visualization of four-phase flow using micromodels

VISUALIZATION OF FOUR-PHASE
FLOW USING MICROMODELS
A REPORT SUBMITTED TO THE DEPARTMENT OF PETROLEUM
ENGINEERING
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
By
Ashwini A. Upadhyaya
September 2001
I certify that I have read this report and that in my opinion it is
fully adequate, in scope and in quality, as partial fulfillment of
the degree of Master of Science in Petroleum Engineering.
__________________________________
Prof. Franklin M. Orr, Jr.,
(Principal Advisor)
iii
Abstract
The presence of four-phase flow has been acknowledged in numerous papers
available in the literature on oil recovery by gas injection. In the reservoir these
phases can include an oil-rich liquid phase, a lighter liquid phase comprising
primarily of components of the injected gas, a vapor phase rich in carbon dioxide
and methane, and water which is always present in most reservoirs (connate or
injected). Though most of the issues dealt with in the past are related to
thermodynamic calculations and compositional simulation, very little attention is
given to the problem of flow behavior.
In this work, we have studied the actual fluid configuration and connectivity under
the four-phase flow condition with the aid of micromodels. The fluids used are
“analog” fluids as the reservoir conditions are difficult to create in the laboratory
apparatus. Capillary tubes have also been used to study the phenomenon of
corner flow and to ascertain spreading of fluid phases in pore like conditions.
The results indicate that in most cases, four phases never seem to occupy a
pore space simultaneously. The order in which the phases are injected does
seem to have an influence on their configuration and connectivity. This has an
important implication on the way the values for their relative permeabilities are
calculated. It was also observed that in case of four-phase flow, there was
mobilization of discontinuous phases in some cases and this leads to some
ambiguity while using relative permeability data. Based on the observations of
the configuration and connectivity of the phases, it would be practical to treat
four-phase flow as a combination of two three-phase systems or as a pseudo
three-phase system, in which the two liquid hydrocarbon-rich phases are lumped
together.
v
Acknowledgments
The support of Stanford University Petroleum Research Institute - C (SUPRI-C)
and its industrial affiliates towards the funding of this project is gratefully
acknowledged.
I would like to thank my advisor, Prof. Franklin M. Orr, Jr., for his guidance and
assistance in research and for the remarkable support shown by him in face of
experimental difficulties. I would like to also thank Prof. Anthony Kovscek and
Prof. David DiCalro for extremely valuable suggestions and assistance in the
laboratory. Also the help provided by Dr. Louis Castanier in the preparation of
the micromodels and Mr. Will Whitted in the laboratory was indispensable. I
would also like to acknowledge the training and support provided by the people
from the Stanford Nanofabrication Facility (SNF).
I would finally like to thank my family, friends and colleagues for their support,
without which the completion of this project would have been impossible. I would
also like to thank the staff of the department of Petroleum Engineering,
especially Yolanda Williams, Ginni Savalli, Stephanie Sorensen and Edi Carrick
for assistance rendered.
vii
Contents
Acknowledgments
vii
Contents
ix
List of Tables
xi
List of Figures
xiii
1.1. Four-Phase Flow
1.2. Micromodels
1.3. Drainage along Corners of Noncircular Capillaries
2.1. Apparatus : General Description
2.1.1. Injection Pump, Inline Filters and Pressure Gauge
2.1.2. Microscope
2.1.3. Recording and Image Capture Assembly
2.2. Micromodel Holder and Micromodels
2.2.1. Micromodel Pattern Preparation
2.2.2. Wafer Imaging
2.2.3. Wafer etching and Finishing
2.3. Micromodel Holder
2.4. Injection Apparatus
3.1. Analog Fluid System
3.2. Micromodel Apparatus Procedure
3.3. Capillary Tube Apparatus
4.1. Tools to Interpret the Flow Behavior
4.2. Injection of Fluids in Order of their Wettabilities
4.2.1. Injection of Water in a Micromodel Filled with Gas.
4.2.2. Injection of Alcohol in a Micromodel Containing Water.
4.2.3. Injection of Oil in a Micromodel Filled with Water and Alcohol.
4.2.4. Injection of Gas in a Micromodel Previously Filled with Water,
Alcohol and Oil, in that order.
4.3. Injection of four phases into the micromodel in changed order
of wettabilities.
4.3.1. Injection of Water and Oil in the Micromodel.
4.3.2. Injection of alcohol in a micromodel containing water and oil.
4.3.3. Injection of Gas in a Micromodel Containing Water, Oil
and Alcohol.
4.4. Interaction in system with higher interfacial system between alcohol
and oil.
Nomenclature
References
A. Relative Permeability Models
19
26
28
31
32
32
33
33
35
37
38
40
41
43
44
45
47
48
48
49
51
60
67
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67
73
80
96
97
103
ix
x
List of Tables
Table 3.1: Properties of the fluid system
44
Table 4.1: Physical properties of the isobutanol, hexadecane, squalane
and air system.
80
xi
List of Figures
Figure 1.1 : A qualitative pressure-composition (P-X) diagram for
CO2/C3/C16 system at 70oF
18
Figure 1.2 : Experimental Apparatus showing the three hydrocarbon
phases: Wasson crude oil plus dissolved gas (312 scf/bbl) at 90oF with
80 mole % CO2.
20
Figure 1.3: Pressure-composition diagram of binary mixtures of
Maljamar seprator oil and CO2. (Orr, personal communications)
21
Figure 1.4: Ternary composition diagram of a CO2-C1-C16 system at
various pressures and 70oF (Orr, personal communications)
22
Figure 1.5: Schematic of Capillary Drainage Setup
28
Figure 1.6: Configuration of fluids in a corner.
29
Figure 2.1: Schematic diagram of micromodel apparatus.
31
Figure 2.2 (b): Repeated pattern of the unit cell, to complete the
micromodel.
34
Figure 2.3: Schematic of micromodel with inlet and outlet channels.
36
o
Figure 2.4 (a): View of a micromodel mounted at an angle of 45 on
an SEM.
38
Figure 2.4 (b): View of the micromodel at an angle and 75X magnification.
Note the rough edges of the micromodel.
39
Figure 2.5: Schematic of the micromodel holder.
40
Figure 3.1: Schematic of the capillary tube experiments
46
Figure 4.1: Schematic of a triangular cross section of a pore
47
Figure 4.2: Schematic view of the pore and throat model in a porous
medium
48
Figure 4.4: Possible locations of alcohol in water filled pore.
50
Figure 4.5: Configuration of alcohol and water in a micromodel.
51
xiii
Figure 4.6: Schematic representation of possible fluid configurations in
a pore containing water, alcohol and oil, injected in that order.
52
Figure 4.7: Micromodel photograph of a pore containing water, alcohol
and oil, injected in that order.
53
Figure 4.8: Schematic of capillary tube experiments with water, alcohol
and oil.
54
Figure 4.9: Photographs illustrating the importance of flow in layers
to the drainage process.
57
Figure 4.10: Simultaneous flow of oil and alcohol when oil in
injected in a micromodel containing water and alcohol.
60
Figure 4.11: Possible configuration of the four phases in the pore body.
61
Figure 4.12: Gas injection in a micromodel containing water, alcohol
and oil.
65
Figure 4.13: Schematic of a four-phase capillary tube experiment.
66
Figure 4.14 : Schematic of a four-phase capillary tube experiment.
67
Figure 4.15 : Schematic showing possible fluid configurations
in a pore containing water and oil, injected with alcohol.
68
Figure 4.16: Schematic view of alcohol injected in a pore body
containing water and oil, based on the pore body and throat model.
69
Figure 4.17: Micromodel photographs of alcohol injected in a
micromodel containing water and oil.
72
Figure 4.18: Schematic view of gas injected in a pore body containing
water, oil and alcohol.
73
Figure 4.19: Gas injection in a micromodel containing water, oil and
alcohol.
79
Figure 4.20: Micromodel filled with oil and water.
81
Figure 4.21: Possible configurations of fluid phases upon injection of
alcohol in a pore containing oil and water, based on the pore and
throat model.
82
Figure 4.22: Injection of alcohol in a micromodel containing oil and water.
88
xiv
Figure 4.23: Schematic representation of fluid configurations in the pore,
upon injection of air in a micromodel containing water, oil and alcohol.
89
Figure 4.24: Gas injection in a micromodel containing water, alcohol
and water.
93
xv
Chapter 1
1. Introduction
Carbon dioxide is not first-contact miscible with most reservoir oils and develops
miscibility upon multiple contacts under conditions found in several reservoirs.
The criteria for miscibility have been reviewed by various authors. (Grigg, et al. ,
1997; Wu and Batycky,1990; Eakin and Mitch, 1998). In certain portions of the
reservoir, particularly in low temperature CO2 reservoir fluid systems, complex
phase behavior has been observed. This phase behavior is characterized by the
presence of a third non-aqueous phase whose composition is rich in the injection
gas, especially in reservoirs where the temperature and pressure is near the
critical point of CO2 (Shelton and Yarborough, 1977; Orr, et al., 1980; Turek, et
al.,1988; Creek and Sheffield,1993). Hence we have the existence of four
phases which include an oil-rich liquid phase, injection-gas-rich liquid phase,
injection gas and water present in the reservoir. Since water is always present,
this phenomenon is also described as L-L-V (liquid-liquid-vapor) behavior. The
existence of this situation complicates the calculation of displacement efficiency
due to the complex phase equilibrium and also to the uncertainty in flow behavior
that arises from complexities in treating the relative permeability values for the
various phases. Fig. 1.1 shows the phase diagram of such a condition
representing the L-L-V behavior. Shelton and Yarborough (1977) observed
similar behavior for a system in which the injection gas was rich in ethane.
