Large-scale properties for two-phase flow in random porous media

Journal

Journal of Hydrology 183 (1996) 69-99

Large-scale properties for two-phase flow in random porous media

A. Ahmadi*, M. Quintard L.E.P.T.-ENSAM (UA CNRS 873), Esplanade des Arts et Mitiers, 33405 Talence Cedex, France

Received 31 January 1994; revision accepted 14 November 1994

Abstract

Hydrocarbon contaminants in the subsurface are important sources of pollution in aquifers. Hence, mathematical models of these flows have become key tools in environmental studies. In this paper we are interested in the flow in the saturated zone. Simulation of two-phase flow in large, complex heterogeneous domains often requires an unacceptably large number of computational grid blocks. Despite recent progress in computational methods and tools, we must call upon special techniques in order to use larger grid blocks while compensating for intracell variations in rock properties and fluid saturations. The use of pseudo-functions is one way of increasing grid dimensions to a more tractable level with a minimal loss of simulation representativeness. This change of scale problem has also been treated theoretically by different scaling-up techniques, such as large-scale averaging. This method calculates the transport equations and the effective properties at a given scale by an averaging process over the equations corresponding to a lower scale. This procedure leads to a closure problem which is very complex in the general case. Previously a first solution of the large-scale averaging problem was proposed in the quasi-static case corresponding to local capillary equilibrium. In the general case of a heterogeneous medium with a complex geometry, this boundary value problem (closure problem) can be solved by numerical methods. For this purpose, after having chosen a grid-block description of our system in accordance with the description used in reservoir simulators, we have implemented a three-dimensional (3-D) numerical resolution of the closure problem. The most important variations in the rock properties are associated with the permeability. We have therefore generated porous media with a random permeability distribution using different methods. Other multiphase properties are chosen to depend on the permeability. The properties of the closure problem in the case of randomly generated porous media are investigated. In particular, we give the conditions under which the general form of the local capillary pressure and relative permeability curves are recovered at the large scale. Particular properties related to each generation method are stated. The equivalent properties are calculated using averages over the results of many realizations of a given medium. The

* Corresponding author.

0022-1694/96/$15.00 0 1996 - Elsevier Science B.V. All rights reserved

XSDI 0022-1694(95)02819-6

70 A. Ahmadi, M. Quintard / Journal of Hydrology 183 (1996) 69-99

influence of the size of the averaging surface for a given correlation length as a function of the variance of the permeability is studied. We have therefore established some general rules for the calculation of the large-scale properties of random porous media.

Nomenclature

Ajk 4, baj Lx, b,, L C(h) E(f) h h

J(X) K K’ K’

KC2

K, KO Kaj KZ

IKeI*

KB Kh Kmin K

K:: Kw Kr, Krz

K:, , K;y , G K; MW N

“jk

NX

4

NZ

4

Ii

4

log, In

PO PC Pcj

5

PC2

interface between the j and k regions (m’) closure variable that maps (V{P,}” - peg) on to the pressure deviation k,, (m) closure variable b, for block j (m) components of the closure variable, b,, in the x, y and r directions (m) covariance function of a given random function f mathematical expectation of a given random functionj length of the distance vector h(m) distance vector (m) Leverett function defining the capillary pressure local-scale intrinsic permeability in the isotropic case and in its principal axes (m’) large-scale intrinsic permeability in the isotropic case and in its principal axes (m’) large-scale intrinsic permeability (m’) arithmetic mean permeability (m’) local-scale permeability for the o-phase in homogeneous region (ml) local-scale permeability tensor for the o-phase (m2) local-scale permeability tensor for the a-phase in block j (m’) large-scale permeability for the o-phase (m’) large-scale volume-averaged permeability for the a-phase (m’) geometric mean permeability (m2) harmonic mean permeability (m’) minimal permeability of the random homogeneous blocks (m2) maximal permeability of the random homogeneous blocks (m’) cross-coupled permeabilities (m2) local scale a-phase relative permeability large-scale a-phase relative permeability two coefficients of the diagonalized large-scale intrinsic permeability (m2) coefficients of the diagonalized large-scale intrinsic permeability (m’) Moissis and Wheeler Method dimension of the flow system considered unit normal vector directed from the medium j toward the medium k (m) number of voxels in the x direction contained in the unit cell number of voxels in they direction contained in the unit cell number of voxels in the .z direction contained in the unit cell harmonic mean fluctuation scale or correlation length (m) i = 1,2,3, fluctuation scale or correlation length in the i direction (m) i = 1,2,3, lattice vectors for a unit cell (m) nepcrian logarithm function local volume averaged pressure in the o-phase (N me2) local-scale capillary pressure (N m-‘) local-scale capillary pressure for the homogeneous block j, (Nmm2) large-scale capillary pressure (Nme2) P,-{P,},, large-scale spatial deviation for the o-phase pressure (Nmm2)

A. Ahmadi, M. Quintard / Journal of Hydrology 183 (1996) 69-99 71

position vector of a current point of the large-scale averaging volume (m) residual oil saturation large-scale residual oil saturation local-scale reduced water saturation local-scale water saturation large-scale water saturation irreducible water saturation large-scale irreducible water saturation local scale water saturation for the homogeneous block j Turning Bands Method local-scale velocity vectors for the water and oil phases (m SC’) large-scale averaging volume (m3) position vector locating the center of a voxel (m) components of the vector x (m)

r

Sor Sor’

& SW $.I swi Swi’

&vj

TB

VW VO VW x

xi>xj,xk

Greek Letters

lengths of the edges of a voxel in the x, y and z directions (m) large-scale volume-average porosity local porosity small parameter indicating on which side of the interface between two voxels the point under consideration lies rotation angle between the principal axes and the original reference frame considered variance

1. Introduction

Mathematical models of fluid flow in geological structures are not only important tools for the study of hydrocarbon contamination in aquifers, they are also commonly used in the petroleum industry by reservoir engineers for testing different operating strategies, comparing various recovery methods and predicting reservoir performance. There is therefore an extensive background concerning the problem in both the hydrology and petroleum literatures.

Simulation of large, complex aquifers or reservoirs often requires an unacceptably large number of computational grid blocks. Despite recent progress in computational methods and tools, we must call upon special techniques in order to use larger grid blocks while compensating for variations in rock properties and fluid saturations at any scale smaller than the grid size.

The use of pseudo-functions is one way of increasing grid dimensions to a more tractable level with a minimal loss of simulation representativeness. Practically, for reservoir simulations, Darcy’s law and the continuity equation are used at the large scale and the properties such as the multiphase effective permeability tensors and the capillary pressure are replaced by the corresponding pseudo-functions which account for the heterogeneity (Coats et al., 1967; Martin, 1968; Hearn, 1971; Dake, 1978; Jacks et al., 1973; Kyte and Berry, 1975; Thomas, 1983; Muggeridge, 1991; Stone, 1991). The most common and general pseudo-function approach is based on numerical


simulations (Huppler, 1970; Jacks et al., 1973; Emanuel and Cook, 1974; Kyte and Berry, 1975; Woods and Khurana, 1977; Thomas, 1983; Kortekaas, 1985; Kossack et al., 1990; Stone, 1991). In these methods, a two-dimensional (2-D) simulation of the flow in a representative reservoir cross-section is performed with fine grids, and the results are used to calculate the pseudo-functions necessary for the numerical simulation of the whole reservoir with coarse grids. Although these methods give satisfactory results, they require a great deal of numerical calculation and are therefore rather expensive.

This change of scale problem has also been treated theoretically by different scaling- up techniques, such as homogenization (Bourgeat, 1984; Amaziane and Bourgeat, 1988; Seaz et al., 1989) and large-scale volume averaging (Quintard and Whitaker, 1988, 1990a, b). Stochastic analysis applied to filtration problems (Mantoglou and Gelhar, 1989) can also be extended to the case of multiphase flow in porous media. Assuming large-scale equations are similar to local-scale equations, large-scale multiphase permeabilities have been calculated by renormalization techniques for some types of heterogeneous systems (King, 1989).

