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PAPER 2001-002

Determination of the RelativePermeability Matrix Coefficients

J.-Y. YuanAlberta Research Council

D. CoombeComputer Modelling Group Ltd.

D.H.-S. Law, A. BabchinAlberta Research Council

This paper is to be presented at the Petroleum Society’s Canadian International Petroleum Conference 2001, Calgary, Alberta,Canada, June 12 – 14, 2001. Discussion of this paper is invited and may be presented at the meeting if filed in writing with thetechnical program chairman prior to the conclusion of the meeting. This paper and any discussion filed will be considered forpublication in Petroleum Society journals. Publication rights are reserved. This is a pre-print and subject to correction.

ABSTRACT

This paper summarizes the recent development of the

generalized matrix formulation and its incorporation into

a reservoir simulator as collaborative efforts conducted

at Alberta Research Council (ARC) and Computer

Modelling Group (CMG). The focus of this paper is on

how the matrix coefficients, which are the generalization

of relative permeability curves, can be determined from

experimental data, in two-phase and three-phase systems.

We studied different types of experimental conditions

available in the literature to obtain these matrix

coefficients for two phase flow. These include

experiments by Bentsen and Manai and Boubiaux and

Kalaydjian. A more general description was derived,

leading to the disclosure of intrinsic connections between

the above two types, and all other possible types. This

allows the choice of various independent pairs of relative

permeability curves as input for a reservoir simulator

(CMG's STARS model). Selected experiments were then

re-simulated with this matrix formulation. The concept

was, for the first time, further generalized systematically

to three-phase flow. A scheme and detailed formulations

have been developed, allowing a reservoir simulator to

deal with a matrix formulation for three-phase flow.

These relations were also implemented in CMG's

simulator STARS.

INTRODUCTION

In the past few years, persistent efforts were made

collaboratively at ARC and CMG in developing a

PETROLEUM SOCIETYCANADIAN INSTITUTE OF MINING, METALLURGY & PETROLEUM

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generalized description of coupled multiphase flow in porous media, in terms of relative permeability matrix coefficients [1-6]. The fundamental concept of this new description can find its root in Prof. S. T. Yuster’s profound work on multiphase flow in a capillary [7] and has been promoted by Prof. W. Rose [8] since 1969. Although non-traditional, this concept has been accepted and well discussed in the area of fundamental research [9-20]. However, it has not been accepted for practical use because it is more complicated and has not yet proven more effective. Our focus, therefore, was to demonstrate that models developed based on the phase-coupling concept could be practical and used for reservoir simulation. Work done at ARC and CMG on this subject consists of several components:

(a) Development of the concept. For example, capillary coupling in addition to the commonly discussed viscous coupling has been considered as another possible mechanism. We have developed several schemes of relating experimental data to the relative permeability matrix coefficients, generalization of relative permeability curves.

(b) Identification of areas for field applications. For example, SAGD process has been identified as one of these applicable areas because it consists of a combination of co- and counter-current scenarios.

(c) Incorporation of the concept into CMG’s STARS.

We believe that all these components are essential especially the last one, which is the key to the success of finally bringing this not-so-new-but-important concept into a practically useful tool for reservoir engineering problems.

BACKGROUND Traditionally, in a porous medium with a two-phase

flow, one expresses flow rates for each phase in the following way:

11 1

1

Kkµ

= − ∇ΦJ , ...................................................... (1)

22 2

2

Kkµ

= − ∇ΦJ , ..................................................... (2)

where, k1, k2 are relative permeabilities and µ1, µ2 are viscosity for each phase. Φ1 and Φ2 are potentials that sum up pressures and gravity heads, for each phase.

The matrix formulation for the same system can be written as:

1 11 1 12 2A A= − ∇Φ − ∇ΦJ , ..........................................(3)

2 21 1 22 2A A= − ∇Φ − ∇ΦJ . .........................................(4)

Comparing Equation (3) with Equation (1), we see that an extra term that involves 2∇Φ appears in Equation (3).

This extra term represents the effects of driving force in fluid 2 on the flow rate of fluid 1. The opposite interaction is represented by the similar extra term appearing in Equation (4). These two terms are called phase-coupling terms. They account for coupling between two fluids in the porous media. The focal point of the continuing discussions over the past decades is not whether these terms exist, but whether they are important. In other words, whether Equations (1) and (2) are good enough in dealing with practical reservoir engineering problems.