Additional precipitation of a solid phase, rich in asphaltene was also observed.
17
Figure 1.1 : A qualitative pressure-composition (P-X) diagram for CO2/C3/C16 system at
o
70 F (Orr, personal communications)
Micromodels were utilized for visualization of four-phase flow, in order to
examine the flow behavior on a pore scale, to study the configuration and the
orientation of the fluids. A micromodel pattern was designed to represent the
cross-section of a porous medium and was used to visualize flow patterns on the
pore scale. Capillary tube experiments were performed in conjunction with the
above experiments to corroborate and interpret observations made for the
micromodels.
18
A brief review of the four-phase flow question is presented, followed by a
description of the development of micromodel and capillary tube experiments.
Also empirical equations used for relative permeability calculations from literature
are presented and their relevance in light of the above observations is presented.
1.1. Four-Phase Flow
As already discussed above the existence of four phases in reservoir conditions
is an established phenomenon. In certain cases this phenomenon is further
complicated by the deposition of a solid phase or a non-soluble phase which
does not flow well (Huang, 1992). The solid phase deposited is also known to
affect the wettability of the reservoir rock. The additional liquid phase formed has
a high CO2 concentration, usually over 90% (Turek, et al., 1988; Creek and
Sheffield, 1993) .The formation of this phase is also observed to occur around
o
the critical point of CO2, which is 87.9 F and 1070 psia. The CO2-rich liquid
phase formed also has a density typically in excess of 0.7 gm/cc and has its
viscosity in the range of 0.05 to 0.1 cP.
Before discussing the complexities of this phase behavior and its influence on
operations, it would be instructive to have a look at the phase diagrams and
experimental PVT data of this region.
Figure 1.2 shows a series of photographs in which these phases are formed in a
PVT cell under series of varying conditions (Orr, 2000).
19
Vapour phase
1249 psia
CO2 rich liquid
phase
1284 psia
1307 psia
1328 psia
Crude oil rich
phase
Figure 1.2 : Experimental Apparatus showing the three hydrocarbon phases: Wasson
o
crude oil plus dissolved gas (312 scf/bbl) at 90 F with 80 mole % CO2.
Figure 1.3 represents the pressure-composition diagram of binary mixtures of
Maljamar seprator oil and CO2. The L-L-V region marked in the diagram
represents our region of interest. (Orr, et al., 1980).
Figure 1.4 represents the ternary composition diagram of a CO2-C1-C16 system
at various pressures and at a temperature of 70oF. The formation of the L-L-V
region is again observed under a certain set of conditions. (Orr and Jensen,
1984).
An observation of the these phase diagrams indicates that the formation of this
four-phase region is restricted to a relatively narrow pressure region ranging from
a few psi to about 200 psi. The reservoirs under which such phase behavior has
been observed are typically low temperature reservoirs and having a high CO2
concentration. Another interesting feature is the fact that this phenomenon has
not yet been reported to occur at temperatures in excess of 120oF.
20
Figure 1.3: Pressure-composition diagram of binary mixtures of Maljamar seprator oil
and CO2. (Orr, personal communications)
21
Figure 1.4: Ternary composition diagram of a CO2-C1-C16 system at various pressures
o
and 70 F (Orr, personal communications)
This phenomenon has been observed to occur in field applications. For several
Permian Basin reservoir fluid and CO2 systems, the pressure and temperature
conditions occur in this phase envelope (Fong, et al., 1992). It has also been
observed at the conditions that occur in the Schrader Bluff oil field located on
Alaska’s North Slope (McKean, et al., 1999). Besides this, several West Texas
floods have also shown a similar phase behavior. The existence of such phase
22
behavior presents two complexities, one associated with the flash calculation of
the phase compositions and the other associated with assigning values of
relative permeabilities to model the flow behavior of this system.
In the past due to limited computing resources and relatively less knowledge of
such systems, several approximations were made while performing flash
calculations. Nghiem and Li (1984) were among the first to incorporate a threephase flash calculation model. Fanchi (1987) suggested that by approximating
the L-L-V system (a three-phase hydrocarbon system) to an L-V system (a twophase hydrocarbon system), there was only a small percentage change in the
recovery factor. With improving computing power and a better understanding of
such systems, handling the thermodynamics of this system seems to be a lesser
concern currently.
However, little has been done to understand the flow behavior phenomenon
associated with such four-phase systems. While considering flow in such
systems the number of phases can be as many as five: aqueous, liquid
hydrocarbon-rich phase, liquid CO2-rich phase, gaseous CO2-rich phase and the
solid precipitates. In most cases the influence of the solid phase on the flow
process is ignored as it is assumed not to flow, but it can have significant
influence on the flow behavior by changing the wettability of the rock surface on
which it deposits (Grigg and Ucok, 1998). In multiphase flow behavior, the
saturation, saturation history and the flow properties of each phase affects the
values of that of the other phases. The problem of measuring relative
permeabilities of the four phases also presents other significant challenges. In
such a flow system measuring the saturation, pressure drops and the fluxes of all
the phases is not adequate, as there are infinite number of displacement paths
given the fact that the displacement involves the variation of three independent
saturations and their corresponding saturation history. It is important to note that
the relative permeabilities are not just a function of the saturations of the various
phases but also depend on the saturation history. Hence it is impractical to
measure the values of relative permeabilities for different displacement
23
processes at varying values of saturation, flux and pressure drops, as the
number of experiments required for the same would be far too large. A more
practical solution is to try to develop an empirical correlation to calculate the
relative permeability of the flow system under consideration. In the past attempts
to model three-phase relative permeabilities empirically have relied on data from
the two-phase experiments, and it would seem to be logical to make such an
endeavor in the case of four-phase systems. However the pore occupancy in the
case of four-phase flow need not be the same as that for the two-phase systems,
and hence an accurate modeling is not possible for relative permeability values
of such systems. Hence it is very important to gain an understanding of the flow
process at the pore scale and to study the configuration and the connectivity of
the fluids at the pore scale. The work done here endeavors to address the
problems of fluid distribution on a pore scale and the connectivity of the fluids
with the aid of micromodel and capillary tube experiments.
Before proceeding with the pore level analysis of this fluid system, it would be
useful to review some work already done to address the effect of four-phase flow
in low temperature CO2 floods. McKean, et al. (1999), while addressing the issue
to the Schrader Bluff CO2 EOR evaluation, have lumped the three hydrocarbon
phases (hydrocarbon-rich liquid phase, CO2-rich liquid phase and CO2-rich vapor
phase) into two pseudo phases while performing simulation runs. However they
noted the limitation of such an approach given that the liquid CO2 phase is very
mobile and occupies an appreciable volume fraction in the reservoir. Grigg and
Ucok (1998) performed slim tube tests on Sulimar Queen reservoir oil under
four-phase conditions to determine the minimum miscibility pressure (MMP) and
also showed the effect of temperature and pressure on the development of
miscibility.
Wang and Strycker (2000) have performed an evaluation of CO2 injection with
three hydrocarbon phases. They have attempted to address issues concerned
with both phase equilibrium and flow behavior. Their results suggest that
commercial simulators such as VIP and GEM are unstable and unable to
24
converge to solutions. This is because of their inability to handle the above
phase conditions. They have suggested the use of UTCOMP, which can predict
fluid flow performance under the three hydrocarbon phases. To tackle the issue
related with the usage of the relative permeability models, they have used
theoretical models such as the Baker model, modified Stone II model, Corey
model and the modified Corey model. These models and their expressions are
listed in Appendix A, and their applicability in light of the experimental
observations is discussed subsequently. Based on comparison with slim tube
tests it was observed that the modified Corey’s model gave the best fit. Implicit in
the above models is the fact that the three-phase water and gas relative
permeabilities are the same as those of two-phase flow since they use the twophase oil-water and oil-gas data to predict three-phase oil relative permeabilities.
When extending their usage to the four-phase flow situation the three-phase oil
relative permeability is shared between the liquid hydrocarbon-rich phase and
the liquid CO2-rich gas phase in proportion of their relative volumes.
Fong, et al. (1992) in their studies made relative permeability approximations by
coupling the two liquid hydrocarbon phases into one pseudo-liquid phase. Based
on their algorithm, the percentage variation in recovery is limited to only a few
percent.
Another important aspect concerning this phenomenon to reservoir flow behavior
and recovery is the extent of this zone in the reservoir. In most of fields the
pressure and the temperature conditions within the reservoir are sufficient to
keep the injected CO2 in the liquid phase, but in the region of the producer wells
there is a drawdown of pressure and this might lead to the formation of the fourphase flow situation. If the extent of this four-phase region is not deep within the
reservoir then the flow properties in this region will be eclipsed by the large
amount of flow coming from the reservoir. However in cases where the above
region does have a sufficient extent into the reservoir, the flow properties of this
region will become dominant and hence their evaluation is important.
25
1.2. Micromodels
Micromodels can be used to study the flow behavior on a pore scale. They are
patterns of a porous medium etched on a silicon or glass surface and hence are
representative of the two dimensional structure of the porous medium.
Mircomodels have been extensively used to study the flow behavior in multiphase flow, oil-foam interaction studies, solution gas drive, contaminant hydrogeology, etc. The patterns used in the construction of the porous medium may
be prepared from thin sections of the porous medium to actually represent the
medium or in several cases are geometrically constructed as series of
repeatable simple of complex geometric figure aggregates. However as the
micromodels represent a two dimensional porous medium flow problem,
extrapolation of results to the three dimensional flow problem occurring in the
real porous medium needs to be done with certain amount of caution. Also it has
been observed that a nonuniform etch depth in the micromodels may lead to
snap-off situations not consistently predictable with the flow behavior (Rossen,
1999). Another constraint of the micromodels regarding dimensionality is the
lower macroscopic connectivity and co-ordination number (Nguyen, et al., 2000).