One of our major aims, in this paper, is to treat the 3-D upscaling of two-phase flow using the large-scale volume averaging method. Using this method, the transport equations and the effective properties at a given scale are calculated by an averaging process over the equations corresponding to a lower scale. An advantage of this approach is that it provides a comprehensive mathematical background to define large-scale variables. The associated large-scale equations are not assumed to be similar to generalized Darcy laws but are derived through the averaging process instead. In addition, large-scale properties also are derived through this averaging process in a general way. This procedure leads to a closure problem which is very complex in the general case, but can be solved numerically for all geometries and anisotropies in the quasi-static case. The quasi-static case corresponds to local capillary equilibrium within the averaging volume. Quintard and Whitaker (1988) have shown that in the quasi-static case, the large-scale equation are similar to the local ones and the large-scale properties are functions only of the large-scale saturation. While in the dynamic case, i.e. when the pressure gradients and the transient effects create significant changes in the saturation over the large-scale averaging volume, the large-scale averaging scheme gives more complex large-scale equations and closure problems featuring time-dependent, directional, dynamic behavior (Quintard and Whitaker, 1990a). In particular, large-scale properties also depend on the large- scale pressure gradients, the gravity orientation and the time derivative of the large-scale saturation. At this point it is clearly understood that solving these closure problems represents an extremely complex task and some work has been done to determine under which condition it is possible to avoid this problem (Ahmadi et al., 1990, 1993).

In this paper, we will present the numerical procedure developed to solve the quasi- static closure problem. We note that this numerical treatment is much more rapid and less costly than the calculation of pseudo-functions which requires a full simulation of the flow on a fine-grid model of the reservoir. At the same time, it is much more general than simple algebraic estimators of flow properties and gives us the large-scale


effective permeability tensors Kk(Sc) and Kz(St) and the large-scale capillary pressure Pz(Sc). Its drawback is evidently associated with the quasi-static assumption for which the conditions will be given in the next paragraph. However, we can show that this assumption is valid in a number of realistic cases (Ahmadi et al., 1993).

In our study, we neglect the existence of inactive zones (regions where one of the phases is immobile). This is verified when we have the same maximal values of the capillary pressure functions, P,(S,), for the different homogeneous media included in the random heterogeneous medium. In this manner, we are avoiding the trapping of a phase in a certain part of the medium which calls upon the theory of percolation. For a study of the invasion-percolation phenomenon owing to inactive zones see the paper of Yortsos et al. (1993).

In many applications, aquifer or reservoir properties are taken from a geostatistical model of the sedimentary deposits (see, for example, Dubrule and Haldorsen, 1991; Fogg et al., 1991; Payne et al., 1991). Pseudo-functions or effective properties are often associated with a small number of realization. In addition, grid block size for the large-scale numerical model is generally chosen independently of the medium’s statistical properties. Therefore, questions can be raised on the representativeness of such a procedure. So another aim of the paper is to use our numerical procedure to calculate the large-scale properties for 3-D and 2-D random porous media. We will study the influence of the size of the averaging surface, as well as the variances of the multiphase properties. These results will give us valuable information concerning the choice of the grid block size and the representativeness of a number of realizations.

2. Large-scale averaging method

Generally, a scaling-up procedure is used to take into account heterogeneities at a smaller scale and is therefore directly related to the heterogeneous nature of natural porous media. In fact, it is now generally accepted that all reservoirs are heterogeneous with different spatial correlations in the three space directions and have properties whose variability is closely related to the observation scale chosen for the study (Fayers and Hewett, 1993). A porous medium can be observed at different scales starting from the pore scale up to the regional scale (Fig. 1).

As we mentioned above, in the large-scale averaging method, the transport equations and the effective properties at a given scale are calculated by an averaging process over the equations corresponding to a lower scale. By applying the volume averaging method to the Stokes equations at the pore scale we can obtain (under certain conditions corresponding to the validity of Darcy’s law) Darcy’s lawa for describing

a The theoretical developments lead to a formulation of the multiphase Darcy’s law that features cross- coupling terms in the equations (for example Kwo’ V. in the equation for oil or Kow. VW in the equation for water). The importance of these coupling terms has been the subject of many investigations (Kalaydjian, 1988; Rose, 1988, 1989). While we believe this is still an open question, the lack of definitive arguments forces us to accept a local-scale description of multiphase flow under the form of the classical generalized Darcy’s law.


Zinszner and Meynot (1

L&C&scale

Dupuy et al. (1964)

Fig. 1. Different observation scales of a porous medium.

the fluids’ flow at the local scale (Raats, 1965; Whitaker, 1986). Similar results have been obtained by using homogenization theory (Auriault, 1987).

In this study, the lower scale considered is the local scale of the porous medium where generalized Darcy’s law is applied. The theory and mathematical developments concerning this upscaling are rather long and have been presented in Quintard and Whitaker (1988). In this paper, we will just outline the major steps associated with the large-scale averaging technique. The local scale equations are as follows Darcy‘s law

v, = -$yVP, -p,g) (cx=o,w) (1)

Continuity

&+v.v,=o (cl! = 0, w) (2)

Capillary pressure and saturation relations

P, = PO - Pw; s, + SW = 1 (3)

where ea is the volume fraction of the o-phase given by ecr = cS, with e being the porosity of the porous medium. These equations are written for each homogeneous

A. Ahmadi. M. Quintard / Journal of Hydrology 183 (1996) 69-99 75

region. In the interface of two homogeneous regions we have the continuity of pressures and conservation of flux. The basis of the method is to write these local- scale equations in terms of averaged quantities and their deviations (Quintard and Whitaker, 1988) using the following decompositions

v, = {V,}’ + Pa

K, = {Ka}* + ri, P, = {Pa}* + r’,

where for a given local variable $J, the large-scale volume average is defined by

(4)

(5)

4 is the deviation and V, is the large-scale averaging volume. The large-scale averaging of the local equations leads to terms in which the deviations remain. Deviations obey the local-scale equations. Therefore, solving the problem using the decomposi- tion in terms of averaged values and deviations requires solving the set of coupled equations formed by large-scale averaged equations and the local-scale ones. It is possible to obtain a closed form of the large-scale equations by representing the deviations as a function of the large-scale averages. The condition to be satisfied by these representations is to provide a good approximation of the coupled local-scale and large-scale equations. We will refer to these representations as the closure problem or closure equations. In the next section, after presenting the conditions associated with the quasi-static assumption, we will state the corresponding closure problem and we will focus our attention on the numerical procedure we have used to solve this closure problem.

3. The quasi-static closure problem and its numerical resolution

We consider a two-phase flow in a heterogeneous porous medium composed of different homogeneous media. The two fluid phases are oil and water (denoted o and w). The different homogeneous media are denoted by the subscriptj = 1,2, . . . , n. The variables with an asterisk (E, Kt , . . .) are the large-scale variables. We note that they are not necessarily equal to the large-scale averages of the corresponding local variables ({PC}*, {K,}*, . . .).

A first solution of the large-scale averaging problem applied to two-phase flow in porous media was proposed by Quintard and Whitaker (1988) in the quasi-static case based on the following assumptions.

(1) The closure problem associated with the local pressure and velocity deviations to the large-scale quantities is quasi-steady, which means that the time dependency of

16 A. Ahmadi, M. Quintard / Journal of Hydrology 183 (19%) 69-99

the local-scale deviations is entirely determined by their representations in terms of the large-scale averaged values. This requires that some time-scale constraints which are not detailed here be satisfied.

(2) The local saturation field in the closure problem is obtained by a local capillary equilibrium condition. The precise mathematical expression of this local capillary equilibrium condition is given below.

For a given large-scale capillary pressure, c, the saturation field in a representative unit cell is obtained by using the local capillary equilibrium assumption, i.e.

Swj=PG'(l$) forj= I,...,?I (6)

The water and oil phase large-scale permeabilities, KG and Ki are obtained by using the result of the following boundary value problem (closure problem) for (I! = o, w

V*(K;VQ,) = -V*K, in each block j (7)

njk*K,.Vb, + njk*Kaj = njk*Kd.Vb,k + njk.Kak at Ajk (8)

{b,}* = 0; b,(r + ZJ = b,(r) (9)

where Ajk is the interface between two homogeneous regions j and k and {ba}* corresponds to the volume average of the vector b, over the large-scale averaging volume V,. Kaj and bai (for i = j or k) are the tensor K, and the vector b, evaluated in the block i. The vector b, is the closure variable used to represent the pressure deviation $a as a function of the large-scale source term

*a = b;(V’a)* - /UT) (10) The second part of Eq. (9) represents periodicity boundary conditions which are

associated with a periodic unit cell representative of the heterogeneous porous medium. The vectors li are the unit lattice vectors of this periodic unit cell. While this concept of a periodic representative unit cell is obviously attractive for a truly periodic system, this idea is questionable in the case of random media. However, periodicity boundary conditions do not impose drastic constraints on the b,-field, while allowing for unconstrained anisotropy effects with any orientation of principal axes. Our results together with other studies (Anguy and Bernard, 1992) suggest this approach does not generally introduce any bias. Similar observations have been made by using periodic boundary conditions for the calculation of effective properties by other scaling-up techniques (Durlofsky, 1991; Pickup et al., 1994). The closure problem (Eqs. (7) and (8)) can also be written in a more general form in terms of a distribution b, (Schwartz, 1950), continuous over the considered domain, obeying the following equation

V.(K;Vb,) = -V.K, (a! = 0, w) (11)

This problem can be solved independently for the two phases giving us the closure variables 6, and 6,. Once the closure problem is solved, large-scale permeabilities Kz can be found by using

K:, = {K,}* + {K;Vb,}* (Q = 0, w) (12)

A. Ahmadi. 44. Quintard / Journal of Hydrology 183 (1996) 69-99 77

Fig. 2. The geometry of a unit cell composed of N, x NY x N, voxels.

