ) Qh , H s1 ! Z s
Mc
Qsp
0 , H s 1 d Z s
(7)
(8)
Where Kcz , Mc , Asp, Qh and Hs+1 are factors similar to that in considering of
river-groundwater interaction as shown in Eq.(6). Two springs (S3 and S6) with
long-time measured data are typically studied to identify the parameters of
spring-nodes. However, the other springs are lack of observation data and have been
81
treated as normal spring nodes in the mode.
In WEP-GW Model, it is not clear how springs are processed.
3.3 Wells
There are two types of wells drilled in the middle reaches area of Heihe River
Basin: shallow agricultural wells that are spread in irrigation districts; deeper
industrial wells to supply water for a factory or drinking water for a city.
It is impossible to simulate each agricultural wells since the number of
agricultural wells is very large (more than 6000). Hence in numerical modeling, they
are usually assumed uniform distributed in an irrigation district and the total pumping
rate is apportioned to each node in the area. A trouble for multi-layer model (Hu-Chen
Model) is to apportion the total pumping rate for every associated layer. Depth of
agricultural wells was roughly investigated statistically. The data is used in
determining the proportion of pumping rate for a model-layer with a certain range of
depth.
Industrial wells are located in several pumping sites. Each pumping site has
several wells pumping groundwater at the same time. However, distance between the
wells is small and can not be distinguished in a regional numerical model. So a
single-well is normally arranged at a node within the pumping site and is assumed to
be equal to the group of wells. Another problem is, these wells are drilled deeply
(100~200 m) across several aquifer layers that are called multi-layer wells. In
multi-layer model, the total pumping rate of a multi-layer well is a result of
groundwater contribution for each layer. It is typically considered in Hu-Chen Model
with coupling seepage-pipe flow model that developed by Chen, et al (1999). An
equivalent vertical hydraulic conductivity is calculated for a multi-layer well
dependent on the flow rate and is introduce into the integrated vertical conductance of
the aquifer-well system. Also this coupling seepage-pipe flow model is applied to
obtain groundwater level within several deep observation wells that can be regarded
as multi-layer pumping well with zero pumping rate.
4. Expectations of Further Development
Recently, it is a challenge for the middle reaches area of Heihe River Basin to
find a sustainable way in utilizing water resources. This way should take care of water
demand in the lower reaches area to maintain a well condition of ecological
environment. To achieve the goal, it is necessary to understand the role of
groundwater as well as the relationship of surface water and subsurface water.
The behavior of groundwater has been studied at the basin scale via numerical
modeling that developed previously. A principle finding is that the vadose zone, the
rivers and channels, the springs and the pumping wells do effect on and be affected by
the regional groundwater flow in an integrated way. It indicates the further
82
development of groundwater flow model in the future. Physical based variable
saturation model may give a more reasonable understand of flow in thick vadose zone
instead of empirical functions. Coupling of groundwater flow model with models of
surface flow system such as rivers and channels is required while the whole
hydrological processes should be quantitative evaluated. When pumping wells and
springs provide irrigation water at the same time, an inner water cycle may produce
and will increase the complexity of groundwater flow locally. Numerical model
should be carefully designed to configure the feature. Currently, limited data is a
difficulty in built up an basin-scale integrated hydrological model in which a
three-dimensional groundwater model is coupled with models of surface water system
and soil-vegetation-atmosphere system, especially with a reasonable parameterized
scheme of human activities. Solving of the problem is dependent on the development
of regional field investigation and data collection of hydrological and ecological
conditions.
Acknowledgement
The author is grateful to the support of Cold and Arid Regions Environmental and
Engineering Research Institute, Chinese Academy of Sciences, Lanzhou. The author
also appreciates Prof. Chen C.X. and Dr. Hu L.T for suggestions in understanding
their work on Heihe River Basin.
References
[1] Chen CX, Jiao JJ. 1999. Numerical simulation of pumping test in multilayer wells with
non-Darcian flow in the wellbore. Ground Water, 37(3): 465–474.
[2] Chen CX, Tang ZH. 1994. Numerical Modeling of Groundwater Flow. Wuhan: China
University of Geosciences Publishing House. [in Chinese]
[3] Chen CX. 1998. Delayed recharge function: A method to treat delay recharge of rainfall to
unconfined aquifer. Hydrogeology and Engineering Geology, (6): 22-24. [in Chinese]
[4] Hu L.T, Chen C.X., Jiao J.J, et. al., 2007. Simulated groundwater interaction with rivers and
springs in the Heihe river basin. Hydrological Processes, in Press, DOI: 10.1002/hyp.6497.
[5] Hu Li Tang, Chen Chong Xi, 2006. Dynamical Simulation for Multilayer Aquifer System at
Middle Reach of Heihe River Basin. Journal of System Simulation, 18(7): 1966-1975. [in
Chinese]
[6] Jia Yang Wen, Wang Hao, Yan Deng Hua, 2006. Distributed model of hydrological cycle
system in Hihe River Basin I: Model development and verification. Shuili Xuebao, 37(5):
534-543. [in Chinese]
[7] Jia Yang Wen, Wang Hao, Yan Deng Hua, 2006. Distributed model of hydrological cycle
system in Hihe River Basin II: Applications. Shuili Xuebao, 37(6): 655-662. [in Chinese]
83
[8] Zhang Guang Hui, Liu Shao Yu, Xie Yue Bo, et. al., 2004, Water Cycle and Development of
Groundwater in the Inland Heihe River Basin in West-North China. Beijing: Geology
Publication House. [in Chinese]
[9] Zhou Xing Zhi, Zhao Jian Dong, Wang Zhi Guang, et. al., 1990, Survey of groundwater
resources and analysis on its application in the middle reaches area of Heihe River in Gansu.
Zhangye: The Second Hydrogeoloical and Engineering Geology Team, Gansu Bureau of
Geology and Mineral Exploitation and Development, Technical Report-1990. [in Chinese]
84
Variable density groundwater flow:
From modelling to applications
Craig T. Simmons1, Peter Bauer-Gottwein, Thomas Graf, Wolfgang Kinzelbach, Henk
Kooi, Ling Li, Vincent Post, Henning Prommer, Rene Therrien, Clifford Voss, James
Ward & Adrian Werner
1
School of Chemistry, Physics and Earth Sciences
Flinders University, GPO Box 2100, Adelaide, Australia
([email protected])
DRAFT Book Chapter for GWADI Workshop, Beijing, China, June 11-15, 2007.
85
1. Introduction
Arid and semi-arid climates are mainly characterised as those areas where precipitation is
less (and often considerably less) than potential evapotranspiration. These climate regions are
ideal environments for salt to accumulate in natural soil and groundwater settings since
evaporation and transpiration remove essentially freshwater from the system, leaving residual
salts behind. Similarly, the characteristically low precipitation rates reduce the potential for salt
to be diluted by rainfall.
Thus arid and semi-arid regions make ideal “salt concentrator”
hydrologic environments. Indeed, salt flats, playas, sabkhas and saline lakes, for example, are
therefore ubiquitous features of arid and semi-arid regions throughout the world [Yechieli and
Wood, 2002]. In such settings, variable density flow phenomena are expected to be important,
especially where hypersaline brines overlie less dense groundwater at depth. In contrast, seawater
intrusion in coastal aquifers is a global phenomenon that is not constrained to only arid and
semi-arid regions of the globe and is inherently a variable density flow problem by its very
nature. These two examples make it clear that variable density flow problems both occur in, but
importantly extend beyond, arid and semi-arid regions of the globe. The intention of this book
chapter is therefore not to limit ourselves to modelling of arid zone hydrological systems, but
rather to present a more general treatment of variable density groundwater flow and solute
transport phenomena and modelling. The concepts presented in this chapter are therefore not
climatologically constrained to arid or semi-arid zones of the world, although they do apply
equally there. The objective of this book chapter is to illustrate the state of the field of variable
density groundwater flow as it applies to both current modelling and applications. Throughout,
we highlight approaches, current resolutions, and explicitly point to future challenges in this area.
86
2. Importance of variable density flow
It has long been known that the density of a fluid can be modified by changing its solute
concentration, temperature and, to a lesser extent, its pressure. Indeed, researchers in the area of
fluid mechanics have studied how fluid density affects flow behaviour for well over a century,
both in fluids only and in fluid-filled porous media. It was understood that areas of application
for these studies was wide and varied including applications to astrophysics, chemical reactor
engineering, energy storage and recovery, geophysics, geothermal reservoirs, material science,
metallurgy, nuclear waste disposal and oceanography, to name just a few. More recently, there
has been a massive explosion in the field because of worldwide concern about the future of our
energy resources and the pollution of our environment. These have been a principal catalyst for
the application of earlier developments made in traditional fluid mechanics to the study of
groundwater hydrology. In the preface to their popular text Convection in Porous Media, Nield and
Bejan [1999] state that “Papers on convection in porous media continue to be published at a rate of over 100
per year”. In groundwater hydrology in particular, Table 1 summarises some key areas in variable
density flow research. The listed references are intended to be illustrative rather than exhaustive.
This table clearly demonstrates the widespread importance, diversity, extent and interest in
applications of variable density flow phenomena in groundwater hydrology. As noted in
Simmons [2005] some key areas include seawater intrusion, fresh-saline water interfaces and
saltwater upconing in coastal aquifers, subterranean groundwater discharge, dense contaminant
plume migration, DNAPL studies, density driven transport in the vadose zone, flow through salt
formations in high level radioactive waste disposal sites, heat and fluid flow in geothermal
systems, palaeohydrogeology of sedimentary basins, sedimentary basin mass and heat transport
and diagenesis, processes beneath sabkhas and salt lakes and buoyant plume effects in applied
tracer tests. The modelling of these physical systems is central to hydrogeologic analyses.
Simmons [2005] presented an overview of variable density flow phenomena, including
current challenges, and future possibilities for this field of groundwater investigation. The
widespread importance and interest in variable density flow phenomena in groundwater
hydrology are also illustrated in exhaustive review articles on this topic by Diersch and Kolditz
[2002] and Simmons et al. [2001]. In addition, the text by Holzbecher [1998] is an excellent
source of material relating to the modelling of density dependent flows in porous media. The
reader is referred to these texts for an exhaustive review of this subject matter. In particular,
Diersch and Kolditz [2002] recently reviewed the state of the art in modelling of density-driven
flow and transport in porous media. They discuss conceptual models for density driven
convection systems, governing balance equations, critical phenomenological laws and
87
constitutive relations for fluid density and viscosity, and the numerical methods employed for
solving the resulting nonlinear problems.
Furthermore, these authors discuss the major
limitations of current model-verification (benchmarking) test cases used in the testing of
numerical models and methods. Diersch and Kolditz [2002] and Simmons [2005] provide
detailed treatments of the current challenges that face the field of variable density groundwater
flow phenomena, from both physical and modelling viewpoints. The reader is referred to those
discussions as an accompaniment to this text.
3. A brief historical perspective
Nield and Bejan [2006] provide an excellent discussion of the topic of convection in
porous media including historical and new conceptual developments within this field. Some of
the earliest work in this field was carried out in the field of traditional fluid mechanics. A logical
trend emerged in the way this field evolved. Earliest work in the early 1900’s involved heated fluids
only (Benard, [1900]; Rayleigh, [1916]), followed by a move to combined heat and porous media
studies in the 1940’s (Horton and Rogers [1945]; Lapwood, [1948]), solute and porous media studies
in the 1950’s and 1960’s (Wooding [1969, 1963, 1962, and 1959]) and finally combined
thermohaline studies in porous media in the 1960’s [Nield, 1968]. Most of these studies were
designed to discover the most fundamental aspects of the behaviour of variable density flow
behaviour and were typically either laboratory or analytically based. In comparison, as groundwater
hydrologists, our interest in energy and solute transport is relatively recent (post 1950’s in general).
Therefore, it is challenging to find much evidence of interest and exploration in variable density
flow groundwater applications at all throughout the earlier historical development of groundwater
hydrology. But, there is one clear exception – the problem of seawater intrusion. Some of the
earliest pioneering work traces back to that of Badon-Ghyben [1888] and Herzberg [1901] whose
analyses could be used to determine the steady position of saltwater-freshwater interfaces in coastal
aquifers. Some of the earliest analytical solutions for the sharp saltwater-freshwater interface in an
infinitely thick confined aquifer emerged in the 1950’s [Glover, 1959; Henry, 1959]. In the problem
of seawater intrusion, Hele-Shaw cell analogs [Bear, 1972] were used in some of the earliest
exploratory studies of seawater intrusion and clearly predated numerical solutions to these
problems. Some of the earliest numerical analyses emerged in studies by Pinder and Cooper [1970],
Segol et al., [1976], Huyakorn and Taylor [1976] and Frind, [1982a,b]. Table 1 illustrates that the
interest in variable density groundwater flow applications has grown to include many other
important areas of hydrogeology. In more recent times, groundwater modelling codes have begun
to employ a range of new and improved numerical techniques and this coupled to faster computers
88
with larger memories has allowed for the simulation and solution of more complex problems, with
fully-coupled flow and solute transport in even 3D cases. Recent developments in modelling efforts
have included simultaneous heat and transport in thermohaline convection problems [Diersch and
Kolditz, 2002; Graf and Therrien, 2007b], adaptation of these to heterogeneous systems including
fractured rock hydrology [Graf and Therrien, 2005; Graf, 2005] and even chemically reactive
transport [Freedman and Ibaraki 2002; Post and Prommer, 2007]. Indeed, it may be argued that one
major limitation of the current field of variable density flow is not necessarily the availability of
numerical models with the inherent capacity to solve these sorts of very complex problems, but is
perhaps the availability of real data to occupy, test and verify the emerging new generation of
computer models.
4. Variable density flow physics
4.1 Overview
Field, laboratory and modelling studies demonstrate that fluid density gradients caused by
variations in concentration and temperature (and to a much lesser degree fluid pressure) are
recognised to play a significant role in solute and/or heat transport in groundwater systems.
Figures 1 and 2 show how fluid density varies as a function of concentration and temperature. It
is easy to accept that fluid density should be at least one factor influencing groundwater flow
processes because in many systems, groundwater concentrations can vary by several orders of
magnitude (e.g., salt lakes, seawater intrusion) and in all cases, the earth’s geothermal gradient has
the potential to set up circulations where warm fluid at depth can rise and shallower cooler fluids
can sink. Simmons [2005] considered a typical groundwater hydraulic gradient of say a 1m
hydraulic head drop over a lateral distance of a kilometre i.e., a gradient of 1 in a 1000 and
showed that the equivalent “driving force” in density terms would be a density difference of 1
kg/m3 relative to a reference density of freshwater whose density is 1000 kg/m3. This is a
solution whose concentration is only about 2g/L (about 5% of seawater!) and this is quite dilute
in comparison to many plumes one may encounter in groundwater studies.
4.2 Possible manifestations of variable density flow
Whether density driven flow is important in a physical hydrogeologic setting is
determined by a number of competing factors. These involve a complex interplay between fluid
properties, the interaction within mixed convective flow of fluid motion driven by both density
differences (free convection) and hydraulic gradients (advection or forced convection) and
89
finally, the properties of the porous medium. The ability to maintain a density gradient and a
convective flow regime is dissipated by dispersive mixing processes. The density contrast and its
stable or unstable configuration (e.g., light fluid overlying dense fluid as in a stable case of
seawater intrusion, or dense fluid overlying light fluid as in the potentially unstable case of a
leachate plume migrating from a landfill) is also of critical importance. These will directly affect
solute transport processes and the way in which variable density flow processes are subsequently
manifested. For example, in a steady state coastal setting, density effects create a seawater wedge
that penetrates inland for some distance. Although tidal effects do create mixing, a thin
interfacial mixing-layer is maintained by a slow flow of less saline groundwater over the wedge.
This sort of rotational flow associated with seawater intrusion phenomena is the manifestation of
variable density flow physics and may be referred to as the ‘‘Henry circulation’’ [Henry, 1964]. In
some specialised cases, where there are large unstable density inversions, transport can be
characterised by rapid instability development where lobe-shaped instabilities (fingers) form and
sink under gravitational influence resulting in enhanced solute transport and greater mixing
compared to diffusion alone [Simmons et al., 2001]. The complexity of the problems involved
generally increases as one moves from situations where the light fluid overlies the dense fluid to
potentially unstable situations where the dense fluid overlies the less dense fluid. The challenges
for numerical simulation appear to grow in the same way, as will be discussed in later sections of
this chapter.
4.3 Dimensionless Numbers
It is important to note that, in all cases, the propensity for density driven flow is largest
where there are large density gradients and the porous media is of high permeability. In contrast,
dispersion acts to reduce the density gradient and therefore to dissipate density driven flow. Text
books such as Nield and Bejan [2006] and Holzbecher [1998] provide a comprehensive overview
of the theoretical treatment of density driven flow in porous media. Critical dimensionless
numbers such as the Rayleigh Number (Ra) and Mixed Convection ratio (M), are important
parameters used in the characterisation of such systems.
In simple free convective systems where mechanical dispersion is assumed independent
of convective flow velocity, the onset of instability is determined by the value of a
nondimensional number called the Rayleigh number (Ra). This dimensionless number is the
ratio between buoyancy driven forces and resistive forces caused by diffusion and dispersion.
Ra is defined by (e.g., Simmons et al., [2001]):
90
Ra =
U c H gk ( C max - C min )H Buoyancy and Gravitatio n
=
=
Diffusion and Dispersion
T Q 0 D0
Do
(1)
E
where Uc is the convective velocity, H is the thickness of the porous layer, D0 is the molecular
diffusivity, g is the acceleration due to gravity, k is the intrinsic permeability, E UwUwC is
the linear expansion coefficient of fluid density with changing fluid concentration, Cmax and Cmin
are the maximum and minimum values of concentration respectively, T is the aquifer porosity,
Q0=PU is the kinematic viscosity of the fluid. Simmons et al., [2001] and Simmons [2005]
discuss some of the difficulties encountered in the application of the Rayleigh number to
practical field based settings. For sufficiently high Rayleigh numbers greater than some critical
Rayleigh number Rac, gravity induced instability will occur in the form of waves in the boundary
layer that develop into fingers or plumes. This critical Rayleigh number defines the transition
between dispersive/diffusive solute transport (at lower than critical Rayleigh numbers) and
convective transport by density-driven fingers (at higher the critical Rayleigh numbers).
In many cases, the system we are studying is a mixed-convective system where both free
and forced convection operate to control solute distributions. In those cases, it is interesting to
examine the relative strengths of each process in controlling the resultant transport process. The
ratio of the density-driven convective flow speed to advective flow speed determines the
dominant transport mechanism. In the case of an isotropic porous medium the mixed
convection ratio may be written as (see for example, Bear [1972]):
M
§ 'U ·
¸¸
¨¨
U
o
¹
©
§ 'h ·
¨
¸
© 'L ¹
Free convective speed
Forced convective speed
(2)
If M>>1, then free convection is dominant. If M<<1, then forced convection is dominant.
Where M~1 they are of comparable magnitude. Here, ¨U is the density difference, URis the
lower reference density [both free convection parameters], ¨h is the hydraulic head difference
measured over a length ¨L [both forced convection parameters].
4.4 The use of Equivalent Freshwater Head in variable density analyses
Despite being one of the seemingly simplest methods of analysing variable density flow
systems, the correct application of the equivalent freshwater head is actually more complicated
than is often understood. Post et al., [in press] present a detailed analysis on the use of hydraulic
91
head measurements in variable density flow analyses. By way of introduction, freshwater head at
point i may be defined as:
h f ,i
zi Pi
(3)
Ufg
where zi (elevation head) represents the (mean) level of the well screen, Pi is pressure of the
ground water at the well screen, ȡf is fresh water density and g is acceleration due to gravity. A
common error that occurs in practice is for equivalent freshwater head to be used in the standard
non-density dependent version of Darcy’s Law that does not include a buoyancy gradient term
critical in vertical flow calculations. This leads to errors in predicting flow rates and directions
where vertical flow effects are of interest in vertically stratified layers (with respect to density) of
fluid. In a recent issue paper by Post et al., [in press], it is noted that the use of EFH without
care, can be particularly problematic. Indeed, Lusczynski [1961] who originally formulated the
concept recognized that equivalent freshwater heads cannot be used to determine the vertical
hydraulic gradient in an aquifer with water of non-uniform density. The term Environmental
Water Head (EWH) was established for this purpose [Lusczynski, 1961]. However, it can be
demonstrated that this Environmental Water Head is equivalent to using Equivalent Freshwater
Head where the appropriate density dependent version of Darcy’s Law is used (as was actually
shown in Appendix 3 of the original Lusczynski study). Post et al., [in press] argue that, provided
that the proper corrections are taken into account, fresh water heads can be used to analyze both
horizontal and vertical flow components. To avoid potential confusion, they recommended that
the use of the so-called environmental water head, which was initially introduced to facilitate the
analysis of vertical groundwater flow, be abandoned in favour of properly computed freshwater
head analyses.
Indeed, some variable density groundwater flow simulators such as FEFLOW and
SEAWAT (see Table 2) employ the equivalent freshwater head formulation in the variable
density version of Darcy’s Law in the familiar form (for the vertical flow component):
qz
ª wh
§U Uf
K f « f ¨
¨
¬« wz © U f
·º
¸»
¸
¹¼»
(4)
Where Kf is the fresh water hydraulic conductivity, hf is the freshwater head, ȡ is fluid density and
ȡf is fresh water density. Here the term
U Uf
, represents the relative density contrast, and
Uf
accounts for the buoyancy effect on the vertical flow. In the case of horizontal flow
92
components, this term is neglected in the respective horizontal flow velocity formulations in
Eqn. 3.
5. Modelling Variable Density Flow Phenomena
An excellent treatment on this subject was presented in a review article by Diersch and
Kolditz [2002] and the reader is referred to that material for an exhaustive discussion on this
topic. The limited number of analytical solutions for variable density flow problems creates a
heavy dependence on numerical simulators. A large and growing number of numerical simulators
now exist for the simulation of variable density flow phenomena, as illustrated in Table 2. Several
of these codes will be demonstrated in later sections of this book chapter where it will be seen
that there are numerous formulations that are used for governing equations and their numerical
solution. In this section, we demonstrate the approach to variable density flow modelling by
using the specific case of the widely used SUTRA [Saturated-Unsaturated TRAnsport Model]
numerical model [Voss, 1984] to demonstrate key issues in variable density flow simulation.
SUTRA is one of the earliest codes that emerged for the simulation of variable density flow
phenomena, and is in wide use today. A summary of the history of SUTRA and its use may be
found in Voss [1999].
5.1 Physical Processes, Governing Equations and Numerical Solutions
Using the example of the SUTRA (Saturated-Unsaturated-TRAnsport) code [Voss,
1984], we demonstrate the general form of the governing equations used by variable density
numerical models. The SUTRA code is a numerical solver of two general balance equations for
variable-density single-phase saturated-unsaturated flow and single-species (solute or energy)
transport based on Bear [1979]. Further details on this code may be found in Voss [1984]. The
summary presented below is originally found in the texts by Voss [1999] and Voss and Provost
[2003].
5.1.1. SUTRA Processes
Voss and Provost [2003] provide a detailed discussion on this subject. Simulation using
SUTRA is in two or three spatial dimensions. A pseudo-3D quality is provided for 2D, in that
the thickness of the 2D region in the third direction may vary from point to point. A 2D
simulation may be done either in the areal plane or in a cross sectional view. The 2D spatial
coordinate system may be either Cartesian (x,y) or radial-cylindrical (r,z). Areal simulation is
93
usually physically unrealistic for variable-density fluid and for unsaturated flow problems. The
3D spatial coordinate system is Cartesian (x,y,z).
Ground-water flow is simulated through numerical solution of a fluid mass-balance
equation. The ground-water system may be either saturated, or partly or completely unsaturated.
Fluid density may be constant, or vary as a function of solute concentration or fluid temperature.
SUTRA tracks the transport of either solute mass or energy in flowing ground water through a
unified equation, which represents the transport of either solute or energy. Solute transport is
simulated through numerical solution of a solute mass-balance equation where solute
concentration may affect fluid density. The single solute species may be transported
conservatively, or it may undergo equilibrium sorption (through linear, Freundlich, or Langmuir
isotherms). In addition, the solute may be produced or decay through first- or zero-order
processes. Energy transport is simulated through numerical solution of an energy-balance
equation. The solid grains of the aquifer matrix and fluid are locally assumed to have equal
temperature, and fluid density and viscosity may be affected by the temperature.
Most aquifer material, flow, and transport parameters may vary in value throughout the
simulated region. Sources and boundary conditions of fluid, solute and energy may be specified
to vary with time or may be constant. SUTRA dispersion processes include diffusion and two
types of fluid velocity-dependent dispersion. The standard dispersion model for isotropic media
assumes direction-independent values of longitudinal and transverse dispersivity. A flowdirection-dependent dispersion process for anisotropic media is also provided. This process
assumes that longitudinal and transverse dispersivities vary depending on the orientation of the
flow direction relative to the principal axes of aquifer permeability.
5.1.2 SUTRA Governing Equations
The general fluid mass balance equation that is usually referred to as the 'ground-water flow
model' is:
§
wS
¨¨ S w US0p HU w
wp
©
ª§ kk U ·
º
· wp §
wU · wU
¨ HS w
¸¸
’ ˜ «¨¨ r ¸¸ ˜ ’p Ug»
¸
wU ¹ wt
¹ wt ©
¬© P ¹
¼
where:
Sw
is the fractional water saturation [1]
H
is the fractional porosity [1]
p
is the fluid pressure [kg/m•s2]
94
Qp
(5)
t
is time [s]
U
is either solute mass fraction, C [kgsolute/kgfluid], or temperature, T [°C]
k
is the permeability tensor [m2]
kr
is the relative permeability for unsaturated flow [1]
P
is the fluid viscosity P(T ) [kg/ m•s]
g
is the gravity vector [m/s2]
Qp
is a fluid mass source [kg/m3•s].
Additionally,
is the fluid density [kg/m3], expressed as:
U
U
U0 wU
U U0 wU
(6)
where:
U0
is the reference solute concentration or temperature
U0
is fluid density at U0
wU
wU
is the density change with respect to U (assumed constant)
and,
S 0p
is the specific pressure storativity [kg/m•s2]-1
S 0p
1 HD HE
(7)
where:
D
is the compressibility of the porous matrix [kg/m•s2]-1
E
is the compressibility of the fluid [kg/m•s2]-1.
Fluid velocity, v [m/s], is given by:
v
§ kk U ·
¨¨ r ¸¸ ˜ ’p Ug
© HS w P ¹
(8)
95
The solute mass balance and the energy balance are combined in a unified solute/energy balance
equation usually referred to as the 'transport model':
>HSwUc w 1 HUsc s @ wU HSwUc w v ˜ ’U - ’ ˜ ^Uc w >HSw VwI D 1 HVsI@ ˜ ’U`
wt
*
Qpc w U U
HSw UJ1wU 1 H
Us J1sUs
HSw UJ0w
(9)
1 H
Us J 0s
where:
cw
is the specific heat capacity of the fluid [J/kg·°C]
cs
is the specific heat capacity of the solid grains in porous matrix [J/kg·°C]
Us
is the density of the solid grains in porous matrix [kg/m3]
Vw
is the diffusivity of energy or solute mass in the fluid (defined below)
Vs
is the diffusivity of energy or solute mass in the solid grains
(as defined below)
I
is the identity tensor [1]
D
is the dispersion tensor [m2/s]
U*
is the temperature or concentration of a fluid source (defined below)
J1w
is the first-order solute production rate [s-1]
J1s
is the first-order sorbate production rate [s-1]
J 0w
is an energy source [J/kg·s], or solute source [kgsolute/kgfluid·s],
within the fluid (zero-order production rate)
J 0s
is an energy source [J/kg·s], or solute source [kgsolute/kgfluid·s],
within the fluid (zero-order production rate)
In order to cause the equation to represent either energy or solute transport, the following
definitions are made:
for energy transport
U{T
U* { T *
Vw {
Ow
Uc w
96
Vs {
Os
Uc w
J1w { J1s { 0
for solute transport
U{C
Us { C s
U* { C *
V w { Dm
Vs { 0
c s { N1
cw { 1
where:
Ow
is the fluid thermal conductivity [J/s·m·°C]
Os
is the solid-matrix thermal conductivity [J/s·m·°C]
T
is the temperature of the fluid and solid matrix [°C]
C
is the concentration of solute (mass fraction) [kgsolute/kgfluid]
T*
is the temperature of a fluid source [°C]
C*
is the concentration of a fluid source [kgsolute/kgfluid]
Cs
is the specific concentration of sorbate on solid grains [kgsolute/kggrains]
Dm
is the coefficient of molecular diffusion in porous medium fluid [m2/s]
N1
is a sorption coefficient defined in terms of the selected equilibrium sorption isotherm
(see Voss (1984) for details).
The dispersion tensor is defined in the classical manner:
Dii
v
Dij
v
2
d v
(10a)
2
dL dT v iv j (10b)
2
L i
dT v 2j
for i
x, y and j
but, i z j
x, y
with:
dL
DL v
(10c)
dT
DT v
(10d)
where:
dL
is the longitudinal dispersion coefficient [m2/s]
dT
is the transverse dispersion coefficient [m2/s]
97
DL
is the longitudinal dispersivity [m]
DT
is the transverse dispersivity [m]
A very useful generalization of the dispersion process is included in SUTRA allowing the
longitudinal and transverse dispersivities to vary in a time-dependent manner at any point
depending on the direction of flow. The generalization is:
DL
DL max DL min
(11a)
DL min cos 2 Tkv DL max sin2 Tkv
DT
D T max D T min
(11b)
D T min cos 2 Tkv D T max sin2 Tkv
where:
DL max
is the longitudinal dispersivity
for flow in the maximum permeability direction [m]
DL min
is the longitudinal dispersivity
for flow in the minimum permeability direction [m]
D T max
is the transverse dispersivity
for flow in the maximum permeability direction [m]
D T min
is the transverse dispersivity
for flow in the minimum permeability direction [m]
Tkv
is the angle from the maximum permeability direction
to the local flow direction [degrees].
These relations (11a,b) provide dispersivity values that are equal to the square of the radius of an
ellipse that has its major axis aligned with the maximum permeability direction. The radius
direction is the direction of flow.
In the case of variable density saturated flow with non-reactive solute transport of total dissolved
solids or chloride, and with no internal production of solute, equations (5), (8) and (9), are greatly
simplified. The fluid mass balance is:
98
US0p wwpt §¨ H wwUU ·¸ wwUt ’ ˜ ª«§¨¨ kPU ·¸¸ ˜ ’p Ugº»
©
¹
©
¹
¼
Qp
(12)
The fluid velocity is given by:
v
§ kU ·
¨¨ ¸¸ ˜ ’p Ug
© HP ¹
(13)
and the solute mass balance is:
HU wU HUv ˜ ’C - ’ ˜ >HUDmI D ˜ ’U@
wt
Qp C* C
(14)
5.1.3 SUTRA Numerical Methods
Again, using the example code SUTRA, we demonstrate how one popular variable
density numerical model solves the above governing equations. A previous summary of the
discussion by Voss [1999] on this subject is repeated below.
The numerical technique employed by the SUTRA code to solve the above equations
presented in Section 5.1 uses a modified two-dimensional Galerkin finite-element method with
bilinear quadrilateral elements. Solution of the equations in the time domain is accomplished by
the implicit finite-difference method. Modifications to the standard Galerkin method that are
implemented in SUTRA are as follows. All non-flux terms of the equations (e.g. time derivatives
and sources) are assumed to be constant in the region surrounding each node (cell) in a manner
similar to integrated finite-differences. Parameters associated with the non-flux terms are thus
specified nodewise, while parameters associated with flux terms are specified elementwise. This
achieves some efficiency in numerical calculations while preserving the accuracy, flexibility, and
robustness of the Galerkin finite-element technique. Voss [1984] gives a complete description of
these numerical methods as used in the SUTRA code.
An important modification to the standard finite-element method that is required for
variable-density flow simulation is implemented in SUTRA. This modification provides a
velocity calculation within each finite element based on consistent spatial variability of pressure
gradient, ’p , and buoyancy term, Ug , in Darcy’s law, equation (13). Without this ‘consistent
velocity’ calculation, the standard method generates spurious vertical velocities everywhere there is
a vertical gradient of concentration within a finite-element mesh, even with a hydrostatic
99
pressure distribution [Voss, 1984; Voss and Souza, 1987]. This has critical consequences for
simulation of variable density flow phenomena. For example, the spurious velocities make it
impossible to simulate a narrow transition zone between fresh water and seawater with the
standard method, irrespective of how small a dispersivity is specified for the system.
The two governing equations, fluid mass balance and solute mass (or energy) balance, are
solved sequentially on each iteration or time step. Iteration is carried out by the Picard method
with linear half-time-step projection of non-linear coefficients on the first iteration of each time
step. Iteration to the solution for each time step is optional. Velocities required for solution of
the transport equation are the result of the flow equation solution (i.e. pressures) from the
previous iteration (or time step for non-iterative solution).
SUTRA is coded in a modular style, making it convenient for sophisticated users to
modify the code (e.g. replace the existing direct banded-matrix solver) or to add new processes.
Addition of new terms (i.e. new processes) to the governing equations is a straightforward
process usually requiring changes to the code in very few lines. Also to provide maximum
flexibility in applying the code, all boundary conditions and sources may be time-dependent in
any manner specified by the user. To create time-dependent conditions, the user must modify
subroutine BCTIME. In addition, any desired unsaturated functions may be specified by usermodification of subroutine UNSAT.
Two numerical features of the code are not often recommended for practical use:
upstream weighting and pinch nodes. Upstream weighting in transport simulation decreases the
spatial instability of the solution, but only by indirectly adding additional dispersion to the
system. For full upstream weighting, the additional longitudinal dispersivity is equivalent to onehalf the element length along the flow direction in each element. Pinch nodes are sometimes
useful in coarsening the mesh outside of regions where transport is of interest by allowing two
elements to adjoin the side of a single element. However, pinch nodes invariably increase the
matrix bandwidth possibly increasing computational time despite fewer nodes in the mesh.
5.2 Testing variable-density aspects of variable density numerical codes
Diersch and Kolditz [2002] provide a very comprehensive summary of model test cases
and benchmarks, including details on the successes and limitations of each case. As summarised
from that article, the major test cases that are used for numerical simulators currently include:
1. The Hydrostatic Test: This type of benchmark is quite simple, but is very instructive. It is
a test of the velocity consistency under hydrostatic and sharp density transition
100
conditions as originally proposed by Voss and Souza [1987]. A closed rectangular region
containing fresh water above highly saline water, separated by a horizontal transition
zone one element wide and with a hydrostatic pressure distribution [Voss and Souza,
1987]. No simulated flow may occur in the cross section under transient or steady state
conditions.
2. The Henry seawater intrusion problem: This test case has been performed on coarse
meshes [Voss and Souza, 1987] and on a very fine mesh [Segol, 1993]. Note that until
Segol [1993] improved Henry’s approximate semi-analytical solution [Henry, 1964], this
test was merely an inter-comparison of numerical code results. No published results had
matched the Henry solution. Segol’s result provides a true analytic check on a variabledensity code. Some confusion has existed in the literature over the correct choice of
diffusion to use in the model test case and apparently two diffusivity values have been
employed that differ by a factor of porosity. In addition to this problem, the Henry
problem has some other deficiencies. An unrealistically large amount of diffusion is often
introduced which results in a wide transition zone. This assists with the numerical
solution, making the solution smooth and less problematic. However, as pointed out by
Diersch and Kolditz [2002], the Henry problem is not appropriate for testing purely
density-driven flow systems. This is because a large component of the flow system
response is actually driven by forced convection and not free convection. As a result,
additional benchmark tests are necessary to test numerical models for free convection
problems (such as the Elder problem) and for cases with very narrow transition zones
(such as the salt dome problem).
3. The Elder natural convection problem: [Voss and Souza, 1987], an unstable transient
problem consisting of dense fluid circulating downwards under buoyancy forces from a
region of high specified concentration along the top boundary. The Elder problem
serves as an example of free convection phenomena, where the bulk fluid flow is driven
purely by fluid density differences. Elder [1967a, 1967b] presented both experimental
results in a Hele-Shaw cell and numerical simulations concerning the thermal convection
produced by heating a part of the base of a porous layer (what we often call now the
“Short Heater Problem”). The Elder problem is a very popular benchmark case and its
convective flow circulation patterns (see Figure 3) are still the subject of scientific
discussion. No exact or qualified measurements exist for the Elder problem and as such
101
the test case relies on an intercode comparison between numerical simulators. The main
issue that has been discussed widely in relation to this problem is the nature of both the
number of fingers and either the central upwelling or downwelling patterns, which are
now recognised to be highly dependent on the nature of the grid discretisation (i.e.,
coarse, fine, extremely fine grids all lead to very different spatiotemporal patterns of
behaviour).
It is likely that convergence has been achieved at extremely fine grid
resolutions (15,000-1,000,000 nodes) as reported in Diersch and Kolditz [2002].
4. The HYDROCOIN salt-dome problem: [e.g. see Konikow et al., 1997], a steady state
and transient problem, initially consisting of a simple forced fresh-water flow field
sweeping across a salt dome (specified concentration located along the central third of
the bottom boundary). This generates a separated region of brine circulating along the
bottom. The nature of the circulation patterns has been shown to be sensitive to the way
in which the bottom boundary condition is represented and implemented, and the
importance choice of mechanical dispersion. These are seen to affect the circulation
patterns that arise in the system.
5. The Salt Lake problem: First introduced by Simmons et al., [1999], this is a very
complicated, transient system in which an evaporation-driven density driven fingering
process evolves downwards in an area of upward-discharging ground water in a HeleShaw cell experiment (see Figure 4). Numerical results are compared with laboratory
measurements (see Figure 5). There has been much discussion on this test case in recent
literature. In particular, the number of fingers and their temporal evolution appear to be
intimately related to the choice of numerical discretisation and the numerical solution
procedure. Interestingly, and paradoxically, several authors have now found that better
agreement between numerical results and experimental results is obtained using a coarser,
rather than finer, mesh [Mazzia et al., 2001; Diersch and Kolditz, 2002]. Numerical grid
convergence has not been achieved for this problem. The higher Rayleigh number of this
test case problem (about 10 times larger than for the Elder problem) is clearly in a range
where oscillatory convection flow regimes are expected. In comparison to the Elder
problem, the salt lake problem is much more complicated. Grid convergence for the
Elder problem required extremely fine numerical resolution, and we therefore expect this
to be increased further again for the case of the salt lake problem. The major challenge
here is that this problem is extremely dynamic and that it is unstable. Moreover,
102
numerical perturbations and dispersion can both trigger fingering and control associated
dispersion (creating either fatter or skinner fingers!) and these are all intimately related to
both the patterns of finger formation as well as their subsequent growth/decay
processes. It is clear that the numerical simulation of such phenomena is extremely
challenging and appears to be currently unresolved.
6. The Salt Pool Problem: The saltpool problem was introduced by Johannsen et al., [2002].
It represents a three dimensional saltwater upconing process in a cubic box under the
influence of both density driven flow and hydrodynamic dispersion. A stable layering of
saltwater below freshwater is considered in time for both a low density case (1% salt
mass fraction) and a high density case (10% salt mass fraction). A laboratory experiment
was used as part of the model testing process. As freshwater sweeps over the stable
saltier water at the bottom of the cube, the breakthrough curves at the outlet point are
measured. The position of the salt water – freshwater interface was measured using
nuclear magnetic resonance (NMR) techniques. Numerous authors have attempted to
model this problem and have achieved variable success. The challenge appears to lie in
the small dispersivity values and the large density contrasts, particularly in the high
density cases. The experiment and test case are resolved for low density contrasts and for
early time behaviour. Some questions do remain about the simulation of high density
contrasts and late time behaviour. Previous studies have emphasised the need for a
consistent velocity approximation in the accurate simulation of the salt pool problem. As
noted by Diersch and Kolditz [2002], the saltpool experiment does provide reliable
quantitative results for a three-dimensional saltwater mixing process under density effects
which were not available prior to its introduction. It is seen that extreme mesh
refinement is required in order to accurately simulate the problem.
More recently than the publication of the work by Diersch and Kolditz [2002], a very new test
case suite has been proposed by Weatherill et al., [2004] that has an exact and irrefutable
solution based upon an analytic stability theory. This is:
7. Convection in infinite, finite, and inclined porous layers: A body of work was developed
in the 1940’s originally by Horton and Rogers [1945] and independently by Lapwood
[1948] to examine the convective stability of an infinite layer with an unstable density
configuration applied across it. The work has an exact analytical solution for both the
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onset of convection (a critical Rayleigh number of 4ư2) and the wavelength of the
convection pattern. Unlike all previous test cases, the availability of analytically based
stability criteria offers an excellent way of testing a numerical code, which does not rely
on a comparison with other numerical simulators. Additionally, these stability criteria
have been applied in traditional convection problems in the field of fluid mechanics for
many decades both within numerical and laboratory based frameworks – from a
scientific viewpoint they are well understood, robust and irrefutable. They are therefore
ideal for testing numerical models. In their study, Weatherill et al., [2004] compared
numerical results using SUTRA with these previous studies and showed that SUTRA
results (for both the onset conditions for instability and its resultant wavelength) were in
excellent agreement with the traditional stability criteria. Numerical solutions to
extensions of this original problem, namely cases of finite layers and inclined layers
(where analytical stability criteria also exist), were seen to be in excellent agreement with
stability criteria and previous results reported in the traditional fluids mechanics
literature. Given the availability of exact and irrefutable stability criteria for these “box
problems”, it is expected that they will become commonplace in future numerical model
testing. They also have solutions which can be extended into 3D benchmark test cases.
5.3
Future challenges for numerical modelling
Simmons [2005] and Diersch and Kolditz [2002] highlighted some current challenges that
face the field of variable density groundwater flow and these clearly have critical implications for the
modelling of such phenomena. Simmons [2005] identified a number of emerging challenges and
suggested that there is little, if any, evidence in current literature that points to the future extinction
of this field of hydrogeology. Future drivers for both research and practical application in this area
will drive numerical modelling needs. Just some future drivers for variable density flow research,
and hence modelling, identified by Simmons [2005] included, amongst others, : (i) the need to better
understand the relationship between variable density flow phenomena and dispersion, (ii) better
geological constraints on variable density flow analyses using lineaments, sedimentary facies data
and structural properties of fractured rock aquifers, (iii) improving the resolution of geophysical,
geoprospecting and remote sensing tools for non-invasive characterisation of dense plume
phenomena and heterogeneity, (iv) linking the fields of tracer and isotope hydrogeology and variable
density flow phenomena, (v) double-diffusive and multiple species transport problems in variable
density flow phenomena, (vi) links between climate change phenomena and the response of variable
density flow phenomena, such as sea level and coastline position change, and the impact of
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transgression and/or regression cycles on aquifer salinisation will be critical, (vii)the gas-liquid phase
chemistry of carbon sequestration processes and its efficiency, safety and long term viability as a
storage option will necessarily involve an understanding of variable density transport dynamics with
multiple phases, (viii) links between variable density flow phenomena and a number of
hydroecological applications (e.g., how does variable density flow effect baseflow accessions and
other surface-groundwater interactions, or salt budge profiles under vegetation, or the
spatiotemporal distribution of stygofauna and biota in subsurface groundwater ecosystems?), (ix)
Some of the above issues point to the need for more detailed and accurate coupling of surface
water, vadose zone and groundwater models – and which will drive an inherent increase in the
complexity of the modelling approach. The numerous future possibilities outlined above suggest
both short term and long term prognoses for variable density flow research and its applications are
very good. It is also therefore logical to conclude that there are many issues that remain largely
unexplored and poorly understood in the numerical modelling of variable density phenomena.
Some major future challenges for the field of variable density flow phenomena are
summarised below. Here, those matters of particular importance to the modelling of variable
density flow systems are highlighted. This extended analysis is based on the treatment presented
recently by Simmons [2005]. Some major challenges identified in that study include:
1. Large mix of spatial and temporal scales: Variable density flow phenomena may be triggered,
grow and decay over a very large mix of different spatial and temporal scales. The particular
challenge lies in the fact that information on very small spatial scales and short time scales is
needed to feed into long time and large spatial scale processes and simulations. Measuring field
scale parameters across this large range of spatial and temporal scales as input for modelling
approaches is a major challenge. Additionally, many current groundwater systems are highly
transient and steady assumptions may be an oversimplification, especially where solute transport
is concerned. For example, tidal oscillations and sealevel rise create the need for transient
analyses in seawater intrusion phenomena. In unstable problems, the onset and growth/decay
of instabilities requires a truly transient analysis of the system. Furthermore, the source of dense
plumes (e.g., leachate from a landfill) are often represented by simplified constant head, flux or
concentration boundary conditions and yet the style of loading is expected to be important
[Zhang and Schwartz, 1995]. There are challenges in better understanding how transient
boundary conditions will affect variable density flow phenomena (e.g., continuous or
intermittent sources).
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2. Heterogeneity and dispersion: It is only more recently that heterogeneity has been considered in
the study of dense plume migration [e.g., Schincariol and Schwartz, 1990; Schincariol et al.,
1997]. Schincariol et al., [1997] explain that heterogeneity in hydraulic properties can perturb
flow over many length scales (from slight differences in pore geometry to larger
heterogeneities at the regional scale), triggering instabilities in density stratified systems. This
is particularly true for physically unstable situations but is also important in other situations.
For example, the mixing process along a saltwater-freshwater interface can be significantly
modified by nonuniform velocities. This can lead to local physical instabilities resulting in a
corrugated interface that at larger scales is interpreted as increased dispersion. Modelling
narrow transition zones and low dispersion situations is challenging, most often requiring
very fine numerical grids and increased computational power. Current research is suggesting
that heterogeneity may serve as a physical perturbation in fingering processes and is therefore a
critical feature controlling the onset, growth and/or decay in variable density flow processes.
For example, low-permeability lenses can effectively dampen instability growth or even
completely stabilize a weakly unstable plume. Dense plume migration, particularly at high
density differences and in highly heterogeneous distributions, is therefore not easily
amenable to prediction [Schincariol et al., 1997; Simmons et al., 2001; Nield and Simmons,
2006]. Whilst some studies have examined dense plume migration in fractured rock using
numerical models [Shikaze et al., 1998; Graf and Therrien, 2005], there is still a need to explore
how more complex and realistic fracture geometries affect variable-density flow processes and
to better understand the links with macroscopic dispersion [e.g., Welty et al., 2003; Kretz et al.,
2003; Schotting et al., 1999].
3. Fluids, solutes and fluid-matrix interaction: Previous studies have typically been limited to single
species conservative transport but some have examined heat and solute transport
simultaneously within thermohaline convection regimes [see Diersch and Kolditz, [2002] for a
discussion on this topic in their exhaustive review of variable density modelling] which may be
important in understanding many processes including mineralization and ore formation in
sedimentary basins, geothermal extraction processes and flow near salt domes. The
simultaneous interaction of heat and solute creates challenges of its own and this complexity is
exemplified where multiple species with differing diffusivities interact. In a subset of these
“double-diffusive” processes, instabilities can occur even where the net density is increasing
downwards. Here, it is the difference in the diffusivities of the heat and solute that drives
transport and it should be realised that the simultaneous interaction of heat and solute creates
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many challenges of its own and that this complexity will be exemplified where multiple species
with differing diffusivities interact. Multiple species studies (beyond the two species heat and
salt) are fairly limited in groundwater literature. However, more recent studies [e.g., Zhang and
Schwartz, 1995] suggest that the chemical composition and reactive character of a plume can
greatly influence plume dynamics. Other studies have begun to examine how chemical reactions
can be coupled with density-dependent mass transport. For example, Freedman and Ibaraki
[2002] incorporated equilibrium reactions for the aqueous species, kinetic reactions between the
solid and liquid phases, and full coupling of porosity and permeability changes that result from
precipitation and dissolution reactions in porous media and showed that complex concentration
distributions result in the variable density flow system. Recently, Ophori [1998] suggested that
the variable viscosity nature of fluids is typically ignored in variable density groundwater flow
problems and suggest that plume migration pathways and rates are inaccurately predicted in the
absence of the variable-viscosity relationship. Other studies have suggested that dense plume
migration is significantly affected by the extent of porous medium saturation. Thorenz et al.,
[2002] and Boufadel et al., [1997] demonstrate that significant lateral flows and coupled densitydriven flow may take place in the partially saturated region above the water table and at the
interface between the saturated and partially saturated zones. All these studies suggest that
further investigation in relation to multiple species transport, non-conservative solute transport,
fluid-matrix interactions and a more complex representation of the chemical and physical
properties of the dense fluid than has previously been assumed is warranted. The coupling of
variable density flow phenomena with more complex chemical and fluid property
characterisation is an area that is still in its infancy and warrants further exploration. As
additional complexity is added to the numerical simulator in this area, there will be a need to
identify new test cases that actually test those newly added features of the code.
4. Numerical modelling: In general, variable density flow problems do not lend themselves easily
to analytical study except under the most simplified of conditions. However, a number of
numerical codes such as those outlined in Table 2 now exist to simulate variable density flow
phenomena. There has been good success simulating variable-density flow at low to
moderate density contrasts and in benchmarking the codes that are used. As was seen in the
analysis of test case problems, numerical modelling efforts were most successful in the cases
of stable problems (e.g., the Henry salt water intrusion case), but increasing problems are
encountered in the case of unstable problems (e.g., confusion over downwelling/upwelling
patterns and the massive computational power required for grid convergence in the case of
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the Elder problem, and the currently unresolved nature of the Salt Lake problem where grid
convergence has not currently been achieved). But it appears that even seemingly simple
matters in numerical modelling remain unresolved and appear to be currently overlooked.
For example, Kooi [submitted], noted that Fickian-type dispersive flux equations are
implemented in models where the dispersive flux is linearly related to the gradient of solute
concentration. He argued that this classical approach, in combination with solute boundary
conditions, fails to properly represent back-dispersion at outflow boundaries and produces
spurious, unphysical dispersive boundary layers. It is clear from Kooi’s [submitted] analysis
that greater insight in the exact nature of dispersive transport at outflow boundaries and
practical ways to represent this in models.
It is clear that variable density problems increase in modelling complexity as one moves
from stable to unstable configurations. The modelling complexity and computational
demand is compounded for large density difference in both stable (e.g., Salt Pool problem)
and unstable problems (e.g., Salt Lake Problem) and in cases where physical dispersion is
very small. Additionally, artefacts of the numerical scheme itself can mimic physically
realistic but perhaps completely inaccurate results (e.g., dispersion enhancing the transition
zones in stable problems, or enhancing the width and reducing their speed of unstable
fingers). The problem is compounded at much higher (unstable) density differences and
hence Rayleigh numbers where the physical problem itself is characterised by chaotic and
oscillatory regimes such as that seen in the recently proposed “Salt Lake Problem” [Simmons
et al., 1999]. What is particularly problematic in the simulation of such unstable phenomena
is that small differences in dispersion parameters or spatial and temporal discretization can
cause very different types of instabilities to be generated [Diersch and Kolditz, 2002]. The
nature of physical perturbations (c.f. numerical perturbations) is almost impossible to
measure and simulate. When using standard numerical approaches, one may attempt to
control numerical perturbations by reducing numerical errors through the use of very fine
meshes and small timesteps but the nature of what physical perturbation is present in the
system in order to generate particular patterns of instabilities is never known. It is almost
impossible to resolve, therefore, what are numerical artifacts and what are physical realities.
Furthermore, as noted by Diersch and Kolditz [2002], one must then ask what perturbation
is no longer significant for a convection process, and what error in the numerical scheme can
be tolerated. These authors note that there are two possible dangers in simulating such highly
nonlinear convection problems within a numerical framework. Firstly, a scheme which is
only conditionally stable, but of higher accuracy, is prone to create non-physical
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perturbations due to small perturbations in the solutions unless the element size and the
timestep are sufficiently small. Conversely, an unconditionally stable numerical scheme but
with lower accuracy, smooths out physically induced perturbations, unless the element size
and the timestep are sufficiently small. It is clear that a very fine spatial and temporal
resolution are essential if high density contrast convection phenomena are to be modelled.
One may conclude that the simulation of unstable phenomena, particularly highly unstable
cases, remains a challenge for further research.
In addition to the lack of control over numerical perturbations and the inability to
quantify the physical perturbation in real convective systems, it is also questionable whether
oscillatory or chaotic regimes such as those encountered at high Rayleigh numbers can be
simulated and predicted by a deterministic modelling approach. In more general terms, it
can be seen that variable density flow problems are currently limited by the need to simulate
large density contrast and low dispersion systems. Both are massively computationally
demanding. As noted by Diersch and Kolditz [2002], large-scale variable density simulations
have an inherent need for expensive numerical meshes. Some of the most promising
developments in the numerical modelling of real-field based problems include adaptive
techniques, unstructured meshes and parallel processing. Diersch and Kolditz [2002] note
that adaptive and unstructured irregular geometries are prone to uncontrollable perturbations
and that robust, efficient and accurate strategies are required, particularly for threedimensional applications. Given these numerical complexities and demands, a critical matter
that warrants attention in modelling of such systems is the level of simplification that is
possible without loss of physical information and accuracy. In many cases the science in
relation to this question is ambiguous, unresolved or is simply ignored. For example, when can
homogeneous equivalents for heterogeneous systems be employed? Or, when is a 2D analysis
of a 3D system permissible? What are the implications of such a simplification? The complexitysimplicity matter must be considered carefully in the modelling of any variable density flow
system. In many cases, it appears that the decision is made implicitly, often with little or no
justification. Without a systematic evaluation of modelling complexity and the subsequent
implementation of a parsimonious modelling framework, it is readily apparent that the
implication of taking either a complex or simple approach may often not be properly
understood. Finally, and perhaps most importantly, there is also a real challenge for greatly
improved field measurements and observations for field truthing of models, minimising their
non-uniqueness and to inspire greater confidence in model predictive output.
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6. Applications and case studies
In this section, we demonstrate a range of case studies and applications, where variable
density flow is considered important. These examples include: saltwater intrusion and tidally
induced phenomena, salinisation associated with transgression and regression cycles, variable density
processes in aquifer, storage and recovery, flow and transport modelling in fractured rock aquifers,
and, chemically reactive transport modelling. Many of these examples are relevant to arid zone areas
but importantly extend beyond them. These applications and case studies are designed to illustrate
important areas where the field of variable density flow modelling is developing. The applications
and case studies are designed to be illustrative rather than exhaustive.
6.1 Seawater Intrusion and Tidally Induced Phenomena
6.1.1 Background
Seawater intrusion (or saltwater intrusion) is the encroachment of saline waters into
zones previously occupied by fresh groundwater. Under natural conditions, hydraulic gradients
in coastal aquifers are towards the sea and a stable interface between the discharging
groundwater and seawater exists, notwithstanding climatic events or sea-level rise [e.g.
Cartwright et al., 2004; Kooi et al., 2000]. Persistent disturbances in the hydraulic equilibrium
between the fresh groundwater and denser seawater in aquifers connected to the sea produce
movements in the position of the seawater-freshwater interface, which can lead to degradation of
freshwater resources. The impacts of seawater intrusion are widespread, and have lead to
significant losses in potable water supplies and in agricultural production [e.g. Barlow, 2003;
Johnson and Whitaker, 2003; FAO, 1997].
Seawater intrusion is typically a complex three-dimensional phenomenon that is
influenced by the heterogeneous nature of coastal sediments, the spatial variability of coastal
aquifer geometry and the distribution of abstraction bores. The effective management of coastal
groundwater systems requires an understanding of the specific seawater intrusion mechanisms
leading to salinity changes, including landward movements of seawater, vertical freshwaterseawater interface rise or “up-coning”, and the transfer of seawater across aquitards of multiaquifer systems [e.g., van Dam, 1999; Sherif, 1999; Ma et al., 1997]. Other processes, such as relic
seawater mobilisation, salt spray, atmospheric deposition, irrigation return flows and water-rock
interactions may also contribute to coastal aquifer salinity behaviour, and need to be accounted
for in water resource planning and operation studies [e.g., Werner and Gallagher, 2006; Kim et
al., 2003; FAO, 1997].
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The investigation of seawater intrusion is a challenging undertaking, and requires careful
consideration of groundwater hydraulics and abstraction regimes, the influence of water density
variability, the hydrochemistry of the groundwater and the inherent dynamics of coastal systems
(i.e. tidal forcing, episodic ocean events such as storm surges and tsunamis, salinity and water
level variation in estuaries and other tidal waterways, amongst others). The challenges of
seawater intrusion investigation have led to the development of specific analytical methods and
mathematical modelling tools that enhance the interpretation of field-based observations, which
are invariably too sparse to provide a complete representation of seawater intrusion processes
and trends.
6.1.2 Analytical Solutions
The simplest analyses of seawater intrusion adopt the assumption of an sharp seawaterfreshwater interface, giving rise to analytical solutions to interface geometry. The GhybenHerzberg approximation (developed independently by B. Ghyben in 1888 and by A. Herzberg in
1901; FAO, 1997) for the depth of the interface below the watertable was the first such solution
and was based on hydrostatic conditions, given as:
z
Uf
Us U f
hf
(15)
where z is the depth below sea level of the interface (m), Uf and Us are the freshwater and
saltwater densities (kgm-3), and hf is the height of the watertable above sea level (m). Ghyben and
Herzberg, working independently, showed that sea water actually occurred at depths below sea
level equivalent to approximately 40 times the height of freshwater above sea level. Glover
(1964) modified the solution to account for a submerged seepage face, as:
z
Uf
q'
Us U f K
Uf
2q ' x
Us U f K
(16)
where q’ is the discharge per unit width of aquifer (m2/s), K is the hydraulic conductivity (ms-1)
and x is the distance from the ocean boundary (m). Further advances in analytical solutions to
sharp interface problems include the works of Kacimov and Sherif [2006], Dagan and Zeitoun
[1998], Naji et al. [1998], amongst others.
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Sharp interface approaches are only applicable to settings where interface thicknesses are
small in proportion to the total aquifer depth. Alternately, the dispersive nature of salt transport
needs to be considered and numerical modelling approaches are therefore ultimately required in
most practical applications in order to solve the coupled groundwater flow and solute transport
equations.
6.1.3 Numerical Models
A major limitation of the sharp-interface approach is that the movement of brackish
groundwater (i.e. diluted seawater) is not able to be analysed in the transition zone. Where the
diffusive behaviour of solute transport is a necessary component of seawater intrusion analyses,
models are applied that adopt numerical methods to resolve the groundwater flow and solute
transport equations. The computational effort of simulating density-coupled groundwater flow
and solute transport is usually an imposition that influences the spatial and temporal resolution
of simulations, and model grid design usually involves trade-offs between model run-times and
the resolution of simulation. This is particularly important for investigations of large areas
and/or where simulations are carried out in a three-dimensional domain.
The U.S. Geological Survey’s finite-element SUTRA code [Voss, 1984] is arguably the
most widely applied simulator of seawater intrusion and other density-dependent groundwater
flow and transport problems in the world [Voss, 1999]. Several other finite-element codes are
capable of seawater intrusion modelling, including the popular FEFLOW model [Diersch, 2005],
FEMWATER [Lin et al., 1997], etc. SUTRA has been applied to a wide range of seawater
intrusion problems, ranging from regional-scale assessments of submarine groundwater
discharge (SGD) [e.g., Shibuo et al., 2006] to riparian-scale studies of estuarine seawater intrusion
under tidal forcing effects (e.g. Werner and Lockington, [2006]). Gingerich and Voss [2005]
demonstrate the application of SUTRA to the Pearl Harbour aquifer, southern Oahu, Hawaii in
analysing the historical behaviour of the seawater front during 100 years of pumping history.
The list of seawater intrusion simulators includes hybrids of the popular finite-difference
MODFLOW [Harbaugh, 2005] including the groundwater flow code SEAWAT [Langevin et al.,
2003] and MODHMS [HydroGeoLogic Inc., 2003]. The advantage of these codes is that they
can be applied to existing MODFLOW models through relatively straightforward modifications
to MODFLOW input data files, thereby reducing model development effort. Further, existing
MODFLOW pre- and post-processors can be utilised. Robinson et al. [2007] applied SEAWAT
to the beach scale problem of groundwater discharge under tidal forcing, and predicted the
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temporal behaviour of submarine groundwater discharge and the geometry of salt plumes at the
coastal boundary.
Werner and Gallagher [2006] demonstrate the application of MODHMS to seawater
intrusion at the regional scale, by adapting an existing hydro-dynamically calibrated MODFLOW
model of the Pioneer Valley coastal plain aquifers. They used the model to demonstrate that
seawater intrusion had not reached an equilibrium state, and that alternative salinisation
processes (e.g., basement dissolution, relic seawater mobilisation, sea salt spray, irrigation salinity)
produced observed salinity patterns in some parts of the study area. Figure 6 shows the original
groundwater model design, Figure 7 provided results from the calibrated model process and
finally concentration distributions are provided in Figure 8. The modelling suggested that the
salinity in the Pioneer Valley is not all caused by classic seawater intrusion (i.e. seawater entering
the aquifer from the ocean). Werner and Gallagher [2006] described the development of a three
dimensional seawater intrusion model from a two dimensional groundwater flow (MODFLOW)
model, and describe some of the practical modelling problems that were encountered along the
way. Some practical problems included: 1. converting from 2D planar to 3D heterogeneous
model frameworks (while maintaining the same depth-averaged groundwater hydraulics between
both 2D and 3D representations), 2. The adoption of 3D boundary conditions (i.e. the
representation of partially penetrating estuaries, amongst others) and their influence on the
results compared to 2D groundwater flow; 3. Converting from the Dupuit assumption to
variably saturated flow (and the conversion from MODFLOW to MODHMS modelling
frameworks). The modelling was used to provide an analysis of the “susceptibility to seawater
intrusion” throughout the aquifer. Susceptibility was assessed using various information sources
– the modelling provided an indication of areas susceptible to future seawater intrusion by
running long-term simulations (80+ years) that used predicted water resource operational
scenarios. Various simulations of water resource management, including “same as existing”,
“worst case”, “severe cutbacks”, were tested. The overall study described a detailed analysis of
seawater intrusion – including model conceptualisation, numerical modelling, and the final
management options for seawater intrusion.
6.1.4 Future Challenges
In recent times, seawater intrusion modelling has become a more straightforward
undertaking, given advancements in computer processing capabilities, improvements in
modelling codes, and the development of user-friendly pre- and post-processors. Further, the
advent and evolution of MODFLOW-based seawater intrusion simulators facilitates a relatively
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uncomplicated transition from MODFLOW model to seawater intrusion simulator. However,
the interpretation of modelling results is still an extremely challenging task, especially where the
aim of the investigation is to develop practical solutions to the mitigation and remediation of
seawater intrusion.
Despite an abundance of bench-marking studies for variable-density codes, the results of
these have not translated into guidelines for interpreting and applying seawater intrusion models.