Though both etched glass and silicon micromodels have been used, glass
micromodels because of the nature of their fabrications have pores whose sizes
are several times larger than the actual size. A brief summary of the use of
micromodels for various applications has been listed below (George, 1999).
Mattax and Kyte (1961) developed the first etched-glass micromodel. This
micromodel comprised of a network of straight, interconnected flow channels.
This was a good tool for viewing interfaces in porous media and was used to
study the effect of the wettability on waterflood oil production.
Davis and Jones(1968) worked on the limitations in etching the micromodels
being produced. They used a photosensitive resist, which was resistant to
several solvents after exposure to ultraviolet light. As a result of this construction
26
technique more complex micromodels were produced to represent the complex
geometry of the pores of the flow media.
Owete and Brigham (1987) developed silicon-wafer micromodels which allowed
a better control on the etch depth and capture the finer details of complex
geometry more accurately. It is important to note that silicon as such is not water
wet. Hence to produce water wet micromodels, they are oxidized in air to
produce a thin film of silicon dioxide on the surface which is water wet. The flow
area of the micromodel is sealed with a glass plate, which is anodically bonded
to the silicon wafer, and inlet and outlet ports communicate the flowing fluids.
Hornbrook, et al. (1991) produced micromodels that were almost identical
replicas of a thin section of a Berea sandstone on a silicon wafer. These models
possessed almost identical properties of wettability and roughness as the original
porous medium. A limitation of this micromodel was the extent of capture of the
thin section of Berea sandstone. A scanning-electron microscope (SEM) image
of this thin section is used to produce the pattern on the silicon wafer. Since the
SEM image can only cover a limited area, the pattern of the micromodel
comprised of a repeated unit cell of the scanned SEM image. This also leads to
lack of continuity at the edges of the unit cells. Typically images are “digitally
treated” before repeatable unit cell patterns are produced to ensure connectivity
at the edge of the unit cells.
Keller, et al. (1996) used micromodels to observe the role of oil layers in threephase flow in porous media.
Though experiments performed in the micromodels for this study have been
done at low to moderate pressures (up to 35 psi pressure), micromodels have
also been used for high pressure experiments. The usage of micromodels at
elevated pressures requires housing the micromodels in a pressure vessel.
Campbell and Orr (1985) were among the first to perform high pressure
micromodel experiments. The performed a high pressure visualization study of
the displacement of crude oil by CO2. Bahralolom and Orr (1986) also performed
27
some of the earlier high pressure glass micromodel experiments while
comparing
N2
and
CO2
flow
mechanisms
in
multi-contact
miscible
displacements. George (1999) has provided a good review of micromodels used
in high-pressure conditions.
1.3. Drainage along Corners of Noncircular Capillaries
Though the micromodels serve as an excellent tool for micromodel visualization
certain features associated with the flow behavior remain ambiguous. The layers
of fluids formed in some cases are extremely thin and not clearly visible under
magnification. To ensure the connectivity and the spreading of layers of fluids,
drainage experiments using noncircular capillaries are performed. The concept is
simply illustrated in Figure 1.5.
Air
Oil
Water
Figure 1.5: Schematic of Capillary Drainage Setup
As shown in the above figure, the fluids are loaded in the capillary tube in the
order of water, oil, air and oil again, starting from below. If oil spreads as a layer
between water an air, it will drain from the region above the air on the top. It has
been observed that all oils having a positive spreading coefficient spread as a
layer between water and air and hence drainage occurs. In many cases, even
nonspreading oils have been observed to spread between the water and air,
28
because of the corner pore geometry (Fenwick, D.H. and Blunt, M.J., 1996;
Zhou, et al., 1997). Spreading is generally ascertained with a calculated property
called the spreading co-efficient (Cs). This is a function of the interfacial tensions
of the various phases at the interface and for the oil-water-gas system is defined
as:
Cs = σgw – (σow + σgo)
Cs – spreading coefficient of oil between water and gas
σij - interfacial tension between phases I and j
It is instructive to consider the connection between corner flow in capillary tubes
and flow in porous media. Porous media are often made up of grains having
sharp corners and surfaces. These corners are extremely important in helping
fluids connect in numerous cases. The use of noncircular capillary tubes is an
attempt to represent the sharp corners present in the porous medium, and hence
draw useful analogies of the spreading of fluids, based on the drainage behavior
in the capillary tubes. Fig. 1.6 shows an example of this spreading behavior.
Pore Corner
Pore Body
Oil spreads as a
layer
between
water and alcohol
Oil
Alcohol
Water
Figure 1.6: Configuration of fluids in a corner.
The spreading of fluids in the corner is not just governed by their spreading
coefficient but also by the corner angle of the pore. Zhou, et al. (1997) have
29
provided an excellent review of hydrocarbon drainage along corners of
noncircular capillary tubes. They have derived mathematical expressions to
calculate the rate of drainage along the corners and drawn useful analogies
leading to the calculation of three-phase relative permeabilities.
30
Chapter 2
2. Experimental Apparatus
Micromodel and capillary tube experiments were performed to the study the
phenomenon of four-phase flow on a pore scale. This section of the report
discusses the details and the procedures used in setting up the apparatus, its
calibration and operation.
2.1. Apparatus : General Description
Figure 2.1 represents a schematic of the setup of the micromodel apparatus. All
the experiments were carried out under constant volumetric flow rates for the
liquids and at constant pressure for the gas. All the tubing used in the apparatus
was of polyethylene and 1/16” in diameter. A brief description of the apparatus is
given below, while important aspects related with the fabrication and the
preparation of the micromodels are listed in section 2.2.
Inlet from ISCO
Pump or Gas
Cylinder
Video Imaging
Assembly
Coiled Tube
Assembly
Micromodel Holder
Outlet
Pressure Gauge
Micromodel
Micro-filter
Figure 2.1: Schematic diagram of micromodel apparatus.
31
2.1.1. Injection Pump, Inline Filters and Pressure Gauge
All the liquids injected in the apparatus were injected at constant volumetric rate.
This was regulated by the use of a ISKO pump (model no. 100DM). The least
count of this pump for constant volumetric rate control was 0.0001 ml/minute.
Inline filters are essentially swagelock lock filters with a cartridge of 2 micron
pores, to filter out small particles which can clog the micromodel. The pressure
gauge is used to record the pressure of the injected fluid and is used only during
the injection of the gas. In case of injection of the liquids, all conditions are
monitored from the injection pump.
2.1.2. Microscope
A Nikon Optiphot-M with a phototube allowing for imaging was used in the
apparatus. The properties of the optical lens used in the objective are listed in
table 2.1. The working distance is the distance between the tip of the lens and
the focal plane of the objective. The lens of 5X magnification is used to track the
motion of the fluids in the micromodel, while the other lenses are used to focus
on a specific portion of a micromodel to study fluid motion there.
Model
Magnification
Working
Numerical
Viewable
Aperture
Diameter
Distance
(µ
µm)
(mm)
5X
5X
20.0
0.1
3000
0.4LWD
20X
6.0
0.4
800
0.5ELWD
40X
10.1
0.5
375
Table 2.1 : Properties of optical lenses.
32
2.1.3. Recording and Image Capture Assembly
The recording assembly consists of an output from the video camera to a video
cassette recorder (VCR), recorded at a speed of 30 frames a second. The
images from the VCR could be transferred to a Macintosh computer with the aid
of a Sipgot II tape video capture board.
2.2. Micromodel Holder and Micromodels
The micromodel holder used in this case was a simple assembly for low
pressure systems and was fabricated from aluminum.
The micromodels are essentially a two dimensional representation of the porous
medium on a silicon wafer. Initially micromodels were etched on a glass surface
to visualize the flow patterns, but because of limitations discussed previously,
glass micromodels have been replaced with silicon micromodels for experimental
studies involving the visualization of flow behavior on the pore scale.
Typically micromodels are made up of a repeated pattern of an SEM image of a
reservoir rock thin section. However such a pattern needs some digital
modification at the edges to ensure continuity in the porous medium. In our
study, the pattern etched on the surface of the micromodel was made from a
random hand drawn pattern to represent the two-dimensional structure of a
porous medium. This pattern has grains ranging from the size of 30 to 200 µm.
All the features of a porous medium were incorporated in the unit cell, which
include small and large pores, channels and very narrow throats. Fig 2.2 (a)
shows the unit cell which was repeated several times over in the preparation of a
pattern, whose dimensions are 5cm X 5cm. Figure 2.2 (b) shows the repeating
unit cells used in the micromodel.
33
Figure 2.2 (a): Pattern of the unit cell used in the construction of the micromodel. The
region within the box represents one unit cell.
Figure 2.2 (b): Repeated pattern of the unit cell, to complete the micromodel.
34
An important feature of the above pattern is the arrangement of the edges, which
sit together like a jigsaw puzzle when the unit cell is repeated. Another important
feature of the micromodel is the presence of a channel at the inlet and the outlet
ports. These channels ensure that flow in the micromodel is linear along the
edges and not like a five-spot pattern. Figure 2.3 represents a schematic of the
micromodel employed in our experiments with the inlet and outlet channels.
There are various stages to the construction of this micromodel, which are
presented in reasonable detail to facilitate refabrication in the future. The
micromodels were constructed in the Stanford Nanofabrication Facility (SNF),
where equipment and raw material necessary for their construction is readily
available.
