A study of regional temperature and thermohydrologic effects of an

Transcription

JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 86, NO. B5, PAGES 3759-3770, MAY
10, 1981
A Studyof RegionalTemperatureand ThermohydrologicEffectsof an
UndergroundRepositoryfor Nuclear Wastesin Hard Rock
J. S. Y. WANG, C. F. TSANG, N. G. W. COOK, AND P. A. WITHERSPOON
Earth Sciences
Division,LawrenceBerkeleyLaboratory,Universityof California,Berkeley,California94720
Heat released
by the radioactivedecayof nuclearwastesin an underground
repositorycauses
a longterm thermaldisturbancein the rock mass.The natureof this disturbancefor a planar repository3000 m
in diameterat a depthof 500m belowsurfaceis investigated.
Loadedinitiallywith a powerdensityof 10
W/m 2 of spentfuel assemblies
10yearsafterdischarge
froma reactor,the maximumincrease
in temperatureof therepository
in graniteis about50øCandtheepicentralthermalgradientabout70øC?km.Differentwasteformsand periodsbeforeburial havesignificant
effectson the thermaldisturbance.
The effectsof temperature
changeson the groundwater
flow are evaluatedwith simplemodelsof a vertical
fractureconnectedto a horizontalfracturein the rockmass.The buoyancygroundwaterflow throughthe
vertical fracture is a function of both the vertical and the horizontal fracture transmissivities, as well as
thechanges
in densityandviscosity
of groundwater
causedby the temperature
changes.
Finite hydraulic
rechargefrom the surrounding
rock massaffectsthe thermohydrologic
disturbance.
INTRODUCTION
Significantquantitiesof nuclearwastesexist already and
continue to be produced[Departmentof Energy, 1978]. At
present,mostof thesewastesare storedin near-surface
facilities. Although every precautionis taken to protect man and
the environment from the potential hazard posed by these
wastes,near-surfacestorageis not regardedas a satisfactory
long-termproposition.Disposalof the nuclearwastesby deep
burial in suitablegeologicformationsis generallyfavored [Interagency
ReviewGrouponNuclearWasteManagement,1978].
The principalattractionof disposalby deepburial is that it
providesa high degreeof physicalisolationof the nuclear
wastefrom the biosphere.The principalconcernof deep burial is that at no stageshouldtoxic componentsof thesewastes
find their way back to the biosphereat levelswhich are not
completelyharmless.
A wealth of scientificknowledge and engineering experience existsconcerningthe excavationof undergroundopenings in a wide variety of geologicmedia. Unfortunately, no
comparableexperienceexistsconcerningthe effectsof the
generationof heat within suchopeningsor of the leakageof
materialsfrom suchopeningsback to the biosphere.
Salt formationshave generallybeenpreferredas candidates
for deepgeologicaldisposalfor a numberof reasons[National
Research Council, 1957]. The relatively high thermal conduction of salt facilitatesthe dissipationof heat releasedby
the radioactivedecayof nuclearwastes.The viscousplasticity
of salt,particularlyat elevatedtemperatures,
assuresthat the
excavationwithin which the wastesare disposedwill, in the
long term, seal themselvesby deformationof the salt itself,
and the presenceof the salt atteststo very slow dissolution
and transportby movementof groundwater.
However, other geologicalmedia may be equally or even
better suited for constructionof deep underground repositoriesfor nuclear wastes,providedthat the openingsand accesswaysto the repositorycan be sealedadequately.The permeabilityof intact piecesof many crystallineand argillaceous
rocks is at least as low as that of salt. However, the permeability of massesof suchrocksarisesmainly from the hyThis paper is not subjectto U.S. copyright.Publishedin 1981 by
the American GeophysicalUnion.
Paper number 80B1825.
draulic conductivity of joints and fracturespervading them
but may still be sufficientlysmall to retard the movement of
groundwaterbetweenthe repositoryand the biosphereto an
adequate degree.
The purposeof this paper is to studythree important aspects
concerningthe designand performanceof such underground
repositoriesfor the disposalof nuclear wastesin hard rock.
First, the heat releasedby the radioactivedecayof the nuclear
wastescauseschangesin the spatial and temporal distributions of temperaturein the rock masswithin which the repository is located. These changesin temperatureproduce thermally induced componentsof compressivestressin the heated
portions of this rock mass and tensile componentsof stress
outside of this zone. These thermally induced changes in
stressmay affect the performanceand designof a repository.
The distributionof temperaturesaround a repositoryis necessary to evaluate the thermally induced changesin stress.However, the evaluationof the stresschangesis not part of this paper. Second, the magnitude and temporal changes of the
temperaturesin the rock masswithin which the repositoryis
located are affectedby the kind of waste and the time after removal from the reactor at which it is buried.
The character-
isticsof different wasteson the temporal changesin temperature are examined. Finally, changesin the temperatureof the
groundwaterin the rock masscontainingthe repositoryaffect
both the densityand viscosityof the water significantly.These
changesin densityand viscositymay resultin perturbationsof
the original hydrologicflow, which could affect the performance of the repositoryin isolating toxic componentsof the
wastesfrom the biosphere.A model of buoyancygroundwater
flow through a simple fracture systemis used to assessthe
magnitude of this phenomenon.
THERMAL
DISTURBANCE
RepositoryModel
To study the long-term regionalchangesin temperaturein
the rock mass,the repositoryis idealized to be a flat circular
disk loaded uniformly with nuclear waste at time t -- 0. The
repositoryis assumedto be a depthD of 500 m below surface
in granite and to have a radius R of 1500 m. The principal
mode of heat transfer from the nuclear waste to the rock mass
is assumedto be by heat conduction.This has been proved to
3759
3760
WANG ET AL..' REPOSITORY TEMPERATURE AND BUOYANCY FRACTURE FLOW
TABLE 1. Thermal Propertiesof Rocks
Rock
K,
PR,
CR,
W/m/øC
kg/m3
J/kg/øC
Granite*
2.5
2600
Stripagranite'•
Basalt:•
Shale{]
3.2
1.62
0.90
2600
2865
2300
•
10-6 m2/s
836
1.15
837.36
1164
1000
1.47
0.486
0.391
* Kappelmeyerand Haenel [1974].