The reasons for this are probably because of differences in scale and in vertical-to-horizontal
aspect ratios between benchmark problems and real field-scale seawater intrusion problems. For
example, both the Henry problem and the benchmark problem recently described by Goswami
and Clement [2007] involve horizontal-to-vertical ratios of about 2, while the seawater intrusion
problem of Werner and Gallagher [2006] involved a horizontal-to-vertical ratio of about 500.
Another significant challenge of practical seawater intrusion modelling is determining the degree
that aquifer heterogeneities are represented in simulations of real-world systems. It is generally
accepted that preferential flow paths play an important role in the behaviour of contaminant
plumes; however the degree of representation of the variability in aquifer properties has not been
adequately explored for the seawater intrusion problem. The problem of selecting the level of
representation of aquifer heterogeneities needs to be considered in combination with the
selection of the appropriate grid resolution. Other potential issues include the inclusion of
transient information in models (irrigation cycles, groundwater extraction regimes) and over large
spatial scales that influence intrusion phenomena. All of these challenges point to the growing
need for further 3D variable density and transient analyses, often covering very large spatial and
temporal scales. The additional inclusion of heterogeneity (whether it be in boundary conditions
in space or time or in geologic properties of the aquifer) poses additional complexity. Resolving
narrow transition zones further compounds the need for greater computational effort. Increased
efficiency in numerical solvers and other numerical efficiencies will be required to meet these
growing computational demands. Additionally, the large computational effort associated with
seawater intrusion modelling of regional-scale systems generally precludes the application of
automated calibration techniques, such as the gradient methods adopted by standard
groundwater model calibration codes like PEST [Doherty, 2004]. Further research is needed to
develop methods of calibrating seawater intrusion models, and to analyse the model uncertainty
of making predictions with these models.
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6.2 Transgression-Regression salinisation of coastal aquifers
6.2.1 Importance of transgression-regression cycles
Tidal forcing, seasonal changes in recharge and groundwater abstraction are key factors
responsible for variability and change in coastal groundwater systems. These drivers act on
human time scales. However, when viewed on longer time scales from centuries to millions of
years, coastal zones are even more dynamic. Sea-level change, erosion and sedimentation have
caused coastlines to shift back (i.e., seaward: regression) and forth (i.e., landward: transgression)
over distances of up to several hundreds of kilometres. Groundwater systems responded to these
conditions by adjustment of flow fields and redistribution of fresh and saline water. Although
the geological changes in boundary conditions generally tend to occur at a slower pace than more
recent man-induced changes or diurnal and seasonal forcing, their impact on current
groundwater conditions is often enormous.
More noticeable, or even catastrophic, retreats of the coastline occur in arid regions
when droughts, river diversion or irrigation schemes decrease the inputs of (fresh) water to
inland seas or lakes. Well documented cases include the Dead Sea, the Aral Sea, Lake
Corangamite and Lake Chad [Nihoul et al., 2004]. Conversely, shoreline advance is also
associated with decreased river discharge to coastal zones when decrease in sediment loads
causes land loss due to erosion and subsidence. Decreased discharge of the River Murray in
Australia and the Nile in Egypt are key examples [Frihy et al., 2003]. Besides enormously adverse
economic and ecological consequences, the hydrogeological effects of such coastline shifts are of
paramount importance as well. These include the relocation of discharge and recharge zones,
changes in groundwater-surface interaction processes and water quality degradation. Variable
density flow also plays a crucial role in these settings.
6.2.2 Observational evidence and conceptual models
It is essential to realize that the textbook conception of the fresh and saline groundwater
distribution, which is classically conceived of as a fresh water lens overlying a wedge of saline
groundwater, is seldom encountered in real field settings due to the dynamic nature of
shorelines. The most conspicuous manifestations of transient effects are offshore occurrences of
fresh groundwater and onshore occurrences of salt water.
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Offshore fresh and brackish paleo-groundwater
Over the years, evidence has been growing in support of the observation that subseafloor fresh and brackish groundwater are common features of continental shelves and shallow
seas around the world. Figure 9 shows the known global occurrences of offshore fresh and
brackish groundwater. Studies from New Jersey, USA [Hathaway et al., 1979; Kohout et al.,
1988], New England, USA [Kohout et al., 1977], Suriname, South America [Groen, 2002] and
the Dutch sector of the North Sea [Post et al., 2000] have shown that groundwater with salt
contents between 1 and 50% of seawater occur up to 150 km from the present coastline and at
depths up to 400 m below the seafloor. Additional, less conclusive cases have been reported for
Port Harcourt, Nigeria, Jakarta, Indonesia and for the Chinese Sea [Groen, 2002].
In many instances these waters occur too far offshore to be explained by active sub-sea
outflow of fresh water due to topographic drive. Moreover, lowest salinities often occur at
substantial depths beneath the seafloor and are overlain by more saline pore waters, suggesting
absence of discharge pathways. These waters therefore are considered paleo-groundwaters that
were emplaced during glacial periods with low sea level. During subsequent periods of sea-level
rise, salinization was apparently slow enough to allow relics of these fresh waters to be retained.
Onshore saline groundwater
In many flat coastal and delta areas the coastline during the recent geologic past was
located further inland than today. As a result, vast quantities of saline water were retained in the
subsurface after the sea level retreated. Such occurrences of saline groundwater are sometimes
erroneously attributed to seawater intrusion, i.e., the inland movement of seawater due to aquifer
over-exploitation (see section 6.1). Clearly, effective water resource management requires proper
understanding of the various forcing functions on the groundwater salinity distribution on a
geological timescale [Kooi and Groen, 2003].
High salinities are maintained for centuries to millennia, or sometimes even longer, when
the presence of low-permeability deposits prevents flushing by meteoric water [e.g., Groen et al.,
2000; Yechieli et al., 2001]. Rapid salinization due to convective sinking of seawater plumes
occurs when the transgression is over a high-permeability substrate. This process is responsible
for the occurrence of saline groundwater up to depths of 400 m in the coastal area of the
Netherlands [Post and Kooi, 2003].
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6.2.3 Numerical modelling
In the analysis of transient effects of coastline migration on the salinity distribution in
groundwater, variable density groundwater flow and solute transport modelling plays a vital role
in developing comprehensive conceptual models.
The large spatial and time scales involved pose special challenges. In particular, the high
resolution model grid required to capture convective flow features imposes a severe
computational burden that limits the size of the model domain. Other complications include the
lack of information on past boundary conditions, insufficient data for proper parameterization,
especially for the offshore domain, and unresolved numerical issues with variable-density codes.
Mathematical modelling applications addressing offshore paleo-groundwater have shown a
typical progression in model complexity. Indeed, all modelling studies to date are cross-sectional
only (2D). Earliest steady-state, sharp interface, homogeneous permeability models [Meisler,
1984] could not capture the marked disequilibrium conditions that are so apparent. The Meisler
[1984] model was only able to simulate the steady-state position of the (sharp) seawater-fresh
groundwater interface. Different simulations were carried out for different sea level low stands.
The model was not capable of simulating the effect of the subsequent sea level rise. With a nonsteady, sharp interface, single aquifer model, Essaid [1990] made a convincing case that the
interface offshore Santa Cruz Country, California, likely has not achieved equilibrium with
present-day sea level conditions, but is still responding to the Holocene sea-level rise. Kooi et al.,
[2000] used a variable density flow model that included a moving coastline to study the transient
behaviour of the salinity distribution in coastal areas during transgression (Figure 10). They
found that Holocene transgression rates were often sufficiently high to cause the fresh-salt
transition zone to lag behind coastline migration.
Person et al., [2003] and Marksammer et al., [2007] have conducted variable density
modelling for the New England continental shelf, simulating sea-level change over several
millions of years. Their results indicate that the first-order continental shelf topographic gradient
is insufficient to cause freshening of aquifer systems to several hundreds of metres depth far out
on the continental shelf. Continental ice sheet recharge [Marksammer, 2007] and/or secondary
flow systems associated with incised river systems in the continental shelf during low-stands
therefore appear to be required to account for distal (several tens of kms off the coast, well
beyond the reach of active, topography driven fresh water circulation) offshore paleo-waters.
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6.2.4 Future challenges
Knowledge of the “anomalous” salinity distributions discussed above is not only relevant
from an academic perspective. Offshore fresh and brackish groundwater occurrences are
expected to become a viable alternative water resource, notably in arid climates and in places
with high population densities. Moreover, in the development of conceptual hydrogeologic
models of coastal aquifers, the relevance of transient phenomena should be considered more
routinely than at present. Such an analysis has major implications for the numerical models based
on these conceptualizations, such as the selection of appropriate boundary conditions and the
choice of the timescales that need to be considered.
The much needed improvements in our understanding of the impact of coastline
migration on hydrogeologic systems rely on more and better field observations. Age dating of
groundwater, and especially coastal groundwater, is extremely difficult but is an essential
component of paleohydrologic analyses. Exploration techniques needed to better delineate the
subsurface salinity distribution require refinement and more versatility, especially in order to
investigate offshore aquifers.
Finally, numerical models need to be improved as well as has been outlined in an earlier
section. Difficulties currently arise in the accurate and physically realistic simulation of density
driven fingering due to grid discretization requirements. Resolving the nature of the relationship
between numerical features of a code and the physical fingering process is crucial since this
unstable fingering process appears to be of prime importance in many hydrogeologic settings.
Greater flexibility in boundary conditions are also required to allow for both heads and
concentrations changes to occur due to changes in sea level. Land subsidence and sedimentation
are among the additional complications that currently remain largely unexplored and poorly
understood. The addition of these additional physical processes into a quantitative modelling
framework will undoubtedly provide challenges for numerical models.
6.3 Aquifer storage and recovery
6.3.1 Importance of Aquifer Storage & Recovery (ASR)
Aquifer storage and recovery (ASR) simply involves the injection and storage of
freshwater in an aquifer. The theory was introduced by Cederstrom (1947) as an alternative to
surface water storage such as dams and reservoirs. The main advantages of ASR over surface
water storage include that it requires a relatively small area of land, potentially reduces the water
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lost to evaporation (e.g., from the surface of a dam), and to a certain extent the aquifer can
behave as a water purification medium.
The original, idealised concept of ASR involved the injection of a cylindrical “bubble” of
freshwater out of a well, into a confined aquifer containing groundwater of a poorer quality (such
that the groundwater itself would typically not be used). The bubble of water would then be
stored in the aquifer until needed, at which point it would be pumped back out of the same well.
In this situation, the volume of recoverable freshwater could be reduced by mixing between the
injected and ambient waters, and by the injected plume migrating down-gradient (in the case
where the water was injected into an aquifer where a background hydraulic gradient existed as is
typical in most cases). In both cases, the recoverable volume will be smaller than the injected
volume. Situations with minimal mixing and little migration down-gradient can allow almost
100% of the injected volume to be recovered. However, situations with large amounts of mixing
and/or a significant migration of the plume down-gradient can lead to a very small (even zero)
percentage of recoverable freshwater. In each specific situation, the percentage of recoverable
freshwater (called the “recovery efficiency”) will determine whether an ASR operation will be
economically viable, and whether it actually offers any advantages over surface water storage
technology.
6.3.2 Variable density flow in ASR
As well as mixing and down-gradient movement, density effects can cause a reduction in
ASR recovery efficiency. It appears that this phenomenon was first documented by Esmail and
Kimbler [1967]. If freshwater is injected into a saltwater aquifer, the density contrast between
injected and ambient water leads to an unstable interface at the edge of the injected plume. The
denser ambient water tends to push towards the bottom of the interface and the lighter injected
water tends to float towards the top. The result is tilting of the interface and a transition from
the idealised cylindrical plume to a conical-shaped plume (see Figure 11). Upon recovery, the
ambient water is much closer to the well at the bottom, and breakthrough of salt occurs, causing
premature termination of the recovery before all of the freshwater has been extracted. In this
way, it is possible to have situations with minimal mixing and minimal background hydraulic
gradient, but still low recovery efficiency due to density effects.
6.3.2 Analytical modelling
Bear & Jacobs [1965] presented an analytical method for predicting the position of the
freshwater/saltwater sharp interface under the influence of pumping (either injection or
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extraction) and a background hydraulic gradient. The analytical solution assumed a homogeneous
aquifer and neglected density effects. Esmail & Kimbler [1967] produced an analytical/empirical
formula to address the density-induced tilting of a freshwater/saltwater “gradient” interface (i.e.
accounting for a mixed zone) in radial geometry. The presence of a mixed zone was found to be
significant. The influence of a background hydraulic gradient was neglected by Esmail &
Kimbler [1967], and the process assumed that the interface would only tilt during storage, not
during either the injection or extraction phase. Peters [1983] presented a semi-analytical solution
for an injected bubble under the influence of density effects, but the presence of a mixed zone
was neglected and the background hydraulic gradient was not considered. When considering the
influence of pumping (especially multiple cycles), background hydraulic gradients, heterogeneity
and density-dependent flow phenomena, analytical solutions become far too complicated to
derive and is often the case, a numerical model is required.
6.3.3 Numerical modelling
Numerical modelling of ASR has been described by Merritt [1986], Missimer
[2002], Pavelic et al., [2006], Yobbi [1996] and Ward et al., [2007]. Brown [2005] provides a
comprehensive summary of ASR modelling (p. 170). Merritt [1986], Missimer et al., [2002] and
Ward et al., [2007] specifically included results of numerical modelling with density effects
explicitly accounted for. Yobbi [1996] included density effects in a modelling study but did not
report on the specific influence and effects of density on the simulation results. Ward et al.,
[2007] performed numerical modelling of density dependent ASR and quantified the influence of
various parameters on the tilting interface.
6.3.3.1 Choice of model
An ASR process can be modelled in 2 dimensions as a horizontal plane [Merritt, 1986;
Pavelic et al., 2004]. Being a radial flow system, it can also be modelled as a 2-dimensional
“radial-symmetric” projection, that is, a 2-dimensional vertical projection that is symmetrical
about the injection well [Merritt, 1986; Ward et al., 2007]. Of course, ASR can also be modelled
in 3 dimensions but this obviously increases computational burden. Consequently, 2D
simplifications (in radial flow coordinates) are often preferred where symmetrical flow can be
assumed.
Largely hypothetical ASR cases where there is no background hydraulic gradient, and
where the aquifer structure is homogeneous or consists of infinite horizontal layers, can be
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modelled as a 2-dimensional radial-symmetric projection [Ward et al., 2007]. Sample model
output comparing density-invariant, small and large density contrasts are shown in Figure 12.
These results were obtained using the FEFLOW model (Diersch, 2005). FEFLOW is a
numerical finite element system that allows various two-dimensional (2D) projections as well as
three-dimensional (3D) flow and transport processes in porous media. For this study a 2D
axisymmetric projection was chosen to model the ASR system. A transient, density-dependent
flow and mass transport regime was chosen. Cases involving a background hydraulic gradient
(more realistic) are inherently asymmetrical and must be modelled either as a 2-dimensional
horizontal or 3-dimensional simulation.
To simulate ASR with variable density, flow must be considered in the vertical
dimension, meaning that a 2-dimensional horizontal plane model will not suffice. One concludes
that density-dependent ASR systems must be modelled either as a 2-dimensional radialsymmetric projection (which is incapable of simulating background hydraulic gradient or
asymmetrical geological structures) or a fully 3-dimensional model. Because in real systems a
background hydraulic gradient is typical, a simple 2-dimensional radial-symmetric projection is
generally insufficient. However, the time and computational power required to establish and
execute a 3-dimensional model that is sufficiently discretized to accurately simulate the densitydependent flow of an injected freshwater plume can lead without robust quantification to the
simpler choice of a density-invariant simulation model. This decision should not be made lightly,
as judging the likely influence of density effects is rather complicated [Ward et al., 2007].
6.3.3.2 When can density effects be ignored?
The concentration difference between injected and ambient water is the most obvious
parameter driving the density-induced tilting of the interface. Indeed, in many cases the
concentration difference is (explicitly or implicitly) assumed to be the only significant parameter
that influences variable-density flow, and modellers make the decision to use a densitydependent or density-invariant numerical model based on this parameter alone. However, Ward
et al. [2007] demonstrated that an ASR system must be considered as a mixed-convective regime
and derived the following mixed convection ratio M as an approximate ratio between the
respective influences of free and forced convection in a simple (radial-symmetric) ASR system.
In the case of an ASR system, they showed that the appropriate formulation of the mixed
convection ratio (equation 2) is given by:
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M
§ U U0 ·
2SrB
¸¸
KV ¨¨
Q
© U0 ¹
(17)
where r is the hypothetical injected bubble radius (L), B is the aquifer thickness, Q is the injection
or extraction flow rate (L3/T), and KV is the vertical hydraulic conductivity (L/T). When M <<
1, forced convection dominates the system (e.g., fast pumping at small radii) and interface tilting
is expected to be relatively insignificant. As M approaches 1, the relative influences of free and
forced convection are comparable (e.g. slow pumping at large radii) and interface tilting is
expected to become significant in this transition zone. Clearly during the storage phase, Q = 0
and M is infinite, indicating that the flow in system is totally dominated by free convection [Ward
et al., 2007]. As can be seen from the mixed convection ratio give in equation 17, a low density
contrast in a highly permeable medium may have the same “density effect” (i.e., the same rate of
tilting) as a higher density contrast in a less permeable medium. Equation 17 brings together
several other potentially significant parameters, such as aquifer thickness, bubble radius and
pumping rate. A highly anisotropic aquifer (in which the vertical hydraulic conductivity is a very
small proportion of the horizontal) may retard the vertically-acting density effects, thereby
attenuating density-induced tilting. After injection finishes, the duration of the storage period is
critical: depending on the density contrast and hydraulic conductivity, a short period of storage
(e.g., <1 month) may lead to minimal tilting, but longer periods may cause significant tilting, and
therefore significant reduction in recoverable freshwater. If ASR is to be promoted as a longterm drought mitigation strategy then long storage durations may be expected [Bloetscher and
Muniz, 2004]. In such cases, the effects of variable density flow are likely to be important in
general hydrogeologic analyses and in numerical modelling efforts.
6.3.3.3 Density effects in layered aquifers
Heterogeneous aquifers that consist of sedimentary strata can lead to density-induced
fingering. Injected freshwater will readily flush ambient saltwater out of layers of high
permeability, but saltwater will remain in layers of low permeability as the freshwater will not be
able to penetrate those layers during injection. Then, during storage, the dense saltwater can
migrate downwards out of the low permeability layers and “bleed” into the freshwater, and
buoyant freshwater can also migrate upwards, flushing salt out of low conductivity layers (see
Figure 13). This can potentially lead to a very significant reduction in recovery efficiency as large
volumes of freshwater stored in the high permeability layers essentially become contaminated
with salt from low permeability layers above and below.
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6.3.4 Future challenges
When attempting to simulate ASR processes, one is faced with several problems. First, to
model a realistic system with a background hydraulic gradient and density effects, a 3D model is
required. Setting up a detailed 3D density-dependent model can be time-consuming and is
generally computationally expensive. Furthermore, if the aquifer is heterogeneous, then the finite
element mesh or grid may need to be highly discretized (perhaps in all 3 dimensions) at
interfaces between regions of different hydraulic conductivity and the run-times are likely to
become excessively large (several days or more). The additional requirement to place model
boundaries sufficiently far away from the injection/extraction activity will also increase
simulation run-times. Large field scale modelling applications of variable density flow in 3D are
therefore limited because of heavy computational requirements.
Ward et al., [2007] described a method for simulating the injection/extraction well that
considered the injection and extraction well to be a column of highly conductive elements (like a
fracture), rather than a uniform flux across the boundary. The reason for this method of
implementation was that the uniform flux method would fail to account for the dense water
initially in the aquifer, which may in fact lead to a non-uniform velocity profile (low velocities at
the bottom of the well and large velocities at the top of the well). Figures 12 and 13 demonstrate
that the density-induced tilting of the interface can actually drive saltwater back up the well
during the storage phase (when the well is not operational) where it can then circulate upwards
into the upper part of the aquifer. The likelihood of observing this phenomenon in the field is
unknown and requires further research.
6.4 Fractured rock flow and transport modelling
6.4.1 Importance of variable density flow in fractured rock aquifers
Fractures may have a major impact on variable density flow because they represent
preferential pathways where a dense plume may migrate at velocities that are several orders of
magnitude faster than within the rock matrix itself. Studying dense plume migration in fractured
rock is especially important in the context of hazardous waste disposal in low-permeability rock
formations. Variable density flow in a vertical fracture has previously been studied by Murphy
[1979], Malkovsky and Pek [1997, 2004] and Shi [2005], showing that two-dimensional
convective flow with a rotation axis normal to the fracture plane can occur. Shikaze et al., [1998]
numerically simulated variable-density flow and transport in a network of discrete orthogonal
fractures. They found that vertical fractures with an aperture as small as 50 Pm significantly
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increase contaminant migration relative to the case where fractures are absent. It was also shown
that dense solute plumes develop in a highly irregular fashion and are extremely difficult to
predict. However, Shikaze et al., [1998] represented fractures by one-dimensional segments, thus
inhibiting convection within the fracture. Importantly, Shikaze et al., [1998] limited their studies
to a regular fracture network consisting of only vertical and horizontal fractures, embedded in a
porous matrix. Recently, Graf and Therrien [2005] included inclined fractures in the simulation
of dense plume migration and showed that variable density flow in a 1D inclined fracture
initiates convection in the 2D porous matrix and that convection cells are independent and
separated by the high-permeability fracture (Figure 14). In a recent study, Graf and Therrien
[2007a] investigated dense plume migration in irregular 2D fracture networks. Simulations
indicated that convection cells form and that they overlap both the porous matrix and fractures.
Thus, transport rates in convection cells depend on matrix and fracture flow properties. A series
of simulations in statistically equivalent networks of fractures with irregular orientation showed
that the migration of a dense plume is highly sensitive to the geometry of the network. If
fractures in a random network are connected equidistantly to the solute source, few equidistantly
distributed fractures favor density-driven transport while numerous fractures have a stabilizing
effect.
6.4.2 Verification of variable-density flow in fractured rock
Few test cases exist to verify a new code which simulates variable-density flow in
fractures. Results presented by Shikaze et al. [1998] can be used to verify dense plume migration
in a set of vertical fractures by comparing isohalines at certain simulation times. The inclinedfracture problem introduced by Graf and Therrien [2005] can be used in both a qualitative (using
isohalines) and a quantitative (using detailed information on breakthrough curves, mass fluxes,
maximum velocities etc.) manner to verify variable density flow in an inclined fracture.
Caltagirone [1982] has presented the analytical solution for the onset of convection in
homogeneous media. The solution can be applied to vertical and inclined fractures by assuming
constant fracture aperture (homogeneity) and by introducing a cosine weight to account for
fracture incline [Graf, 2005]. Although the solution by Caltagirone [1982] is the only available
analytical solution on variable-density flow applied to box type problems of finite domain
dimensions, it is not truly an analytical solution, but rather uses analytical procedures to derive
the critical Rayleigh number for different aspect ratios in order to determine the conditions that
govern the stable and unstable states of convection in box type problems. It is these solutions
that also form the basis for newly proposed benchmark test cases presented in Weatherill et al.,
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[2004]. We conclude that, to date, a robust and well accepted benchmark problem for variable
density flow in fractures oriented in 3D and embedded in a 3D porous matrix does not exist.
Thus, there is a clear need to further explore how to best test and benchmark 3D variable density
flow in fractured media.
6.4.3 HydroGeoSphere code capability
The HydroGeoSphere (HGS) model simulates variable-density flow in 3D porous media
and in 2D fractured media where fractures are planar and parallel to at least one coordinate axis
[Therrien and Sudicky, 1996; Therrien et al., 2004; Graf and Therrien, 2005; 2007a; 2007b].
There are, in principal, two methods to calculate density: (i) Using Pitzer’s ion interaction model
in the VOPO model [Monnin, 1989; 1994], which is physically based but slow because it iterates
between density and molality. The VOPO algorithm calculates the density of water from the
concentrations of the solutes Na+, K+, Ca2+, Mg2+, SO2î4 , HCO3î and CO2î, based on Pitzer’s
ion interaction model. It allows for the accurate calculation of the density of natural waters over
a wide range of salinities (i.e., from fresh water to brine). However, the speed of the computation
often makes it impractical for numerical modeling. (ii) Using an approximate but faster empirical
model. HGS can use both methods and salinity is calculated from concentrations of individual
ions found in natural water.
6.4.4 Future challenges
Assuming 2D spatial dimensions is a common simplification made in numerical
simulations. Because density-driven convection can naturally occur in multiple ways in 3D,
representing non-planar fractures in 3D is essential. Prior attempts have shown that this is a
tedious task but has been addressed successfully [Graf and Therrien, 2006; 2007c]. Variabledensity flow in inclined fractures embedded in a 3D porous matrix is a great future challenge and
subject to ongoing investigation (Figure 15). Subsequently, benchmark problems for variabledensity flow in 3D fractured porous media are important and should be defined, followed by
further 3D analyses.
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6.5 Chemically reactive transport modelling
6.5.1 Importance of chemical reactions in variable density flow
Mixing of different waters, whether driven by the existence of salinity/temperature
gradients or other forces, may trigger geochemical reactions. Well-documented salt-water related
problems include the dissolution of calcite in fresh-salt water mixing zones [e.g. Wigley and
Plummer, 1976; Sanford and Konikow, 1989] and cation exchange due to salt water intrusion or
freshening [Valocchi et al., 1981; Appelo and Willemsen, 1987; Appelo et al., 1990; Appelo,
1994]. Precipitation of (carbonate) minerals has also been extensively studied in geothermal
problems due to its relevance for aquifer thermal energy storage [e.g., Brons et al., 1991].
Moreover, ore formation in relation to thermal convection is a topic that has received ample
attention in basin-scale geological studies [e.g., Person et al., 1996].
6.5.2 Governing Equations
Reactive transport of dissolved solutes in groundwater is described by the advectiondiffusion equation, which is a statement of the conservation of mass of a chemical component i:
wnCi
wt
’ ˜ ( nD ˜ ’Ci ) ’ ˜ ( qCi ) rreac
(18)
where Ci denotes the chemical concentration of component i (M/L3), n is the porosity
(dimensionless), D is the dispersion coefficient (L2/T) and q is the specific discharge (L/T), i.e.
the volumetric flow rate per unit of cross-sectional area. The third term on the right-hand side,
rreac, accounts for the change in solute concentration due to chemical reactions. A similar
expression holds for the temperature in heat-transport problems. Darcy’s law in a variable
density formulation is given in Eqn (14), and the general equation of state is given in Eqn (6).
The concentration depends on the flow field (through q in equation 18), the flow field depends
on the density (through ȡ in equation 13) and ȡ is influenced by C and hence also by q.
Therefore, in variable-density flow problems, the governing equations for groundwater flow and
solute transport need to be solved simultaneously.
For reactive transport modelling, it is important to pay careful attention to the equation of state.
This expression relates fluid density and dissolved solute concentration can take different forms,
for example a simple linear expression:
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U
U 0 HC
(19)
where U 0 is the density when C = 0, and İ is the slope of the ȡ-C curve. In multicomponent
systems, U is not a function of a single solute but depends on all the different solutes that are
present in the solution. A more appropriate form of equation (19) then becomes:
U
U 0 ¦ H i Ci
(20)
i 1,n
where n is the number of components and the subscripts i refer to the individual components.
Relations of this type were employed by Zhang and Schwartz [1995] and Mao et al. [2006].
Alternatively fluid density might be computed (slightly) more accurately on the base of a
thermodynamic framework [Monnin, 1994; Freedman and Ibaraki, 2002; Post and Prommer,
2007].
6.5.3 Development and applications of numerical models
The development of numerical codes that address reactive transport under variable
density conditions was driven by a range of different motivations. On one hand, for example,
large-scale transport phenomena over geological time-scale are simulated to understand the
evolution and spatial distribution of ore-bodies [e.g., Raffensperger and Garven, 1995] while, on
the other hand, models and simulations that focus on the movement of groundwater and the
related water quality evolution are typically dealing with much smaller spatial- and shorter timescales. The latter, for example, was the case in the studies by Zhang and Schwartz [1995], who
have developed a numerical simulator to assess the fate of contaminants in a dense leachate
plume and by Christensen et al. [2001], who studied hydrochemical processes during horizontal
seawater intrusion. Furthermore, Mao et al. [2006] have developed a model and applied it to
simulate the results of a tank simulation involving chemical reactions in an intruding seawater
wedge. Sanford and Konikow [1989], Freedman and Ibaraki [2002] and Rezaeia et al., [2005]
developed codes and used them for generic simulations that evaluated the effect of chemical
reactions on dynamic changes of the hydraulic properties of the aquifer material. Post and
Prommer [2007] developed a variable-density version of the reactive multicomponent transport
model PHT3D [e.g., Prommer et al., 2003; Prommer and Stuyfzand, 2005; Greskowiak et al.,
2005; Prommer et al., 2006]. Their model combines the previously existing and widely used tools
SEAWAT, which accounts for multicomponent transport under variable-density conditions, and
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the geochemical reaction simulator PHREEQC-2. They used the model to study the relative
importance of reaction-induced density changes (resulting from cation exchange and calcite
dissolution) on the groundwater flow field during free convection. To assess and quantify the
potential effects, they reformulated the classic, well-studied Elder problem, in which flow is
solely driven by density gradients (free convection), into a reactive multicomponent transport
problem. The main outcome of the modelling study was that the flow field appears to be altered
only in cases when the fluid density is strongly affected by changes in solute concentration due to
chemical reactions (Figure 16). Perturbations smaller than approximately 10% did not result in
differences in the flow pattern that could visually be detected, but an effect was noticed in the
rate of salt plume descent that became higher as the density of the downward migrating plumes
increased. The critical result here is that when low density-contrasts drive groundwater flow,
chemical reactions are more likely to be important and hence need to be incorporated in the
analysis, unlike when density contrasts are high, such as in seawater intrusion type of problems.
The PHT3D model was also applied in the study of Bauer-Gottwein et al., [2007],
simulating shallow, unconfined aquifers underlying islands in the Okavango Delta (Botswana).
These islands are fascinating natural hydrological systems, for which the interplay between
variable density flow, evapotranspiration and geochemical reactions has a crucial effect:
Astonishingly, the combination of these mechanisms keeps the Okavango delta fresh, as salts are
transported from the freshwater body towards the island’s centre, where evapoconcentration
triggers (i) mineral precipitation and (ii) convective, density-driven downward transport of the
salt into deeper aquifer systems. The model simulations evaluated the conceptual model and
quantified, using measured hydrochemical data as constraints, the individual processes and
elucidated their relative importance for the system. They found that the onset of density-driven
flow was affected by mineral precipitation and carbon-dioxide degassing as well as by
complexation reactions between cations and humic acids.
6.5.4. Future challenges
Numerical modelling of reactive transport under variable density conditions is a complex
undertaking. As with transport under density-invariant conditions, numerical approximations
introduce artificial dispersion and oscillations which pose special difficulties for chemical
calculations. These are especially felt when redox problems are simulation as these require a high
accuracy. The need therefore remains to search for mathematical techniques that combat these
undesirable artefacts. In many cases, it should be noted that model results are extremely
sensitive to spatial discretization and that grid convergence can be a problem. There are also
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limitations faced by the assumption of local chemical equilibrium and how this relates to the
choice of spatial and temporal discretisation (e.g., the time taken to equilibrium could be longer
that the time step size).
The capabilities of the codes developed so far are already quite impressive but existing
limitations warrant further improvement and refinement. Multiphase systems, interaction with
hydraulic properties and deforming porous media can already be handled by some codes but the
development of more comprehensive models is warranted due to the complexity of geological
systems.
Despite these limitations, the level of sophistication of the reactive transport codes has
already outrun our capability to collect the appropriate input data. Parameterization of even the
simplest models, such as well controlled laboratory column experiments, is by no means trivial.
For example, challenges relate to the translation of thermodynamic constants for ideal minerals
to more amorphous phases in natural sediments or batch-determined kinetic laws to field
conditions. Unless new techniques emerge (e.g., nanotechnology?) that allows us to change the
way we collect field data, the reliable application of sophisticated codes will be seriously limited.
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Variable density groundwater system
Sea water intrusion, fresh-saline water interfaces in
coastal aquifers
Subterrenean groundwater discharge
Infiltration of leachates from waste disposal sites, dense
contaminant plumes
DNAPL flow and transport
Density driven transport in the vadose zone
Flow through salt formations in high level disposal sites,
heat and solute movement near salt domes
Heat and fluid flow in geothermal systems
Sedimentary basin mass and heat transport processes,
diagenesis processes
Palaeohydrogeology of sedimentary basins
Processes beneath playas, sabkhas and playa lakes
Operation of saline (and irrigation) water disposal basins
Density affects in applied tracer tests
Relevant papers
Yechieli et al., (2001)
Kooi et al., (2000)
Post and Kooi (2003)
Underwood et al., (1992)
Voss and Souza (1987)
Huyakorn et al., (1987)
Pinder and Cooper (1970)
Werner and Gallagher (2006)
Langevin (2003)
Kaleris et al., (2002)
Liu and Dane (1996)
Zhang and Schwartz (1995)
Oostrom et al., (1992a,b)
Koch and Zhang (1992)
Schincariol and Schwartz (1990)
Pashcke and Hoopes (1984)
Le Blanc (1984)
Frind (1982)
Li and Schwartz (2004)
Lemke et al., (2004)
Oostrom et al., (2003)
Ying and Zheng (1999)
Ouyang and Zheng (1999)
Jackson and Watson (2001)
Williams and Ranganathan (1994)
Hassanizadeh and Leijnse (1988)
Oldenburg and Pruess (1999)
Gvirtzman et al., (1997)
Garven et al., (2003)
Sharp et al., (2001)
Raffensperger and Vlassopoulous
(1999)
Wood and Hewett (1984)
Senger (1993)
Gupta and Bair (1997)
Yechieli and Wood (2002)
Sanford and Wood (2001)
Simmons et al., (1999)
Wooding et al., (1997a,b)
Duffy and Al-Hassan (1988)
Simmons et al., (2002)
Barth et al., (2001)
Zhang et al., (1998)
Istok and Humphrey (1995)
Le Blanc et al., (1991)
Table 1. Some key physical systems where variable density groundwater flow
phenomena are important in groundwater hydrology.
130
Simulator
References for code development and verification
CFEST
Gupta et al., [1987]
FAST-C
Holzbecher [1998]
FEFLOW
Diersch [1988], Diersch [2005]
FEMWATER
Lin et al., [1997]
HEATFLOW
Frind [1982]
HST3D
Kipp [1987]
HYDROGEOSPHERE
Therrien et al., [2004]
METROPOL
Leijnse and Hassanizadeh [1989]
MITSU3D
Ibaraki [1998]
MOCDENSE
Sanford and Konikow [1985]
MODHMS
HydroGeoLogic Inc., [2003]
NAMMU
Herbert et al., [1988]
PHT3D/SEAWAT-2000 Post and Prommer [2007]
ROCKFLOW
Krohn [1991], Kolditz et al. [1995]
SEAWAT-2000
Langevin and Guo [2006]
SUTRA
Voss [1984]
SWIFT
Reeves et al., [1986]
TOUGH2
Oldenburg and Pruess [1995]
VapourT
Mendoza and Frind [1990]
VARDEN
Kuiper [1983, 1985], Kontis & Mandle [1988]
Table 2. Some numerical simulators for variable-density groundwater flow and solute
transport problems. This list is illustrative not exhaustive.
131
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138
Figure 1. Brine density as a function of NaCl mass fraction, computed for the conditions of
atmospheric pressure at temperature of 25oC. Source: Adams and Bachu [2002].
Figure 2. Brine density calculated at conditions characteristic of some typical sedimentary basins.
Source: Adams and Bachu [2002].
139
Figure 3. Effect of spatial discretization on the computed salinity distribution at 4, 10, 15, and
20 years simulation time. Positions of the 20% and 60% isochlors are shown for: (a) coarse mesh
(1170 nodes, 1100 quadrilateral elements), comparable with the discretization used originally by
Elder [1967] and Voss and Souza [1987], (b) fine mesh (10,108 nodes, 9900 quadrilateral
elements). It is clear that the nature of the central upwelling and downwelling is very sensitive
to grid discretisation. Source: Diersch and Kolditz [2002].
140
Figure 4. Developmental stages of unstable plume behaviour in Hele-Shaw cell laboratory
experiment. The effective Rayleigh number for this system is Ra=4870. Elapsed times (a,b,....,h)
given in dimensionless times are T=2.02, 4.03, 8.00, 10.01, 12.02, 13.98, 15.99, 18.00, measured
from a virtual starting time of 12 hours and 15 minutes (the actual experimental starting time was 11
hours and 32 minutes). One unit in dimensionless time is equivalent to 21.4 minutes real time [after
Wooding et al., 1997b; Simmons et al., 1999].
141
Figure 5. Developmental stages of unstable plumes in the SUTRA solute transport model. Contours
represent the concentration of the plumes. Contour interval is 2,000 mgL-1. The effective Rayleigh
number for this system is Ra=4870. Elapsed times (a,b,...,h) given in dimensionless times are
T=2.01, 4.02, 8.03, 10.05, 12.06, 14.06, 16.07,18.08, measured from start of numerical simulation.
One unit in dimensionless times is equivalent to 21.4 minutes real time. Source: Simmons et al.,
[1999].
142
Figure 6. Numerical model of salt water intrusion in a regional scale 3D model, Pioneer Valley,
Australia.
Based upon Kuhanesan [2005] groundwater flow model. [Source: Werner and
Gallagher, 2006].
Figure 7. A comparison between observed salinity contours (grey lines) and predicted salinity
contours (black lines) from the a uncalibrated model and b calibrated model. The coastline is not
shown. [Source: Werner and Gallagher, 2006].
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Figure 8. Profiles of salinity distributions (relative salinity units used, whereby 1.0 is sea water).
The vertical axis scale is m AHD. [Source: Werner and Gallagher, 2006].
144
Figure 9. Inferred occurrences of offshore fresh to brackish paleo-groundwater (modified from
Groen, 2002) and continental shelf areas that were exposed during the last glacial maximum (2515 ka BP).
145
Figure 10. Variable density flow simulation showing salinization lags behind coastline migration
during transgression on a gently sloping surface and development of offshore brackish
groundwater (after Kooi et al., 2000). Highly unstable convective fingering is seen as dominant
vertical salinization mechanism.
146
Figure 11. Basic concept of interface tilting leading to conical plume in the case of variable
density analysis of aquifer storage and recovery.
147
Figure 12. An illustrative comparison between FEFLOW numerical model results for the cases
of no density, small density and large density contrasts. Results are given for: end of injection
phase, end of storage phase, and end of recovery phase. Each image is symmetrical about its lefthand boundary, which represents the injection/extraction well where fresh water is injected into
the initially more saline aquifer. The effects of variable density flow are clearly visible. Increasing
the density contrast results in deviation from the standard cylindrical “bubble” most, as seen in
the no density difference case.
148
Injection
50 days
100 days
Storage
150 days
200 days
250 days
300 days
350 days
Figure 13. Time series plot of a 100-day injection and 250-day storage, in an aquifer consisting of
6 even horizontal layers, with the hydraulic conductivity varying by a ratio of 10 between high
and low K layers. Note the fingering processes.
149
Figure 14. Variable-density flow in an inclined fracture embedded in a 2D porous matrix. Red
colors refer to high concentration and blue colors refer to low concentration. Arrows represent
groundwater flow velocities.
150
Figure 15. Variable density flow in an inclined fracture embedded in a 3D porous matrix. Arrows
represent groundwater flow velocities.
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Figure 16. Chemically reactive adaptation of the Elder short heater natural convection problem.
Width of the model domain is 600 m. Top: Plume configuration after 2920 days for the adapted
Elder problem modelled by Post and Prommer [2007] when no chemical reactions are
considered. Bottom: Plume configuration after 2920 days for the adapted Elder problem when
cation exchange and calcite equilibrium are included in the simulation. Darker shading represents
higher density.
152
Chapter 1
Development and Application of Seawater Intrusion Models: theory,
analytical, benchmarking and numerical developments, applications
Abdelkader Larabi
*Professor, LIMEN, Ecole Mohammadia d’Ingenieurs, B.P. 765, Agdal, Rabat –
Morocco, E-Mail : [email protected]
Abstract
The management of groundwater resources involves the allocation of
groundwater supplies and water quality to competing water demands and uses. The
resource allocation problem is characterized by conflicting objectives and complex
hydrologic and environmental constraints, especially in coastal aquifers. The
development of mathematical simulation models provides groundwater planners with
quantitative techniques for analyzing alternatives groundwater resources management.
Coastal aquifers involve varying conditions in time and space, owing to the
occurrence of seawater intrusion, leading to upconing due to intensive pumping near
the coast. An overview of modelling seawater intrusion in coastal aquifers is
presented, including the physical approaches that assume the occurrence of a sharp
interface or solute transport mixing zone. Some analytical solutions have been also
outlined as to serve benchmark problems for testing developed numerical algorithms
for sharp and solute transport interfaces. Based on the two physical approaches,
numerical models have been developed and using conventional finite differences,
finite elements and boundary integral techniques. Results of some field applications
are presented to show the utility and the capability of these tools of predicting
accurate information with regard to seawater intrusion, its extent, interface upcoming,
critical pumping rates and other information on management issues.These information
are required to assist managers, planners and decision makers for an integrated water
resources management as demonstrated by the regional models developed for some
coastal aquifers.
Keywords: Coastal
management.
aquifers,
Seawater
intrusion,
Modelling,
Groundwater
1. Introduction
Management of a groundwater system means making such decisions as to the
total quantity of water to be withdrawn annually, the locations of wells for pumping
and for artificial recharge and their rates and control conditions at aquifer boundaries
in addition to decisions related to groundwater quality. In order to assist the planner,
or the decision maker, to compare alternative models of action and to ensure that the
153
constraints are not violated, a tool is needed that will provide information about the
response of the system to various alternatives. Good management requires the ability
to predict the aquifer’s response to planned operations, that may take the form of
changes in the water levels, changes in water quality, or land subsidence (Bear and
Veruijt, 1987).
The necessary information about the response of the system can be provided by a
model that describes the behavior of the considered system in response to excitation.
During the past 40 years, the field of aquifer management modeling has developed as
a distinct discipline. It has provided a framework which replaces trial and error
simulations. Modern aquifer management models combine simulation tools with
optimization techniques. This combination of the optimization and aquifer simulation
provides a framework that forces the engineer or hydrogeologist to formulate
carefully the groundwater management problem. The problem must contain a series of
constraints on heads, drawdowns, pumping rates, hydraulic gradients, groundwater
velocities, and solute concentrations. In addition, an objective, usually involving costs,
cost surrogates, or risk, must be seated.
This is the case of aquifers in islands and coastal areas which are prone to
seawater intrusion. Because seawater is denser than freshwater, it will invade aquifers
which are hydraulically connected to the sea. Under natural conditions, freshwater
recharge forms a lens that floats on top of a base of seawater. This equilibrium
condition can be disturbed by changes in recharge and/or induced conditions of
pumping and artificial recharge. Seawater intrusion must be addressed in managing
groundwater resources in islands and coastal areas. The freshwater lens is very
susceptible to contamination, and once contaminated, it can be very difficult to restore
to pristine conditions. To plan counter measures, it is necessary to predict the
behaviour of groundwater under various conditions, such as to determine the extent to
which ingress of seawater occurs. As seawater intrusion progresses, existing pumping
wells, especially close to the coast, become saline and have to be abandoned, thus
reducing the value of the aquifer as source of freshwater. Also, the area above the
intruding seawater wedge is lost as source of water.
The contact between freshwater and seawater in islands and coastal aquifers,
requires special techniques of management, because of varying conditions in time and
space, such as pumping near the coast. In response to pumping from a well in the
fresh-water zone, the fresh-saltwater interface moves vertically toward the well
causing the upconing phenomenon. The problem of saltwater intrusion can be
analysed by two methods. The first method considers both fluids miscible and takes
into account the existence of transition zone (Voss and Souza, 1984). The second
method is based on an abrupt interface approximation (Bear and Verruijt, 1987)
where it is assumed that (i) the freshwater and saltwater do not mix and (ii) are
separated by a sharp interface. This method is considered as appropriate
approximation in the case where the extent of the transition zone is relatively small
compared to the aquifer thickness. The first attempts to solve the saltwater intrusion
were based on the assumption of the existence of a sharp interface between freshwater
and saltwater (Strack, 1976; Collins and Gelhar, 1971).
154
In this paper, we discuss different aspects of seawater intrusion in coastal
aquifers, since the famous works of Badon-Ghyben (1888). Some analytical solutions
available for both sharp interface and transition zone approaches are also reviewed, as
they serve as benchmark examples for testing developed numerical models and codes.
Also, some 3-D numerical models are presented which enables the prediction of the
fresh-saltwater interface displacement in response to varying conditions in time and
space. Applications of these codes to some field cases are also given assuming both
approaches of the abrupt interface and the mixing zone. The freshwater-saltwater
interface is determined, including its location, shape and extent. Examples of the
regional models, presented in this work, are based on the Galerkin finite element
approach (Larabi and De Smedt, 1994) and the finite difference technique (Mcdonalds
and Harbaugh, 1988). These models are capable to simulate groundwater flow, solute
transport and the transient interface displacement due to groundwater exploitation,
especially under upconing conditions due intensive pumping. Although simulation
models provide the resource planner with important tools for managing the
groundwater system, the predictive models do not identify the optimal groundwater
development, design, or operational policies for an aquifer system. Hence, they
should be combined to groundwater optimization models to identify the optimal
groundwater planning or design alternatives in the context of the system’s objectives
and constraints (Willis and Yeh, 1987).
2. Approaches to modeling seawater intrusion
2.1 Occurrence of seawater intrusion
The occurrence of seawater intrusion in coastal aquifers can be visualised with a
simplified conceptual model (Figure 1). The essential elements are freshwater
recharge from inland sources, seawater intrusion from the ocean and an interface or
transition zone between the two types of water. The transition zone is a zone of
mixing between the freshwater and the saltwater (Figure 2). Under equilibrium
conditions, the interface will remain fixed, and freshwater will discharge along the
seepage face. In coastal aquifers, recharge is predominantly due to lateral inflow,
whereas in island aquifers, it is due to vertical recharge. These conceptual models are
analysed in different ways. Coastal aquifers are usually modelled in the vertical plane.
Island systems are also analysed in the vertical plane when the thickness of the
freshwater lens is a significant fraction of the horizontal width of the lens, a situation
common in atoll islands. When the lens thickness is small compared with its
horizontal extent, the system can often be treated in the horizontal plane.
155
Figure 1. Schematic sketch of saltwater intrusion, sharp interface model.
2.2 Approaches for sharp interface and miscible flow models
The first question in developing a simulation model is whether it is appropriate
to deal with the problem as miscible or immiscible flow. If the transition zone is thin
relatively to the thickness of the freshwater lens and it is immobile, then it is
appropriate to assume that the freshwater and saltwater don’t mix (immiscible), and
the transition zone is considered to be a sharp interface.
Figure 2. Salt/fresh water transition zone in multi-layered aquifer.
There are several methods for dealing with seawater intrusion as two-phase,
immiscible problem. These methods are generally known as interface methods. Under
very dynamic conditions, or in case where the freshwater lens is relatively thin
compared with the transition zone, it may be necessary to adopt miscible flow models.
Reilly and Goodman (1985) presented a historical perspective of quantitative analysis
of the saltwater freshwater relationship in groundwater systems. Custodio (1992)
reported that both approaches produce good results if applied to the right
156
circumstance. Pinder and Stothoff (1988) pointed out that the sharp interface
assumption is appropriate for steady state conditions. Additionally, if conditions of
variable recharge and/or significant pumping exist, the interface will move in
response to the non equilibrium conditions. For this problem, the hydraulic head and
fluid velocity in both the fresh and salt portions of the aquifer must be considered.
Since these quantities are unknown, the problem becomes one of coupling differential
equations for motion of the freshwater and saltwater, and solving for the unknown
position of the interface between them.
In fact when the sharp interface approximation is inappropriate owing to the
large thickness of the transition zone relative to the freshwater lens, it is advisable to
treat the freshwater and saltwater as miscible. There are two elements to the problem:
1) a single fluid phase with variable density; and 2) a solute representing the total
dissolved solids in seawater, which affects the fluid density. As a result, there are four
unknown dependent variables: fluid specific discharge, fluid pressure, salt
concentration, and fluid density, requiring the solution of four equations (Frind and
Mangata, 1985). Often the most disruptive influence is the effect of pumping
activities. Withdrawal of water from the freshwater lens via pumping wells creates
saltwater upconning, which is the migration of the fresh-salt interface towards a well
in response to pumping withdrawals. However, Bear (1979) has given some examples
of real aquifers in which experimental measurements have shown the salt
concentration to vary sharply at a determined location, clearly establishing a region of
low salt concentration (freshwater) and a region of high salt concentration (saltwater).
3. Analytical solutions and benchmarking
3.1 Key role of analytical solutions
In fact, analytical solutions can not solve ‘real-world’ problems, due to their
simplified physical assumptions and geometry. But they play an important role to
serve a number of important purposes such as instructional tools to present
fundamental insights to clearly understand the mechanical trend of the groundwater
flow in coastal aquifers, while numerical solutions are often difficult to provide it.
In addition and despite their simplifying assumptions, analytical solutions can
also be used as a tool for first-cut engineering analysis in a feasibility study. More
sophisticated models normally require hydrological and hydrogeological information
that is either not available or not known to the required solution at the time of initial
investigation. Hence, without the support of reliable input data, these sophisticated
models cannot provide more accurate results. That is why before a large scale site
study, or a comprehensive numerical modelling, it is often interesting to perform
some preliminary calculations based on basic information and simplifying
assumptions.
However, still the most important contribution of the analytical solutions is to
serve as benchmark problems for testing numerical algorithms. Numerical models,
especially newly developed ones based on numerical techniques such as FE, FD, BE
157
or mixed techniques, need to be verified and validated against programming and other
errors before distribution for use and/or for real world applications.
3.2 Interface models
An exact mathematical statement of the saltwater intrusion problem, assuming
that an abrupt interface separates fresh water and saltwater, was presented by Bear
(1979). However, in order to simplify the problem, the Ghyben-Herzberg relationship
can be introduced. The vertical position of the saltwater interface measured from the
sea level, zi, is given by
zi =
Uf
z
zw = w
Us - U f
G
(1)
where G = (Us - Uf)/Uf and zw is the position of the water table measured above sea
level. Assuming a sea water density Us of 1025 kg/m3, and a fresh water density Uf of
1000 kg/m3, this relationship predicts that there will be approximately 40 times as
much fresh water in the lens beneath sea level as there is fresh water above sea level.
Bear and Dagan (1964) investigated the validity of the Ghyben-Herzberg
relationship. The hodograph method was used to derive an exact solution and it was
shown that close to the coast the depth of the interface is greater than that predicted
by the Ghyben-Herzberg relationship.
Analytical solutions for the interface problem based on the Dupuit assumption
can be found in the works of Strack (1987), Van Dam (1983), and Bear (1979). These
analytical models give reasonable results if the aquifer is shallow, but lack accuracy
in the region of significant vertical flow components. This assumption is also
incapable to handle anisotropy and nonhomogeneity of the aquifer. Only a few
analytical solutions are available for layered aquifers (Mualem and Bear, 1974;
Collins and Gelhar, 1971).
The simplest stationary interface solution based on this assumption was proposed
by Glover (1959) for confined flow, later extended by Van der Veer (1977) to include
phreatic flow. In the Glover solution the aquifer is assumed to be confined, with a
bounding layer positioned at sea level. The origin of the coordinate system is chosen
at the contact point of the sea and the continent. The saltwater interface can be
considered as a streamline and, its position can be determined as a function of the
distance from the shoreline, (Glover, 1959)
2
2Q
§ Q ·
zi = ¨
x
¸ +
© GK ¹ GK
(2)
where Q is the constant total freshwater flow per unit length of shoreline (L2T-1). The
width of the region through which freshwater discharges to the ocean is given by
x0 = -
Q
2GK
158
(3)
If the fresh water flow Q can be estimated or measured, then equation (2)
provides a simple method to calculate the depth to saltwater in the coastal zone.
Alternatively x0 can be estimated.
Detournay and Strack (1988) derived an approximate analytical solution
technique to solve groundwater flow which involves the determination of a phreatic
surface, an interface separating flowing fresh water from saltwater at rest, and seepage
and straight outflow faces where fresh water discharges into the sea, by application of
conformal mapping techniques using the hodograph method. Results are presented in
a table, where the coordinates of the phreatic surface and the interface are calculated
for the flow case in which the coast and the sea bottom slopes have angles of
respectively 30o and 15o with the x-axis.Van der Veer (1977) has proposed an
analytical solution for the steady interface flow in coastal aquifer flow systems
involving recharge on the top phreatic surface.
Strack (1976), developed an analytical solution for regional interface problems in
coastal aquifers based on single-valued potentials and the Dupuit assumption and the
Ghyben-Herzberg formula, in the context of steady state. The critical pumping rate
has been also computed in the case of a fully penetrating well. Motz (1992) proposed
an analytical solution for calculating the critical pumped flow rate in an artesian
aquifer for a critical interface rise supposed to be 0.3(b – l), where b denotes the
aquifer thickness and l the distance from top of aquifer to bottom of well screen
(Figure 2). Bower et al.(1999) modified the critical interface rise based on an
analytical solution, which allows the critical pumping rate to be increased.
Figure 3. Saltwater upconing beneath a well
Solutions for the miscible fluid model with dispersion have also been obtained
by a number of investigators. Henry (1964) used a Fourrier-Galerkin method, but was
restricted in the choice of the parameters by convergence requirements of the method.
This was the first solution for predicting the steady state salt distribution in a cross
section of a confined coastal aquifer.
159
3.3 Benchmark models
For the sharp interface benchmarking, a 3-D sand model (Sugio, 1992) has been
designed to study saltwater intrusion into freshwater in a sand box. This laboratory
model consists of a 3-D sand box, 163.8cm long and 47.5cm high, while the width of
the model has two values : 63.2cm from the downstream end until the length 82.3cm,
and 30cm in the remaining part. Salt was added to the fresh water and thoroughly
mixed up to the density of 1030kg/m3 to model sea water, and then was colored by
dye in order to observe the position of the interface separating fresh water and
saltwater. Sand with 0.76mm mean diameter was used for which the hydraulic
conductivity was obtained from measurements (Sugio, 1992) as 1.293cm/s. The
upstream and downstream water levels are respectively h1 = 44.15cm and zs =
40.67cm. Fresh water flows through the sand model for about three hours to achieve
steady state conditions. The behaviour of the fresh water-saltwater interface was
measured on the front, side and bottom sections of the box model. This sand model is
used to test the numerical model capability to accurately simulate 3-D groundwater
flow problems with a fresh water-saltwater interface as it was demonstrated by the
GEO_SWIM code (Larabi, 1994).
Three well known benchmark problems in the literature have been selected to be
presented for validation purposes of the developed numerical models based on
variable density dependent flow and solute transport in coastal aquifers:
The Henry’s problem: This test (Henry, 1964) deals with the advance of a salt water
front in a uniform isotropic aquifer limited at the top and the bottom by impermeable
boundaries recharged at one side by a constant freshwater i0nflux, and faced the sea
on the other side with hydrostatic pressure distribution. For the mass fraction
conditions, the influx side is at c= 0, the top part of the sea side is treated in a manner
to permit convective mass transport out of the system (outflow face) and the remained
part of the seaside is at c=1. Many numerical codes have been evaluated and tested
with the Henry’s solution. Segol (1993) showed, however that the Henry’s solution
was not exact because Henry (1964) eliminated, for computational reasons,
mathematical terms from the solution that he thought to be insignificant. When Segol
(1993) recalculated Henry’s solution with the additional terms, the improved answer
was slightly different from the original solution. With the new solution, Segol (1993)
showed that numerical codes, such as SUTRA (Voss, 1984), could reproduce the
correct answer for the Henry problem.
The Elder’s problem: This case serves as an example of free convection phenomenon
originally used for numerical analysis of thermally driven convection. After
adaptation, this test benchmark is used to verify the accuracy of a model in
representing fluid flow purely driven by density differences. The large density
contrast (1.2 to 1) constitutes a strong test for coupled flow problems. The domain is
represented by a rectangular box with a source of solute at the top taken as a specified
concentration boundary (c=1), whereas the bottom is maintained at zero concentration
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(c = 0). For the flow conditions, zero value of equivalent freshwater head is attributed
to the upper corners. Several authors studied this case, among others, Voss and Souza
(1987), Oldenburg and Pruess (1995), Kolditz et al.(1998) Aharmouch and Larabi
(2004). This problem was originally designed for heat flow (Elder, 1967), but Voss
and Souza (1987) recast the problem as a variable density groundwater problem in
which fluid density is a function of salt concentration.
The salt dome (HYDROCOIN level 1, case 5): The purpose of the Hydrologic Code
Intercomparison (HYDROCOIN) project was to evaluate the accuracy of selected
groundwater modelling codes. One of the problems used for testing is the
HYDROCOIN problem. This test is used to model variable density groundwater over
a salt dome. The boundary conditions are: the hydraulic head varies linearly on the
top of the aquifer whereas the other sides are impervious. The mass fraction (c = 0) on
the top at the inflow part while the central base of the aquifer is at c =1. For this
problem, Herbert et al.(1988) presented a steady state solution based on the parameter
stepping technique.
More literature on these benchmark examples can be found in several papers
such as those of Voss and Souza (1987), Oldenburg and Pruess (1995) and Kolditz et
al. (1998), among other authors.
4. Numerical models
4.1 Numerical methods for groundwater management
Numerical modeling has emerged over the past 40 years as one of the primary
tools that hydrologists use to understand groundwater flow and saltwater movement in
coastal aquifers. Numerical models are mathematical representations (or
approximations) of groundwater systems in which the important physical processes
that occur in the systems are represented by mathematical equations. The governing
equations are solved by mathematical numerical techniques (such as finite-difference,
finite-element, boundary finite element, or finite volume methods) that are
implemented in computer codes. The primary benefit of numerical modeling is that it
provides a means to represent, in a simplified way, the key features of what are often
complex systems in a form that allows for analysis of past, present, and future
groundwater flow and saltwater movement in coastal aquifers. Such analysis is often
impractical, or impossible, to perform by field studies alone. Numerical models have
been developed to simulate groundwater flow solely or groundwater flow coupled to
solute transport (the movement of chemical species through an aquifer). For a number
of reasons, numerical models that simulate groundwater flow and solute transport are
more difficult to develop and to solve than those that simulate groundwater flow alone.
Coastal aquifers are particularly difficult to simulate numerically because the density
of the water and the concentrations of chemical species dissolved in the water can
vary substantially throughout the modeled area. Numerical models based on the sharp
interface approximation assumption are reported in Mercer et al. (1980), Polo and
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Ramis (1983), Essaid (1990), Larabi and De Smedt (1994a, 1994b, 1997). Huyakom
et al. (1987), Galeati et al. (1992), Das and Datta (1995, 1998), Putti and Paniconi
(1995), Langevin and Guo (2002), and Aharmouch and Larabi (2004) present some of
the recent developments in modeling of seawater intrusion in coastal aquifers for
density dependent miscible flow and transport.
Two and Three-dimensional, dynamic numerical models handle more complex
situations under realistic representation of the mechanisms involved and account for
variability in physical situations. Three-dimensional models are more realistic and
yields better insight. They are particularly effective in handling irregularities in
coastal aquifer geometry, water use; heterogeneity in aquifer formations; temporal
variations in boundary conditions; and spatial variations in salt concentrations.
Therefore, these models can provide greater detail and accurate results. The only
drawback is related to the intensive computational cost (memory and CPU) and the
amount of data needed to feed up these models.
4.2 Governing equations for groundwater flow and solute transport models
Analysis of saltwater intrusion, assuming that mixing occurs at the transition
zone between seawater and freshwater, because of hydrodynamic dispersion, requires
the solution of two partial differential equations representing the mass conservation
principle for variable-density fluid (flow equation) and for the dissolved solute
(transport equation). The mathematical model of density-dependent flow and transport
in groundwater is expressed here in terms of an equivalent freshwater total head (for
more details see Langevin, 2003): The variable density groundwater flow is described
by the following partial differential equation:
§
·
(U U f )
’ ˜ U K f ¨ ’h f ’z ¸
¨
¸
Uf
©
¹
US f
wh f
wU wC
n
Uq
wt
wC wt
(4)
Where:
x,y, and z are coordinate directions; z is aligned with gravity (L).
ȡ is fluid density (ML-3).
Kf is the equivalent fresh water hydraulic conductivity (LT-1).
hf is the equivalent fresh water head (L).
ȡf is the density of fresh water (ML-3).
Sf is equivalent fresh water storage coefficient (L-1).
t is time (T).
n is porosity (L0).
C is the concentration of the dissolved constituent that affects fluid density (ML-3).
U is the fluid density of a source or a sink (ML-3).
q is the flow rate of the source or sink (T-1).
To solve the variable density ground water flow equation, the solute-transport
equation also must be solved because fluid density is a function of solute
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concentration, and concentration may change in response to the groundwater flow
field. For dissolved constituents that are conservative, such as those found in seawater,
the solute transport equation is:
wC
wt
’ ˜ D’C ’ ˜ Q C qs
Cs
n
(5)
Where:
D is the dispersion coefficient (L2T-1).
Q is the groundwater flow velocity (LT-1).
qs is the flux of a source or sink (T-1).
Cs is the concentration of the source or sinks (ML-3).
4.3 Recent Numerical Codes
Currently, several solute transport models suitable for the simulation of seawater
intrusion and up-conning of saline water beneath pumping sites are available. These
include SUTRA (Voss, 1984), which is a two-dimensional, vertical cross-sectional
model; SEAWAT (Guo and Bennett, 1998), CODESA3D (Gambolati et. al., 1999),
GEO_SWIM (Aharmouch and Larabi, 2004) and FEFLOW (Diersch, 2004) are
examples of three-dimensional models. These models provide solution of two
simultaneous, non-linear, partial differential equations that describe the conservation
of “mass of fluid” and “conservation of mass of solute” in porous media; as described
above.
SUTRA
SUTRA (which is named from the acronym Saturated Unsaturated TRAnsport)
was published by the United States Geological Survey (Voss, 1984). The model is
two-dimensional and can be applied either aerially or in a cross-section to make a
profile model. The equations are solved by a combination of finite element and
integrated finite difference methods. Although SUTRA is a two-dimensional model, a
three-dimensional element is provided since the thickness of the two-dimensional
region may vary over the solution domain. The coordinate system may be either
Cartesian or radial which makes it possible to simulate phenomena such as saline upconing beneath a pumped well. SUTRA permits sources, sinks and boundary
conditions of fluid and salinity to vary both spatially and with time. The dispersion
processes available within SUTRA are particularly comprehensive. They include
diffusion and a velocity-dependent dispersion process for anisotropic media. This
means that it is possible to model the variation of dispersivity when the flow direction
is not along the principal axis of aquifer transmissivity.