Once we have a pattern design of a micromodel on a glass-chrome plate called
the mask (see section 2.2.1 below), the fabrication involves the following steps:
•
Imaging : The silicon wafer is coated with a photosensitive chemical which is
then exposed to ultraviolet light through the mask on top. This produces an
impression of the mask on the exposed wafer, which is then developed.
•
Etching: This developed wafer is then etched with hydrofluoric acid to
produce a micromodel with an etch depth of around 25-30 µm on the silicon
wafer.
•
Cleaning and Bonding: The silicon wafer with the etched pattern is then
cleaned in a sulfuric acid cleaning solution and then bonded to a glass plate
to create a flow medium.
The following sections discuss the above procedure in greater detail and list the
precautions, which need to be taken to manufacture good micromodels.
2.2.1. Micromodel Pattern Preparation
The unit cell used, which is repeated in the micromodel, is made by a random
hand-drawn pattern, incorporating various features of porous media and is also
of comparable dimension. The preparation of the pattern is done using the
software, L-EDIT, available at the SNF lab. This produces a file, which is
35
compatible with the mask-making machine. In the preparation of this pattern,
there are two tasks:
•
Design of the repeatable unit cell having the features of the porous medium
on it.
•
Design of the inlet and outlet ports and channels of the entire micromodel
assembly.
The file that is compatible for the preparation of this micromodel is available with
the SUPRI-C group and can be opened using the software L-EDIT. Subsequent
modifications to the flow pattern or the micromodel ports and channels is a
relatively easy task.
Inlet
Outlet
Figure 2.3: Schematic of micromodel with inlet and outlet channels.
To communicate fluids through the micromodel, inlet and outlet ports are
provided. These are represented by the small square boxes in Figure 2.3 at the
four corners of the micromodel. Though only two diagonal ports are necessary
for these experiments, the four-corner design was done to incorporate the need
of future experiments, which could require a combination of four inlet and outlet
36
ports. Once the pattern is prepared with the aid of the software, it is reproduced
on a glass chrome plate and is called the mask. The process of mask
manufacturing is a complicated one and is best handled by people in the
fabrication facility.
2.2.2. Wafer Imaging
The silicon wafers used for the manufacture of micromodels are of the type KTest. Another variety of wafers commonly available is K-Prime, but this just
differs in the value of resistivity and is substantially more expensive. The wafers
are coated with a 1.65 µm thick layer of Shipley 3612 photoresist without
edgebead removal (edgebead removal is the process in which a small ring of the
photoresist around the edge of the wafer is removed). Before coating the wafer
with photoresist it is important to make sure that the wafers are dry and do not
contain any moisture in them. If the wafers are new and out of a sealed box, they
should be used for coating immediately. If the wafers are old and being used a
long time after opening the seal of the box, before coating them with photoresist,
they need to be put in a singe oven at a temperature of 150oC for 20 minutes. It
is also important to note to use the special polymer cartridge in the singe oven to
prevent melting of the regular PVC cartridges.
Once the wafers are coated with photoresist they should be exposed to the
ultraviolet light within a period of one to two hours. At no point of the process
should they be taken out of the room with special lighting. This is done to prevent
the pre-exposure of the resist-coated wafers. The mask used in the process of
exposing the wafer to the ultraviolet light needs to be cleaned every time before
use, to prevent exposing dirt marks and dust particles. The cleaning procedure
involves cleaning the mask with acetone, methanol and isopropanol, strictly in
that order. Subsequent to this the mask needs to be washed with de-ionized
water and a N2 jet in a special mask cleaning apparatus.
During the process of exposing the resist coated wafer to ultraviolet light it is
important to make sure that the mask and the wafer are aligned, so as to prevent
37
the pattern for being off center on the wafer. The exposure time of the wafer to
the ultraviolet light should vary between 3 to 4 seconds. These exposed wafers
are then developed in Shipley MF-319 and LDD-26W developing chemicals.
2.2.3. Wafer etching and Finishing
The developed wafers are then etched using hydrofluoric acid. Given the
hazardous chemicals involved in the process, it is best handled by the welltrained technicians at the fabrication facility. The wafers are typically etched to a
depth between 20 to 30 µm. Figure 2.4 (a) and Figure 2.4 (b) show the
photographs of the etched wafers taken from an SEM (Scanning Electron
Microscope).
Figure 2.4 (a): View of a micromodel mounted at an angle of 45o on an SEM.
38
25 microns
Figure 2.4 (b): View of the micromodel at an angle and 75X magnification. Note the
rough edges of the micromodel.
The etched wafers should be washed before further use with a cleaning solution
comprising of NoChromix and sulfuric acid for 15 minutes. After washing them
o
in water they are heated on a hot plate at 700 F for 30 minutes. This is done to
oxidize the silicon to form a thin film of silicon dioxide on the surface. Silicon is
not water wet, but silicon dioxide is water wet. The wafers are they bonded to a
glass plate with drilled holes for the inlet port to the silicon wafer to complete the
flow system. A process called anodic bonding does this bonding. The glass plate
o
(pyrex) placed on the silicon wafer is heated to 700 F. A wire gauze with a weight
is placed on the assembly of the wafer and glass on the hot plate and a potential
of 1000 volts is applied for 1 hour. At these elevated temperatures the positive
sodium ions on the glass plate become very mobile and are attracted to the
negative electrode on the glass surface where they are neutralized to form a
bond. Terry (1975) provides a good description of the bonding process. Pyrex is
well suited for this application because of the close proximity of its thermal
expansion co-efficient (3.25 x 10-6 / oC) compared to that of silicon (2.56 x 10-6 /
o
C). Detailed properties of pyrex are present in Appendix B (Sagar and
Castanier, 1997).
39
2.3. Micromodel Holder
The micromodel is placed in a holder which has conduits connected to the
plumbing of the setup as show in Figure 2.1. This holder is fabricated from
aluminum and consists of an inlet and an outlet port, which communicate with
the micromodel at the drilled sections of the pyrex glass plate. The ports are
sealed with viton O rings. These rings need replacement after a few runs. Figure
2.5 shows a schematic of the micromodel holder.
Upper portion of the micromodel holder
Lower portion of the micromodel holder
Outlet port
Groove for
O ring
Viewing window
Inlet port
Figure 2.5: Schematic of the micromodel holder.
40
Inlet port to
micromodel
2.4. Injection Apparatus
The fluids used in the process are pre-equilibrated fluids. These fluids are
injected into the apparatus using an ICSO syringe pump (model 100DM, with
ISCO pump controller), using water as the pushing fluid. Since there is a need
to keep the injected fluids in pre-equilibrated form, we need to avoid the mixing
of the water of the pump and the injected fluids. One way to achieve this is by
using a piston cylinder in-between, which separates the pushing fluid and the
injected fluid. Another technique adopted in our apparatus is to use a spirally
wound thin tube of a large length in the plumbing. Because of the near plug flow
nature (parabolic flow profile) of the flow in the thin tubing, given the low injection
rates and the small rates of diffusion at the interface of contact, this method
ensures that fluid at the other end of the spirally wound tube remains in its preequilibrated form.
41
Chapter 3
3. Experimental Procedure
3.1. Analog Fluid System
In the experiments performed, the fluids used were not the fluids found in
reservoirs at severe conditions of temperature and pressure. To facilitate and
easier handling of the fluids used to observe four-phase flow, an analog fluid
system was used. The fluids used were water, isobutanol (alcohol), hexadecane
and carbon dioxide. Table 3.1 lists the properties of these fluids. This fluid
system has typically very low values of interfacial tension between the oil and the
alcohol phases, analogous to the oil and liquid carbon dioxide phase found in the
reservoir. In another system of fluids used, squalane was added to adjust the
interfacial properties of the fluid system. For both these systems, it was assumed
that interfacial properties of the injected gas (CO2) could be approximated to that
of air (which was used in the measurement of the interfacial tensions). The
properties of this fluid system is also listed in Table 3.1
43
Property
Water
Isobutyl Alcohol
Hexadecane
Air
Initial Solution
5.0 ml
10.0 ml
10.0 ml
-
Water
-
3.47
3.54
30.7
Isobutyl
-
-
0.1
24.9
-
-
-
22.4
Composition
Interfacial
Tensions
(dynes/cm)
Alcohol
Hexadecane
Table 3.1: Properties of the fluid system
3.2. Micromodel Apparatus Procedure
After the micromodel is placed in the holder and the plumbing of the apparatus
done, carbon dioxide is first passed through the micromodel. This is done to
drive out the air present in the apparatus, so that subsequent filling of the
apparatus with liquids will leave fewer bubbles in the micromodel due to the
solubility of CO2 in the liquids. It is very important to have an inline swagelock
filter of cartridge specification of 2 µm in place to prevent the clogging of the
micromodel. At this point the flow rate of carbon dioxide through the apparatus is
measured by using an inverted cylinder in a trough of water, and an approximate
calculation of the permeability of the micromodel is done. However given the
compressible nature of the gas more reliable permeability calculations are
performed by using a single-phase fluid through the micromodel. These
44
micromodels have a permeability value of around 165 mD based on the singlephase experiments involving water. Woody, et al. (1996) have presented a good
review of computation of permeability for micromodels which have a quarter five
spot pattern system.
After passing carbon dioxide through the micromodel for 1 to 2 hours, the fluids
can be introduced. Again it is important to use the inline filter while passing fluids
through the micromodel and ensuring to use a different cartridge for each fluid to
avoid interfacial tension effects of the various fluids. The fluids are passed
through the apparatus at a predetermined flow rate. Typically this flow rate was
maintained at a value of around 0.001 ml/minute. It is extremely important to set
an upper pressure limit of around 30–35 psi for the low-pressure micromodel
apparatus, to prevent the micromodels from breaking in case of an unexpected
pressure rise caused by the obstruction of the micromodel or the plumbing. The
fluids are then passed through the micromodel in various orders to study the
effect of the various phases. Water, the wetting phase is always introduced in
the beginning.