The large-scale equations are coupled via a large-scale capillary pressure relationship which is readily obtained by taking the large-scale average of Eq. (6), we get

.s;. = {d&}*/(E)* = {dJ;‘(P,)}*/{~}* (13)

We need to invert this result in order to get the capillary pressure relationship under a classical form. In this quasi-static case the large-scale properties can be calculated in a relatively easy and rapid way and they are a function of the large-scale saturation only.

The quasi-static closure problem can be solved analytically in certain simple cases such as homogeneous or stratified media (Quintard and Whitaker, 1988, 1990b, Ahmadi et al., 1993). However, in the general case of a heterogeneous medium with a complex geometry, this problem can be solved by numerical methods. In this purpose, we have chosen a grid-block description of our system in accordance with the description used in reservoir simulators. The geometry of the unit cell (large- scale averaging volume in our case) is described by voxels as illustrated in Fig. 2.

The unit cell is therefore composed of N, x NY x N, voxels. The voxel (i,j, k) is a parallelepiped centered at x = (xi, xi, xk) with assigned permeability tensors K, (o = o, w). We suppose that the permeability tensors considered for each homogeneous region are isotropic and expressed in their principal axis and can therefore be represented by scalar effective permeabilities K, and K,.

The approach adopted here is based on a finite volume discretization of the boundary value problem. Let xi, Xi, xk be the coordinates of the voxel (i,j, k) with edges given by the three values 6x, 6y, Sz. The numerical scheme is given here for the x-component of the closure vector b,, denoted b,,. The value of b,, at point

78 A. Ahmadi, M. Quintard 1 Journal of Hydrology 183 (1996) 69-99

(xi, xj, xk) is denoted b,(i,j, k). Integrating Eq. (11) over the (i,j, k) voxel gives

SyGzK,(i + l/2 - v,j,k) 6%,,(i + l/2 - v,j, k)

dX - 6JdzK,(i - l/2 + V,j, k)

X db,,(i - l/2 + v,j,k) %,,(i,j + l/2 - V, k)

8X + 6xxszK,(i,j + l/2 - V, k)

ay

- GxSzK,(i,j - l/2 + V, k) ab,,(i,j - i/2 + V, k)

ay + GxGyK,(i,j, k + l/2 - v)

x ab,,(i,j, k + l/2 - v)

az - GxSyK,(i,j, k - l/2 + v) auu, k - l/2 + 4 = ()

az (14

The small parameter v indicates on which side of the interface between two voxels lies the point under consideration. Because of the source term in the right-hand side of Eq. (11) (condition given by Eq. (8)) the variable b, features jumps in the first derivative at the interfaces between homogeneous regions (in our case between the voxels). It is necessary to take into account in the discretization the jumps in the effective permeability as well as in the derivative of b,,.

Using Taylor’s expansions of db,,.ax on the right- and left-hand sides of the interface i + l/2, as well as the condition given by Eq. (8) we can approximate terms such as

&(i + l/2 - v,j,k) ab,,(z’ + l/2 - v,j, k)

ax in the following manner

K,,(i + l/2 - v,j, k) ab,,(i + l/2 - v,j, k) 2KAi + lJ, W,(i,j, k)

ax = K,(i+ l,j,k) +K,(i,j,k)

x L(i + 1 J, k) - bax(Lj, k) + &(i,j, k) (15) 6X &(i + l,j, k) + fG(i,j, k)

x [K(i + 1 ,j, k) - K,(i,j,k)l Using the same procedure, the expressions of the derivatives with respect to y and z variables can be-derived

K,(i,j + l/2 - V, k) ab,,(i,j + l/2 - v, k) 2&(&j + l,k)&(G, k)

ay = K,(i,j + 1, k) + K&j, k)

x b,,(i,./ + l, 4 - b,,(u, 4

SY

and

K,(i,j, k + l/2 - V) a&&j, k + l/2 - V) 2&(iJ, k + l)&(V, k)

az = K,(i,j, k + 1) + K,(i,j, k)

x b,x(U,k + 1) - b&&j, k)

sz

(16)

(17)


These results are used in Eq. (14) to provide discretized equations associated with the voxel (i,j,k). Periodicity boundary conditions are taken into account explicitly. This finally leads to a linear system with respect to all b,,(i,j, k). This linear system is solved using a conjugate gradient method with conditioning taking into account the sparseness and the relative complexity of the related matrix, owing to the periodic boundary conditions. A computer program has been developed for treating this problem and we have performed numerical tests on random porous media for both 2-D and 3-D media.

4. Random porous media

Geological structures are, in general, very heterogeneous and have spatial variations in their geometry and petrophysical properties at different observations scales. On the other hand, the aquifer or reservoir structure remains poorly known because the information comes from rather sparse drilling wells, while geology and seismic data provide only rough guidelines about stratifications, heterogeneities, etc. Owing to the uncertainty associated with the description of the porous media, it is often suitable to characterize them by their statistical properties and therefore resort to stochastic models and simulations (Matheron, 1967; Gelhar et al., 1974; De Marsily, 1987; Dagan, 1989). In order to study the properties of a flow in a heterogeneous porous medium, we generate a ‘synthetic’ porous medium compatible with the statistical information provided (Galli et al., 1990; Guerillot et al., 1990). To obtain statistically significant results, we average the properties obtained from simulations for different realizations of the same porous medium.

The most important variations in the rock properties are associated with the permeability. We have therefore generated porous media with a random permeability distribution. Other multiphase properties are chosen to depend on the permeability. For a given stochastic ensemble the adopted procedure is as follows: (1) equivalent properties are calculated for each realization, (2) statistical properties such as mean and variance are deduced from a large number of these calculations. This procedure can be repeated for different sizes and different positions of the large-scale averaging volume (or surface, depending on the dimension of the problem). General rules are then extracted from these calculations.

5. Media with no correlations between their different petrophysical properties

We have first considered the simple case of a heterogeneous medium composed of n homogeneous blocks of the same size whose permeabilities are chosen randomly following any given probability distribution. The other petrophysical properties (Swi, Sor, e, PC(&), K,.,(S,) and K,(%)) are supposed independent of the permeability. The Swi, Sor and the porosity (E) are chosen randomly. The capillary pressure curves are chosen to be of the same form for all blocks

PC(&) = F(S,) with S, = (S,,, - Swi)/(l - Swi - Sor) (18)


where S, is the reduced saturation and F is an invertible function. In the same manner, the relative permeabilities of all blocks have the same form

Kr,(&) = G&Q (cr = 0, w) (19)

By a theoretical study based on the properties of the quasi-static closure problem (see Appendix A), we can show that whatever the number of blocks included in the averaging surface and whatever the probability laws considered for Swi, Sor and 6, the large-scale capillary pressure and relative permeability curves have the same form as the local ones. In this particular case, the anisotropic nature of the medium can be taken into account by introducing a tensorial absolute permeability and a scalar relative permeability.

/CL = K*KrL (CX = 0, w) (20)

We will therefore study the behavior of the mathematical expectation of K*, Swi”, Sor* and E* obtained by the numerical resolution of the quasi-static closure problem over a great number of realizations. Our numerical results have shown that in all cases studied the values of E(E*), E(Swi*), and E(Sor*) are very close to the mathematical expectation of E, Swi and Sor calculated analytically using the corresponding probability density functions. These values are, in the general case, independent of the variability of permeability. Their difference in comparison with the analytical values and their standard deviations decreases, however, as size of the averaging volume (or averaging surface in the 2-D case) increases. So we will focus our interest on the calculation of the absolute permeability tensor. First of all, let us note that the tensors K* depend obviously on the number of grid blocks considered in each homogeneous block and optimization tests have been performed, in order to make the proper choice. Secondly, in each case we have calculated K” in its principal axes and have therefore obtained the diagonal terms Kzx, K;Y, Kf, and the angles 8 and $J between these principal and the original reference frame considered.