SEAWAT
The original SEAWAT code was written by Guo and Bennett (1998) to simulate
ground water flow and salt water intrusion in coastal environments. SEAWAT uses a
modified version of MODFLOW (McDonald and Harbaugh 1988) to solve the
variable density, ground water flow equation and MT3D (Zheng, 1990) to solve the
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solute-transport equation. To minimize complexity and runtimes, the SEAWAT code
uses a one-step lag between solutions of flow and transport. This means that MT3D
runs for a time step, and then MODFLOW runs for the same time step using the last
concentrations from MT3D to calculate the density terms in the flow equation. For the
next time step, velocities from the current MODFLOW solution are used by MT3D to
solve the transport equation. For most simulations, the one-step lag does not introduce
significant error, and the error can be reduced or evaluated by decreasing the length of
the time step.
CODESA3D
CODESA-3D (COupled variable DEnsity and SAturation 3-Dimensional model)
can solve the density-dependent miscible flow and transport equations and predict
solute migration. It is a three-dimensional finite element simulator for flow coupled to
solute transport in variably saturated porous media on unstructured domains
(Gambolati et. al., 1999; Lecca, 2000). The numerical model is a standard finite
element Galerkin scheme, with tetrahedral elements and linear basis functions, and
weighted finite differences are used for the discretization of the time derivatives.
Model parameters that are spatially dependent are considered constant within each
tetrahedral element. Parameters that depend on pressure head and/or concentration are
evaluated using values averaged over each element and are also element-wise
constant. The CODESA-3D code is born from the integration and extension of parent
codes developed, during the last ten years, by the Department of Applied Mathematics
of the University of Padova in collaboration with the Environment Group of CRS4
(Center for Advanced Studies, Research and Development).
GEO_SWIM
GEO_SWIM (GEOprofessional SaltWater Intrusion Model) is a 3-dimensional
finite element model for variable density flow and solute transport in variably
saturated porous media. The Galerkin technique is used for the spatial discretization,
using hexahedral elements, with a finite difference for time discretization. The
mathematical model is made of a set of nonlinear coupled PDE’s of flow and solute
transport to be solved for hydraulic head and mass fraction of the solute component.
The flow equation simulates the water movement in the porous medium whereas the
transport equation calculates the displacement of the dissolved salt. The accuracy of
the model has been checked via the most commonly used benchmark tests
(Aharmouch and Larabi, 2004 and 2005), either for seawater intrusion (Henry’s
problem) or brine transport (Elder’s problem and Hydrocoin test). Application of the
model had been realized in the case of seawater intrusion in some coastal aquifers
situated in Morocco (Aharmouch and Larabi, 2004). A version of the code also exists
for handling the sharp interface case (Larabi and De Smedt, 1997).
FEFLOW
FEFLOW (Diersch, 2004) is a finite element groundwater modelling commercial
code developed by WASY AG (Germany). The code solves groundwater flow, solute
transport, density dependent solute transport, coupled heat and flow, saturated and
unsaturated flow. Different types of licenses are available at different prices
164
depending on the type of problems that the user needs to solve. The cheapest version
includes only 2D groundwater flow, while the most expensive includes 3D flow,
transport, and heat processes. The code includes different types of elements that can
be combined. This allows for example to model 2D planar structures such as a fault
plane within a porous matrix or linear elements such as a drain or a stream in which
the flow equation may be different than within the porous matrix. The code solves
both steady state and transient equations.
Other codes have been also developed in USA such as SHARP, which is a quasithree-dimensional, numerical finite-difference model to simulate freshwater and
saltwater flow separated by a sharp interface in layered coastal aquifer systems; and
FEMWATER which is a 3D finite element, saturated / unsaturated, density driven
flow and transport model.
5. Verification of the models and Applications (GEO_SWIM)
5.1 The Glover problem:
To test the validity and the accuracy of the proposed numerical scheme, several
benchmark problems with available analytical solutions (Glover, 1959; Strack, 1976;
Van der Veer, 1977a; Motz, 1992) and numerical solutions (Larabi and De Smedt,
1997) are solved and compared.
Figure 4. Glover (1959) problem.
This example treats a steady state two-dimensional seawater intrusion as
depicted in Figure 4. Glover (1959) presented an analytical solution for this case. For
the numerical analysis, an arbitrary length of the flow domain is taken as 420 m in the
x -direction. The flow domain is discretized with hexahedral elements of size 4 u 4 u 3
m3. This yields grids with 106, 2 and 10 nodes respectively in the x -, y - and z directions. The domain contains 2,120 nodes and 945 elements. For the upstream
freshwater outflow values, it is taken to be 3.9 m2/day for case 1 and 18.8 m2/day for
case 2. Other parameters of the problem are U f 1 g/cm3 , U s 1.029 g/cm3 , and
K
69 m/day .
In the present problem, the aquifer is confined; hence there is no unsaturated
zone. In Figure 5, we present the interface location found from the numerical solution
165
in solid lines for the two cases with different outflow rate. For comparison, Glover’s
analytical solution is plotted as dash lines. We observe excellent agreement.
Figure 5. Comparison of numerical solution and Glover (1959) analytical solution.
5.2 Upconing beneath a pumping well in an unconfined aquifer (Strack, 1976)
This example deals with an unconfined aquifer, meaning that there exist a
phreatic surface (water table). Also, there is a pumping well located in the freshwater
zone, which draws down the water table, causing the saltwater/freshwater interface to
rise. This phenomenon is called upconing. This problem is modeled after Strack
(1976), which not only uses the Ghyben-Herzberg assumption, but also the Dupuit
assumption of nearly horizontal flow. The problem becomes two-dimensional in the
horizontal plane. In the present work, however, the Dupuit assumption is not taken
and the problem is solved as a three-dimensional one.
The domain is of the size 1, 000 m u 4, 000 m u 21m respectively along the x , y
and z directions. Due to the symmetry with respect to the y -axis, only half of the
domain is discretized. A non-uniform grid is used for the xy -plane, where a minimum
grid spacing of 5 m along the x- and y-axis is used to account for the steep hydraulic
gradients in the vicinity of the pumping well, whereas a maximum of 100 m is used
elsewhere (39 columns u 31 rows). The vertical axis is subdivided into 11 planes. A
constant spacing is used from mean sea level to the aquifer bottom and another plane
is taken at z = 0.89 m, above the sea level. The total number of nodes is 13,299 with
11,400 elements.
Table 1 shows the parameters used in the simulation, which follows those in
Huyakorn, et al. (1996). For the boundary condition, the imposed flux q 1 m 2 /day is
for two-dimensional planar simulation, which is modified to a specific discharge of
q 0.048 m/day for the present case. For the coastal boundary condition we use a
fixed hydraulic head ( h 0 at x 0 ) as taken by Strack (1976).
166
Table 1. Parameters used for the Strack (1976) problem.
Parameters
Values
Fresh water flux Q f
1 m2/day ( at x = 1000 m)
Well pumping rate Qw
400 m3/day
Well location ( xw, yw )
(600 m, 0)
Hydraulic conductivity K
70 m/day
Depth from mean sea level to aquifer base
20 m
Figure 6. Interface locations provided by GEO-SWIM Model
and the analytical solution, for the Strack’s problem.
5.3. Saltwater intrusion in an unconfined aquifer with recharge (Van Der Veer,
1977)
Van Der Veer (1977a) proposed an analytical solution for steady state flow in an
unconfined coastal aquifer where both freshwater-saltwater interface and water table
are present. This solution differs from Strack (1976) solution in that the flow in
vertical direction is fully considered. This consideration of the vertical direction
167
allows the existence of a freshwater outflow face at where the aquifer meets the sea
(Figure 7).
The solution region is 2,000 m in length and 410 m in thickness. For the
numerical solution, 202u2u51 hexahedral elements are used. The portion above the
sea level is refined more than the portion below, in order to determine the water table
location more accurately. The spacing in the x-direction is 10 m for the first 1960 m
and 8 m for the remainder, and 5 m for the y-direction. In the z-direction, it varies for
from 1 m above sea level to 10 m below it. The other characteristics of the problem
are U f 1 g/cm 3 , U s 1.020 g/cm3 and K 10 m/day . The boundary conditions used
are as follows: at x=2000, where the aquifer is exposed to the sea, the constant head
condition due to hydrostatic sea water is applied. On the top of aquifer, there is a
replenishment of 0.004 m/day .
Figure 7 shows the steady freshwater-saltwater interface and the phreatic surface
for both the Van der Veer (1997a) analytical solution and the present numerical
solutions.
0
Dash: Present model solution (run 1)
Solid Present model solution (run 12)
Dash-dot: Analytical solution (Van der Veer, 1977a)
Z (m)
-100
-200
-300
-400
0
500
1000
1500
2000
X (m)
Figure 7. Water table and interface positions produced by GEO-SWIM and Van der
Veer (1997a) analytical solution (dash-dot line).
5.4. Upconing in a confined aquifer overlain by an aquitard (Motz, 1992)
This example involving a confined aquifer overlain by an aquitard with a
pumping well in the aquifer was investigated by Motz (1992, 1997). The data is based
on a field case in Pasco County, Florida (Motz, 1992), and is summarized in Table 2.
For the numerical solution, the aquifer is represented by a square as described by
Huyakorn, et al. (1996). Due to symmetry, only the first quadrant of the threedimensional domain was simulated, with pumping well located at x y 0 (Figure
6). The size of the x-y plane is 15000 m u15000 m . The mesh is non-uniformly spaced:
168
from the origin until ( x, y ) (2000 m, 0 m) and ( x, y ) (0 m, 2000 m) , a regular
spacing of 50 m is used. Beyond these two points, greater spacing of 1000 m is used.
Thus the mesh consists of 54 rows × 54 columns. The z -direction is subdivided into
21 irregularly spaced planes. The first two planes are located at z 3 m and
z 39 m respectively. Constant spacing ( 'z 24.56 m ) is used along the well
screen ( z 39 m to z 260 m ). Another constant spacing ( 'z 17.9 m ) is used
from z 260 m to z 439 m . The hexahedral element mesh is composed of
61,236 nodes with 56,180 elements. Several steady state simulations are performed
for different pumping rates: Q 25, 50, 75, 100, 120, 140 and 180 kg/s.
Figure 8 shows the saltwater-freshwater interface below the pumping well for
both the numerical solution and the Motz (1992) analytical solution. For low pumping
rates (25, 50 and 75 kg/s), the comparison between the two solutions shows very good
agreement. With increasing pumped rates (100, 120 and 140 kg/s), the agreement is
still good, except in the vicinity of the pumping well. For these pumping rates, the
differences between the numerical and analytical solution for the maximum interface
rise are 1.8%, 4.1%, 6.8%, 9.8%, 12.3%, and 14.8%, respectively, with reference to
the freshwater lens thickness of 439 m.
It should be mentioned that the above flow rates are chosen to be less or equal to
the critical pumping rate Qc 140 kg/s as found by Motz (1992), which allows the
interface to rise up a distance about 30% between the pumping well and the
undisturbed interface. For the last pumping rate (180 kg/s) simulated here, the
numerical and the analytical solutions show significant difference. As pointed out in
Motz (1992), the analytical solution is limited to small interface rises; hence the
discrepancy with the present three-dimensional solution is anticipated.
Table 2. Parameters for the Motz (1997) problem.
Parameters
Value
Initial elevation of the saltwater-freshwater interface
-438.8 m
Initial freshwater head
10.9 m
Horizontal intrinsic permeability k x , k y
9.26 u10-12 m2
Leakance of the confining bed k zc / bc
1.2 u10-3 /day
Freshwater head in overlying constant head source bed
10.97 m
Freshwater head at constant head boundary nodes
10.97 m
Bottom elevation of well screen
260.0 m
Length of well screen
221.0 m
Distance from well bottom to the initial interface
178.8 m
169
-300
Solid line: Present model solution
Dashdot line: Analytical solution (Motz,
-325
1992)
Z(m
)
-350
-375
-400
-425
Initial
-450
0
500
interface
1000
1500
2000
X (m)
Figure 8. Interface position for both numerical and Motz (1992) analytical solutions,
respectively for pumping rates of 25, 50, 75, 100, 120, 140 and 180 kg/s.
5.5 The Henry’s problem:
This test (Henry, 1964) deals with the advance of a saltwater front in a uniform
isotropic aquifer, limited at the top and the bottom by impermeable boundaries,
recharged at one side by a constant freshwater influx, and faced the sea on the other
side. For the flow conditions the seaside is considered with hydrostatic pressure
distribution whereas an imposed flux of q0 6.6 u 10 5 m / s is applied to the upstream
side. For mass fraction conditions, the influx side is at c 0. , the top part of the
seaside is treated in a manner to permit advective mass transport to flush out of the
system (outflow face condition) and the remained part is at c 1. The results in
steady state is given in Figure 10, 11 and 12 respectively for the isochlor distribution,
the streamlines and the equivalent freshwater heads and velocity field.
The boundary conditions used in this test are the same used previously by Frind
(1982); Putti and Paniconi (1995). The studied domain is represented by a twodimensional refined mesh consisting of 41 columns and 21 rows ( 'x 'z 5 cm )
yielding 861 nodes and 800 quadrilateral finite elements.
Figure 9. Description and boundary condition for the Henry’s problem
Simulation runs are performed with either isotropic dispersivity
( D L D T 0.035 ) or constant dispersion tensor ( D 6.6 u10 6 m 2 / sec ) for the cases
of density ratio of 0.0245, 0.0500 and 0.1000. The results in steady state are given in
170
Figure 10, 11 and 12 respectively for the isochlor distribution, the streamlines and the
equivalent freshwater heads and velocity field.
Figure 10. Steady state solution of the Henry’s problem. The lines indicate the
isochlors.
Figure 11. Steady state solution of the Henry’s problem. The stream lines show the
re-circulating saltwater.
171
Figure 12. Steady state solution of the Henry’s problem. Equivalent freshwater heads
and Velocity field.
5.6 Elder’s problem:
This case serves as an example of free convection phenomenon. The large
density contrast (1.2 to 1.) constitutes a strong test for coupled flow problems.
The domain is represented by a rectangular box with a source of solute at the top
taken as a specified concentration boundary (c=1.), whereas the bottom is maintained
at zero concentration (c=0.). For the flow conditions, zero value of equivalent
freshwater head is attributed to the upper corners. Parameters necessary for running
this test are the same used previously by Voss and Souza (1987) and Kolditz et al.
(1998).
Figure 13. Description and boundary conditions of Elder’s problem
The mesh used is relatively fine and consists of 80 elements horizontally and 50
elements vertically (Kolditz et al., 1998). For time discretization we used dynamic
stepping strategy, starting with low time steps which are increased or decreased
depending on the convergence behaviour, in conjunction with fully implicit Euler
method. Simulations are performed for salinity patterns at 2, 4, 10, 15 and 20 years
for which Figure 14 presents 2 and 4 years salinity patterns.
172
Figure 14. Salinity patterns for 2 and 4 years elapsed time
5.7 Hydrocoin test:
This test is used to model variable density groundwater over a salt dome. The
modeled region is a vertical section of 300m height and 900m width. The salt dome is
located at the central section base at 300m depth. The boundary conditions used and
the parameters necessary to this test case are those taken by (Herbert et al., 1988;
Oldenburg and Pruess, 1995; Kolditz et al., 1998). A fine mesh consisting on 91
columns and 31 rows uniformly spaced is used
Two simulation sets are performed until dynamic equilibrium was reached.
These sets consider purely dispersive (DL = 20 m, DT =2 m, D0=0 m²s-1) and purely
diffusive (DL = DT =0 m, D0=5u10-7 m²s-1) systems. For each system, simulations are
conducted for salinity patterns around the salt dome respectively for the two systems
of flow are determined (Figure 15 and 16)
For the case of the purely dispersive system, the salinity patterns (Figure 15)
seems to reproduce those obtained by Kolditz et al. (1998) and more or less those of
Herbert et al. (1988). In this case, the flushing water, coming from the aquifer top,
seems to be unable to carry away the dissolved salt. Instead, a stagnation zone is
taking place.
Figure 15. Results for purely-dispersive flow (DL =20 m, DT =2 m, D0 = 0 m²s-1)
In the case of a purely diffusive system (constant dispersion tensor), we perform
long term transient simulations until reaching the equilibrium. The obtained results
173
show practically the same salinity patterns (Figure 16).
Figure 16. Results for purely diffusive flow (D0 = 5u10-7 m²s-1)
6. Case studies
6.1 The Martil coastal aquifer (GEO_SWIM code)
The Martil aquifer is situated in the North-Western part of Morocco on the
Mediterranean coast (Figure 17). It is the important local aquifer for potable water
supply, irrigation and industry in the Martil region. In accordance with geology, the
aquifer system consists of two aquifer units, separated by an aquitard. The upper unit
is a Quaternary alluvial deposits of Martil river, the lower unit is composed of
sandstone-limestone Pliocene formation, whereas the aquitard is generally marly and
clayey. The aquifer thickness is very irregular in different directions. Figure 18a
shows a cross-section from West to East.
Figure 17. Location map of the Martil aquifer
174
The conceptual model used in this study is a multi-layer aquifer system of
variable thickness. The finite element mesh used for the numerical simulation of the
Martil aquifer, shown in Figure 3b, is adjusted to fit the structure and the extension of
the hydrogeological layers and the wells location. The 3-D mesh contains 56232
nodes, and 48288 hexahedral elements. The saturated hydraulic conductivity for
different aquifers are Ks = 3.5 m/day in the alluvial material (upper), Ks = 4.5 m/day
for the sandstone-limestone (lower), and Ks = 0.05 m/day for the marl/clay aquitard
[9]. The other parameters are held constant (Ts = 0.45, Tr = 0.1).
Figure 18a. Martil West-East cross sectional view
Figure 18b. Martil 3-D structured mesh.
The boundaries are treated as impervious, except at the eastern part, which is in
direct contact with the sea (outflow face). Its extent is not known and constitutes a
part of the problem. We also consider Dirichlet conditions at nodes belonging to the
Martil and Alila rivers. In order to take into account the dry seasons these values are
not set necessarily to be equal to the elevation. For the effective rainfall, we use a
non-uniformly distribution taking into account the variably material distribution on
the top layer. In a first step steady state simulations are performed under natural
conditions such that natural groundwater flow can be reproduced efficiently. The
175
purpose is to adjust conductivities and effective recharge. The obtained results are
compared to observed piezometric map of 1966. The calibration is done using the trial
and error technique. This procedure is repeated until a satisfactory pattern is found.
Computed groundwater potential heads, shown in Figure 19, are compared to
recorded values at the observation wells in 1966. The correspondence is relatively
good. The 1966 interface location is also illustrated in Figure 20.
Figure 19. Computed (left) and observed (right) steady state groundwater potentials
in the phreatic aquifer for 1966.
A second set of simulations is performed starting from the 1966 head distribution,
to predict the seawater intrusion in the aquifer. A transient simulation is made of over
30 years, considering variable pumping rates used for water public supply shown in
Table 4. For irrigation, an amount of 7949 m3/day is pumped from several wells to
irrigate 300 ha of agricultural area. The recharge is maintained constant along the
simulation time.
Table 4. ONEP pumping rates
Well number
Pumping rate (m3/day)
992/2
780
994/2
1351.56
995/2
1181
996/2
418.75
1153/2
1260
1174/2
248.4
The results shown in Figure 21 show that the saltwater interface starts to move
until 1982 and then remains stable along the time, although pumping wells withdraw
upstream in the aquifer. This simulation scenario shows that there is no significant
displacement of the interface due to the ONEP pumping wells upstream of the Martil
river.
176
Figure 20. Plane view of the 1966 interface to the location.
Figure 21. Cross-sectional view of the moving interface positions from 1966 to 2014
(a new steady state regime is reached since 1982).
6.2 The Oued-Laou coastal aquifer (GEO_SWIM code)
The Oued-Laou coastal aquifer is located 48 km to the southeast of Tétouan city,
in the northwestern part of Morocco, on the Mediterranean coast. This 18 km² alluvial
aquifer is the region’s major groundwater source for potable water supply and
irrigation. The aquifer is widely opened on the sea and it is composed of plioquaternary material (El Amrani et al., 2004).
177
Z
Y
M
e
d
i
t
e
r
r
a
n
e
a
n
S
e
a
X
Figure 22. 3-D structured mesh of Oued-Laou aquifer
The conceptual model used in this study is a one-layer aquifer system of variable
thickness. Figure 22 shows the 3-D structured mesh of Oued-Laou aquifer. The mesh
is adjusted to fit the aquifer geometry and the well locations. The GEO_SWIM code
(abrupt interface version) was applied to the aquifer system to describe groundwater
flow to determine the seawater intrusion interface behavior.
Figure 23. Plan view of the aquifer with pumping well locations.
Figure 24. Position of the interface toe (scenario 2).
Present model (dashed and solid lines for sets 1 and 2 respectively)
Two wells are located as shown in Figure 27, one close to the coastline with
pumping rate of 661.3 m3/day for potable water supply, and one inland with pumping
rate 148.4m3/day for irrigation. For management purposes, two scenarios are
investigated for the extent of the interface location. The first scenario considers the
same set of data as described above, except that the rainfall rate is reduced to half
( q0 0.0006 m / day ). In the second scenario, in addition to the reduction of rainfall,
178
the pumping rates are amplified to 2,800 m3/day for the water supply well and 600
m3/day for the irrigation well. The simulated results of these two scenarios are
compared to those published by Larabi and De Smedt (1997). Figures 25 and 26 show
the interface position in cross section and plane views. The numerical results obtained
by the two numerical models are in good agreement. For scenario 1, the pumped rates
don’t represent a threat for the freshwater inland, instead the reduced replenishment.
For the second scenario however, the supposed pumping rates constitute the critical
exploitation feature. Indeed, upconing is developed beneath the pumping well w1 near
the coast, however, the interface don’t reach the well bottom. On the other hand,
freshwater can be safely pumped from well W2.
25
0
-25
-50
-75
6500
528000
529500
531000
.
Figure 25. Vertical cross-section through pumping well W1 showing the interface
position (scenario 2). Present model (dashed and solid lines for sets 1 and 2
respectively)
Transient simulations have been also performed to analyse the solute transport
interface motion. The initial condition for our simulations is static saturated domain
without salt. The two wells implanted near the coastline are also considered. Figures
26 and 27 show the concentration distribution respectively on a vertical section close
to the pumping wells and on the bottom of the aquifer, at time simulation t = 8 years.
Figure 26. Concentration profile (c=0.1 ; 0.3 ; 0.5 ; 0.7 ; 0.9)
Figure 27. Isochlors and velocity field on the bottom of the aquifer (convergence of
velocity areas indicate the pumping well locations).
179
6.3 The Rmel coastal aquifer (SEAWAT code)
Introduction
The R’mel coastal aquifer is located north of (Morocco, and considered among
the main sub-atlantic coastal aquifers in Morocco with an area of 240 km2. the
availability of ground water and the good hydrodynamic parameters of the aquifer
have favourably increased irrigation by pumping, and contributed to the agriculture
development in this plain; in addition to supplying potable water to tree large urban
centers (Larache, Ksar lakbir, etc.) and rural agglomerations.
But, this development of pumping has induced environmental problems, such as:
over-exploitation of the aquifer with a groundwater table decrease; salt water
intrusion; degradation of the groundwater quality. The aim of the study is to improve
Understanding of its hydrodynamic, to estimate the water balance taking into account
the contribution of the invaded seawater; to characterize the evolution of seawater
intrusion in time and space and the use of the model as a tool for sustainable
quantitative and qualitative management of the R' Mel aquifer.
For this, a mathematical model in transient conditions, based on the SEAWAT
code, has been developed to study groundwater flow and salt water intrusion in this
aquifer. Different management scenarios have been simulated to provide the
managers with a prediction tool that could help in decision making. The simulation
results with respect to development of water resources in this coastal aquifer showed
that surface water is required to protect the irrigated area and to improve water quality
in the area which is contaminated by seawater intrusion, on the other hand.
180
Figure 28. Location map of the Rmel aquifer
The Rmel plain is caracterised by a moderated climate with oceanic influence.
The average monthly temperature ranges from 12ć in January to 24ć in August; the
average annual precipitation is 695 mm. (90% are recorded between November and
April) and the average evapotranspiration is 467 mm (DRPE, 1987).
Sea water intrusion simulation in steady state and transient conditions
In order to study the seawater intrusion effect on the aquifer management, a
mathematical model has been designed, calibrated in study state and in transient
conditions. The objective of this model is to simulate the extent of the saltwater
intrusion. The model results will be of great importance for the management policy at
the local level.
Figure 29 shows the calibrated, simulated groundwater heads in the Rmel aquifer
for the reference 1972. The results show good agreement between calculated and
measured heads, and the established freshwater-saltwater interface was far away from
inland. However, the calibration is more reliable in the North-western sectors and
181
center-East where we have piezometric data. Table 5 shows also the resulting water
balance with different components, including the discharge to the sea.
Figure 29. Calibration of the Rmel aquifer
The results of the transient calculations are prescribed in Figure 30 which shows
satisfactory agreement between measured and calculated heads in different target
observation wells for the considered simulation period. These transient simulations
have led to answer some questions raised by the decision makers and local managers
such as:
- To determine the date of the beginning of the seawater intrusion in the aquifer and to
follow up its evolution;
- To identify invaded zones by this intrusion, as well as their degree of contamination;
and
- To quantify the seawater intrusion volume, as well as the other components of the
hydraulic mass balance.
Figure 30 illustrates the sea water intrusion evolution versus abstractions and
recharge of the aquifer. This shows that seawater intrusion started since 1992
following the increased development of ONEP’s withdrawals for the drinking water
supply of Larache.
182
Table 5. Calculated mass balance in 1960 under the steady state conditions
Figure 30. Quantitative evolution of seawater intrusion, natural recharge and
withdrawals in the R’mel aquifer.
183
Table 6 illustrates the results of different water balance terms between 1972 and
2003, as obtained from the transient simulations. This table shows a reserve increase
between 1972-1980: 2.93 Mm3/year between 1972-1975 and of 1.615 Mm3/year
between 1975-1980. As a consequence, seawater intrusion advances inland and
continues to progress until 1995.A pronounced destocking of the aquifer is developed
between 1980-1985 of 4.22 Mm3/year, which continued until 1990 but with an
average of 0.2 M m3/year.
Table 6. Calculated mass balance for the transient simulations.
Figure 31. Calculated salinity map for 1996 and 2000
184
Figure 31 shows the areas intruded by sea water which is located immediately
close to the ONEP pumping well field for which abstractions increased since1992.It
shows also that the salinity exceeds 2 g/l in layer 2 where the well filters are located.
Extension of the invaded zone by seawater intrusion becomes 2 times between 1996
and 2000.
Simulating provisional scenarios for water resources management of the aquifer
3 scenario schemes have been applied based on the water authority planning in
order to assess quantitative and qualitative impact on the aquifer:
Scenario 1 consists on maintaining the present aquifer exploitation, which
means that the exploitation volume stands as 13.24 M m3/year until 2020.
Scenario 2 consists on satisfying the regional water demand until 2020. This
demand should come from the ONEP’s withdrawals from the existing wells field
supplying Larache and its agglomerations projected for 2020;
Scenario 3 is expected to satisfy the maximum demands projected for the
drinking water and supply for Larache and Ksar El Kebir cities until 2020.
Table 7 gives predicted volumes of seawater intrusion for the 3 scenarios. It also
shows that sea water intrusion volume would decrease in scenario 1 since 2003 to
reach a steady state beyond the year 2020, while it continues to increase for scenario 2
and 3.
Table 7. Predicted volumes of seawater intrusion for the 3 scenarios
The predicted results for scenario 1 are illustrated in Figure 32.This shows that
the contaminated zone by seawater intrusion continues to extend more (Figure 32a
and b). The water salinity is much higher on the level of layer 2 than layer 4 and this
is due to the overexploitation in layer 2 (location of the filters). Also the ONEP’s well
field will be reached by seawater intrusion and the concentration would be of 1g/l
(Figure 32c).
185
Figure 32. Predicted salinity distribution for Scenario1
186
Figure 33. Predicted salinity distribution for Scenario 2
Figure 34. Predicted salinity distribution for Scenario 3
Figure 33 shows considerable increase of salinity in layer 2 (values generally
range between 15 and 25 g/l) for scenario 2.The ONEP’s well field would be already
reached by the seawater intrusion front in 2010 and the concentration would be of 1.5
g/l;
Figure 34 shows that the plume distribution is almost identical for scenario 2 and
3, but the RADEEL’s well field, located at the south of the ONEP wells, would not be
187
contaminated by sea water intrusion, although with the pumping increase, because it
is far from the coast.
In conclusion, for scenario 1, the ONEP well field would be already
contaminated by seawater around 2020; and the groundwater quality varies badly in
the West Northern coastal part of the aquifer. In all, the impact of scenarios 2 and 3
on the groundwater quality would accelerate seawater intrusion advance, which would
reach the ONEP well field in 2010; and would more accentuate the deterioration of
groundwater quality to higher concentrations.
6.4 The Gaza Coastal aquifer (Palestine) (SEAWAT)
The Gaza coastal aquifer is a dynamic groundwater system, with continuously
changing inflows and outflows. The equilibrium condition that once may have existed
between fresh and saline water has been disturbed by large scale pumping (Figure35).
The aquifer has been overexploited for the past 40 years; this has induced seawater
flow towards the major pumping centers in the Gaza Strip to the north of the Gaza
City and near Khan-Younis City. A typical hydrogeological section of the Gaza strip
aquifer is given by Figure 36.
Figure 35. Spatial distribution of the existing pumping wells
Figure 36. Typical hydrogeological section of the Gaza strip aquifer
The coupled flow and transport finite difference code (SEAWAT) was applied to
examine how far inland seawater transition zone has moved since intrusion began.
188
Across section finite difference grid along the first 5 km of Khan-Younis is illustrated
in Figure 37. The SEAWAT model result was compared to the previous SUTRA
model result which are in a good agreement (Figure 38).
Figure 37. Finite difference grid along the first 5 km of Khan-Younis cross section
Figure 38. Calculated salinity by SEAWAT and SUTRA codes versus observed
values
The model gave good results for evolution of salinity in the aquifer. The
preliminary model results suggested that the seawater intrusion began in the 1960s
which is in agreement with the available information about general pumping and well
information. Most of the seawater intrusion is happened to the north of Gaza city and
also near Khan-Younis city in the south as it is demonstrated in Figure39, which
illustrates the isolines of calculated TDS concentration (kg/m3) by SEAWAT in year
1996 along Deir El Balah geoelectrical cross section; the isolines of calculated TDS
concentration (kg/m3) by SEAWAT in year 2003 along Jabalya cross section; and
189
isolines of calculated TDS concentration (kg/m3) by SEAWAT in year 2003 along
Khan Younis cross section.
Figure 39. Isolines of calculated TDS concentration (kg/m3) for Deir El Balah,
Jabalya and Khan Younis cross sections
190
The numerical model is applied to test the overall regional impact on the aquifer
with two future scenarios of pumping. The first scenario is pumping from the aquifer
continuously until year 2020 to reach 200 Mm3/yr; and the second scenario is to
decrease the pumping from the aquifer year 2020 to reach 110 Mm3/yr from the
existing pumping wells. It is predicted that between years 2003 and 2020, the first
scenario will induce a considerable quantity of seawater intrusion especially in the
northern part (Figure 40). Model results indicate that the extent of the isoline (TDS
concentration = 2.0 kg/m3) at the base of sub-aquifer A will move about an additional
1.5 km in the northern part. On the other hand, the results of comparison indicate that
the second scenario will prevent any further seawater intrusion after year 2003. In
year 2020 the total inflow from the sea is estimated to be 72 Mm3/yr and 32 Mm3/yr
for the first scenario and second scenario respectively, where the discharge to the sea
for the same year is estimated to be about 3 Mm3/yr and 18 Mm3/yr for the first and
second scenarios.
Figure 40. Simulated extent of TDS concentration (less than 2.0 kg/m3) in subaquifers A in 2003 and 2020
191
Figure 41. Predicted contour lines of groundwater levels for second scenario in 2020
Given the uncertainties in the available data, additional refinement of the model
grid at this stage does not provide more accuracy. However, in general the model
reasonably simulates the position of saltwater transition zone, particularly near the
coast. The current model is a reasonable representation of the aquifer in an overall
regional context. In future as new data become available, the model should be updated
periodically to refine estimates of input parameters, and simulate new management
options
6.5 Numerical modeling of seawater intrusion in the Sahel aquifer of the Atlantic
coast of Morocco (CODESA-3D code)
The Sahel region, of about 100 km in length, is an important agricultural plain
where groundwater is heavily used as an irrigation supply. In 1994 a study to evaluate
the impact of pumping on the salinization of the aquifers and to define a proper
exploitation scheme, was completed by the regional hydraulic department in
collaboration with FAO. A hydrogeochemical database was built up using data
collected from a network of 30 wells. These data were used to implement a 2D
groundwater flow model using the USGS MODFLOW software (McDonald and
Harbaugh, 1988). To overcome some of the limitations of this earlier study, among
which the assumption of a static salt- freshwater interface coinciding with the aquifer
bottom which over-estimated the freshwater outflow to the sea, simulations using
CODESA-3D model was undertaken. The model implementation allows a more
realistic representation of the hydrodynamics of the aquifer system, considering also
the presence of a mixing zone between the freshwater and the seawater.
This study describes simulations being conducted on a series of cross-sections
perpendicular to the coast and corresponding to the geophysical profiles performed in
192
a previous study, have been analyzed. Along these profiles measures of the depth of
the salt/freshwater interface were available. The first simulation, covering the period
1950-1992, is calibrated to the water table levels and to the observation data and flow
simulations. Figure 42 shows the comparison results in 1950. The Darcy velocity field
is also shown, which denotes at the ocean side on the left boundary of the domain an
upper outlet of freshwater into the sea and, correspondingly, a lower inlet of seawater
into the aquifer. Figure 43 compares the measured saltwater-freshwater interface with
the computed relative concentration isocontours of a 42-year transient coupled flow
and transport simulations.
The study on these selected cross-sections has pointed out the important role of
an adequate grid density (20 m) and a good estimation of the unsaturated zone and
transport dispersivity parameters for the characterization of the saltwater intrusion in
the Sahel aquifer system.
Figure 42. Comparison between observed water table level (black dashed line) and
computed pressure head ȥ (m) isocontours for profile 4w. The computed Darcy
velocity field (m/s) is also shown (blue arrows).
193
Figure 43. Measured salt freshwater interface (black line, from geophysical
investigations) and computed isosaline c [/] contours.
7. Conclusion
An overview of modelling seawater intrusion in coastal aquifers is presented.
Some analytical solutions and laboratory model, either based on sharp or variable
density approaches, have been also outlined as to serve benchmark problems for
testing developed numerical codes. However, the question of which type of physical
approach is appropriate to deal with seawater intrusion in coastal aquifer, must be
clarified, especially when studying field cases. Sharp interface model or the disperse
interface model? This depends on the scale of interest: if near-coast, the effects and
variations in salt concentration are important, then dispersion features of the interface
are important; if far-coast and the regional monitoring aspects are important, then
more interested to investigate the distance and depth of the interface, and so the sharp
interface approximation is preferred.
Recent numerical codes based on a 3-D finite elements and finite differences,
developed to predict the location, the shape and the extent of the moving interfaces in
coastal aquifers involving unconfined or confined flow have been also presented.
Some field applications have been studied using recent numerical codes in order to
assess groundwater resources and to test the groundwater system response to planning
scenarios to help the managers in making decision for rational groundwater
management. Examples are taken from Morocco and Palestine, where models were
applied on some coastal aquifers to simulate groundwater flow and to study the
impact of irrigation and increasing pumping well rates used for water public supply.
The site description and the simulation results are presented in separate papers
published previously (Aharmouch and Larabi 2004, Hilali and Larabi 2004, Lakfifi et
al., 2004, Qahman and Larabi 2006).
194
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Hydraulics and Hydrology, edited by J. Gayer, O. Starosolszky and C. Maksimovic, Water
Resource Centre, Budapest, pp. 81-88, 1992.
Van Dam, J. C. (1983), The shape and position of the salt water wedge in coastal aquifers. In:
Proc.Hamburg Symposium of Relation of Groundwater Quantity and quality. IAHS,
Publication No. 146.
Van der Veer, P. (1977), Analytical solution for steady interface flow in a coastal aquifer
involving a phreatic surface with precipitation, J. of Hydrol., 34, 1-11.
Voss, C.I. (1984), A finite element simulation model for saturated-unsaturated, fluid density
dependent groundwater flow with energy transport or chemically reactive single species
solute transport : USGS Water Resources Investigation Report 84-4369, 409 pp.
Voss, C.I. and WR. Souza, Variable density flow and solute transport simulation of regional
aquifers containing a narrow freshwater-saltwater transition zone: Water Resou. Res., V. 23,
no. 10, pp. 1851-1866, 1987.
Voss, C.I. and W. R. Souza, AQUIFE-SALT : A finite element model for aquifer containing a
seawater interface. Water Resources Investigations Report 84-4369, US Geological Survey
(1984) 37.
Willis, R. and Yeh W. W-G (1987). Groundwater Systems Planning and Management. PrenticeHall, Inc, New Jersey.
Zheng, C. (1990). MT3D: A modular three-dimensional transport model for simulation of
advection, dispersion and chemical reactions of contaminants in groundwater systems.
Report to the U.S. Environmental Protection Agency, Ada, Oklahoma.
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Chapter 2
Coastal Aquifer Management: Management Model For Seawater
Intrusion In The Chaouia Coastal Aquifer (Morocco)
Abdelkader LARABI* and Latifa LAKFIFI**
*LIMEN, Ecole Mohammadia d’Ingénieurs, B.P. 765, Rabat, e-mail:
[email protected]
**Direction de la Recherche et de la Planification des Eaux, Rabat
Abstract:
Management of coastal aquifers involves seawater intrusion problem, which is a
special issue in groundwater resources management, and needs a special treatment to
study the seawater-freshwater interface. A full detailed field case has been studied,
and a particular attention has been done to seawater intrusion problem and how to
control it, using mathematical modelling. The Chaouia coastal aquifer is located south
of the Casablanca city (Morocco), and considered among the main sub-Atlantic
coastal aquifers in Morocco with an area of 1200 km2. The only water resource
available in the Chaouia plain comes from the shallow groundwater. This favourable
situation has increased irrigation by pumping, and contributed to the agriculture
development in this plain. But, this development of pumping has induced
environmental problems, such as: over-exploitation of the aquifer with a groundwater
table decrease of 0.6 m/yr; progressive dry out of some aquifer levels; salt water
intrusion; degradation of the groundwater quality; and significant increase of the
abandoned pumping wells. For this, a mathematical model in transient conditions,
based on the SEAWAT code, has been developed to study groundwater flow and salt
water intrusion in this aquifer. Different management scenarios have been simulated
to provide the managers with a prediction tool that could help in decision making for
quantitative and qualitative groundwater management in this groundwater system.
The simulation results with respect to development of water resources in this coastal
aquifer showed that surface water is required to protect the irrigated area and to
restore the abandoned exploitation on one hand, and to improve water quality in the
area which is already contaminated by seawater intrusion, on the other hand.
Keywords: Coastal aquifer, groundwater, modeling, seawater intrusion, overpumping. Morocco.
1. INTRODUCTION
Morocco is extended on a coast of more than 3500 km, on the Atlantic Ocean
and the Mediterranean Sea. As for the remainder of the world, the majority of urban
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and farming agglomeration activities (e.g. fishing, industry, harbors, agriculture and
tourism) are located on the coasts and the inshore plains. These intensive
socioeconomic developments have leaded to extensive use of freshwater resources in
the coastal zones, and especially the groundwater resources. Groundwater has some
advantages if it is compared to the surface water, because of its spatial distribution,
regularity, easiness of exploitation and low cost of mobilization.
The coastal aquifers in Morocco are considered very important source for water
supply and very essential for socioeconomic development. High rates of urbanization
and increased agricultural and economical activities have required more water to be
pumped from the aquifers. This pumping has continually increased the risk of
seawater intrusion and deterioration of freshwater quality of the coastal aquifers. The
contamination of groundwater by seawater intrusion threatens the agricultural
development. In recent years, the effect of seawater intrusion is more remarkable on
many coastal areas in Morocco, especially, in some aquifers located on the Atlantic
coast.
Currently, the agricultural activity is in a critical situation, because of the
progressive water resources shortness and due to the deterioration of groundwater
quality, due to the overexploitation of the aquifer and to successive drought seasons.
Groundwater quality in the Chaouia aquifer is generally poor. Over-exploitation
has resulted in saltwater intrusion and upconing. In some parts of the study area, a
slow, continuing decline in groundwater levels has been observed. Saltwater intrusion
presently poses important threat to water supply, and especially with future water
demand as it is expected to increase. Hence, it is necessary to develop efficient and
fast tools for predicting the response of coastal aquifers to different pumping schemes
while taking into account as constraint the risk of saltwater intrusion. In practice, the
spatial relationship between freshwater and saline groundwater in coastal and inland
aquifers is complex, and management of the freshwater resources can be difficult. The
aquifer system is rarely near equilibrium, and the fresh and saline water bodies are
normally separated by a transition zone, the result of chemical diffusion and
mechanical mixing. Under these conditions, the response of the saline water body to
pumping is difficult to predict and depends on various factors, including aquifer
geometry and properties; abstraction rates and depths; recharge rate; and distance of
pumping wells from the coastline. Efficient tools such as numerical groundwater
models are required to quantify the aquifer response to these excitations. This is the
case of the Chaouia coastal aquifer, where a mathematical model that considers
seawater intrusion has been implemented to help the managers in sustainable water
resources planning and management. Hence, the objectives of this study are:
1- To analyze the phenomenon of seawater intrusion and the definition of conditions
that govern the behavior of freshwater/saltwater transition zone in the coastal aquifer
under different physical approaches.
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2- To test management scenarios based on various economic projects; and select the
best scenario for the regional water resources authorities in order to implement the
corresponding economic projects which improve water resources development.
For this, a specific hydrogeological study of seawater intrusion must be done,
focusing on the hydrodynamics and hydrochemical behavior of the seawater intrusion
in the aquifer and to assess its impacts on freshwater resources using the variabledensity SEAWAT code. The model output will show the seawater intrusion
occurrence and predict its future behavior along the Chaouia coastal aquifer.
2. DESCRIPTION OF THE CHAOUIA AQUIFER
The Chaouia is a part of the Moroccan coastal plain located to the south-west of
Morocco. Its area is about 1200 km2 and its length is approximately 25 km along the
Atlantic coast and located between the cities of Casablanca to the North and
Azemmour near the city of El Jadida to the South-West (Figure 1).
Figure 1. Location map of the Chaouia aquifer.
The average mean temperature ranges from 25 °C in summer and 10 °C in winter.
The climate is semi-arid, but influenced by the Atlantic Ocean. The summer is hot;
especially on July and August; and this leads to make more pressures on groundwater
abstractions for irrigation from the aquifer system. The coldest months on the year are
December and January. The average annual rainfall varies from 500 mm/yr in the
north at Casablanca to 300 mm/yr in the south near Azemmour. Most of the rainfall
occurs in the period from October to April, the rest of the year being completely dry.
201
The annual mean of the evapotranspiration in the study area is estimated to 840
mm/yr.
The hydrogeology of the Chaouia coastal aquifer consists of Paleozoic shale
which forms the aquifer bottom, but the aquifer system is filled with the following
formations:
x
altered shale in the part between Tnine Chtouka and Casablanca;
x
limestone and Cenomanian marine limestone with 60 m of average thickness,
in the south-western zone;
x
sandy dunes of 10 m thickness at the coastal strip,
x
chalky hills that constitute mainly the quaternary deposits and cover the wadi
area;
x
conglomerates, alluvia and silts of small thickness, which crop out just on the
wadi Oum Er Rbia banks.
The altered Paleozoic shale constitutes the main aquifer, with 90% of the total
area. Toward the south-west of the area, the Paleozic bottom rises up and allows a
water divide to occur, separating circulations of water in the chalky horizons of the
Cenomanian to the west of the one of the quaternary deposits and altered shale to the
East. The Quaternary deposits generally have a maximum saturated thickness of 10 m
except, in the inshore zone where it can reach about 20 m.
Schematization of a hydrogeological cross section of the Chaouia aquifer system
is shown in Figure 2.
Figure 2. Schematic hydrogeological cross section of the Chaouia aquifer.
202
The groundwater system characteristics
The regional groundwater flow is mainly SE-NW discharging towards the
Atlantic Ocean, except in the south-western where a part of the outflow discharges
toward the Oum Er Rbia river. To the southwest the hydraulic gradients range
between 0.1 and 1‰. Between Tnine Chtouka and Casablanca they vary from 2 to
3‰, with lower values toward the uphill limit of the zone, of the order of 0.5‰. This
difference in the hydraulic gradient is due to the variation of the permeability. The
depth of water table varies between 10 m along the coastline and can exceed 40m
along the south-west part in the Cenomanian marine-chalk. The comparison of the
piezometric situations of 1971 and 1995 shows that the decrease of the groundwater
level reached 10 m in the coastal strip and 15 m in the rest of the aquifer.
The groundwater productivity of the aquifer varies from 0.5 to 2 L/s in the
coastal zone; 1 to 2 L/s in the zone between Tnine Chtouka and Casablanca and 2 to 4
L/s in the part between Tnine Chtouka and Azemmour.
From a hydrological point of view, the Chaouia plain is not crossed by any
permanent river. Wadi Oum Er Rbias and Wadi Bousskouras, which are located
respectively on the western and eastern limits of the study area, don't contribute to the
replenishment of the aquifer. Hence, the major source of renewable groundwater in
the aquifer is rainfall. The total rainfall recharge to the aquifer is estimated to be
approximately 53 Mm3/yr, while the lateral inflows to the aquifer is estimated to 6.6
Mm3/yr. The groundwater abstraction from the pumping wells was estimated to 34
Mm3/yr, this is based on a field investigation in 1995. This investigation has also
shown that 54% of the total pumping wells used for irrigation have been abandoned,
mainly due to the high salinity in the pumped groundwater, which exceeds 3 g/L. The
outflow towards the Atlantic Ocean is estimated to approximately 41 Mm3/yr in 1949.
The recent results of a hydrogeochemical analysis show that the groundwater
salinity values are very high in some inshore sectors, exceeding 6 g/L. However, in
1971 the salinity reached only 2 g/L in these sectors, with a maximum of 4.5 g/L
locally. This salinity increase in those places is due to the seawater intrusion resulting
from overpumping and conjugated to reduced recharge of the aquifer.
3. MODELING SEAWATER INTRUSION IN THE CHAOUIA AQUIFER
3.1 Simulation code SEAWAT
The original SEAWAT code was written by Guo and Bennett (1998) to simulate
ground water flow and salt water intrusion in coastal environments. SEAWAT uses a
modified version of MODFLOW (McDonald and Harbaugh, 1988) to solve the
variable density, ground water flow equation and MT3D (Zheng, 1998) to solve the
solute-transport equation. The SEAWAT code uses a one-step lag between solutions
of flow and transport. This means that MT3D runs for a time step, and then
MODFLOW runs for the same time step using the last concentrations from MT3D to
calculate the density terms in the flow equation. For the next time step, velocities
from the current MODFLOW solution are used by MT3D to solve the transport
203
equation. For most simulations, the one-step lag does not introduce significant error,
and it can be reduced or evaluated by decreasing the length of the time step.
3.2 Model Set-up
Mesh discretization and boundary conditions
The developed model simulates transient variable density groundwater flow
coupled to solute transport for a period from 1960 to 2002. This model is based on the
hydrogeological description presented in the previous sections. Regular spaced finitedifference cells of 1 km × 1 km on the horizontal plane are constructed (Figure 3).
Hence, the grid consists of 24 rows, 71 columns and 8 regular spaced layers on the
vertical direction.
In a first step, the model developed is expected to simulate the global aquifer
behavior. However, for studying local variations of the seawater intrusion, a refined
grid is necessary, as will be developed in the further sections.
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No flow boundary
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Figure 3. Finite-difference grid and boundary conditions for the Chaouia regional
model.
Boundary conditions and aquifer parameters
The hydraulic aquifer parameters assigned to the model are based on those
produced from the pumping tests performed in the aquifer system. The hydraulic
conductivity distribution, for tests carried out in the area, shows that values vary from
7×10-4 m/s to 2×10-6 m/s on the western area, with a mean value in general of 2×10-4
204
m/s. At the Eastern part the average value of the hydraulic conductivity is around
6×10-4 m/s.
Only six values of the storage parameter are available for the Chaouia coastal
aquifer. Measurements that carried-out, were concentrated only along the aquifer
coastal part, and the measured values range from 0.13% to 7%. In the rest of the
aquifer, parameters were obtained from Moroccan hydrogeological literature for
similar type of soils (DRPE, 1995).
Constant head and concentrations are specified to the cells along the Atlantic
coast, respectively 0.0 m and 35 kg/m3. The head for each cell is converted to
freshwater head using the specified salt concentration of 35 kg/m3 at the center
elevation of the cell. Constant head along the Oum Er Rabia river presents at layer 3,
expressed by a slope of 0.1% from the sea. The concentration is assumed to be 0.0.
The lateral flow of groundwater upstream is represented by a general head boundary
based on the groundwater level record. The aquifer bottom is a no flux boundary,
while on the top of the aquifer a recharge influx is assigned. The northern and
southern west boundaries are assumed to be no flow boundaries, due to the existence
of impervious layers of schist.
3.3 Calibration and model results
First the numerical model is established and tested against steady state
groundwater flow on 1949. During calculation, measured and calculated groundwater
heads are compared, and the difference referred to as the residual. Figure 4 shows the
calibrated, simulated groundwater heads in the Chaouia aquifer for year 1949
conditions. The results show good agreement between calculated and measured heads,
and the established freshwater-saltwater interface was far away from inland. Table 1
shows also the resulting water balance with different components, including the
discharge to the sea, which is estimated to 43.85 Mm3/yr.
It is clear from Table 1 that the main component of the input corresponds to the
recharge from precipitation and that the ocean boundary constitutes the main outlet of
the aquifer.
Transient simulations are conducted starting from the initial heads and
concentration with reference to 1960 steady state conditions assuming that there was
no change in groundwater abstraction between 1949 and 1960. The initial
concentrations in the aquifer were assumed to be 0 kg/m3. Transient calculation was
performed for the 1960-2001 target period (41 years).
The major pumping started from 1960 and was changed with lateral flows for the
specified stress periods, based on the field investigations and the recorded data. The
results of the transient calculations are prescribed in Figure 5 which shows
satisfactory agreement between measured and calculated heads in different target
observation wells for the considered simulation period. These transient simulations
have led to answer some questions raised by the decision makers and local managers
such as:
205
- To determine the date of the beginning of the seawater intrusion in the aquifer and to
follow up its evolution;
- To identify invaded zones by this intrusion, as well as their degree of contamination;
and
- To quantify the seawater intrusion volume, as well as the other components of the
hydraulic mass balance.
Table 1. Calculated mass balance in 1960 under the steady state conditions.
Volumes in Mm3 /yr
Water balance component
Inflow
Outflow
Recharge from precipitation
52.67
Lateral flow from upstream aquifer.
6.24
Total
58.91
Discharge to Atlantic Ocean.
43.85
Discharge to Oum Er Rbia river
1.20
Evaporation
13.34
Agricultural and domestic abstraction
0.53
Total
58.91
Error %
7 E-6
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Calculated Head
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#
#
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Figure 4. Calibrated and observed water level for the year 1949.
Figure 5. Calibrated and observed water level for observation wells.
207
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Table 2 illustrates the results of different water balance terms between 1965 and
2001, as obtained from the transient simulations. This table shows that the intensive
groundwater pumping, especially between 1965 and 1985, resulted in less storage in
the aquifer which reached 24 Mm3 in 1985. As a consequence, seawater intrusion
advances inland and continues to progress until 1995. The concentration values
calculated by the model give the following invaded zones by seawater intrusion and
the extent:
- The first zones invaded by the seawater intrusion in the beginning of the 1980’s are
located north of Azemmour south west. Afterward, this invaded zone extended along
the inshore line, on about 20 km of length and 2 km of width as shown by Figures 6
and 7.
- The aquifer is strongly contaminated on the S-W coastal zone, in which the toe
interface does not exceed the strip of 1km of large. Beyond this zone the
contamination of the aquifer is limited;
- Generally, the calculated concentration values exceed 10 g/L in the aquifer bottom
as demonstrated by several profiles of Figure 8.
The relative vulnerability of the south-west part of the aquifer to the seawater
intrusion in comparison with the rest of the coast is explained by the following:
x
x
x
the relatively high hydraulic conductivity of the fissured limestone located in
this sector of the coastal aquifers;
the structure of the aquifer bottom that is deeper in this zone and reaches its
maximum depth (between -60 m to -90 m) in this sector; and
the high concentrations of the pumping wells in this sector.
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Salinity in g/l
S.Saîd
M'achou
<2
2-6
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>30
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Figure 6. Simulated salinity distribution in 1985.
208
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Table 2. Calculated mass balance for the transient simulations.
209
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Figure 7. Simulated salinity distribution in 1995.
Figure 8. Salinity evolution in profiles of the model layers.
In order to study the seawater intrusion effect on the south-western part, a model
of transverse section with a refined mesh has been developed. The objective of this
local model is to simulate the extent of the saltwater intrusion with a finer spatial
210
resolution in two dimensions. The model results will be of great importance for the
management policy at the local level.
The mesh cells dimension is reduced to 100 m along the x-axis for the same
simulation period adopted for the global model.
Figure 9 illustrates the results of this transverse model in transient conditions.
These show that seawater intrusion started in 1980-85 by contaminating the aquifer on
100 m from the coast line; and the estimated extent of the seawater intrusion wedge
reaches 1300 m in 2001.
Figure 9. Result of the cross sectional simulation model in 2001.
4. PREDICTED RESULTS FOR RATIONAL WATER MANAGEMENT
The calibrated model can be exploited as a management tool for simulating the
response of the aquifer, including the quantitative and qualitative aspects (variation of
concentration in the Chaouia aquifer) based on proposed planning schemes by the
local managers.
As mentioned in the previous sections, measured withdrawal of groundwater for
irrigation supply, especially at the south-west sector, is the main cause of seawater
intrusion into the aquifer, conjunctive decrease in groundwater withdrawal with
artificial recharge is an important alternative to be considered in the future to prevent
any further intrusion of saltwater inland. For this purpose, planning scenarios schemes
were designed to use the calibrated model for simulating the future changes in
drawdown and salinity concentrations in a period of 40 years:
- The first scenario consists on the design of an irrigation project from surface water
pumped from the Oum Er Rbia river. Pumping from the aquifer has to be stopped on
the inshore strip to the East of Azemmour along an area of 3 km of width and of about
20 km of length. A brutal reduction of withdrawals occurs when the irrigation project
starts in 2005; this leads to a quasi-stabilization withdrawal of about 22 Mm3/y until
the end of the simulation period.
211
- The second scenario (the worst) assumes that the irrigation project would not be
achieved and the pumping from the aquifer of about 30.5 Mm3/y continues to occur.
The evolution of the groundwater quality of the aquifer is analyzed in order to define
the affected area by seawater intrusion.
- The third scenario assumes maintaining the present exploitation and improves the
overexploitation by artificial recharge of the aquifer through a series of injection wells
upstream of the highly contaminated zone (300 l/s).
In all cases, conditions of average annual rainfall infiltration rate (normal
conditions denoted as case 1, 3, and 5) and minimal annual rainfall infiltration rate
about 70% of the normal infiltration rate (drought conditions, denoted as case 2, 3 and
6), are considered.
Results of scenario simulations
The corresponding scenario simulations are performed for a period of 40 years
(2001-2040). For scenario 1 (case 1) (Table 3), although the seawater intrusion
volume decreases and the equilibrium is established after the year 2010 in response to
the irrigation project from surface water, the groundwater quality in the south-western
sector would remain of high salinity concentrations. The improvement of the water
quality is a slow process, and would only be felt after the year 2020, when the width
of the invaded zone by seawater intrusion would be reduced 1 km and the salinity
would decrease significantly, as it is shown in Figure 10 (1) and 11.
Table 3. Calculated volumes of seawater intrusion for scenario 1 (in Mm3/yr)
Year
2005
2010
2015
2020
2030
2040
case 1
1.45
0.2
0
0
0
0
case 2
2
0.42
0.27
0.26
0.31
0.5
The results given also in Table 3 for scenario 1 (case 2) shows that the seawater
intrusion volume decreased progressively until the year 2020; afterwards it continues
to increase. The expected groundwater quality is not improving as shown by Figure
10(2) 12. The north-eastern sector of the coastal part would be also concerned by this
intrusion. The improvement of the groundwater quality would remain very slow by
2020, due to the reduced discharge toward the sea, which is 30% less (drought
conditions) compared to the discharge in case 1 (normal conditions).
The results of the first worst scenario 2 (case 3) (Table 4), show that the
groundwater quality in the coastal zone is alarming, because the salinity concentration
increases much as shown by Figure 10(3) and 13. Indeed scenario 2 (case 3), with the
average annual rainfall infiltration rate, leads to reach after the year 2005 a new
equilibrium with a practically a steady state volume of seawater intrusion (1.4
Mm3/yr). However, the salinity in groundwater would continue to increase and would
exceed 15g/L by 2020 in the deep aquifer levels.
212
Table 4. Calculated volumes of seawater intrusion for scenario 2 (in Mm3/yr)
Year
2005
2010
2015
2020
2030
2040
Case 3
1.45
1.42
1.40
1.41
1.43
1.47
Case 4
2
2.3
2.6
3
3.4
4
For the second worst scenario 2 (case 4) (Table 4), the salinity and the seawater
intrusion volume would increase continuously and no stabilization would be
established. The seawater intrusion volume would reach 3 Mm3/yr by 2020; the
salinity would exceed 20 g/L in the deepest aquifer levels. In addition, the invaded
zone by the seawater intrusion would spread over 3 km and the northern sector of the
coastal part would be also affected significantly by this intrusion (Figure 10(4) and
14).
For scenario 3 (case 5) the seawater intrusion volume decreases and the
equilibrium is restablished after the year 2005 in response to the artificial recharge
project from surface water (Table 5), the groundwater quality in the south-western
sector improves significantly only after 2010, due to the slow process, Figure 10 (5)
and 15 illustrate this situation.
Table 5. Calculated volumes of seawater intrusion for scenario 3(in Mm3/yr)
Année
2005
2010
2020
2030
2040
Case 5
1.45
0
0
0
0
Case 6
2
0.06
0.26
0.31
0.5
The results given also in Table 5 for scenario 3 (case 6) shows that the seawater
intrusion volume increases progressively after 2005; the expected groundwater quality
is improving slowly as shown by Figure 10(6) and 16. The north-eastern sector of the
coastal part would be concerned by this intrusion.
In conclusion, we can say that the improvement of the groundwater quality in
the aquifer is conditioned by the withdrawal decrease, and by increasing the discharge
toward the ocean which mainly depend on the aquifer recharge/discharge conditions.
Hence, the predicted results from planning scenarios show that a sustainable
management of the aquifer, especially in its coastal part, requires the use of surface
water which will contribute first to the economic development of the irrigated area by
restoring the abandoned exploitation; and secondly to improve significantly the
groundwater quality in the invaded sectors of the aquifer by seawater intrusion.
213
Figure 10. Salinity variation in profiles for the model layers (Scenario 1, 2, and 3)
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S
340000
340000
Casablanca
#
O.Merzeg
N
EA
OC
o
O.H
IC
NT
LA
T
A
320000
O. Bou Skoura
Dar Bouazza
#
uara
#
#
#
300000
O.
Ou
m
#
Soualem Trifia
Médiouna
#
#
#
Derroura
#
Nouacer
Od. Harriz
Berrechid
LEGEND
Bir Jdid
Er
Rb
ia
Haouzia
300000
Salinity in g/l
Rhnimiyne
Chtouka
320000
Bouskoura
#
Azemmour
<5
#
S.Saîd
M'achou
5 - 10
10 - 20
20 - 30
280000
280000
> 30
220000
240000
260000
280000
300000
Figure 11. Results of salinity distribution for scenario 1(1) in 2020.
214
220000
240000
260000
280000
300000
N
W
E
S
340000
340000
Casablanca
O.Merzeg
320000
#
O.H
AN
CE
O
IC
NT
LA
T
A
O. Bou Skoura
Dar Bouazza
#
o ua
ra
#
O.
Ou
m
#
#
#
#
#
300000
Médiouna
#
Soualem Trifia
320000
Bouskoura
Derroura
#
Nouacer
Od. Harriz
Berrechid
LEGEND
300000
Bir Jdid
Er
Rb
ia
Haouzia
#
Salinity in g/l
Rhnimiyne
Chtouka
#
Azemmour
<5
5 - 10
S.Saîd
M'achou
10 - 20
20 - 30
280000
280000
> 30
220000
240000
260000
280000
300000
Figure 12. Results of salinity distribution for scenario 1(2) in 2020.
220000
240000
260000
280000
300000
N
W
E
S
340000
340000
Casablanca
IC
NT
o ua
O.H
A
T LA
#
N
EA
OC
O.Merzeg
320000
O. Bou Skoura
Dar Bouazza
#
ra
#
#
#
300000
O.
Ou
m
#
Soualem Trifia
Médiouna
#
#
320000
Bouskoura
#
Derroura
#
Nouacer
Od. Harriz
Berrechid
300000
Bir Jdid
Er
Rb
ia
LEGEND
Haouzia
Rhnimiyne
Chtouka
#
Azemmour
#
Salinity in g/l
<5
S.Saîd
M'achou
5 - 10
10 - 20
20 - 30
280000
> 30
220000
240000
260000
280000
300000
Figure 13. Results of salinity distribution for scenario 2(3) in 2020.
215
280000
220000
240000
260000
280000
300000
N
W
E
S
340000
340000
Casablanca
IC
NT
O.H
LA
AT
#
N
O.Merzeg
320000
EA
OC
O. Bou Skoura
Dar Bouazza
#
o ua
ra
#
#
#
300000
O.
Ou
m
#
Soualem Trifia
Médiouna
#
#
320000
Bouskoura
#
Nouacer
Derroura
#
Od. Harriz
Berrechid
300000
Bir Jdid
Er
Rb
ia
LEGEND
Haouzia
Rhnimiyne
Chtouka
#
Azemmour
#
Salinity in g/l
<5
S.Saîd
M'achou
5 - 10
10 - 20
20 - 30
280000
> 30
220000
240000
260000
280000
300000
Figure 14. Results of salinity distribution for scenario 2(4) in 2020.
Figure 15. Results of salinity distribution for scenario 3 (5) in 2020
216
280000
Figure 16. Results of salinity distribution for scenario 3(6) in 2020
5. CONCLUDING REMARKS
This study constitutes a methodological approach to generalize modeling of
seawater intrusion to coastal aquifers in Morocco. In fact, this phenomenon remained
until now far from being well known for these aquifers. The present case study of the
Chaouia coastal aquifer is an example of a pilot study to develop also for several
coastal aquifers threatened by this phenomenon, as Morocco is extended over more
than 3500 km of coasts.
All data used in this application have been analyzed, structured and organized in
a database built-in via the GIS “Arc View " code and integrated into the conceptual
model.