3.3. Capillary Tube Apparatus
The utility of the capillary tube experiments in studying the spreading nature of
various fluids has been discussed in the previous sections. The capillaries used
for the process are triangular with 0.9mm side length. The fluids in the capillary
are loaded in various combinations to study the spreading behavior of the fluids
in this system. The capillaries used for this process need to be completely water
wet and hence are thoroughly cleaned with a mixture of sulfuric acid and
NoChromix. The capillary tube is completely injected with the equilibrated
water, and then the various phases are loaded from the other end by injecting
them with a syringe having a thin needle. It is important to note that the capillary
tube is initially injected with water throughout its length to ensure that water is
always present in the corners of the capillary tube. Subsequent phases injected
push out the water from the other end of the capillary tube as they fill in. Once
45
the phases are loaded in the capillary tube, it is put in the vertical position and
drainage of the various layers is observed. Figure 3.1 shows a schematic of the
above process.
Air
Oil
Alcohol
Water
Figure 3.1: Schematic of the capillary tube experiments
46
Chapter 4
4. Experiments Performed and Observations
4.1. Tools to Interpret the Flow Behavior
In the course of the explanations, it will be convenient to represent the
observations about fluid configuration and connectivity with the aid of the
following diagrams. Figure 4.1 represents a triangular cross section of a
hypothetical pore. Since the pores found in a porous medium have grooves and
corners, a triangular cross section is a reasonable approximation of this physical
aspect.
Pore Corner
Pore Body
Oil
Alcohol
Water
Figure 4.1: Schematic of a triangular cross section of a pore
Figure 4.2 represents the porous medium from the perspective of the pore body
and throat model. The larger spaces in the porous media are called the pore
bodies or pores, while the narrower constrictions are called the throats. A pore is
typically surrounded by a number of throats leading to it. In this conceptual
picture only two throats are connecting to a pore.
47
Pore
Throat
Figure 4.2: Schematic view of the pore and throat model in a porous medium
Besides this we also make use of capillary tube experiments to interpret and
corroborate the observations from the micromodels. The schematic for this has
been explained in the previous section.
4.2. Injection of Fluids in Order of their Wettabilities
The first set of experiments performed in the study was done by introducing the
phases in order of their wettabilities, namely water, alcohol, oil and gas. The
wettability of the liquids is established by studying the curvature of the interface
of various combinations of the phases, in absolutely clean glass capillary tubes.
The wettability of the fluids was ascertained to be water, alcohol, oil and air, in
decreasing order of wettability. The most important aspect of the study was to
study the orientation or the configuration of the various phases and their
connectivities. This behavior has an influence of determining theoretical and
practical models for calculating the relative permeabilities of such system.
4.2.1. Injection of Water in a Micromodel Filled with Gas.
Initial injection of water in the micromodel showed certain important features
related with the wettability of the micromodel and the importance of corner flow.
Figure 4.3 shows a picture representing the entry of water into a micromodel
48
containing CO2 initially. As can be seen from the figures, the water moves ahead
from the physical front in the form of thin films (see the box region in Figure 4.3),
in the corners of the pore structure of the micromodel. This is indicative of corner
flow, a mechanism that arises due to capillarity and has significant influence in
maintaining the connectivity of phases and their drainage from a porous medium.
Grain
Gas in
Pore Body
Water In
Film
Throat
Water
Figure 4.3: Corner flow during imbibition of water in the micromodel.
4.2.2. Injection of Alcohol in a Micromodel Containing Water.
Subsequent injection of alcohol in a micromodel filled with water presents a
choice of configurations for the location of the alcohol. Figure 4.4 represents the
possibilities of the location of the alcohol in the pore body. These possible
configurations include water being completely displaced with alcohol, alcohol
occupying the corners of the throat or alcohol occupying the center of the
micromodel leaving water in the corners. Given that the medium is water wet, the
most probable option is the last one mentioned above.
49
Water Injection into
an empty pore
Alcohol Injection
Alcohol
Water
Figure 4.4: Possible locations of alcohol in water filled pore.
Figure 4.5 represents a condition of the micromodel after alcohol injection and
indicates that alcohol, which is the non wetting phase compared to water, pushes
water out of the larger space of the pore and occupies the central portion of the
pore body. Water is still present in the pore body, but is pushed towards the
edge of the pore and occupies these spaces. Water is also present in the
narrower throats of the medium. In the Figure 4.5 it is almost impossible to locate
the water film present in the corner of the pore body. This is because of the very
thin film of water present in the corners. However water is clearly visible in the
smaller throats.
50
Grains of porous
medium,
lighting
effect causes darker
appearance.
Water in
smaller
throats
Water in
Alcohol
film
Alcohol
Water
Figure 4.5: Configuration of alcohol and water in a micromodel.
4.2.3. Injection of Oil in a Micromodel Filled with Water and Alcohol.
Subsequent injection of oil into this micromodel containing water and alcohol
presents a plethora of possible situations within the pore structure, which are
represented schematically in Figure 4.6. Because oil is a nonwetting phase, it is
possible that it would not displace water from the corners and the narrower
throats. However, it is uncertain whether the oil would completely displace the
alcohol, spread as a film between alcohol and water or occupy the central portion
of the pore space, thus resulting in a film of alcohol between oil and water.
51
Alcohol
Water
Oil
Figure 4.6: Schematic representation of possible fluid configurations in a pore containing
water, alcohol and oil, injected in that order.
The observations in this case are inferred from looking at the micromodel
apparatus and corroborated by the capillary tube experiments. Figure 4.7 shows
a photograph from the micromodel assembly showing the presence of three
layers of fluids within a pore body in the micromodel. This indicates that oil, the
least wetting phase among water, alcohol and oil, occupies the position in the
center of the pore and pushes the alcohol as a layer between the oil and water.
Though the spreading coefficient of alcohol between oil and water would seem to
indicate nonspreading behavior, spreading of alcohol is observed. This is
because fluids are know to spread even in face of adverse spreading coefficient,
due to the geometry of the pore spaces and the corners in particular (Keller, et
al., 1997).
52
Oil
Alcohol
Water
Figure 4.7: Micromodel photograph of a pore containing water, alcohol and oil, injected
in that order.
Simultaneously the capillary tube experiments also corroborate this observation.
Figure 4.8 represents a schematic of a capillary tube experiment loaded with
fluids in the order of water, alcohol, oil and alcohol, from bottom-up. It was
observed that after some time, the alcohol layer above the oil drained, which
indicates that alcohol forms a layer between water (always present in the corners
of the capillary tube given the way the experiments are conducted) and oil even
in face of an unfavorable spreading coefficient.
53
Air
+
Oil
Alcohol
Water
Oil
Alcohol
Water
Alcohol drains
as a layer
between oil and
water
Figure 4.8: Schematic of capillary tube experiments with water, alcohol and oil.
Figure 4.9 indicates another set of observations observed during the three-phase
flow. The series of photographs indicates the drainage of alcohol in layers, as oil
is injected into the micromodel. It is also very important to note the thinness of
the layers of alcohol, which highlights the importance of flow in layers, and the
contribution of this flow to the drainage process. In the series of photographs, as
oil is injected into the system, simultaneous displacement of water and alcohol
(already in the micromodel) is observed. It can be seen in the photographs that
alcohol flows from region A to region B (marked in the photographs) through a
thin film. Subsequently alcohol is also drained from region B by another thin layer
of alcohol.
54
Grains of porous
medium
Water (present
in
sharp
corners)
Region A
Alcohol
Thin
Region B
of
layer
alcohol,
draining
Oil
Figure 4.9 (a)
Water pushing
alcohol out of
pore
Alcohol
being
pushed out from
the pore space
(region A)
Alcohol
Region B
Thin film of alcohol
connecting
the
alcohol
being
displaced
from
region A to B.
Oil
55
Figure 4.9 (b): Alcohol forms a thin film between water(present in corners) and oil.
Alcohol
pushed out by
water
from
Region A
Alcohol Region B
Thin alcohol film
between
water
and oil, draining
alcohol
from
region B
Direction of alcohol drainage
Oil
Figure 4.9 (c): Alcohol drained through connecting film demonstrating importance of flow
in layers and the connectivity of fluids through layers.
Region A
Region B
56
Figure 4.9 (d): Alcohol draining from region B.
Region A now
completely
filled
with
water.
Alcohol drained
from region A
and B, as a thin
layer between
water and oil.
Oil
Figure 4.9 (e): Alcohol drained out from regions A and B.
Figure 4.9: Photographs illustrating the importance of flow in layers to the drainage
process.
Figure 4.10 shows another phenomenon observed during the displacement of
alcohol by oil, as oil is injected into the micromodel. The interfacial tension
between the alcohol and the oil is relatively low, and the flowing oil at the center
of the pore body drags the alcohol film along with it. This leads to a succession
of snap-off events of the oil phase by the alcohol phase.
57
Oil flowing at
the center of the
pore
Alcohol film
between oil
and water
Water
Figure 4.10 (a): Oil injection into micromodel containing water and oil.
Alcohol draining
from
upper
regions
being
dragged
by
draining oil due
to low interfacial
tension
Oil
Figure 4.10 (b)
58
Alcohol
draining
from upper
regions
being filled
with oil now.