Because in this particular case the calculation of two-phase flow properties reduces to the calculation of one-phase flow properties and therefore more particularly the large-scale permeability tensors, we capture this occasion, to present first of all some background on this subject. Then we will study the variations of the large-scale permeabilities obtained by our procedure with the variance of the local permeability and the size of the averaging volume and finally we will compare our results with those given in the literature. We note that, owing to the linear form of the one-phase flow problem, for the calculation of one-phase flow properties the quasi-static assumption is not necessary.

6. Background on the calculation of effective absolute pemeabiity

The problem of composition of permeabilities for random porous media has been widely studied (Warren and Price, 1961; Matheron, 1967; Kirkpatrick, 1973; Gutjahr et al., 1978; Ababou, 1988, 1992; King, 1989; Dagan, 1989; Noetinger, 1994). In one of the first works on the subject, Warren and Price (1961) used 3-D numerical


simulations to carry out a comparative study of the arithmetic, harmonic and geometric means. They concluded that the geometric mean gives the closest estimation of the large-scale single-phase permeability for isotropic random porous media with different probability distributions of the individual permeabilities.

Matheron (1967) has shown the importance of the number of dimensions of space on the macroscopic permeability for a random porous medium being a realization of an ergodic and stationary stochastic function. He shows that the macroscopic permeability K’ is always between the arithmetic and harmonic averages of the local permeabilities

[E(K_‘)I_’ <K* <‘E(K) (21)

We know that in the particular case of stratified media the harmonic and arithmetic averages can be applied to flows normal and parallel to the strata. In the general case, on the basis of 2-D numerical experiments, it was proposed (Warren and Price, 196 1) to use the geometric mean as a good estimate of the large-scale permeability, i.e.

1ogK’ = E(logK) (22)

for a class of probability density functions close to log-normal distribution. Matheron (1967) showed that this empirical rule can only be exact in the 2-D case.

In particular, it can be applied to the interesting case of a macroscopically isotropic 2-D medium with log-normal or log-uniforma permeabilities. Based on an approximation proposed by Schwydler (1963), he showed that for a subisotropic medium, we have

K* = YE(K) +; [E(K--‘)I-’ (23)

where N is the number of dimensions of the space considered. We will not go into detail on the conditions imposed in order to have a subisotropic

medium (see Matheron, 1967). However, there is at least one case for which the subisotropic assumption is verified (the case studied by Schwydler himself); it is the case where the permeability tensor at the local scale is spherical (K = K(x)Z) and has a covariance function, C(h), independent of the direction of the distance vector, h, and depends only on the length of the vector h = (ir].

Even though the result given by Eq. (22) is based on an approximation, we can deduct the following general qualitative rule:

“for media having the ‘necessary’ isotropic properties, the macroscopic permeability is half-way between or two thirds-way between the harmonic and the arithmetic averages for 2D or 3D cases respectively.”

In particular, for a 3-D subisotropic log-normal or log-uniform medium, Matheron

a Although Matheron mentioned the case of log-normal permeability fields, the necessary condition to be fulfilled in order to obtain the result given by Eq. (23) is that K/E(K) and H/E(H) have the same spatial laws (His the resistivity equal to l/K) and that the law be invariant by rotation. This condition is satisfied in the case of a log-uniform distribution as well.


proposes the following expression of the effective permeability

K* = [E(K)]N-"NIE(K-*)]-l~~ = Kg exp [%(l -a]

where Kg is the geometric mean and CY&,(~~ is the variance of the log(K)-field. This expression has been proposed by many other authors (Gutjahr et al., 1978; Dagan, 1979; Noetinger, 1994).

Ababou (1988) proposes an analytical model for the calculation of the macroscale permeability tensor in terms of the microscale random permeability field in the general case of a anisotropic medium. He has derived expressions of the principal components of the large-scale permeability tensor in the case of a log-normal local permeability distribution as

Ki = (K,)“i(Kh)l-ai (i= l,...,N) (25)

where N is the dimension of the flow system and oi = (N - fh/li)/N. K, and Kh represent the arithmetic and harmonic mean permeabilities, respectively. I, is the fluctuation scale in the i direction (or the correlation length) and f,, is the N-dimensional harmonic mean fluctuation scale given by

(26)

In a more recent paper (Ababou, 1992), he gives the following expression which is equivalent to Eq. (25)a

Kz=Ksexpp(l--$!)I (i=l,...,N) (27)

where Kg is the geometric mean and &,sCKJ is the variance of the log(K)-field. These results are in agreement with the ones presented above for the particular case of isotropic media. In particular, the geometric mean is obtained for the two principal components of the large-scale permeability tensor in the case of a 2-D isotropic medium. While in the 3-D case, the expressions obtained for the three components of the permeability tensor are different from the geometric mean and are in agreement with Eq. (24). These equations yield effective permeabilities identical with the extrapolated perturbation solutions of Gelhar and Axness (1983) except in the case of full 3-D anisotropy (Ii # I2 # 13) w h ere the expressions are close but not identical in form.

’ Eq. (27) is slightly different from the equation given in Ababou (1988). We think that the difference probably is due to a typing error in the above mentioned paper.

83 A. Ahmadi. M. Quinrard / Journal of Hydrology 183 (1996) 69-99

7. Large-scale intrinsic permeabilities for random porous media

7.1. No spatial correlation

7.1 .l. Influence of permeability variance and the size of the averaging volume Owing to great simulation costs, the following tests have been restricted to the 2-D

case only without loss of generality. We have therefore considered the case of an isotropic heterogeneous porous medium composed of homogeneous blocks of the same size whose permeabilities follow a log-uniform distribution. We have studied the influence of the variance of the local permeability (by choosing different RK = Kmax/Ktin) as well as the size of the averaging surface (given by N x N number of homogeneous blocks it includes) on the large-scale properties. A series of optimization tests have led us to 6 x 6 grid blocks in each homogeneous block. We must note that in the particular case of quasi-static assumption, the closure problem can be put in dimensionless form and the dimensions of the medium have no consequence on the large-scale properties we obtain. Only the ratio of the size of the homogeneous blocks to the size of the large-scale averaging volume (or surface in the 2-D case) is important. For the general dynamic case, where the gradients of the large-scale properties intervene, the dimensions are important and have a great influence on the results (Quintard and Whitaker, 1990a). The test cases are listed in Table 1.

We have made the following observations.

(1) In Fig. 3, we represent the expected values of the two coefficients K& and K$ of the diagonalized permeability tensors as a function of the log-permeability variance (c&~)) for an averaging surface of 5 x 5 homogeneous blocks. Each point on the

pcrmcabilily (lo-l2 m2) .5-

0 I 2 3 4

variance of In(K) Fig. 3. Expectation of the two coefficients of the diagonalized large-scale permeability tensor as a function of the log-permeability variance for a random heterogeneous medium composed of 5 x 5 homogeneous blocks.

84

Table 1

A. Ahmadi, M. Quintard / Journal of Hydrology 183 (1996) 69-99

Description of cases considered

Case Blocks &Ii, K (10-‘2m2) ( l?l’12 m2)

RK c? In(K)

1 5x5 0.1 0.2 2 0.04 2 5x5 0.1 1 10 0.44 3 5x5 0.1 5 50 1.28 4 5x5 0.1 10 100 1.71 5 5x5 0.1 100 1000 3.98 6 10 x 10 0.1 0.2 2 0.04 I 10 x 10 0.1 1 10 0.44 8 10 x 10 0.1 5 50 1.28 9 10 x 10 0.1 10 100 1.77

10 10 x 10 0.1 100 1000 3.98

curves is obtained from 100-300 realizations. For a given averaging surface, as a&,) increases, the standard deviations increase and we observe an artificial anisotropy (Fig. 4). This means that for large variances of permeability, one must use a larger averaging surface in order to take into account the stochastic properties of the medium. This behavior is more pronounced for smaller averaging surfaces. For low values of ~Y&Q, a small averaging surface is sufficient to give significant results, while for greater c&:,(,) utilization of a larger averaging surface is necessary.