The coupled flow and transport code (SEAWAT) was applied to analyze
seawater intrusion in the Chaouai coastal aquifer in both qualitative and quantitative
issues. The model results showed that seawater intrusion started by 1980-1985 in the
south-western sector of the coastal part and developed within the time, due to
intensive pumping from the wells and drought conditions.
The numerical model is also applied to test the response of the aquifer of three
planning scenarios for a period of 40 years. The first one is the design of an irrigation
project from surface water and stopping groundwater withdrawal in the south-west
sector; the second scenario is a continuous pumping from the aquifer, and the third
one is the design of a series of artificial recharge wells along the south western coast.
It is predicted that the first scenario will reduce considerably the quantity of seawater
217
intrusion and improve the groundwater quality, although in a slow manner in the
south-western sector. However, the results from the second scenario will induce a
considerable quantity of seawater intrusion, and its extent progressed more into the
inland aquifer and will reach the north sector of the coastal part. The third scenario
will also reduce considerably the seawater intrusion volume and improve the water
quality in a similar way as scenario 1.
Given missing data with respect to the aquifer, more recommendations are
necessary to improve the results of this model:
- Measurements of salinity on the vertical profiles are necessary in the invaded zone
by the seawater intrusion, the measurement values will contribute to improve the
calibrated model.
- Some tracing element tests in the coastal part have to be carried out in order to
estimate the dispersivity parameters of the aquifer;
- Refinements of the model cells have to be performed, especially near the shore
where the area is more threatened by the seawater intrusion.
Acknowledgments :
The authors are very thankful to the hydraulic department (Direction Générale de
l’Hydraulique in Morocco) for the fruitful and successful collaborations between the
University and the Administration in conducting this study.
References:
Bentayeb, A, (1972). Etude hydrogéologique de la Chaouia côtière avec essai de
simulation mathématique en régime permanent. Thèse Univ. Scien. Tech. du
Languedoc, Montpellier.
Direction de la Recherche et de la Planification des Eaux (DRPE) (1995). Etude
hydrogéologique et modélisation mathématique de la nappe de la Chaouia côtière,
Direction Générale de l’Hydraulique, Rabat (Internal report).
DRPE, (1994). Reconnaissance par prospection électrique dans la Chaouia côtière
(project carried out by LPEE) Rabat (Internal report).
Guo, W., Bennett. G.D. (1998). Simulation of saline/fresh water flows using
MODFLOW In: Proc MODFLOW ‘98 Conference. International Ground Water
Modeling Center, Golden, Colorado, 1: 267–274.
Langevin, C.D. and Guo W. (2002). User’s Guide to SEAWAT, A computer program
for simulation of three-dimensional variable density groundwater flow. U.S
Geological Survey, Open-File Report 01-434, Tallahassee, Florida.
218
McDonald M.G., and Harbaugh. A.W. (1988) A modular three dimensional finitedifference ground-water flow model. U.S. Geological Survey Techniques of
Water Resources Investigations, Book 6.
Zheng, C., Wang. P.P (1998) MT3DMS, A modular three dimensional multi-species
transport model for simulation of advection, dispersion and chemical reactions of
contaminants in groundwater systems: Vicksburg, Mississippi, Waterways
Experiment Station, U.S. Army Corps of Engineers.
219
Performing Unbiased Groundwater Modeling: Application of The
Theory of Regionalized Variables
Shakeel Ahmed, Aadil Nabi Bhat and Shazrah Owais
Indo-French Centre for Groundwater Research
National Geophysical Research Institute
Hyderabad-500007, India
Abstract:
Importance of groundwater is extremely obvious in arid and semi-arid regions.
However, groundwater, an important constituent of hydrological cycle and major source
of water supply in various sectors, is getting depleted due to over-exploitation, because of
high population growth as well as extensive agricultural uses and urgently requires a
reliable management option. It is essential to have a thorough understanding of complex
processes viz., physical, chemical and/or biological occurring in the system for
groundwater assessment and management. To understand these complexities groundwater
model plays an important role. Groundwater models are simplified, conceptual
representations of a part of the hydrologic cycle. They are primarily used for hydrologic
prediction (hydraulic head, flow rates and solute concentration) and for understanding
hydrologic processes (contaminant migration, solute transport etc.). However, in most
regional studies the modeler has only a few measurements of head, the results of some
pumping tests, and a vague idea of the boundary conditions, leakage, or rate of recharge.
Because of this models are calibrated with fewer available data, which would
significantly lead to erroneous results. Geostatistics, based on theory of regionalized
variables has now-a-days found application in almost all the domains of hydrogeology
from parameter estimation to predictive modeling for groundwater management. It could
be applied at each step of hydrogeological modeling studies viz., from data collection
network designing, parameter estimation for fabrication and calibration of aquifer
modeling. Kriging, best linear unbiased estimator of the regionalized variables, is a
potential tool in increasing the accuracy of model calibration as it provides the best
estimates of the system parameters with a measure of confidence (estimation error) that
can be used in performing simulations with an ease for the modeler to investigate the
model uncertainties. Thus with the help of kriging, equally probable possibilities can be
generated that conforms the available data. The proposed methodology allows the users
to attain a better picture of the aquifer system and would assess the accuracy and
reliability of model predictions.
Keywords: Geostatistics; Kriging; Simulations; Estimation error.
221
INTRODUCTION
Models are nowadays widely used in hydrogeologic studies mainly for
understanding and the interpretation of the issues having complex interaction of many
variables in the system. These are used to calculate the rate and direction of movement of
groundwater through aquifers and confining units in the subsurface and for the
predictions about the hydraulic heads, flow rates and solute concentrations. These
calculations are referred to as simulations. The simulation of groundwater flow requires a
thorough understanding of the hydrogeologic characteristics of the site. The
hydrogeologic investigation should include a complete characterization of the subsurface
extent and thickness of aquifers and confining units (hydrogeologic framework),
hydrologic boundaries (also referred to as boundary conditions), which control the rate
and direction of movement of groundwater, hydraulic properties of the aquifers and
confining units, and the description of the horizontal and vertical distribution of hydraulic
head throughout the modeled area for both beginning (initial conditions), equilibrium
(steady-state conditions) and transitional conditions when hydraulic head may vary with
time (transient conditions), and distribution and magnitude of groundwater recharge.
These models are computer based numerical solutions to the boundary value problems
and thus can be summarized in few steps as ‘Specified hydrogeological parameters or
inputs, given initial and boundary conditions and lastly the solution of the differential
equations’. However, when the predictions have to be made for a particular area the
available data is certainly not sufficient or the modeler has only a few measurements of
head, the results of some pumping tests, and a vague idea of the boundary conditions,
leakage, or rates of recharge. Because of this models are calibrated with fewer available
data, which would significantly lead to erroneous results.
There is no doubt that the mathematical and computational aspect of groundwater
modeling has reached a satisfactory level of development. The focus of research in recent
years has shifted to the problems of parameter identification and uncertainty
quantification. Hydrogeological parameters display a large spatial heterogeneity, with
possible variations of several orders of magnitude within a short distance. This spatial
variability is difficult to characterize in a deterministic way. However, a statistical
analysis shows that hydrogeological parameters do not vary in space in a purely random
fashion. There is some structure to this spatial variability that can be characterized only in
a statistical way.
During the last few decades numerous mathematical approaches have been used to
estimate hydrogeological parameters from scattered or scant data. The study of
regionalized variables starts from the ability to interpolate a given field starting from a
limited number of observations, but preserving the theoretical spatial correlation. This is
accomplished by means of a geostatistical technique called kriging. The advantage of the
geostatistical estimation technique is that the variance of estimation error can be
222
calculated at any point without knowing the actual measurement on that point. Compared
with other interpolation techniques, kriging is advantageous because it considers the
number and spatial configuration of the observation points, the position of the data point
within the region of interest, the distances between the data points with respect to the area
of interest and the spatial continuity of the interpolated variable.
THE THEORY OF REGIONALIZED VARIABLES: GEOSTATISTICS
Geostatistics, a statistical technique based on the theory of regionalized variables, is
used to describe spatial relationship among the data sets. It was initially used by mining
industry as high cost of drilling made the analysis of the data extremely important. The
most important problem faced in the mining is the prediction of the ore grade in the
mining block from observed samples that are irregularly spaced. The basic tool in
geostatistics, the variogram, is used to quantify spatial correlations between observations.
The estimation procedure Kriging is named after D. Krige, who and his colleague’s
applied statistical techniques to ore reserves in 1950’s. Due to lot of advantages of
kriging, the application of kriging can be found in very different disciplines ranging from
mining and geology to soil science, hydrology, meteorology, environmental sciences,
agriculture etc. Kriging can be applied wherever a continuous measure is made on a
sample at a particular location in space and/or time (Cressie, 1991; Armstrong, 1998).
Kriging is best to adopt for spatially dependent variables, where it is necessary to use the
limited data in order to infer the real nature as precisely as possible.
The application of kriging to groundwater hydrology was initiated by Delhomme
(1976,1978,1979), followed by a number of authors, viz., Delfiner and Delhomme (1983),
Marsily et al (1984), Marsily (1986), Aboufirassi and Marino (1983, 1984), Gambolti and
Volpi (1979) and so many else. In geostatistics, the geological phenomenon is considered
as a function of space and or time. If z(x) represents any random function with measured
values at “N” locations in space z(xi) i = 1,2,3,…N and if the value of the function z has
to be estimated at the point xo. Each of z1, z2, z3,….zN contributes a part of it to zo and
taking this in to account, therefore, the expected value of zo is zo*, which is equal to
z*(xo) = Ȝ1z1 + Ȝ2z2 +Ȝ3z3+Ȝ4z4+………. ȜNzN
The kriging estimate can be defined as:
N
z*(xo) =
¦ O z(xi)
i
(1)
i 1
Where z*(xo) is the estimation of function z(x) at the point xo and Ȝi are the weighting
factors.
For the purpose of making this predictor to be unbiased the following conditions
need to be satisfied.
223
On an average the difference between the estimated value and the true (unknown)
value that is the expected value of estimation error should be zero. This un-biased
condition is also sometimes called as universality condition.
E [z*(xo)- z(xo) ] = 0
(2)
The condition of optimality means the variance of the estimation error should be
minimum.
ıK2 (xo)= var [(z*(xo)- z(xo)] is minimum.
(3)
Using Equation 1 and 2, we get
N
¦O
i
=1
(4)
i 1
expanding Equation 3, we obtain
N
ıK2(xo)= ¦ O i
i 1
N
N
j 1
i 1
¦ O j E[z (xi) z (xj)@ + E [z*(xo)2 @ – 2 ¦ O i E [z (xi) z*(xo)]
(5)
The best unbiased linear estimator is the one which minimizes ıK2(xo) under the
constraint of Equation 4. Introducing the Langrange multipliers and adding the term -2 P
N
( ¦ O i -1), we obtain
i 1
N
Q = ıK2(xo) - 2 P ( ¦ O i -1)
i 1
N
N
=
¦O ¦O
i
j
C (xi , xj) + C (O) –
j 1
i 1
N
N
2 ¦ O i C (xi , xo) - 2P
¦O
i 1
i
+ 2P
(6)
i 1
to minimize the above equation, making partial differentiation of Q w.r.t. Ȝi and μ and
equal to zero, we obtain the following kriging equations:
N
¦O
j
C (xi , xj) - P = C (xi , xo)
(7)
j 1
i = 1,2,3,…..N
N
¦O
j
1
(8)
j 1
where C (xi , xj) is the covariance between points xi and xj. Substituting Equation 7 into
Equation 5, we obtain the variance of estimation error.
ıK2(xo) = C (O) -
N
¦O
i 1
224
i
C (xi , xo) + P
(10)
The square root of this equation gives us standard deviation ıK(xo), which means that
with the 95% confidence, the true value will be within z*(xo) r 2 ıK(xo) range.
In case the covariance cannot be defined, we can derive the following kriging
equations:
N
¦O
j
J (xi , xj) + P = J (xi , xo)
(11)
j 1
i =1,2,3,…..N
N
¦O
j
1
(12)
j 1
and the variance of the estimation error becomes as:
N
Vk2 = ¦ O i J (xi , xo) + P
(13)
i 1
Equation 12 and 13 are a set of (N+1) linear equations with (N+1) unknowns and on
solving them we obtain the value of Ȝi, which are used to calculate the Equation 1 and 10
or 13.
Kriging is the best linear unbaised estimator (BLUE) as this estimator is a linear
function of the data with weights calculated according to the specifications of
unbaisedness and minimum variance. The weights are determined by solving a system of
linear equations with coefficients that depends only on the variogram which describes the
structure of the family of the functions. The weights are not selected on the basis of some
arbitrary rule, but infact depends on how the function varies in space. A major advantage
of kriging is that it provides the way to estimate the magnitude of the estimation error,
which is the rational measure of the reliabilty of the estimate. Since the variance of
estimation error depends on the variogram and the location of measurements, therefore,
before deciding a new location for measurement or deleting an existing measuremnet
point, a variance can be calculated and a better network of data can be designed based on
the minimum varaince even proir to any measurement.
The Variogram
It has been observed that all the important hydrogeological properties and
parameters such as piezometric head, transmissivity or hydraulic conductivity, storage
coefficient, yield, thickness of aquifer, hydrochemical parameters, etc. are all functions of
space. According to De Marsily (1986) these variables (known as the regionalized
variables) are not purely random, and there is some kind of correlation in the spatial
distribution of their magnitudes. The spatial correlation of such variables is called the
structure, and is normally defined by the variogram. The experimental variogram
measures the average dissimilarity between data separated by a vector h (Goovaerts,
1997). It is calculated according to the following formula.
225
2
J h 1 N h ¦ >z x h zx @
2 N h i 1
h = separation distance between two points, also called the lag distance.
A generalized formula to calculate the experimental variogram from a set of
scattered data can be written as follows (Ahmed, 1995):
1 Nd
J( d,T )= ¦[z( xi +dˆ,Tˆ )- z(xi ,Tˆ )]2
2Nd i 1
(14)
where,
d - 'd d dˆ d d +'d ,T - 'T dTˆ dT + 'T
with
d=
1
Nd
Nd
¦dˆi , T =
i 1
1
Nd
(15)
Nd
¦Tˆ
i
(16)
i 1
Where d and T are the initially chosen lag and direction of the variogram with 'd and 'T
as tolerance on lag and direction respectively. d and T are actual lag and direction for the
corresponding calculated variogram. Nd is the number of pairs for a particular lag and
direction. The additional Equation 16 avoids the rounding-off error of pre-decided lags
and directions (only multiples of the initial lag and fixed values of T only are taken in
conventional cases). It is very important to account for every term carefully while
calculating variograms. If the data are collected on a regular grid, 'd and 'T can be taken
as zero and d and T becomes d and T respectively. Often, hydrogeological parameters
exhibit anisotropy and hence variograms should be calculated at least in 2 to 4 directions
to ensure existence or absence of anisotropy. Of course, a large number of samples are
required in that case.
The variogram model is the principal input for both interpolation and simulation
schemes. However, modeling the variogram is not a unique process. Various studies
(Dagan and Neuman, 1997, Cushman, 1990) tried to correlate the mathematical
expressions normally used to describe variogram models to the physical characteristics of
the parameters.
Variograms obtained by modeling an experimental variogram are often not unique.
It is therefore, necessary to validate it. Cross-validation is performed by estimating the
random function at the points where realizations are available (i.e. at data points) and
comparing the estimate with data (Ahmed and Gupta, 1989). In this exercise, measured
values are removed from the data set one by one and the same is repeated for the entire
data set. Thus, all the measurements points, the measured value (z), the estimated value
(z*) and the variance of the estimation error (ı2) become available. This leads to
computing following statistics:
226
1
N
N
¦ (z
*
(17)
*
(18)
i
z i ) | 0 .0
i
z i ) 2 | min
i
z i )2 /V i
i 1
1
N
¦ (z
1
N
¦ (z
N
i 1
N
*
2
1
(19)
i
(20)
i 1
zi zi
Vi
*
d 2.0
Various parameters of the variogram model are gradually modified to obtain
satisfactory values of Equation 17 to 20. During the cross-validation many important
tasks are accomplished, such as:
ƒ
Testing the validity of the structural model.
ƒ
Deciding the optimum neighbourhood for estimation.
ƒ
Selecting suitable combinations of additional information particularly in case of
multivariate estimation.
ƒ
Sorting out unreliable data.
APPLICATION OF GEOSTATISTICS IN GROUNDWATER MODELING
Theoretical Aspects
The formulation of the general groundwater flow equation in porous media is based
on mass conservation and Darcy’s law, and can be represented as follows in threedimensions (see e.g. Bear, 1979; Mercer and Faust, 1981; Marsily, 1986):
w §
wh · w §
wh · w §
wh ·
¨ . x ¸ ¨¨ . y ¸¸ ¨ . z ¸
wx ©
wx ¹ wy ©
wy ¹ wz ©
wz ¹
r R' S s
wh
..........(21)
wt
where Kx, Ky and Kz [LT-1] are the hydraulic conductivities (or permeability) along the x,
y and z coordinates, respectively, assumed to be the principal directions of anisotropy, h
[L] is the hydraulic head given by h=p/ȡg+z (p is the pressure, ȡ the density of water, g
the gravity constant, and z the elevation); h is also the water level measured in
piezometers; R c [T-1] is an external volumetric flux per unit volume entering or exiting
the system and S s [L-1] is the specific storage coefficient.
227
Equation 21 can be integrated over the vertical (two-dimensional flow) and
simplified when the flow takes place in confined conditions in the aquifer, and the head is
assumed constant over the vertical:
wh ª wh º wh ª wh º
T
»
«T
wx «¬ x wx »¼ wy ¬ y wy ¼
rR S
wh
.......... ...
wt
(22)
where Tx and Ty [L²T-1] are the transmissivities (product of the hydraulic conductivity and
saturated thickness of the aquifer) in the x and y directions, assumed to be the principal
directions of anisotropy in the plane, respectively; R [LT-1] is an external volumetric flux
per horizontal unit area entering or exiting the aquifer, in practice, it represents recharge,
or sometimes evapotranspiration from a shallow aquifer, or else leakage fluxes between
aquifers; S [-] is the storage coefficient (product of S s , the specific storage coefficient,
by the thickness of the aquifer). In case of unconfined conditions, Equation 21 can be
integrated over the vertical and simplified as follows:
wh º
wh ª
wh º wh ª
K
h
K
h
x
y
wy »¼
wx «¬
wx »¼ wy «¬
rR Sy
wh
.......... ........( 23)
wt
where S y [-] indicates the specific yield of the water-table aquifer. In this case, the
hydraulic head is expressed taking the bottom of the aquifer (assumed horizontal) as the
z=0 reference plane, so that the head h is also the saturated thickness of the aquifer. Since
this equation is non linear, and difficult to deal with, in many instances, Equation 22 is
used as an approximation for unconfined aquifers as well, making the simplifying
assumption that the product K*h of the saturated thickness h of the aquifer by its
permeability K is equal to a constant, the transmissivity T, even if the saturated thickness
varies with time. In that case, the storage coefficient S in Equation 22 is replaced by the
specific yield S y .
These equations with the appropriate boundary and initial conditions can be solved
by standard numerical methods such as Finite Differences or Finite Elements. A large
body of literature is available on the analytical or numerical formulation, development
and solution of the groundwater flow equations. Some useful references are: Freeze
(1971); Remson et al (1971); Prickett (1975); Trescott et al (1976); Pinder and Gray
(1977); Cooley (1977, 1979, 1982); Brebbia (1978); Mercer and Faust (1981); Wang and
Anderson (1982); Townley and Wilson (1985); Marsily (1986).
The major steps involved in performing aquifer modeling of an aquifer are as
follows :
1. Conceptualization of the aquifer system and establishment of various flows.
228
2. Preparing a database of the essential and available parameters with their
georeference.
3. Analyzing the variability of various paramters in the area.
4. Discretization of the aquifer system into meshes with theoretical and practical
constraints.
5. Fabrication of the aquifer model by assigning various parameters to each mesh.
6. Execution of model under steady and transient conditions and comparing the
output with filed values.
7. Calibration of aquifer model to eleminate the mismatches.
8. Performing sensitivity analyses to determine the sensitivity and priority of various
parameters
9. Finalization of the calibrated model simulating the flow in the aquifer system.
10. Building the futuristic scenarios and predcting the waterlevels using the
calibrated model.
The entire procedure involves a large number of decisions and hence the risk of high
biasness. The present work analyses the pratcial problems and constraints faced at all
levels and to solve them in an unbiased way using the theory of regionalized variables.
Discretization
This is the first step in the aquifer modeling. Here the modelers have to descritize
the area into large number of uniform grids or meshes over which it is supposed that
there is no variation in aquifer properties. Although with the invent of advanced
computers, the computation of large number of grids/meshes is not a difficult task but the
preparation of data to be assigned to each grid is time consuming. The main question
arises here is how to decide the number of grid and their sizes? Here the use of
geostatistics lies in the modeling. Initially the area is divided in to very finer grids and
with the help of geostatistical estimation procedure, the variance of estimation error over
each grid is calculated. As we know that the closer the spatial difference, the closer will
be the values and thus final grids sizes can be decided by uniting the grids having same or
close values of the variance of estimation error. Thus the grid size is decided on the basis
of geostatistical estimation which provide unbiased, minimum variance and with a unique
value over the entire area of the mesh as block kriging is used.
Model Fabrication
After discretization of the area in to optimal grid sizes, the hydrogeological
parameters are required over each grid. It is often very difficult to determine these
parameters accurately in the field over such a large number of points either due to
economic reason or because of access to the area. The data is always available only at
few places in the area. Now it is necessary to link the two situations, and to supply the
parameter values for all the discretized volumes from the available observed data.
229
It has been observed that the deterministic approaches are not much applicable due
to great spatio-temporal variability of the hydrogeological parameters. The field
heterogeneity of the groundwater basins is often inextricable and very difficult to analyze
with deterministic methods (e.g., Bakr et al., 1978; Delhomme, 1979; Ganoulis and
Morel-Seytoux, 1985). In-situ measurements at the basin scale have shown that physical
properties of the hydrogeological variables are highly irregular. However, this spatial
variability is not, in general, purely random and the variables exhibit some kind of
correlation in their spatial distribution.
It is more convenient and common to introduce geostatistics in a probabilistic
framework, for performing estimation considering the spatio-temporal variability of the
parameters. By using geostatistical estimation techniques, it is not only possible to
evaluate the values of parameters at each and every grid but the level of accuracy in terms
of variance of estimation error can also be obtained prior to estimation.
Calibration
Calibration is the process of modifying the input parameters to a groundwater model
until the output from the model matches an observed set of data with in some acceptable
criteria. This criterion is decided with the help of geostatistics. A confidence interval is
given by the standard deviation of the estimation error that provides a useful guide to
parameter modification at each mesh and to check that the calculated value falls inside
the confidence interval of the observed values.
The calibration process typically involves calibrating the steady state and transient
state conditions. In steady state condition unbiased calibration of aquifer model is
obtained by using kriging estimation variance and the result lies within the 95% of
confidence interval. Mathematically it can be stated as,
him hi
Vi
<2
(24)
Where,
him = modeled water level.
hi* = estimated water level.
V i = standard deviation of the estimation error.
Similarly in case of transient condition, the calibration of the aquifer model in an
unbiased way is performed using hydraulic conductivity within 95% confidence interval
of estimation variance. Mathematically it can be stated as,
K 2V k d K T t K 2V k
where,
K* = estimated hydraulic conductivity
KT = observed hydraulic conductivity
230
(25)
V k = standard deviation of the estimation error.
Prediction
After the model has been succesfully calibrated it is now ready for predictive
simulation. Thus the model now can be used for predicting some future groundwater flow
or contaminant transport conditions. It can also be used for evaluating different
remediation alternatives. However errors and uncertainities in a groundwater flow
analysis and solute transport analysis make any model prediction no better than
approximation unless all the model predictions should be expressed as a range of possible
outcomes that reflects the assumptions involved and uncertainity in the model input data
and parameter values. In spite of all unbiased and best efforts, still a large number of
calibrated models could be avialable for prediction. Thus to make them unique, a double
sum is calculated using the statnadard deviation of the water level estimation error also
for all the time step and for all the meshes. Thus a calibrated model providing minimum
value of this sum could be used for prediction.
A CASE STUDY FROM THE MAHESHWARAM WATERSHED, ANDHRA
PRADESH, INDIA
The Maheshwaram watershed of about 53 km² in the Ranga Reddy district (Figure 1)
of, Andhra Pradesh, India, is underlain by granitic rocks. This watershed is a
representative Southern India catchment in terms of overexploitation of its weathered
hard-rock aquifer (more than 700 borewells in use), its cropping pattern, rural socioeconomy (based mainly on traditional agriculture), agricultural practices and semi-arid
climate. The granite outcrops in and around Maheshwaram form part of the largest of all
granite bodies recorded in Peninsular India. Alcaline intrusions, aplite, pegmatite, epidote,
quartz veins and dolerite dykes traverse the granite. There are three types of fracture
patterns in the area, viz. (i) mineralised fractures, (ii) fractures traversed by dykes, and (iii)
late-stage fractures represented by joints. The vertical fracture pattern is partly
responsible for the development of the weathered zone and the horizontal fractures are
the result of the weathering. Hydrogeologically, the aquifer occurs both in the weathered
zone and in the underlying weathered-fractured zone. However, due to deep drilling and
heavy groundwater withdrawal, the weathered zone has now become dry. About 150 dug
wells were examined and the nature of the weathering was studied. The weathered-zone
profiles range in thickness from 1 to 5 m below ground level (bgl). They are followed by
semi-weathered and fractured zones that reach down to 20 m bgl. The weathering of the
granite has occurred in different phases and the granitic batholith appears to be a
composite body that has emerged in different places and not as a single body. One set of
pegmatite veins displacing another set of pegmatitic veins has been observed in some
well sections. Joints are well developed in the main directions – N 0° – 15°E, NE-SW,
and NW-SE that vary slightly from place to place.
231
Figure 1. Geographical location of the study area and the watershed
The groundwater flow system is local, i.e. with its recharge area at a topographic
high and its discharge area at a topographic low adjacent to each other. Intermediate and
regional groundwater flow systems also exist since there is significant hydraulic
conductivity at depth. Aquifers occur in the permeable saprolite (weathered) layer, as
well as in the weathered-fractured zone of the bedrock and the quartz pegmatite intrusive
veins when they are jointed and fractured. Thus only the development of the saprolite
zone and the fracturing and interconnectivity between the various fractures allow a
potential aquifer to develop, provided that a recharge zone is connected to the
groundwater system.
Mean annual precipitation (P) is about 750 mm, of which more than 90% falls
during the monsoon season. The mean annual temperature is about 26 °C, although in
summer ("Rabi" season from March to May), the maximum temperature can reach 45 °C.
The resulting potential evaporation from the soil plus transpiration by plants (PET) is
1,800 mm/year. Therefore, the aridity index (AI = P/PET = 0.42) is 0.2 < AI < 0.5,
typical of semiarid areas (UNEP 1992). Although the annual rainfall is around 750 mm
and the recharge around 10-15%, the water levels have been lowered by about 10 m
during the last two decades due to intensive exploitation. The transmissivity of the
fractured aquifer was measured by seven pumping tests in wells and it varies
considerably from about 1.7×10-5 m²/s to about 1.7×10-3 m²/s, and a low storage
232
coefficient (0.6%) indicates a weak storage potential of the aquifer (Maréchal et al.,
2004). The withdrawal which increases year by year has to be controlled to allow
recharge by rainfall to maintain or restore the productive capacity of the depleted aquifer.
Our data show that the weathered-fractured layer is conductive mainly from the
surface down to a depth of 35 m, the range within which conductive fracture zones with
transmissivities greater than 5×10-6 m2/s are observed (Maréchal et al. 2004). The lower
limit corresponds to the top of the unweathered basement, which contains few or poorly
conductive fractures (T < 5×10-6 m2/s) and where only local deep tectonic fractures are
assumed to be significantly conductive at great depths, as observed by studies in Sweden
(Talbot and Sirat, 2001), Finland (Elo, 1992) and the United States (Stuckless and Dudley,
2002). An unpublished study of the same area has statistically confirmed this result by the
analyses of airlift flow rates in 288 boreholes, 10 to 90 m deep, where the cumulative
airlift flow rate ranges from 1.5 m3/h to 45 m3/h, and the mean value increases drastically
in the weathered-fractured layer at depths between 20 and 30 m. Below 30 m, the flow
rate is constant and does not increase with depth. In practice, drilling deeper than the
bottom of the weathered-fractured layer (30 – 35 m) does not increase the probability of
improving the well discharge. The data confirm that the weathered-fractured layer is the
most productive part of the hard-rock aquifer, as already shown elsewhere by other
authors (e.g. Houston and Lewis 1988, Taylor and Howard 2000).
Two different scales of fracture networks are identified and characterized by
hydraulic tests (Maréchal et al., 2004): a primary fracture network (PFN) which affects
the matrix at the decimetre scale, and a secondary fracture network (SFN) affecting the
blocks at the borehole scale. The latter is the first one described below.
The secondary network is composed of two conductive fractures sets - a horizontal
one (HSFN) and a sub-vertical one (VSFN) as observed on outcrops. They are the main
contributors to the permeability of the weathered-fractured layer. The average vertical
density of the horizontal conductive set ranges from 0.15 m-1 to 0.24 m-1, with a fracture
length of a few tens of meters (10 - 30 m in diameter for the only available data). This
corresponds to a mean vertical thickness of the blocks ranging from |4 m to |7 m. The
strong dependence of the permeability on the density of the conductive fractures indicates
that individual fractures contribute more or less equally to the bulk horizontal
conductivity (Kr = 10-5 m/s) of the aquifer. No strong heterogeneity is detected in the
distribution of the hydraulic conductivities of the fractures, and therefore no scale effect
was inferred at the borehole scale. The sub-vertical conductive fracture set connects the
horizontal network, ensuring a vertical permeability (Kz = 10-6 m/s) and a good
connectivity in the aquifer. Nevertheless, the sub-vertical set of fractures is less
permeable than the horizontal one, introducing a horizontal-to-vertical anisotropy ratio
for the permeability close to 10 due to the preponderance of horizontal fractures.
As discussed earlier, the horizontal fracture set is due to the weathering processes,
through the expansion of the micaceous minerals, which induces cracks in the rock.
These fractures are mostly sub-parallel to the contemporaneous weathering surface, as in
233
the flat Maheshwaram watershed where they are mostly horizontal (Maréchal et al.,
2003). In the field, the bore wells drilled with a fairly homogeneous spacing throughout
the watershed confirm this conclusion: of 288 wells, 257 (89%) were drilled deeper than
20 meters and 98% of these are productive. In any case, the probability of a vertical well
crossing a horizontal fracture induced by such wide-scale weathering is very high.
The primary fracture network (PFN), operating at the block scale, increases the
original matrix permeability of Km = 10-14 – 10-9 m/s to Kb = 4×10-8 m/s. Regarding the
matrix storage, it should also contribute to the storage coefficient in the blocks of Sb = 5.7
x 10-3. The storage in the blocks represents 91% of the total specific yield (Sy = 6.3×10-3)
of the aquifer; storage in the secondary fracture network accounts for the rest. This high
storage in the blocks (in both the matrix and the PFN) and the generation of the PFN
would result from the first stage of the weathering process itself. The development of the
secondary fracture network (SFN) is the second stage in the weathering process: that is
why the words “primary” and “secondary” have been chosen to qualify the different
levels of fracture networks. The obtained storage values are compatible with those of
typical unconfined aquifers in low-permeability sedimentary layers.
The universal character of granite weathering and its worldwide distribution
underline the importance of understanding its impact on the hydrodynamic properties of
the aquifers in these environments.
Several input parameters were analysed for their variability to assess the areas of
high, low and medium variability. Finally the water level gradient have been kriged all
over the area by taking initially a reasonable fine grids of 200m by 200m size. The
estimation error of the estimates of the water level gradient in a pre-monson season when
the pumping has been least and water levels were not disturbed by the anthropogenic
activity, were taken as guiding parameter and by joining the grids with uniform values the
meshes were made courser and courser. Figure 2 shows the experimental and theoretical
variogram of the water level gradient.
Figure 3 depicts a reasonable discretization of the aquifer system with variable
meshe sizes in a nested square fashion. The discretization thus are made based on the
criteria of parameters varibility and distribution of the estimation error. However, this has
to be refined as the software MODFLOW does not permit a nested square grids and due
to this constraints, the meshes were further modified but this change was made only to
suit the software. There are software that may permit the use of originally obtained mesh
discretization. The meshe sizes thus used are 200m by 200m, 200m by 400m, 400m by
200m and 400m by 400m with a total of 764 meshes in the watershed (Figure 4).
The model discreized thus has reasonable logics and have been obtained following
criteria laid down mainly on the variability of the paramters. The system parameter
mainly K values obtained from the hydraulic tests on 25 wells fairly distributed in the
area, were kriged using a suitable variogram and obtained the values over all the 764
meshes of the model. Of course, the field values were log-transformed and then the
estimated values were backword transformed by taking anti-logs. The Specific Yield
234
values were not available so two zones were assigend. The eastern and western
boundaries are taken as no flow boundaries as the major flow direction is from south to
north. The remaining nodes at the boundaries are taken as fixed head boundaries as the
water levels are well known, Other input paramters such as rainfall recharge,
groundwater withdrawal, groundwater evaporation, Irrigation return flow from irrigated
fields etc. are treated as regionalized variables and they have all beenestimated using
theory of regionalized variables. Figures 5 and 6 show the variogram of recharge
percentage and the map of the estimated recharge percentage over the area
Figures 7 and 8 show the variogram and the estimates of groundwater withdrawals.
In this case the number of meshes and number of pumping wells are more or less
conciding and then values were estimated at each grids using block estimation. The
model with these paramter values was run for the steady and transient conditions. Figures
9 show the estimated values of initial water level that was used for steady state simulation.
Figure 2. Experimental and Theoretical Variogram of the waterlevel gradient
235
Figure 3. The discretization of the aquifer into Nested Square meshes
236
Figure 4. Variable size mesh discretization suitable to MODFLOW
Figure 5. Variogram of the percentage of rainfall recharge
237
1902000
1901000
1900000
1899000
1898000
1897000
1896000
1895000
1894000
1893000
224000 225000 226000 227000 228000 229000 230000 231000 232000 233000
Figure 6. The map showing variation of recharge percentage to rainfall
Figure 7. Variogram of the groundwater withdrawal with a high nugget effect.
238
1902000
1901000
1900000
1899000
1898000
1897000
1896000
1895000
1894000
1893000
224000 225000 226000 227000 228000 229000 230000 231000 232000 233000
Figure 8. Estimated values of the groundwater withdrawal for a typical period
1902000
1901000
1900000
1899000
1898000
1897000
1896000
1895000
1894000
1893000
224000 225000 226000 227000 228000 229000 230000 231000 232000 233000
Figure 9. Piezometric map in the area for the initial conditions (June 2000)
239
The most important aspect in deaaling with and removing biasness from the aquifer
modeling is during calibration and acceptance of the model. The Equation 24 has been
used for testing the acceptability and zones of calibration if required and the changes
during the calibration was based on the Equation 25. The values of Equation 24 is thus
plotted in a map at Figure 10 that very intrestingly shows the meshes needed caibration
chnages and these chnages were carried out using Equation 25.
The comparison during the transient state was also made using the geostatistics.
1902000
1901000
1900000
1899000
1898000
1897000
1896000
1895000
1894000
1893000
224000 225000 226000 227000 228000 229000 230000 231000 232000 233000
Figure 10. Map showing the areas that need calibration and the areas where the
comparison of water-levels are satisfactory.
CONCLUSIONS
It is often very difficult to analyse field hetrogeneties of groundwater basins with
deterministic methods. Thus stochastic methods has been used especially for the
estimation of the hydrogeological parameters. The theory of regionalized variables
provides a sound stochastic model for the study of spatial phenomena and its applicaion
in the field of hydrogeology has been found to be extremely fruitful. The advantage of
240
geostatistical methods is that it provide the variance of estimated error together with the
estimated value. Of course there are tremendous applications of this method particularly
in groundwater modelling:
¾ The closer the values of aquifer parameters to reality the faster will be the model
calibration. Better estimated values with lower estimation variance are initially
assigned to the nodes of aquifer model using geostatistical estimation.
¾ An assumption is made in aquifer modeling that a single value of system parameter
represents the entire grid(although, very small). Averaging over a block in two or
three dimension can be obtained through block estimation.
¾ An optimal grid size and number of nodes in discretizing aquifer system, can be
obtained and best locations for the new control points can be predicted.
¾ A confidence interval given by the standard estimation of the estimation error
provides a useful guide to K/T modification over each grid and to check that the
calculated heads fall inside the confidence interval of the observed heads.
¾ A performance analysis of the calibrated model can be achieved to decide the best
calibrated model using variance of the estimated error which can be used for
prediction.
The main handicap in applying geostatistical techniques to the hydrogeological
problems is the scarcity of the observed data. Careful estimation of the variogram and
cross-validation of the variogram models are therefore essential. Thus, a few
modifications and improvement to the existing techniques permits to utilize
hydrogeological data successfully in prediction of aquifer parameters. One such
modification which has found much use is ˝kriging with external drift˝.
In geostatistics assumptions are made on the nature of the underlying random
function (such as the intrinsic hypothesis or second order stationarity). These assumptions
are critical to the success of the application of these techniques.
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243
WATER RESOURCES RESEARCH, VOL. 34, NO. 5, PAGES 1011–1025, MAY 1998
An evaluation of Jacob’s method for the interpretation
of pumping tests in heterogeneous formations
Peter M. Meier, Jesús Carrera, and Xavier Sánchez-Vila
Departament Enginyeria del Terreny i Cartogràfica, ETS Enginyers de Camins, Canals i Ports de Barcelona
Universitat Politècnica de Catalunya, Barcelona, Spain
Abstract. Most pumping tests are interpreted using the classical Theis assumption of
large-scale homogeneity with various corrections to account for early time behavior of
drawdown curves. When drawdowns are plotted versus log time, late time data often
delineate a straight line, which is consistent with Jacob’s approximation of Theis’ solution
but may seem surprising in view of the heterogeneity of natural media. The aim of our
work is to show that Jacob’s method leads to a good approximation of the effective
transmissivity of heterogeneous media when constrained to late time data. A review of
several multiwell pumping tests demonstrates that when drawdown curves from each
observation well are interpreted separately, they produce very similar transmissivity T
estimates. However, the corresponding estimates for storativity span a broad range. This
behavior is verified numerically for several models of formation heterogeneity. A very
significant finding of the numerical investigation is that T values estimated using simulated
drawdown from individual observation wells are all very close to the effective T value for
parallel flow. This was observed even in nonmultiGaussian T fields, where high T zones
are well connected and where the effective T is larger than the geometric average of point
values. This implies that Jacob’s method can be used for estimating effective T values in
many, if not most, formations.
1.
1.1.
Introduction
Basic Concepts and Previous Work
Formation transmissivity is the most important parameter to
be determined in many hydrogeological problems. Transmissivity is often determined in the field using pumping tests.
Hydrogeologists have to rely on an interpretation model for
the evaluation of pumping test data. Constant rate pumping
tests are most often interpreted using the Theis [1935] method
or Jacob’s semilogarithmic approximation [Cooper and Jacob,
1946]. Both of these techniques use the temporal evolution of
pumping-induced drawdown to obtain estimates of transmissivity and storativity under the assumptions of homogeneity, a
two-dimensional domain, and confined conditions. Because of
the importance of Jacob’s method for this work, we review it
briefly here (further details can be found in most hydrogeology
text books; see, e.g., Freeze and Cherry [1979]). Jacob’s method
is based on the fact that the Theis well function plots as a
straight line on semilogarithmic paper at large dimensionless
times. Hence the method consists of drawing a straight line
through the late time data points and extending it backward to
the point of zero drawdown (time axis intercept), which is
designated t 0 . The drawdown per log cycle is obtained from
the slope m of the line. Values for transmissivity T and storativity S can then be found from
2.3Q
4␲m
(1)
2.25Tt 0
r2
(2)
T⫽
S⫽
Copyright 1998 by the American Geophysical Union.
Paper number 98WR00008.
0043-1397/98/98WR-00008$09.00
where Q is the constant pumping rate and r is the radial
distance to the observation well; if drawdowns are measured at
the pumping well, r is equal to the effective radius of the well.
The method is considered a valid approximation of the Theis
solution if the slope can be derived from data points with u ⫽
r 2 S/(4Tt) smaller than 0.03 for which the approximation error
is ⬍1%.
The analytical solution underlying the Theis and Jacob techniques is based on the assumption of aquifer homogeneity.
Other analytical solutions assume that the aquifer can be subdivided into, at most, two or three regions with uniform parameters [e.g., Streltsova, 1988; Butler, 1988, 1990; Butler and
Liu, 1991, 1993]. Although such configurations are simplifications of the complex structure of real heterogeneous formations, they can provide insight into the behavior of drawdown
in heterogeneous formations. For example, Butler [1990] explains how the Theis and Jacob methods lead to different T
values in heterogeneous aquifers because they apply different
weightings to different portions of the drawdown curve. As
shown in (1), pumping test analysis using Jacob’s method is
based on the rate of drawdown change. When T only depends
on radial distance (axial symmetry), such change reflects aquifer properties only within a ring of influence through which the
front of the pressure depression passes within the considered
time interval. Therefore estimated transmissivity is independent of the aquifer properties between the inner radius of the
ring of influence and the pumping well. However, storage estimates depend on the variations in T between the pumping
well and the front of the cone of depression [Butler, 1988].
Butler and Liu [1993] conceptualized the nonuniform aquifer
as a uniform matrix into which a disk of anomalous properties
has been placed. They found, among other things, that Jacob’s
method can be used in any laterally nonuniform system to
estimate matrix transmissivity if the flow to the pumping well is
245
1012
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approximately radial during the period of analysis. Oliver
[1990] used a perturbation approach to determine the weighting function according to which formation permeabilities are
averaged in the determination of apparent values from Jacob’s
method for pumping test analysis. He found that the averaging
area is an annular region of the formation and that both the
averaging area and the radius of investigation of a pumping
test increase with time. Oliver [1993] used a perturbation approach to study the effect of two-dimensional spatial variations
in transmissivity and storativity. He concluded that the area of
investigation of a pumping test is bounded by an ellipse that
encloses the pumping well and the observation well and that
observation well drawdown is relatively sensitive to near-well
transmissivity, whereas the observation well drawdown derivative is influenced by the near-well properties for a finite period only.
Previous numerical investigations of transient radial flow in
heterogeneous systems resembling real formations by Warren
and Price [1961], Lachassagne et al. [1989], and Butler [1991]
have been concerned with estimates of transmissivity from
pumping tests and their relationship to statistical parameters
of the heterogeneous transmissivity field. These authors conclude that transmissivity values obtained from late time data
using standard analysis techniques and especially Jacob’s
method are close to the geometric mean of the transmissivity
fields and are a reasonable representation of transmissivity on
the regional scale (assuming boundary effects are negligible).
Gómez-Hernández et al. [1995] simulate pumping tests in nonstationary anisotropic T fields and conclude that the simulated
drawdown curves can be misinterpreted as typical responses
from a variety of homogeneous aquifer conceptual models
(e.g., leaky aquifers, double porosity systems, Theis with
boundary effects, and others). Herweijer [1996] simulates
pumping and tracer tests in T fields corresponding to both a
deterministic sedimentological facies model and a Gaussian
geostatistical model. He concludes that early time portions of
pumping test data reveals high conductivity interwell pathways,
which dominate solute transport.
1.2.
Working Hypothesis and Scope of the Work
We have frequently observed that when applying Jacob’s
method to late time drawdown data at several observation
wells, transmissivity estimates T est tend to be fairly constant,
while the storativity estimates S est display a great spatial variability. Similar observations have previously been reported by
Schad and Teutsch [1994] and Herweijer and Young [1991] in
studies in which the Theis method is used to analyze drawdown
data from observation wells in heterogeneous alluvial aquifers.
This behavior is surprising in most cases because independent information (such as short term hydraulic tests, geological
information, etc.) often suggests that the actual formation
transmissivity T is strongly heterogeneous. Conversely, the actual aquifer storativity S is not thought to be highly variable.
The variability of the storativity estimates is sometimes attributed to radial flow in homogeneous anisotropic formations
[e.g., Streltsova, 1988]. However, methods accounting for homogeneous anisotropic transmissivity [e.g., Neuman et al.,
1984; Hantush, 1966; Papadopulos, 1965] did not provide consistent results when applied to our field cases. Storativity depends on porosity and the rock and water compressibility. The
latter are believed to be fairly constant at the local scale.
Porosity usually varies within a small range compared with
hydraulic conductivity, as reported by Guimerà and Carrera
[1997] for fractured rocks, Ptak and Teutsch [1994] for a porous
aquifer, Bachu and Underschulz [1992] for sandstone cores,
and Neuzil [1994] for clays and shales. Because of this, we have
assumed storativity to be homogeneous for our numerical
work.
The above paragraphs appear paradoxical: Long-term hydraulic tests lead to small variability of T est and large variability
of S est, while the opposite should be expected for point values
of T and S. We conjecture that this paradox can be attributed
to the fact that methods developed for homogeneous media
are being used for interpretation of tests performed in heterogeneous formations.
Our work is centered around the above conjecture. Hence
we first provide some field observations to show that this situation indeed exists. Second, we use a series of numerical
simulations to show that Jacob’s method yields nearly constant
T est and spatially variable S est in aquifers with variable T and
constant S. Third, we investigate how the spatial correlation
structure of T and the variance of the input log T field affect
the results. Finally, we compare the value of transmissivity
obtained using Jacob’s method with the effective transmissivity
derived for parallel flow.
2.
Field Observations
We have obtained similar T est and strongly varying S est values when using Jacob’s method at several field studies in very
different types of heterogeneous media, including a singleshear zone in granitic rock, a fractured gneiss formation, and
alluvial aquifers. In what follows, we first present field data
from the migration experiment of the joint project of Nagra
(Swiss National Cooperative for the Disposal of Radioactive
Waste) and PNC (Power Reactor and Nuclear Fuel Development Corporation, Japan) at the Grimsel underground rock
laboratory, located in the Swiss Aar Massif within granitic
rocks. Second, we show the results from the interpretation of a
long-term pumping test in a fractured gneiss formation at El
Cabril, which is the site for storage of low- and intermediatelevel radioactive waste in southern Spain. Third, we present
two other cases from the literature in which different authors
have reported a similar behavior of T est and S est values in
alluvial aquifers.
2.1.
Migration Experiment at the Grimsel Test Site
The Grimsel rock laboratory consists of galleries lying 450 m
below the surface within granitic rock. A migration experiment
[Frick et al., 1992] was conducted within an almost vertical
shear zone that dips parallel to the foliation of the host rock.
This shear zone, henceforth designated as the migration shear
zone, is more or less isolated from other shear zones and
consists of a series of subparallel fractures. Mylonite, cataclastite, and other gouge materials are observed within the subfractures, influencing the permeability within the shear zone.
The migration shear zone can be conceptualized as a vertical
zone of intersecting fractures with a thickness of ⬃0.5 m and
can be considered to be a two-dimensional feature at the scale
of the tests described here. Hydrological exploration involved
drilling eight boreholes into the shear zone. Locations of the
boreholes and the laboratory gallery within the shear zone
plane are shown in Figure 1. The intervals in the boreholes
were isolated with packers for the performance of hydraulic
and tracer tests.
Single-hole short-term pumping tests of 5–10 s in duration
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(rates between 10 and 200 mL/min) were conducted to obtain
point estimates for the transmissivity. These tests were analyzed with the MARIAJ code [Carbonell and Carrera, 1992],
which uses an automatic optimization procedure coupled with
the general analytical solution of Barker [1988]. Transmissivity
estimates range between 10⫺8 and 5 ⫻ 10⫺6 m2/s, indicating
strong local heterogeneity, while storativity estimates vary from
5 ⫻ 10⫺10 to 10⫺5 m2/s [Meier, 1997]. The large variation in the
S estimates is believed to reflect the uncertainties in estimating
S from single-well tests but may also be a product of an analysis
method based on an analytical solution for homogeneous aquifers. Cross-hole constant-rate pumping tests of ⬃2 hours in
duration were performed in boreholes 4, 6, 8, 9, and 10 using
pumping rates of 200 mL/min. Analysis of the late time data at
both the pumping and the observation wells with Jacob’s
method resulted in a virtually constant T est value of ⬃10⫺6
m2/s for all tests and in a broad range of S est values extending
from 4 ⫻ 10⫺7 to 5 ⫻ 10⫺2 m2/s [Meier, 1997]. Transmissivities
and storativities are displayed in Figures 2a and 2b for the
single-hole tests and for the cross-hole test performed by
pumping in borehole 9. A semilog plot of the drawdown curves
for the cross-hole pumping test in borehole 9 is shown in
Figure 3 as an example of the drawdown behavior seen in all
the cross-hole tests. Straight lines with similar slopes start
developing after about 3500 s for all drawdown curves. Boundary effects due to the influence of the gallery are not observed
during these tests. Note that inverse geostatistical modeling of
the steady state hydraulic head field and the transient drawdown data reveals a variation in transmissivity of several orders
of magnitude and the existence of highly transmissive channels
within the shear zone [Meier et al., 1997]. Such channels may be
responsible for the relatively large drawdown at borehole 5
shown in Figure 3.
1013
Figure 2. Results from single-hole and cross-hole tests conducted in the migration shear zone for (a) transmissivity estimates and (b) storativity estimates. Note that transmissivities
derived from short-term single-hole tests display a large variability in contrast to transmissivities from cross-hole tests,
which remain virtually constant. Storativity estimates span a
wide range of values for single-hole and cross-hole tests.
The El Cabril site is located in a very sparsely populated area
west of Cordoba in southern Spain. The underlying formations
are composed of biotitic gneiss and metaarkoses. These formations were affected by the compressive Hercynian deformation, so the formations follow a predominantly WNW-ESE and
NW-SE trend and are characterized by primarily vertical dips.
Several sets of transverse fractures (directions 60⬚N, 90⬚N, and
N-S) are clearly indicated on the surface [Carrera et al., 1993].
A large number of slug and short-term pumping tests were
carried out at this site. The transmissivities obtained from
these tests vary over 5 orders of magnitude. A long-term pumping test was also performed [Bureau de Recherches Geologiques
et Minières, 1990]. The locations of the pumping (S33) and
observation wells are shown in Figure 4. The drawdown curves
from individual observation wells were interpreted separately
using the MARIAJ code [Carbonell and Carrera, 1992] under
the assumption of a homogeneous and isotropic medium. The
estimated parameters obtained for each well are shown in
Figure 5. The T est values range from 0.38 to 0.52 m2/d, while
the S est values range from 10⫺4 to 3 ⫻ 10⫺2 m2/d. Thus the T est
values are almost constant, while the S est values range over
Figure 1. Locations of the boreholes and the laboratory gallery in the migration shear zone at the Grimsel test site.
Figure 3. Semilog plot of the drawdown observed at the
pumping well and the observation points during a cross-hole
test in the migration shear zone.
2.2.
El Cabril
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Figure 4. Location of pumping and observation wells at the
El Cabril site.
more than 2 orders of magnitude. This test was later interpreted by means of a finite element model automatically calibrated with the approach of Carrera and Neuman [1986].
Drawdowns could only be closely matched by assuming that
S227 (the observation well at which the smallest S est was obtained) was connected to S33 through a fracture, a likely situation given the geology at the site. Note that a convergent flow
tracer test was performed at this site by injecting a different
tracer into each of the observation wells displayed in Figure 4
while well S33 was pumped. Porosities were estimated under
the assumption of homogeneity using the TRAZADOR code
[Benet and Carrera, 1992]. The pattern of estimated porosities
is very similar to that of S est in Figure 5 [Sanchez-Vila et al.,
1992; Carrera, 1993]. This suggests that relatively high transmissivity pathways are reflected not only in S est but also in
estimated porosity when homogeneous media solutions are
used.
2.3.
Field Observations Reported in the Literature
Schad and Teutsch [1994] provide an extensive data set from
a series of pumping tests performed in a highly heterogeneous
aquifer at the Horkheimer Insel field site in the Neckar Valley
of Germany. Because of its relevance to our work, we review
briefly some parts of their study. The unconfined aquifer in
which the tests were performed consists of ⬃4 m of poorly
sorted sand and gravel deposits of a braided river origin. This
aquifer is overlain by 5– 6 m of mostly clayey flood plain deposits and underlain by a hydraulically tight clay. Hydraulic
conductivity data derived from grain-size distributions had an
overall variance of log conductivity equal to 2.35, indicating
significant heterogeneity. Schad and Teutsch performed 26
pumping tests (duration of 2 hours) at different wells at the
site. During each of these tests, up to four observation wells
were monitored at radial distances of 2 to 36 m from the
pumping well. The analysis of the early time data, using the
Theis method, resulted in a broad range of T est values (1.6 ⫻
10⫺2 to 3.2 ⫻ 10⫺1 m2/s) and in a very broad range of S est
values (5.8 ⫻ 10⫺4 to 1.0 ⫻ 10⫺1 m2/s). The analysis of the late
time data, using the Theis method, resulted in a small range of
T est values (2.4 ⫻ 10⫺2 to 5.0 ⫻ 10⫺2 m2/s) and in a broad
range of S est values (1.7 ⫻ 10⫺2 to 1.3 ⫻ 10⫺1 m2/s). They also
conducted a long-term (96 hour) pumping test in which drawdown was monitored at 23 observation wells. The analysis of
this test with the Theis method resulted in a very small range
of T est values (2.9 ⫻ 10⫺2 to 3.5 ⫻ 10⫺2 m2/s) and in a broad
range of S est values (2.6 ⫻ 10⫺2 to 1.1 ⫻ 10⫺1 m2/s). Schad and
Teutsch also mention that S est values show a large variability at
short observation distances and a progressively smaller variability for larger distances. They point out that the variations in
S est should be a function of the relative transmissivity of the
flow path between the pumping and the observation well. If
this path has a high relative transmissivity, S est should be small
and vice versa.
Herweijer and Young [1991] present a qualitative model of
aquifer heterogeneity for the interpretation of spatial and temporal variability of hydraulic parameters derived from pumping tests in a heterogeneous fluvial aquifer at Columbus Air
Force Base, Mississippi. Their results from pumping tests analyzed with the Theis method cover a wide range for storativity
estimates and a relatively small range for transmissivity estimates. They conclude that the storage estimates show a strong
dependence on the flow pattern induced by pumping and the
degree of interconnection of high hydraulic conductivity lenses
between the pumping and observation wells.
3.
Numerical Simulation Methodology
We performed a series of numerical simulations to corroborate our field observations and to further develop those of
Schad and Teutsch [1994]. Our methodology, which is similar
to that used by Lachassagne et al. [1989], consists of the following steps: (1) generation of two-dimensional (2-D) heterogeneous T fields, (2) simulation of transient radial flow toward
a pumping well in the generated T fields using a finite element
code, (3) analysis of the drawdown curves simulated at all
nodes of the grid using Jacob’s method, (4) generation of maps
showing the spatial distribution of the T est and S est values and
drawdown, and (5) analysis of the results. Regarding the analysis of the results, we first compare the spatial S est distribution
with the spatial structure of the input T fields. Second, we
check whether the average of the T est values (designated as
T avg) is somewhat representative of the full domain by comparing it with the effective transmissivity obtained from theoretical considerations and from numerical analysis of parallel
flow in the same T fields.
Figure 5. Transmissivity and storativity estimates obtained
from the interpretation of drawdown curves at each observation well for the long-term pumping test in borehole S33 at the
El Cabril site. Again, T estimates remain virtually constant,
while S estimates span 3 orders of magnitude.
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1015
Figure 6. Schematic drawing of the model setup indicating the discretization scheme in the top left quadrant, the pumping well location, the full model domain, the heterogeneous domain, and the heterogeneous
subdomain for which our methodology is considered valid.
We first describe in detail the methodology and then examine the influence of the variance and spatial correlation structure of T on the results. Our investigations include multilognormal, nonmultilognormal, and fractal T fields.
The simulated random transmissivity fields are square in
shape and are discretized into 500 ⫻ 500 square elements of
unit area. We assign T values, which are generated using the
computer code GCOSIM3D [Gómez-Hernández and Journel,
1993], to each of the elements and consider these as point
values.
To minimize boundary effects, the heterogeneous domain of
500 ⫻ 500 square elements of unit area is bounded on all sides
by 20 rectangular elements of increasing size as shown schematically in Figure 6. The T values of these elements are set
equal to the geometric mean transmissivity of the random T
field (Tg ⫽ 1.0). Constant head conditions are applied to the
external boundaries of the full domain. This full domain is
square in shape and consists of 291,600 elements. A homogeneous input storativity of 1.0 is used for all the simulations.
Although 1.0 is a physically implausible value for storativity,
this assignment has no effect on the results (S ⫽ 1.0 was
chosen so that the average diffusivity (T/S) of the heterogeneous domain was 1.0). The pumping well is located at the
center of the domain. The transient flow equation is solved
with the finite element computer code FAITH [Sánchez-Vila et
al., 1992], which is especially designed to work with a large
number of elements. Drawdown values are calculated every 10
time units and are written to a file every 100 time units for all
nodes of the finite element mesh. The T est and S est values are
calculated with Jacob’s method for all nodes by fitting a
straight line through the last three drawdown points. Special
care is taken to make sure that drawdown curves are not
sensitive to the homogeneous outer zone surrounding the het-
erogeneous domain. Several preliminary simulations showed
that this influence starts at ⬃30,000 time units. Our simulations
are therefore terminated at 20,000 time units, and our methodology is considered valid in a square domain of 100 units on
a side centered on the well. This small heterogeneous domain
is schematically indicated in Figure 6.
Note that this methodology involves applying Jacob’s
method over the same time range for all observation points.
This results in Jacob’s method being applied at different dimensionless times (1/u ⫽ 4Tt/(r 2 S)) at different points in the
domain. Since the appropriateness of the approximation underlying Jacob’s method is a function of the dimensionless
time, the appropriateness of this approximation varies as a
function of location within the domain. For example, the dimensionless time is 15.79 for a radius of 71 units, corresponding to the distance between the pumping well and the corners
of the domain of valid methodology, for t ⫽ 19,900, which is
the average time of the small time period from 19,800 to 20,000
time units for which Jacob’s method is applied, for an average
T ⫽ 1.0 and for S ⫽ 1.0. For 1/u ⫽ 15.79 the error in using
Jacob’s method instead of the Theis method is about 3.0%,
assuming that these criteria are valid for heterogeneous conditions. The errors are considerably smaller at the center of the
domain (about 0.2% for a radius of 30 units and the same
parameters used above). Therefore we expect to obtain a slight
radial trend with smaller T values in the center and larger T
values at further distances from the pumping well as a consequence of applying Jacob’s method over the same time range
for all observation points. However, we want to stress that the
above considerations are strictly valid only for homogeneous
conditions and may not be appropriate for highly heterogeneous T fields.
The arithmetic mean of the 10,000 T est values within the
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precisely through pumping tests in wells that the transmissivity
of an aquifer is measured in situ.” It should be stressed that the
problem of an effective transmissivity for radial flow conditions
is not examined here, because we are primarily interested in
the relationship between the transmissivity estimates provided
by an interpretation model for pumping test analysis (in our
case Jacob’s method) and the effective transmissivity for parallel flow conditions (T effp), which is a representative parameter for most problems in hydrogeology.
In section 4 we examine the relation between T effp and T avg
for three conceptual models for aquifer heterogeneity: (1) a
multilognormal T field, (2) a nonmultilognormal T field with
connected high T zones without a large-scale trend, and (3) a
fractal field with a large-scale trend.
Figure 7. Boundary conditions used to compute effective
transmissivity for parallel flow conditions in a heterogeneous T
field.
small domain, which are obtained using Jacob’s method, will
be designated as T avg throughout the remainder of this work.
T avg is compared (1) with the geometric sample mean (T g ) of
the input T fields and (2) with the effective transmissivity
obtained from parallel flow simulations (T effp) using the same
T fields as in the radial flow simulations. The motivation for
this twofold comparison is based on the work of Sánchez-Vila
et al. [1996]. This work suggests that the assumption of multilognormality may not be valid in many real field cases, even if
point T values display a lognormal distribution and, furthermore, that the appearance of scale effects in T may be attributed to departures from multilognormality. Here the term
“scale effect” is used to describe the apparent increase of T
with the size of the definition domain and, more specifically,
the increase of large-scale effective T with respect to the geometric average of point values. They performed 2-D flow simulations with a mean parallel gradient in nonmultilognormal T
fields and showed that the effective transmissivity T effp does
not necessarily agree with T g , contrary to T eff derived for
multilognormal distributions. Following their approach, we obtain T effp for T fields by solving the steady state flow equation
under very simple boundary conditions as shown in Figure 7.
When applied to our heterogeneous T fields (large domain of
500 ⫻ 500 cells), we obtain a mean gradient that is constant
and parallel to the x coordinate axis. Under such conditions,
the effective transmissivity in the x direction is given by
T effpx ⫽ Q T ⫻ L/⌬H, where Q T is the total flux per unit width
into (and out of) the domain and L is its length. A change of
boundary conditions is used to find the effective transmissivity
in the y direction T effpy . The geometric mean T effp ⫽
公T effpxT effpy is considered an appropriate quantity for the
comparison with our simulation results (T avg). The T g , T effp,
and T avg values for all our test cases are given in Table 1.
It should be stressed that a single effective transmissivity,
defined as the tensor that relates the expected values of flow
and head gradient, can only be defined for parallel flow conditions, for which it is equal to T g under relatively broad
conditions [Matheron, 1967]. De Marsily [1986, p. 82] states, “If
the flow is not uniform (converging radial, for example), there
is no law of composition, constant in time, that makes it possible to define a mean darcian permeability. This problem is
quite worrying from the conceptual viewpoint in so far as it is
4.
Test Cases
4.1. Test Case 1: Multilognormal Random Field,
Effect of log T Variance ␴2Y
The objective of the first test case is to corroborate the field
observations and to investigate the influence of ␴ 2Y on T est and
S est. We simulate three T fields, assuming Y ⫽ ln T is a
stationary multinormal random function characterized by the
mean 具Y典 (⬇0.0 for the three T fields), the variances ( ␴ 2Y ⫽
0.25, 1.0, and 4.0), and the variogram (a spherical isotropic
model with a range of 26 unit lengths, which corresponds to an
integral scale of ⬃10 unit lengths). The T fields are generated
by conditioning on the four square elements surrounding the
node located at the center, which serves as the pumping well,
fixing their values to T g ⫽ 1.0, T g being the geometric mean
of the point T values. The effect of varying the transmissivity at
the well (T w ) will be investigated in test case 3. The T field
with ␴ 2Y ⫽ 1.0 is shown in Figure 8a, where the black areas
represent the most transmissive quartile and the white areas
represent the least transmissive one. Table 1 indicates that
T effp agrees well with the sample T g of the input T fields, as
expected for multilognormal fields.