Oil
Figure 4.10 (c)
Oil
Snap-off type appearance
of flow. Oil phase is
surrounded alcohol phase,
thus giving an impression
of a snap-off.
Alcohol
Figure 4.10 (d)
59
Oil
Alcohol
Figure 4.10(e)
Figure 4.10: Simultaneous flow of oil and alcohol when oil in injected in a micromodel
containing water and alcohol.
4.2.4. Injection of Gas in a Micromodel Previously Filled with Water, Alcohol
and Oil, in that order.
When gas is injected into the three-phase system described in the previous
section, an interesting question regarding the configuration of the four phases in
the pore body arises. Figure 4.11 shows the possible configuration of fluids
within the pore body in such a scenario. Since gas is the least wetting phase, it
needs to be seen whether the gas displaces all of the other nonwetting phases,
or only one of them preferentially based on their wettability, or whether the gas
occupies the center of the pore, thus giving rise to the possibility of the existence
of four phases within a single pore.
60
Alcohol
Gas
Water
Oil
Figure 4.11: Possible configuration of the four phases in the pore body.
Figure 4.12 shows a series of photographs of the micromodel containing water,
alcohol and oil as gas is being injected in it. The photographs indicate that gas,
which is the least wetting phase, occupies the central portion in the pore body.
The alcohol forms a film between oil regions (marked by an oval in the
photographs below) and also forms a layer between the gas and oil (marked by
the rectangular regions in the photographs). These rectangular regions are
interesting because, one could argue the presence of four phases within the
pore, as water is always assumed to be present as a thin film in the corners.
However four phases are not explicitly visible within the pore structure. As gas
flows for a longer period of time, these alcohol films vanish as the alcohol is
almost completely displaced from the porous medium. Thus the primary phases
flowing through the porous medium are oil and gas. The alcohol present as a film
between water and oil, connects as a film and is displaced from the porous
medium to a lower saturation.
61
Oil
Alcohol
Water
Figure 4.12 (a)
Alcohol
Gas
Oil
Figure 4.12 (b)
62
These
rectangular
marked regions
show a transient
four-phase zone
in
a
pore,
assuming a thin
water
film
is
always present at
the wall and not
explicitly visible.
However
these
alcohol
films
eventually vanish.
Oil
Alcohol
Gas
Figure 4.13 (c)
Alcohol
Gas
Oil
Figure 4.12 (d)
63
Gas
Figure 4.12 (e)
Oil
Figure 4.12 (f)
64
Alcohol
Oil
Alcohol
Gas
Figure 4.12 (g)
Figure 4.12: Gas injection in a micromodel containing water, alcohol and oil.
The capillary tube experiments also support the interpretation given above. Oil
does not seem to spread as a layer between alcohol and gas. Figure 4.13 shows
a schematic of a capillary tube experiment with the tube filled with water, alcohol,
oil, air, alcohol and oil, in the order from bottom up. It is observed that the alcohol
drains first and then the oil drains second. The drainage process is not
simultaneous. This illustrates that oil does not spread between alcohol and air,
and hence four phases in the order of gas, oil, alcohol and water are not
observed in the pore body. Also alcohol and oil both spread as layers between
water and air, which is observed from three-phase flow.
65
Air
Oil
Alcohol
Water
Alcohol
Water
Gas
Oil
Figure 4.13: Schematic of a four-phase capillary tube experiment.
Since alcohol is denser than oil, one might argue that the oil has no additional
impetus to flow before the alcohol drains in the above experiment. Hence the
above experiment was performed in a slightly reversed order of fluid
configurations as illustrated by the schematic diagram in figure 4.14. It was again
observed that the alcohol layer on the top drains first, followed by the oil layer.
This clearly indicates that oil does not spread as a layer between air and alcohol.
Both these capillary tube experiments suggest that the four-phase flow situation
is composed of two three-phase systems. Ahead of the gas front, there is a flow
of water, oil and alcohol. In the gas region, there seems to be a flow dominated
by gas and oil, with some of the alcohol connecting up and being displaced by
the injected gas.
66
Air
Oil
Alcohol
Water
Alcohol
Water
Gas
Oil
Figure 4.14 : Schematic of a four-phase capillary tube experiment.
4.3. Injection of four phases into the micromodel in changed order of
wettabilities.
4.3.1. Injection of Water and Oil in the Micromodel.
In the next set of experiments performed, the fluids were injected into the
micromodel in changed order of wettabilities. The order of injection of the fluids
was water, oil, alcohol and gas. As expected, injection of oil into a micromodel
containing water pushed the water to the corners of the pores and the narrower
throats.
4.3.2. Injection of alcohol in a micromodel containing water and oil.
The injection of the alcohol now presents a range of possibilities as illustrated by
Figure 4.15. Among the more probable configurations, the alcohol can spread as
a film between the water and oil or occupy the central portion of the pore pushing
the oil as a film between water and alcohol.
67
Alcohol
Water
Oil
Figure 4.15 : Schematic showing possible fluid configurations in a pore containing water
and oil, injected with alcohol.
It was observed before that the alcohol spreads between oil and water, and
hence it would seem to be logical to expect that the alcohol would demonstrate a
similar behavior in this case. Since water is always present in the corners, Figure
4.16 shows a schematic of the possible configurations of the fluids based on the
pore body and throat model. Based on the previous set of observations (section
4.2), the expected configuration would be the one in which the alcohol spreads
as a layer between water and oil.
68
Alcohol
Water
Oil
Figure 4.16: Schematic view of alcohol injected in a pore body containing water and oil,
based on the pore body and throat model.
Figure 4.17 shows a series of photographs from the micromodel experiment
describing the process of the injection of alcohol in a micromodel containing
water and oil. It is observed that the alcohol flows through the central portion of
the pores and pushes the oil aside. However the alcohol does not flow as a
continuous phase. This discontinuous flow behavior of alcohol is because of the
unstable configuration in which the alcohol occupies the center of the pore. All
the capillary tube experiments indicate the spreading of alcohol between oil and
water, and oil was never observed to spread as a layer between alcohol and
water. Hence this flow behavior is assumed to stem from the low interfacial
tension between the alcohol and oil, because of which the alcohol is
preferentially able to flow through the center of the pores which present a lower
flow resistance to the alcohol compared to its flow in the form of a film, where the
flow resistance is higher.
69
+
Oil
Alcohol
Oil
Water
Alcohol
Note the
discontinuity
between the
phases
Figure 4.17 (a)
Oil
Alcohol
Water
Figure 4.17 (b): Succession of snaps showing flow of alcohol in a micromodel containing
water and oil. Note that alcohol flows as a discontinuous phase.
70
Alcohol
Oil
Figure 4.17 (c)
Figure 4.17 (d)
71
Oil
Alcohol
Figure 4.17 (e)
Figure 4.17 (f)
Figure 4.17: Micromodel photographs of alcohol injected in a micromodel containing
water and oil.
72
4.3.3. Injection of Gas in a Micromodel Containing Water, Oil and Alcohol.
Figure 4.18 represents a schematic of the possible configurations, when gas is
injected into the above system. It is important to ascertain whether four phases
exist in the pore or if the gas displaces one of the phases preferentially based on
the wettability. The two configurations on the right, present the most probable
configurations based on previous observations.
Water
Oil
Alcohol
Gas
Figure 4.18: Schematic view of gas injected in a pore body containing water, oil and
alcohol.
Figure 4.19, represents the photographs of the micromodel with gas being
injected into the above system. It is observed that as gas is injected into the
system, simultaneous drainage of oil and alcohol occurs. It is important to note
that the alcohol, which is present as a discontinuous phase in the micromodel,
continues to flow as a discontinuous phase through the oil. The gas breaks
through the higher permeability channels of the micromodel as seen in the
photographs. This essentially leaves three phases in the pore body. Water, the
most wetting phase, remains closest to the pore surface, and oil spreads as a
73
layer between the water and the gas. Most of the alcohol is drained out in a
discontinuous phase flow, but a small fraction of the original alcohol remains
trapped in a few isolated pores. This largely simplifies the problem to that of a
three-phase situation involving water, oil and air. In the photographs below,
simultaneous displacement of oil and alcohol is observed. It is also seen that for
some time, alcohol appears to be a continuous phase at the center of the pore.
However this is because of all the alcohol draining from the upstream region and
the large amount of alcohol drainage gives it the appearance of a continuous
phase flow. The flow of the alcohol phase is still discontinuous. By the time gas
enters the pore, most of the alcohol is displaced.
74
Water
Oil
Alcohol
(discontinuous)
Figure 4.19 (a): Displacement of water, oil and alcohol by gas.
Oil
Figure 4.19 (b): Alcohol is present as a discontinuous phase surrounded by water.
75
Oil
Alcohol
Figure 4.19 (c): Note the flow of alcohol as a discontinuous phase, contrasted to the
previous photograph (Fig. 4.19 (b)).
Figure 4.19 (d)
76
Alcohol
Oil
arrows indicate continuous flow path of alcohol
Figure 4.19 (e): Due to drainage of large quantity of alcohol from the upstream region,
alcohol appears to flow as a continuous phase at the center of the pore.
Trapped
Alcohol
Oil
Gas
77
Figure 4.19 (f)
Rectangular
regions:
Here
oil
spreads as a
thin
layer
between
water
and
gas.
Trapped
Alcohol
Gas
Figure 4.19 (g)
Oil
Gas
Figure 4.19 (h)
78
Water
Gas
Oil
Gas
Figure 4.19 (i)
Gas
Water
Oil
Gas
Figure 4.19 (j)
Figure 4.19: Gas injection in a micromodel containing water, oil and alcohol.