'•Pratt et al. [1977].
•Martinez-BaezandAmick [1978].
õFairchildet al. [ 1976].
be a good assumptionbased on recent Stripa data analysis
[Hood, 1979].
The temperature field T(r, z, t) resulting from heat conductionin the rock massis given by a solutionto the diffusion
equation:
10I OT}
02T
lOT
r Orr-•r + Oz
2-gr Ot gr--K/pRCn(1)
t
gr
K
pn
Cn
ß
4(,rgr3t3)
'/2
exp-
4grt
Io
r'dr' (3)
where Io is the zeroth-order modified Bessel function of the
first kind.
For the purposesof this paper, (2) and (3) have beensolved
numericallyfor a number of expressions
qfft').Temperatures
on the z axis for simple functionsof qfft'), such as constant
power, exponentialdecayof power, and a decreasein power
inverselyas the squareroot of time can be expressed
in terms
of tabulated functions.These are given in Table 2 and can be
usedto checkthe accuracyof the numericalsolutionsto (2)
and (3). In this paper,numericalresultsreproducetheseanalytical solutionsto within 0.01øC.
where
r
z
Y,•>(r, z, t) =
radial coordinate;
vertical coordinate;
time after loading;
thermal diffusivity;
thermal conductivity;
density of the rock;
specificheat of the rock.
Values for the thermal properties of granite and other hard
rocks are given in Table 1.
When heat is releasedfrom the nuclearwasteat an average
rate qfft')by the disklike repositoryfrom time t' -- 0 to time t'
-- t, the temperaturechangeat any point (r, z) is
DifferentFormsof Waste
At present,considerationis being given to the disposalof
two principalformsof nuclearwastegenerallyknownashighlevel waste.They are spentfuel, as dischargedfrom a reactor,
and the productsfrom reprocessing
spentfuel to recoverthe
uranium and plutonium. Figure I illustratesthe power densitiesof the two principalwasteforms from a pressurizedwapnCRck(t')
V_•>(r,
z,t- t')dt'
ter reactor[Kisneret al., 1978].Either form of high-levelwaste
is assumedto be buried 10 yearsafter dischargefrom the reacpnCnck(t')
V+•,(r,
z,t- t')dt' (2) tor at an initialloadingdensityof 10W/m 2in theplaneof the
repository.The curvesin Figure I give, numerically,the valThe first term in this equationrepresents
the changein tem- uesof rk(t')which are usedin (3) to calculatethe changesin
perature from the disklike sourcein the infinite medium and temperaturefield. The regionalcontoursof thesechangesin
the secondterm from an image sourceis a correctionfor the temperature 100 yearsand 1000years after burial are as illuspresenceof a boundaryat constanttemperatureat the ground trated in Figures 2 and 3. It is important to note how much
surface,z -- 0. The functionsV,•, in (2) are the instantaneous greater are the magnitudesof these changesfor spent fuel
unit-strengthdisk sourcesat t' of radiusR in the planesz
than for reprocessed
waste,especiallyafter 1000yearsof burial.
-T-D[Carslawand Jaeger, 1959]'
AT(r,
z,t)- 1
TABLE 2. Analytic Solutionsof TemperatureRise on the z Axis of Disk Source
PowerForm
PowerDensity•t)
SolutionFormf(x, t)*
Constant
•(0)
•b(O)Ocrt)
•/2ierfc
(x)
K
Exponentialdecay
•0) exp (-ht)
•J(0)
(KT/•x)l/2
Im[•l(•kt)
1/2
4'ix]
K
Inversesquare
root
2ToK/(•r•zt)
•/e
Toerfc(x)
*AT(O,z, t) -• f {[(z+ D)e/4tcrt]
l/e,t} - f {[(Re + (z + D)2)/4grt]
1/2,t} - f {[(z- D)2/4•crt]
'/2,t} +
f {[(Re+ (z - D)2)/4•rt]1/2,t}' ierfc,firstintegral
ofcomplementary
errorfunction;
erfc,complementary
errorfunction;Im W, imag;'xary
part of the errorfunctionof complexargument.
WANG
ET AL.: REPOSITORY
TEMPERATURE
FRACTURE
FLOW
3761
Near-SurfaceCoolingPeriods
_
_:
AND BUOYANCY
'--...,
•
-'_
The power output of all nuclearwastesdecaysvery rapidly
immediately after discharge from the reactor. Accordingly,
the effectsof a period of near-surfacecoolingof thesewastes
before burial in a repository may be expectedto be significant. Table 4 and Figure 7 illustrate the effectson the maximum repositorytemperaturesand epicentral thermal gradients with short periods of near-surfacecooling. From these
data, it is concludedthat the period for which the wastesare
coolednear surfacebefore burial is a very important factor in
determining the effect of a repositoryon the temperature of
surroundingrock mass.
RepositoryDepthand Radius
The effects of different repository depths and radii have
been analyzedfor situationswhere the ratio of depth to radius
is lessthan 1. The maximum temperatureis not significantly
10
10
•
10
•
10
•
10
•
Timeofterdischorge
tw (years)
AT:5øC •,_5øC
Fig. 1. Areal power densityof heat generatedby the storednuclear
-500
wastes.
E -IOOO
N
The maximum averagetemperaturein a repositoryoccurs
at its centerand is plotted as a functionof time after burial in
Figure 4. For both spentfuel and reprocessed
wastethis temperaturereachesa maximumafter a period of lessthan 100
yearsand thereafterdecaysvery slowlyovera periodof many
thousandsof years.The maximum temperatureand all other
temperaturesare of courseproportionalto the powerdensity
with which the repositoryis loaded.
To studythe far-field effectsand, in particular,thoseat the
surface,the maximum ground surfacetemperaturegradient at
the epicenterabove the repositoryhas been calculatedfor
both forms of wasteand is illustratedin Figure 5. This gradient and the corresponding
heat flux throughthe surfacereach
maximum valuesat about 2400 yearsafter burial for the spent
fuel and 1300yearsafter burial for the reprocessed
waste.The
maximum value of the temperature gradient for the spent
fuel, 70øC/km, is about 3 timesgreaterthan that of the reprocessedwaste,22øC/km. The total thermal gradientis the sum
of the original geothermal gradient, typically of 30øC/kin,
and the thermal gradient induced by the repository.