Figures 8c and 8d show the simulated drawdown curves at
several observation points located in different directions (1 to
8) but at the same distance R from the pumping well (see
Table 1. Comparison Between Effective Transmissivity for
Parallel Flow Conditions and Average Transmissivity
Estimates Obtained From Jacob’s Method for Different
Integral Scales, Pumping Well Transmissivity, log T
Variances, and Geometric Mean Transmissivity
Test
Case
Tw
I
␴ 2Y
Tg
T effp/T g
T avg/T g ,
⫾Range
1
1
1
3
3
4
4
4
5
1.00
1.00
1.00
1.65
0.61
1.00
1.00
1.00
1.00
10
10
10
10
10
25
25
25
䡠䡠䡠
0.25
1.01
4.04
1.01
1.01
1.03
2.06
4.12
0.74
1.00
1.03
1.00
1.03
1.03
1.00
1.02
1.00
0.97
1.00
1.00
0.99
1.00
1.00
1.10
1.24
1.58
0.97
1.01 ⫾ 2.1E-4
0.97 ⫾ 2.6E-4
0.96 ⫾ 8.5E-4
0.97 ⫾ 2.5E-4
0.97 ⫾ 2.7E-4
1.13 ⫾ 4.2E-4
1.23 ⫾ 6.3E-4
1.48 ⫾ 7.6E-4
1.07 ⫾ 1.2E-3
T w , pumping well transmissivity; I, integral scale; ␴ 2Y , log T variance;
T g , geometric mean transmissivity; T effp, effective transmissivity for
parallel flow conditions; T avg, average transmissivity estimate. Notice
that T effp and T avg are very close except when T is nonstationary (test
case 5). Notice also the small range of T avg. Read 1.01 ⫾ 2.1E-4 as
1.01 ⫾ 2.1 ⫻ 10⫺4.
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1017
Figure 8. Test case 1. (a) Multilognormal random T field of 500 ⫻ 500 units with ␴ 2Y ⫽ 1.0 and (b)
subdomain of 100 ⫻ 100 units for which the methodology is considered valid are shown here. “W” indicates
the well location. The numbers indicate the observation points at a radial distance R ⫽ 30 units. The solid
triangles indicate the observation points at R ⫽ 10 units. (c) and (d) Semilog plots of the simulated drawdown
at the observation points shown in Figure 8b for ␴ 2Y ⫽ 0.25 and 4.0 are shown. Note that late time slopes are
approximately equal for all curves.
Figure 8b), for the T fields with ␴ 2Y ⫽ 0.25 and 4.0. Results are
presented for R ⫽ 10 and R ⫽ 30. The corresponding drawdown curves for a homogeneous T field with T ⫽ T g ⫽ 1.0
and S ⫽ 1.0 are shown for comparison. Note that only a small
number of drawdown points are shown on each curve to keep
the figures clear. All drawdown curves exhibit late time slopes
which are similar to that corresponding to the homogeneous
case. In Figure 8d the drawdown curves at observation points
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2 and 5 at radial distances of 30 unit lengths from the well
illustrate the variability in early time pressure response in heterogeneous media. Observation point 2 and the well are connected through a zone of low T, which can easily be recognized
in Figure 8b. The drawdown response is therefore delayed with
respect to the homogeneous case. Jacob’s method yields a S est
greater than the formation S because t 0 , the intercept with the
time axis of the straight line fitted through the last data points,
is greater than that of the homogeneous case. On the contrary,
observation point 5 and the well are connected through a zone
of high T. The drawdown response is therefore earlier in time
than the homogeneous case, resulting in a S est smaller than the
formation S. A high ␴ 2Y results in greater differences between
the drawdown curves at the same radial distance to the well,
which in turn results in a higher variability of S est and vice
versa. The differences between the drawdown curves decrease
with increasing radial distance to the well, causing a decrease
of the S est variance with the radial distance from the well, as
the flow process samples a larger number of heterogeneities.
This is consistent with the field observations of Schad and
Teutsch [1994] discussed in section 2. Note that for R ⫽ 10 the
solid line (homogeneous case) is below all responses in Figure
8c and all but one in Figure 8d for times later than 200 time
units at which the pressure front has already passed the observation points at R ⫽ 10. This is due to the increased slopes of
the drawdown curves (with respect to the homogeneous case)
between ⬃100 and 500 time units. The increased slopes reflect
the relatively low geometric average of ⬃0.8 of the point T
values within the annular ring between radial distances of ⬃10
and 30 units from the well. The increased slopes of the drawdown curves between 100 and 500 time units result in a general
decrease of the S est values for the observation points at R ⫽
10. In summary, S est values seem to depend not only on the
transmissivity connecting the well and the observation points
but also on transmissivities within a certain catchment area
surrounding both the well and the observation points.
The spatial distribution of S est and T est and the drawdown
for the T fields with ␴ 2Y ⫽ 0.25 and 4.0 are shown in Figures
9a–9f. The following can be concluded from these figures: (1)
The T est values obtained from Jacob’s method are very close to
T g ⫽ 1.0, which is equal to T effp in multilognormal T fields (as
discussed in section 3, the radial trend of the T est distributions
may be attributed to applying Jacob’s method over the same
time range to all points, regardless of the distance to the
pumping well); (2) the S est values vary strongly in space despite
the fact that the formation S was homogeneous; (3) the spatial
structures of S est resemble each other for different values of
␴ 2Y (see Figure 10c for ␴ 2Y ⫽ 1.0), and the variance of S est
depends on ␴ 2Y ; and (4) a dependence between the heterogeneous input T field (Figure 8b) and the spatial distribution of
S est does exist, although the S est field is smoother. Note that we
have also found that results are independent of the integral
scale (I) by performing the same test with different values of I.
4.2. Test Case 2: Multilognormal Random Field,
Effect of Support Scale
The idea behind test case 2 is to examine what happens when
we sample the field more coarsely and assign the data as block
values. A new multilognormal random field with increased
support scale is generated using the T field with ␴ 2Y ⫽ 1.0
corresponding to the first test case, while maintaining 具Y典 and
changing only slightly ␴ 2Y and the variogram. The generation
procedure is the following: The T value of one element of the
original T field is assigned to its surrounding elements, forming
blocks of 3 ⫻ 3 unit lengths. This procedure is repeated for
every third element in the x and y directions for the T field
shown in Figure 8a. The central domain of the resulting T field
with the increased support scale is shown in Figure 10a. Comparison with Figure 8b indicates that the main features of the
original T field are maintained. The spatial distributions of S est
obtained for the T field with increased support scale and for
the original T field are shown in Figures 10b and 10c, respectively. The change of support scale results only in minor
changes of the spatial S est distribution. The increase of support
scale has virtually no effect on the spatial distribution of T est,
which is therefore not shown.
4.3. Test Case 3: Multilognormal Random Field,
Effect of Transmissivity at the Well
In the T fields presented in the first two cases the transmissivity at the well (T w ) was selected to be equal to T g . In test
case 3 we are interested in examining the effects that changing
the T w value would have on the S est and T est values. For this
purpose, two new fields were generated by conditioning the
simulations on T w values different from T g ⫽ 1.0. The two
new values were T w ⫽ 0.61 and T w ⫽ 1.65. The statistical
parameters and the random number seed corresponding to the
first test case were maintained. Comparison of the subdomains
of the fields generated with T w ⫽ 1.0, T w ⫽ 0.61, and T w ⫽
1.65 (Figures 8b, 11a, and 11b, respectively) indicates that only
the T values at the central domain of ⬃20 ⫻ 20 elements are
significantly changed.
The spatial distributions of S est and T est are shown in Figures 11c–11f. The T avg values for the two fields remain practically unchanged with respect to the first test case (see Table
1). This is in conceptual agreement with the results of Butler
[1988], Butler and Liu [1993], and Oliver [1993], who state that
methods of transmissivity estimation that use the late time rate
of change of drawdown, instead of the total drawdown, are less
likely to be influenced by near-well nonuniformity. Note that
the radial trend of the T est distributions, which is similar to the
one observed for test case 1, may be attributed to applying
Jacob’s method over the same time range to all points, regardless of the distance to the pumping well.
Noticeable differences between the two simulations are restricted to the S est in the central domain where the input T
fields are different. High T w results in high S est next to the well
and vice versa. Note that the results within a radial distance of
⬃5 unit lengths around the well are considered to be affected
by numerical errors due to a poor spatial discretization in the
vicinity of the well. However, our qualitative interpretation of
the simulations seems to be in agreement with the common
knowledge that S est obtained from single-hole hydraulic tests
can disagree considerably with formation S (a detailed discussion about this topic is given by Butler [1988, 1990]). For the
case of S est that is considerably greater than formation S we
expect that the pumping well is situated within a zone of T ⬎
T est or vice versa. The formation S has then to be calculated
from rock and fluid properties, which is a common practice in
the field of reservoir engineering. Note that differences between S est and real aquifer S are generally attributed to skin
effects [e.g., Earlougher, 1977], which are conceptually equivalent to heterogeneities near the well. The fact that the S est
values at the well are highly sensitive to near-well nonuniformity may potentially be utilized to characterize small-scale
heterogeneity using S est values from single-hole tests at differ-
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1019
Figure 9. Test case 1: (a) and (b) spatial distribution of S est for ␴ 2Y ⫽ 0.25 and 4.0, (c) and (d) spatial
distribution of T est for ␴ 2Y ⫽ 0.25 and 4.0, and (e) and (f) drawdown distribution for ␴ 2Y ⫽ 0.25 and 4.0. A
noteworthy feature of Figures 9c and 9d is the very small range of T est values, suggesting that T estimates in
this field are independent of the location of the observation points.
ent locations. However, the applicability of these findings to
real field data will strongly depend on several factors frequently encountered in field situations. These factors include,
among others, skin effects due to well drilling and development, well losses, well bore storage effects, and delayed drainage.
4.4. Test Case 4: Nonmultilognormal Field,
Effect of ␴2Y
Test cases 1–3 show that Jacob’s method yields values very
close to T g and T effp when applied to pumping test data from
multilognormal T fields. The objective of this test case is to
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Figure 10. Test case 2: (a) subdomain (100 ⫻ 100 units) with
increased support scale (T blocks of 3 ⫻ 3 units) and with the
geostatistical parameters of test case 1, (b) spatial distribution
of S est for test case 2, and (c) spatial distribution of S est for test
case 1 with original support scale (T blocks of 1 ⫻ 1 units) and
␴ 2Y ⫽ 1.0.
extend our study to nonmultilognormal T fields. For this purpose, we use field number 4 of Sánchez-Vila et al. [1996]. This
field is shown in Figure 12a. It was generated by connecting
high T values of an originally multilognormal field while maintaining the histogram for the In T field and the normality of the
univariate distribution of T values. The variogram and the
multilognormality of the original field were not retained. The
generation procedure is explained by Sánchez-Vila et al. [1996].
We believe that nonmultilognormal T fields like the one shown
in Figure 12a, having well-connected high T zones, are better
conceptual models for fractured rocks or alluvial aquifers than
multilognormal T fields. Sánchez-Vila et al. [1996] have shown
that T effp may depend on ␴ 2Y in nonmultilognormal T fields.
Here we examine whether the good agreement between T effp
and T avg obtained for multilognormal T fields holds also for
nonmultilognormal T fields.
Radial flow was simulated in three T fields with 具Y典 ⫽ 0.0
and ␴ 2Y values of 1.0, 2.1, and 4.1. The spatial T est distributions
are shown in Figures 12c, 12e, and 12f. Note that the radial
trend of the T est distributions, which is similar to the ones
observed for test cases 1–3, may be attributed to applying
Jacob’s method over the same time range for all observation
points, regardless of the distance to the pumping well.
Table 1 shows that T avg agrees well with T effp but is quite
different from T g at large ␴ 2Y . These results indicate that
performing a pumping test and using the slope of the drawdown curve in the analysis allow a value of transmissivity to be
obtained that is very close to the effective transmissivity of the
area influenced by the test, regardless of whether the T field is
multilognormal. We have observed this type of behavior on
several occasions. The most dramatic is the one reported by
Carrera et al. [1990], who found T derived from a long-term
pumping test to be 20 times larger than the geometric average
of short-term single-hole tests performed in a fractured medium at nine wells at the Chalk River site in Canada. Note that
highly transmissive channels as conceptualized in Figure 12a
are frequently observed within fractured zones. Hence we are
lead to the conclusion that transmissivities derived from longterm pumping tests are frequently larger than those derived
from the geometric mean of short-term single-hole tests (see
Sánchez-Vila et al. [1996] for a further discussion of this issue).
The main finding from this section is that an effective transmissivity for parallel flow is that obtained from long-term
pumping tests, rather than the geometric average of local values.
The spatial S est distribution is shown for ␴ 2Y ⫽ 1.0 in Figure
12d. Unlike the other test cases, it is difficult to recognize here
a correlation between the input T field and the spatial S est
distribution. The fine structure of the input T field is not
reflected by the S est distribution, which seems to be the result
of a spatial integration. Therefore we conjecture that S est depends not only on the nature of the connection between the
well and the observation points but also on all the Y values in
a certain catchment area surrounding both the well and the
observation points. This aspect is further explored by X.
Sánchez-Vila et al. (Pumping tests in heterogeneous auquifers:
An analytical study of what can be really obtained from their
interpretation using Jacob’s method, submitted to Water Resources Research, 1998). Note that the spatial S est distributions
for the other ␴ 2Y qualitatively resemble the one shown in Figure 12d and are therefore not presented.
4.5.
Test Case 5: Fractal Field
A log transmissivity field was generated having a power law
semivariogram ␥ (h) ⫽ c h 2 ␻ with ␻ ⫽ 0.25 and c ⫽ 0.027,
as proposed by Neuman [1990, 1994]. This T field, shown in
Figure 13a, is generated by conditioning on the transmissivity
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MEIER ET AL.: JACOB’S METHOD AND HETEROGENEOUS FORMATIONS
1021
Figure 11. Test case 3: (a) and (b) Subdomains for T fields with T w ⫽ 0.61 and 1.65 ( ␴ 2Y ⫽ 1.0) (the
pumping well locations are in the center of the subdomain), (c) and (d) spatial distributions of S est for T w ⫽
0.61 and 1.65, and (e) and (f) spatial distributions of T est for T w ⫽ 0.61 and 1.65.
value (T w ⫽ 1.0) at the well. The field has a sample mean
具Y典 ⫽ ⫺0.03 and a sample variance ␴ 2Y ⫽ 0.74.
The spatial T est distribution, shown in Figure 13c, may be
affected by the spatial distribution of the extreme transmissivity values which are located at the corners of the large domain,
as the radial pattern observed in test cases 1– 4 is distorted. In
this case, the original T field is markedly nonstationary. Hence
it is not surprising that some trend can be observed in T est.
However, the range of T est is relatively small. The fact that
T avg ⬎ T g ⬎ T effp (see Table 1) may indicate a problem for the
definition of an effective T value for our fractal field.
According to the relation given by Neuman [1994], T eff(L) ⫽
T g exp (cL 2 ␻ [0.5 ⫺ ␤ /E]), where L is the characteristic
length of a rock volume which acts as the support for a per-
255
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MEIER ET AL.: JACOB’S METHOD AND HETEROGENEOUS FORMATIONS
Figure 12. Test case 4: (a) nonmultilognormal random T field with ␴ 2Y ⫽ 1.0 (500 ⫻ 500 units), (b)
subdomain (100 ⫻ 100 units) for which our methodology is considered valid, (c) spatial distribution of T est for
␴ 2Y ⫽ 1.0, (d) spatial distribution of S est for ␴ 2Y ⫽ 1.0, and (e) and (f) spatial distribution of T est for ␴ 2Y ⫽
2.1 and 4.1.
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MEIER ET AL.: JACOB’S METHOD AND HETEROGENEOUS FORMATIONS
1023
Figure 13. Test case 5: (a) fractal T field of 500 ⫻ 500 units, (b) subdomain (100 ⫻ 100 units) for which our
methodology is considered valid, (c) spatial distribution of T est, and (d) spatial distribution of S est.
meability test, E is the dimension of the flow condition
(1, 2, or 3), and ␤ ⫽ 1 for infinite isotropic media, 0 ⱕ ␤ ⱕ 1
for bounded isotropic media, and 1 ⱕ ␤ ⱕ E for infinite
anisotropic media. Using the above relation with our T avg,
T effp, T g , and L ⫽ 700 unit lengths, we obtain ␤ ⫽ 0.82 for
T avg, indicating a bounded isotropic medium, and ␤ ⬎ 1.0 for
T effp, indicating an anisotropic medium. Inspection of Figure
13a reveals the somewhat anisotropic nature of this field. However, because the well is located at the center of the field and
the extreme T values are concentrated at the corners, the
drawdown behavior is similar to the one for a bounded domain. The high T zones at the bottom left and the top right
corners act similar to constant head boundaries, whereas the
low T zones at the other corners act similar to no-flow boundaries.
The spatial distribution of S est (Figure 13d) seems to be
dominated by the heterogeneities of the small domain pre-
sented in Figure 13b. High S est correspond to zones with a low
connectivity to the well, for example, the bottom right corners
of Figures 13b and 13d, and vice versa.
5.
Summary and Conclusions
1. Our numerical simulations of pumping tests in aquifers
with heterogeneous transmissivity T and homogeneous storativity S show that applying Jacob’s method to late time drawdown data at observation points results in a strong spatial
variability of storativity estimates S est and in an almost homogeneous spatial distribution of transmissivity estimates T est.
This is in agreement with field observations.
2. Our simulations indicate that S est depends mainly on the
connectivity between the well and the observation points. This
was suggested by Schad and Teutsch [1994] on the basis of field
observations and is consistent with the discussions of Butler
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MEIER ET AL.: JACOB’S METHOD AND HETEROGENEOUS FORMATIONS
[1990] and Butler and Liu [1993]. Furthermore, S est values
seem to depend also on the transmissivities within a certain
catchment area surrounding both the well and the observation
points.
3. As has been shown in previous investigations, nonuniformities near the well have no effect on T est. However, S est
values vary considerably in the near-well region. A well located
within a zone of T ⬎ T est results in S est values considerably
greater than the formation S and vice versa. This result has
been documented in the petroleum engineering and groundwater literature [e.g., Barker and Herbert, 1982; Butler, 1988,
1990]. However, these results were mostly considered within
the context of artificially induced heterogeneities around a well
(e.g., skin effects due to drilling and development). We conjecture that these results could potentially be used to characterize natural small-scale heterogeneity (still much larger than
the well radius), using S est values from single-hole pumping
tests at different locations, provided that artificially induced
heterogeneities at the well are either measured by other means
or are negligible in comparison with natural small-scale heterogeneity.
4. T est values show an almost homogeneous spatial distribution for stationary input T fields. The smooth radial trend of
the T est distributions obtained for all test cases may largely be
attributed to applying Jacob’s method over the same time
range for all observation points, regardless of the distance to
the pumping well. However, in the case of the fractal T field
the spatial distribution of T est may also be affected by the
spatial trend of the input T field, although the enormous transmissivity contrasts of the input field are smoothed out dramatically. Furthermore, the arithmetic average of the T est values
does not coincide with the effective transmissivity for parallel
flow conditions in the fractal field (T effp/T avg ⫽ 0.9), although
it should be noted that the concept of effective transmissivity is
only valid for stationary fields.
5. Transmissivity estimates obtained from Jacob’s method
under radial flow conditions are close to the effective transmissivity for parallel flow conditions for the area of influence
of the pumping test in multilognormal and nonmultilognormal
T fields. We consider nonmultilognormal T fields, with wellconnected zones of relatively high T, to be realistic conceptual
models for many, if not most, heterogeneous formations.
Therefore the above conclusion bears two significant implications. First, Jacob’s method does indeed work very well in most
heterogeneous formations. Second, the problem that a single
effective transmissivity value depending only on the statistical
properties of a T field cannot be derived for radial flow conditions does not affect the pumping test interpretation. Jacob’s
method directly provides a representative transmissivity which
can be used for most hydrogeological problems dealing with
parallel flow.
6. Our numerical work is based on the assumption of twodimensional and confined flow conditions and might therefore
idealize complexities associated with field tests, especially in
unconfined aquifers. Three-dimensional flow conditions might
also be of concern in cases where the aquifer thickness is of a
similar magnitude to the distance between the pumping well
and the observation wells. The importance of such effects
should be investigated in further studies.
Acknowledgments. Comments by G. Teutsch of the University of
Tübingen, S. Young of Hydrogeologic Inc., and J. J. Butler of the
Kansas Geological Survey have helped improve the final text and are
gratefully acknowledged. The first author acknowledges financial support from SATW (Swiss Academy of Engineering Sciences) and Nagra
(Swiss National Cooperative for the Disposal of Radioactive Waste).
The work of the remaining authors was funded by ENRESA (Spanish
Nuclear Waste Management Company), 1995SGR00405 (CIRIT), and
AMB95-0407 (CICYT). The data from the Grimsel test site were
kindly supplied by Nagra and PNC (Power Reactor and Nuclear Fuel
Development Corporation, Japan).
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259
WATER RESOURCES RESEARCH, VOL. 41, W11410, doi:10.1029/2005WR004056, 2005
A procedure for the solution of multicomponent
reactive transport problems
M. De Simoni
Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Politecnico di Milano, Milan, Italy
J. Carrera and X. Sánchez-Vila
Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia, Barcelona, Spain
A. Guadagnini
Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Politecnico di Milano, Milan, Italy
Received 23 February 2005; revised 16 July 2005; accepted 1 August 2005; published 8 November 2005.
[1] Modeling transport of reactive solutes is a challenging problem, necessary for
understanding the fate of pollutants and geochemical processes occurring in aquifers,
rivers, estuaries, and oceans. Geochemical processes involving multiple reactive species
are generally analyzed using advanced numerical codes. The resulting complexity has
inhibited the development of analytical solutions for multicomponent heterogeneous
reactions such as precipitation/dissolution. We present a procedure to solve groundwater
reactive transport in the case of homogeneous and classical heterogeneous equilibrium
reactions induced by mixing different waters. The methodology consists of four steps:
(1) defining conservative components to decouple the solution of chemical equilibrium
equations from species mass balances, (2) solving the transport equations for the
conservative components, (3) performing speciation calculations to obtain concentrations
of aqueous species, and (4) substituting the latter into the transport equations to evaluate
reaction rates. We then obtain the space-time distribution of concentrations and
reaction rates. The key result is that when the equilibrium constant does not vary in space
or time, the reaction rate is proportional to the rate of mixing, rT u D ru, where u is the
vector of conservative components concentrations and D is the dispersion tensor. The
methodology can be used to test numerical codes by setting benchmark problems but
also to derive closed-form analytical solutions whenever steps 2 and 3 are simple, as
illustrated by the application to a binary system. This application clearly elucidates that in
a three-dimensional problem both chemical and transport parameters are equally important
in controlling the process.
Citation: De Simoni, M., J. Carrera, X. Sánchez-Vila, and A. Guadagnini (2005), A procedure for the solution of multicomponent
reactive transport problems, Water Resour. Res., 41, W11410, doi:10.1029/2005WR004056.
1. Introduction
[2] Reactive transport modeling refers to the transport of
a (possibly large) number of aqueous species that react
among themselves and with the solid or gaseous phases. It is
relevant because it helps in understanding the fate of
pollutants in surface and groundwater bodies, the hydrochemistry of aquifers, and many geological processes [e.g.,
Gabrovšek and Dreybrodt, 2000; Freedman et al., 2003;
Salas and Ayora, 2004; Emmanuel and Berkowitz, 2005].
Unfortunately, it is complex, both conceptually and mathematically. Modeling reactive transport involves two coupled ingredients: (1) the mass balance of all participating
species, which is expressed by the solute transport equation
for mobile species; and (2) a set of equations describing the
reactions among species. The nature of participating species
and reactions leads to a broad range of possible behaviors of
Copyright 2005 by the American Geophysical Union.
0043-1397/05/2005WR004056$09.00
261
the system [Rubin, 1983; Molins et al., 2004]. Moreover,
reactions such as dissolution can also modify the aquifer
properties (i.e., porosity and hydraulic conductivity) [e.g.,
Wood and Hewett, 1982; Philips, 1991; Kang et al., 2003;
Singurindy and Berkowitz, 2003; Singurindy et al., 2004].
Therefore it is not surprising that numerous general mathematical formulations to solve reactive transport problems
are available [e.g., Rubin, 1990, 1992; Yeh and Tripathi,
1991; Friedly and Rubin, 1992; Lichtner, 1996; Steefel and
MacQuarrie, 1996; Clement et al., 1998; Saaltink et al.,
1998, 2001; Tebes-Stevens et al., 1998; Robinson et al.,
2000; Molins et al., 2004]. The resulting sets of governing
equations have been included in a large number of reactive
transport codes that can handle several species with different types of reactions (see Saaltink et al. [2004] for a list of
codes).
[3] As opposed to numerical solutions, very few analytical results are available. These are of general interest, as
they provide insights on the nature of the solution and allow
evaluating the relative importance of the involved parame-
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
ters and processes. Analytical and semianalytical solutions
are available in the case of linear and nonlinear equilibrium
sorption reactions [e.g., Serrano, 2003, and references
therein]. Analytical solutions are also available to solve
reactive transport problems in case of (networks of) firstorder kinetic reactions [Sun et al., 1999, 2004; Clement,
2001; Serrano, 2003; Quezada et al., 2004]. These methodologies are appropriate for radioactive decay chains and
for many biochemical processes. However, they are difficult
to extend to geochemical processes whose rates are nonlinear functions of the concentrations of the dissolved species,
or to cases in which equilibrium conditions can be assumed
and the reaction rates cannot be written explicitly.
[4] Because transport times are generally large in aquifers, most aqueous reactions and many dissolution/precipitation reactions can be modeled with the assumption of
chemical equilibrium conditions at each point of the
domain. As equilibrium is reached instantaneously, the
amount of reactants evolving into products depends on
the rate at which they mix, which is, in turn, governed by
transport. To illustrate the concept, Figure 1a depicts a
reaction of pure dissolution/precipitation in the mixing of
two waters. The concentrations of the dissolved species, c1
and c2, in both mixing waters satisfy the equilibrium
condition c1c2 = K, where K is the equilibrium constant.
For conservative species, concentrations of the mixture
would be a linear function of the mixing waters concentrations, while equilibrium concentrations for reactive
species are governed by a number of factors, including
pH, salinity and temperature [e.g., Wigley and Plummer,
1976]. As a result, it is generally difficult to predict whether
the mixture will precipitate or dissolve, and to estimate the
amount of precipitated/dissolved mass.
[5] In this paper we propose a concise methodology to
analyze groundwater reactive transport in the presence of
general homogeneous and classical heterogeneous equilibrium reactions (according to the classification of Rubin
[1983]) caused by mixing of different waters. We analyze
situations where all reactions in the system can be assumed
at equilibrium conditions at all times. Emphasis is placed on
evaluating not only concentrations, but also reaction rates.
The method is first applied to a binary system, providing an
original analytical expression for the reaction rate in the
case of mixing-driven precipitation. The methodology is
then extended to deal with all kinds of classical heterogeneous and homogeneous equilibrium reactions. The complete methodology allows the mathematical decoupling of
the problem into four steps: (1) definition of mobile
conservative components of the system, (2) transport of
these components, (3) speciation, and (4) evaluation of
reaction rates and mass change of constant activity species.
The method is proposed as a useful tool in deriving further
analytical solutions describing reactive transport processes.
It can also be of practical use in developing, adapting, or
testing numerical codes to solve solute transport problems
involving multiple reactions.
[6] The outline of the paper is as follows. In section 2 we
present the mathematical formulation of the problem.
Section 3 is focused on the reactive transport problem in
the presence of a pure dissolution/precipitation reaction.
Specifically, we analyze in detail a three-dimensional
porous medium with uniform flow carrying two dissolved
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Figure 1. Schematic representation of a pure dissolution/
precipitation process during the mixing of two waters. (a)
Concentrations of the dissolved species, c1 and c2, in the
mixing waters satisfy the equilibrium condition c1c2 = K, K
being the equilibrium constant. From the conservative
mixing point (at which the concentrations are weighted
arithmetic averages of the input c values), the mixture
evolves toward the equilibrium curve following a straight
line (whose slope is given by the ratio of the stoichiometric
coefficients and is equal to 1 in the simple binary case
presented in the text). (b) Example of the evaluation of
(22a). Function u (arbitrarily chosen, for illustrative
purposes) and the two factors in (22a) are displayed versus
the distance from the location at which u reaches its
maximum value. Note the different scales: @ 2c2/@u2 is
nearly constant, while the term accounting for mixing
displays the largest variations.
constituents at equilibrium with a solid mineral. We derive a
closed-form solution for the mixing-driven precipitation
process induced by an instantaneous point-like injection
of water containing the same constituents of the resident
water, but at a different equilibrium state. Finally, in
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
section 4 we generalize the formulation to solute transport
problems involving multiple reactions.
2. Preliminary Concepts
2.1. Chemical Equilibrium
[7] Equilibrium reactions are described by mass action
laws, which in the case of multiple species can be written in
a compact form as [Saaltink et al., 1998]
Se log a ¼ log K0 ;
ð1Þ
where Se is the stoichiometric matrix, that is a Nr Ns
matrix (Nr and Ns being the number of reactions and the
total number of chemical species, respectively) containing
the stoichiometric coefficients of the reactions, a is the
vector of activities of all species and K0 is the vector of
equilibrium constants. In general, the activity of aqueous
species, aa, is a mildly nonlinear function of aqueous
concentrations, ca, which can be written as
log aa ¼ log ca þ log gðca Þ;
ð2Þ
where the activity coefficient, g, may be a function of all
aqueous concentrations and can be computed in several
ways, the Debye-Huckel equation being a frequent choice
for moderate salinities [Helgeson and Kirkham, 1974]. For
dilute solutions, which are frequently observed in many real
groundwater problems, the activity coefficients can be
assumed as unitary. If some reactions involve pure phases
or, in general, mixtures with fixed proportions of constituents or minerals or gas phases, the activities of the
corresponding species can be assumed to be fixed and
constant (e.g., activity is unity for pure minerals and water).
[8] Let Nc be the number of constant activity species. We
split vector a (equation (1)) into two parts, ac and aa,
containing the constant activity and the remaining aqueous
species, respectively. Here and in the following, subscripts c
and a are used to identify the constant activity and the
remaining aqueous species, respectively. Matrix Se is also
divided into two parts, that is, Se = (SeajSec), where Sec and
Sea contain the stoichiometric coefficients of constant
activity species and of the aqueous species, respectively.
Since Nc activities are fixed and (1) can be used to eliminate
Nr activities, we can now pose the problem in terms of Ns Nc Nr independent activities (conforming with Gibbs
phase rule). Once these are calculated, we would obtain the
remaining Nr activities on the basis of the chemical equilibrium relations (equation (1)). We call as primary those
species corresponding to the independent Ns Nc Nr
activities; the remaining Nr species are called secondary. We
0
00
jSea
), where
then split Sea into two parts, that is, Sea = (Sea
0
00
Sea and Sea contain the stoichiometric coefficients of the
primary and secondary species, respectively.
[9] It is convenient to redefine the chemical system so that
the matrix of the stoichiometric coefficients of the secondary
species coincides with the opposite of the identity matrix, I.
Mathematically, this is equivalent to multiplying equation (1)
by (S00ea)1 (a proper choice of primary and secondary species
00
leads to Sea
invertible). As activities ac are fixed and known,
we can rewrite the mass action law (1) as
log a00a ¼ S0ea log a0a log K;
ð3Þ
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where a0a and a00a are the activities of primary and secondary
0
are the redefined
species, respectively; S00ea = I and Sea
matrices of stoichiometric coefficients of secondary and
primary species, respectively; K is the redefined vector of
equilibrium constants, log K = (S00ea)1 (log K0 Sec log ac).
With these definitions and upon assuming unitary activity
coefficients, the mass action law (1) reduces to
Sea log ca ¼ log K:
ð4Þ
When the assumption of unitary activity coefficients is not
justified, one needs (2) to relate activities and concentrations.
However, it is still possible to write the mass action law in
terms of concentrations. This is accomplished by substituting K with an equivalent equilibrium constant, K*, defined
as
log K* ¼ log K Sea log gðca Þ;
ð5Þ
where K* is a function of ca, besides other possible independent state variables, such as temperature.
[10] In the following we present all chemical equations in
terms of concentrations, using (4). Appendix A is devoted
to the discussion of an example to facilitate understanding
of the meaning of the vectors and matrices appearing in the
mass action law.
2.2. Mass Balances
[11] Mass balance of each species can be written in a
concise vector notation as
@ ðmÞ
¼ MLt ðcÞ þ f:
@t
ð6Þ
Here, vector m contains the mass of species per unit volume
of porous medium; it can be split into two parts, mc and ma,
respectively related to the constant activity species and to
the remaining species. Vector c contains the concentrations
of species (mi = nci for mobile species, where n is porosity).
Matrix M is diagonal and its diagonal terms are unity when
a given species is mobile and zero otherwise; f is a general
source/sink term. The linear operator Lt(ci) appearing in (6)
is defined as
Lt ðci Þ ¼ r ðq ci Þ þ r ðnDrci Þ;
ð7Þ
where D is the dispersion tensor and q is Darcy’s flux.
[12] Assuming that the source/sink terms are only due to
chemical reactions and that the system is always at chemical
equilibrium, we can express f as
f ¼ STe r;
ð8Þ
where r is the vector of reaction rates (expressed per unit
volume of medium) and Se is defined after (1). We notice
that reactive transport processes also affect immobile species. For instance, while a mineral at equilibrium with the
solution is not transported, its mass is changed in order to
allow the system to attain equilibrium.
[13] A reactive transport process is completely described
by the concentrations of the Ns species, together with the
definition of the Nr (unknown) reactions rates, including the
Nc rates of mass change of the constant activity species. To
DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
W11410
this end, one has to solve the Ns mass balance equations (6)
together with the Nr equilibrium equations (4). While the
problem is very complex because of the nonlinearity of the
governing system, it can be significantly simplified upon
introducing the concept of components [e.g., Rubin, 1990,
1992; Friedly and Rubin, 1992; Steefel and MacQuarrie,
1996; Saaltink et al., 1998, 2001; Molins et al., 2004], as
described below.
2.3. Components
[14] Components are linear combinations of species
whose mass is not affected by equilibrium reactions. Their
introduction is convenient since it allows eliminating the
chemical reactions source term in the transport equations.
Most of the solution methods for reactive transport problems mentioned in the introduction are based on this
concept. We introduce the components matrix, U, defined
such that
USTea ¼ 0:
ð9Þ
0
Recalling that the system was defined so that Sea = (Sea
jI),
a widely used expression for U is
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For this system, Se can be split as
Sec ¼ ð1Þ
ð14aÞ
Sea ¼ S0ea j S00ea ¼ ð1 j 1Þ:
ð14bÞ
Matrices Sec and Sea contain the stoichiometric coefficients
related to the constant activity species and to the remaining
aqueous species, respectively. Equation (14b) implicitly
identifies B1 as primary species and B2 as secondary.
[17] Thus the mass action law (equation (4)) for the
considered system is expressed as
log c1 þ log c2 ¼ log K:
ð15Þ
The equilibrium constant, K, is strictly related to solubility
of the solid phase, S3s, and usually depends on temperature,
pressure and chemical composition of the solution [e.g.,
Philips, 1991; Berkowitz et al., 2003].
[18] Recalling (8), the mass balance equations for the
three species are
ð10Þ
@ ðnc1 Þ
Lt ðc1 Þ ¼ r
@t
ð16aÞ
where N u = N s Nc N r. The expression for the
components matrix is not unique. The motivation of
expressing U by (9) is that multiplying (6) by U, yields
the equation
@ ðnc2 Þ
Lt ðc2 Þ ¼ r
@t
ð16bÞ
@ ðnuÞ
¼ Lt ðuÞ;
@t
@ ðm 3 Þ
¼ r;
@t
ð16cÞ
U ¼ INu j S0T
ea ;
ð11Þ
where u = Uc is a vector of Nu components. Equation (11)
represents Nu transport equations where no reaction sink/
source terms appear. That is, components are indeed conservative quantities in systems at equilibrium. A general
formulation for decoupling transport equations in the presence of kinetic reactions is given by Molins et al. [2004].
[15] In the following we first solve a simple binary
system and then generalize the methodology to deal with
classical heterogeneous and homogeneous chemical reactions at equilibrium.
3. Binary System
3.1. Problem Statement
[16] We consider a reaction of pure dissolution/precipitation [Philips, 1991] at equilibrium, where an immobile solid
mineral S3s dissolves reversibly to yield ions B1 and B2:
B1 þ B2
!
S3s :
ð12Þ
We further assume that the mineral, S3s, is a pure phase, so
that its activity equals 1. With the notation of section 2,
vector ma contains the mass per unit volume of medium,
m1 = nc1 and m2 = nc2, of the aqueous species B1 and B2
respectively, while mc contains the mass per unit volume of
medium, m3, of the solid mineral, S3s. The stoichiometric
matrix of the system described by (12) is
Se ¼ ð1 1 1Þ:
ð13Þ
where r is the rate of the reaction (12). The reaction rate
expresses the moles of B1 (and B2) that precipitate in order
to maintain equilibrium conditions at all points in the
moving fluid (as seen by (16a) and (16b)) and coincides
with the rate of change of m3 (as seen by (16c)).
[19] In order to obtain the space-time distribution of the
concentrations of the two aqueous species and the
reaction rate, we need to solve the nonlinear problem
described by the mass balances of the aqueous species
(equations (16a) and (16b)) and the local equilibrium
condition (equation (15)). The rate of change in the solid
mineral mass is then provided by (16c).
3.2. Methodology of Solution
[20] The general procedure to solve a multispecies transport process will be detailed in section 4. Here, we start by
illustrating the application of the method to the binary
system described in section 3.1 and provide a closed-form
analytical solution.
[21] In essence, the solution procedure develops according to the following steps:
[22] 1. Definition of mobile conservative components of
the system: Using (10) for U and (14b) for Sea, yields
U ¼ ð1 1Þ;
ð17Þ
this implies that one needs to solve the transport problem for
only one conservative component u = Uca = c1 c2. Notice
that simply subtracting (16b) from (16a) leads to the
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
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equation which governs the transport of the conservative
component u = (c1 c2). In other words, dissolution or
precipitation of the mineral S3s equally affects c1 and c2 so
that the difference (c1 c2) is not altered.
[23] 2. Transport of the conservative components: Since
only one component is evidenced in this case, one needs to
solve (11) for u.
[24] 3. Speciation: Here one needs to compute the concentrations of (Ns Nc) mobile species from the concentrations of the components. In our binary system, this
implies solving
c1 c2 ¼ u
ð18aÞ
log c1 þ log c2 ¼ log K:
ð18bÞ
[25] Assuming that K is independent of c1 and c2, the
solution of (18a) and (18b) is
c1 ¼
uþ
c2 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2 þ 4 K
2
u þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2 þ 4 K
:
2
ð19aÞ
ð19bÞ
[26] 4. Evaluation of the reaction rate: Substitution of the
concentration of the secondary species, B2, into its transport
equation (16b) leads to (see Appendix B for details)
r @c2
@K 1
@ 2 c2
@ 2 c2 T
r uDrK
¼
þ Lt ð K Þ þ 2 rT uDru þ 2
@u
@u@K
n @K
@t n
2
@ c2 T
r KDrK:
ð20Þ
þ
@K 2
When K is a function of conservative quantities such as
salinity, s, equation (20) can be rewritten (see Appendix C
for details) as
r @ 2 c2 T
@ 2 c2
@ 2 c2 T
r uDrs:
¼ 2 r sDrs þ 2 rT uDru þ 2
@s
@u
@u@s
n
ð21Þ
Moreover, in the case of constant K, equation (20) reduces
to
r @ 2 c2 T
r u D ru
¼
n @u2
ð22aÞ
@ 2 c2
2K
¼
:
@u2
ðu2 þ 4 K Þ3=2
ð22bÞ
In general, the reaction rates should be used to compute the
mass change of solid mineral. In turn, this would cause a
modification in the medium properties. Here, however, we
will neglect such changes assuming that modification in
the solid mass due to transport involves very thin layers of
the matrix [Rubin, 1983] and no significant variations of the
pore system occur.
[27] The results encapsulated in (20), (21) and (22)
deserve some discussion. First of all, under chemical
265
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equilibrium conditions, they provide a way to compute
directly the rate of dissolution/precipitation as a function
of quantities such as the concentrations of components, the
equilibrium constants, and the dispersion coefficients, without the need to actually evaluate the concentrations of the
dissolved species. We note that (21) includes the model of
Philips [1991] as a particular case.
[28] Furthermore, equation (22) shows that the reaction
rate is always positive (i.e., precipitation occurs) in systems
where K is constant. This is consistent with the comments of
Rubin [1983], who points out that, in the case of the reaction
described by (12), reactive transport processes cannot result
in dissolution of the solid mineral. This is also evident from
Figure 1a. Figure 1a displays another interesting feature.
The equilibrium point can be obtained by drawing a line
from the conservative mixing point toward the equilibrium
line. The slope of this line is equal to the ratio of the
stoichiometric coefficients and is equal to 1 in our example.
[29] Equations (20) and (22) clearly demonstrate that the
reaction rate depends on chemistry, which controls @ 2c2/@u2
(equation (22b)), but also on transport processes, controlling
the gradient of u. Figure 1b qualitatively shows a synthetic
example which allows evaluating the relative importance of
the two factors (chemistry and transport) in the reaction rate.
We note that it would be hard to make general statements
about which one is more important, which is consistent with
the sensitivity analysis of Tebes-Stevens et al. [2001]. One
of the most paradoxical features of Figure 1b is that reaction
does not necessarily take place where concentrations attain
their maximum values. In fact, the reaction rate equals zero
when u is maximum or minimum (i.e., when c1 or c2 reach
their maximum value, respectively). In general (see equation (20)), the reaction rate depends on both the gradients of
u and K. A nonzero gradient of K, for example, can occur
when the mixing of different waters induces spatial variability in temperature or salinity [Berkowitz et al., 2003;
Rezaei et al., 2005].
[30] It is interesting to notice that, in the absence of the
first contribution on the right hand side of (20), all terms are
proportional to the dispersion tensor, D, thus strengthening
the relevance of mixing processes to the development of
such reactions. In particular, the term rT u D ru can be
used as a measure of the mixing rate, which is consistent
with the concept of dilution index, as defined by Kitanidis
[1994] on the basis of entropy arguments. This result also
suggests that evaluating mixing rates may help to properly
identify not only the sources of water [Carrera et al., 2004],
but also the geochemical processes occurring in the system.
[31] In general, the amount of S3s that can precipitate is
controlled by the less abundant species. Thus we expect
precipitation to be highest when mixing induces similar
values of c1 and c2. Mathematically, this is evidenced by the
dependence of the reaction rate on @ 2c2/@u2, that reaches a
maximum when u = 0 (see equation (22b)). We also notice
that the rate (22a) increases with decreasing K, since
solubility also decreases (in a system described by (12),
solubility is the square root of K).
[32] The methodology we have presented can easily be
extended to deal with solute transport in the presence of
multiple reactions. This is shown in section 4. It is then
relevant to stress the point that analytical solutions for these
types of systems are possible whenever an analytical solu-
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
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tion is available for the conservative transport problem (11).
One such case is discussed below.
where r is the normalized radial distance from the center of
the plume, defined as
3.3. Analytical Solution: Pulse Injection in a
Binary System
[33] In this section we apply the general methodology to
derive a closed-form solution for a mixing-driven precipitation reaction. Closed-form solutions provide basic means
to investigate the physical underlying processes and to
analyze the relative importance of the parameters involved.
Moreover, analytical solutions are potentially useful as
benchmark for numerical codes and can be of assistance
in developing methodologies for the setup of laboratory
experiments and procedures for data analysis and/or
interpretation.
[ 34] We consider a three-dimensional homogeneous
porous formation, of constant porosity, n, under uniform
flow conditions. The system is affected by an instantaneous
point-like injection of water containing the same constituents as the initial resident water. This problem can be used
as a kernel for other injection functions. We consider the
case of constant K. We assume that the velocity is aligned
with the x coordinate, V = q/n = Vix (ix being the unit vector
parallel to the x axes), and that the dispersion tensor is
diagonal, DL and DT respectively being its longitudinal and
transverse components. We further assume that the time of
the reaction is small as compared to the typical time of
transport (i.e., equilibrium condition).
[35] The reactive transport system is governed by (12),
(15), and (16). The boundary conditions are
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð x Vt Þ2 y2 þ z2
:
r¼
þ
2tDL
2 tDT
ci1 ¼ ci ðx ! 1; t Þ ¼ ci0 ;
i ¼ 1; 2:
ð23Þ
Initially, we displace resident water by injecting a volume Ve
of solution with concentration ciext = ci0 + cie. In order to
find an analytical solution for small Ve, it is mathematically
convenient to write the initial concentration condition, after
equilibrium is reached at the injection point, as
ci ðx; t ¼ 0Þ ¼ cie Ve dðxÞ=n þ ci 0 ;
i ¼ 1; 2;
ð24Þ
where equilibrium must be satisfied at all points in the
aquifer. This implies that (1) c10c20 = K, and (2) (c10 + c1e)
(c20 + c2e) = K. From these two equilibrium conditions, it
should be clear that c1e and c2e would have different signs.
[36] With this in mind, we then follow the steps detailed
in section 3.2. The (conservative) component u = c1 c2
satisfies (11) with boundary and initial conditions
u1 ¼ uðx ! 1; t Þ ¼ c1 ðx ! 1; t Þ c2 ðx ! 1; t Þ ¼ u0 ð25aÞ
uðx; t ¼ 0Þ ¼ ue Ve dðxÞ=n þ u0 :
ð25bÞ
With this definition, ue is then the excess of the injected
component u that remains in the aquifer immediately after
injection. The solution of (11), subject to boundary and
initial conditions (25a) and (25b) is [Domenico and
Schwartz, 1997, p. 380]
uðx; t Þ ¼ u0 þ
18
ð2pÞ1=2
ue
1
exp r2 ;
ed
2
V
ð26Þ
ð27Þ
e d is the ratio between the
The dimensionless quantity V
volume containing about the 99, 7% of the excess of
injected mass [Domenico and Schwartz, 1997] and the
injection volume
pffiffiffi pffiffiffiffiffiffi
72pn 2t3=2 DL DT
e
Vd ¼
Ve
ð28Þ
and is a measure of the temporal evolution of the dispersive
effect.
[37] Substituting (26) into (19a) and (19b), we obtain the
concentrations, c1 and c2, of the dissolved species. Finally
the expression of the local mineral mass precipitation rate, r,
per unit volume of medium is derived from (22) as
nK
rðx; t Þ ¼
"
t u0 þ
2
ue
18
ð2pÞ1=2 e
Vd
ue
18
ð2pÞ1=2 e
Vd
r2 exp½r2 exp 12 r2
#3=2 :
2
ð29Þ
þ4K
We notice that the rate, r, vanishes in the trivial case of ue =
0, since the concentrations of the two injected species are
the same as the resident ones and there is no excess of
injected mass in the solution right after the injection. The
e d ! 1. The latter situation
rate vanishes also when V
describes a scenario where gradients of the component u are
negligible because of large dispersion effects or because a
large amount of time has elapsed since injection.
[38] To illustrate the features of this solution we consider
the following problem:
water
pffiffiffiffi we start with a president
ffiffiffiffi
characterized
by c10/ K = 0.25 and c20/ K = 4.0 (u0/
pffiffiffiffi
K = 3.75); we then inject
pffiffiffiffi water from an external
pffiffiffiffi source,
characterized
by
c
/
K
=
0.184,
c
/
K = 5.434
1ext
2ext
pffiffiffiffi
/ ffiffiffiffiK = 5.25). Equilibrium
condition
is
satisfied
when
(uextp
pffiffiffiffi
pffiffiffiffi
c 1e/ K = 0.066, c 2e/ K = 1.434 (u e/ K = 1.5).
Figure 2a depicts thepffiffiffi
dependence
of
ffi
pffiffiffiffithe dimensionless
concentrations ~c1 = c1/ K and ~c2 = c2/ K on the normalized
distance
pffiffiffiffiffiffiffiffiffiffi from the center of the (moving) plume, (x Vt)/
e d = 3.5. Dimensionless
2tDL (while z = y = 0), and V
concentrations (~c1NR, ~c2NR) for the corresponding nonreactive system are also shown for comparison. Notice that while
both concentrations are higher than the initial ones in the
nonreactive case, in the reactive case ~c1 decreases while ~c2
increases (in agreement with the fact that c1e < 0 and c2e > 0).
[39] The spatial distribution
pffiffiffiffi of the (local) dimensionless
reaction rate, ~r = rt/(n K ), for the same conditions of
Figure 2a, is depicted in Figure 2b. A comparison of
Figures 2a and 2b clearly elucidates that the system is chemically active (i.e., the reaction rate is significant) at locations
where concentration gradients of both species are relevant.
This implies, in turn, that strong gradients of the component u
give rise to significant reaction rates, in agreement with (22a).
In general, no reactions occur within the system when concentration gradients vanish. As a consequence, no reaction occurs
266
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From (29) we also observe that a change in DT has stronger
effects rather than a modification in DL. This is so since DT
e d and is also clear when
is not squared in the definition of V
considering that dispersion is enhanced along two directions
by increasing DT, while an increase in DL affects only one
spatial direction. We will comment further on this topic in
the following.
[41] The sensitivity of the reaction rate to ue for a given
with
toffiffiffiffia resident
u0 is presented
pffiffiffiffiin Figure 4, p
ffiffiffiffi reference p
water with u0/ K = 20 (c10/ K = 0.05; c20/ K = 20.05).
pffiffiffiffi
We start by considering the case with a negative ue (ue/ K =
e d = 3.5.
30). The dimensionless mixing volume is set as V
The shape of the reaction rate function is displayed in
Figure 4a at the plane z = 0. From the plot it is clear that, at
any given time, precipitation would concentrate in a (three-
Figure 2. Dependence of (a) dimensionless concentrations
~c1, ~c2 and (b) dimensionless reaction rate, ~r, on the
normalized
distance
from the center of the (moving)
plume,
pffiffiffiffi
pffiffiffiffiffiffiffiffiffi
ffi
e d = 3.5, ue/ K = 1.5, and
(x Vt)/
2tDL , for z = y = 0, V
pffiffiffi
ffi
u0/ K = 3.75. In Figure 2a, the dimensionless concentrations ~c1NR, ~c2NR are those of a nonreactive system; the
dashed line indicates the initial concentrations of the two
species within the system.
at the (moving) center of the plume, that is, at the points of
highest (or lowest) concentration values.
[40] Figures 3a and 3b display the same quantities of
e d = 16. A comparison between
Figures 2a and 2b but for V
Figures 2a and 3a reveals that the concentration profiles
e d increases. A larger value
display a lower amplitude when V
e d (equation (28)) can be seen as an increase either in
for V
elapsed time or in the effects of dispersion for a given time.
Thus it is expected that more dispersed plumes are associated to weaker gradients of u and thus to reduced precipitation rates. This behavior is not observed in the range of
e d 1), where an increase of
small dispersion effects (V
dispersion phenomena enhances the mixing process and the
related reaction. This feature can be observed mathematie d tends to zero.
cally by taking the limit of (29) when V
267
Figure 3. Dependence of (a) dimensionless concentrations
~c1, ~c2 and (b) dimensionless reaction rate, ~r, on the
normalized
distance
from the center of the (moving)
plume,
pffiffiffiffi
pffiffiffiffiffiffiffiffiffi
ffi
e
,
for
z
=
y
=
0,
V
=
16,
u
/
K
=
1.5,
and
(x Vt)/
2tD
L
d
e
pffiffiffiffi
u0/ K = 3.75. In Figure 3a, the dimensionless concentrations ~c1NR, ~c2NR are those of a nonreactive system; the
dashed line indicates the initial concentrations of the two
species within the system.
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Figure 4. Dependence of the dimensionless
reaction rate, ~r, on the dimensionless distancepffiffiffiffi
from the
pffiffiffiffi
e d = 3.5: (a) three-dimensional view for ue/ K = 30
center of mass of the plume forpuffiffiffi0ffi/ K = 20 and V
and (b) radial sections for ue/ K = 20, 30, 40, 20, 30, and 40.
dimensional) aureole around the moving center of the plume.
The actual location and shape of this aureole is governed
by (29), and would depend on DL, DT and ue. One should
note that Figure 4a displayspan
artificial
symmetry, as
ffiffiffiffiffiffiffiffiffi
ffi
coordinate x is normalized
pffiffiffiffiffiffiffiffiffiffi by 2tDL, while the y direction
is normalized by 2tDT . Figure 4b depicts radial profiles
of the dimensionless reaction rate, ~r, for different ue.
Because of the symmetry of the solution with respect to
the normalized coordinates, we display only radial profiles
starting from the plume center. Figure 4b is organized in
such a way that while the curves resulting from positive
values of ue are displayed on the larger scale, those arising
by negative values of ue are displayed within the insert.
Figure
p
ffiffiffiffi 4b reveals
pffiffiffiffi that the reaction rate is larger when ue/
K and u0/ K have opposite signs. An explanation of
this is provided with the aid of Figure 5 where, p
forffiffiffiffi the
sake of discussion, we consider the effect of jue/ K j =
30. From the points reached on the equilibrium curve
pffiffiffiffi
right after injection, characterized by (cie + ci0)/ K ,
concentrations would evolve until they reach the asymptotic condition characterized by the same concentrations
as the resident water. In the case of conservative solutes,
the paths would be the straight lines depicted in Figure 5
as dashed lines. For reactive solutes the path would be
along the hyperbola (equilibrium line: ~c1~c2 = 1). From
Figure 5 we see that there is no symmetry: in the case of
ue < 0 reactive and conservative solutes paths are quite
Figure 5. Scheme of a pure dissolution/precipitation
reaction induced by a point-like injection. The equilibrium
pffiffiffiffi
condition at the injection point is characterized
pffiffiffiby
ffi jue/ K j =
30, while the initial water is identified by u0/ K = 20 (the
initial point is also the final equilibrium point). Dashed lines
outline the path of the reaction in the case of conservative
solutes. For reactive solutes, the path would develop along
the hyperbola (equilibrium line).
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
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similar, which is not the case when ue > 0. As a
consequence, in the latter case a larger amount of precipitation is needed in order to reach equilibrium at all
locations and times. Going back to Figure
pffiffiffiffi4b, we observe
that the largest absolute value of ue/ K produces the
largest reaction rate, as a consequence of the mixing of
increasingly different waters, thus inducing significant
gradients of u.
[42] Knowledge of the local rate is essential to evaluate
the global reaction rate
Z
rW ð t Þ ¼
rðx; t Þ dW;
ð30aÞ
W
which is an integral measure of the rate of precipitation of
the mineral mass of the system at any given time. Here, W is
the entire volume of the medium. This information is of
practical interest for experimental applications devoted to
analyze precipitation/dissolution processes, such as those
reported by Berkowitz et al. [2003] and Singurindy et al.
[2004].
[43] In general a closed form solution for the integral
(30a) does not exist. Upon using spherical coordinates, the
three-dimensional integral (30a) can be written as the
following one-dimensional integral,
nK Ve =n e
rW ¼
Vd
t 36 p
2
Z1
0
18
ue
1=2 e
ð2pÞ Vd
!2
r2 exp r2
33=2
!2
u
1
e
4 u0 þ
exp r2
þ4K 5
4p r2 dr:
ed
2
ð2pÞ1=2 V
18
ð30bÞ
As evidenced by (30b) all information related to dispersion is
e d. Thus the sensitivity of rW on dispersion
concentrated in V
can be analyzed from the following expression,
ed
ed
@rW
@rW @ V
@rW V
DT @rW
¼
¼
¼
:
ed @DL @ V
ed 2DL 2DL @DT
@DL @ V
[44] An asymptotic expression for (30b) (see Appendix D
for details) can be derived analytically in the presence of
e d) as
large dispersion effects (i.e., large V
rW ð t Þ ¼
Ve K u2e
27
:
ed 4p1=2 ðc10 þ c20 Þ3
t V
Z1
rðx; t Þ dt
ert ðxÞ ¼
269
ð32Þ
e d (equation (28)), expression
Recalling the definition of V
e
(32) reveals that, for large V d, the global reaction rate, rW,
decreases with time proportionally to t5/2. Figure 6 depicts
the dependence
pffiffiffiffiof the dimensionless global
pffiffiffiffi reactionprate,
ffiffiffiffi
e d for different ue/ K and u0/ K =
~rW = rW t/(Ve K ), on V
20; bold lines are the asymptotic solution (32), while thin
lines with symbols correspond to ~rW as obtained by
numerical solution of (30b). We observe that ~rW increases
e d < 9),
e d in the range of small dispersion effects (V
with V
since an increase of dispersion effects enhances the mixing
process and the related reaction is enhanced. On the
e d for larger
contrary, ~rW is a decreasing function of V
e d > 9), as the prevailing effect of the
dispersion effects (V
enhanced dispersion effects is now a reduction of
concentration gradients and thus of precipitation. Consistently with our previous comments,
forpsmall
dispersion
pffiffiffiffi
ffiffiffiffi
opposite
effects ~rW is larger when ue/ K and u0/ K have p
ffiffiffiffi
signs,
pffiffiffiffi while it is independent of the sign of ue/ K and
u0/ K when dispersion effects are large.
[45] We note that the asymptotic solution (32) coincides
with the exact results of the numerical integration of (30b)
e d 100. Considering, as an example, a medium
when V
characterized by DT = 0.21 m2/day and DL = 2.1 m2/day and
setting Ve/n = 1 m3, this means that the precipitation
phenomenon starts decreasing according to a factor t5/2
after 1 day from the injection. This suggests that the bulk of
the reactive process develops at early time and is confined
within a region of the domain which is relatively close to
the injection point.
[46] As a final result, Figures 7a and 7b depict the spatial
distribution of the dimensionless total rate of precipitation
ð31Þ
Equation (31) shows that the overall reaction rate, rW, is much
more sensitive to DT than to DL. Considering that longitudinal dispersion is often taken to be about five to ten times
of transverse dispersion, equation (31) implies that the
overall reaction rate would be ten to twenty times more
sensitive to DT than to DL. Furthermore, the input of
solutes is frequently continuous in time and reactants do
not enter directly into the flow domain, as they are laterally
driven into it. Both factors would tend to enhance the role of
transverse dispersion in reactive transport problems. This is
particularly relevant in view of the uncertainties surrounding
the actual values of transverse dispersion. While some argue
that transverse dispersion tends to zero in three-dimensional
domains for large travel times [Dagan, 1989], others
[Neuman and Zhang, 1990; Zhang and Neuman, 1990;
Attinger et al., 2004] show that this is not the case, thus
corroborating the idea that the actual transverse dispersion is
linked to the interplay between spatial heterogeneity and
time fluctuations of velocity [Cirpka and Attinger, 2003;
Dentz and Carrera, 2003].
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0
pffiffiffiffi
n K
;
ð33Þ
obtained by numerical integration of (29) on the horizontal
plane z = 0 for the entire duration of the process. Figures 7a
and 7b havepffiffiffiffi
been obtained on the basis of the initial
/
K
= 20, while the boundary conditions are
condition
u
0
pffiffiffiffi
ue/ K = ±20.
[47] Figures 7a and 7b corroborates the findings that the
system is active within a region close to the injection point.
We note that the reaction can be considered exhausted when
Vx/DL > 0.5 and Vy/DT > 2. Considering, as an example, a
medium characterized by DT = 0.21 m2/day and DL =
2.1 m2/day, setting Ve/n = 1 m3 and V = 0.5 m/day, this
means that the precipitation process is negligible for x >
2.1 m and y > 0.85 m.
4. Generalization of the Methodology of Solution
[48] We now extend the methodology illustrated in
section 3 to describe multispecies transport processes in
the presence of generic homogeneous and classical hetero-
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
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geneous reactions. We assume that all the variable activity
species are mobile, so that mass balances (equation (6)) for
dissolved and constant activity species are expressed as
@ ðnca Þ
¼ Lt ðca Þ þ STea r
@t
@ ðmc Þ
¼ Mc Lt ðmc Þ þ STec r;
@t
Figure 6. Dependence of the dimensionless global reace d,
tion rate, ~rW, on pthe
ffiffiffiffi dimensionless
pffiffiffiffi dispersive volume, V
for different ue/ K and u0/ K = 20. Thin lines with
symbols correspond to ~rW as obtained by numerical
integration of (30b); thick lines correspond to the
asymptotic solution (32).
ð34aÞ
ð34bÞ
where matrix Mc is diagonal and its diagonal terms equal 1
when a given constant activity species is mobile or zero
otherwise. Our aim is to evaluate the concentrations of
dissolved species, ca (Ns Nc unknowns) and the rates of
the reactions, r (Nr unknowns). Once r is known, (34b)
provides the rates of mass change of the constant activity
species. In order to calculate ca and r, we need to solve the
algebraic-differential system given by the Ns Nc partial
differential equations of transport for ca (equation (34a))
and the Nr nonlinear algebraic equilibrium conditions
(equation (4)). The general methodology of solution of this
system of equations is detailed in the following.
4.1. Step 1: Definition of Mobile
Conservative Components
[49] The first step of the methodology consists of defining the chemical system (species and reactions, that is, the
stoichiometric matrix of (1)) and reducing the number of
equations describing transport of dissolved species by
eliminating the source term in (34a). To this end, we use
the components matrix, U, defined by (9).
4.2. Step 2: Transport of Components
[50] Since components are conservative, equation (11)
can be used to describe their transport. The initial and/or
Figure 7. Spatial distribution of the dimensionless total rate of precipitation, ~rt, as obtained
by
pffiffiffiffi
numerical
pffiffiffiffi integration of (29)
pffiffiffion
ffi the horizontal plane z = 0 during the entire process, for u0/ K = 20,
(a) ue/ K = 20 and (b) ue/ K = 20.
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
boundary conditions for (11) are given by u0 = Uc0, c0
being the vector of initial and/or boundary species concentrations. Since the reaction sink/source terms have been
eliminated, all components are independent. Moreover, the
resulting transport equation is identical for all components,
the only difference being in the boundary conditions. Only
in some particular cases (such as in the example of
section 3.3) a closed-form analytical solution is possible.
Otherwise, any conventional transport simulator can be
used. The underlying hypotheses are that (1) the mobile
species are subject to the same advective flow field and
(2) the same dispersion processes apply to all species, with
equal dispersion coefficients. Other than that, the formulation does not impose any additional restrictions on the
nature of flow. That is, flow can be steady or transient,
saturated or unsaturated, single phase or multiphase, and
temperature may be constant or variable.
4.3. Step 3: Speciation
[51] This step is devoted to evaluating the concentrations
of the Ns Nc aqueous species, ca. The components,
computed in step 2, provide N u = Ns Nc Nr equations.