It is not difficult to realize that there should be coupling between the two fluids when they have to flow in the same porous medium. In fact, there are two types of coupling, one due to viscous forces [7] and the other due to capillary interactions [2, 20]. Reflecting these couplings are Equations (3) and (4). This sort of linear response between driving forces and flow rates (or fluxes) can be seen in many other physical systems that consist of multiple driving forces and multiple fluxes [9, 21].

The main disadvantage of using Equations (3) and (4) is that the determination of these coefficients may be more complicated. The main advantage is that these equations are correct descriptions of a multi-phase flow. In a simplified way, Rose [12] showed that a small coupling could make a big difference. He demonstrated a significant change in saturation relaxation rate, if the matrix formulas are used instead of the traditional relative permeabilities. We believe a more complete investigation on whether the correct descriptions would

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lead to better results in terms of accuracy and applicability is needed.

It is well known that relative permeability curves vary with experimental conditions, even for the same porous medium. Examples include the differences in co-current relative permeability curves and counter-current ones [10, 13-14]. On the other hand, the mobility matrix coefficients, Aij (i, j = 1, 2) in Equations (3) and (4), should be independent of experimental procedures.

DETERMINATION OF RELATIVE PERMEABILITY MATRIX COEFFICIENTS

Formulations for Two-Phase Flow The mobility matrix coefficients, Aij, can be

represented in terms of the relative permeability matrix coefficients, kij. Their relations can be seen when Equations (3) and (4) are re-written as:

( )1 11 1 12 21

K k kµ

= − ∇Φ + ∇ΦJ , ..................................(5)

( )2 21 1 22 22

K k kµ

= − ∇Φ + ∇ΦJ . .................................(6)

Obviously, we have

( ), , =1,2ijij

i

KkA i j

µ= . .............................................(7)

Direct experimental determination of these relative permeability matrix coefficients may be tricky and still under development [19]. Here, we focus on a more practical scheme suggested in a previous CIM paper [3]. This scheme consists of two steps: (a) experimental determination of relative permeabilities under independent experimental conditions, normally co-current and counter-current conditions; and (b) calculation of relative permeability matrix coefficients using observed relative permeability curves specific to these experimental conditions.

Typically, an experiment under a given set of conditions determines a set of relative permeability curves. For two-phase case, the number of curves is two. On the other hand, a relative permeability matrix contains four coefficients. Therefore, it is in general needed to obtain two sets of relative permeability curves in order to

fully determine the matrix coefficients. This means one needs to conduct two experiments under independently different conditions.

We use Equations (5) and (6) to describe the first experiment and use the following equations to describe the second experiment:

( )* * *1 11 1 12 2

1

K k kµ

= − ∇Φ + ∇ΦJ , ................................. (8)

( )* * *2 21 1 22 2

2

K k kµ

= − ∇Φ + ∇ΦJ . ................................ (9)

The essence of this approach is that the conventional relative permeability concept is still honored, i.e., Equations (1) and (2) are still valid for describing the first experiment, provided that the relative permeability curves are obtained specific to the first experimental conditions. Similarly, we can have the following equations to describe the second experiment:

** *11 1

1

Kkµ

= − ∇ΦJ , ................................................... (10)

** *22 2

2

Kkµ

= − ∇ΦJ . ................................................... (11)

In a one-dimensional case, by eliminating 1∇Φ and

2∇Φ in Equations (1), (2), (5) and (6), we obtain:

1 2 1 1 11 2 1 1 12 1 2 2k k J k k J k k Jµ µ µ= + , ........................... (12)

1 2 2 2 21 2 1 1 22 1 2 2k k J k k J k k Jµ µ µ= + . .......................... (13)

Similarly, Equations (8)-(11) yield:

* * * * * * *1 2 1 1 11 2 1 1 12 1 2 2k k J k k J k k Jµ µ µ= + , ......................... (14)

* * * * * * *1 2 2 2 21 2 1 1 22 1 2 2k k J k k J k k Jµ µ µ= + . ........................ (15)

Equations (12)-(15) are then used to obtain the relative permeability matrix coefficients:

( )* * * *1 1 2 1 2 2 1 2

11 * * * *1 2 1 2 1 2 1 2

k k k J J k J Jk

k k J J k k J J−

=−

.................................. (16)

( )* * *2 2 1 1 1 11

12 * * * *2 1 2 1 2 1 2 1 2

k k k k J Jk

k k J J k k J Jµµ

−=

−................................ (17)

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( )* * *1 1 2 2 2 22

21 * * * *1 1 2 1 2 1 2 1 2

k k k k J Jk

k k J J k k J Jµµ

−=

−................................ (18)

( )* * * *2 2 1 1 2 1 1 2

22 * * * *1 2 1 2 1 2 1 2

k k k J J k J Jk

k k J J k k J J−

=−

................................. (19)

These are the general formulation for calculating relative permeability matrix in two-phase flow. As we can see, these formulas are in most cases impractical because they directly require flow rate data. However, since we know how we conduct the experiments, we normally have a good knowledge on how these flow rates are related. In the following, we will discuss a few special sets of one-dimensional experimental conditions under which we can determine these relationships and hence relative permeability matrix coefficients.

Case 1: B-conditions Experimental conditions in this case are:

1 2∇Φ = ∇Φ ........................................................... (20)

for co-current experiment, and

* *1 2∇Φ = −∇Φ ........................................................ (21)

for counter-current experiment. These conditions were specified in Bentsen’s experiments [16]. The oil-water co-current and counter-current relative permeability curves measured are shown in Figure 1. Under these conditions and using Equations (1)-(2) and (10)-(11), we obtain:

*1 1

11 2k kk += , ......................................................... (22)

*1 1

12 2k kk −= , ......................................................... (23)

*2 2

21 2k kk −= , ......................................................... (24)

*2 2

22 2k kk += .......................................................... (25)

Figure 2 shows the relative permeability matrix coefficients obtained from Bentsen’s experiments.

Case 2: K-conditions While the co-current condition, Equation (20), remains

the same, the counter-current condition in this experiment is different:

* *1 2J J= − . ...............................................................(26)

These conditions are believed to be satisfied in Kalaydjian’s experiment [13-14]. Although Kalaydjian did not describe these experimental conditions in his publication, these conditions were confirmed in the numerical history matching of his experiments as will be discussed later in this paper. The oil-water co-current and counter-current relative permeability curves obtained by Kalaydjian et al. are shown in Figure 3. Under these conditions, we obtain [4]:

( )* *1 2 1 1 2

11 * *2 1 1 2

k k kk

k k

µ µ

µ µ

+=

+

% % %

% % , ...........................................(27)

( )* *1 2 1 1

12 * *2 1 1 2

k k kk

k k

µ

µ µ

−=

+

% % %

% % , ...............................................(28)

( )* *2 1 2 2

21 * *2 1 1 2

k k kk

k k

µ

µ µ

−=

+

% % %

% % , ..............................................(29)

( )* *2 1 2 2 1

22 * *2 1 1 2

k k kk

k k

µ µ

µ µ

+=

+

% % %

% % ............................................(30)

Figure 4 shows the relative permeability matrix coefficients obtained from Kalaydjian’s experiments.

Case 3: α-conditions This is a generalization of the case 2. The condition

for the counter-current in this experiment is: * *1 2J Jα= − ,.............................................................(31)

while co-current condition remain the same as in case 1. The relative permeability matrix coefficients under α-conditions are:

( )* *1 2 1 1 2

11 * *2 1 1 2

ˆ ˆ ˆ

ˆ ˆk k k

kk k

µ αµ

µ αµ

+=

+, .........................................(32)

( )* *1 2 1 1

12 * *2 1 1 2

ˆ ˆ ˆ

ˆ ˆk k k

kk k

αµ

µ αµ

−=

+, .............................................(33)

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( )* *2 1 2 2

21 * *2 1 1 2

ˆ ˆ ˆ

ˆ ˆk k k

kk k

µ

µ αµ

−=

+, ..............................................(34)

( )* *2 1 2 2 1

22 * *2 1 1 2

ˆ ˆ ˆ

ˆ ˆk k k

kk k

αµ µ

µ αµ

+=

+..........................................(35)

Obviously, when 1α = the α-conditions reduce to the K-conditions. It is also worthwhile noticing the symmetry under the transformation (α, 1, 2) ⇔ (α−1, 2, 1). The value of α can be the density ratio in some cases.