The decay in power output of nuclear wasteswith time is
determinedby their compositionand age.Different fuel cycles
yield wasteswith differentcompositions.
To evaluatesomeof
theseeffects,it is assumedthat the repositoryis loaded with
0.01 MTHM/m 2 (MTHM = metric ton of uranium fuel
chargedto the reactors).The maximumtemperaturesand epicentral thermal gradientsfor differentwastesfrom a pressurized water reactor(PWR) and a boiling water reactor(BWR)
are givenin Table 3. The samedata are illustratedin Figure 6
Spent fuel
-IO0-year storage
-1500
-20000 20 40 coo
AT (øC)
0'
I
500
I
I000
1500 2000
r(m)
5oc% AT:5øC
-500
•
E -I000
N
_ ReprocessedHLW
IO0-year storage
-1500
zo
_20000,..
•.,.',,
AT (øC)
•4o
as a function of time. In Table 3, data are also given for a re-
positoryloadedat an initial powerdensityof 10 W/m 2 for
, , I ,
, I , ,
comparison.It can be seenthat the differencesin temperature
500 ooo 5oo$oo
for the differentfuel cyclesarisemainly from variationsof inir(m)
tial power loading density. However, significantdifferences
still exist for the resultsof constantpower loading density of Fig. 2. Isothermsand profilesof temperaturerisearoundrepository
10 W/m 2.
after 100 yearsof wastestorage.
3762
WANG
ET AL.: REPOSITORY TEMPERATURE
-iooo
-15oo
r- Spentfuel
[ IOO0-year
storage
' .... ' .... ' .... "'
&T (øC)
AND BUOYANCY
FRACTURE
FLOW
this rock mass. Increasing the groundwater temperature results in decreaseddensity and viscosityof the water. In this
sectionthe changesin buoyancygroundwaterflow inducedby
these temperature changesare evaluated.
Two typesof model geometryare assumed.The first model
comprisesa simple horizontal fracture at the depth of the repositoryconnectinga rechargezone to a dischargezone and
intersectinga vertical fracture containing the axis of the repositoryas illustratedin Figure 10a.Flow from the repository
to the surfaceoccursthrough the vertical fracture. The effective hydraulic apertureof the horizontal fractureis bx, and its
permeabilitykx -- bx2/12[Lamb, 1932;Snow,1965;Iwai, 1977;
Witherspoonet al., 1980]. The aperture and permeability of
the verticalfractureare bzand kz -- b•2/12,respectively.
The
vertical fracture is of distanceL,c from the rechargezone and
of distanceLdcfrom the dischargezone. For the caseswith the
0'
I
500
I
10•
15•
2000
o
-500
-1000
-1500
-20000 20 4O 60
AT (øC)
f
500
I000
W of the fractures much less than the diameter
of the
repository,the flows are assumedto be unidirectionalalong
the fractures.The geometryas illustratedin Figure 10awill be
referred to as the linear flow model. Before the repositoryis
loaded and the rock mass subjectedto changesin temperature, it is assumedthat the original groundwaterflow is horizontal from rechargezone to dischargezone. As the rock mass
heats up, so will the groundwaterin the vertical fracture containingthe centerof the repository.This will perturbthe original flow pattern.
The secondtype of geometrystudiedis that of a radial flow
model as illustrated in Figure 10b. This correspondsto the
first model with the shapeof the horizontal fracture changed
to be that of a circulardisk of radiusL. The apertureand permeabilityof the radial horizontalfractureare b, and k, = b,•/
12, respectively.For both models,the flow of groundwateris
assumedto be confinedwithin the fractures.In practice,the
flow of groundwaterthrough most hard rock, which generally
has a very low intrinsic permeability [Brace et al., 1968], is
largely through fractures,so that the modelsapproximatethe
mechanicsof groundwater flow through hard rock masses,
differing only in their simplicity.
r(m)
oo. ,
width
1500 2000
r(m)
Fig. 3. Isothermsand profilesof temperaturerise around repository
after 1000 yearsof wastestorage.
8O
6O
affectedby this range of depths and radii, but changesin farfield, long-term temperaturemust be expected.Figure 8 illustratesthe dependenceof the epicentralthermal gradientwith
depth and radius.
•
I--
Rock Properties
The maximum repositorytemperatureand epicentral thermal gradient have been calculated for a repository located
within different hard rocks,usingthe valuesof thermal propertiesgiven in Table 1. From the resultsillustratedin Figure 9
it can be seenthat significantchangesin temperaturefield result from different rock properties.
THERMOHYDROLOGIC
Fracture
DISTURBANCE
Flow Models
Changesin the temperature of the rock masscontaininga
repositorywill changethe temperatureof the groundwaterin
'
-
Average temperature rise
at repository center
500m deep in granite
_
IOw/m2 at loading
40
-
_
2O
I
I01
102
I03
104
105
Storage time t (year)
Fig. 4. Temperaturerise at the centerof a disk-shapedrepositoryin
granitestoreduniformly with nuclearwastes.
WANG ET AL.: REPOSITORY TEMPERATURE AND BUOYANCY FRACTURE FLOW
IO0
6
8O
5
3763
nesqapproximation[Wooding,1957;Combarnous
and Bories,
1975]that variationsin the densityof water with time can be
neglected except for the effects on the buoyancy of the
groundwater.As a resultof this, (4) reducesto
V. q = 0
(5)
For low velocities, inertial forces are much less than viscous
b 60
forces,so that by means of Darcy's law the equation of toomentum
can be written
• 40
as:
p
VP+ kq- pg= 0
•
(6)
2• where
2O
I
P
0
kinematic viscosityof water;
permeability;
g gravitationalacceleration.
pressure;
p
k
0
0
2000
4000
6000
8000
I0000
Equations (5) and (6) describethe incompressibleflow of
groundwater.
Storogetime t (yeor)
Fig. 5. Ground surfaceepicentralthermal gradient rise above a
disk-shapednuclear waste repositorywith radius of 1500 m and 500
m deep in granite.