The mass action law for each reaction provides the remaining Nr equations. Thus one needs to solve the nonlinear
algebraic system
u ¼ Uca
ð35aÞ
log c00a ¼ S0ea log c0a log K:
ð35bÞ
It is clear that the system is nonlinear, both explicitly
(simultaneous linear dependence on ca and log ca) and
implicitly (in general, K may depend on activity coefficients
and thus be a nonlinear function of ca, as discussed in
section 2.1). The solution of this system can be very
complex. However, in some geochemical problems (such as
the binary system of section 3) it is possible to derive an
analytical solution. Formally, one can substitute (35b) into
(35a), obtaining Nu nonlinear algebraic equations for c0a.
Their solution renders c0a, as a function of u and K, which is
then substituted in (35b) to obtain c00a. In summary, we can
write
c0a ¼ c0a ðu; KÞ
ð36aÞ
c00a ¼ c00a ðu; KÞ:
ð36bÞ
4.4. Step 4: Evaluation of Reaction Rates and
Mass Change of Constant Activity Species
[52] Here, we substitute the concentrations computed in
step 3 into the mass balance equations to obtain the Nr
reaction rates. We recall (section 2) that each equilibrium
reaction yields a secondary species and that the
corresponding transport equation solely depends on its
reaction rate, as the redefined matrix of the stoichiometric
coefficients of the secondary aqueous species coincides with
the opposite of the identity matrix, I. Therefore, considering
the Nr transport equations of the secondary species leads to
@ n c00a
r ¼ Lt c00a :
@t
ð37Þ
271
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Substituting the functional dependences of (36b) into (37),
it is possible to evaluate the rates of the reactions (see
Appendix E for details). The rate of the mth reaction (we
switch temporarily to index notation to avoid ambiguities) is
given by
Nr
@Kp
rm X
@c00am
¼
V rKp þ r DrKp
n
@Kp
@t
p¼1
þ
Nu X
Nu
Nu X
Nr
X
X
@ 2 c00am T
@ 2 c00am T
r ui Druj þ 2
r ui DrKq
@u
@u
@u
i
j
i @Kq
i¼1 j¼1
i¼1 q¼1
þ
Nr X
Nr
X
@ 2 c00am
rT Kp DrKq :
@Kp @Kq
p¼1 q¼1
ð38Þ
This equation can be simplified when K is a function of
state variables satisfying nonreactive transport equation
(recall equation (21)) and, especially, when K is constant.
The latter condition leads to
Nu X
Nu
rm X
@ 2 c00am T
¼
r ui Druj :
n
@ui @uj
i¼1 j¼1
ð39Þ
Reverting to vector notation, (39) can then be expressed as
r ¼ nHrT u D ru;
ð40Þ
where H is the vector of Hessian matrices (a third-order
tensor) of the reactions (as represented by the corresponding
secondary species) with respect to the components.
[53] The rate of mass change of constant activity species
is then obtained by substituting the reaction rates into the
corresponding mass balance equations (34b). In the case of
immobile species (minerals subject to precipitation or dissolution), the rate of mass change is given directly by the
reaction rate. Knowledge of these rates of solid mass change
is of particular interest, since it is a prerequisite to estimate
the characteristic time of changes in medium properties
[Wood and Hewett, 1982; Philips, 1991].
[54] When the constant activity species are mobile (e.g.,
dissolved gases or some colloids), the formulation is still
valid but more complex, as one would need to use the
complete equation (34b) to obtain the reaction rate. If the
species is water, then one is rarely interested in finding its
concentration, other than in special cases, such as in the
presence of osmotic effects. In such cases our formulation is
still valid, but water would not be a constant activity species
anymore.
4.5. Additional Considerations
[55] As previously pointed out, a key feature of our
methodology is that it allows obtaining concentrations of
solutes and reaction rates independently of constant activity
species.
[56] As compared to existing formulations proposed in
the literature to solve multispecies transport processes in the
presence of a generic number of homogeneous and classical
heterogeneous reactions [Rubin, 1990, 1992; Friedly and
Rubin, 1992; Steefel and MacQuarrie, 1996; Saaltink et al.,
1998, 2001; Molins et al., 2004], our method is simpler and
more concise. It allows eliminating constant concentration
species from the beginning, thus simplifying the formulation from the outset.
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
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[57] The method leads to general expressions for reaction
rates. The latter should then be used to compute the mass
change of solid mineral, which may induce modifications in
the medium properties. Here we neglect such changes and
assume that modifications in solid mass due to transport
involve very thin layers of the matrix [Rubin, 1983] and no
significant variations of the pore system occur. In many
practical problems minerals may appear (in response to the
changes caused by mixing and reactions) or disappear (by
being totally dissolved) in zones of the flow domain. This
can be handled by defining different chemical systems in
each zone. Unfortunately, this would lead to severe mathematical complications.
[58] The first three steps of the proposed method are
similar to those implemented in many computer codes, such
as HYDROGEOCHEM [Yeh and Tripathi, 1991],
PHREEQC [Parkhurst, 1995] and RETRASO [Saaltink et
al., 2004] (which also includes a detailed list of codes). The
evaluation of reaction rates using (38) or (40) is no less
cumbersome than simply substituting the concentrations of
secondary species into the transport simulator to solve for r,
as implied by (37). In fact, conventional speciation codes do
not yield the second derivatives of secondary species. Our
equations yield valuable insights into the basic processes
controlling the rate of equilibrium reactions and allow for
explicit evaluations of these rates in cases when an analytical solution is feasible.
[64] 5. The general expression provided for the reaction
rates highlights that mixing processes control equilibrium
reaction rates and evidences the possibility of inducing
reaction by simply mixing waters at different equilibrium
conditions. This type of mixing-driven chemical reactions
can help in explaining the enhancement of reaction processes observed when different solutions mix in carbonate
systems [Gabrovšek and Dreybrodt, 2000; Corbella et al.,
2003; Rezaei et al., 2005].
[65] 6. We applied the general methodology to a binary
system and derived an original closed-form analytical
solution of a mixing-driven precipitation reaction induced
by a pulse injection in a three-dimensional homogeneous
porous medium in the presence of uniform flow. Our results
prove that in binary systems the mixing process of two
waters, both of which are at equilibrium, induces precipitation processes throughout the system. The solution also
demonstrates that (1) the presence of precipitation significantly modifies the concentrations of aqueous species, when
compared to a nonreactive situation; (2) the features of the
reactive process are strongly dependent on flow characteristics and on the difference in the concentrations of both the
initial and the injected water; (3) the bulk of the reactive
process develops at early time, thus remaining confined
within a region of the domain which is relatively close to
the injection point; and (4) the overall reaction rates are
more sensitive to transverse than to longitudinal dispersion.
5. Conclusions
Appendix A: Illustrative Example of the
Chemical Equilibrium Formulation
[59] Our work leads to the following major conclusions:
[60] 1. The methodology we propose allows obtaining the
local concentrations of dissolved species and the rates, r, of
the reactions occurring in the system. It also allows evaluating
the rate of change in the mass of the solid minerals involved in
the phenomena as a function of r; this is a prerequisite to
estimate the characteristic timescale of changes in medium
properties [Wood and Hewett, 1982; Philips, 1991].
[61] 2. An appealing feature of our method is that it
allows separating the solution of chemical equations to the
mass balance equations of dissolved species. This leads to a
reduced number of transport equations to be solved, and,
more importantly, deconstructs the problem to the solution
of a set of independent nonreactive advection-dispersion
equations.
[62] 3. From a practical standpoint, our methodology is a
very powerful tool in deriving analytical solutions for
multispecies system where homogeneous or classical heterogeneous reactions occur [Rubin, 1983] on the basis of
known classical solutions describing nonreactive processes.
It can also be used to develop and/or simplify numerical
codes for solving reactive transport problems.
[63] 4. As compared to formulations previously proposed
in the literature to solve multispecies transport phenomena
in the presence of a generic number of homogeneous and
classical heterogeneous reactions [Rubin, 1990, 1992;
Friedly and Rubin, 1992; Steefel and MacQuarrie, 1996;
Saaltink et al., 1998, 2001; Molins et al., 2004], our method
is simpler and more concise. Moreover, it permits not only
to evaluate concentrations of dissolved species, but also
provides general expressions for the rates of the system
reactions and for the rate of change in solid mass involved
in the phenomena.
[66] As an example to help understanding the meaning of
vectors and matrices employed within the mass action law
(1), we analyze the system of reactions leading to dedolomitization [Ayora et al., 1998]. The system is governed by
four equilibrium reactions (Nr = 4)
þ
CO2
3 ¼ HCO3 H
ðA1aÞ
þ
CO2 ¼ HCO
3 þ H H2 O
ðA1bÞ
CaMgðCO3 Þ2s ¼ Ca2þ þ Mg2þ þ 2CO2
3
ðA1cÞ
CaCO3s ¼ Ca2þ þ CO2
3 :
ðA1dÞ
We identify H2O, CaCO3(s) and CaMg(CO3)2s as constant
activities species (Nc = 3; one aqueous and two solid
species). As the total number of species involved is Ns = 9,
we need to define Ns Nr Ns = 2 primary species. We
2
then define H+ and HCO
3 as the primary species and CO3 ,
2+
2+
CO2, Mg and Ca as the secondary species.
[67] We write the vector of activities, a, as
2
2þ
a ¼ Hþ ; HCO
3 ; CO3 ; CO2 ; Mg ;
Ca2þ ; CaMgðCO3 Þ2s ; CaCO3s ; H2 OÞT ;
ðA2Þ
which is written respecting the order presented in section 2,
first the primary species, then the secondary, and last the
constant activity species. The stoichiometric matrix is
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DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
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0
HCO
CO2
CO2
3
3
1
1
0
1
0
1
0
2
0
0
1
0
0
00
¼ ðSea j Sec Þ ¼ Sea Sea Sec :
Hþ
B 1
B
Se ¼ B
B 1
@ 0
0
Mg2þ
0
0
1
0
Note that negative and positive coefficients in Se are related
to reactants and products of the reactions, respectively.
Multiplying S0ea and S00ea by (S00ea)1 we rewrite the
stoichiometric matrix so that the matrix of the stoichiometric coefficients of the secondary species coincides with
the opposite of the identity matrix, I, and redefine the
matrices Sea as
0
Hþ
B 1
0
B
Sea j I ¼ B
B 1
@ 1
1
HCO
3
1
1
1
1
CO2
3
1
0
0
0
CO2
0
1
0
0
Mg2þ
0
0
1
0
1
Ca2þ
0 C
C
0 C
C:
0 A
1
ðA4Þ
This last operation has simply implied rewriting the last two
reactions (A1c) and (A1d) in terms of primary species (i.e.,
eliminating CO2
3 ), so that the last two reactions now read
Mg2þ ¼ CaMgðCO3 Þ2s CaCO3s þ Hþ HCO
3
Ca2þ ¼ CaCO3s þ Hþ HCO
3:
ðA5aÞ
ðA5bÞ
Appendix B: Reaction Rate for a Binary System
[68] Here we detail the steps leading to a general expression for the rate of the reaction of pure precipitation/
dissolution processes in a binary system (given by
equation (20)). We start from the transport equation of the
species B2 (equation (16b)) and assume n to be constant.
Recalling (19b), the time derivative of c2 can be expressed
as
@ ½n c2 ðu; K Þ
@c2 @u @c2 @K
¼n
þ
:
@u @t @K @t
@t
ðB1Þ
We introduce the advective and diffusive linear operators,
Lt_adv and Lt_d, defined as
Lt ðcÞ ¼ n½Lt
adv ðcÞ
þ Lt d ðcÞ;
ðB2Þ
where
Lt
adv ðcÞ
¼ V rc
Lt d ðcÞ ¼ r ðDrcÞ
ðB3aÞ
ðB3bÞ
and V = q/n.
273
Ca2þ
0
0
1
1
CaMgðCO3 Þ2s
0
0
1
0
CaCO3s
0
0
0
1
W11410
1
H2 O
0 C
C
1 C
C
0 A
0
(A3)
[69] Applying the advective operator to c2 leads to
Lt
adv ðc2 Þ
¼
@c2
@c2
ðV ruÞ þ
ðV rK Þ;
@u
@K
ðB4Þ
while applying the diffusive operator to c2 leads to
@c2
@c2
Lt d ðc2 Þ ¼ r ðDruÞ þ r ðDrK Þ
@u
@K
@c2
@c2
@ 2 c2
r ðDruÞ þ
r ðDrK Þ þ 2 rT uDru
¼
@u
@K
@u
@ 2 c2 T
@ 2 c2 T
r uDrK þ
r KDrK:
ðB5Þ
þ2
@u@K
@K 2
Substituting (B4) and (B5) into the transport operator (B2),
applied to c2, leads to
@c2
½V ru þ r ðDruÞ
Lt ðc2 Þ ¼ n
@u
@c2
þ
½V rK þ r ðDrK Þ
@K
@ 2 c2
@ 2 c2 T
r uDrK
þ 2 rT uDru þ 2
@u
@u@K
@ 2 c2 T
þ
r KDrK :
@K 2
ðB6Þ
Finally, substituting (B1) and (B6) into (16b) and recalling
(11), yields
r
@c2
@K 1
@ 2 c2
@ 2 c2 T
r uDrK
¼
þ Lt ð K Þ þ 2 rT uDru þ 2
@K
@u
@u@K
n
@t n
@ 2 c2 T
r KDrK:
ðB7Þ
þ
@K 2
Appendix C: Reaction Rate for a Binary
System When the Equilibrium
Constant Depends on Conservative Quantities
[70] This Appendix is devoted to finding the expression for
the rate of the reaction of pure precipitation/dissolution, when
the equilibrium constant, K, is a function of a given quantity,
s, satisfying a nonreactive form of the advection-dispersion
equation. This could be, for example, the case of salinity.
[71] We start from the general expression of the reaction
rate (20)
r
@c2
@K 1
@ 2 c2
@ 2 c2 T
r uDrK
¼
þ Lt ð K Þ þ 2 rT uDru þ 2
@K
@u
@u@K
n
@t n
@ 2 c2 T
þ
r KDrK:
ðC1Þ
@K 2
DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
W11410
As K is function of s, the time derivative of K can be
expressed as
@ ½ K ðsÞ @K @s
¼
:
@t
@s @t
e d is large
[73] The derivation relies on the fact that when V
we can write an approximate expression for the local rate
e d) as
upon disregarding terms of order larger than 2 (in 1/V
ðC2Þ
"
#2
r
2K
1
18 ue
r2 exp r2 ;
¼
3=2
1=2 e
2
n
2t
ð2pÞ Vd
u0 þ 4K
Recalling the definition of advective and diffusive linear
operators, Lt_adv and Lt_d, (equations (B3a) and (B3b)), we
split the operator Lt (K) as
1
Lt ðK Þ ¼ Lt
n
adv ð K Þ
þ Lt d ð K Þ:
ðC3Þ
Z1
3 pffiffiffi
r4 exp r2 dr ¼
p;
8
ðD2Þ
0
ðC4Þ
integration of (D1) over the domain renders rW (equation
(32)) as
The diffusive operator acting on K provides
Lt d ð K Þ ¼ r ðD1Þ
where r is defined by (27).
[74] Recalling that
Applying the advective operator to K leads to
@K
Lt adv ð K Þ ¼
ðV rsÞ:
@s
W11410
@K
@K
@2K
ðDrsÞ ¼
r ðDrsÞ þ 2 rT sDrs:
@s
@s
@s
rW ¼
Ve K u2e
27
:
ed 4p1=2 ðc10 þ c20 Þ3
t V
ðD3Þ
ðC5Þ
By virtue of (C4) and (C5), (C3) results
1
Lt ð K Þ ¼
n
@K
@2K
½V rs þ r ðDrsÞþ 2 rT sDrs : ðC6Þ
@s
@s
Substituting (C2) and (C6) into (C1), and recalling that s
satisfies a nonreactive format of the advection-dispersion
equation, leads to
r
@c2 @ 2 K T
@ 2 c2
r sDrs þ 2 rT uDru
¼
2
@K @s
@u
n
@ 2 c2 T
@ 2 c2 T
r uDrK þ
r KDrK:
þ2
@u@K
@K 2
@ 2 c2 T
@ 2 c2 @K 2 T
r
KDrK
¼
r sDrs:
@K 2
@K 2 @s
[75] Here we detail the steps needed to obtain (38),
expressing the reaction rates for the general case of multicomponent transport. We start from the transport equations
for the secondary species (equation (37)) and recall that c00a is
solely a function of u and K (equation (36b)). Let us first
consider a single secondary species’ concentration, c00am.
Using the chain rule, we express the time derivative of c00am
as
Nu
Nr
X
X
@ n c00a m ðu; KÞ
@c00a m @ui
@c00a m @Kp
þn
;
¼n
@ui @t
@Kp @t
@t
i¼1
p¼1
ðC7Þ
Since K is a function of s, the last two terms in (C7) can be
written as
@ 2 c2 T
@ 2 c2 T
r uDrK ¼
r uDrs
@u@K
@u@s
Appendix E: General Expression for the
Reaction Rate
where Nu = Ns Nc Nr is the number of primary species.
[ 76 ] Application of the advective operator, L t_adv
(equation (B3a)), yields
ðC8Þ
Lt
adv
Nu
Nr
X
00 X
@c00a m
@c00a m ca m ¼
ðV rui Þ þ
V rKp :
@ui
@Kp
i¼1
p¼1
ðC9Þ
Substituting (C8) and (C9) into (C7) and noticing that
,
@ 2 c2 @c2 @ 2 K @ 2 c2 @K 2
we finally obtain
¼
þ
@s2
@K @s2 @K 2 @s
ðE1Þ
ðE2Þ
The diffusive operator, Lt_d (equation (B3b)), acting on c00am
is expressed as
Lt
r @ 2 c2 T
@ 2 c2
@ 2 c2 T
r uDrs: ðC10Þ
¼ 2 r sDrs þ 2 rT uDru þ 2
@s
@u
@u@s
n
d
00 Nu
Nu
X
X
@c00a m
@ca m
c00a m ¼
r ðDrui Þ þ
r
ðDrui Þ
@u
@ui
i
i¼1
i¼1
00 Nr
Nr
X
X
@c00a m
@ca m
DrKp :
r DrKp þ
r
þ
@K
@K
p
p
p¼1
p¼1
ðE3Þ
Appendix D: Analytical Solution for the
Overall Reaction Rate in a Binary System
Upon noting that
00 X
Nu
Nr
X
@ca m
@ 2 c00a m
@ 2 c00a m
¼
r
ruj þ
rKq ;
@ui
@u
@u
@u
i
j
i @Kq
j¼1
q¼1
[72] Here we provide the analytical expression of the
overall rate, rW (equation (30b)), in the case of large
e d).
dispersion effects (large dimensionless volumes, V
274
ðE4Þ
DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS
W11410
equation (E3) can be rewritten as
Lt
d
Nu
Nr
X
X
@c00a m
@c00a m
c00a m ¼
r ðDrui Þ þ
r DrKp
@u
@K
i
p
i¼1
p¼1
þ
Nu X
Nu
X
@ 2 c00
am
rT ui Druj
@ui @uj
i¼1 j¼1
Nu X
Nr
X
@ 2 c00a m
þ2
þ
rT ui DrKq
@ui @Kq
i¼1 q¼1
Nr X
Nr
X
@ 2 c00a m T
p¼1 q¼1
@Kp @Kq
r Kp DrKq :
ðE5Þ
Using (E2) and (E5), the linear operator (7) acting on c00am
can be written as
(
Nu
X
00 @c00a m
Lt ca m ¼ n
½V rui þ r ðDrui Þ
@ui
i¼1
þ
þ
Nr
X
@c00a m V rKp þ r DrKp
@Kp
p¼1
Nu X
Nu
X
@ 2 c00
am
i¼1 j¼1
@ui @uj
rT ui Druj
Nu X
Nr
X
@ 2 c00a m T
r ui DrKq
@ui @Kq
i¼1 q¼1
)
Nr X
Nr
X
@ 2 c00a m T
þ
r Kp DrKq :
@Kp @Kq
p¼1 q¼1
þ2
ðE6Þ
Substituting (E1) and (E6) into (37), when written for a
00
, and recalling (11), leads to
single species, cam
Nr
X
@Kp
rm
@ca00m
¼
V rKp þ r DrKp
n
@Kp
@t
p¼1
þ
Nu X
Nu
X
@ 2 ca00m
i¼1 j¼1
þ
@ui @uj
rT ui Druj þ 2
Nu X
Nr
X
@ 2 ca00m T
r ui DrKq
@ui @Kq
i¼1 q¼1
Nr X
Nr
X
@ 2 ca00m T
r Kp DrKq :
@Kp @Kq
p¼1 q¼1
ðE7Þ
Finally, the rate of reaction of the entire system is
Nr
X
@Kp
1
@c00am
V rKp þ r DrKp
r ¼
@Kp
@t
n
p¼1
þ
Nu X
Nu
Nu X
Nr
X
X
@ 2 c00am T
@ 2 c00am T
r ui Druj þ 2
r ui DrKq
@ui @uj
@ui @Kq
i¼1 j¼1
i¼1 q¼1
þ
Nr X
Nr
X
@ 2 c00am T
r Kp DrKq :
@Kp @Kq
p¼1 q¼1
ðE8Þ
[77] Acknowledgments. The second and third authors acknowledge
financial support by the European Union and Enresa through project
FUNMIG. The authors are thankful to three anonymous reviewers for their
comments on a preliminary version of this paper.
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276
Stream-Stage Response Tests and Their Joint
Interpretation with Pumping Tests
by Tobias S. Rötting1,2, Jesus Carrera2, Jose Bolzicco2, and Josep Maria Salvany2
Abstract
Hydraulic head response to stream-stage variations can be used to explore the hydraulic properties of streamaquifer systems at a relatively large scale. These stream-stage response tests, also called flooding tests, are typically interpreted using one- or two-dimensional models that assume flow perpendicular to the river. Therefore,
they cannot be applied to systems that are both horizontally and vertically heterogeneous. In this work, we use the
geostatistical inverse problem to jointly interpret data from stream-stage response and pumping tests. The latter
tests provide flow data (which are needed to resolve aquifer diffusivity into transmissivity and storage coefficient)
and may supply supplementary small-scale information. Here, we summarize the methodology for the design,
execution, and joint numerical interpretation of these tests. Application to the Aznalcóllar case study allows us to
demonstrate that the proposed methodology may help in responding to questions such as the continuity of aquitards, the role and continuity of highly permeable paleochannels, or the time evolution of stream-aquifer interaction. These results expand the applicability and scope of stream-stage response tests.
Introduction
and Singh 2003). Some authors have also interpreted the
tests in a plane perpendicular to the stream (Loeltz and
Leake 1983; Carrera and Neuman 1986c; Moench and
Barlow 2000) to account for two-dimensional (2D) heterogeneity.
However, these interpretation methods suffer from
three limitations. (1) They assume that flow is perpendicular to the river, thus ignoring heterogeneity (irregular
aquifer geometry, paleochannels, silt layers, etc.).
(2) They use only head data, which only reveal information about aquifer diffusivity (Carrera and Neuman
1986b). Accurate flow rate data, which are difficult to
obtain during high-flow events, are needed to resolve
aquifer diffusivity into transmissivity and storage coefficient. (3) They do not allow direct incorporation of other
types of hydraulic data (e.g., cross-hole pumping tests).
These additional tests can yield the aforementioned flow
rate data and supplementary small-scale information.
They also create additional flow conditions to better identify the transmissivity distribution (‘‘multi-directional
aquifer stimulation’’: Snodgrass and Kitanidis 1998;
‘‘hydraulic tomography’’: Yeh and Liu 2000, Liu et al.
2002). Joint interpretation of different tests or of different
data types provides an integrated picture of the system
and can narrow the range of likely parameter sets (e.g.,
Naturally occurring or controlled stream-stage variations have been used successfully to study large-scale
hydraulic properties of fluvial aquifers (e.g., Pinder et al.
1969; Grubb and Zehner 1973; Loeltz and Leake 1983;
Carrera and Neuman 1986c; Reynolds 1987; Sophocleous
1991; Tabidian et al. 1992; Genereux and Guardiario
1998; Bolster et al. 2001). Aquifer reaction to tidal fluctuations has also been employed to calculate aquifer parameters (e.g., Erskine 1991; Jha et al. 2003). A large number
of one-dimensional (1D) and mostly analytical solutions
are available for interpreting the tests (e.g., Ferris 1951;
Rorabaugh 1960; Rowe 1960; Hantush 1961; Cooper and
Rorabaugh 1963; Pinder et al. 1969; Hornberger et al.
1970; Singh and Sagar 1977; Onder 1997; Mishra and
Jain 1999; Zlotnik and Huang 1999; Singh 2003; Swamee
1Corresponding author: Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia, 08028
Barcelona, Spain; [email protected]
2Department of Geotechnical Engineering and Geosciences,
Technical University of Catalonia, 08028 Barcelona, Spain
Received October 2004, accepted July 2005.
Copyright ª 2005 National Ground Water Association.
doi: 10.1111/j.1745-6584.2005.00138.x
277
Carrera and Neuman 1986c; Anderman et al. 1996; Weiss
and Smith 1998; Saiers et al. 2004). This can be best achieved by automatic calibration of a numerical aquifer
model (Poeter and Hill 1997; Hill et al. 1998; Carrera et
al. 2005).
When data from several tests are interpreted jointly,
some parameters may change between tests (e.g., severe
floods may erode streambed sediments, thereby altering
stream-aquifer interaction). The flexibility of a numerical
model should allow accounting for these changes.
The aim of this article is to summarize a methodology
for the design and execution of stream-stage response
tests and their joint interpretation with pumping tests. A
quasi–three dimensional (3D) geostatistical inversion
code is used for parameter estimation. The methodology
is illustrated through application to the Aznalcóllar case
study. Here, an alluvial aquifer became contaminated by
heavy metal–laden acidic water from a mine tailings
spill. Pumping tests did not suffice to characterize the
aquifer on a sufficiently large scale, so that a number of
questions were left unanswered regarding (1) the continuity of a silt layer that was found while building a permeable reactive barrier (PRB) across the floodplain; (2)
the role and extent of deep gravel channels; (3) the time
evolution of the stream-aquifer connection; and (4) the
impacts of the PRB construction. A stream-stage response test was performed to answer these questions.
approximated for each observation well as (e.g., Crank
1975)
teq ’
SL2
T
ð1Þ
SL2
ð2Þ
T
where L is the distance between well and stream and T
and S are estimates of aquifer transmissivity and storage
coefficient, respectively. Ideally, test duration should
be longer than the teq of all the wells involved in the test.
Estimates of T and S can be obtained from pumping tests
or from textural data and geology. However, room should
be left for surprises as deep observation wells may
respond faster than derived from Equations 1 and 2
(Carrera and Neuman 1986c) and T may grow with scale
(e.g., Sanchez-Vila et al. 1996).
The network of piezometers should cover the whole
area of interest. Well screens should be short wherever
vertical variations in aquifer parameters or responses
are expected. All relevant features (e.g., abrupt lateral
changes in aquifer characteristics, formation boundaries,
layers) should be monitored. Preliminary information on
the approximate position of such boundaries may be obtained from standard hydrogeological exploration methods (boreholes, trenches, geophysical methods, hydraulic
tests, etc.) and from the observation and qualitative interpretation of piezometric surfaces under different flow
conditions (especially aquifer response to previous
stream-stage variations, i.e., flood events, if such data are
available).
The river stage should be measured at several points,
especially if stream width or section changes along the
test reach. In general, the propagation of the hydraulic
pulse is much faster in the channel than in the aquifer.
However, if the test reach is long, upstream wells close to
the river will have equilibrated by the time the pulse reaches the downstream end of the reach. Therefore, it is
necessary to measure stream stage along the river. Moreover, actual stage variations are sensitive to river width.
Therefore, stage measurement points should be located
so as to make sure that river width and section variability
are properly sampled.
The test may be performed by recording the response
to a naturally occurring flood or to an artificial variation
in stream stage (e.g., by opening or closing the gates of
a dam). However, Equation 2 implies that measurements
may have to be made at short intervals. Continuous
recording of heads at such intervals may saturate the
recording devices. Therefore, if possible, an artificial
stream-stage variation should be preferred. In this way,
not only the heads but also other parameters (e.g., concentrations or temperature) may be monitored. Here, we
will deal only with head measurements.
Test data are best interpreted in terms of head variations (rather than actual values). Head variations are
defined as changes in head caused by the stream-stage
response test compared to the natural evolution of heads
in the aquifer system, similar to drawdowns in pumping
tests. River-stage variations are defined analogously. The
tr ’ 0:1
Methodology
Stream-Stage Response Test Design and Execution
A stream-stage response test is a hydraulic test that
consists of observing the head response of an alluvial
aquifer to a controlled change of the water level in
a stream or channel. This response is measured in piezometers and can be used to determine aquifer characteristics and parameters. As the source of stress is a line and
not just a point, the area of influence is much greater than
that of pumping tests. The aquifer can be stressed either
by raising or by lowering the water level in the stream.
The methods presented here can be applied to both kinds
of situations.
The three key parameters in the design of a streamstage response test are (1) the magnitude of water level
change in the stream; (2) the test duration; and (3) the
measurement intervals. They are discussed subsequently.
Stream stage should change as much as possible to
maximize head response, but the water level should
remain within the channel (i.e., should not inundate the
floodplain) to simplify interpretation. The water level
change depends on river channel characteristics and flow
rate, which can be specified if the stream-stage variation
is generated by opening the gates of a dam.
Appropriate test duration and measurement intervals
depend on aquifer parameters. Test duration should be
long enough for most wells to react significantly to the
stage change (i.e., longer than equilibration time, teq).
Measurement intervals should be short enough to define
the transient part of the test (i.e., shorter than the well
reaction time, tr). For planning purposes, teq and tr can be
278
advantages of this procedure are explained subsequently.
Measurements must start prior to the test in order to be
able to filter out natural trends of head variability.
It may be advisable to use automatic recording devices in piezometers and river-stage measuring points, particularly if many points have to be monitored at short
intervals.
Aquifer heads and stream stage are expressed in
terms of variation with respect to the natural evolution of
the system. A hydraulic test can be considered as a signal
that is added on the natural evolution of the studied system. Therefore, when the test data are expressed in terms
of variations, the model only needs to simulate the
changes induced by the test (in our case, pumping rates
or variations in river stage and their effect on aquifer
heads) but not, for example, the natural base flow in the
aquifer. This simplifies boundary and initial conditions.
Only the boundary conditions that are driving variables
of the test (pumping rates, river-stage variations) have to
be expressed as time functions. All other boundary conditions of parameters that are not changed by the test are
homogeneous (zero values). Initial aquifer head variations and stream-stage variations are also zero because by
definition, before the start of the test, aquifer heads and
steam stages are equal to the ‘‘natural’’ values of the system. Additionally, systematic measurement bias (e.g., due
to erroneous reference levels) is eliminated by working
with head variations.
Numerical Interpretation
Emphasis is placed on estimation of spatially varying
parameters. Otherwise (i.e., homogeneous medium),
a model perpendicular to the river should suffice. Moreover, many conceptual models need to be tested. Therefore, the use of geostatistical inversion methods is
recommended (Meier et al. 2001). This reduces the effort
devoted to calibration and allows the modeler to concentrate on conceptual issues. Here, we used the quasi-3D
finite-element code TRANSIN II (Medina et al. 1995).
This code calculates both direct and inverse flow and
transport problems in a stationary or transient regime.
The inverse problem is solved by minimizing a maximum
likelihood criterion (Carrera and Neumann 1986a), which
includes prior information on aquifer parameters. For
a geostatistical interpretation, it can include the spatial
correlation structure (covariance matrix) of the hydraulic
conductivity of the aquifer system. The code also computes several model identification criteria (Akaike, modified by Akaike, Hannan, and Kashyap; details are given
by Carrera and Neuman 1986a), which allow one to evaluate the quality of different model structures. According
to Carrera and Neumann (1986c), Kashyap’s (1982) is the
best criterion because it takes into account the goodness
of fit while penalizing overparameterization.
The modeling procedure consists of five steps: (1)
definition of the conceptual model; (2) definition of a geostatistical model for transmissivity; (3) discretization; (4)
nonlinear maximum likelihood estimation of the model
parameters; and (5) revision of the conceptual model and
repeated calibration until obtaining a good fit between
observed and modeled data. The general principles of this
methodology have been described by Meier et al. (2001).
The details specific to the joint interpretation of streamstage response and pumping tests with parameters varying
in time are described subsequently.
Geostatistical Model for Transmissivity
Transmissivity can be treated deterministically or
stochastically. In the latter case, one must specify the
covariance structure of transmissivity. In layered aquifers,
the different alluvial layers that were deposited at different times can often be assumed as statistically independent of each other. The same applies to any other
component of the aquifer (i.e., anthropogenic structures
such as PRB). This must be reflected in the covariance
matrix, where each formation is represented by a block
that describes intercorrelation within the formation or
structural unit, while the different blocks are not correlated with each other.
Discretization
In the quasi-3D approach implemented in TRANSIN
II, aquifer layers are represented by 2D triangles and rectangles, while aquitards are represented by 1D linear elements. If equilibration time in the aquitard with respect
to head changes in the aquifers is quick compared to the
duration of the hydraulic tests and time steps of the
model, then transient head variations within the aquitard
can be neglected and the aquitard can be represented by
linear elements. Otherwise, the linear elements have to be
refined by intermediate nodes (Chorley and Frind 1978).
In order to include both data of small-scale pumping tests
and those of larger-scale stream-stage response tests,
numerical accuracy demands that the mesh be highly
refined between sources of stress (rivers and pumping
wells) and observation points. However, the mesh can be
coarser far away from these zones.
Different events that occurred during a period where
aquifer parameters did not change can be interpreted
within the same mesh. To do so, the data sets and time
functions of each event have to be separated by
‘‘dummy’’ periods with zero stress that allow aquifer
head and stream-stage variations to recover to initial
levels (Figure 1).
Conceptual Model
To simulate stream-aquifer interaction, a ‘‘mixedtype’’ boundary is used, relating flow across the riverbed
to the difference between external head He (river stage)
and aquifer head h according to:
q ¼ aðHe 2hÞ
ð3Þ
where q is flux (m/d), i.e., flow rate per unit area, and a is
the leakance (d21), also called conductance, which represents the hydraulic conductivity of the riverbed divided
by its thickness. More complex approaches to describe
stream-aquifer interaction have been developed (for
a review, see Sophocleous 2002). Nevertheless, here we
apply this simple approach in order to keep the number of
parameters low.
279
Head variation (m)
Stage variation (m)
Pumping rate (m3/day)
0.8
whether computed parameters are reasonable) for preliminary screening of conceptual models. For quantitative
comparison, we use overall model fit, as measured by the
heads objective function (sum of squared weighed residuals; for details, see Carrera and Neuman 1986a and
Meier et al. 2001) and, better, Kashyap’s model structure
identification criterion (equation 30 in Carrera and
Neuman 1986a).
a)
0.4
0.0
-0.4
-0.8
0.8
b)
0.4
The Aznalcóllar Case Study
0.0
-0.4
Site and Geology
The aim of the Aznalcóllar case study was to characterize the River Agrio alluvial aquifer situated in the
Aznalcóllar mining district in the Seville province of
southwest Spain (Figure 2a). This aquifer had become
contaminated by heavy metal–laden acidic water from the
mine tailings spill that occurred on April 25, 1998 (e.g.,
Grimalt et al. 1999). As one of the remediation measures,
a PRB was built across the floodplain (Carrera et al.
2001). As the pit for barrier emplacement was excavated,
the aquifer turned out to be more complex than expected
(Salvany et al. 2004) in several aspects, which will be
explained subsequently.
The River Agrio is a tributary of the River Guadiamar (Figure 2a) and forms an alluvial valley made up of
four Quaternary terraces (Salvany et al. 2004):
-0.8
2000
c)
1000
0
0
5
Time (days)
10
1000
1005
Time (days) + 1000
Figure 1. Example of input processing in order to interpret
data from one pumping test and one stream-stage response
test in one mesh: (a) observed head variations in an observation well (‘‘S-5’’ of the case study), (b) stream-stage variation, (c) pumping rate of a pumping well (‘‘S-3’’ of the case
study).
d
Calibration
Calibration is best performed automatically to avoid
the need for performing a large number of model runs to
attain a manual fit by trial and error. In essence, one has
to specify the covariance matrix of all the data. Here, we
assume head errors to be independent, so that one only
needs to define their standard deviation. As for the model
parameters, the covariance matrices of transmissivity are
derived from the geostatistical model. All other parameters are assumed independent, so that one only needs to
define their standard deviation.
All model parameters (aquifer transmissivities, aquitard hydraulic conductivities, storage coefficients, and
leakance) are calibrated simultaneously using the data of
all the tests. As a result, the problem may be ill posed,
resulting in instability, high uncertainty, and poor convergence. To deal with these problems, one may choose to
fix some parameters or to artificially increase the weight
assigned to prior estimates. Here, we propose the
approach of Carrera and Neuman (1986c), which consists
of initially assigning a high weight to prior information
in order to facilitate convergence. In successive runs,
prior information weight is lowered stepwise in order
to give the model more freedom in adjusting parameter
values.
When different model configurations are tested in
order to explore the structure of the stream-aquifer system, they can be compared using a number of criteria
(Carrera et al. 1993). We use residual analysis (examining
the fit between measured and computed heads at every
observation point) and parameter assessment (evaluating
d
d
d
The upper terrace (T3) is hydraulically separated from the
other terraces by the bedrock and will therefore not be
considered in this study.
The intermediate terrace (T2) forms a wide flat area below
the hills. It is composed of three layers (Figure 3): a lower
layer of coarse gravels and sands, an intermediate layer
of silts containing subordinate levels of sands and small
gravels, and an upper layer of sandy gravels, which always
lies above the water table and will not be considered hereinafter. The deepest gravels form a lower paleochannel
that flows obliquely to the surface terrace and river trends.
The lower terrace (T1) is a single deposit of sandy gravels
a few meters below the T2 terrace. It forms an upper paleochannel that follows the river trend and occasionally cuts
the T2 deposits and even the bedrock.
The current floodplain (T0), 1 to 1.5 m below the T1
terrace, is an erosive terrace that cuts through T1.
The Miocene Blue Marls Formation forms the impervious base of the aquifer system.
To sum up, in the vicinity of the PRB (Figure 3,
cross section B-B9), the hydrogeologic system consists
of the rivers Agrio and Guadiamar and of three layers: a
confined aquifer formed by the lower T2 deposits, an
aquitard consisting of the T2 intermediate silts, and
a phreatic aquifer formed by T1 deposits. In this study,
we suppose that the phreatic aquifer is cut off ~400 m upand downstream of the PRB because the silt layer outcrops right on top of the Blue Marls in the channel of the
River Agrio (crosshatched areas in Figure 2b). In the
following, the T1 deposits will be referred to as ‘‘upper
gravels’’ and the T2 lower gravels as ‘‘lower-layer
gravels’’ or ‘‘paleochannel.’’
280
Figure 2. (a) Geological map of the studied site showing the River Agrio terraces and their bedrock. The dotted-line rectangle
shows the extent of the numerical model. (b) Location of observation wells used during the flooding test (filled circles) and
of other wells and trenches (open circles). The crosshatched areas show where the Blue Marls appear in the bed of the River
Agrio (a and b adapted from Salvany et al. 2004). (c) Observation wells around the permeable reactive barrier (‘‘PRB’’). Suffixes ‘‘-u’’ and ‘‘-l’’ stand for wells screened only in the upper or lower gravel layer, respectively. The stars represent piezometer nests inside the three reactive PRB modules (‘‘RMB, CB’’, and ‘‘LMB’’ stand for right margin, central, and left margin
barrier module).
The Aznalcóllar PRB spans the floodplain 2 km
downstream of the tailings pond. It consists of three reactive modules (Carrera et al. 2001). Each module is 30 m
wide, 1.4 m thick, and penetrates into the Blue Marls
(on average 6 m deep). The three modules are separated by
two 10-m wide nonreactive sections of low permeability.
Uncertainties remain about the degree of hydraulic
connection between the two aquifers and about the lateral
extension of the aquifer layers. It is not clear if the silt
layer is a continuous feature in the area where the upper
gravels overlie the lower gravels. Likewise, the course of
the T2 paleochannel is only known in the area where
wells have been drilled.
The hydraulic connection of the two aquifer layers
to the River Agrio is also unclear: remediation of the
toxic spill led to an artificial river channel, which was
merely a few meters wide and very shallow. Therefore,
it is believed that it was only connected to the upper
water table aquifer. However, between December 2000
and January 2001, severe floods reshaped the floodplain
morphology, increased the width of the channel to
>10 m, and deeply eroded the riverbed. This may have
enhanced stream-aquifer interaction between River Agrio
and the upper aquifer by changing the hydraulic conductivity of the streambed sediments. The lower aquifer
may have also become connected to the upper aquifer
and to the river by partial erosion of the silt layer.
As all these features might severely affect PRB efficiency, a number of questions about this hydrogeologic
system were raised, regarding (1) the continuity of the
silt layer; (2) the geometry of the lower gravel layer and
its hydraulic connection to the river; (3) the evolution of
stream-aquifer interaction; and (4) the impact of PRB
Figure 3. Cross sections of River Agrio deposits (adapted
from Salvany et al. 2004). See Figure 2b for location.
281
construction on the flow system. A stream-stage response
test promised to be an adequate instrument to study these
issues.
River stage (m a.s.l.)
35.2
Stream-Stage Response and Pumping Tests
Prior to stream-stage response test execution, reaction
and equilibration time were calculated for several wells in
the upper and lower gravel layers using Equations 1 and 2
and prior estimates of T and S as obtained from pumping
tests (see four examples in Table 1). Reaction and equilibration times are quite short, especially for wells in the
confined layer. Therefore, automatic sensors were
installed in 14 wells and at one point in the river. Due to
technical limitations of the data-logging system, the measuring intervals used during the test were somewhat larger
than the lowest calculated reaction times (5 min in five
wells connected to the water table aquifer, 1 min in all
other wells). All other points were measured manually.
The Aznalcóllar drinking water dam was opened at
12:00 h on June 5, 2002 and closed at 12:00 h on June 6,
2002. The flow rate was 11 m3/s, a value that had produced
a large rise in river stage without causing excessive floodplain inundation in previous high-flow events. Water level
rose by up to 64 cm near the PRB (Figure 4) and up to 97
cm in a narrow part of the riverbed near well A-1bis (not
shown). The evolution of water levels was monitored in 54
wells (Figures 2b and 2c) and at 11 river-stage measurement points. Manual measurements were continued until
the evening of June 7; the automatic sensors kept on measuring in 15-min intervals until July 22. Almost all the wells
in the floodplain and beneath the T2 terrace responded very
quickly (within <1 h) to changes in the river stage.
Prior to the stream-stage response test, three series of
cross-hole pumping tests had been performed at the PRB
site. Wells S-1, S-3, and S-6 (cf. Figure 2c) were pumped
sequentially in each series, and drawdowns were measured in the surrounding wells. Two such series were performed prior to PRB construction (in January and March
2000). They differ somewhat in pumping rates and duration. One test series (in March 2001) was performed
after PRB construction and after the riverbed had been
affected by the winter 2000/2001 floods. During this last
test series, the stream stage of the River Agrio varied
twice due to changes in the water release rate of the Aznalcóllar drinking water dam, so this undesired variation
had to be taken into account for the interpretation of the
test (cf. Figure 1a).
35.0
34.8
34.6
34.4
5 jun 02 6 jun 02 7 jun 02 8 jun 02 9 jun 02 10 jun 02 11 jun 02
0:00
0:00
0:00
0:00
0:00
0:00
0:00
Date + time
Figure 4. River Agrio stage close to the PRB during the
stream-stage response test.
Numerical Interpretation
The numerical model simultaneously interprets the
stream-stage response test and the three series of crosshole pumping tests.
Conceptual Model
The aquifer system consists of two aquifer layers
(Figures 3 and 5a) separated by a silt aquitard as
described in Site and Geology. The lower confined aquifer consists of the lower gravels in the area of the T2 paleochannel and of T2 intermediate silts in the remaining
area between the margins of the alluvial valley. The
course of the paleochannel beyond the area where wells
had been drilled was extrapolated parallel to the river
(Figure 5a). The model area extends up to the mining
compound upstream of the PRB (dotted-line rectangle in
Figure 1a). Downstream of the PRB, the model boundary
is formed by the River Guadiamar. All other boundaries
are assumed to be of prescribed flow type and will be
treated as no-flow because we are working with head variations (see Methodology).
In order to explore the structure of the aquifer system
and to test the benefits of the geostatistical approach, different model configurations were tested. After preliminary screening model runs, three conceptual models were
believed reasonable. They differ in leakance zone structure and treatment of transmissivity (geostatistical or
deterministic), as summarized in Table 2.
Model 1 contains four leakance zones (‘‘Upstream a,’’
‘‘Upper two-layer a,’’ ‘‘Downstream a,’’ ‘‘Guadiamar a’’;
cf. Figure 5b). It assumes that river-aquifer interaction
does not change in time. The rivers are hydraulically
Table 1
Estimated Reaction and Equilibration Time for Two Wells in the Upper Gravel Layer and for
Two Wells in the Lower Gravel Layer Using Prior Estimates of T and S
Upper Gravel Layer (water table aquifer)
Lower Gravel Layer (confined aquifer)
300 m2/d
0.2
3000 m2/d
0.0001
T (estimated)
S (estimated)
Well
S-3-u
S-6-u
S-23
S-25
Distance to river (L)
Reaction time (tr)
Equilibration time (teq)
7m
5 min
50 min
50 m
4h
40 h
150 m
0.1 min
1 min
400 m
0.8 min
8 min
282
geostatistical inversion, transmissivity is treated deterministically: the transmissivity fields of the upper aquifer
layer, the lower aquifer paleochannel, and the three barrier modules are replaced by single zones, resulting in
a model with only 11 independent transmissivity zones
that are all statistically independent from each other.
As no quantitative data on leakances were available,
prior estimates and initial values of all leakances of the
three model configurations were set to an intermediate
value of 1 d21. The variances were set to 5 orders of magnitude (r2 log a ¼ 5) in order to allow the models to
assign both very good and very poor hydraulic connections to the different leakance zones.
Discretization
The two aquifer layers are represented by 2D
triangles and squares (Figures 6a through 6c). The aquitard is represented by 1D linear elements without intermediate nodes because preliminary evaluation of the
equilibration time in the silt aquitard indicated that
equilibration is fast. Intermediate nodes are used only at
wells with continuous screens that connect both aquifers.
These nodes distribute the total pumping rate between
the aquifers.
The total model domain is 5120 by 1536 m. The
largest 2D elements have a side length of 256 m, the
smallest elements, 0.7 m. Close to the PRB, the River
Agrio is represented as an area having the real width of
the river. Further away it is represented as a line. Streamaquifer interaction is represented using Equation 3, so
that total inflow at each node is equal to the flux, given by
Equation 3, times the nodal area. The latter is taken as the
length of river represented by the node times a fixed
width of 12 m in the portions where the river is represented as a line.
Two unconnected meshes are used for simulating the
four tests: one (‘‘2000 mesh’’) representing conditions
prior to the PRB construction and the winter 2000/2001
floods, and one (‘‘2001/2002 mesh’’) representing conditions after both events. The latter is a bit more refined in
order to adequately represent the barrier and the additional wells. Two tests series are interpreted in each
mesh. The first test series is separated from the second
series by a 1000-d dummy period to allow heads to
recover to zero (cf. Figure 1).
Time discretization is variable and was adjusted until
further refinement in time did not change the solution.
Figure 5. (a) Transmissivity zones (the subdivisions of the
geostatistical transmissivity fields are not shown) and (b) leakance zones of the conceptual and numerical model. Only
the downstream half of the model area is shown; zones continue homogeneously upstream.
connected only to the uppermost layer in each segment.
Therefore, in the two-layer zone, the River Agrio has no
hydraulic connection to the lower-layer gravels (‘‘Lower
two-layer a’’ is not assigned).
Model 2 acknowledges that the 2000/2001 floods
may have changed the aquifer connection with the River
Agrio. This is represented by assigning different leakances to each segment of the River Agrio in the 2000 and
2001/2002 tests. Additionally, the River Agrio may also
be connected to the lower layer in the two-layer zone
(Lower two-layer a is also assigned in both meshes).
River Guadiamar is assumed to remain unchanged by the
floods. This totals nine leakances (four leakances for
River Agrio in 2000, four leakances for River Agrio in
2001/2002, and one leakance for River Guadiamar, which
is identical in both periods).
Model 3 has the same structure of leakance zones as
model 2, but in order to examine what is gained by
Table 2
Overview of Tested Model Configurations: Treatment of Transmissivity and Number of Estimated
Parameters in the Three Model Configurations
Treatment of transmissivity
Transmissivity zones
Storage zones
Leakance zones
Sum of estimated parameters
Model 1
Model 2
Model 3
Geostatistical
102
4
4
110
Geostatistical
102
4
9
115
Deterministic
11
4
9
24
283
was obtained from boreholes and trenches. The silt layer
was assumed to be continuous, with a constant thickness
of 1 m. From these point values, the transmissivity values
were interpolated by block kriging using exponential
variograms. The kriging parameters (Table 4) were estimated assuming longer ranges parallel to paleochannels
than laterally and choosing relatively short ranges to
avoid prior information to force too much continuity.
The transmissivity field of the upper layer has 35
zones, that of the lower aquifer paleochannel has 49
zones, and the lower layer silts are divided into three
independent zones. The silt layer is represented by a single zone. For the PRB, each reactive module is divided
into four correlated zones, plus two zones for the separating low-permeability modules. This totals 102 transmissivity zones. Each aquifer (cf. Figure 5a) is divided into
transmissivity zones that are correlated according to the
geostatistical model (cf. Table 4). The size of the zones is
adapted according to distance to test zones and density of
wells, with zones being smaller where the density of
observation points is higher.
Calibration
Standard deviations of head data of the different
wells (required for defining the relative weights of head
data) were 5 cm for the six fully penetrating wells close
to the barrier where the mesh and T zones were refined
and 15 cm elsewhere because of increased discretization
errors. Pumping well drawdowns were excluded from calibration to prevent discretization and in-well effects such
as skin from affecting the aquifer calibration.
Figure 6. Finite-element mesh in plane view, showing only
the 2001/2002 mesh. (a) Total modeled area, (b) zoom of
the zone with two layers. In a and b, the two-layer zone with
the smaller upper layer is shaded in gray. (c) Zoom of the
surroundings of the PRB.
Results
Estimated parameters of the three model configurations are represented in Tables 5 and 6. Calibrated heads
(Figures 7 and 8) of most wells are shown for model 2,
which achieves the smallest residuals (Table 5). Fits are
good for most wells during the stream-stage response test
(Figures 7 and 8d), except at the end of the recovery
period. Fits for the pumping test data are reasonable
(Figures 8a through 8c). Since fits are quite similar in the
Geostatistical Model for Transmissivity
As stated in the Methodology, the different alluvial
layers and each module of the PRB are statistically independent. Prior information about aquifer transmissivities
was calculated as saturated layer thickness times an initial estimate of hydraulic conductivity (Table 3) that was
obtained from grain size analyses. Saturated thickness
Table 3
Prior Estimates and Variances of Hydraulic Conductivity (m/d) and Storage Coefficient (2) or
Storativity (m21, for the silt layer) of the Different Model Zones
Upper gravel layer
Silt layer
Lower-layer gravels
Lower-layer silts
Right margin barrier module
Left margin barrier module
Central barrier module
Nonreactive sections
1Variance
Prior Estimate
of Hydraulic
Conductivity, K (m/d)
Variance or
Sill r2 of Log T or
Log K
200
0.05
1000
0.1
0.5
1
1
0.2
11
2
11
1
41
41
41
1
Prior Estimate of
Storage Coefficient, S (2), or
Storativity, Ss (m21)
for model 3; sill for models 1 and 2, where covariances are determined by block kriging (see text and Table 4).
284
0.2
5 3 1023
1023
1023
0.2
0.2
0.2
0.2
Variance r2
of Log S
or Log Ss
1
1
1
1
1
1
1
1
Table 4
Geostatistical Parameters Used for Block Kriging of the Different Transmissivity Regions of the Model
Upper Layer
Number of zones
Direction of anisotropy (clockwise relative
to the river axis near the PRB)
Range in principal direction (m)
Range in secondary direction (m)
Sill (r2log T)
25
0
Lower-Layer
Gravels (Paleochannel)
49
45
100
30
1
300
100
1
three PRB modules, only results from the central barrier
module are shown. The residuals of model 1 are similar to
those of model 2 in most wells as expressed by the heads
objective function (Table 5). The major difference can be
found in well S-2 during the second pumping test before
PRB construction (Figure 9), where model 1 performs
much worse than model 2, and model 3 actually achieves
the best fit of all models. However, residuals of model 3
are much larger in wells S-3, S-5, S-17 (not shown), and
A-1bis (Figure 10). This causes the heads objective function to be more than twice as high as those of the other
models. Kashyap’s model identification criterion is best
(most negative) for model 2 (Table 5). Despite the smaller
number of calibrated parameters (cf. Table 2), models 1
and 3 achieve poorer scores.
Reactive Modules
of the PRB
4 zones/module
Isotropic
10
10
4
Calibrated upper-layer transmissivities are similar for
models 1 and 2 (Figures 11b and 11c, right-hand side)
and generally higher than prior estimates (Figure 11a),
except at the upstream end. In the lower-layer paleochannel, the calibrated transmissivities of models 1 and 2
(Figures 11b and 11c, left-hand side) tend to be lower
than the prior values, especially close to the southern limits of the displayed area. Toward the northern limits,
model 1 assigns transmissivities that are higher than prior
estimates, while model 2 assigns lower values. Model 3
calibrates lower transmissivities than the geometric
means of models 1 and 2 (Table 5) for both the upper
gravel layer and the lower-layer paleochannel. The
hydraulic conductivities of the PRB modules in the three
models remain similar to their prior estimates (Table 5),
Figure 7. Measured (dots) and calculated (continuous line) head variations of model 2 for the stream-stage response test. For
clarity, only selected wells and part of the measured data points are shown.
285
Table 5
Results of the Model Evaluation Criteria, Prior Estimates and Calibrated Values of Transmissivities (T) of
Upper Gravel Layer and Lower-Layer Gravels, Hydraulic Conductivities (K) of PRB Modules and
Silt Layer, and Storage Coefficients for the Different Model Configurations (for models 1 and 2
geometric mean of each transmissivity field or PRB module)
Calibrated Values
Heads objective function
Kashyap (1982)
Transmissivities and hydraulic
conductivities
Upper gravel layer T (m2/d)
Lower-layer gravels T (m2/d)
Right margin barrier K (m/d)
Central barrier K (m/d)
Left margin barrier K (m/d)
Silt layer K (m/d)
Storage coefficients (2)
Upper gravel layer
Silt layer
Lower-layer gravels
Lower-layer silts
1For
Prior Estimate
Model 1
Model 2
Model 3
—
—
224.0
25093
209.5
25209
535.0
24643
166.3 ¼ 102.22
1718 ¼ 103.24
0.50
1.0
1.0
0.20
492 ¼ 102.69
1634 ¼ 103.21
0.47
11
1.5
0.066
416 ¼ 102.62
1451 ¼ 103.16
0.46
1.1
2.2
0.11
266 ¼ 102.42
1399 ¼ 103.15
0.0066
0.84
1.2
0.19
0.20
0.005
0.001
0.001
0.23
0.034
0.0004
0.0007
0.26
0.049
0.0059
0.0008
0.20
0.084
0.0011
0.0009
the silt layer: storativity (m21)
Figure 8. Measured (dots) and calculated (continuous line) head variations of model 2 in the wells of the central PRB module
for (a) the first and (b) the second pumping test before PRB construction, (c) the pumping test after PRB construction, and
(d) the stream-stage response test (only wells not shown in Figure 7). For clarity, not all measured data points are shown.
286
a in both meshes. The biggest difference between the two
models is found in Guadiamar a, which is lower than the
prior estimate in model 2 (similar to model 1) but higher
in model 3.
Head variation (m)
0.0
-0.1
Discussion
Measured data
-0.2
Model 1
Silt Layer Continuity
The hydraulic conductivity of the silt layer is much
lower than that of the upper and lower gravel layers in
the three models. This shows that the models need to separate these layers hydraulically in order to reproduce the
observed data. In the pumping tests (especially before the
floods), the presence of the silt layer enables hydraulic
pulses to pass underneath the river through the lower
gravel layer, while the very good connection between the
river and the upper gravel layer leads to a fast recovery
after the pumping has stopped or the stream-stage peak
has passed. Therefore, the silt layer seems to be a continuous feature.
Model 2
Model 3
-0.3
0
1
2
3
4
Time (days)
Figure 9. Measured and calculated head variations of the
different model configurations in well ‘‘S-2’’ for the second
pumping test before PRB construction.
with two exceptions: the hydraulic conductivity of the
right margin barrier module of model 3 is 2 orders of
magnitude lower than the prior estimate, and in the central barrier module of model 1, the hydraulic conductivities of the two K zones next to the river increase to 214
and 40 m/d, raising the geometric mean of this module to
11 m/d. In all the three models, the silt layer hydraulic
conductivity is slightly lower than the prior estimate and
the silt layer storage coefficient is 1 order of magnitude
higher, while the remaining storage coefficients are similar to the prior values (Table 5).
Many calibrated leakances (Table 6, cf. Figure 5b)
change considerably with respect to their prior estimates.
In model 1, the calibrated leakance for the connection of
River Agrio with the upper layer (Upper two-layer a)
rises 6 orders of magnitude, while Guadiamar a drops by
1 order of magnitude. In models 2 and 3, the general patterns in the leakance zones are similar, even though actual
values differ. In the two-layer zone, the highest leakance
is assigned to the upper layer in 2001/2002 and a moderate value is assigned to the lower layer during the same
period, while both values are lower in 2000. Upstream a
in the 2001/2002 mesh is very low, while it remains
almost unchanged in the 2000 mesh, as does Downstream
Stream-Aquifer Interaction
Models 2 and 3 consistently display a marked difference between the leakances of the 2000 and the 2001/
2002 meshes. In 2001/2002, they assign higher leakances
than in 2000 to both layers of the two-layer zone. At least
in the 2001/2002 mesh of both models, the River Agrio
also has a moderate connection to the lower layer in this
area. Even model 1, which was not allowed to assign
a leakance to the lower-layer gravels within the two-layer
zone, yields high hydraulic conductivities in two sections
of the central PRB module, probably to hydraulically
connect the river to the lower layer.
This shows that the winter 2000/2001 floods changed
stream-aquifer interaction in this aquifer and that at least
since then, the River Agrio is hydraulically connected to
the lower aquifer close to the PRB. This is consistent
with the observation that the River Agrio deepened and
broadened its channel during the winter 2000/2001
floods. The river increased its hydraulic connection to the
upper layer and probably excavated the silt layer in some
parts, creating or enhancing the connection to the lower
layer. As a result, pumping during the test performed
after the winter 2000/2001 floods is barely noticed across
the river.
0.8
Head variation (m)
Measured data
Role and Geometry of the Lower Gravel Layer
Many wells at a considerable distance from the river
react quickly to the stream-stage variation. This implies
that (1) the gravels found beneath the western T2 terrace
are well connected to the River Agrio and that (2) a confining layer exists in this part of the aquifer (low storage
coefficients assigned by all models, cf. Table 5). This
explanation is consistent with the silt layer continuity and
river-aquifer interaction discussed previously.
As to paleochannel geometry, the proposed eastern
and western boundaries can satisfactorily explain the
observed data, especially the high contrast in reaction to
the stream-stage response test between wells that are only
Model 1
0.6
Model 2
Model 3
0.4
0.2
0.0
0
1
2
3
4
Time (days)
Figure 10. Measured and calculated head variations of the
different model configurations in well ‘‘A-1bis’’ for the
stream-stage response test.
287
Figure 11. Transmissivity fields (left-hand side: lower layer; right-hand side: upper layer) of the geostatistical models. For
the lower layer, only the central part is shown; transmissivity zones continue homogeneously up- and downstream. (a) Prior
estimate of models 1 and 2. (b) Calibrated values of model 1. (c) Calibrated values of model 2.
short distances apart (Figure 7, i.e., S-26, S-24, S-28 vs.
S-25, S-26 at the western border and S-5, S-6 vs. S-7 at
the eastern border). At the southern end of the paleochannel, both models 1 and 2 assign a low leakance to
River Guadiamar and lower paleochannel transmissivity. Model 3, which has to assign a homogeneous
transmissivity to the whole paleochannel, cannot selectively lower transmissivity close to Guadiamar River.
Instead, it even raises Guadiamar a, achieving fits in the
wells of the southern paleochannel (S-23, S-24, S-28; not
shown), which are similar to those of models 1 and 2.
This seems to contradict the results of the other models,
Table 6
Prior Estimates and Calibrated Leakances (d21) for the Different Model Configurations
Leakance Zone
Upstream a
Upper two-layer a
Lower two-layer a
Downstream a
Guadiamar a
Prior Estimates
All Models and Meshes
1.0
1.0
1.01
1.0
1.0
Model 1
Both Meshes
Model 2
2000 Mesh
0.87
121,200
na
1.0
0.098
1.2
7.6
0.014
1.0
na ¼ not assigned.
1Except model 1 (leakance zone not assigned).
288
2001/2002 Mesh
0.0033
129
7.8
0.99
0.088
Model 3
2000 Mesh
2001/2002 Mesh
0.97
0.39
0.67
1.0
0.00027
24
5.8
1.0
8.1
but presumably, model 3 uses River Guadiamar as an
injection zone (analogously to an image well) in order to
imitate the effect of the low-permeability zone assigned
by models 1 and 2. Therefore, we conclude that the paleochannel is only poorly connected to River Guadiamar.
This conclusion is consistent with independent observations not used during modeling and not shown here.
First, well
S-28 was not contaminated, as it should if the paleochannel had been connected to River Guadiamar. Second,
natural head gradients in the paleochannel point to the
River Agrio, supporting that this area is not well connected to River Guadiamar.
Nevertheless, it is less clear how well and where the
lower paleochannel is connected to River Agrio upstream
of the studied area. The situation is ambiguous in two aspects. First, it appears contrary to the aforementioned theory on stream-aquifer interaction that Upstream a is
higher before the winter 2000/2001 floods in models 2
and 3. This is probably only an artifact of the lack of data
in this zone during the pumping tests (no measurements
were taken in well A-1bis). Therefore, the models are not
very sensitive to this leakance in the 2000 mesh and
maintain its value close to the prior estimate. Second,
none of the three model configurations achieves a satisfactory fit for well A-1bis (Figure 11). All models
overestimate both peak height and peak arrival time.
Additional wells and river-stage measurement points during the stream-stage response test would have helped to
increase model sensitivity in this area.
Models 1 and 2 obtain almost equal fits for most
observation points. Model 2 uses somewhat more plausible parameter values that are consistent with independent
observations (changes in stream channel geometry due to
the winter 2000/2001 floods) so it appears more realistic,
but other models could be developed that can explain the
observed data equally well.