(2) In Fig. 5, the expected values of the two coefficients KzX and K;Y of the diagonalized permeability tensor are represented for a given log-permeability variance as a function of the size of the averaging surface. These properties tend to asymptotic values with decreasing standard deviations. These results show that it is possible to include some statistical information about the porous medium under investigation in the framework of the large-scale averaging method. This can be done by specifying

1 2 3 4

variance ofln(K)

Fig. 4. Expectation of the ratio I&./K&, as a function of the log-permeability variance for two different averaging surfaces (N x N).

A. Ahmadi, M. Quintard 1 Journal of Hydrology 183 (1996) 69-99 85

pemeability (10 -13rn2) 3.5 r I

I

100 200 300 400

number of blacks (NxN)

Fig. 5. Calculation of effective properties for a random heterogeneous medium for a log-permeability variance of 0.44 as a function of the size of the averaging surface.

random properties at the closure level, i.e. the unit cell, however, getting a representative set of large-scale properties may require a large size of the averaging surface. The meaning of large depends essentially on the permeability ratio; the larger the log- permeability variance, the larger the size of the averaging surface required in the computation if one wants to assign a large-scale property to the medium as a whole.

7.1.2. Comparison with literature

7.1.2.1. 2-D case. We have compared our results obtained using the large-scale averaging method under the quasi-static assumption in the 2-D case with theoretical results found in the literature. As we saw in Section 6, the accepted result for the effective permeability of 2-D random porous media is the geometrical mean. We must note that besides the conclusions of Warren and Price (1961) which were based on

dMWM&Od b) TB Method

Fig. 6. Example of a 80 x 80 field generated by the two methods with a correlation length of eight blocks.

86 A. Ahmadi, M. Quintard 1 Journal of Hydrology 183 (19%) 69-99

Table 2 Results obtained for an averaging surface of 10 x 10 homogeneous blocks

I3 W) E(G) -V;,J (lO_” m’) (10-‘2m2)

EW;eo) (lo-l2 m2)

0.04 1.41 1.41 1.41 1.41 0.44 3.20 3.07 3.14 3.16 1.28 7.25 6.51 7.02 7.07 1.77 10.17 8.79 9.91 10.00 3.98 32.49 24.62 31.80 31.62

numerical results, the theoretical developments were all based on the ergodicity and stationarity assumptions. Therefore, in our case of limited domain, the closeness of the results found with the theoretical results is highly related to the variance of the log-permeability field and the size of the averaging surface. In Tables 2 and 3, we have listed the expected values of effective permeabilities obtained by the large-scale averaging method (E(K&) and E(K&)), as well as the expectation of the geometrical means calculated over each realization, E(KP), and the analytical value exp[E(ln K)].

We observe that for cases where the necessary assumptions (ergodicity and stationarity) are more closely approached, i.e. large number of blocks and small c&(~), the large-scale averaging techniques gives results in excellent agreement with the theoretical results. That is, the expectations of two coefficients of the diagonalized permeability tensor Kz, and K;Y are always very close to the geometrical mean and therefore halfway between the arithmetic and harmonic averages. As the number of blocks (size of the averaging surface) decreases or a& increases, we obtain artificial anisotropy and a discrepancy with the theoretical results. These correspond to cases for which, with the given c&(K) the size of the averaging surface is not large enough to allow us to take into account the stochastic properties of the medium.

7.1.2.2. 3-O cases. In the 3-D case, we have carried out a number of realizations of a heterogeneous medium composed of 4 x 4 x 4 homogeneous blocks. The permeability of the locks followed a log-uniform distribution with the minimum and maximum values of 1 O- I2 m2 and lo- * ’ m* , respectively, giving a log-permeability variance equal to 0.44. We have focused our attention on the expectation of one-phase flow permeabilities

Table 3 Results obtained for an averaging surface of 5 x 5 homogeneous blocks

r3 Wf) W:,) (lo-l2 m*)

-W;y) (lo-‘* m’)

-G&o) (lo-l2 m*)

0.04 1.40 1.38 1.39 1.41 0.44 3.14 2.89 3.00 3.16 1.28 7.15 5.74 6.57 7.07 1.77 10.28 7.65 9.22 10.00 3.98 34.96 20.45 28.76 31.62


which are given below

1 -0.4 x 1O-3 0.8 x 1O-3

E(Ki) = 3.39 x lo-‘* m* -0.4 x 10e3 0.997 0.6 x 1O-3

0.8 x 1O-3 0.6 x 1O-3 0.998

-0.4 x 1O-3 0.8 x 1O-3 (28)

1

E(Kz) = 3.29 x 10-‘2m2 -0.4 x 10~~ 0.997 0.6 x 1O-3

0.8 x 1O-3 0.6 x 1O-3 0.999

The matrix of the standard deviations associated with each component are given below

u -339x10-12m2 w- .

(29)

We note that the large-scale permeabilities obtained (Eq. (28)) are nearly isotropic. The extra diagonal terms are three orders of magnitude smaller than the diagonal terms which are very close to one another. In this particular case the following analytical results can be found

arithmetic average: E(K) = 3.91 x 10-‘2m2

harmonic average: [E(K-‘)I-’ = 2.56 x lo-‘* m*

geometric average: exp[E(log(K))] = 3.16 x lo-l2 m*

K’ found by Eq.(24):K* = 3.39 x lo-‘*m2

The values we have found (Eq. (28)) conform to the analytical results in the literature and given by Eq. (24) above. It is interesting to note that although the size of the averaging volume compared with the correlation length (one block) seems rather small, the result obtained is satisfactory.

7.2. Spatially correlated isotropic and anisotropic media

In order to study more realistic geological structures, we have generated 2-D porous media with a spatially correlated permeability distribution by two different methods. Owing to the high simulation costs, we concentrate for this particular case on the calculation of the large-scale intrinsic permeability tensor only. Two methods from the many available have been tested.


E(K*d (1 O-l* m*)

standard deviation for variance I 9.21

6-e 1

4., . . . . . . . . ;;va/nw= 9.21

standard deviation for variance = 0.223 variance = 0.223

5 10 15 20 25

side of averaging surfacehc

Fig. 7. E(K&) for two different variances of In(K) as a function of the size of the averaging surface.

(1) Moissis and Wheeler Method (MW) (Moissis and Wheeler, 1990). (2) Turning Bands Method (TB) (Mantoglou and Wilson, 1982).

Both methods allow generation of isotropic log-normal fields with a given mean, variance and correlation length. Examples of permeability fields composed of 80 x 80 homogeneous blocks and with a correlation length of eight blocks generated by the two methods are given in Fig. 6.

7.2.1. Influence of permeability variance and the size of the averaging volume For a number of realizations of a heterogeneous porous medium generated by the

MW Method, we have studied the influence of the size of the averaging surface on the large-scale permeability tensor (Fig. 7). The expected values of the two coefficients of the large-scale permeability tensor tend to asymptotic values with decreasing standard

101 permcobility (lo-‘3 m2)

___------

9’ --- --

~-x.y_ - _ _ _ - - -_*_+K-x-x --

8’ ~~_x_*_m-%M-”

_-_---- _ _ _ _ - - - -

_---

7

0 5 10 15

side of the averaging swfac&

20 25

Fig. 8. E(K&) as a function of the size of averaging volume for a ensemble of realizations by the TB method.

A. Ahmadi, M. Quintard / Journal of Hydrology I83 (1996) 69-99 89

deviations. The results seem to stabilize for an averaging surface of a side equal to ten correlation lengths. The expectation of the ratio of the two coefficients tends to unity as the size of the averaging surface increases. We can verify from Fig. 7 that in this case of media with finite correlation lengths, the permeability variance plays again an important role.

Concerning 2-D porous media generated by the TB method, the fields obtained have a more ‘natural’ aspect compared with those obtained by the MW method (Fig. 6). But the behavior of the large-scale permeability is rather different. The evolution of this property computed using our numerical model as a function of the large-scale averaging surface is represented on Fig. 8. The two coefficients B(K&) and E(J$,) do not tend to asymptotic values even when we consider a surface of a side equal to 25 correlation lengths. This might be due to the fact that for the TB method, the range of the covariance function is three times the correlation length and therefore a larger averaging surface is necessary in order to have stabilized results. Although the results concerning these tests with the TB method do not lead us to general conclusions, we draw attention to the fact that different methods of generating random porous media might lead to different behaviors of large-scale properties. For example, using a spectral approach for the calculation of effective properties, Dykaar and Kitanidis ( 1992) found that an averaging surface of more than 80 integral scales is necessary in order to obtain asymptotic values in the 2-D case. Their generation method is different from the two methods we have used. It is therefore, important to point out the nature of media generated by different methods and its consequence on the calculated properties.