The first case consideredis the linear flow model with zero
original groundwater flow and with the repository located
midway between the rechargeand dischargezones,Lrc = Lac
= L. The flow is symmetricaboutx -- 0 with boundaryconditions
Flow Equations
P(t)--O
For any temperaturefield the flow of the groundwatermust
satisfythe equationsof conservationof massand momentum.
The equation for the conservationof massis
Ot+ V. q--0
(4)
whereq is the massflux that is the productof the densityand
velocityof the water in the fracture,p is densityof water, and
t is time.
at x=0
z--0
bxqx(t)+«b•qfft)=O at x--0
(7)
z---a
(8)
P(t)--Po-/_•po(z)gdz
atx=Lz=-D(9)
The water table is assumed to be at the surface z --- 0 with zero
hydraulic pressurein (7). Equation(8) describes
Kirchoff's
law for thejunctionbetweenverticaland horizontalfractures.
Poin (9) is the hydrostaticpressureimposedat the inlet of the
In most studies of thermal convection, because of the small
horizontal
fracture.
In accordance
with (5), the unidirectional
flowsare inde-
compressibilityof water, it is usualto make useof the Boussi-
TABLE 3. Effectsof Fuel Cycles
PowerDensityat Loading
(AT).... t
Fuel Cycle*
(A(VT)z)max,t
øC
Nuclear
Spentfuel
HLW + PuO2:U recycle
HLW: U + Pu recycle
HLW: U recycle
HLW: no recycle
60 (50)
65 (52)
90 (40)
43 (42)
43 (41)
Spentfuel
HLW + PuO2:U recycle
HLW: U + Pu recycle
HLW: U recycle
HLW: no recycle
50 (51)
57 (54)
71 (40)
35 (42)
35 (42)
Nuclear
øC/km
W/m 2
kW/Canister
Reactor P WR
83 (70)
86 (69)
68 (30)
22 (22)
22 (22)
Reactor
11.9
12.5
22.7
10.2
10.3
(10)
(10)
(10)
(10)
(10)
0.548
2.61
4.75
2.14
2.16
9.96 (10)
10.5 (10)
17.5 (10)
8.30 (10)
8.35 (10)
0.182
2.20
3.65
1.74
1.75
B WR
73 (74)
77 (73)
59 (34)
19 (23)
19 (23)
Valueswithoutparentheses
correspond
to 0.01-MTHM/m: wastecapacity.Valueswith parentheses
correspond
to 10-W/m: powerdensity.
*Spentfuel--fuel assemblydischargedirectlyfrom the reactor.HLW + PuO2:U recycle•reprocessed
wastewith U and Pu removedfrom the dischargefuel, U recycledin the reactor,and Pu storedtogether
with the water. HLW: U + Pu recycle--reprocessedwaste with U and Pu removed from the discharge
fuel and recycledin the mixed oxide reactor.HLW: U recycle---reprocessed
wastewith U and Pu removedfrom the dischargefuel and U recycledin the reactor.HLW: no recycle--reprocessed
wastewith
U and Pu removedfrom the dischargefuel.
t(AT)max:maximum value of the averagetemperaturerise at the centerof repository.(A(•T)z)max:
maximum value of the groundsurfacethermal gradientrise at the epicenterabovethe repository.
3764
pendentof the spatialcoordinates
alongthe fractures.With
100
/'""x
/
8O
'\\
/
/
t
,.,
/
60
to the boundaries:
0.01MTHM/m
2 capacity
ß
x\•,.HLW:
U+Purecycle
/
•HLW
+PuOa:
U-recycle
i
(D
constant
flow,(6)canbeintegrated
fromtherepository
center
PWR nuclear waste storag
,,'
'•-.
q,(t)
0- P(O,-D,
t)= -•
v(0, z, t) dz -
(0, z, t) g dz
(10)
Spentfuel
Po
- P(O,
-D,t)---qx(t)
k,
Lv(x,-D,t)dx (11)
4O
Equations(8), (10), (11) can be solvedfor the unknownsq:(t),
qx(t),and P(0, -.D, t). The resultfor q,(t) is
k•pog
2O
•
ß
eL(t)
= Vo 1+Fb•kffb•k
•
(1•)
_
where
_
I
I01
102
I0•
104
k•pog/Vo hydraulicconductivityof the verticalfracturewith
constantdensity and viscosityat surfacetemper-
05
Storagetime t (year)
ature;
is buoyancyhydraulic gradient;
F recharge
retardation
parame.
ter;
I00f
'''I'''I'''Ii''I''
PWR nuclear waste storage
bzk: verticaltransmissivity(aperture-permeability
prod-
0.01MTHM/m
2 capacity
Uctof verticalfracture);
b•k• liorizontal transmissivity(aperture-permeability
8O
productof horizontalfracture).
The buoyancyhydraulic gradient is dependsonly on the
vertical variationsof densityand viscosity:
• 40 HLW':
U+Pu
recycle--'•'"'--....
....
(ApgD)
pogD
Vo
(13)
wheretYzis the averagedviscosityover the flow path alongthe
vertical fracture from z = -D to z -- 0 and (ApgD) is the
buoyancy-drivingpressure
20
1I,$t'%''
.•.•.....-HLW
:no-recycle
O0
is(t) ....
.,
(a•gD) =
Do(Z)- p(O,z, t)]g dz
/_o
(14)
D
,,,,,
2000 4000
60'00 8000 I0000
Storoge
•imet(yeor)
The densityp(O,z, 0 in (14) is calculatedat temperatureTo(z)
+ AT(O, z, t), and po(Z)at To(z). The ambient temperature
Witha surface
temFig.6. Effects
of ciifferent
PWRfuelcycles
fromthe same maybegivenby To(z)= (20- 0.03z)øC
ß
amount of fuel on the repository temperatureand ground surface
thermal gradient.
perature of 20øC and an original geothermal gradient of
30øC/km.