Conclusions
A methodology for the design, execution, and joint
numerical interpretation of stream-stage response tests
was summarized and successfully applied to a test performed in the River Agrio (southwest Spain). The objective of the test was to complement three sets of cross-hole
pumping tests in the characterization of the layered
stream-aquifer system. Two geostatistical models and one
‘‘conventional’’ model with homogeneous transmissivities
were tested in order to investigate the trade-off between
model complexity and achieved fit.
The results show that
d
d
Impact of PRB Construction
The PRB construction has created a low-permeability
feature in the aquifer. Observation wells now react much
less to pumping in wells on the opposite side of the PRB
than before PRB construction. Head variations in these
wells during the last pumping test series are due to
changes in river stage rather than to pumping (cf. Figures
6a and 6b and 8a through 8c).
d
d
The combination of pumping and stream-stage response
test data allows calibrating both transmissivities and storage coefficients. The pumping tests yield small-scale
information about the barrier and its surroundings and provide the flow data necessary to resolve diffusivity into T
and S, while the stream-stage response test gives information on a larger scale.
River-aquifer interaction can be simulated using leakances.
The geostatistical approach obtains better overall results
than the more conventional deterministic approach.
The numerical model used in the case study allowed us to
explore (1) aquitard continuity; (2) degrees and temporal
changes of stream-aquifer interaction; (3) the impact of the
construction of a PRB on the hydraulic system; and—to
some degree—(4) aquifer geometry.
In summary, the stream-stage response test provides
hydraulic information at a relatively large scale and is
relatively easy to perform. Therefore, its use is recommended. However, a thorough interpretation requires coupling to pumping tests or other types of flow data, which
can be achieved satisfactorily using geostatistical inversion codes.
Modeling Approach
The present numerical model allows us to simultaneously interpret pumping and stream-stage response test
data and to resolve aquifer diffusivities into transmissivities and storage coefficients. In our case study, the
stream-stage response test alone would not reveal information about PRB characteristics because ground water
flow during the stream-stage response test is parallel to
the barrier.
A large number of observation points were used in
the calibration. Therefore, large errors in single points
lead to relatively small changes of the heads objective
function. As a result, the fits have to be verified individually in all observation points in order to evaluate the conceptual model (analysis of residuals). This means that
automatic calibration helps to test alternative conceptual
models rapidly, but it does not eliminate the need for
thorough, and often tedious, qualitative analysis of results. This applies to any model with many observation
points, not only to the interpretation of stream-stage
response tests.
Acknowledgments
This study was funded by the EU PIRAMID project
(EVK1-1999-00061P) and by the Spanish CYCIT program (HID99-1147-C02). The authors wish to thank the
Hydrographical Confederation of the Guadalquivir and
C. Mediavilla from the Geological and Mining Institute
of Spain in Seville for making possible the execution of
the stream-stage response test, and J. Jodar and C. Knudby for their help in the initial design of the numerical
model. T. Rötting’s work was supported by the German
Academic Exchange Service DAAD (grant number D/01/
29764) and the Gottlieb-Daimler and Karl Benz Foundation (grant number 02-18/01). The authors also thank
289
Steffen Mehl, Marios Sophocleous, and one anonymous
reviewer for their constructive comments that helped to
improve the present paper.
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Dealing with spatial heterogeneity
Gh. de Marsily · F. Delay · J. Gonalvs · Ph. Renard ·
V. Teles · S. Violette
Abstract Heterogeneity can be dealt with by defining
homogeneous equivalent properties, known as averaging,
or by trying to describe the spatial variability of the rock
properties from geologic observations and local measurements. The techniques available for these descriptions
are mostly continuous Geostatistical models, or discontinuous facies models such as the Boolean, Indicator or
Gaussian-Threshold models and the Markov chain model.
These facies models are better suited to treating issues of
rock strata connectivity, e.g. buried high permeability
channels or low permeability barriers, which greatly affect flow and, above all, transport in aquifers. Genetic
models provide new ways to incorporate more geology
into the facies description, an approach that has been well
developed in the oil industry, but not enough in hydrogeology. The conclusion is that future work should be
focused on improving the facies models, comparing them,
and designing new in situ testing procedures (including
geophysics) that would help identify the facies geometry
and properties. A world-wide catalog of aquifer facies
geometry and properties, which could combine site genesis and description with methods used to assess the
system, would be of great value for practical applications.
Received: 17 April 2004 / Accepted: 15 December 2004
Published online: 25 February 2005
Springer-Verlag 2005
G. de Marsily ()) · J. Gonalvs · S. Violette
Universit Pierre et Marie Curie, Laboratoire de Gologie
Applique,
Paris VI, Case 105, 4 place Jussieu, 75252 Paris Cedex 05, France
e-mail: [email protected]
Tel.: +33-144-275126
Fax: +33-144-275125
F. Delay
Universit de Poitiers, Earth Sciences Building,
40 Avenue du Recteur Pineau, 86022 Poitiers cedex, France
P. Renard
University of Neuchatel, Centre of Hydrogeology,
Rue Emile Argand 11, 2007 Neuchatel, Switzerland
V. Teles
Laboratoire des Sciences du Climat et de l’Environnement,
UMR CEA-CNRS,
Orme des Merisiers, b
t. 709, 91191 Gif sur Yvette Cedex, France
Rsum On peut aborder le problme de l’htrognit
en s’efforant de dfinir une permabilit quivalente
homogne, par prise de moyenne, ou au contraire en dcrivant la variation dans l’espace des proprits des roches partir des observations gologiques et des mesures
locales. Les techniques disponibles pour une telle description sont soit continues, comme l’approche Gostatistique, soit discontinues, comme les modles de facis,
Boolens, ou bien par Indicatrices ou Gaussiennes
Seuilles, ou enfin Markoviens. Ces modles de facis
sont mieux capables de prendre en compte la connectivit
des strates gologiques, telles que les chenaux enfouis forte permabilit, ou au contraire les facis fins de barrires de permabilit, qui ont une influence importante
sur les coulement, et, plus encore, sur le transport. Les
modles gntiques rcemment apparus ont la capacit de
mieux incorporer dans les modles de facis les observations gologiques, chose courante dans l’industrie ptrolire, mais insuffisamment dveloppe en hydrogologie. On conclut que les travaux de recherche ultrieurs
devraient s’attacher dvelopper les modles de facis, les comparer entre eux, et mettre au point de nouvelles
mthodes d’essais in situ, comprenant les mthodes gophysiques, capables de reconnatre la gomtrie et les
proprits des facis. La constitution d’un catalogue
mondial de la gomtrie et des proprits des facis
aquifres, ainsi que des mthodes de reconnaissance utilises pour arriver la dtermination de ces systmes,
serait d’une grande importance pratique pour les applications.
Resumen La heterogeneidad se puede manejar por medio de la definicin de caractersticas homogneas equivalentes, conocidas como promediar o tratando de describir la variabilidad espacial de las caractersticas de las
rocas a partir de observaciones geolgicas y medidas locales. Las tcnicas disponibles para estas descripciones
son generalmente modelos geoestadsticos continuos o
modelos de facies discontinuos como los modelos Boolean, de Indicador o de umbral de Gaussian y el modelo
de cadena de Markow. Estos modelos de facies son mas
adecuados para tratar la conectvidad de estratos geolgicos (por ejemplo canales de alta permeabilidad enterrados
o barreras de baja permeabilidad que tienen efectos importantes sobre el flujo y especialmente sobre el transporte en los acuferos. Los modelos genticos ofrecen
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nuevas formas de incorporar ms geologa en las descripciones de facies, un enfoque que est bien desarollado
en la industria petrolera, pero insuficientemente en la
hidrogeologa. Se concluye que los trabajos futuros deberan estar ms enfocados en mejorar los modelos de
facies, en establecer comparaciones y en disear nuevos
procedimientos para pruebas in-situ (incuyendo la geofsica) que pueden ayudar a identificar la geometra de las
facies y sus propiedades. Un catlogo global de la geometra de las facies de los acuferos y sus caractersticas,
que podra combinar la gnesis de los sitios y descripciones de los mtodos utilizados para evaluar el sistema,
sera de gran valor para las aplicaciones prcticas.
Introduction
The word “Hydrogeology” can be understood as a combination of “hydraulics” and “geology.” Hydraulics is a
relatively simple science; we know, at least in principle,
the governing hydraulic equations and can solve them,
analytically or numerically, given the geometry of the
system, boundary conditions, etc. Geology is more complex: it refers not only to the description of what the
system looks like today, its properties in space, etc., but
also to the history of its formation, because geologists
have been trained to accept that one needs to understand
the succession of complex processes involved in the
creation and modification through time of the natural
objects that one is trying to describe. To understand this
complexity, geologists have a limited number of clues or
data, whose interpretation requires several assumptions
and may lead to alternative solutions. Geology also includes a set of disciplines whose contribution is needed to
study or describe the system: sedimentology, tectonics,
geophysics, geochemistry, age dating, etc. “Hydrogeology” is thus the science where the two are combined:
finding the solution of the flow (and transport) equations
in a complex, only partly identified, geologic system.
If the world were homogeneous, i.e. if the rock properties were constant in space, and/or easy to determine,
hydrogeology would be a rather boring job: solving wellknown equations in a perfectly identified medium. Fortunately, the world is heterogeneous, with highly nonconstant properties in space, and “dealing with heterogeneity” is what makes the work fascinating. Everybody
has heard of in situ experiments conducted in the field by
experimentalists trained to work in the lab.: the first thing
they do is to physically “homogenize” the site, by mechanically mixing the superficial horizons. “Otherwise it
is too complex and one cannot understand what is going
on,” they say, referring to Occams’razor. The first thought
that comes to mind is that they miss part of the fun by
ignoring spatial heterogeneity, the second one is that their
results are mostly useless, because the world is heterogeneous, and to understand “hydrogeology,” one has to
acknowledge this and deal with it.
Hydrogeology has been too much inclined towards
“hydraulics” and the solving of the flow equations, and
not enough towards “geology” and understanding/describing the rock structure, facies and properties in a geologically realistic manner, thus proposing “exact” solutions, but to poorly posed problems.
This article first provides a brief history of how hydrogeologists have dealt with heterogeneity so far, and
then an attempt is made to give a personal view of how
hydrogeologists may be dealing with it in the future.
Making predictions is quite difficult, and a French saying
adds “especially for the future”! These predictions are
very likely to be wrong, but it is hoped that these suggestions may trigger additional work, foster discussions,
generate controversy, and that, in the long term, better
methods will be developed to deal with heterogeneity.
Four important issues are not addressed here: (i) the
transition from Navier-Stokes’ equations to Darcy’s law,
which, at the pore scale, is the first scale of heterogeneity
of the velocity vector in natural media; (ii) the multiplicity of scales of heterogeneity, discussed in Carrera
(this issue), and also Noetinger (this issue); and (iii) the
multiple processes involved in flow and transport in
natural media (flow, transport, diffusion, biogeochemical
reactions, etc.). The focus will rather be on the methods
used to account for heterogeneity for any of these processes, to represent it and to model it. Finally, (iv) the
reasons why the Earth is heterogeneous will not be examined either, i.e.: sedimentation processes, the formation
of crystalline rock, tectonics, diagenesis, etc, because it is
assumed that all geologists are familiar with this.
Brief history of the methods used to deal
with heterogeneity
The early approach
The first practical field hydrogeologists dealt with heterogeneity by trying to locate “anomalies” and make use
of them. Water is not present everywhere in the ground;
only those who can locate the highly porous and permeable strata in the ground are able to decide on well locations or discover springs. For instance, Brunetto Latini
(1220–1295) wrote: “The earth is hollow inside, and full
of veins and caverns through which water, escaping from
the sea, comes and goes through the ground, seeping inside and outside, depending on where the veins lead it,
like the blood in man which spreads through his veins, in
order to irrigate the whole body upstream and downstream” (in Poire 1979). Famous hydrogeologists, like
Paramelle (1856), used their geologic knowledge to detect
heterogeneities and discover springs. Others use divining
rods... Darcy (1856) however, who was probably the first
to attempt to quantify flow, developed his theory for artificial homogeneous sand filters and did not have to deal
with heterogeneity. He produced however cross sections
of an “aquifer,” tapped by an artesian well, therefore
implicitly differentiating between an aquifer and an
“impervious layer.” It was Dupuit (1857, 1863) who first
tried to apply Darcy’s law to natural media, which were of
course heterogeneous. It is not clear if Dupuit was con-
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scious of the magnitude of heterogeneity. To extend
Darcy’s law to natural media, his approach was intuitive:
assume the existence of a “homogeneous” layer and use
Darcy’s law. He did not discuss in detail how he would
define the permeability of this homogeneous layer, but
since Darcy had developed an apparatus to measure the
permeability, he would most likely have taken samples on
which to measure it. Had he taken several samples, and
measured different values, nothing is there to give a clue
on how he would have averaged them, in order to define
an “equivalent” homogeneous layer. In fact, Dupuit was a
mathematician, he developed the steady-state flow equation, and proceeded to find analytical solutions in various
settings (like the classical Dupuit–Forscheimer solution
around a well, Dupuit 1863). Had he been aware of heterogeneities, his analytical tools would not have allowed
him to take them into account, and he would have ignored
them altogether. Theis (1935) was the first to produce a
simple tool to directly “estimate” the equivalent homogeneous property of an aquifer; this was the new well test
that automatically “averages” the local (variable) properties, such as permeability and storativity. Before Theis,
it was already possible to derive in situ values for permeability, by means of the steady-state solutions around a
well (Thiem 1906), or local injection tests (Lefranc, Lugeon, etc) and the local values were in general averaged
without any consideration for the type of average to use.
This quotation is by Morris Muskat (1949), speaking of
oil reservoirs: “it appears extremely unlikely that actual
underground strata will be of strictly uniform permeability over distances or areas associated with oil-producing reservoirs. However, as such random lateral variations as undoubtedly occur are literally impossible of
exact determination, they must be considered as averaged
to give an equivalent uniform-permeability stratum.
Moreover, even if the nature of these variations were
known, the difficulties of exact analytical treatment
would still force the use of averaging procedure and reduction to an equivalent permeability system.” He then
goes on to mention the Cardwell and Parsons (1945)
bounds for averaging, and treats some special cases where
simple geometric trends in permeability are known.
The simple distinction between an aquifer and a substratum or confining layers seems to have emerged relatively early. It is a first recognition of heterogeneity, but
with a very simple “layered cake” conceptual model,
which can be traced back to many early earth scientists,
like the British geologist Whitehurst in 1778, or the
French chemist Lavoisier in 1789 (who was interested in
geology). This conceptual model is deeply inscribed in
our discipline, and hard to abandon. It has been the cause
of the use of transmissivity rather than permeability. The
classification of the confining layers into aquitards and
aquicludes came much later, when leakage was studied,
and is probably due to Paul Witherspoon in the 1960s, see
e.g. Neuman and Witherspoon (1968).
The rules for averaging permeability in hydrogeology
were discussed in detail by Matheron (1967), although the
issue of averaging had already been addressed earlier in
other fields of physics, e.g. by Landau and Lifschitz
(1960), or by Cardwell and Parsons (1945). Matheron
dealt with the issue by considering permeability as a
magnitude varying randomly in space according to a
specified distribution. He showed that for two-dimensional parallel flow, in steady state, for a spatial distribution that is statistically invariant by a rotation of 90
and identical for the permeability k/E(k) and the resistivity k1/E(k1), the correct average is the geometric
mean. An invariance of the spatial distribution implies
that not only the univariate (pdf) and bivariate (covariance) statistics are identical, but the complete spatial
distribution function must remain the same for k and k1.
In particular, this applies to the multi-Gaussian log-normal distribution, which is often mentioned as correctly
representing the distribution of permeabilities in natural
media. Note that the physical reason why permeabilities
are distributed log-normally has never been explicitly
stated, and, on the contrary, examples are found where
this assumption is not correct. One has to remember,
however, that the flow equation is a diffusion equation
which can be written, for heterogeneous media:
@ ½lnðKÞ=@x @h=@x þ @ 2 h=@x2 ¼ . . .
Assuming that the parameter ln(K) has a Gaussian distribution is quite frequent for the same equation in other
fields of physics. Matheron also recalls the arithmetic and
harmonic bounds for averaging, which apply to all media,
already published by Cardwell and Parsons in 1945.
Surprisingly, he also states (without complete demonstration) that there is no averaging in radial flow in
steady-state, which is against the standard practice of
steady-state well testing; he argues that the average permeability can vary anywhere between the arithmetic and
harmonic means, depending on the value at the well, and
on the boundary conditions. It has however been shown
empirically that, in transient state at least, the geometric
mean is the long-term average that a well test produces
(see e.g. Meier et al. 1999). This issue of averaging will
be discussed again below, and its role and some of its
properties will be specified. It must be emphasized that
pumping tests are still a key research topic, since they
remain the most useful way to investigate the properties
of the underground. In other words, hydraulic tests in
boreholes still have a long life ahead; for instance, new
analytical solutions such as those by Barker (1988) or
Chang and Yortsos (1990), Acuna and Yortsos (1995), for
non-integer or fractal spatial dimensions, which do not
apply only to fractured rocks, show that new pumping test
analyses can help to better characterize heterogeneity,
even if the fractal dimension itself is not a predictive
parameter that can yet be linked with the geometry,
spatial permeability distribution or observable connectivity of the medium. But other techniques, as suggested
below, may be developed.
Did this simple equivalent homogeneous medium approach work, with the tools available at the time (analytical solutions, the image theory, conformal mapping,
etc...)? Yes, somehow it did, since early hydrogeologists
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were able to develop well fields, to adapt the exploitation
to the resources and to solve local engineering problems
without major catastrophes.
Heterogeneity and numerical modeling
The breakthrough in quantitative hydrogeology came in
the early sixties with numerical modeling. It is hard to
pinpoint who built the first digital model of an aquifer, it
came after several attempts (started in the oil industry) to
use electric analogs, the digital models being initially
numerical versions of the analog, see e.g. Walton and
Prickett (1963), Prickett (1968). The names that come to
mind are Tyson and Weber (1964) in Los Angeles, Zaoui
(1961) at Sograh and the Chott-el-Chergui in Algeria,
Prickett (1975) for a review, and others.
The analogs and, better still, the digital models, offered
the possibility of making the aquifer properties variable in
space while adapting the model to the geometry of the
domain and changing the boundary conditions, the recharge, etc. Heterogeneity and the way in which it was
accounted for in models will be the focus here. Assuming
that there are a few pumping test values for the aquifer
permeability, they are nevertheless much fewer than the
number of cells in the model: the model offers a possibility of “dealing” with heterogeneity in a much more
precise way than the well test data alone would permit.
The initial approach was to assume that a permeability (or
transmissivity) given by a pumping test was a “local average” of the true local value, over an area equivalent to
that of the mesh of the model. In other words, the issue of
“support” in Geostatistics, which concerns the size of the
domain on which a measurement is made, was not yet
clearly understood, but the idea was firmly established
that the permeability measured by a well test was to be
assigned to the mesh where that well was located. Between wells, the value assigned to each mesh was to be
interpolated from the adjacent wells. This could be done
by zoning (e.g. Vorono, or Thiessen, polygons surrounding each well), or by polynomial trend fitting, or by
hand contouring, etc. Heterogeneity was thus defined by a
set of “measured” averaged values at local wells, interpolated in space. This did not work very well: since the
model could calculate heads and flow rates, these values
could be compared with observations, and the model
could be “verified.” Model calibration was the answer,
initially through trial-and-error and soon to be followed
by automatic inverse procedures (see Carrera, this issue).
The important point here is that heterogeneity is represented (at a certain scale) in the models, but heterogeneity
is neither defined nor prescribed: the heterogeneity description is based on the local (averaged) measurements,
and its distribution in space is inferred by the inverse
procedure. As the modelers soon realized that the inverses
have an infinite number of solutions, they tried to restrict
the range of “inferred heterogeneities” by assigning
constraints to the inverse: regularity, zoning, algorithmic
unicity, etc, see Carrera (this issue).
Did that work? Yes, it did, and it is still often used
today, a calibrated model is a reasonable tool for aquifer
management and decision making, see e.g. a discussion
with Konikow and Bredehoeft (1992a, 1992b) in Marsily
et al. (1992). Where the approach started to fail or at least
to show its limitations was first in the oil industry, where
the predictions of oil recovery and water-cut (i.e. waterto-oil ratio in a producing well) made by calibrated
models proved very unreliable. Had such models been
used to predict groundwater contamination problems,
their limitations would also have become apparent, but
such problems were still new in hydrogeology at that time
(seventies–eighties). The reason is that oil production and
water-cut, as well as contaminant transport, are very
sensitive to conductive features such as high-permeability
channels, faults or, on the contrary, low permeability
barriers that are by definition intrinsically “averaged” in
the well-testing method and in the inverse inference approach. In a comparison of seven inverse methods, Zimmerman et al. (1998) tried to “embed” in artificially
generated media such highly conductive channels, and
asked the seven inverses to identify them, based on calibration on head measurements. The results showed that
most missed such heterogeneous features.
In fact, the head variations due to heterogeneity are
small, whereas those of velocities and travel time are
large, as can be seen intuitively when considering a layered aquifer with flow parallel to the bedding: the vertical
heterogeneity does not produce any variation in the head
distribution over the vertical, while velocities can vary
tremendously from layer to layer. Trying to infer the
heterogeneity from the head data alone was a quasi-impossible task. At present, head and concentration data
(e.g. environmental tracers or contaminants) are more
often available and used jointly in the inversion.
In the oil industry, to overcome these limitations, important efforts were devoted to “reservoir characterization” in the 1980s, following the introduction of “sequential stratigraphy,” see e.g. Vail et al. (1991). This
involved advanced geologic analysis of depositional environments, combined geophysics and well logging, detailed characterization of modern outcropping depositional analogs of deep buried reservoirs, in order to learn
how to describe and represent them. This advanced geological analysis of reservoirs and of organics-rich sourcerocks, to characterize their heterogeneity, was at the base
of the development of new tools, such as Geostatistics and
Boolean methods, which will now be described. Unfortunately, these “reservoir characterization” efforts did not
really catch on in hydrogeology, which was (and still is)
lagging behind reservoir engineering in this matter.
Stochastic methods and geostatistics
Considering an aquifer property as a random variable was
discussed, e.g. by Matheron already in 1967, but for the
purpose of averaging, not to describe heterogeneity. In the
oil industry, Warren and Price used a stochastic approach
as early as 1961 to assign permeability values to a 3-D
reservoir model, also with the problem of averaging in
mind, but this did not evolve into significant changes in
the treatment of heterogeneity. The origin of stochastic
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hydrogeology can be traced to two schools: Geostatistics,
developed by Matheron (1963, 1965), for mining estimates, and first applied to hydrogeology by Delhomme
(1976, 1978, 1979), and to Freeze (1975), Gelhar (1976)
and Smith and Freeze (1979). In 1975, Freeze assigned
random values of permeability to cells of a 1-D model,
drawn from a log-normal distribution, without considering any spatial covariance between the random values, as
Warren and Price (1961) had done. Delhomme (1976,
1978, 1979) and Smith and Freeze (1979) included such a
covariance. At the same time, Gelhar (1976, 1993) was
pioneering stochastic hydrogeology, taking the covariance
into account; his work was more theoretical and used
analytical methods, as did the subsequent works by Dagan
(1985, 1989).
Geostatistics, and in particular the definition of the
covariance (or the variogram) of the permeability distribution, is the first appearance of the new concept that
heterogeneity can be described by a “structure,” i.e., that
the geological processes that created the medium have
imposed a pattern on the spatial distribution of the inhomogeneous values. It is no longer possible to produce a
“plausible” map of the heterogeneity of a medium without
having some underlying “structure” in mind. This structure is, in Geostatistics, defined by a new function, the
spatial covariance (or variogram) which is a tool to
characterize the heterogeneity. The inference of this
structure from the data is the compulsory first step of a
Geostatistical analysis, with the tacit assumption that all
relevant heterogeneities and “structures” can be captured
or represented by the covariance, which will be challenged later. See e.g. Chiles and Delfiner (1999), or
Marsily (1986).
The Geostatistical approaches to dealing with heterogeneity were initially considered as continuous processes.
These can produce three results:
1. The first is a better estimate of the spatial distribution
of the parameter values in the cells of a model, based
on a set of “measured” values. Kriging provides an
“optimal” (in the sense of minimum variance of the
estimation error, not necessarily “best approach”)
method for assigning permeability (or transmissivity)
values to the meshes of a model, significantly superior
to zoning, or to arbitrary interpolation (see e.g.,
Marsily 1986; Chiles and Delfiner 1999). However,
when highly correlated structures such as buried
channels or fractures are present, it is not possible to
directly represent them with a single variogram, and,
if they have been identified, geologically informed
zoning is preferable. Furthermore, an extension of
Kriging, co-Kriging, can be used to estimate permeability based on both well-test results and additional
measurements (e.g. specific capacity, electrical resistivity, etc.). See e.g. Aboufirassi and Marino (1984),
Ahmed and Marsily (1987, 1988, 1993). Lastly,
Kriging provides a rigorous framework to address the
“support” issue, i.e. to consider data that are representative of different averages over space, e.g., slug-
test data that are only local values and well test data,
which are averages over areas depending on the duration of the test. Kriging can incorporate these different data and produce estimates that are representative of averages over the precise area of a mesh, that
can vary in size inside the domain, see e.g., Chiles and
Delfiner (1999), Roth et al. (1998), Rivoirard (2000).
In several instances, it has been shown that a Kriged
distribution of the permeability or transmissivity will
produce a model that is almost calibrated and need not
be adjusted, see e.g. Raoult (1999), or to some extent,
in basin modeling, Gonalvs et al. (2004b). It is clear
however that Kriging does not always perform well,
as will be seen later. One also needs to apply Kriging
with some rigor on the log-transformed transmissivity
values, in order to estimate geometric mean values
and not arithmetic means. Back-transforming the
Kriged lnT into T values must also be done correctly,
i.e. simply as T= exp[lnT] without any additional term
using the variance of the estimation error to supposedly correct an assumed estimation of the median
rather than of the mean, as is sometimes erroneously
done: see e.g. Marsily (1986).
2. The second is that Geostatistics provides a clear
concept to constrain an inverse calibration procedure,
to infer the spatial distribution of the parameter value.
This is discussed at length in Carrera (this issue) and
was the basis for the “Pilot Point” inverse approach
(see e.g. Marsily 1978; Certes and Marsily 1991;
Ramarao et al. 1995; Lavenue et al. 1995; Marsily et
al. 2000; Lavenue and Marsily 2001).
3. Monte-Carlo simulations of aquifer models where the
parameters are conditional realizations of the (unknown) spatial distribution of the parameters is one
method to assess the consequences on flow and
transport of the uncertainty on the heterogeneity of the
system, as represented by the Geostatistical approach.
Assuming that the conditional realizations display the
full range of uncertainty on the heterogeneity of the
aquifer, and assuming that the covariance is actually
capable of capturing all of the relevant heterogeneity,
the Monte-Carlo flow simulations provide an estimate
of the resulting uncertainty on the flow and transport
(head, flow rate, concentration, etc); see e.g., Delhomme (1979), Ramarao et al. (1995), Lavenue et al.
(1995), Zimmerman et al. (1998).
Geostatistics has helped greatly, and the method is widely
used to deal with heterogeneity. It is conceptually simple,
the additional degree of freedom is very small (a variance,
a range and a type of covariance model), the tools are
available and the data requirements are standard ones: to
build a groundwater model, well tests are necessary, and
Geostatistics make better use of the data without asking
for more. Additional data such as specific capacity,
electric resistivity, etc., can also be used jointly. To the
question: “how do we know, in a very practical sense, that
the generated statistics are correct?,” the answer is that
this description of reality is a model, not an exact de-
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Fig. 1 Examples of Boolean models: (left) sand lenses, from Haldorsen and Damsleth (1990), and (right) circular discrete fractures,
from Billaux (1990)
scription; natural heterogeneity is much more complex
than any model can account for. The only value of using
such a model is if it provides a better description (closer
to the unattainable reality) than previous models, and thus
gives better predictions than its competitors. Through the
examples outlined above, simple geostastistics has shown
that it provides better descriptions of continuous parameter fields than any alternative method. This is not to say
that it is correct, only that it is useful.
Stochastic shales
Geostatistics represented a first attempt at describing the
structure (in a statistical sense) of heterogeneities. The
tool used for that, the variogram (or the covariance), has
two major components: the range, which is proportional
to the average size of heterogeneous bodies, and the sill,
which is a measure of the magnitude of the changes (e.g.,
in permeability) from one heterogeneous body to another.
These “bodies” were thought of as “hills” and “holes,”
circular or ellipsoidal (in the case of an anisotropic variogram), distributed randomly in space. On a Kriged map,
or better on a conditionally simulated one, it is easy to see
the range of the structure, which is close to half the average distance between two “hills” or two “holes” in the
map. The sill is close to half the average squared difference between the values measured on the top of the hills
and those measured at the bottom of the holes. The map
was representing a continuous function. It is clear however that this description of reality is only a model.
In 1986, Haldorsen and Chang initiated a breakthrough
in the oil industry with their “stochastic shales” approach,
where the “objects” are discontinuous. Their concept of
heterogeneous bodies is also statistical, but founded on
geometrical concepts, not covariances. They consider
heterogeneous sand bodies as “objects,” whose shape is
perfectly known (e.g., shoe boxes, see Fig. 1) but the size
of the box and the position of its center are drawn randomly from a prescribed distribution. These objects are
embedded in a continuous matrix, which for Haldorsen
and Chang, was a shale, the “objects” being sand lenses.
This type of model is called Boolean. The shape of the
object is extremely flexible and left to the decision of the
modeler, see many examples in e.g. Haldorsen and
Damsleth (1990). The required statistics concerning the
shapes and positions of the objects are estimated from
available data, such as borehole logs, outcrop mapping,
seismic surveys, etc. Several sets of objects with different
shapes and properties can be used together (e.g. sand
lenses and limestone bodies inside a shale matrix, or
different types of sand structures). Once the distribution
in space of the objects is drawn, they are sometimes called
facies, or “flow units”; it is still necessary to discretize the
domain into cells and to assign properties to each facies:
permeability, porosity, etc. This can be done by a direct
relation (each facies is described by one set of parameters), or by a random sampling of parameter values, defined by a distribution function and sometimes by a covariance, for each facies. The model is by definition
stochastic, and many realizations can be produced.
This approach allowed abrupt changes in facies and
permeability when Geostatistics was generating continuous fields. This is a fundamental step toward modeling
actual geological media, where discontinuities and abrupt
changes are widely present (e.g. sand bars vs. matrix,
channels vs. floodplain, turbidites vs. continental margin
deposits, etc). Note that the concept of roughness defined
here for a surface could be defined for a 3-D medium as a
“texture” parameter. This parameter has not been considered so far, since it is not relevant for the flow problem, but it might be interesting to investigate it when
dealing with dispersion.
Did this approach work? It was indeed a breakthrough,
essentially in the oil industry, as it empirically popularized the underlying concept of connectivity. If two sand
lenses are not hydraulically connected, fluids will not
readily flow from one to the other. This lack of connectivity happens in nature, and so far the models had been
inept at dealing with this concept, Geostatistics introduced quite smooth transitions in space, not abrupt
changes, whereas in reality nature does not vary
smoothly. So the stochastic shales concept was very
useful, although Boolean models addressing the connectivity issue had been developed much earlier, e.g. by
Matheron (1967); Marsily (1985) and Fogg (1986) also
emphasized the connectivity issue. In hydrogeology, the
“stochastic shale” concept has not been widely used (see
however e.g. Desbarats 1987; Pozdniakov and Tsang
2004) except that the same type of Boolean model was
captured by the fractured rock community, where the
“objects” were discrete fractures (Fig. 1). The concept of
fracture connectivity (sometimes named percolation
threshold, in the framework of the percolation theory)
makes a lot of sense. Many versions of the discrete
fracture model are available and have been tested, validated and compared, see e.g., Marsily (1985), Long and
Billaux (1987), Cacas et al. (1990a, b), Long et al. (1991).
They also completely renovated the vision of how fractured systems behave by introducing this concept of
connectivity, which will be discussed later together with
the upscaling issue. These Boolean tools are available and
used, but require a new type of data on the geometry of
the “objects” which they represent. Some of these data are
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available (well logs) and were in fact ignored in earlier
approaches. But some are not easily accessible, e.g., the
fracture shapes and sizes, and have to be “guessed” or
assumed constant in space, so that what is seen on an
outcrop can be taken as representative of what lies kilometers below the surface, in the real aquifer. One additional drawback of the Boolean models is that they are
extremely difficult to condition to observed data. In other
words, it is very hard to make the stochastically generated
“objects” occupy the exact positions of the observed ones,
e.g., along a borehole (sand lenses, fractures, etc.). Some
attempts at conditioning have been made (see e.g. Andersson 1984; Chiles and Marsily 1993; Lantujoul
1997a, b, 2002, for fractured rocks and facies). Conditioning to objects without a volume (i.e. a fracture) is
easier than to an object with a volume (i.e. a facies of a
given thickness). Conditioning may be iterative, and is a
priori only possible for a finite number of conditioning
points. For facies, it is therefore necessary to discretize
the facies, e.g. along a borehole. More fundamentally,
some scientists question the utility of these models and
claim that continuous Geostatistical models can manage
just as well as discontinuous Boolean ones (e.g. Anda et
al. 2003), but the authors of the present paper strongly
disagree. Boolean models have made an extremely valuable contribution in dealing with heterogeneity by introducing the facies concept and addressing the connectivity
issue.
Geostatistics fights back:
discontinuous facies models
With their stochastic shales concept, Haldorsen and
Chang (1986) in fact started a controversy with the continuous Geostatistical approach. The tenants of Geostatistics immediately fought back, and introduced discontinuous Geostatistical models. The concept of disjunctive Kriging was introduced by Matheron as early as
1973 (Matheron 1973, 1976), but not commonly used;
Journel (1983), Journel and Isaaks (1984), Journel and
Alabert (1990) and Journel and Gomez-Hernandez (1993)
developed the Indicator Kriging approach, where an indicator can take (at any given point in space) the value
zero or one, depending on whether the point is inside or
outside a given facies. Matheron and the Heresim Group
at the French Institute of Petroleum developed the
Gaussian Threshold model, where a continuous Gaussian
random function in space generates a given facies if the
function value at a point in space falls between two
successive prescribed thresholds (Matheron et al. 1987,
1988; see also Rivoirard 1994; Chiles and Delfiner 1999;
Armstrong et al. 2003, or Marsily et al. 1998, for a review
of such stochastic models). With the new method, “objects” are also produced as in the Boolean one, but these
objects are defined by applying a threshold value to the
result of a continuous Geostatistical simulation. This description of the various methods is a drastic simplification
of the complexity of these models and ignores the significant differences between them. But the final outcome
of the simulations is a series of “facies” in 3-D space, the
shapes of which are indirectly defined by the covariance
function of the underlying continuous Geostatistical
function (indicator or Gaussian).
An alternative is the multiple-point Geostatistical approach developed by Journel, see Strebelle (2002). The
approach allows simulations of facies maps using conditional probabilities that describe the exact geometry of the
surrounding data. In practice, the conditional probabilities
are calculated from a training image that can be derived
either from outcrop observation, expert knowledge or
geophysics (see an example in Caers et al. 2003). The
method makes it possible to simulate complex geometries
such as channels, meanders or lenses and can preserve the
relations between the facies.
An infinite number of simulations of a given aquifer
can be generated with these methods. Both the Gaussian
threshold and the multiple-point methods are very powerful, and the 3-D facies structures that they generate are
consistent with what geologists have observed on outcrops or inferred from imaging the subsurface, as illustrated by Fig. 2 (Beucher personal communication 2004).
They look realistic, much more so than the Boolean ones
and have many features of real geologic structures.
Contrary to the Boolean ones, these models can easily be
conditioned to observations and thus provide the correct
generated facies at the location where the true facies has
been observed, e.g., along a borehole. When a series of
facies distributions in 3-D space has been defined, it is
still necessary to assign properties to each facies (permeability, porosity, etc). As in the Boolean models, this
can be done deterministically or by sampling from a
prescribed distribution. The fitting of such models is
discussed below.
Datawise, this approach offers the hydrogeologist who
carries out the study the opportunity to use the geologic
information that is, in general, available, or that could be
collected at little additional cost by examining the geology (adequate description of the boreholes, better interpretation of the geophysics, comparison of the site with
existing well-known sites, etc...). By contrast, these data
are more appreciated and used in the oil industry, where a
good geological description is made of each borehole,
sometimes with cores taken from the formations, and always including well logging. In that respect, hydrogeologists are clearly lagging way behind the reservoir engineers in terms of use of geologic data. The fitting of the
facies covariance function for the truncated Gaussian
model is thus possible, and this approach is increasingly
being used. Seismic data can also be used, either to define
seismic facies that are correlated with “real” facies, or as
indicators of the proportion of each facies in a vertical
profile (Beucher et al. 1999; Fournier et al. 2002), or as a
training image for the multiple-point approach. Making
use of additional data is certainly a better way to deal with
heterogeneity than trying to extract new results from the
same old data.
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Fig. 2 Example of a four-facies
Gaussian Threshold model using the HERESIM code
(Matheron et al. 1987, 1988).
a Cross-section; b proportion
of each facies over the vertical;
c 3-D representation. From
(Beucher, personal communication, 2004)
Markov chain models
In the same vein as the Indicator facies models, a third
approach was developed in the 1990s to address the same
problem of building facies distributions from the geologic
information, to describe the heterogeneity. It is based on
the use of transition probability and Markov chains, to
describe facies discontinuous in space. Following Carle et
al. (1998), the transition probability from a facies k to a
facies j is defined as:
tjk ðhÞ¼ Probability that k occurs at x
þ h given that j occurs at x
where x is a point in space and h a lag vector. It can be
shown that the transition probability can be related to the
indicator cross-variograms gjk(h) of facies j and k through
the relation:
2gjk ðhÞ ¼ pj 2tjk ð0Þ tjk ðhÞ tjk ðhÞ
where pj = E[Ij(x)] is the volumetric proportion of facies j,
assuming stationarity in space, and Ij(x) is the indicator
variable of facies j, i.e. Ij=1 if point x is inside facies j,
and zero otherwise. Note that the transition probability is
not symmetric in h and –h, whereas the cross-variogram
is. Defining such a transition probability is, by definition,
a Markovian approach: the probability of k occurring at
location x+h is only dependent of what happens at location x, and nothing else; by contrast, a non-Markovian
approach would say that the transition probability may
also depend on what happens at other points in the
vicinity of x. For those more familiar with Markovian
processes in time, the assumption is that the probability of
what happens at time t+Dt is entirely defined by the initial
condition, i.e. what happens at t.
Given this definition, the continuous-lag Markov chain
model simply defines the function which relates the
transition probability tjk(h) to the lag h. This function is
assumed exponential with distance in the example given
by Carle et al. (1998), which can be seen as choosing a
given functional form for a set of variograms and cross
variograms in the Indicator approach: tjk ðhÞ¼ exp rjk h .
The coefficients of the exponential, rjk, are the unknowns
to be calibrated, called the conditional rates of change
from category j to category k per unit length of the lag
distance h. Furthermore, this rate of change can be made a
function of the direction of the lag vector h. The exact
Markov chain is in fact written in matrix form, since there
are n2 transition probabilities if there are n facies; these n2
rates are however not independent, and only (n1)2 coefficients have to be calibrated on the available data. In
practice, these coefficients can be directly related to
fundamental interpretable properties of a geologic medium, such as proportions of each facies, mean length,
asymmetry and juxtapositional tendencies. The first step
of the Markov model is thus a calibration of the parameters on the data, just as the first step in Geostatistics is to
calibrate the variogram models on the data. Then, conditional simulations can be generated, as in the indicator
or Gaussian threshold approach.
Despite its simplicity, the Markov model seems to be
quite flexible and to better account for spatial cross-correlation, such as juxtapositional relationships, e.g. finingupward tendencies of different facies, than the indicator
approach. The approach seems also internally consistent,
as is the Gaussian threshold method, since the full matrix
of the transition probabilities between all facies is consistently calibrated simultaneously from the data, whereas
with the Indicator method, the direct and cross-vari-
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ograms are generally calibrated one by one and may not
always be consistent. Their authors claim that it is easier
to fit to the data than the Indicator variograms, and that it
can produce simulations by the sequential simulation
method and simulated annealing. It seems that the major
advantage of the Markov transition probability approach
may be to better insure that a given facies is found close
to another one, as occurs in nature, due to sedimentological principles.
Additional presentations of the Markov chain model,
and its comparison with the Indicator approach, can be
found in Chiles and Delfiner (1999). Additional development and applications to real facies systems in hydrogeology can be found in Carle and Fogg (1996, 1997),
Carle et al. (1998), Fogg et al. (1998, 2000), Weissmann
and Fogg (1999), Weissmann et al. (1999, 2002), LaBolle
and Fogg (2001).
Upscaling
Both the Boolean models and the facies models require
upscaling. Upscaling is in fact the new terminology for
averaging, a concept already discussed and initially used
to find an equivalent property for an entire aquifer, when
analytical methods made it necessary to have one single
parameter for an entire system. But here the issue is different. The Boolean and facies models are pixel or vauxel
models, i.e. they represent reality as a series of small cells
or volumes. Typically, several tens or hundreds of millions of cells are generated by the models, still far too
many to use them directly as calculation cells for the flow
models, even if parallel computing and the continuous
increase in computing power have today raised the
number of cells acceptable in a model to millions. Furthermore, uncertainty evaluation requires that many stochastic realizations of the same problem are run, thus
prohibiting the treatment of very large problems with the
flow models.
Upscaling is the grouping of these elementary cells, to
which a permeability has been assigned, as discussed
above, into larger blocks that define the upscaled grid of a
flow model. There are numerous methods of upscaling,
generally with the objective that the flow and transport
calculations made at the upscaled level, with the upscaled
parameters, provide a calculated solution as close as
possible to the one which could have been calculated, if
the small scale meshes had been kept, with their original
parameter values. Upscaling is not a problem for porosity
(except if the upscaled volume includes aquitards in
which solute can diffuse), as porosities add up arithmetically, but the real issue is the permeability, see e.g. a
review in Renard and Marsily (1997). For infinite media,
averaging has been resolved at least theoretically, as
shown above (e.g. the geometric mean in 2-D, Matheron
1967, or a power mean in 3-D, e.g. Noetinger 1994, 2000;
Abramovich and Indelman 1995; Indelman and Rubin
1996). But these theoretical results are only valid if rigorous assumptions are satisfied, on the distribution function of the permeability (in general, log-normal), on the
type of flow (in general, parallel), etc. One particularly
striking example of how wrong such generally unquestioned assumptions can be is given by Zinn and Harvey
(2003), who compare three random permeability fields in
2-D, all with nearly identical log-normal distributions and
isotropic covariance functions. The fields differ in the
pattern by which the high- or low-conductivity regions are
connected: the first one has connected high-conductivity
structures; the second is multivariate log-Gaussian and,
hence, has connected structures of intermediate value; and
the third has connected regions of low conductivity. The
authors find substantially different flow and transport
behaviors in the three different fields, where only the logGaussian case behaves as predicted by the stochastic
theory. This example stresses again the importance of
connectivity in dealing with heterogeneity, as pointed out
by e.g. Fogg (1986), or Western et al. (2001). Given these
results, one must consider that upscaling is still a very
important research subject, where the detailed properties
of the elementary cells must be taken into account in a
much more precise way than just by their average values.
However, the need to upscale may diminish with the
constant increase in computing power of modern equipment. On the other hand, an important issue is the increasing use of unstructured grids within the numerical
models while Geostatistics requires a regular grid (support effect). To preserve the coherence of the Geostatistical model, it will still be necessary to transfer the heterogeneity model onto the unstructured grid using both
upscaling techniques, when the mesh of the numerical
grid is larger than the Geostatistical one (He et al. 2002);
downscaling techniques will also be needed when the
opposite situation occurs e.g. around flow singularities.
Upscaling of geochemical parameters, such as distribution
coefficients, kinetic constants, etc, is an almost virgin
issue, see e.g. some preliminary work by Pelletier (1997),
Glassley et al. (2002) or discussion in Delay and Porel
(2003), Carrayrou et al. (2004).
Fitting a facies model
When a facies model has been built, either with a Boolean
or a Geostatistical approach, and when permeability values have been assigned to each small-scale cell, and upscaled to the flow model cells, the ideal situation would
be that the model is (by chance) perfectly calibrated.
Unfortunately, this almost never occurs, and the model
has to be calibrated. There are two possible ways of doing
this: keep the geometry of the facies, and change the
values of the permeabilities within the facies. Or alternatively, keep the values of the permeabilities, and
change the geometry of the facies. The first option is
standard: the facies geometry can be seen as zoning in a
classical inverse (see Carrera, this issue) and the permeabilities can be adjusted within each facies. However,
some recent work on the adjustment of the geometry has
been published, called the “gradual deformation” (Hu
2000, 2002; Hu et al. 2001a, b). This offers the possibility
of gradually and continuously changing the shape of both
Boolean objects (e.g. the position of a discrete fracture, its
size, etc, or the shape of a sand lens) and Geostatistical
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Fig. 3 Observed and simulated
cross-section of the San Francisco Bay, using the genetic
approach. From Kolterman and
Gorelick (1992)
facies. This is a breakthrough in the calibration of facies
models that could be combined with a more classical
inverse on the parameter values.
Genetic/genesis models
The next breakthrough in dealing with heterogeneity
came in 1992 with Kolterman and Gorelick. Despite some
early attempts at recreating the spatial distribution of rock
properties by modeling the rock formation processes (e.g.
by basin modeling, which will be discussed later), this
paper was the first real attempt at generating sediment
facies not by Boolean or Geostatistical methods, but by
using a sedimentation model, that of Tetzlaff and Harbaugh (1989), and simulating river flow and sediment
transport, deposition and erosion over a period of 600,000
years, for the San Francisco Bay. This required considerable computing power and CPU time, as well as the
reconstruction of the climate history of the region over the
same period of time, day by day, and of the evolution of
the streams, the bay and the movement of the Hayward
Fault, which crossed the area. Fig. 3 is a cross-section of
the outcome of this effort, taken from the authors, and
compared with observations.
The authors of this paper believe this is a breakthrough
because, for the first time, the major effort is put into
building a model whose sole purpose is to represent the
actual geologic processes that created the sediments; from
the outcome of this model, the properties of these sediments are derived, in particular their heterogeneity. Of
course this is a very challenging task, with immense
difficulties: which process is represented, how to reconstruct the necessary data (e.g. climate records), how can
the embedded uncertainty be quantified, are there random
components in the modeling (there are some random
decisions made by the model through time in Kolterman’s
and Gorelick’s 1992 approach). But the power of the
genetic approach is that the modeling of processes can in
principle represent features that statistical methods would
never have captured. For instance, a meandering depositional environment would never be amenable to a simple
bivariate Geostatistical approach, only multiple-point
Geostatistics (still under development, see e.g. Strebelle
2002; Krishan and Journel 2003) could approach that, or
bivariate Geostatistics where the correlation structure itself is a correlated random field function of the meander
properties, see e.g. Carle et al. (1998). Although a first
attempt at comparing a genetic and a Geostatistical ap-
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proach on the same site has been published recently
(Teles et al. 2004, see below), there is still much work to
be done to compare the various genetic and Geostatistical
models; this is a very important area of new research.
Another striking example can be seen in Fig. 3: Kolterman and Gorelick introduced the movement of the Hayward fault in their model; the result is that the mouth of
the sediment outlet in the bay (where the coarse sediments
accumulate) moves from North to South during the simulation, as can be seen on the North–South cross-section.
This would never have been found in a statistical approach.
It is worth noting that very early attempts at developing genetic models had been launched by Matheron in
the late 1960s, he called them at the time “the Ambarzoumian processes,” see e.g. Matheron (1969) or Jacod
and Joathon (1971, 1972). But they had not been used in
hydrogeology. A number of genetic models have been
developed since, mostly for the oil industry, where sediment transport and deposition are approximated by a
diffusion equation, which can be shown to approximately
represent sediment transport, in a marine or deltaic environment; see e.g. Paola et al. (1992), Heller and Paola
(1992), or Grandjeon (1996), Grandjeon et al. (1998),
Grandjeon and Joseph (1999), Doliguez et al. (1999),
Euzen et al. (2004), or Quiquerez et al. (2000). Webb
(1995), Webb and Anderson (1996), have developed a
sediment model, partly empirical, partly genetic, to represent fluvial deposits. Kolterman and Gorelick (1996)
present a review of such models. Teles et al. (2001) have
built a genetic/genesis model of fluvial sedimentation,
representing both meandering and braiding patterns,
which uses empirical rules to represent sediment transport
and erosion. The model does not represent water flow and
sediment transport per se, but rather the result of the alluvial processes by using geometric rules. It is considered
that these geometric empirical rules embed the fluvial
processes that create the structures. Some of these geometric rules can be seen as in part Boolean. A series of
different facies are thus assembled, see e.g. Fig. 4.
As for Boolean or Geostatistical facies models, each
facies is then assigned a property value, e.g. a permeability, to be introduced, after upscaling, into a flow
model. Teles et al. (2004) showed that the genetic model
is able to represent the connectivity of buried channels in
the alluvium (or the role of barriers of such channels,
when they are filled with clay), whereas the bivariate
Geostatistical approach ignores such features, with significant influence on the prediction of solute transport. So
far, these genetic models have not been conditioned to
hard data. They cannot generate a facies at the location
where it has been observed or, at best, with great difficulty. It is conceivable to combine multiple-point Geostatistics and genetic models where the outcome of the
latter would be used as training images for the former;
other ideas have also been suggested, e.g. to prevent a
non-observed facies from being deposited at a given location; this approach has been used to generate conditional Boolean fracture sets by not keeping random
fractures which do not appear at the conditioning points,
see e.g. Andersson (1984), Chiles and Marsily (1993).
At a different scale, basin models can also be considered as genetic models: Betheke (1985), Ge and Garven
(1992), Garven (1995), Burrus (1997), Person et al. (1996,
2000). Indeed, they represent the succession of sediment
deposition, they can include high resolution sediment
types for a given facies and they explicitly model compaction and porosity–permeability reduction as a function
of the effective stress. They also represent thermal evolution, organic matter maturation and sometimes, some
temperature-dependent diagenetic processes (porosity
clogging). The 3-D Paris basin model is perhaps one of
the more detailed examples of this type. It is based on
more than 1,100 litho-stratigraphic well data (Gonalvs
et al. 2004a), see Fig. 5.
Some genetic models try to produce faults as a result of
tectonic stresses, see e.g. Quiblier et al. (1980), Renshaw
and Pollard (1994, 1995), Taylor et al. (1999), Bai and
Pollard (2000), Wu and Pollard (2002), Person et al.
(2000), Revil and Cathles (2002). Karstic systems can also
be studied with a genetic approach, by trying to model the
carbonate dissolution mechanism; a number of attempts
have been made in this area, see Bakalowicz (this issue).
So far, the genetic approach has been focused on detrital
or alluvial sediments rather than on calcareous deposits,
although some attempts have been made to represent the
evolution of limestone (Grandjeon and Joseph 1999). But
much work remains to be done on various types of rocks.
Does this approach work? The answer is yes, and it is
developing rapidly. In terms of data, it requires a solid
scrutiny and interpretation of the litho-stratigraphic record. Sedimentologists have to study, in detail, the
available sediment samples, sometimes to date them
(with14C on organic debris for recent alluvial sediments,
or with micropaleontology, isotopic geochronology...), to
investigate the depositional environment, and interpolate
the information in space keeping a genetic concept in
mind. For the Aube valley, as described in Teles et al.
(2004), for instance, a series of 44 auger holes was sufficient to build a reasonable genetic model of the plain,
bearing in mind the existence of studies of the succession
of climate periods during the Holocene, where each period is characterized by a type of sediment, a deposition
pattern (braided or meandering), which is valid for a large
area, not only for a given alluvial plain. Generic or regional geologic knowledge is thus available, and vastly
increases the value of the collected data. An example of
the value of such knowledge can be found in Herweijer
(1997, 2004). He studied the famous MADE experimental
site in the Mississippi valley, where very well characterized tracer tests had been made (see e.g. Harvey and
Gorelick 2000). His approach was to construct a conceptual model of the stratigraphy of the site, based on the
available geologic observations and the results of the
pumping tests. Previous authors had tried to use the
Geostatistical approach at MADE, and to infer the spatial
covariance structure of the alluvial sediments in order to
describe its heterogeneity, without paying much attention
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Fig. 4 An example of sediment
pattern in the Rhne Valley in
France, as created by a genetic
model (Teles et al. 2001).
a Plan view of the sediments in
the plain. Meshes are
200200 m, the alluvial domain
is 20 km long. b Cross section
(North-West–South-East at
Villeurbanne, as marked on a.
The colors show the sediment
units ages from 15,000 years to
present, as given in b. The
lithology of each sediment is
associated with each episode.
From Teles et al. (2001)
to the geology (“all this is alluvial material”). What
Herweijer (1977, 2004) observed is that within the alluvium, two successive sedimentation structures existed;
one from a braided stream, one from a meandering
stream. Geostatistics cannot easily incorporate such information. Calibrating a unique covariance structure for
the entire thickness of the alluvial aquifer was just like
trying to find a compromise between apples and oranges,
whereas incorporating some genetic knowledge would
definitely have improved the “dealing with heterogeneity,” even with the Geostatistical approach and without a
genetic model. An alternative approach could have been
to use “variogram maps” (or covariances) which can incorporate anisotropy and statistical non homogeneity.
Much more complex structures than those achievable with
a simple variogram can be obtained in this way, see e.g.
Anguy et al. (2001, 2003).
Very recently, ANDRA, the French Nuclear Waste
Disposal Agency, presented some on-going work on the
construction of a flow and transport model for two carbonate aquifers, in the Dogger and Oxfordian of the Paris
basin, where these two aquifers surround a clay layer
whose feasibility as a repository is being studied. The
model is to be used to represent the transport to the outlets
of radionuclides that would eventually leak out of the clay
formation. To calibrate the flow model, ANDRA and its
contractor, the French Institute of Petroleum, used the
genetic model DIONISOS (Grandjeon 1996; Euzen et al.
2004), which solves the diffusion equation to represent
transport and deposition of sediments, to define the
properties of the two aquifers. This work will be published in 2005 (Houel et al. 2005). Genetic models are no
longer only theoretical research tools for academics!
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Fig. 5 Basin modeling of the Paris basin: calculated horizontal
permeability and storativity distributions of three aquifer layers at
present. The heterogeneity is the result of the sediment facies description, kriged from the borehole lithostratigraphic data, defined
by the percentage of three poles: 1. clay; 2. limestone; 3. sand. The
maps include the effect of the compaction history with different
compaction rules for each pole, and different porosity–permeability
relations. The calculated permeability in each cell is the geometric
weighted average of the permeability of each pole. The depth of the
center of each layer is given on the right. From Gonalvs et al.
(2004b)
Different genetic models may be more appropriate for
different scales. Those representing small-scale processes,
such as Webb (1995) or Teles et al. (2001, 2004) are
better suited to account for stratigraphic details that are
not generally recorded in classical hydrogeologic studies,
e.g. an alluvial meandering plain over a small distance.
On the contrary, the genetic models based on the diffusion
equation to represent transport and deposition are better
suited for large scale problems, like marine sedimentation, alluvial fans, even marine carbonates. The Kolterman and Gorelick (1992) approach may lie in between.
Ignoring heterogeneity
It is impossible to completely ignore heterogeneity, but
one can use methods that “deal” with heterogeneity automatically and allow the user to ignore it. A simple example is permeability measurements. Any hydrogeologist
knows that if he/she needs to know the permeability of an
aquifer, making measurements on core samples will not
be a good solution. One should rather perform a pumping
test, of a reasonable duration, and extract from it the
average permeability of the aquifer, which can then be
used to make predictions, or build a model. It is the “tool”
(the pumping test) that “deals” with the heterogeneity,
and automatically produces a reasonable and useful value
for practical purposes.
Again and again, it has been found that if a watershed
model is built, to represent surface and subsurface flow,
making local measurements of the soil permeability, on
cores or even with double-ring infiltrometers, produces a
permeability value much too low to represent the observed behavior of the watershed (flow, water levels,...
see e.g. Carluer and Marsily 2004). The same observation
has been made e.g. on a deep sandstone aquifer, the Pierre
Shale, see Neuzil (1994). Is this due to an insufficient
number of measurements? To an inappropriate measurement system? To unidentified processes acting at a different scale than that of the measurements? If the flow in
The future
When does heterogeneity matter?
The first question is: “when does heterogeneity matter?”
As has been shown earlier, methods to deal with heterogeneity exist and will certainly be improved in the future,
but they generally need additional data, on top of the
usual ones, or at the very least, increased scrutiny of the
geologic data; they also need a good deal of thinking...!
An attempt to list the areas where “dealing with heterogeneity” is obligatory, and those where ignoring it will do
just as well is given below.
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the aquifer is to be modeled accurately, and the only
available information is a set of local values measured on
cores, then it is indeed necessary to “deal with heterogeneity” and derive a method that can construct the spatial pattern of the local permeabilities, and identify the
connectivity of special features and channels that govern
the flow, which are, it is believed, not accessible at the
local measurements scale.
But, in general, when the appropriate measurements
are available at the appropriate scale, then it is indeed
possible to ignore, to some extent, heterogeneity and use
the global measurements directly to make predictions.
This would apply first to flow, not to transport. The
conclusion is thus that where regional aquifer management is concerned, for optimization of well discharge,
estimates of available resources, then a large-scale view
of the heterogeneity is sufficient, and aquifer property
estimates can be derived by means of the usual tools:
pumping tests and aquifer model calibration by trial-anderror or inverse procedures. Of course, as emphasized
earlier, selecting an inverse which adopts a given parameterization of the permeability field already amounts
to making some assumption on the nature of the heterogeneity (e.g. zoning, Pilot Points based on Kriging, etc),
but again, the tool (inverse) will “deal” with heterogeneity, and most likely two different inverses will, in the
end, provide predictions that are not vastly different, even
if the permeability fields are far from identical. Just to
give an example, Besbes et al. (2003), see also OSS
(2003), recently built a flow model of the multilayer
aquifer system of northern Sahara, extending over three
countries (Algeria, Libya, Tunisia) and an area of more
than 1 million km2, probably the largest flow model ever
built. The objective was to agree on a multinational policy
of exploiting this huge aquifer, which receives little recharge and is currently being mined by increasing withdrawal. How long can the situation last, are there risks
linked with over-exploitation, is pumping in one country
affecting its neighbors? The distance between “adjacent”
wells is sometimes several hundred kilometers; even if the
geology is well-known in terms of horizons, local heterogeneity is obviously not a concern. Average transmissivity, vertical permeability of aquitards, local discharge into the sea and into playas, present level of
withdrawal, present recharge, and finally values of the
specific yield in the unconfined sections of the aquifers
are the most important parameters that will govern the
long-term response. The precise nature of the boundary
conditions, present and future recharge (with climate
changes) are also very important issues to be resolved in
such large aquifers (see e.g. Jost et al. 2004). Where
should the money be spent? Most likely on large scale
reconnaissance: geology, pumping tests when available or
feasible, collection of seismic surveys (made for oil exploration) that give the thickness of each layer, use of
Geostatistics (Kriging) to interpolate the scarce data in
space, and then model calibration using the 50 years of
existing records. Depending on which lithologies are
present, large-scale sedimentation models or basin models
could be used to determine the general trend in the
properties of the system, after a state-of-the-art hydrostratigraphic conceptual model has been built. This would
be the recommendation. There is a small incentive to
bracket the uncertainty in this example. If a withdrawal
limit is given to a particular country, not to jeopardize
another country’s resources, is this known with a 5%, or
50% bracket? To address that issue, it is necessary to
recognize that the basic parameters of the model are uncertain, and to do one of the following:
i
Simple sensitivity study. Arbitrarily assume that the
calibrated parameters have a given estimated range of
uncertainty (e.g. 20%, 100%...) everywhere and run
the model with different sets of parameters, and see if
the consequences are different.
ii Post-optimal residual sensitivity study. This can be
done if a stochastic inverse has been run, which gives,
after calibration and linearization, the residual uncertainty on the parameters, which can be different in
each area of the aquifer. This defines the range of
parameter uncertainty, which was “guessed” and assumed uniform in approach (i). See e.g. Cooley
(1997).
iii Full-scale post-optimal uncertainty analysis using a
stochastic inverse and Monte-Carlo simulations, see
e.g. Ramarao et al. (1995), Lavenue et al. (1995).
In this huge aquifer, there may be some local problems,
however, where local heterogeneity can be important.
One is the exact situation around playas. At present, the
playas are outlets for the aquifers. In the future, it has
been shown by the huge model that the lowering of the
piezometric surface will dry up this flux: the playas,
containing salt brines, may start salt water infiltration into
the aquifers, when they receive storm flow, or return flow
from irrigation. Predicting the rate of salinity increase in
the aquifers, the time needed for near-by wells to become
brackish, or possibly engineering measures to prevent this
from occurring, will require a better understanding of the
flow and transport at a small scale around the playas, thus
taking local heterogeneity into account. Another instance
is the vertical upward leakage from deep brackish aquifers
into superficial freshwater aquifers. Assigning a single
permeability to one aquitard is likely to be much too
simplistic, and a detailed study of the aquitard heterogeneity would be necessary. But this is unlikely to happen in
practice today, as (i) it would be quite expensive; (ii)
there is no well established method so far to deal with
aquitard heterogeneity, and (iii) the bad consequences
would be felt in tens of years, thus reducing the incentive
to assemble funds. However, this problem of aquitard
heterogeneity should be considered as an area for future
work, see below.
Taking heterogeneity explicitly into account
One opposite example when “dealing with heterogeneity”
is obligatory is in transport. Consider a remediation case,
where surfactants have to be injected to dissolve and re-
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cover DNAPL’s in a superficial aquifer. A large amount
of money will be spent on injecting, recovering and
treating the water. Having an optimally functioning system is economical, and spending money to decipher the
heterogeneity will pay off, as this will make it possible to
optimally locate the wells, adjust the pumping rates, and
insure that the system is hydraulically closed and that no
surfactant coupled with DNAPL’s will leak out of it. The
scale of the problem is meters, at most hundreds of meters. What can be done? It would be useless to apply the
“ensemble stochastic approach,” and try to characterize
the heterogeneity by a covariance function: in order to
apply the stochastic theories, some averaging must be
done. The pollution plume becomes “ergodic” when the
travel distance is several times (on the order of 10) the
correlation length. One is not interested in an average or
ensemble behavior, but in a detailed local description of
the real system. One needs imaging of the underground.