79
4.4. Interaction in system with higher interfacial system between alcohol
and oil.
In the previous section it was observed that though the capillary tube
experiments suggested the spreading of alcohol between water and oil, when
alcohol was injected in a micromodel containing oil and water, the alcohol flowed
as a discontinuous phase through the central portion of the pore body, and did
not form a layer between the oil and water. This behavior could perhaps be
attributed to the low interfacial tension between alcohol and oil. To determine
whether the flow behavior of alcohol observed in the previous section was due to
the low interfacial tension between alcohol and oil, the experiment in the
previous section was repeated with a fluid system having a higher interfacial
tension between the oil and the alcohol. The system of fluids used is the same
as before, but addition of squalane modified values of interfacial tension. The
system of fluids used had the following composition and properties:
Property
Water
Isobutyl Alcohol
Hexadecane +
Air
Squalane
Initial Solution
5.0 ml
14.0 ml
2.0 ml
-
Composition
+ 4.0 ml
Interfacial Tensions
(dynes/cm)
Water
-
2.91
3.94
30.6
Isobutyl Alcohol
-
-
1.28
22.4
Hexadecane +
-
-
-
26.2
Squalane
Table 4.1: Physical properties of the isobutanol, hexadecane, squalane and air system.
80
The fluids were injected in the micromodel in an order similar to that in the
previous section, i.e., water, oil, alcohol and air. Water injection was again
characterized by flow in corners and the presence of a thin film of water ahead of
the waterfront. Subsequently oil is injected into the water filled micromodel. The
injected oil, which was nonwetting, occupied the central portion of the pore body
and pushed the water as a film against the pore walls and in the smaller throats
of the porous medium as shown in Figure 4.20.
Oil
Water
Figure 4.20: Micromodel filled with oil and water.
Subsequent injection of alcohol into the system presents the range of fluid
configurations shown in Figure 4.21. The alcohol might either displace the oil
completely, flow through the central portion of the pore pushing the oil aside as a
film or flow as a film between water and oil. It is also important to note that
though there is a change in the values of the spreading coefficients, their signs
remain the same.
81
Alcohol
Water
Oil
Figure 4.21: Possible configurations of fluid phases upon injection of alcohol in a pore
containing oil and water, based on the pore and throat model.
Based on spreading behavior previously discussed, the configuration on the
extreme left seems to be the most probable one. However because of the low
interfacial tensions between alcohol and oil in the previous experiments, it was
observed that alcohol flowed through the center of the pore in the form of
discontinuous drops. However in this system, where the interfacial tension
between alcohol and oil is substantially higher, it was observed that alcohol does
flow as a film between the oil and the water, resulting in some oil being trapped
in the center of the pore. In certain regions of the porous medium, piston-like
displacement of the oil by the alcohol was also observed, but the flow was
largely comprised of flow of alcohol in thin films.
Figure 4.22 shows a series of photographs of the micromodel during the injection
of alcohol. The photographs were taken seconds apart and can be considered
as a series of snapshots of the video (separated apart by frames of appx.1
second). It can be seen that as alcohol was injected into the porous medium,
some amount of alcohol and water is drained from the upstream regions. Ovoid
82
marked regions in the photographs clearly show the flow of water in layers, while
rectangular marked regions show the flow of alcohol in layers. It would be
interesting to note the importance of flow in layers, highlighted by the abrupt
appearance of a phase in a certain region of the micromodel.
Oil
Water
Figure 4.22 (a): Micromodel containing water and oil.
83
Oil
Abrupt
appearance of
water phase
due to flow of
water in a thin
layer close to
the pore body.
(contrast
to
previous
photograph
which
does
not show this
water region)
Figure 4.22 (b): Flow of water in layers.
Water
draining from
upstream
regions
displaces oil
and occupies
the narrower
throats of the
porous
medium.
Figure 4.22 (c)
84
Oil
Water
Note
abrupt
appearance of
water
phase
(compare
to
previous
photograph).
This is due to
flow of water in
thin layers.
Figure 4.22 (d)
Oil
Alcohol
Water
Figure 4.22 (e)
Note sudden
appearance of
alcohol
between
oil
and
water,
suggesting
flow of alcohol
in a layer.
85
Water
Trapped
oil,
surrounded by
alcohol.
Bypassed
region
Alcohol
Figure 4.22(f)
Appearance
of
alcohol
phase due to
flow
in
layers.
Figure 4.22 (g)
86
oil
Alcohol
Water
Oil
Figure 4.22 (h)
The
alcohol
region,
slowly
gets displaced
through layers
not
clearly
visible. Note the
alcohol region
in upper snap,
but
displaced
out in the lower
picture.
Figure 4.22 (i)
87
Trapped
oil
Water
Alcohol
Bypassed oil
Figure 4.22: Injection of alcohol in a micromodel containing oil and water.
Capillary tube experiments also corroborate these observations. In a capillary
tube filled with water, alcohol, oil and alcohol in that order, it was observed that
the upper alcohol region drains below, flowing as a layer between oil and water.
When the photographs in Figure 4.22 are compared to those in Figure 4.19, the
difference in the two systems in obvious. In all the photographs in Figure 4.19,
the alcohol phase (system with low interfacial tension between alcohol and oil) is
seen to flow at the center of the pore as a discontinuous phase. However in
Figure 4.22 photographs, the alcohol phase is seen to clearly spread as a layer
between the oil and water and forms a continuous phase throughout the porous
medium.
Injection of gas into the micromodel containing alcohol, oil and water presents
some possible configurations of fluid phases as illustrated in Figure 4.23. It is
important to verify at this point about the existence of four phases within the pore
88
structure. Thus it is important to observe if the gas occupies the central portion of
the pore with the remaining liquid hydrocarbon phases or if it drains both of them
or either of them preferentially based on their wettabilities.
Water
Oil
Alcohol
Gas
Figure 4.23: Schematic representation of fluid configurations in the pore, upon injection
of air in a micromodel containing water, oil and alcohol.
Figure 4.24 is a series of photographs from the micromodel that shows the fluid
configuration and connectivities after injection of gas. Ahead of the gas front,
there is three-phase flow of oil, water and alcohol. Oil present as a discontinuous
phase trapped by layers of alcohol is mobilized and flows as discontinuous drops
(contrast this to previous fluid system in which alcohol was the discontinuous
phase and flowed in a similar manner). By the time the gas front reaches a
certain portion of the porous medium, most of the oil is displaced from there and
subsequent flow is characteristic of three-phase flow involving water, alcohol and
gas. The following photographs illustrate the above observations. The initial
photographs show the three-phase flow of water, oil and alcohol while the latter
photographs show the invasion of the pores by gas, and associated flow
behavior. The ovoid marked regions represent oil flowing in a discontinuous
manner.
89
Alcohol
Trapped
Oil
Water
Figure 4.24 (a): Micromodel containing water, oil and alcohol.
Oil being
mobilized
due to gas
injection
upstream
Figure 4.24 (b)
90
Oil
being
drained from
upstream
connecting
up
and
draining.
Figure 4.24 (c)
Figure 4.24 (d)
91
Just before gas invasion of this pore space.
Oil
lcohol
Figure 4.24 (e)
Gas
Gas
Gas
Oil
Alcohol
Figure 4.24 (f)
92
Gas
Gas
Gas
Alcohol
Oil
Figure 4.24 (g)
Figure 4.24: Gas injection in a micromodel containing water, alcohol and water.
In the photographs in Figure 4.24, one cannot see four phases existing in the
pore space simultaneously. After gas invasion of the pores, here exist three
phases within the pore space. These phases are water, alcohol and gas. This is
clearly evident from the rectangular region marked Figure 4.24 (g).
93
Chapter 5
5. Discussion and Conclusions
An experimental apparatus for the visualization of four-phase flow was set up,
and experiments were performed at a constant flow rate of injection of the fluids.
Pore scale flow events related with the configuration and the connectivity of the
fluids were observed and certain conclusions regarding the four-phase flow
phenomenon can now be made.
It was observed that the saturation history is very important in determining the
orientation and connectivity of the fluid phases in the porous medium, ie., the
order of filling of the fluids in the porous medium is important in determining the
configuration of the fluids in the pore space. In the system in which the interfacial
tension between the oil and the alcohol was very low, it was observed that
though spreading of the alcohol between water and oil is the favorable situation
(as illustrated from the capillary tube experiments and the micromodel
experiments with order of filling of fluids being water, alcohol and oil); when the
fluids are injected in a different order of their wettabilities, the alcohol phase can
occupy the central portion of the pores as a discontinuous phase. Subsequent
injection of gas into the system causes this discontinuous alcohol phase to be
mobilized and the alcohol phase is displaced ahead of the gas front. However in
the system where the interfacial tension between the alcohol and oil phase is
higher it is observed that the alcohol phase spreads between the water and the
oil for the same experiment as above. This highlights that for the system with
lower interfacial tension between the hydrocarbon phases, the order of injection
of the phases is important. In most reservoir systems where four-phase flow is
observed, a liquid carbon dioxide phase exists, which has a relatively low
interfacial tension with respect to the oil phase in the reservoir. This could
94
perhaps correspond to our analog system in which the interfacial tension
between oil and alcohol is relatively low. This assumption is based on the
observation that when liquid carbon dioxide is injected in a micromodel
containing oil, one can see the flow of liquid carbon dioxide as a discontinuous
phase under certain conditions. Hence we could relate the injection of alcohol in
section 4.3 to the liquid CO2-rich phase, which was also observed to flow as a
discontinuous phase. The four-phase analogies could be made, given the above
assumptions.