7.2.2. Cornpurison with literature

7.2.2.1. Isotropic random porous media. Again, we compared the results obtained by our procedure with the ones given in the literature. For all cases of 2-D porous media

5 10 15 __ ZU 25

side of averaging surface/k

Fig. 9. Comparison between the expectation of K& and K$, calculated by the large-scale averaging method and the expectation of the geometric mean for 2-D realizations of a random medium composed of 120 x 120 blocks by the MW method. The correlation length is equal to five blocks and the variance of In(K) is 9.21.

A. Ahmadi. M. Quintard 1 Journal of Hydrology 183 (1996) 69-99

permeability (lo’” rn’)

15 20 25

side of averaging surface I icy

Fig. 10. Comparison between the expectation of K& and K&, calculated by the large-scale averaging method and the expectation of the geometric, arithmetic and harmonic means for 2-D realizations of an anisotropic random medium composed of 150 x 150 blocks by the MW Method I,, = 50 blocks, I, = 5 blocks; variance of In(K) = 0.223.

studied, the expectation of the geometrical mean, E(Ks,,), is situated between the expectations of the two components of the diagonalized permeability tensor (E(K,{,) and I?($)). As an example, let us consider the results corresponding to realizations of a random medium composed of 220 x 120 blocks by the MW method. The correlation length is equal to five blocks and the variance of In(K) is 9.21. In Fig. 9, we present E(KlJ and E(K;U) obtained by the large-scale averaging method as well as E(Kseo) as a function of the size of the averaging surface. The three expectations

O.t+ 0 5 10 15 20 25

side of avenging surface I ky

- E(K*xx), LSA

- E(K*yy), LSA

-‘- E(K*xx), Ababou (b)

- E(K*yy), Ababou (b)

. E(K*xx), Ababou (a)

A E(K*yy), Ababou (a) _

IO

Fig. 11. Comparison between the expectation of K& and KGY calculated by the large-scale averaging (LSA) method and the expectation of Kzx and KJY calculated by the formulas given by Ababou (1988,1992): the curves noted (a) correspond to results calculated using Eq. (25) and the ones noted (b) correspond to ones calculated using Eq. (27) 2-D realizations of an anisotropic random medium composed of 150 x 150 blocks by the MW Method. I,, = 50 blocks, 1, = 5 blocks; variance of In(K) = 0.223.


converge as the size of the averaging surface increases. For small averaging surfaces, we have artificial anisotropy and therefore a more important difference between the three values. An analogous behavior is observed for other correlation lengths and variances for both MW and TB methods.

7.2.2.2. Anisotropic random porous media. By multiplying the horizontal grid sizes by a given factor, we have constructed anisotropic porous media. We have generated realizations of an anisotropic 150 x 150 medium with correlation lengths of 50 and five blocks in the x and y directions, respectively. The value of c&K) considered is 0.223. While in the isotropic cases, the expectation of the two coefficients of the diagonalized tensor is rather close to the expectation of the geometric mean, E(K,,,); in the anisotropic case, the expectation of the coefficient in the horizontal direction is close to the expectation of the arithmetic mean, E(K,), and that of the coefficient in the vertical direction is close to the expectation of the harmonic mean, E(&) (Fig. 10). We have also plotted the analytical values of exp[E(ln(K))] and E(K) and [E(K’)I-’ which are the upper and lower bounds (Eq. (21)) and correspond to stratified formations. In Fig. 11, we compare our results found by the large-scale averaging method (LSA) with those proposed by Ababou (1988). We have plotted results found using Eq. (25) (noted a in the figure) and Eq. (27) (noted b in the figure). It is interesting to note that although the two expressions (25) and (27) are identical theoretically and for an analytical calculation, when the coefficients are calculated numerically over an ensemble of realizations, there is significant difference between the results obtained by the two expressions. Eq. (25) is closer to both E(&) and E(&) and nearly identical to the large-scale averaging results.

8. Multiphase flow properties for random porous media with correlated parameters

In this part of our study, we look at the multiphase properties for the more realistic case of two-phase flow in porous media having correlations between their different petrophysical properties. The local values defined for each homogeneous block are given below

(1) The intrinsic permeability tensor for each block is isotropic and diagonal. Its value follows a log-uniform distribution between two given extrema: Kmin and Kmax.

(2) The porosity is related to the permeability by the following experimental correlation law (Jaquin, 1964)

K = A~4~5

where A is a constant.

(30)

(3) Concerning the irreducible water saturation, it is well known that Swi increases as permeability decreases (see, for example, Bobek, 1990). We have chosen Swi to be proportional to l/a thus giving a correlation similar to the ones observed by Chillingar and Mannon (1972). The irreducible water saturation Swi is therefore

92

defined by

A. Ahmadi, M. Quintard / Journal of Hydrology 183 (1996) 69-99

u/* - VG) SWi = (l/G _ 1,&l (Swim, - Swbd + SW&ii,

(4) The capillary pressure is given by (Rose and Bruce, 1949)

N&) = C&i%%)

.I(&) = u{ 1 - exp[b(l - S,)]} + ~(1 - S,) + d (1 - w d),2

(31)

(32)

with C = 700 000-’ kg s-* and the following values for the dimensionless variables: u = 10000, b = -20, c = 7000, d = 2650 and e = 10.

(5) The relative permeability curves are given by

Kr,(&) = (1 - s*>*, Kr,(S,) = s,’ (33)

In this case, the large-scale relative permeabilities and capillary pressures must be calculated for each realization of the medium and are not necessarily equal or close to the local curves. We have calculated the large-scale properties for a great number of realizations in 3-D and 2-D cases. From these results, we calculate the expectations

E(K)(S), E(JKJ(G), E(JX,)(G). W e note that the calculation of these properties by a pseudo-function approach would have demanded enormous computer time and expenses.

When in practical cases, we are confronted with a scaling-up problem under the assumptions made above, it is often considered intuitively that the large-scale curves will have the same form as the local scale ones. In Fig. 12, we compare the above- mentioned curves with the curves obtained using the same correlation laws as the local curves but with the corresponding large-scale quantities for a great number of realizations in the 2-D case.

capiuary pressure @a) 40000[

10000.

0 0 0.2 0.4 0.6 0.8 1

sr

rdativc permeability 1

- E(Krw*) /

0.6

0.2 0.4 0.6 0.8 Sr

Fig. 12. Comparison of E(c), E(Kr$), E(Krz) with the curves obtained using the same correlations as in the local scale (pE’, Krq, Kl’).

A. Ahmadi, hf. Quintard 1 Journal of Hydrology 183 (1996) 69-99

40000 Capillary Pressure (Pa)

I 30000 8 -

I

20000 -

15000 -

10000 -

5000 -

93

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I

Water Saturation

Fig. 13. Large-scale capillary pressure c obtained for five realizations of a 3-D random porous medium. The local capillary pressures are different for each homogeneous block and are of the form given by Eq. (32). Note that for practical reasons (calculation cost and time), the large-scale properties have been calculated only for a limited number of points.

Owing to considerable calculation times, we have restricted the 3-D calculations to only five realizations of a given random porous medium. The large-scale capillary pressure and relative permeability curves obtained are compared with the rock curves representative of the local correlations (Figs. 13- 15).

1 Oil Relative Pamability

0.6

- !3e&?l

--*----- series2

. . . ...” . . . . . . . . m-3

_..I.._.._.._ .&&&

-Rodccwvc

0.4 0.6

Reduced Water Saturation

0.8 1

Fig. 14. Comparison of the large-scale oil relative permeability Krg obtained for five realization of a 3-D random porous medium with the rock curve. Note that for practical reasons (calculation cost and time), the large-scale properties have been calculated only for a limited number of points.


0 0.2 0.4 0.6

Reduced Water S8turation

U.8 1

Fig. 15. Comparison of the large-scale water relative permeability Kr: obtained for five realizations of a 3-D random porous medium with the rock curve. Note that for practical reasons (calculation cost and time), the large-scale properties have been calculated only for a limited number of points.

These results show that the use of local-scale correlations for the calculation of the large-scale curves is not justified either in the 2-D case or 3-D case.