TABLE 4. Effectsof SurfaceCooling
PowerDensityat Loading
Surface
Cooling
Period,years
(AT).... * øC
(A(VT)z).... *øC/kIEl
W/m2
kW/Canister
SpentFuel of P WR
93 (9)
1
98 (9)
104. (10)
4.81
2
5
76,(14)
67 (33)
90 (16)
86 (43)
56.4(10)
20.1(10)
2.6,0
0.927
10
60(50)
83(70)
11.9(•0)
0.548
1
2
5
10
,:
Reprocessed
HLW.' No Recycleof PWR
95 (9)
35 (3)
103. (10)
21.5
62 (11)
50 (27)
43 (41)
11.5
3.9
2.16
31 (6)
26 (14)
22 (22)
55.0(10)
1,8.7(10)
10.3.(10)
Valueswithoutparentheses
correspond
to 0.01-MTHM/m2 wastecapacity.
Valueswith parentheses
correspond
to 10-W/m2 Powerdensity.
*(AT)m•x:
maximum
valueoftheaverage
temperature
riseat thecenter
ofrepository.
(A(VTL)I•:
maximum
value
ofthegrodnd
surface
thermal
gradient
riseattheepicenter
above
therCJ?ository.
•-Values
without
parentheses
correspond
to0.01-MTHM/m
2waste'c•apacity.
Values
wi• parentheses
correspond
to 10-W/m2 powerdensity.
I00
,
[[[
!
/'•"'x [['[
,
i[[
Equation(12) for qz(t),can be generalizedto the caseswith
[,,[ i [ ,,]t[[ , , [[,[]][ , [
nonzero
originalgroundswater
flowandwithnonequal
dis-
\
[
x
8O
Spent fuel
-
k
/
tances from the repositoryto the rechargeand discharge
zones,L,½• Ld½.By golvingthe whole flow systemwith the
vertical and both horizontalflow pathsin Figure 10a and with
0.01MTH
M/m2copac•ty
x•l
/-•
6O
3765
Yearsurfacecooling
thewatertableheadequalto h,½at therecharge
zoneand
.."'.. x•2 Year
equalto hd½
at the dischargezone,the expression
for q,(t) is
,' /
<• 40
•?•o
Year
qz(t) ---
,'
ß
k,pog
4 -I-4Vo(F.c-'
- Fs½-')[(L.½F.½)-'
+ (L,•½F,•c)-']-'(D6)-' (16)
2o
I + [(L,½•,½)-'
+
i
i
i i iiii
i
i
i i iiii
i0•
i
i
i
iiiii
102
i
i
i
iiiii
i03
I
I
Equation
(16)iss'•nilar
to(12)withminor
modification
inthe
driving
gradien•
byti•eoriginal
hydraulic
gradient
io-- (h,chd½)/(L,•
+ L,t½).
T•e denominator
of (16)isofthesame
form
IIII
i05
104
of (12), but the slightdifferencebetweenthe averagesof viscosityoverthe rechargeand dischargeflow paths,•,½and •ac,
•' '",..
80
Reprocessed
HLW
f• \.,,,
0.01MTH
M/m
2capacity
"'".,¾......I
Year
surface
cooling
I00'
-
'\
,,
• f
.---.
•'5Year
..' ..'
,,v•
,,V\
.'
,'
•o
-
,,\\
60
_
,,\'\
-
•
oi
io•
io2
io3
io4
<1
io5
40
S,oro,e,,me,,,or, I:/
2O
lOOt
' ' ' i , ,, i , ,, i, , , i, ,'
•
•
x,eo.ur,oe
oo,,n
/}•__•
•'•,•2Year
• •/'//•
-••.• :•' 5Year
I0 Y or
2000
4000
, 6000
8000
'
-
.......
--
Storage
time
t(year)
'
•
..-..............
,.......
O0 000 .000 000 000
•
• 40 '
0
-
8o
10000
N
Storoge
time
t(year)
• 40
pcd0ds
of •astcswith dc•sit• of
Fi•. ?. Effectsof suffa• •oli•
0.01MTEM/m• o• tbc•cpositoff
tcmpc•atu•
c a•d
S•0u•d
s•ffa•
2
ß hc •cc•a•sc •cta•datioa pa•amct½•• (12) is
L •
15
r(t)••.•
()
The
geometdc
factor
• Fisthe
ra[i9
L/2D;
.•isihe
average
o
20Od4000
$toro•e
•ime
ofviscosity
overtheflowpathalong
th•h%•ontal
fractureFig.8. Effects
ofdifferent
depths
andradiiofr•posito•
onthe
from x • 0 to x • L.
'•
groundsurfacethermalgradient.
3766
/
/
i
60
.•
g 40[-/I/
•
".•
i
'x
Spentfuel
sumed that the horizontal
'x
i
I
model is the same as that in the linear flow model with upward flow from the repositoryto the surface.However,it is as-
\.
.
/
"'.-,•Shole
x
'\,•
/,,- "-..
'N,.
Basalt
\
'-...-•..,/
L /
/..'
gra
"ite"•",•xx,
<:i . !/ /"' Stripa
,. \
20/,/""/
Grante
"-•',\'\•' t
4
/."
',.\
-
"/"
x
\
J
",
0
i01
102
i05
104
105
.....
,,,,,• ........ •
........ •
........ •
symmetry,the product•'qriS constantin accordance
with (5).
The Darcy equation(6) can be integratedfrom r -- R to r = L
to obtain the pressuredifferencebetweenthe repositoryand
the outer boundary.Within the repositorythe pressuredrop
from the rim of the repositoryto the inlet of the vertical fracture is neglected.This assumptionis justified if the permeabilityof the back-filledmaterialfor the repositoryis much
higher than the permeability of the surroundingfracture. In
the linear flow model, the flow path from x -- 0 to x = R is
treated as part of the horizontal fracture.This portion of the
flow path can be neglectedif the sameassumptionabout the
permeability in the repositoryis made.
With the modificationsto radial geometry,Kirchoff's law
for the connection between the vertical fracture and the radial
80
fracture
........
is
2•rRb/lr(R,t) + Wb,q,(t)--- 0
Reprocessed HLW
ß
60
/
/
/
.