There seems to be no limitation on the number of cases we might consider. In order to determine the relative permeability matrix coefficients, one needs to simply follow two steps: (i) under given experimental conditions, measure two independent sets of relative permeability curves; and (ii) calculate the relative permeability matrix coefficients using Equations (16)-(19) based on the given experimental conditions and the measured relative permeability curves.

It must be emphasized that our principle assumption is that the relative permeability matrix coefficients are independent of experimental conditions, while the relative permeability curves are. In other words, under different experimental conditions, one may obtain different relative permeability curves, but all will lead to a unique set of relative permeability matrix coefficients for a given porous medium and fluids. Therefore, relative permeability curves obtained under different conditions should be related. For example, if we measured relative permeability curves under α-conditions we can calculate relative permeability curves under B-conditions. From Equations (22)-(25) and (32)-(35), we have

1 1̂k k= , ...................................................................(36)

2 2ˆk k= , ..................................................................(37)

( )( )* * *1 1 2 1 1 2* *

1 1 * *2 1 1 2

ˆ ˆ ˆ ˆˆ

ˆ ˆk k k k

k kk k

µ αµ

µ αµ

− −= +

+, .........................(38)

( )( )* * *2 2 2 1 1 2* *

2 2 * *2 1 1 2

ˆ ˆ ˆ ˆˆ

ˆ ˆk k k k

k kk k

µ αµ

µ αµ

− −= −

+. ........................(39)

These relations allow us to convert relative permeabilities obtained from under one set of experimental conditions to another.

General Consideration for Three-Phase Flow In this section, we extend the “B-conditions” in a two-

phase system, i.e., Equations (20) and (21), to a three-phase system.

The driving force in one phase can be parallel or opposite to the driving forces in the other one or two phases. There are four different configurations for an oil-water-gas system:

Configuration (p): all phases are driven in the same direction. This configuration is represented by the following symbol:

( )pw o g

↑ ↑ ↑=

. .................................................... (40)

Configuration (q1): water is driven opposite to the oil and gas driving forces:

( )1qw o g

↓ ↑ ↑=

. .................................................... (41)

Configuration (q2): oil is driven opposite to the water and gas driving forces:

( )2qw o g

↑ ↓ ↑=

..................................................... (42)

Configuration (q3): gas is driven opposite to the water and oil driving forces:

( )3qw o g

↑ ↑ ↓=

..................................................... (43)

It is not difficult to realize that not all of these configurations are independent to the others. In other words, we can easily build one configuration from a linear combination of the other three. The relation is:

w o g w o g w o g w o g ↑ ↑ ↑ ↓ ↑ ↑ ↑ ↓ ↑ ↑ ↑ ↓

= + +

,......... (44)

or

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( ) ( ) ( ) ( )1 2 3p q q q= + + . ........................................ (45)

Therefore, to construct a three-phase relative permeability matrix, we need only to consider any three of the above four configurations. For example, we shall use configurations (q1), (q2), and (q3).

For each configuration, three-phase relative permeability curves can be traditionally estimated from two pairs of two-phase relative permeability curves, i.e., water-oil and gas-oil, using classical models such as Stone models.

This way, since we have three configurations, we can obtain total nine three-phase relative permeabilities. For convenience, we let 1 represents water phase, 2 oil phase, and 3 gas phase. Then we have nine three-phase relative

permeability curves: ( )jqik , where, , 1,2,3i j = .

The general three-phase equations are:

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1

for 1,2,3.i ij jji

K k iµ =

= − ∇Φ =∑J .................... (46)

Therefore, we have for configuration (q1):

( ) ( )1 11 2 3 1 i q

i i i ik k k kδ− + + = − , .................................. (47)

where ijδ is the Kronecker delta symbol:

0 ;1 .ij

i ji j

δ≠

= =........................................................ (48)

Similarly, for configuration (q2):

( ) ( )2 21 2 3 1 i q

i i i ik k k kδ− + = − ;.................................... (49)

And for configuration (q3):

( ) ( )3 31 2 3 1 i q

i i i ik k k kδ+ − = − . .................................... (50)

Since configuration (p) involves co-current flows only, we simply have:

( )1 2 3 for 1,2,3p

i i i ik k k k i+ + = = . ........................ (51)