9. Conclusions

The numerical resolution of the quasi-static closure problem for the calculation of effective multiphase properties by the large-scale averaging method has been presented. This procedure allows the calculation of the large-scale properties with uncomparably less effort and cost than pseudo-function methods in many practical cases. The method is applicable to any geometry and anisotropy and can therefore be considered as an interesting alternative to classical methods. Using this procedure, we have studied the behavior of the large-scale properties for random porous media and we have arrived at the following conclusions.

(1) We have been able to verify that the results found by the large-scale averaging technique in 2-D and 3-D cases are in agreement with the classical results known in the stochastic theory literature.

(2) Except for very simplistic descriptions of heterogeneous porous media, it is not possible to determine correct large-scale properties using simplified formulations. In general, the use of a more elaborate tool such as the large-scale averaging method is necessary. This point has been verified in 2-D and 3-D cases.

(3) The representativeness of a realization, as measured by the standard deviation of the ensemble of realizations, is closely related to the variation of the permeability in

the medium and the size of the averaging volume. The representativeness increases as

A. Ahmadi. M. Quintard 1 Journal of Hydrology 183 (1996) 69-99 95

the size of the averaging surface increases and as the local-scale permeability variance decreases. Special attention must therefore be paid to the size of the numerical blocks in a reservoir model: if the size of the blocks is of the same order of magnitude as the correlation length, there are great doubts as to the representativeness of one or a few realizations. When choosing the block size, one should take into consideration the permeability variance of the porous medium. Different methods of generation of random porous media lead to very different large-scale property behavior for the same given statistical properties.

In general, average properties can be obtained for a spatially correlated medium only if the characteristic dimension of the averaging surface is one order of magnitude larger than the correlation length. If this characteristic dimension is of the same order of magnitude as the correlation length, one is not sure of the representativeness of one or a few realizations.

Acknowledgment

Financial support from Elf Aquitaine Production is gratefully acknowledged.

Appendix A

We consider a heterogeneous medium composed of n homogeneous blocs of the same size whose permeabilities are chosen randomly following any given probability distribution. The other petrophysical properties (Swi, Sor, e, PC(&), K,(S,) and K, (S,)) are supposed independent of the permeability. The Swi, Sor and the porosity (E) are chosen randomly. The capillary pressure curves are chosen to be of the same form for all blocks

PC(&) = F(S,) with S, = (S, - Swi)/(l - Swi - Sor) (Al)

We recall that S, is the reduced saturation and F is a given invertible function. In the same manner, the relative permeabilities of all blocks have the same form

Kr,(&) = G-J&) (d!=o,w) (AZ)

Capillary pressure curves

Based on the local capillary equilibrium assumption, for a given large-scale capillary pressure, PE, the saturation in each homogeneous region is found by using the relation

Pz = F(S,) = F[(Swj - Swij)/(l - Swij - Sorj)] j= l,...,n (A3)

where the subscript j is related to the block considered and n is the total number of homogeneous blocks included in the large-scale averaging volume. The saturation in

96 A. Ahmadi, M. Quintardl Journal of Hydrology 183 (1996) 69-99

each block is

Swj = (1 - SWij - SOrj)F-'(Pl)+ SWij j= l,...,n (A4)

We define the large-scale saturation S$ and the large-scale porosity E* as follows

(A5)

1 n c*=- ej

c n. ]=I

We can therefore write

S,tf* =tc[(l - SWij - Sorj)F-‘(Pz) + SWi/]~j J=l

Using the linearity of the sum and dividing by e*, we obtain

-f ($‘jsorj)/‘*]

Using Eq. (AS) for S, = Swi and S, = Sor, we have

e=F S$ - Swi’

1 - Swi’ - Sor* >

646)

(A?

(A8)

(~49)

So, whatever the number of blocks considered in V,, and for any probability law used for E, Swi and Sor, the large-scale capillary pressure curve will be of the same form as the curves considered for the elementary homogeneous blocks.

Relative permeability curves

The theoretical analysis of the properties of the tensors Kk and KE obtained using the large-scale averaging method is a difficult task. In fact, these tensors are obtained by a numerical resolution of the closure problem. We insist on the fact that the use of a scalar relative permeability defined by

KL = K’Kri (a=o,w) (A 10) is restricted to isotropic systems only (Bear et al., 1987; Quintard and Whitaker, 1988) and that in the general case, it is necessary to keep a tensorial relative permeability. For a given large-scale capillary pressure, pE, the saturation field is known. We can therefore calculate the local-scale effective permeability field. For a given block j,


assumed isotropic and homogeneous, we can write

&j = KjKraj(&j) (cY=o,w) (All) We see that, for a given large-scale capillary pressure, the reduced saturation is the same in all blocks. Based on the Eq. (A2), we have the same relative permeability values as well. For any block, j, we have

Koj = KjGa(SrIc) (a=o,w)

So, in this very particular case, our problem is reduced to the one-phase permeability calculation case.

References

Ababou, R., 1988. Three-dimensional flow in random porous media. Ph.D. Thesis, Department of Civil Engineering, MIT, Cambridge, MA.

Ababou, R., 1992. Random porous media flow on large 3-D grids: numerics, performance and application to homogenisation. In: Proc. IMA Summer Prog. on Environmental Studies. Institute for Mathematics and its Applications (IMA), Minneapolis, July.

Ahmadi, A., Labastie, A. and Quintard, M., 1990. Large-scale properties for flow through a stratified medium: a discussion of various approaches. In: D. Guerillot and 0. Guillon (Editors), 2nd European Conf on the Mathematics of Oil Recovery. Technip, Paris, pp. 91-98.

Ahmadi, A., Labastie, A. and Quintard, M., 1993. Large-scale properties for flow through a stratified medium: various approaches. SPE 22188, Sot. Pet. Eng. J., August: 214-220.

Amaziane, B. and Bourgeat, A., 1988. Effective behaviour of two-phase flow in heterogeneous reservoir. In: M.F. Wheeler (Editor), Numerical Simulation in Oil Recovery, IMA Volumes in Mathematics and its Applications. Springer, pp. l-22.

Anguy. Y. and Bernard, D., 1992. Numerical computation of the permeability tensor evolution versus microscopic geometry changes. In: T.F. Russel, R.E. Ewing, C.A. Brebbia, W.G. Gray and G.F. Pinder (Editors), Computational Methods in Water Resources IX, Vol. 2, Mathematical Modelling in Water Resources. Computational Mechanics, Southampton, Elsevier Applied Science, London, pp. 385-392.

Auriault, J.L., 1987. Nonsaturated deformable porous media: quasistatics. Transport in Porous Media, 2: 45-64.

Bear, J., Braester, C. and Menier, PC., 1987. Effective and relative permeabilities of anisotropic porous media. Transport in Porous Media, 2: 301-316.

Bobek, J.E., 1990. Selecting reservoir-rock and fluid properties data. In: C.C. Mattax and R.L. Dalton (Editors), Reservoir Simulation. Society of Petroleum Engineers, Monograph Series, Vol. 13, pp. 29-41.

Bourgeat, A., 1984. Homogenized behaviour of two-phase flows in naturally fractured reservoirs with uniform fractures distribution. Comput. Meth. Appl. Mech. Eng., 47: 205-216.

Chillingar, G.V. and Mannon, R.W., 1972. Oil and Gas Production from Carbonate Rocks. American Elsevier, New York.

Coats, K.H., Nielsen, R.L., Terhune, M.H. and Weber, A.G., 1967. Simulation of three-dimensional, two- phase flow in oil and gas reservoirs. Sot. Pet. Eng. J., 240: 377-388.

Dagan, G., 1979. Models of groundwater flow in statistically homogeneous porous formations. Water Resour. Res., 15(l): 47-63.

Dagan, G., 1989. Flow and Transport in Porous Formations. Springer, Berlin. Dake, L.P., 1978. Fundamentals of Reservoir Engineering. Elsevier Science B.V., Amsterdam. De Marsily, G., 1987. Modelling and applications of transport phenomena in porous media. Lecture series

1988-01, Von Karman Institute For Fluid Dynamics, Brussels. Dubrule, 0. and Haldorsen, H.H., 1991. Geostatistics for permeability estimation. In: Z.W. Lake, H.B.

Carroll, Jr. and T.C. Wesson (Editors), Characterization II, Academic, San Diego, pp. 223-247.


Dupuy, M., Morineau, Y. and Simandoux, P., 1964. Etude des He&o&r&t& dun echantillon de milieu poreux. Rapport A.R.T.E.P., Ins. Fr. Pet., Rueil-Malmaison.