,/
lar to those in the linear flow model. The result for the vertical
\•,,,Shole
-
\
!
buoyancyflow qfft) is nearly identicalwith (12) of the linear
flow model:
,
k,pog
•<:l40/! ....• \•
I-
(17)
Other boundaryconditionsand solutionproceduresare simi\
/
flow is radial for r _> R. With radial
/
ia
q,(t)
----Vo 1+Fb,kffb•kr
(18)
,'
with radial transimissivity replacing the linear horizontal
transmissivity.The rechargeretardation parameteris differ-
ß Stnpa
_ _ ,,,,•,
'x.•Basalt
oiI ........
, ........
, ........
,
I0•
102
103
104
105
100
' ' • '
' •''
/
•' 4ø•i/ /•' Str'pa
gran'te---•
.............
:......
'
..---...................
o
4000
WIn(L/R)
1•
r
6
BuoyancyFlow
N
2000
F(t)=
The main differencebetween (19) of the radial flow model
and (15) of the linear flow model is the geometricratio factor;
•r in (19) is the averageof viscosity,weightedby r-z, over the
radial flow region from r = R to r = L.
b.-•'
..........
80 Spent
fuel
//'•'-Shale
0
ent:
6000
8000
I0000
Fig. 9. Effectsof rockformationson the repositorytemperatureand
ground surfacethermal gradient.
The verticalbuoyancyflow qfft) in (12), (16), or (18) dependson the buoyancyhydraulic gradient,the vertical hydraulicconductivity,the rechargeretardationparameter,and
ratio of the vertical and horizontal hydraulic transmissivities.
Considerfirst the caseof a linear flow model with equal
aperturesfor the vertical and horizontal fractures,b, = bx -- 1
•m, and with equal distancesto the rechargeand discharge
zones,Lrcm Lac= 5000 m. The vertical buoyancyflow can be
evaluatedby numericalintegrationof the buoyancydriving
pressurealong the vertical flow path and the averagesof viscosityalongthe vertical and horizontalflow paths.To calculate the displacement
of waterin the verticalfracture,qfft) can
be integratedover time. The groundwaterinitially at the
depth of the repositorywill move upward in the vertical fracture as a function of time after burial of the wastes. The re-
sultsof onerepositoryat depthof 500 m are illustratedin Figure 11. For comparison,the resultsof anotherrepositoryat
has been taken into accountin the rechargeretardation parameter.
depth 1000 m are also included.
The verticalflow in the caseof spentfuel is substantially
greaterthan that of reprocessed
waste.The buoyancymovebeyond the repositoryto surroundingrechargezone as illus- ment of groundwateris proportionalto the changein tempertrated in Figure 10b, the rechargeto buoyancyflow comes ature of the rock mass,and the temperaturedisturbanceinfrom all directions. The vertical fracture in the radial flow
ducedby a spentfuel repositoryis significantlyhigher than
For the radial flow model with horizontal fracture extended
3767
A
•/w
b
LDC
ed
flow
.•--•---"•
•
x,,,e--Regiono
I flow
bx
B
Dr
Fig. 10. Two-fracture
models
for simulating
buoyancy
groundwater
movement
through
a repository:
(a) linearflow
model and (b) radial flow model.
that by a reprocessed
wasterepository(seeFigure 3). On the
connections and conditions. The horizontal fracture repre-
other hand, there is little differencefor the caseswith different
sentsthe rechargeflow path. The shorterthe distanceL from
repositoryto rechargezone,the lessthe rechargeresistance,
and the greaterthe flow of water in the verticalfracturefor a
givenbuoyancy.
As examples,
the velocityvt(0 = qt(0/Pofor
depths.Thusit appearsthat the flow of waterin the vertical
fracture dependsupon the integral of the buoyancyforce
throughoutthe lengthof the verticalfracture,whichis a function mainly of the total heat releasedfrom the repository.
L -- 0, 2000, 5000, and 10,000m is plotted in Figure 13. The
The main driving forcefor the buoyancyflow is due to the buoyancyflowisproportionalto an effectiveverticalgradient:
contrastin densitybetweenthe heatedwater near the repository with temperatureTo+ AT and the ambientwaterin the
½"bxkx
(20)
it(t)= 1+ Fbtkt/
rechargeand dischargezoneswith temperatureTo.Figure 12
illustratesthe weak dependenceof buoyancyflow on the natural geothermalgradient.The slightdifferences
for different wherethe rechargeretardationfactorF -• 0.5 L/D for the lingeothermal
gradientsresultmainlyfrom the temperature
de- ear flowmodel(see(15)). The resultsof vt andit for the linear
inFigure
13illustrate
thesensitive
dependence
'of
pendence
of viscosity.
The generalformulae[Meyeret al., flowmodel
1967]of densityandviscosity
of waterasnonlinearfunctions buoyancyflow on the magnitudeof F. From the figureit is
apparentthat the curvesfor a 500-mand a 1000-m-deep
reof temperature
andpressure
areusedin our calculations.
positoryhaveapproximately
thesamemaximumvaluesof the
RechargeResistance
buoyancyvelocitywhenthey havethe sameLID ratio or F
In additionto the buoyancyof the heatedwater in the vertical fracture,the flow of this water ig affectedby the hydrologic
value.
Althoughthe casewith L -- 0, or equivalentlyF-- 0, has a
3768
o
bx:bz: !,um
A
- LRC:
LDC:
L
•
- R=15OOm
-200
/
D= 500m
!
_ Curves
/
(•) L:O
/
- (•) L:2000m:4D /
(•) L:5OOOm:
IOD /
E•-400
•
•
--10-•
\
_-
\
\
\
-
_
½- -8oo
._,,,
--6 E
.-•
-o
-4
-2
'
-
101
-1000
I
E
-8 'O
-
- (•)L:IOOOOm:2OD/ \
' Spent
fuel /
\
-----Reprocessed
/
\
•-600
_
-
10•
10
•
10
•
10
•
10
s
102
0
105
10
4
10
5
IO4
IO5
Storoge time t (yeors)
$foroge lime f (yeors)
10,
Fig. 11. Water movementalongverticalfracturefrom the repository
(linear flow model).