These lead to the following relation:

( ) ( ) ( )3

1

1 ij jqpi i

jk kδ

=

= −∑ .............................................. (52)

This relation is a more explicit equivalency to the symbolic Equation (45). Using Equations (47), (49), (50) and (52), we can evaluate all nine relative permeability matrix coefficients:

( )( ) ( )3

1

1 1 12

il lqij jl i

l

k kδδ=

= − −∑ , ................................(53)

where , 1,2,3i j = ; or, using Equation (52), we obtain an

alternative:

( ) ( ) ( )1 12

ij jqpij i ik k kδ = − − . ....................................(54)

In brief, to calculate all nine relative permeability matrix coefficients in a three-phase system, we first measure relevant two-phase relative permeability curves; then use a classical model to evaluation three-phase relative permeabilities for any three of the aforementioned four configurations; finally, using Equations (52) and (53), we compute the three-phase relative permeability matrix coefficients.

IMPLEMENTATION OF COUPLED MODEL IN STARS

For the STARS implementation, these basic equations are decomposed into “traditional” terms and corrections as

( )pi

i i ii

KkJ Cµ

= − ∇Φ + ,...........................................(55)

in which

( ) ( ) ( )3

1

12

ij jqpi i i j

ji

KC k kδ

µ =

= − − − ∇Φ ∑ . .................(56)

The identity

( )1 2 1ij

ijδ δ− + = .....................................................(57)

has been used to obtain Equation (56). The reason for this decomposition is to only invoke the coupling model calculations if the option is flagged. It is clear, from Equations (52), (53) and (56), that the correction terms in Equations (55) vanish when the cross coefficients of the relative permeability matrix vanish.

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The above equations are written in terms of three-phase relative permeabilities. That is, all permeabilities in all cases (co-current and counter-current) are assumed to be functions of all phase saturation most generally. All possibilities of various wettabilities and spreading conditions are included in this treatment. In STARS, the input for three-phase relative permeabilities consists of two sets of two-phase relative permeabilities (nominally termed water-oil and oil-gas) in which each set is a function of one saturation, plus a keyword such as *STONE1 or *STONE2 indicating how these two-phase relative permeability curves are to be combined in a three-phase model. Because the individual two-phase curves (entered as tables) can have various shapes, various two-phase wettabilities can be accommodated. Additionally STARS has an extra keyword (e.g., *WATERWET) to describe different three-phase wettability situations. In summary, STARS relative permeability options are quite general and all these possibilities are combined into the three-phase relative permeability curves in a separate module BEFORE the general equations given above (“traditional” flows plus corrections) are accessed. Only for user input does one need to be concerned with two-phase relative permeabilities, because all flow calculations are performed with the combined three-phase relative permeabilities.

Similar to this basic structure, the user input for the three-phase co-current and counter-current relative permeabilities is expressed in terms of pairs of two-phase information. Thus for each of the three-phase relative permeability situations (p, q1, q2, q3), we require two sets of two-phase curves to enter:

1. For the three-phase co-current case, we enter two-phase water-oil and two-phase oil-gas, both as their co-current case.

2. For the three-phase water counter-current case, the two-phase water counter-current and the two-phase oil-gas co-current curves are entered.

3. The three-phase oil counter-current case uses the two-phase counter-current curves for both water-oil and oil-gas.

4. Finally, for the three-phase gas counter-current gas, the water-oil co-current and the oil-gas counter-current are entered.

In this way, all of the three-phase situations can be expressed in terms of the two-phase flow information.

Schematically, in terms of the two-phase flow information, the generalized B-conditions of equal or opposite flow potential gradients can be expressed.

Configuration (p): all phases are driven in the same direction. This configuration, as described by Equation (40), is simply expressed as:

( )pw o o g

↑ ↑ ↑ ↑= +

. ........................................... (58)

Configuration (q1): water is driven opposite to the oil and gas driving forces:

( )1qw o o g

↓ ↑ ↑ ↑= +

. .......................................... (59)

Configuration (q2): oil is driven opposite to the water and gas driving forces:

( )2qw o o g

↑ ↓ ↓ ↑= +

........................................... (60)

Configuration (q3): gas is driven opposite to the water and oil driving forces:

( )3qw o o g

↑ ↑ ↑ ↓= +

........................................... (61)

To recover the pure two-phase situations discussed in the first section of this report but within the general three-phase development, it is sufficient that the counter-current flow of the remaining two-phase pair is set equal to the co-current flow of the same pair.