Based on the observations, much of the four-phase flow field can be treated as a
combination of two three-phase flow situations. Ahead of the gas front,
interactions were primarily restricted to the liquid phases (water-oil-alcohol), while
at the gas front and behind the interactions were primarily between water, a
liquid hydrocarbon phase (oil or alcohol) and gas. A closer observation shows
that there are two primary phases flowing during gas injection, alcohol and gas or
oil and gas. However the presence of water occupying the smaller throats and
the corners and crevices of pores has an influence on the area available in the
pore cross-section for flow, and hence it is more prudent to treat four phase flow
as a combination of three phase flows and not as a combination of two phase
flows. It would be necessary to qualify the previous hypothesis given the
limitations of the experimental approach. Since the fluids were injected in
successive order, only one of the two liquid hydrocarbon phases was present in
larger quantity. In a real reservoir development scenario, the saturations of the
two liquid hydrocarbon phases might be appreciable and comparable, and the
hypothesis of approximating the four-phase flow as a combination of two threephase flows might not be applicable. The zones in which four phases were
present simultaneously were small in the experiments reported here. For those
zones, an appropriate relative permeability model is to be determined yet. The
presence of four phases in a pore would imply a smaller area of cross-section of
the pore available to flow for the various phases, and would also impact the
connectivity of the various phases and hence their displacement pattern.
95
One way to tackle the problem of relative permeabilities for reservoir
development would be to break down the four-phase flow problem into a
combination of two three-phase flow problems, one ahead of the gas front and
the other behind it. A possible way of tracking the gas front would be by tracking
the saturations of the gas in the various blocks of the reservoir simulator.
However such an approach would be limited by the presence of a diffuse front
and also an abrupt change in the relative permeability values of the phases at
the front.
It was also interesting to note that ahead of the gas front, there was mobilization
of the discontinuous phase (oil or alcohol). In the system having low interfacial
tension between the oil and the alcohol though alcohol was a disconnected
phase, it was observed to flow freely; even in the three-phase situation of
injection of alcohol in a porous medium containing water and oil. In the system
having a higher interfacial tension between the oil and alcohol, it was observed
that ahead of the gas front, the disconnected oil phase was also mobilized. This
-3
-2
is because the capillary number for the flow is in the range of 10 to 10 in both
the cases, and hence the viscous forces and the capillary forces are almost
equal. When phases flow in a discontinuous manner, the concept of relative
permeability breaks down (relative permeability can be thought of as a fraction of
cross-section of porous medium available to a particular continuous phase to
flow). These observations seem to suggest that the use of a pseudo liquid phase
(combining the two liquid hydrocarbon phases into one) in certain previous
reservoir developments might be a plausible idea. Based on observed flow
behavior it should be possible to do reservoir simulations by using three-phase
relative permeabilities of water, oil and gas, by making certain modifications in
the three-phase relative permeabilities (see Appendix A). These modifications as
discussed by Wang and Strycker (2000), split the relative permeability value of
the oil phase into two fractions, based on the saturations of the two liquid
hydrocarbon phases (eq. A-11, A-22).
96
The presence of four phases within a single pore cannot be ruled out completely,
though it was largely absent for most of the experiments performed. In the
system having a low interfacial tension between the oil and alcohol, four phases
in a pore were observed for a short transition period (when order of injection was
water, alcohol, oil and gas). In that case the four-phase problem was a
combination of two three-phase problems, alcohol spreading between oil and
water, and oil spreading between alcohol and air, both being favorable
conditions. However as the alcohol phase was displaced, these films were no
longer observed. In none of the other cases, were four phases observed in a
pore space simultaneously.
It is important to recognize that most of these observations are strictly restricted
by the regime of flow, capillary dominated or viscous dominated, as well as the
regions of the porous medium in which the observation was made. In our
experiments, four phases were injected in succession into a porous medium,
while one could expect the simultaneous formation of four phases in the regions
of the reservoir where there is a sudden pressure drawdown. Hence when gas is
injected in the experimental setup, there is predominantly a flow of two phases,
which might be different from a reservoir scenario which might involve the flow of
two liquid hydrocarbon-rich phases and a gas-rich phase. Hence the short
transition region of four-phase flow observed in one set of experiments, might
exist in the presence of larger fractions of the two liquid hydrocarbon phases.
Besides this there is also the inherent limitation of the two dimensional nature of
the micromodels.
As a follow up to this work, it would be useful to perform core flood experiments
with the fluid phases and measure the saturations and flow rates along with
pressure drops and compare the observations with a combination of three phase
relative permeabilities, ahead and behind the gas front. It would also be very
useful to be able to develop a technique to distinguish the various liquid phases
in the micromodel, by the use of certain dyes or use of refractive properties of
the phases involved.
97
Nomenclature
98
Cs
= spreading coefficient of oil between water and gas
σij
= interfacial tension between phases I and j
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Appendix A
A. Relative Permeability Models
The various models used for computation of four-phase relative permeabilities by
Wang and Strycker (2000) are listed below. Based on their observation, the
modified Corey Model gives the best fit. In all these models the three-phase oil
relative permeability is obtained from two-phase oil-water and oil-gas data. The
values of water and gas relative permeabilities are the same as the two-phase
data. The two liquid hydrocarbons share their relative permeabilities with the
calculated oil relative permeability in proportion of the relative volumes.
Baker Model
The three-phase oil relative permeability is interpolated as follows,
Kro =
( Sw − Swr ) Krow + ( Sg − Sgr ) Krog
…………………….(A-1)
( Sw − Swr ) + ( Sg − Sgr )
Where, Krow and Krog are the two-phase relative permeabilities calculated as,
o
row
(
o
rog
(
Krow = K
Krog = K
So − Sorw e
) ow ………………….……………(A-2)
1 − Swr − Sorw
1 − S * g − Slrg e
) og ………………………….……..(A-3)
1 − Slrg − Sgr
Where,
Swr – residual water saturation in oil-water two-phase system.
Sorw – oil residual saturation in the oil-water two-phase system.
103
o
o
K row, K rog – end point relative permeabilities of oil in the oil-water and oil-gas
two phase systems respectively. (maximum oil relative permeability).
Sgr – residual gas saturation in the oil-gas two-phase system.
Sorg – residual oil saturation in the oil-gas two-phase system.
Slrg = Swr + Sorg , total residual liquid saturation to gas phase during two phase of
oil and gas.
S*g = 1- So – min(Sw,Swr)
eow – empirical term to theoretically fit oil-water relative permeability curves.
eog - empirical term to theoretically fit oil-gas relative permeability curves.
The water and gas relative permeabilities are calculated by,
(
Sw − Swr
)ew …………………………..(A-4)
1 − Swr − Sorw
Krg = Korg (
Sg − Sgr
)eg ………………..…….(A-5)
1 − Swr − Sorg − Sgr
Krw = K
o
rw
Where,
K
o
rw
- end point water relative permeability value
Korg – end point gas relative permeability value
Modified Stone II Model
This is the Stone’s model presented above which has been normalized by Nolen
and is described as follows,
Kro = Korow K ……………………………………………(A-6)
104
K=(
Krow
Krog
+ Krw )(
+ Krg ) – ( Krw + Krg ) …….(A-7)
Korow
Korow
The oil-water two-phase permeability is calculated as,
Krow = Korow (
1 − Sw − Sorw e
) ow …………………….…(A-8)
1 − Swr − Sorw
The oil-gas two-phase relative permeability is calculated as,
Krog = K
o
rog
(
1 − Sg − Swr − Sorg e
) og ………………....(A-9)
1 − Swr − sorg − Sgr
Krw and Krg, the water and gas relative permeabilities are calculated from the
Baker model.
For the Baker and the modified Stone II model, the two liquid hydrocarbon
phases share the relative permeabilities in proportion to their relative volumes,
i.e. on their saturation weighted basis:
Kro = Kro (
So
) ……………………………….….(A-10)
So + S 4
Kr4 = Kro (
S4
) ……………………………….….(A-11)
So + S 4
Corey’s Model
In this model, the relative permeabilities of the various phases depend on their
own saturations and are computed as follows,
Krj = Korj (
Sj − Sjr
Np
1 − ∑ Slr
)ej ………………..)(A-12)
l =1
Where,
105
Krj – Relative permeability of phase j
Korj – Phase j end point relative permeability at Sj = 1 -
∑ Skr
, (Skr,Sjr, residual
k \= j
saturation of phase k or j, \= is representation for ≠)
ej – Corey exponent for phase j
In absence of three-phase oil exponent and end point relative permeability data,
the following can be used,
Koro = bKorow + (1-b)Korog ………………(A-13)
eo = beow + (1-b) eog ……………………(A-14)
b=1-
Sg
…………….…….(A-15)
1 − Swr − Sorg
Modified Corey’s Model
Dria, et al. (1993) modified the Corey’s model and the three-phase relative
permeabilities are calculated as:
Krw = K
o
rw
(Swe)ew ………………..(A-16)
Where,
Swe – Effective saturation of water phase
o
2
e
o
Kro = K ro(Soe) [ 1 – (1 – Soe)
] ……………….(A-17)
Krg = Korg (Sge)2 [1 – (1-Sge)eg ] …………..(A-18)
Where,
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Swe =
Sw − Swr
……………………..…(A-19)
1 − Swr − Sor
Soe =
So − Sor
…………………………(A-20)
1 − Swr − Sor
Sge =
Sg − Sgr
…………………..(A-21)
1 − Swr − Sor − Sgr
Also,
Kr4 = Kor4 (S4e)2[1 – (1-S4e)e4] ………..…(A-22)
S4e =
S 4 − S 4r
…………….….(A-23)
1 − Swr − Sor − S 4 r
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