'
bz: bx: I/xm
LRC
:LDc:L
R:15OOm
D=lOOOm
large buoyancyflow, it is an unrealisticrepresentationof the
hydrologicconditionof a repository.The geometryof L -- 0
in the linear flow model representsa situation where instant
Curves
(•) L:O
(•) L:2OOOm
:2D
recharge is available to the vertical fracture flow at its inlet
(•) L=5000m=5D
end nearthe centerof the repository.For a low-permeability
formationsuitablefor a repository,the rechargezonesshould
be somedistanceaway from the repository.
In general,F • 0, and the effectiveverticalgradienti• in
(20) dependson the ratio of the vertical and horizontal fracture transmissivities.
The buoyancyflow is no longersimply
proportionalto the verticalpermeabilityk• or bfi. Exceptfor
(•)L:,0000m:
Spenfuel
-----
Reprocessed
o
-.•.
•' 2
0
I
0
I
101
IOz
10
3
.•
Fig. 13. Effectsof rechargedistancesL on the flow velocitiesand
hydraulic gradientsalong the vertical fracture from (a) 500-m-deep
and (b) 1000-m-deeprepository(linear flow model).
-200
© -400
the conditionthat the resistancefactor Fb,k,/bxkx<< 1, q,(t)
doesnot increaseas b, increases.On the contrary,the buoyancyflow q,(t) approacheszero as b, increases
to infinity. The
b,3 dependancein the resistancefactorovercomes
the b,2 dependencein the permeability.With a limited recharge(bx, k•
constant),the buoyancygradient cannot effectivelydrive the
o
u -600
(D
large amount of water in a large vertical fracture. It can
be shown that at a given time, q,(t) is maximum with b, -
9> -8oo
(2/F)•/2b• and
-1000 .........
I
10
•
102
10
a
104
10
•
Storagetime t (years)
Fig. 12. Effectsof differentoriginal geothermalgradientson the
water movementalongthe verticalfracturefrom the repository(linear
flow model).
q•(t)
< 3 Vo
i•
(21)
Thus buoyancy groundwater flow in the vertical fracture is
limited by the permeability of the horizontal fracture rather
than the permeabilityof the verticalfracture.In Figure 14, re-
suitswith a constantapertureb• -- 1/tm and with a rangeof
WANG ET AL..'REPOSITORY
TEMPERATURE
AND BUOYANCYFRACTUREFLOW
0
.' bx:
!/zm
LRC
=LDC:
5000m
bx:bz IFm
. R: 1500m
.-. -200
3769
//
:rn Spent
/;•i••//
R=1500
fuel/,
.-. -200
hx:
0.001
m/m it!/
v
.-
/bz
• -4oo
: ;t
E•-400
o
D:
500m ..._•.'•"
.
....
_
ß
•
l,-
i
LRC=2500m
LDC:
7500m
///Rep,
r,oc,e,
sse•
• -600 ....
__LRc=7500
mLDC=2500
m ;)! •,w .,,•
- --LRc:LDc:
5000m
•/
-600
._
-800
I
D:lOOOm
I0•
I0•
I0•
///
///
,';:2" .
,'//
.///"
.
-I000 ...... • ....... •
I
,'//
-8oo.
.
-IOOO
I04
I0•
IO
s
Fig. 14. Effectsof differentverticalfractureapertureson the water movementalongthe verticalfracturefrom the repository(linear
flow model).
I0•
I0•
10
4
IO
s
Storogetime t (yeors)
Fig. 15. Effectsof differentrechargedistanceLrc and discharge
distanceLac on the water movementalong the verticalfracture from
the repository(linear flow model).
DISCUSSION
aperturesbz -- 10, 1, and 0.1 pm are illustrated. The movement of the groundwaterin the vertical fracture is signifiAlthough the thermohydrologoic
model usedfor the analycantly slowerboth for the case of bz -- 10bxand bz -- 0. lb,, sispresentedin this paper is very simple,it shouldpossess
the
than with bz -- bx.
samephysicalbehavioras that of the more complexsystems
All the precedingresultshave beencalculatedfor the sym- of fractureswhich accountfor the permeabilityof massesof
metricrepositorylocation.In Figure 15 the effectsof the posi- hard rock. Accordingly,it shouldprovide a good insightinto
tion of the repositorybetween the rechargeand discharge the dynamicsof thermally-inducedgroundwaterflow, and ilzonesare illustrated.The original hydraulicgradientio is as- lustrate the sensitivity of this flow to various parameters.
sumedto be 0.001m/m in the calculationof qz(t)in (16). The However, the actual numerical results should be considered as
resultsindicatethat the couplingbetweenthe buoyancyflow no more than order of magnitude estimates. It must be
and the original groundwater flow is small in this model. It
pointedout alsothat the transportof nuclidesfrom the reposishouldbe noted that (16) reducesto (12) for Zrc -- La½even
when io• 0. In otherwords,for the symmetriccase,the buoy-
ancyflowinthevertical
fracture
isindependent
oftheoriginal
groundwaterflow in the horizontal direction.
Considernext the radial flow modelwhich representsa dif-
br :lp.M
L:
5000M
W: 2000 M
ferentrecharge
geometry.The rechargeretardationfactorin
(19)isF = WIn (L/R)/2rrD.
In comparison
withthelinear
Spent fuel
case,the radial model with rechargefrom all directionsis less
,
sensitive
to the distance
L from the repository
to the outer
boundary.The hydrologicparametersof rechargedistanceL
-- 5000
m,vertical
fracture
widthW= 2000
m,andrepository
radius R -- 1500 m are used in the calculations, and the results
areshown
in Figures
16and17.Thebuoyancy
flowisfasterin
theradialmodelthanthecorresponding
result
ofL -- 5000m
bz: Ip.M
o=I000M
_ bz:
IOp-M
/• -
D:,5
-
--
_
in the linear modelin Figures13 and 14. However,the qualitative dependences
of buoyancyflow on the transmissivity
ratio and on the depth of the repository are the same for both
models.
Under the condition that the buoyancy flow is limited by
the recharge,either becauseof long recharge distancesor
small horizontal-verticaltransmissivityratio, the resistance
factorFb,kz/b•k•
orFbzkz/brkr
willbemuchgreater
than1.In
D=IOOOM
•-.-.•
_
-
.....