TWO PHASE EXAMPLE Flow experiments involving co-current and counter-

current spontaneous water/oil imbibition were performed on natural porous media by Bourbiaux and Kalaydjian [14]. Although their numerical simulation of the experiments revealed the difference between the co-current and the counter-current relative permeabilities,

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however, no knowledge was given on how the water and oil flow rates were related. Therefore, their numerical simulation was repeated here in order to shed some light into the experimental conditions.

Detailed experimental descriptions were given by Bourbiaux and Kalaydjian [14]. Very brief descriptions are given here. The first experiment was a dominant co-current imbibition and the second experiment a pure counter-current imbibition. In the first experiment, the bottom face of the porous medium was put in contact with water and the top face in contact with oil. Oil production was possible by the top face because a bypass maintained a hydraulic communication between the two end faces. So oil production by the bottom face was generated by counter-current flow and oil production by the top face by co-current flows. The water/oil interface inside the bypass was situated at the upper face level so that gravity forces hindered counter-current flows at the bottom face and acted as driving forces for the co-current flow from the bottom to the top face. In the second experiment, only the upper end of the porous medium was put in contact with water. The bottom cap was filled with oil and closed to prevent any oil production from this face. Both gravity and capillarity were driving forces for counter-current oil production from the top face.

Numerical Simulation of Kalaydjian’s Experiments

The numerical simulator, STARS, developed by the Computer Modelling Group (CMG) was used in the numerical study to simulate the flow behaviours of the experiments. The co-current and counter-current relative permeabilities determined by Bourbiaux and Kalaydjian [14] were used to predict the oil recovery history for each experiment.

Numerical prediction of the oil recovery history is shown in Figure 5 for the dominant co-current imbibition experiment. It was found that good agreement between the numerical prediction and the experimental data was obtained when the co-current relative permeability curves were used. On the other hand, under-prediction of the oil recovery history was observed when the counter-current relative permeability curves were used. Figure 6-Figure 8 show water saturation contours, water phase and oil

phase velocities as functions of time, respectively. Water saturation contours indicated the upward propagation of the waterfront from the bottom face and phase velocities clearly indicated the co-current nature of the flows.

Numerical prediction of the oil recovery history is shown in Figure 9 for the pure counter-current imbibition experiment. It was found that good agreement between the numerical prediction and the experimental data was obtained when the counter-current relative permeability curves were used. On the other hand, over-prediction of the oil recovery history was observed when the co-current relative permeability curves were used. Figure 10-Figure 12 show water saturation contours, water phase and oil phase velocities as functions of time, respectively. Water saturation contours indicated the downward propagation of the waterfront from the top face and phase velocities clearly indicated the counter-current nature of the flows. Figure 13 shows the values of the phase velocities that satisfied the counter-current experimental conditions of

* *1 2J J= − .

A matrix formulation simulation of these experiments was conducted as follows. The K-condition co-current and counter-current relative permeabilities were converted to the B-condition relative permeabilities through the use of Equations (36)-(39). These were then input into the STARS simulator using the general matrix formulation described above. The simulator then automatically calculated the appropriate flow behaviour for each experiment and reproduced the above matches of the co-current and counter-current experiments. Additional experimental cases could in principle be simulated, e. g., Kalaydjian's total flow case without the use of additional relative permeability curves.

CONCLUSION A general matrix formulation for a two-phase flow has

been developed. Special cases and relationships among them have been studied. These relationships allow the choice of various independent pairs of relative permeability curves as input for CMG STARS simulator. All other possible choices can be converted into the chosen one.

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A scheme and detailed formulations have been developed, allowing a reservoir simulator to deal with three-phase flow. The three-phase computation scheme is a generalization of the two-phase computation scheme. A general implementation scheme for the incorporation of the phase coupling model in the reservoir simulator STARS was developed.

CMG STARS is the first reservoir simulator to incorporate the generalized matrix formulation for two-phase and three-phase flows.

The generalized relative permeability matrix model provides most complete multiphase flow description available, although many fundamental and application issues remain. This general implementation will allow sensitivity analysis of the effects of phase coupling in a wide variety of oil recovery processes, including VAPEX and SAGD.