Durlofsky, L.J., 1991. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res., 27: 699-708.

Dykaar, B.B. and Kitanidis, P.K., 1992. Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach, 2. Results. Water Resour. Res., 28: 1167- 1178.

Emanuel, A.S. and Cook, G.W., 1974. Pseudo-relative permeability for well modeling. Sot. Pet. Eng. J., February: 7-9.

Fayers, F.J. and Hewett, T.A., 1993. A review of current trends in petroleum reservoir description and assessment of the impacts on oil recovery. Adv. Water Resour. 15: 341-365.

Fogg, G.E., Lucia, F.J. and Senger, R.K., 1991. Stochastic simulation of interwell-scale heterogeneity for improved prediction of sweep efficiency in a carbonate reservoir. In: L.W. Lake, H.B. Carroll, Jr. and T.C. Wesson (Editors), Reservoir Characterization II. Academic, pp. 355-381.

Galli, A., Guerillot, D. and Raveme, C., 1990. Combining geology, geostatistic and multiphasic flow for 3D reservoir studies. In: D. Guerillot and 0. Guillon (Editors), Second European Conf. on the Mathe- matics of Oil Recovery, Arles, France. Ed. Technip, Paris, pp. 11-20.

Gelhar, L.W. and Axness, CL., 1983. Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 24: 1491-1500.

Gelhar, L.W., Ko, P.Y., Kwai, H.H. and Wilson, J.L., 1974. Stochastic modelling of groundwater systems. R.M. Parsons Lab. for Water Resources and Hydrodynamics, Rep. 189, MIT, Cambridge, MA.

Guerillot, D., Lemouzy, P., Galli, A. and Ravenne, C., 1990. 3D fluid flow behaviour in porous media characterized by geostatistical methods. In: SPE 21081, Society of Petroleum Engineers Latin American Petroleum Conf., Rio de Janeiro, October 1990.

Gutjahr, A.L., Gelhar, L.W., Bakr, A.A. and MacMilan, J.R., 1978. Stochastic analysis of spatial variability in subsurface flows, 2, Evaluation and application. Water Resour. Res., 14: 953-959.

Hearn, C.L., 1971. Simulation of stratified waterilooding by pseudo relative permeability curves. J. Pet. Technol., July: 805-813.

Huppler, J.D., 1970. Numerical investigation of the effects of core heterogeneities on waterflood relative permeabilities. Sot. Pet. Eng. J., December: 381-392; Trans. Am. Inst. Mech. Eng., 249.

Jacks, H.H., Smith, O.J.E. and Mattax, CC., 1973. The modelling of three-dimensional reservoir simulator-The use of dynamic pseudo-functions. Sot. Pet. Eng. J., 13(3): 175-185.

Jaquin, C., 1964. Correlation entre la permeabilite et les caracteristiques geometrique du grb de Fontaine- bleau. Rev. Inst. Fr. Pet., 7-8: 921-937.

Kalaydjian, F., 1988. Couplage entre phases fluides dans les ecoulements diphasiques en milieu poreux. Doctoral Thesis, Universite de Bordeaux I.

King, P.R., 1989. The use of renormalization for calculating effective permeability. Transport in Porous Media, 4: 37-58.

Kirkpatrick, S., 1973. Percolation and conduction. Rev. Mod. Phys., 45: 574-588. Kortekaas, T.F.M., 1983. Water/oil displacement characteristics in cross-bedded reservoir zones. Paper

No. 18436, Society of Petroleum Engineers. Kortekaas, T.F.M., 1985. Water/oil displacement characteristics in cross-bedded reservoir zones. Sot. Pet.

Eng. J., December: 917. Kossack, C.A., Aasen, 3.0. and Opdal, S.T., 1990. Scaling-up heterogeneities with pseudofunctions. Sot.

Pet. Eng. FE, September: 367-390. Kyte, J.R. and Berry, D.W., 1975. New pseudo-functions to control numerical dispersion. Sot. Pet. Eng. J.,

August: 269-275. Mantoglou, A. and Gelhar, L.W., 1989. Three-dimensional unsaturated flow in heterogeneous systems and

implications on groundwater contamination: a stochastic approach. Transport in Porous Media, 4: 529-548.

Mantoglou, A. and Wilson, J.L., 1982. The turning bands method for simulation of random fields using line generation by a spectral method. Water Resour. Res. 18: 1379-1394.

Martin, J.C., 1968. Partial integration of equations of multiphase flow. SPEJ., December: 370-380.


Matheron, G., 1967. Elements pour une Theorie des Milieux Poreux. Ed. Masson et Cie, Paris. Moissis, D.E. and Wheeler, M.F., 1990. Effect of the structure of the porous medium on unstable miscible

displacement. In: J. Cushman (Editor), Dynamics of Fluids in Hierarchical Porous Media. Academic. Muggeridge, A.H., 1991. Generation of effective relative permeabilities from detailed simulation of flow in

heterogeneous porous media. In: L.W. Lake, H.B. Carroll, Jr. and T.C. Wesson (Editors), Academic, pp. 197-225.

Noetinger, B., 1994. The effective permeability of a heterogeneous porous medium. Transport in Porous Media, 15(2).

Payne, D.V., Edwards, K.A. and Emanuel, A.S., 1991. Examples of reservoir simulation studies utilizing geostatistical models of reservoir heterogeneity. In: L.W. Lake, H.B. Carroll, Jr. and T.C. Wesson (Editors), Reservoir Characterization II. Academic, pp. 497-523.

Pickup, G.E., Ringrose, P.S., Jensen, J.L. and Sorbie, K.S., 1994. Permeability tensors for sedimentary structures. Math. Geol., 26: 227-250.

Quintard, M. and Whitaker, S., 1988. Two-phase flow in heterogeneous porous media, the method of large- scale averaging. Transport in Porous Media, 3: 357-413.

Quintard, M. and Whitaker, S., 1990a. Two-phase flow in heterogeneous porous media I: the influence of large spatial and temporal gradients. Transport in Porous Media, 5: 341-379.

Quintard, M. and Whitaker, S., 1990b. Two-phase low in heterogeneous porous media II: numerical experiments for flow perpendicular to a stratified system. Transport in Porous Media, 5: 429-472.

Raats, P.A.C., 1965. Development of equations describing transport of mass and momentum in porous media. Ph.D. Thesis, University of Illinois, Champaign-Urbana.

Rose, W., 1988. Measuring transport coefficients necessary for the description of coupled two-phase flow of immiscible fluids in porous media. Transport in Porous Media, 3: 163-171.

Rose, W., 1989. Data interpretation problems to be expected in the study of coupled fluid flow in porous media. Transport in Porous Media, 4: 185-198.

Rose, W. and Bruce, W.A., 1949. Evaluation of capillary characters in petroleum reservoir rock. Trans. Am. Inst. Mech. Eng., 186: 127-142.

Saez, A.E., Otero, C.J. and Rusinek, I., 1989. The effective homogeneous behaviour of heterogeneous porous media. Transport in Porous Media, 4: 213-238.

Schwartz, L., 1950. Thiorie des Distributions. Hermann et Cie, Paris. Schwydler, MI., 1963. Sur les caracteristiques moyennes des courants d’koulement dam les milieux a

h&bog&&es aleatoire. Izd. Akad. Nauk SSSR, Merh. Mas. 4: 127-129 (in Russian). Stone, H.L., 1991. Rigorous black oil pseudo-functions. 21207, Society of Petroleum Engineers, pp. 57-68. Thomas, G.W., 1983. An extension of pseudofunctions concepts. 12274, Society of Petroleum Engineers. Warren, J.E. and Price, H.S., 1961. Flow in heterogeneous porous media. Sot. Pet. Eng. J., 153-169. Whitaker, S., 1986. Flow in porous media I: a theoretical derivation of Darcy’s law. Transport in Porous

media, 1: 3-35. Woods, E.G. and Khurana, A.K., 1977. Pseudofunctions for water coning in a three-dimensional reservoir

simulator. Sot. Pet. Eng. J., August: 251-262. Yortsos, Y.C., Satik, C., Bacri, J.C. and Salin, D., 1993. Large-scale percolation theory of drainage.

Transport in Porous Media, 10: 171-195. Zinszner, B. and Meynot, Ch., 1982. Visualisation des propriites capillaires des roches reservoir. Rev. Inst.

Fr. Pet. 37: 337-362.

Large-scale properties for two-phase flow in random porous media

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