/...._-;..---
ß
, .......I ,,
I0•
I0]
I0•
I0•
I0•
thiscase,qz(t)is proportionalto F-' or to the depthD for both
Storoge time t (yeor)
models(see(15) and (19)). Hence the time requiredfor the
water movementfrom depthto the surface,which is roughly Fig. 16. Flow velocitiesalongverticalfracturewith differentvertical
proportionalto D/q•, is insensitiveto the changein depth.
fractureaperturesand depthsfor L -- 5000 m (radial flow model).
3770
WANG
ET AL.' REPOSITORY
TEMPERATURE
br :lp.M
L:
-200 -
FRACTURE
FLOW
fractures,the maximum flow being determinedby the transmissivityof the horizontal fracture.
• ii1]j i
_
AND BUOYANCY
5000M
Acknowledgment. This work is supportedby the U.S. Department of Energy under contract W-7405-ENG-48.
W: 2000M
Spent fuel
N
E
-400
-
bz: I p.M
_
bz: IOp.
M
REFERENCES
•
o
I
D= 500M
E
i
-600
,
/
-
Brace,W. F., J. B. Walsh, and W. T. Frangos,Permeabilityof granite
under high pressure,J. Geophys.Res., 73(6), 2225-2236, 1968.
Carslaw, H. S., and J. C. Jaeger,Conductionof Heat in Solids,2nd ed.,
p. 260, Oxford at the Clarendon Press,London, 1959.
Combarnous,M. A., and S. A. Bories, Hydrothermal convectionin
saturatedporousmedia, A dvan. Hydrosci.,10, 231-307, 1975.
Department of Energy, Report of task force for review of nuclear
waste management, Rep. DOE?ER-OOO4?D,Washington, D.C.,
Feb.
-800
1978.
Fairchild, P. D., G. D. Brunton, and J. F. Cuderman, National waste
terminal storageprogram on radioactivewastestorage,progr. rep.
Y?OWI-8, p. 130, Officeof WasteIsolation,Oak Ridge Nat. Lab.,
-1000
I0
I02
I0
10
4
10
5
Storoge time t (yeor)
Fig. 17. Water movement along vertical fracture from the repository (radial flow model).
tory to surfacedoesnot take placeat the samerate as that of
the groundwater.Nuclide transportis retardedin a certaindegreeas the resultof physicaland chemicalprocesses,
suchas
sorption.
Oak Ridge, Tenn., Nov. 1976.
Hood, M., Someresultsfrom a field investigationof thermo-mechanical loading of a rock masswhen heater canistersare emplacedin
the rock, presentedat the U.S. Symp. Rock Mech. 20th, 1979.
InteragencyReview Group on Nuclear Waste Management, Subgroundreport on alternativetechnologystrategiesfor the isolation
of nuclear waste,Rep. TID-28818, Washington,D.C., Oct. 1978.
Iwai, K., Fundamentalstudiesof fluid flow througha singlefracture,
Ph.D. thesis,Univ. of Calif., Berkeley, 1977.
Kappelmeyer,O., and R. Haenel, Geothermics
with SpecialReference
to Application,GeopublicationAssociates,Berlin, Stuttgart, 1974.
Kisner, R. A., J. R. Marshall, D. W. Turner, and J. E. Vath, Nuclear
wasteprojectionsand source-termdata for FY 1977,Rep. Y?O WI/
TM-34, Officeof Waste Isolation,Oak Ridge Nat. Lab., Oak Ridge,
Tenn., April 1978.
Lamb, H., Hydrodynamics,6th ed., p. 583, Cambridge University
The calculations
reportedin this papersuggest
that under
certain circumstances, thermally induced buoyancy
Press, New York, 1932.
groundwater
flowmaybe a mechanism
by whichtoxicmateri- Martinez-Baez,L. F., and C. H. Amick, Thermal propertiesof Gable
als from a repositorycouldbe transported
to the biosphere. Mountain basalt cores,Hanford Nuclear Reservation,Rep. LBL7038, Lawrence Berkeley Lab., Berkeley, Calif., 1978.
The magnitudeof this flow dependsuponmany factors.Of Meyer,
C. A., R. B. McClintock, G. J. Silvestri,and R. C. Spencer,Jr.,
these,the aggregateincreasein the temperatureof the rock
1967 ASME Steam Tables,American Societyof Mechanical Engineers, New York, 1967.
masscontainingthe repositoryis one of the mostimportant.
This temperature
is affectedby the designof the repository, National Research Council, The disposal of radioactive waste on
land, report, Committee on Waste Disposal,Division of Earth Scithe kind of nuclearwasteburied in it, and the period for
which this waste has been cooled near surface before burial.
Significantdifferencesexist betweenreprocessed
wasteand
spentfuel in respectof the degreeto which the rock massis
heatedandhencethe time takenfor groundwater
to reachthe
surfaceby buoyancyflow.The depthof the repositorybelow
surfaceis of muchlesssignificance.
Coolingof the wastesnear
surfacecan be usedto compensatefor thesedifferencesand
reducesubstantially
the total amountof heatput into the rock
mass.The thermalconductivityof differentkindsof rockshas
significanteffect on thermally induced disturbancetoo. Finally, the buoyancygroundwaterflow dependsupon the ratio
of the hydraulic transmissivities
of the vertical and horizontal
ences,National Academy of Sciences,Washington,D.C., 1957.
Pratt, H. R., T. A. Schrauf, L. A. Bills, and W. A. Hustralid, Thermal
and mechanicalpropertiesof granite, Stripa, Sweden,Rep. TR-7792, Terra Tek, Salt Lake City, Utah, Oct. 1977.
Snow, D. T., A parallel-plate model of permeablefractured media,
Ph.D. thesis,Univ. of Calif., Berkeley, 1965.
Witherspoon,P. A., J. S. Y. Wang, K. Iwai, and J. E. Gale, Validity
of cubic law for fluid flow in a deformable rock fracture, Water Resour.Res., 16(6), 1016-1024, 1980.
Wooding, R. A., Steadyfree thermal convectingof liquid in a saturated permeable medium, J. Fluid Mech., 2, 273-285, 1957.
(Received June 28, 1979;
revisedJuly 11, 1980;
acceptedDec. 16, 1980.)

A study of regional temperature and thermohydrologic effects of an

Transcription

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