ACKNOWLEDGEMENTS The work was supported by the AACI Research

Program. We also would like to thank our colleague, Dr. Dave Cuthiell, for many fruitful discussions throughout the course of this study.

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15. Goode, P., and T. Ramakrishnan: Momentum Transfer across Fluid-Fluid Interfaces in Porous Media: a Network Model, AIChE J., Vol. 39, pp.1124-1134, 1993.

16. Bentsen, R. G., and A. A. Manai: On the Use of Conventional Co-current and Counter-current Effective Permeabilities to Estimate the Four Generalized Permeability Coefficients which Arise in

10

Coupled Two-Phase Flow, Transport in Porous Media, Vol. 11, pp.243-262, 1993.

17. Bentsen, R. G.: An Investigation into Whether the Nondiagonal Mobility Coefficients which Arise in Coupled, Two-Phase Flow are Equal, Transport in Porous Media, Vol. 14, pp.23-32, 1994.

18. Liang, Q., and J. Lohrenz: Dynamic Method of Measuring Coupling Coefficients of Transport Equations of Two-Phase Flow in Porous Media, Transport in Porous Media, Vol. 15, pp.71-79, 1994.

19. Dullien, F. A. L., and M. Dong: Experimental Determination of the Flow Transport Coefficients in the Coupled Equations of Two-Phase Flow in Porous Media, Transport in Porous Media, Vol. 25, pp.97-120, 1996.

20. Bentsen, R. G.: The Physical Origin of Interfacial Coupling in Two-Phase Flow through Porous Media, to be published, 2001

21. Landau, L. D., and E. M. Lifshitz: Statistical Physics, Pergamon Press, 1980.

FIGURES

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Water Saturation

Rel

ativ

e Pe

rmea

bilit

y to

Oil

0

0.1

0.2

0.3

Rel

ativ

e Pe

rmea

bilit

y to

Wat

er

Symbols: MeasuredCurves: Fitted

*rok

rwkrok

*rwk

Figure 1: Oil-water relative permeability curves measured in Bentsen and Manai’s experiments [16].

kww

kwo

kow

koo

Figure 2: Relative permeability matrix coefficients obtained from Bentsen and Manai’s experiments using Equations (22)-(25).

0

0.1

0.2

0.3

0.4

0.5

0.35 0.40 0.45 0.50 0.55 0.60Water Saturation

Rela

tive

Perm

eabi

lity

to O

il

0

0.01

0.02

0.03

0.04

0.05

Rela

tive

Perm

eabi

lity

to W

ater

*rok

rwkrok

*rwk

Figure 3: One set of oil-water relative permeability curves used in Kalaydjian et al.’s experiments [14].

Relative Permeability Matrix Coefficients

0

0.1

0.2

0.3

0.4

0.5

0.40 0.44 0.48 0.52 0.56 0.60

Water saturation

0.000

0.002

0.004

0.006

0.008

0.010

koo

kowkwo

kww

Figure 4: Relative permeability matrix coefficients obtained from the relative permeability curves in Figure 3 using Equations (27)-(30).

1

Figure 5: Oil recovery history match in co-current flow

experiment of Kalaydjian et al. [14].

Figure 6: Water saturation evolution in co-current flow

experiment of Kalaydjian et al.[14].

Figure 7: Water phase velocity map as it evolves in co-

current flow experiment of Kalaydjian et al. [14].

Figure 8: Oil phase velocity map as it evolves in co-

current flow experiment of Kalaydjian et al. [14].

2

Figure 9: Oil recovery history match in counter-current

flow experiment of Kalaydjian et al. [14].

Figure 10: Water saturation evolution in counter-current

flow experiment of Kalaydjian et al. [14].

Figure 11: Water phase velocity map as it evolves in

counter-current flow experiment of Kalaydjian et al. [14].

Figure 12: Oil phase velocity map as it evolves in

counter-current flow experiment of Kalaydjian et al. [14].

3

15 hours, Kalaydjian counter-current0

5

10

15

20

25

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

J (cm/hr)

Oil flux

Water flux

Figure 13: Fluxes of water and oil in counter-current flow

experiment of Kalaydjian et al. [14]. Within reasonable

range of errors, it is clear that they are equal and opposite

to each other.