Any geophysical imaging technique (radar, electric panels, electromagnetic surveys, NMR, etc, see Gurin, this
issue) will be the starting point. An initial borehole survey
will then describe the lithology, and provide clues to
understand the structure of the aquifer, together with the
geophysics. The technique needed here to describe the
heterogeneity is definitely a facies model, one needs a
lithological description (nature and shapes of facies), as
precise as possible, on the basis of which properties will
be assigned. This is not just permeability, but porosity,
clay content, sorption parameters, etc, all properties that
are estimated facies by facies, and extrapolated in space
to represent the site. Modeling must be very detailed, and
upscaling may not be needed, what is still a challenge is
to determine the number of samples that are needed per
facies, their size, measurement method (on samples or in
situ) and the grid size that can represent them. A first cut
of the facies model would be obtained from a genetic
model. A state-of-the-art geologic characterization of the
site must first be completed, as commonly made in petroleum work but less often in hydrogeology. Then, a
genetic model (such as Kolterman and Gorelick 1992;
Webb 1995; or Teles 2001, 2004) should be built. It will
describe which different stratigraphic units are present at
the site, the likelihood of erosion processes, the type of
sedimentation pattern (meandering or braided stream, if in
an alluvial deposit, etc). Since these models are presently
not conditioned to the data, it will be necessary to go to a
Geostatistical facies model, not a Boolean one which is
difficult to condition. How to “fit” the Geostatistical facies model to the genetic one is still unclear (this discussion is about the future!), but the basic idea would be
to select the covariances (in case of bivariate Geostatistics) or the multiple-point pattern by fitting them on the
images produced by the genetic model. Another approach
to fitting a facies model on the data is the transition
probability and Markov chain method, as briefly described above, see e.g. Carle and Fogg (1997), Carle et al.
(1998), Weissmann and Fogg (1999), Weissmann et al.
(1999). The final facies model would then be conditioned
on all the observation points, and the geophysics. The
fine-tuning of the model would be made by calibration on
experiments made on the site (pumping tests, see e.g. an
example of 3-D interpretation of a pumping test with an
inverse in Lavenue and Marsily 2001). Preliminary tracer
tests would also be used to calibrate the model. Multiple
realization models may be necessary to provide an estimate of the uncertainty of the answers, e.g. the duration of
the remediation period.
The grey areas when dealing with heterogeneity
In the two above examples, one real (Sahara) and one
semi-real (DNAPL), the answer to the question “does one
need to address the heterogeneity issue?” is a clear-cut
“no” and a “yes,” the difference is, in part, related to the
difference in scale of the two problems. But there are grey
areas, where the answer is not that simple. There is indeed
no universal answer to the method of treatment of heterogeneity. The transport problems will in general require
that more attention be paid to heterogeneities than the
flow problems since for transport, connectivity is most
important. The scale of the problem must also be considered: a small-scale problem requires detailed understanding of the heterogeneous pathways, whereas for a
large-scale problem, flow and transport will be averaged
by the crossing of several heterogeneous structures, and
an “ensemble” average will emerge (the behavior becomes ergodic, in the stochastic language), averages can
be used, or averaging tools, such as the covariance
function.
Finally, the situation is also influenced by the medium
in which the study is performed. There is a plethoric
amount of work in the recent literature that concludes on
the non-ergodic behavior of fractured rocks regarding
both flow and transport. This would mean that heterogeneity (in this case mostly its influence on connectivity)
should be carefully accounted for. On the other hand,
densely fractured aquifers have been managed successfully for a long time, assuming the continuity of flow and,
to some extent, a pseudo-homogenization scale. For such
media, there is no clear answer except for theoretical
cases or synthetic cases. Further work is needed to classify the behavior of fractured rocks with respect to
topologic parameters such as connectivity, fracture length
distribution, mass density of the network, etc. See initial
attempts in e.g. Aupepin et al. (2001), Rivard and Delay
(2004).
The type of answer to be provided is also a criterion.
Suppose that a leachate plume from a landfill is to be
studied. If the objective is to determine the flux to a river,
the detailed pathway that the plume follows to reach that
river is not important, it is the rate of mass transfer that
matters. Average properties can be used. If, on the contrary, the question is the maximum concentration that can
be attained in a downstream well, the pathways, the dilution and dispersion along that pathway, etc, require a
more detailed analysis of the heterogeneity at the site.
Finally, the cost of data collection must also be considered: deciding to launch a more sophisticated study and
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collect the necessary data requires that a realistic budget
is available.
This question pertains to an on-going controversy in
hydrogeology, the parsimony issue, which is related to
this article through the selection of the method to deal
with heterogeneity: should simple and robust models (e.g.
equivalent homogeneous media, continuous Geostatistics,
zoning...) be preferred, as much as possible, or should one
rather try to represent heterogeneity with the more complex methods? The authors of this paper think that using
simple models has been taken too far in hydrogeology and
is no longer justified, powerful models exist and computer
time is no longer an issue. Making simplifying assumptions such as averaging ignores the connectivity issue;
treating a problem in 2-D may not be justified if the flow
and transport processes include a 3-D component (e.g.
vertical heterogeneity of the aquifer, density effects, etc).
Even if the 3-D structure of the system is poorly known,
making “educated guesses” on this 3-D structure may be
less erroneous than ignoring the 3-D structure, and using
averages. Problems like scale-dependent dispersion, extreme tailings, etc, are in fact the results of homogenization rules that do not apply at the scale of interest, and
thus do not allow the use of the classical governing
equations, which assume ergodic behavior. Many ineffective “pump and treat” operations are due to this misguided approach. This is however not to say that simple
models do not have a role to play, in many instances: see
e.g. a discussion on this issue in Voss (1998). But to
determine if a simple model is a satisfactory answer to a
given problem, there is no alternative than to compare its
output with that of a more complex one!
Areas for future work
In the following, attempts are made to indicate where
future research efforts should be concentrated in order
better to handle heterogeneity.
1. Transport problems generally require a better understanding of heterogeneity than flow problems. Average properties as given by a well test include the
heterogeneities, for flow calculations, but do not describe adequately the transport. And, in most cases,
the time required for a tracer test to provide representative answers is prohibitive. The type of tracer
experiments carried out at Borden in Canada, Cape
Cod in Massachusetts or MADE in the Mississippi
valley is scientifically of great interest for testing assumptions, models and methods, but too costly to be
used regularly for transport problems. Something else
is needed.
2. Even for transport problems, the approaches may
differ if the source term is diffused or at one point. For
diffuse sources, e.g. nitrate or pesticide contamination, an average description of the medium may be
enough; what is of interest here is the mass conservation, possible decay by biological processes (linked
to the average velocity and transfer time), and general
flow direction. The major problem is rather the source
term, the amount used by the farmers, and the modeling of the consumption/release/decay of the products in the plant root zone. The exact concentration in
a given well downstream need not be precise, it is the
order of magnitude with respect to the norms that
matters. This would apply to relatively well-defined
aquifers with simple structures, like the Chalk aquifer
in northern France and Great Britain. Aquifers with
complex multilayered structures or with interbedded
aquitard lenses may require more refined treatment, if
breakthrough concentrations at different horizons
need to be predicted, see e.g. Weissmann et al. (2002).
3. Two-phase flow is probably more demanding than
solute transport, in terms of describing heterogeneity
(see Noetinger, this issue). This is why many methods
summarized above were developed by the oil industry,
for predicting oil recovery and water-cut. This would
also apply, in hydrogeology, for NAPL and DNAPL
studies, and for the unsaturated zone. Many more
parameters than just permeability and porosity are
required (relative permeabilities, wetability, capillary
pressure, sorption, etc). All these must be treated simultaneously, since they are correlated. One of the
facies approaches seems the best option. In soil science, genetic models should be developed to better
describe the structure, in terms of facies or horizons,
of the unsaturated zone. For sea water intrusion, or for
soil salinization by capillary rise, although transport is
not multiphase, the problem is similar with strong
dependence on heterogeneity. Heterogeneity is also a
fundamental issue for superficial hydrological processes occurring in the vadose zone (infiltration, saturation, overland run-off) and their description. It has
long been recognized that the simple Horton model
does not fit all observations, but it is still widely used
for lack of other simple models. Heterogeneity is an
important aspect to consider since it can explain some
of the observations such as “saturation from below.”
This phenomenon might be due to local heterogeneities that lead to local perched saturated zones
and, in the end, to overland flow.
4. Dealing with heterogeneity is a three-part issue: the
problem to solve/the methods to use and models to
build/the data to collect, in that order, and the relevant
scales of each of these. But very often the problem is
ill-posed: the difficulty is there, the data are there, and
the hydrogeologist is asked to select and use the best
method to treat the data and solve the problem! In
general, however, there is some flexibility in designing an additional data collection phase, which provides a measure of freedom.
5. Many research programs have, at times, collected
large amounts of data without any model in mind, or
else an inappropriate one. In general, these data are
useless, even leading to the wrong type of model. A
way to remedy that is to develop experimental sites
where measurements and experiments are set up and
interpreted by joint research groups involved with
both experimental and modeling work. This was the
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case for the famous experiments in e.g. Borden, Cape
Cod, MADE, Mirror Lake in North America, the
Neckar alluvial site in Germany, or the fractured rock
site of Fanay-Augres in France. One may expect a
better bridging because experiments are built to feed
models and models evolve to account for new measurements.
6. Subsurface imaging is obviously an asset, as it can
help describe the geometry of the heterogeneous
system. Geophysics could be used more systematically; so far, good examples of a successful use of
geophysical images in the treatment of practical
problems where the heterogeneity plays a role are
lacking. Additional geological reasoning, as required
by the genetic approach, has the merit of putting the
problem into a broader context, and to bring in sources of information relevant to a site, but generic in
nature: e.g. succession of climate states, from which
the morphology and nature of the sediments can be
derived. On the North American continent, fluvioglacial sediments are probably the first target where
genetic studies would be of interest, see e.g. Anderson
(1989).
7. Except for geophysical imaging, the range of exploration methods to display heterogeneity is poor. Most
of the standard tests are integrating tests, not exploratory tests (pumping tests, tracer tests, etc). New
methods must be developed. In the detailed field
studies for transport (e.g. Borden, Cape Cod, MADE,
etc), one interesting tool was the borehole flow meter
used, to assess the local permeability variations of the
formation. But this tool was applied in a Geostatistical
framework, to assess the spatial variability (covariance) over the vertical and the horizontal of the
aquifer, treated as a continuous stochastic field. It may
be possible to use the same tool in a facies-type of
approach, to identify the local properties of each facies. Tsang and Doughty (2003) used temperature or
salinity changes by logging boreholes which they had
flushed and filled with freshwater to detect flowing
fractures. A related example of assigning facies permeability from the results of in situ tests is given in
Reis et al. (2000). They built a Geostatistical facies
model of a petroleum reservoir. They used this model
at the finest scale grid (without upscaling) to represent
radial flow around a well and thus interpret well tests
in four different wells. Rather than interpreting each
well test independently with a standard Theis approach, which would provide four different integrated
transmissivities, they decided to identify the individual permeability of each facies, assumed to have
identical properties at each well location, so that the
four tests could be simulated with the same permeability values assigned to each facies. Of course, at
each well, the facies distribution was different, but
one nice feature of the Geostatistical facies model is
that it can easily be conditioned at the wells, therefore
at each tested well, the exact vertical distribution of
the observed facies was represented in the radial flow
model. One needs to rethink the interpretation of the
“classical” data in terms of describing the heterogeneity. To quote C.V. Theis (1967): “I consider it
certain that we need a new conceptual model, containing the known heterogeneities of natural aquifers,
to explain the phenomenon of transport in ground
water.” New tests may have to be invented, that emphasize the role of heterogeneity rather than averaging
it (e.g. injection tests in packed sections of a well
rather than a full well pumping test, over the whole
thickness of the aquifer). Today, such tests are not
thought to be very useful because there is no way to
exploit their results. But if one thinks in terms of a
facies model, which has been built for the site, and
needs parameters for each facies, then there are tens of
tests that could be designed and interpreted with the
facies model, just as Reis et al. (2000) did. The experimental sites whose interest was emphasized in
Section 4 above should be used for that kind of development. Major breakthrough in experimental work
has already been achieved in many areas in nuclear
waste Underground Research Laboratories, such as
Lac du Bonnet, Canada, Yucca Mountain, US,
Grimsel and Mont Terri, Switzerland, Stripa and
sp, Sweden, Mol, Belgium, Asse and Gorleben in
Germany, Tournemire in France, etc, thanks to
available funding, creative ideas, and the need to
characterize in detail the formations, even if the design of new measurement methods was not the primary objective.
8. There is a need to create a catalog of aquifer and
aquitard properties and descriptions, with a summary
of the methods used in the reconnaissance, and the
modeling of the site (as well as the purpose of the
study). For one thing, there is a great similarity (even
if there are differences) in geologic objects. In tectonics, the study of the Himalayas has a lot to learn
from the results of the study of the Alps or Rocky
Mountains, and vice versa, as the processes at work
are the same. It should be the same for hydrogeology.
The structure of the sediments in a river along the
coast of Peru has probably a lot of features in common
with a similar river in the Pyrenees. This becomes
particularly true when considered in genetic terms, the
processes are the same everywhere, it is only the local
conditions (climate, geometry, geology...) that make
the two aquifers different. But that difference is secondary to describing the shapes, the texture, and
mostly the methods used for studying the “objects.”
Furthermore, one study in Texas may provide “default” values for a parameter in France that was not
measured. Take the case for instance of a local dispersivity value in a given facies. If a good description
of that facies is available (grain size analysis, type of
sediment, etc), and if there is a similar facies at a
different site, one may start by assuming that the
Texan value is a first guess for the new case. This
“default” value would be all the more credible if, by
assembling the catalog, similar facies are found to
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have similar properties at different locations. The link
between grain size and hydraulic properties needs to
be re-evaluated and used (note that German hydrogeologists make much more use of grain size data than
any others). The creation of a catalog has already been
advocated (e.g. Voss, personal communication 1988;
Dagan 2002), but never put into practice. One reason
may be the cost, and the exact focus of the catalog,
which should be clear: if integrated values are to be
transposed from site to site, this will never work (e.g.
a transmissivity measured in an aquifer in Texas will
be of little use for a case study in France). But if facies
are considered, sedimentary structures, rock history,
methods used to characterize sites, then the need for
the catalog may be more evident. One would need an
International Organization like UNESCO to develop
such a project on a world-wide basis.
9. The upscaling problem may need additional work, see
Noetinger (this issue). The problem that Zinn and
Harvey (2003) emphasized is a major one. When one
does upscaling, the small-scale boxes that need to be
“averaged” into a big box may have internal structures
that do not permit them to be treated as a “homogeneous” small-scale medium defined by one single
value (of permeability, even with anisotropy), as
shown by these authors. The small-scale boxes have
themselves complex internal structures, with different
connectivity properties. In other words, there is a
nested scale problem to consider. The upscaling in the
vertical direction also needs more attention, the scale
of variability is finer and the measurements may not
always provide an adequate description of the true
structure in the vertical direction (there is no easy
method to measure vertical permeability in boreholes). There is a need to look again at upscaling.
Until the large-scale permeability of an aquifer can be
reconstructed from small-scale measurements, there
will be a credibility problem for hydrogeology.
10. The connectivity issue is of utmost importance, not
just for upscaling, but for correct flow and transport
predictions. Just to give an example, the Rhine Valley
aquifer in France has been heavily polluted by salt
brines in the area upstream from Mulhouse by
leaching from potash mine tailings. This started in the
early 1900s, and in the 1980s, the salt plume had
moved several kilometers downstream, and was getting closer to the water supply wells of the city of
Mulhouse. Detailed surveys had been made of the
position of the plume, its spreading and its progression, until it was decided to move the wells laterally to
a non-polluted area. Alas! When the new wells were
drilled, it was discovered that a second plume, totally
undetected, and connected to the first one only in the
source area, was also present, and that the selected
“non-polluted” location was no better than the initial
one! In this case, the distribution and connectivity of
high-permeability facies should have been studied; in
other cases, it is the low-permeability barriers connectivity that would matter. Is the facies approach the
best one to tackle this issue? Or is percolation theory
the right tool? Or the fractal approach? Again, there is
a scale problem; percolation thresholds are valid for
ensemble averages, i.e. large-scale problems, and
detailed facies description is perhaps better suited for
the small scale. This is a very important research
topic. The multiple-point geostatistical approach may
be able to respect the connectivity pattern of a training
image; the definition of a connectivity function which
can be imposed on a simulation by simulated annealing may be another, see e.g. Allard (1994) or
Western et al. (2001).
11. Heterogeneity of low-permeability layers and aquitards is almost an untouched subject, although very
relevant for waste storage or brine or pollutant leakage
through aquitards to aquifers. Heterogeneity can be
linked to sedimentological processes, such as those
studied e.g. by Ritzi et al. (1995) on till and lacustrine
clays of glacio-fluvial aquifers; it can also be linked
with tectonic processes in the underlying or overlying
aquifers. It has indeed frequently been observed, when
a multilayered aquifer model is calibrated, that the
leakage factor needs to be increased in the vicinity of
faults, see e.g. Castro et al. (1998). Other factors
should also be considered.
12. In a genetic approach, rock properties evolution
through time due to geomorphological and mechanical processes, to weathering at the surface, or to dissolutions/precipitation by geochemical processes at
depth are still almost virgin topics, where predictive
modeling is concerned.
13. It has been emphasized here that hydrogeology has so
far failed to adequately apply fundamental geologic
methods and principles to aquifer characterization. In
the future, this will be an important field of activity.
Rather than bringing in merely “static” geological
characterization, it seems appealing to consider
modeling approaches used in other Earth sciences
fields such as geomorphology, marine or continental
sedimentology to generate particular landforms and
features. One interesting approach was developed by
Niemann et al. (2003) to simulate topographies from
sparse elevation data. Their model embeds the effects
of evolutionary processes (tectonic uplift, fluvial incision and hillslope erosion). The simulated surface is
iteratively adapted to observation points by spatially
changing the erodability field. This leads to surfaces
that are consistent with the heterogeneity, roughness
and drainage properties of fluvially-eroded landscapes
as well as being in exact agreement with observation
points. Following this approach, one could consider
embedding empirical laws describing the 3-D structure of sediments or features in Geostatistical methods
or interpolators.
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Conclusion
In conclusion, the authors of this paper believe that the
future of “dealing with heterogeneity” in hydrogeology
depends largely on a conscious decision to better characterize, describe and model the geology of the sites of
interest, similar to what has been done for the last 25
years in the oil industry. In many instances, the homogenization or averaging approach that has been commonly
used so far in hydrogeology has shown its limits. Detailed
studies are needed of outcropping analogs, catalogs of
well-characterized study sites, improved geophysical imaging at the depth of interest with dedicated methods, as
the seismic methods used in the petroleum industry are
poorly suited to superficial aquifers. There are already a
series of facies models that seem adequate to model the
observed structures, they should combine genetic approaches and Geostatistical, Boolean or Markovian
methods. There is certainly room for improvement in
these methods, but what is most needed at this time is a
thorough inter-comparison of these facies modeling approaches (and of simpler models too) on a set of wellcharacterized study sites, where the geologic description
is properly done, and where transport experiments are run,
like in the old Borden, Cape Cod, MADE sites. When a
systematic comparison of methods is done, as for the
inverse in the 1990s (Zimmerman et al. 1998), the outcome can really show the possibilities and limits of each
approach, whereas, at present, most authors have shown
how good their own method is, on theoretical or real
examples, but have not convincingly shown how it
compares with others.
A second area of development is to derive methods to
identify the material properties at the scale of the facies
structures. All the facies need to be identified, from the
highly permeable to the low, including the intermediate
mixed texture facies such as sandy mud, silty clay, etc,
that are presently often ignored. The methods to identify
the facies properties can go from newly designed in-situ
tests at the proper scale, to correlating these properties
with indirect data like grain size, geophysical properties
(electric, mechanic, magnetic, etc) or a combination of all
these (joint geophysical inversion). On that aspect, the
geophysicists need to start thinking about defining the
underground by complex facies models, with which they
would try to identify by inversion the physical properties,
and stop offering “layered cake” structures where they
have identified equivalent properties that are no longer
wanted.
A third area of development is to derive techniques to
assess the connectivity of facies. The geometric properties
of the facies are the key ingredients, which can result
from geometric assumptions (directional transition probabilities, length distribution, etc). But they can also result
from field testing, by interference tests, environmental
tracer studies, temperature measurements, etc. As was
shown here, the upscaling issue, which often remains a
necessity, is very much dependent on the correct assessment of the connectivity of facies.
Acknowledgments The authors wish to express their gratitude to
Pr. Graham Fogg and Pr. Gedeon Dagan for their detailed review
on an earlier draft of this paper, which helped them immensely to
improve the text. A third anonymous reviewer is also thanked for
his/her helpful and wise comments. The comments of two colleagues, Jean-Paul Chiles (Paris School of Mines) and Pierre
Delfiner (Total), are also gratefully acknowledged. Finally, the
authors wish to thank the Editor in Chief, Dr. Clifford Voss, for
inviting them to write this paper, and for numerous comments and
suggestions throughout the preparation of this paper
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315
Stochastic methods for groundwater resource protection zones 1
Dr Adrian Butler
Department of Civil and Environmental Engineering, Imperial College London,
Exhibition Road, London, SW7 2BU, United Kingdon.
E-mail [email protected]
1. Introduction
Groundwater resources in arid and semi-arid regions are extremely important for
the viability of these areas. Therefore, ensuring their long-term protection, and in
particular the protection of abstraction wells is essential. Furthermore, the technical,
legal, social and financial difficulties that can arise from restoring contaminated
groundwater mean that the protection of aquifers is preferable. Regulations for the
protection of drinking water wells usually require the delineation of areas that define a
prescribed minimum groundwater residence time and the entire catchment area of the
well. On the one hand there are existing or planned activities, which represent hazards
or risks for a particular groundwater resource, and on the other hand, the aquifers
exhibit a certain degree of vulnerability to a wide range of chemical and biological
pollutants.
The protection of a groundwater source (Figure 1) is approached by considering
the risks to the quality of that source. Because a source is an active abstraction, it is
presumed that the maximum level of protection is required. The initial approach is to
identify all pathways to the source, irrespective of relative vulnerability or any
potential hazard. The concept is one of reducing risks of contamination to the source
to a minimum. Thus, this concept usually implies the determination of an area, known
as the Zone of Contribution or ZOC (see Table 1 for a list of terms and definitions),
about the source, from which most hazards are to be excluded, through land use
planning. The ZOC represents an ‘all risks’ source protection zone (SPZ).
Figure 1 An illustration of the effect of a pumping well on groundwater flow.
1
Material derived from Stauffer, F., Guadagnini, A., Butler, A.P., Hedricks Franssen, H-J., van de Wiel,
N., Bakr, M.I., Riva, M. and Guadagnini, L., Delineation of source protection zones using statistical
methods, Water Resources Management, (2005), 19, 163–185.
317
However, the SPZ may be modified (i.e. the area reduced) by accepting very low
risks to the source arising from unspecified potential contamination originating from
certain sub-zones of the ZOC (Figure 2). The sub zones are usually delineated by
determination of those pathways with potential travel times to the source above a
certain limit (e.g. 50 days). Thus certain activities, in terms of land use, may be
permitted which represent hazards that do not significantly increase the risk to the
source when combined with long pathways.
Table 1. Terms and definitions commonly used in well capture zones
TERM
DEFINITION
Capture Zone
The area around the well from which water is captured within a certain time t.
Sometimes the time dependent aspect is emphasised by using the word timerelated capture zone, but in this report the word capture zone will
automatically imply time dependence.
Well catchment
The capture zone for infinite time. It encompasses the entire area over which
the well draws the water pumped from it. Another word that can refer to both a
capture zones and a well catchment is the zone of contribution (ZOC).
Isochrone
The boundary of a capture zone. It is a contour line of equal travel time to the
well.
Well head
protection area
(WHPA)
The area around a pumping well, which is associated with a specified level of
protection. Its delineation is a political consideration based on a risk analysis,
which may or may not involve an actual capture zone delineation.
Source Protection zones
around a borehole
III
II
I
I – Inner zone (time related)
II – Outer zone (time related)
III – Total source catchment
Figure 2 Illustration of groundwater source protection zones
Basic elements for the protection of groundwater from the point of view of water
quality are (Figure 2):
x
Well/spring capture zone (time-related): Normally it is required that the residence
time of any groundwater or pollutant should exceed a prescribed value (e. g.,
between and 10 and 50 days according to the regulations on water resources
protection in many countries). The idea behind this regulation is that pathogenic
microbes are generally eliminated within this time span, and that in the case of
hazards there would be enough time for interventions (abstraction of polluted
water, establishment of hydraulic barriers) or other remediation measures.
318
x
Well/spring recharge area: Any groundwater within this zone, and therefore any
pollutant, would eventually reach the well, provided the flow field within this
zone is at steady state. Regulations for the protection of water resources require
the designation of recharge areas of pumping wells, which are endangered by
pollution. The recharge area is related to the well/spring catchment or wellhead
protection zone. Springs are normally treated in a similar way as wells.
Necessary measures and restrictions with respect to land use and restrictions of
human activities within established protection zones are defined by the regulations.
The protection of aquifers should be accompanied by a monitoring of the piezometric
head and of the groundwater quality by an appropriate network of observation wells
and periodical sampling. To assess the yield and potential capacity of groundwater
resources, and to calculate the extent of the capture zones and contaminant travel
times, the flow and transport characteristics of the aquifer are needed. This includes
also an assessment of the temporal development (seasonal development, long term
development) of the groundwater flow.
2. General procedure for the delineation of protection zones
The following types of groundwater protection zones are considered here: the
time-related capture zone of a well, and the recharge area of a well. Well capture
zones for a prescribed groundwater residence time can be determined by evaluating
the corresponding isochrones, which are the contour lines of equal groundwater
residence time. These isochrones can be calculated analytically for a system of wells
in a uniform base flow (e. g. Bear and Jacobs, 1965). They can also be computed
numerically for arbitrarily shaped flow fields (Kinzelbach et al., 1992), for example
by back-tracking a fluid particle starting at the well until the prescribed residence time
is reached. Well catchments can be determined numerically by back tracking a fluid
particle starting near the stagnation point of the well. In both cases the detailed
velocity field is required, assuming that advection is the dominant process. For the
evaluation of the velocity field the following parameters and conditions are generally
required:
(1)
The flow geometry: This information is obtained from hydrogeological
investigations. The prevailing flow field can often be approximated by a
horizontal two-dimensional (2D) flow and transport model. Moreover, compared
to three-dimensional (3D) flow the formulation and numerical implementation
of 2D models is usually much simpler than in the 3D case. Nevertheless, it
should be kept in mind that 3D effects may be important in practice. For
instance, the evaluation of a 3D capture zone or catchment, at least in the
vicinity of the well, is (in principle) required when dealing with partially
penetrating or partially screened pumping wells.
(2)
The pumping rate of the well: the given or planned schedule of the pump should
be taken into account.
(3)
The groundwater recharge rate: the rate is estimated on the basis of hydrological
considerations.
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(4)
The infiltration rate from rivers and creeks: the rate can be estimated on the
basis of hydrological considerations, or by calibration of a flow model using
nearby head and/or concentration data.
(5)
The levels of the bottom and of the top of the aquifer formation: this information
is generally obtained from borehole and/or geophysical investigations.
(6)
The piezometric head of the aquifer: this information is generally obtained from
boreholes and/or geophysical investigations.
(7)
The location of the boundary of the flow domain to be investigated: this
information is obtained from a regional hydrogeological and hydrological
investigation. The boundaries are often chosen in such a manner that a feasible
formulation of the boundary conditions (fixed head or streamline) can be
obtained.
(8)
The boundary conditions: this information consists of the heads at the boundary
(or portions of it) or of the water flux through the boundary (or portions of it).
This information can be obtained from hydrological and hydrogeological
investigations.
(9)
The hydraulic conductivity (or transmissivity) of the aquifer: this information
can be obtained from pumping test evaluation or other procedures.
(10) The porosity of the aquifer: this information is relevant for proper isochrones
prediction and can be deduced, for instance, from tracer tests.
3. Impact of parameter uncertainty
Relatively small capture zones can usually be determined in a reasonable manner
using the above stated principles. However, problems arise for larger capture zones or
recharge areas due to the impact of parameter uncertainty. Evers and Lerner (1998)
asked the question "How uncertain is our estimate of a wellhead protection zone?"
Therefore, the above list of parameters and conditions should be discussed in a
qualitative manner with respect to the associated parameter uncertainty:
(1)
The extent of the flow domain is subject to uncertainty, mainly due to the
extrapolation and interpolation of data.
(2)
The pumping rate of the well is probably the less uncertain of all information.
Often, long term averaged pumping rates can be used. However, the pumping
schedule can affect the capturing mechanism by the well by the time-dependent
velocity field.
(3)
The groundwater recharge rate can, in general, only be indirectly determined. It
depends on the rainfall rate, on the evaporation and transpiration rate, and on the
subsequent flow processes in the unsaturated zone. Overall, the recharge rate is
time dependent and more or less spatially variable. Often these effects can
hardly be assessed precisely. Even the temporally and spatially averaged
recharge rate may show considerable uncertainty.
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(4)
The infiltration rate from rivers, creeks, and lakes is difficult to assess since it
cannot be measured directly, in general. It depends on the local infiltration
conditions, which can be affected by clogging. The rate is in general time
dependent and spatially variable.
(5)
The level of the bottom and the top of the aquifer are usually based on local
borehole information and can be obtained by interpolation. Consequently, some
uncertainty remains. The situation may be improved by a combination of
geophysical techniques.
(6)
The piezometric head of the aquifer is based on local borehole measurements
and represents valuable data used for a calibration of the flow model. The
piezometric head essentially dominates the flow directions. Consequently,
transient effects in the head field can be of utmost importance. It is usually
vertically averaged information while in some cases can be known at different
intervals along a vertical.
(7)
The location of the boundary of the aquifer is based on a regional
hydrogeological and hydrological assessment and is always subject to
uncertainty.
(8)
Fixed head boundary conditions are subject to uncertainty caused by data
interpretation and interpolation. The transient behaviour of these conditions can
hardly be assessed in detail. Flux boundary conditions are also difficult to
estimate. They can often be determined in a satisfactory manner with the help of
flow models, provided that reliable data of hydraulic conductivity and
piezometric head are available. Nevertheless, some uncertainty inevitably
remains. The averaged value and the transient behaviour of both types of
boundary conditions can be important for the flow field, and, therefore, for the
location of the capture zone or catchment.
(9)
Hydraulic conductivity always shows a more or less pronounced spatial
variability due to the heterogeneous nature of aquifers. Therefore a thorough
investigation of the field scale hydraulic conductivity is advisable. The local
values can never be known in detail everywhere. Spatial variability can
considerably affect the uncertainty of the location of the capture zone or
catchment. In addition, the scale at which the measurements have been taken has
to be carefully considered in the evaluation of the measurement.
(10) Aquifer kinematic porosity directly affects the flow velocity and therefore the
residence times, which subsequently determines the location of the capture zone.
Moreover, a spatial variability of field scale and local porosity may exist also in
unconsolidated aquifers. However, the effect of a spatial variability can be
smaller than that of the hydraulic conductivity.
Many of the above listed items concern local information, which is typically
measured in boreholes. Therefore, the quality of the overall information in a particular
flow domain very much depends on the spatial and/or temporal density of the
available data. However, due to economic and logistic reasons, the information is
often sparse. The location of data points is normally restricted to particular regions
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within the aquifer. Similarly, the temporal frequency of measurements is often limited.
Moreover, experimental data are sometimes corrupted by measurement and
interpretive errors. Overall, the combined effects of the uncertainty of all parameters
and conditions can considerably affect the precision of the calculated capture zone or
catchment. For small areas a more simplified and intuitive assessment of the
uncertainty is often possible, which can be taken into account in the delineation of the
protection zone. However, for larger areas the uncertainty can be quite large.
Depending on the economical and ecological importance of the protection zone the
implications of the degree of uncertainty associated with its predicted location can be
prohibitive. Therefore, methods are needed to quantify the uncertainty and provide
guidance in the acquisition of site data to reduce it. In general, uncertainty can be
reduced by increasing investigations (borehole investigations, parameter estimation,
etc.). However, since resources are limited the task should consist of a methodological
approach, which is pragmatic in the sense that it optimises efficiency. Consequently,
there is a need for knowledge and tools, which enable a conceptual and quantitative
assessment of the impact of parameter uncertainty on the location of existing or
planned protection zones.
The consequence of the uncertainty of the essential parameters, which determine
capture zones or catchments of a pumping well, is that the location of these zones
cannot be determined with certainty. Therefore the location of the protection zones
can only be defined in a statistical manner. Consequently, the best we can do is to
offer a probability map, rendering the probability with which a particular location
belongs to the capture zone or catchment. Such concepts can be fed directly into a
risk-assessment of a particular groundwater resource.
4. Importance of data in the analysis of well protection zones
The uncertainty about the location and the extent of protection zones can
generally be reduced by an increase of direct measurements (for example by
increasing the amount of hydrogeological data, such as hydraulic conductivity, or
direct measurements of state variables, such as piezometric head). Furthermore,
geological data from field investigations (such as borehole descriptions, cone
penetration tests, etc.) can, in principle, also be used to constrain predictions, resulting
in a global reduction of uncertainty in the delineation of protection zones. The most
important hydrogeological data are listed in Section 2. These data may be used:
x
To establish deterministic flow and transport models of the aquifer. Here, we use
the wording "deterministic" in order to identify a model, which does not provide,
by its nature, a quantification of uncertainty associated to predictions other than by
means of a traditional sensitivity analysis. Deterministic models, based on
calibrated averaged parameters (using the concept of equivalent parameters for
different aquifer zones), are often obtained using various manual or automated
optimisation procedures.
x
To deterministically delineate protection zones of pumping wells for various
conditions (boundary conditions, pumping rates, recharge conditions, etc.), using
322
the calibrated flow and transport models. However, a considerable uncertainty
may remain; the latter cannot be quantified, by definition, via the use of a
deterministic model.
x
To evaluate the statistical and stochastic parameters characterizing spatial
variability.
x
To establish a stochastic model (Section 5) based on the deterministic (i. e.,
certain, or known with rather reasonable certainty) and stochastic parameters,
forcing terms and boundary conditions.
5.
Modelling spatial variability and uncertainty of variables
The uncertainty of the parameters may be on the one hand due to measurement
errors inherent in a specific evaluation method, and on the other hand due to the more
or less strong spatial variability of many parameters, like hydraulic conductivity K(x),
which can never be known in detail everywhere. A viable way out of the dilemma
may be, for example, to cast the problem in a probabilistic framework and considering
the aquifer as one of many stochastic realizations. Stochastic variables such as
hydraulic conductivity do not behave like a white noise but show a distinct spatial
correlation structure with the correlation between two values, depending on their
distance. This correlation structure can be described by, e. g., a two-point (auto-)
covariance function. A further important feature is the probability density function of
the parameter under consideration.
A common approach in the practical application of prediction models is to
formulate equivalent parameters, thus replacing the real heterogeneous system by a
homogeneous equivalent model (e. g., Renard and de Marsily, 1997). Therefore one
task consists of finding adequate equivalent parameters, such as equivalent hydraulic
conductivity, as a function of quantities investigated.
The investigation of the impact of spatial variability of flow parameters concerns
the evaluation of the expected (mean) location (first statistical moment) of the capture
zone or catchment and its associated variance (second statistical moment). Such
moments can be theoretically based on the ensemble of all equally possible
heterogeneous realizations of the considered aquifer, honouring some sets of
measured data. The formulation of second moments is referred below by the use of
moment equations (Section 7). Widespread numerical procedures to the solution of
this problem are, for example, Monte Carlo – based techniques (Section 6), in which
space-dependent or also time-dependent parameter values of numerical models are
generated in a statistical manner, followed by a subsequent (numerical) solution and
analysis of each of the corresponding deterministic systems. Usually, the following
statistical or geostatistical parameters are required for a stochastic method:
x
The stochastic variable(s), e. g., the hydraulic conductivity K(x), or its natural logtransformed counterpart, Y(x)=ln(K(x)).
323
x
The probability density function, pdf, of the stochastic variable: This is often
approximated by a normal or a log-normal probability distribution. For lognormally distributed variables a log-transform of the variable is applied.
x
The ensemble mean value of the stochastic variable, e. g., <Y(x)>, or the
geometric mean value, Kg.
x
The variance of the stochastic variable, e. g., VY2; and the related standard
deviation.
x
The two-point covariance function CY(x1,x2). Often, a particular and convenient
invariant covariance model is selected to express the spatial correlation, e. g. the
exponential covariance model CY(x1,x2)=VY2 exp(-|x1-x2|/IY), where the correlation
length, IY, has to be evaluated on the basis of available data.
When considering randomness in more than one parameter, cross-correlations or
cross-covariance amongst different parameters might be needed. The parameters have
to be evaluated a priori, based on measurements or on experience. For spatially
distributed variables, like hydraulic conductivity, a variogram analysis (de Marsily,
1986) may be used.
6.
Use of Monte Carlo techniques
Monte Carlo (MC) techniques are general and versatile tools that allow one or
several parameters of a model to be uncertain. The idea is to generate many
realisations of synthetic aquifer flow (and eventually transport) models in such a
manner that they reflect the observed (experimental) parameter uncertainty. The
results are subsequently analysed in a statistical manner to quantify the uncertainty
inherent in the expected result. However, MC techniques are often very time
consuming and it not always clear what number of realizations is necessary for the
convergence of the method (e. g., Ballio and Guadagnini, 2004). Nevertheless, they
represent rather general and versatile tools for the investigation of ranges of spatially
variable parameters in the context of both linear and non-linear problems.
Consider a randomly heterogeneous flow domain, where, for simplicity, we
assume the hydraulic conductivity, K(x), to be the only source of uncertainty, and the
objective is to condition prediction and associated uncertainty to both hydraulic
conductivity and head measurements. For this domain a log-transformed hydraulic
conductivity field Y(x)=ln(K(x)) is generated (realization i), conditional to hydraulic
conductivity measurements in boreholes. Statistics of Y to be employed as input to the
generation process are usually derived from the available data set and are a critical
aspect of the entire procedure. The corresponding flow field is then calculated using a
forward numerical model and the computed hydraulic heads are compared with
available measured hydraulic head values. The misfit between the model predictions
and the measurements can be used to obtain a better estimate of the hydraulic
conductivity field using a numerical inverse modelling technique, thus effectively
conditioning the i-the realization of the MC process on the available hydraulic head
data. The ensemble of the results of all realizations (i =1, …, NMC) is then statistically
324
analysed in order to obtain predictions and quantify the uncertainty of the expected
results. Solving the inverse groundwater flow and/or transport model for each
realization is many times more intensive in computing time than solving the forward
flow and/or transport model and the computational time increases with the number of
conditioning data. A possible procedure for applying conditional MC techniques to
delineate well protection zones (or catchments) under steady state-flow conditions can
be summarized as follows:
(1)
Generation of NMC realisations (e. g., NMC = 500) of a random log-transformed
hydraulic conductivity field Yi(x)=ln(Ki(x)) (i = 1, ..., NMC), conditional to
available Kj(xj) data at locations xj , where j = 1,…, NK (see below), NK being the
number of conditioning K measurements.
(2)
Solution of the groundwater flow problem for each of the realizations Ki(x)
using a numerical inverse modelling technique, conditional to head data hk(xk),
with k=1,…,Nh or other type of data (see below). This results in updated
realizations Ki(x). Alternatively, conditioning on hk(xk) data can also be
performed simultaneously with Kj(xj) data.
(3)
For each flow field hi(x) one or several idealized tracer particle are released at
each grid cell of a regular grid. Numerical particle tracking is performed to
calculate the well capture zone (for the given residence time) or the well
catchment for each realization using an advective transport model. The end
points of all particles are recorded.
(4)
The statistical analysis of the ensemble of particle trajectories over all NMC
realizations provides the probability distribution P(x) of the capture zone or the
catchment. In other words, one obtains a map showing the spatial distribution of
the probability P that a fluid particle (or idealized tracer particle) released at a
particular location x is captured by the well within the requested residence time,
or the probability P that a particular location x belongs to the catchment.
Codes to generate random hydraulic conductivity fields Y(x)=ln(K(x))
investigated by the authors during this work, were SGSIM (Deutsch and Journel,
1998), FGEN (Robin et al., 1993), GSTAT (Pebesma, 1999), or GCOSIM3D (GómezHernández and Journel, 1993). GCOSIM3D is a follow-up version of SGSIM. It
enables a better reproduction of the covariance function CY(x1,x2), especially in case
of long correlation lengths IY.
The groundwater flow equation can be solved, for instance, with the finite
difference code MODFLOW (McDonald and Harbaugh, 1988) and forward particle
tracking can be performed using the computer code MODPATH (Pollock, 1994),
whereby at least one particle is released at each grid cell.
One of the stochastic inverse modelling techniques investigated in this work is
the Sequential Self-Calibrated method (Gómez-Hernández et al., 1997; Hendricks
Franssen, 2001) for the inverse modelling of groundwater flow and mass transport,
conditioned to hydraulic conductivity and head data. The method can also handle (in
principle) transient head data with the joint estimation of spatially variable hydraulic
conductivity and storativity fields and is formulated in three dimensions. The method
325
has been extended to estimate jointly hydraulic conductivities and recharge
(Hendricks Franssen et al., 2004a).
Another investigated inverse modelling technique is the Method of Representers
(Valstar, 2001). This inverse algorithm considers spatially variable parameters
explicitly. It can use both hydraulic head and concentration data to reduce the
uncertainty of model parameters for three-dimensional, quasi-steady flow regimes.
For conditioning on head measurements the Method of Representers has been
implemented into MODFLOW (Van de Wiel et al., 2002) and in a modified version of
the finite element code S-InvMan (Bakr, 2000). It has also been used to examine the
worth of head and concentration data on groundwater remediation using the pumpand-treat method (Bakr et al., 2003).
The accuracy of the numerical estimate of the probability P(x) that a location x
belongs to a capture zone or catchment strongly depends on the number of Monte
Carlo runs, NMC, and therefore on the convergence of the method. A minimum
number NMC for which the estimated value of P(x) is practically independent of NMC
has to be identified. Van Leeuwen (2000) suggests that, at least in two dimensions,
approximately NMC=500 realisations normally result in an acceptable convergence,
and that the convergence after about 1000 realisations hardly improves.
For illustration, the results of an unconditional Monte Carlo analysis aimed at the
evaluation of the catchment of a well is shown in Figure 3. At the western boundary a
constant head boundary condition was applied. The remaining boundaries were
chosen as impermeable. The geometric mean hydraulic conductivity Kg was 10 m/d,
the variance VY2 was 0.1, the correlation length IY=50m, adopting an exponential
covariance function for Y. The pumping rate was 200 m3/d and the recharge rate 1
mm/d. The number of Monte Carlo runs was NMC=1000. The map shows the
probability P(x), that a fluid particle at a given location x reaches the well.
Further numerical or analytical Monte Carlo techniques to deduce capture zones
or catchments in a statistical manner were suggested, e. g., by Kunstmann and
Kinzelbach (2000), Franzetti and Guadagnini (1996), Guadganini and Franzetti (1999),
Van Leeuwen et al. (1999), Wheater et al. (2000), Hunt et al. (2001), Feyen (2001),
Jacobson et al. (2002), or Feyen (2003). In addition, van Leeuwen et al. (1999) have
used Monte Carlo analysis to evaluate the impact of uncertainty in the location of drift
overlying a production aquifer on well capture zones at Wierden, Netherlands.
326
500
0
-500
-500
0
500
1000
Figure 3 Probability map of a well catchment; Monte Carlo results
(well located at x=0, y=0).
7.
Use of moment equations
A major conceptual disadvantage of Monte Carlo approaches is that they do not
provide a theoretical insight into the nature of the solution. Therefore there is a need
for alternative approaches. One method is based on the use of moment equations.
A complete first-order (in the variance of Y) stochastic mathematical formalism
that allows to obtain an estimate of the travel time and the trajectory (rendered by the
first moments) together with the associated prediction errors (rendered by their second
moments) for idealized tracer particles advected in a randomly heterogeneous aquifer
has been derived (Guadagnini et al, 2003; Riva et al., 2004) and compared against
numerical Monte Carlo simulations the formalism, solute particles are injected at a
single point and travel along a (random) trajectory towards a given discharge point or
line. Travel time mean and variance are functions of first and second moments and
cross-moments of trajectory and velocity components. Trajectory mean and variance
are functions of the statistical moments of the components of the velocity field. The
equations were developed from a consistent first order expansion in VY. As such, they
are nominally limited to mildly non-uniform fields, with VY2 < 0.5, or more
heterogeneous fields, in the presence of conditioning. The work has been developed in
two dimensions and the extension to a more general three-dimensional scenario is
straightforward. Furthermore, procedures were developed, which allow conditioning
of the statistical moments of the flow field and, therefore, of travel time moments, on
hydraulic heads measurements and / or aquifer architecture (e.g., Hernandez et al.,
2003; Winter et al., 2003).
Stauffer et al. (2002) investigated the uncertainty in the location of twodimensional, steady state catchments of pumping wells due to the uncertainty of the
spatially variable hydraulic conductivity field by a semi-analytical Lagrangian
technique. For the analysis it is assumed that the aquifer can be modelled as a steadystate horizontal, confined or unconfined system. The well discharge rate and the
recharge rate are constant. The uncertainty bandwidth of the catchment boundary is
327
approximated at first order (in the variance of Y) by formulating the transversal
second moment of the tracer particle displacements along the expected mean
boundary of the catchment, starting at the stagnation point in a reversed velocity field.
A special approach is suggested for the estimation and conditioning of uncertainty in
the location of the stagnation points. For illustration, the results for the estimated
catchment boundary of a well together with its uncertainty bandwidth is shown in
Figure 4. The conditions were the same as in Figure 3. A similar Langrangian
approach for time-related capture zones in heterogeneous aquifers without recharge
was presented by Lu and Zhang (2003).
y [m]
1300
800
300
0
500
1000
1500
x [m]
Figure 4 Uncertainty bandwidth (mean and 95% probability) of the location of a well
catchment boundary estimated by the Lagrangian semi-analytical method
(Stauffer et al., 2002).
Bakr and Butler (2004) have used an alternative numerical approach, in which
the original partial differential equation of flow and advective transport, is first
discretized on a specified grid using finite elements, and then the resulting equation,
the so-called space-time equation, is used to derive the statistical moments of the flow
and mass transport quantities. The approach is based on a Taylor’s series
approximation of the discrete system of equations and is often termed the vector
space-state/adjoint state approach.
8.
Impact of recharge uncertainty
The spatial variability of hydraulic conductivity is usually believed to be the
main contributor to the uncertainty in the estimation of a well catchment and,
therefore, its effects have been intensively studied (see, e. g., references cited above).
The impact of the spatially variable recharge on the estimation of a well catchment is
known to a lesser extent. In this case, the relevant question to be answered relates to
whether recharge uncertainty does contribute significantly to the well catchment
uncertainty in case there is at the same time a considerable spatial variability of
hydraulic conductivity. In addition, another source of uncertainty may be the temporal
variability of recharge.
328
The recharge uncertainty has been subjected to a synthetic study (Hendricks
Franssen et al., 2002). The study focused on a flow regime that is typical for a humid,
temperate climate. The average yearly recharge is chosen as 360 mm and the seasonal
variations in precipitation are not very strong. Furthermore, the aquifer materials have
a relatively high permeability. The main practical conclusions from the Monte Carlo
type study were:
x
The spatial variability of recharge has only a limited impact on the uncertainty of
the well catchment. The impact is larger in case of a larger recharge correlation
length. However, even in case of large recharge correlation length and
unrealistically large spatial recharge variability, its influence on the well
catchment is very limited for moderately heterogeneous hydraulic conductivity
fields (VY2|1). It is expected that for a moderately or strongly heterogeneous
hydraulic conductivity field the spatial variability of recharge is overruled by the
spatial variability of hydraulic conductivity.
x
Nevertheless, the uncertainty in the mean recharge has an important impact on the
well catchment uncertainty, also in case of a moderately or even strongly
heterogeneous hydraulic conductivity field. For recharge it is therefore important
to estimate the uncertain mean, while the detailed estimation of the spatially
variable patterns of recharge is normally not needed. It may be necessary to
estimate the mean areal recharge for multiple zones in case there is evidence of
varying means between different areas (as is the case for differences in land use).
x
The temporal recharge variability can be important to consider in some specific
situations. In case the groundwater residence time in the catchment is not clearly
longer as compared to the time scale on which the recharge fluctuates, it may be
necessary to investigate time series of recharge. In humid, temperate climate zones
the variations between years are normally limited. However, the recharge varies
significantly over a year. Normally a recharge minimum in summer and early
autumn and a recharge maximum in winter and early spring is reached. In case the
expected groundwater residence time in the catchment is not clearly larger than
one year, effects of temporal recharge variability may be expected.
9.
Impact of uncertainty in geostatistical parameters, average hydraulic
conductivity, and boundary conditions
In most of the studies in stochastic subsurface hydrogeology the covariance
function CY(x1,x2) and the geometric mean hydraulic conductivity Kg are assumed to
be known exactly. In practice, they are estimated from a limited number of hydraulic
conductivity data. As a consequence, these estimates are associated with a
considerable uncertainty. A further important source of uncertainty, which normally is
not addressed in hydrogeological studies, is related to the boundary conditions. The
location of the boundaries and the prescribed flux or prescribed head values on the
boundaries are normally assumed known.
The impact of the mentioned sources of uncertainty was tested in a synthetic
study (Hendricks Franssen et al., 2004b). The studied two-dimensional domain had
329
extensions of 5 x 5 km. The northern and southern boundaries were impervious and
along the western and eastern boundaries fixed heads of respectively 0m and 5m were
imposed. A pumping well was located 500m west of the domain centre. The area
received a spatially uniform recharge of 363mm/year. Steady-state groundwater flow
in an aquifer with constant thickness was simulated. A hydraulic conductivity field
was generated with a mean value of 102 m/d and an exponential covariance function
CY with a variance of VY2=1, and a correlation length of IY=500m (1/10 of the domain).
This reference field was considered as unknown reality of the study. As a
consequence, a water divide along the eastern part of the area was present and the
well pumps water from a considerable area located west of the water divide. Figure 5
shows the corresponding reference well catchment. The reference hydraulic
conductivity field was sampled 100 times by selecting 10 measurements each. From
the selected 100 random data sets 100 different mean hydraulic conductivity values
and 100 different covariance functions of Y=ln(K) were estimated. The impact of
these uncertainties on the estimation and estimated variance of the hydraulic
conductivity field, the hydraulic head field and the catchment zone were quantified.
The following practical conclusions could be drawn from this study:
(1)
The uncertainty in the mean hydraulic conductivity, e. g., Kg, had a very limited
influence on the characterisation of the hydraulic conductivity field, the
hydraulic head field and the catchment zone.
(2)
The uncertainty in the hydraulic conductivity covariance function CY(x1,x2) had
a slightly larger impact on the characterisation of the hydraulic head field, the
hydraulic conductivity field and the catchment zone than the uncertainty in the
mean hydraulic conductivity, but was nevertheless quite small. However, this
conclusion is valid only for the ensemble mean over all 100 samples of
measurements. For the case that just one set of measurements is taken, as is of
course the case in practical situations, the impact of the uncertainty of the
covariance function can be much more pronounced.
(3)
The uncertainty with respect to piezometric head values on prescribed head
boundaries had a large effect on the characterisation of the hydraulic head field,
the hydraulic conductivity field and the catchment zone.
(4)
Inverse modelling (conditioning to error-free piezometric head data) was able to
reduce the impact of the above mentioned three sources of uncertainty.
Note that the conclusions from the synthetic study cannot necessarily be
generalized to any other case.
330
5000
4000
y (m)
3000
Well
2000
1000
0
0
1000
2000
3000
4000
x (m)
Figure 5 Reference well catchment of the synthetic study for the investigation of the
impact of uncertainty in covariance function CY, the geometric mean hydraulic
conductivity Kg, and the boundary conditions (Hendricks Franssen et al., 2004).
10. Sampling design and monitoring strategies
Incorporation of measurement data through conditioning is a requirement for
reducing uncertainty in the location of the well capture zone or catchment. However,
the success of this conditioning depends on the type of measurement, the amount and
spatial pattern of the measurements and the measurement error (Bakr and Butler, 2003;
Hendricks Franssen and Stauffer, 2004). Furthermore, the conditioning performance
seems to be dependent on the type of aquifer (confined or unconfined), the correlation
length IY and variance VY2 and the well pumping rate (van Leeuwen et al., 2000; Bakr
and Butler, 2003; Van de Wiel et al., 2002).
Extensive Monte Carlo – based analyses on the effect of hydraulic conductivity
(or transmissivity) data and piezometric head observations (both separately and
combined) in reducing capture zone uncertainty enabled to provide a set of basic rules
about location and type of data and for the development of optimal measurement
strategies for uncertainty reduction. The effect of conditioning is primarily measured
in terms of the reduction in the width of the capture zone’s probability distribution.
In general, uncertainty related to well capture zones or catchments decreases
with increasing incorporation of hydraulic conductivity and hydraulic head data
(conditioning density). However, the conditioning effect for different sampling
schemes with similar amounts of measurement data can be quite different. What is the
relative worth of different types of measurements? Synthetic case studies indicate the
greater importance of hydraulic conductivity measurements over head data in
reproducing the reference well capture zone (Bakr and Butler, 2003; Feyen et al.,
2003). It appears that it is not possible to get satisfactory results using head
measurements alone, especially in highly heterogeneous aquifers (for VY2>1).
331
Although head measurements are capable of estimating hydraulic gradient quite
accurately, they do not contain sufficient information on the variation of hydraulic
conductivity which, in turn, leads to high variability in pore water velocity and hence
the well capture zone location. Furthermore, results show that a combination of both
head and hydraulic conductivity data can reduce the width of the capture zone
distribution more significantly than either type of data alone (Bakr and Butler, 2003).
Feyen et al. (2003) concluded that head observations are more effective in reducing
the width of the capture zone distribution, whereas hydraulic conductivity
measurements are more valuable in predicting the actual location of the unknown
capture zone. Feyen et al. (2003) also incorporated tracer arrival times, hydraulic
conductivity measurements and hydraulic head observations in the stochastic capture
zone delineation. Their evaluation indicates that travel time data seem to be effective
in reducing the overall uncertainty and to some extent in revealing large irregularities
in the shape of the capture zone. In general, the incorporation of tracer and solute
concentration enhances the aquifer characterization and the accuracy of flow and
transport predictions, since such data are complementary to head and conductivity
measurements (Medina and Carrera, 1996; Valstar, 2001; Hendricks Franssen et al.,
2003).
Observation network of head and/or hydraulic conductivity measurements can be
optimised in a systematic manner. Such methodologies can also provide a way to
minimize the number of sampling locations, required to reduce capture zone
uncertainty to an acceptable level. Van de Wiel et al. (2002) investigated several
strategies for selecting the optimal additional location for a piezometric head
observation in a synthetic confined aquifer with a single pumping well. The strategies
are: a) Selecting the location, where the head variance is highest; or b) selecting the
location where the sum of covariances between head at that location and at the other
potential measurement locations is largest; or c) selecting the location that minimises
the head variances summed over the model domain; or d) selecting the location where
the reduction in capture zone uncertainty is highest. The last strategy clearly showed
the best results in reducing the capture zone uncertainty. However, it is also the most
time intensive, whereas the three other criteria do not require Monte Carlo simulations.
Amongst these three design strategies, the strategy that minimised the summed head
variance performed poorly in terms of minimisation of capture zone uncertainty in
case of a relatively small number of selected head measurements.
Hendricks Franssen and Stauffer (2004) proposed optimisation procedures for
selecting new locations for both head and hydraulic conductivity data in a synthetic
confined aquifer with spatially uniform recharge and a single pumping well. The
algorithm enables to place additional measurement locations nearly optimally. The
true optimum can hardly be found since only a limited number of possible
combinations of new measurement locations can be analysed realistically, since
computation of all possible combinations, even for a few additional measurements, is
extremely time consuming. Two selection criteria were implemented, a) the
minimisation of the expected average log-transformed hydraulic conductivity variance,
and b) the minimisation of the average hydraulic head variance. It was found that both
strategies were successful. However, the differences between the optimal strategies
332
and heuristic criteria, where the sampling points are distributed evenly over the whole
aquifer, were small. This indicates that covering the domain of interest regularly with
a measurement network is a close to optimal strategy in characterizing the general
flow field. However, selecting measurement locations in zones with a capture
probability of about 0.05<P(x)<0.95 seems to be a better option for characterizing a
well catchment. Nevertheless, there is a trade-off between continuing adding
measurement points in these zones and placing additional locations in the areas
around the uncertain zones.
11. Application to a field case
Many of the methodologies developed for the stochastic characterization of well
capture zones and catchments have been intensively tested in several synthetic
numerical simulations. The next logical step was to apply these methodologies in a
real world case study. For this purpose the Lauswiesen test site in the Neckar valley
near the city of Tuebingen in southwest Germany was selected. This site has been
intensively studied before and during the W- SAHaRA project (Martac et al., 2003a, b)
including stochastic inverse modelling. For the sake of illustration, Figure 6 shows the
estimated 50 days well capture zone for the Lauswiesen site during the tested
pumping conditions, using a Monte Carlo approach based on the Representer Method.
The main conclusion from this field case is that the stochastic methodologies are
indeed capable of yielding reasonable stochastic estimates of the well capture zone or
catchment. However, as compared with a synthetic case there exist several
implications that point to the need for further research and development. In a synthetic
test case the performance of an algorithm can be easily evaluated in a systematic
manner. In practice one would like to use all available information, but then no data
are available anymore to test how well the model predictions performed. Although
detailed verification is in general not possible in practice, it is desirable to exclude
some data and/or tests from the conditioning procedures and to use them for
comparison with the model predictions. In the present test case we found that the
model predictions do not deviate too much from the experimental results. The peak
arrival time for three tracers was quite well predicted.
333
Figure 6 Lauswiesen field case: Map of ensemble averaged hydraulic head (m) and
the probabilistic 50 days well capture zone of abstraction well F0 calibrated for the
pumping stage. Locations of head measurements (H) and injection wells (F1-F6) are
marked. Dark grey represents a high value for hydraulic head.
It can be stated that the results are generally compatible with the tracer tests. The
most important conclusions were:
x
We found that the role of the river, and its interaction with the aquifer was crucial.
Measured time series of groundwater and Neckar river levels indicate that the
river level has a major impact on the hydraulic head distribution in the aquifer.
Errors in modelling this boundary condition prohibit a more reliable
characterisation of the aquifer hydraulic conductivity and of the capture zone. In
order to directly handle the inverse stochastic modelling of river-aquifer
interactions in an adequate and concise manner, further research is needed. In
addition, in order to enable a successful inverse stochastic simulation of the riveraquifer flow conditions more measurement locations along the river are needed.
x
The estimation of the hydraulic conductivity covariance function had to be based
on sparse data only. This was a further main source of uncertainty.
x
It was shown that inverse modelling is to a certain degree able to correct the
estimation error.
x
Data from sieve analysis (soft information) allowed a more realistic reconstruction
of the spatial distribution of aquifer hydraulic properties (Martac et al., 2003).
334
The practical case of the Lauswiesen field site shows that there is a need for
inverse modelling procedures that can handle three-dimensional aquifers with a large
number of grid cells and transient flow conditions. In this sense, parallelization of
Monte-Carlo type models can lead to an improvement in their performance. Ye et al.
(2004) presented a comparative analysis, in terms of runtimes, of the computational
efficiencies of parallel algorithms used in the context of recursive Moment Equation
and Monte Carlo methods. In addition, the models should be able to condition to
tracer test information on a routinely basis in order to further improve the predictions.
12. Conclusions
A result of incomplete knowledge of the essential parameters that determine well
capture zones or catchment areas is that the location of these zones/areas cannot be
determined with certainty, since the amount of available data is always limited. In
particular, the location of data points is often restricted to specific regions within the
aquifer (due to economic and logistic reasons). Furthermore, experimental data are
always corrupted to some degree by measurement and interpretive errors.
Consequently, the location of the protection zones can only be defined in statistical
manner and should therefore, be represented using a probabilistic catchment/capture
zone map (as illustrated in Figures 3, 4 or 5). This provides us the probability P(x),
with which a particular location x belongs to the capture zone or well catchment
(Section 3). This requires a stochastic analysis for well capture zone and catchment.
The general procedure is depicted in Figure 7. The probabilistic capture zone or
catchment map of a pumping well can then directly be used by the decision maker for
the delineation of the protection zone, based on specific political, ecological, and/or
economical reasons. Moreover, such a probabilistic representation naturally fits with
the requirements for a risk-based assessment that are frequently required in
groundwater management decisions.
The previous sections have described the development and evaluation of
methods and tools to produce stochastic capture zone/catchment maps of pumping
wells. However, it is important to emphasise that the selection of a specific method
for a stochastic analysis of protection zones of pumping wells will depend on specific
site conditions and data, i.e. the dimension of the flow problem, the flow geometry,
the aquifer type (confined, unconfined, multi-aquifer system), the recharge conditions,
the number of wells, the spatial variability of hydraulic conductivity, and the type,
number and location of conditioning data.
The methodologies that characterize capture zones and catchments stochastically
have reached the stage that they can be applied in practice, but further development is
needed so that they can be applied more routinely. Semi-analytical Lagrangian
methods without conditioning on measured data (Stauffer et al., 2002) are in many
practical studies an option to get a quick "idea" of the uncertainty of a well catchment.
They have the advantage that they need relatively little computing time. One of the
few special requirements would be the estimation of a reliable hydraulic conductivity
covariance function. Expert knowledge is needed to estimate such a covariance
function. In case of only few measurements, multiple calculations with different
335
covariance functions are possible, and not very time consuming. More general,
methods based on moment equations, conditional to measurements, have shown their
potential for interesting future applications.
Identify groundwater abstraction
and potential sources of pollution
Have well
capture/ catchment zones
been demarcated?
No
Delineate capture zones
Yes
Method of derivation
Existing numerical model (2/3D)
e. g. MODFLOW, MODPATH
Simple analytical solution
(e.g. Bear and Jacobs)
Is there a
need to quantify model
uncertainty?
No
STOP
Yes
Collect and/or collate
data
Develop new/modified
numerical model (2D/3D)
Probabilistic analysis:
Identify sources of uncertainty (e.g. conceptual model,
hydrogeological parameters , boundary conditions)
Undertake preliminary unconditional simulation
Incorporate data for conditional simulation
Select method
1
Monte Carlo
analysis
Select type and location of
additional input measurements
Yes
2
Moment Equation
analysis
Reduce
uncertainty
No
STOP
Figure 7
Stochastic analysis of well capture zones or catchments.
336
Numerical Monte Carlo type methods are already well developed for flow
problems. They tend to be more flexible and can for example handle cases with a
large log-transformed hydraulic conductivity variance and may also handle strongly
non-linear systems. In addition, in case of inverse modelling they can also treat
systems with various sources of uncertainty jointly, like hydraulic conductivities,
spatial and temporal recharge and various boundary conditions.
The focus of this work has been primarily targeted at unconsolidated porous
media. Although fractured rock systems have not been specifically considered, the
techniques developed here may also be applied where such systems are treated as
equivalent porous media.
Acknowledgements. The material presented in this study was undertaken within the
European research project "Stochastic Analysis of Well Head Protection and Risk
Assessment" (W-SAHaRA), Contract Nr. EVK1-CT-1999-00041.
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