reconstructed porous media and their application to fluid flow and solute transport

Journal of Contaminant Hydrology, 13 (1993) 3-22 Elsevier Science Publishers B.V., Amsterdam

Reconstructed porous media and their application to fluid flow and solute transport

J. Sallrs, J.F. Thovert and P.M. Adler

LPTM, Asterama 2, Avenue du TOlOport, F-86360 Chasseneuil, France

(Accepted for publication January 8, 1993)

ABSTRACT

Sallrs, J., Thovert, J.F. and Adler, P.M., 1993. Reconstructed porous media and their application to fluid flow and solute transport. In: J.I. Kim and G. de Marsily (Editors), Chemistry and Migration of Actinides and Fission Products. J. Contain. Hydrol., 13: 3-22.

It is possible to simulate geological media with the same porosity and correlation function of the pore space as real ones. This method is illustrated by two- and three-dimensional examples and discussed. Then, programs able to solve the local field equations in any geometry are used to study various transport processes such as convection, diffusion and Taylor dispersion in these media whereby macroscopic quantities such as permeability, formation factor and dispersion tensor are obtained; the first two quantities are compared to experimental data in Fontainebleau sandstones.

1. INTRODUCTION

The study of transport in porous media has been spurred on over the years by various industries of vital importance, such as filtration, oil recovery and more recently, hazardous waste repositories. Much has been written on this topic; among the most recent works are: Dullien (1979), de Marsily (1986), Bear and Bachmat (1990) and P.M. Adler (1992).

This paper deals with the derivation of the local properties of porous media. Our analysis has two original features; the first is the great emphasis given to the geometrical representation of the porous media. In order to simulate real media such as geological materials as accurately as possible, the porosity and the correlation function of the pore space are measured on thin sections. Then, random media with the same average properties can be generated numerically.

The second original feature is the determination of the macroscopic transport properties of these media from the numerical resolution of the local equations governing transport in samples of reconstructed media. For instance, the permeability is derived from the velocity field obtained by solving the Stokes equations with the no-slip condition at the solid wall. This

4 I ' ~ \ , ! S } ! ' ,

approach has been systematically used and applied to other transport in porous media. This paper presents a short review of our contributions m this field and is organized as follows.

Section 2 is devoted to the generation of reconstructed media from measurements made on thin sections. Two quantities are measured from the pore space: porosity and autocorrelation function of the pore space. It is then shown how to generate a three-dimensional random porous medium with a given porosity and a given correlation function. The medium is made of elementary cubes which are filled by solid or liquid. It can be generated in two steps starting from independent Gaussian variables X(x). Linear combinations of these variables yield a population Y(x) which is still Gaussian but correlated; the correlation depends upon the set of coefficients a of tile linear combinations. This population is then transformed into a discrete population Z(x) which takes only two values, 0 and 1; this transformation is such that the average value of Z(x) is automatically equal to the porosity.

In order to generate a given porous medium, one must first solve a sort of inverse problem which consists of a two-step determination of the coefficients a. Once these coefficients have been determined, artificial media can be generated at will starting from an arbitrary seed; two- and three-dimensional representations of such a medium are shown.

Section 3 is devoted to the study of transport in these media. Permeability is addressed first and the Stokes equation is solved; the numerical results are then compared to data obtained by Jacquin (1964) on Fontainebleau sandstones, Paris Basin, France; the ratio between them is always smaller than a factor 5.

Studies on diffusion and dispersion are, at the moment, limited to molecules or particles such as colloids whose size is very small compared to a characteristic size of the pores of the porous material. Diffusion is studied by solving the Laplace equation in the fluid phase; comparison with Jacquin's (1964) data is made through the formation factor.

Taylor dispersion is addressed by two different methods, namely by solving the so-called B-equation or by a Monte Carlo calculation which mimics the motion of particles under the action of convection and diffusion. An application to reconstructed media is described and discussed.

In Section 4, some tentative remarks are made on the limitations of this methodology and on its possible extensions to the study of transport and deposition of particles in porous media.

RECONSTRUCTED POROUS MEDIA AND FLUID FLOW AND SOLUTE TRANSPORT

2. RECONSTRUCTED MEDIA

2.1. Measurements

Fontainebleau sandstones were selected here because they are known to have remarkable properties. They are made of a single mineral, quartz; they do not contain any clay, hence there is little physicochemical interaction between the solid phase and aqueous fluids. The geometric structure of these sandstones is quite simple since they display only an intergranular porosity; there is no microporosity inside the particles. Moreover, they are known to be remarkably homogeneous; another important advantage is the fact that porosity may be varied while, globally, the same structure is conserved by a variation in the degree of cementation.

Several thin sections of these sandstones are shown in Fig. 1. The pore space is obtained by injecting a dyed glue into the medium; then the sample is cut and the pore space is replaced by the dye-hardened glue. Sections (~ 20 #m thick) were obtained by abrasion of the samples. The pore space is clearly defined since the glue is dyed red; colour pictures of these section can be taken with the help of a microscope.

The examples in Fig. l a-d clearly show that this material was originally nonconsolidated sand. Then, various geological processes such as the accretion of the sand particles took place resulting in a decrease of the porosity. Porosity in such sandstones ranges roughly from 0.03 to 0.35.

The statistical geometric characteristics of the pore space can be measured when the phase function Z(x) is introduced:

1, if x belongs to the pore space Z(x ) = (1)

O, otherwise

where x denotes the position with respect to an arbitrary origin. The porosity • and the correlation R~(u) can be defined by the statistical

averages (which will be denoted by an overbar):

• = Z(x) (2a)

R~ (u) = [Z(x) - •][Z(x + u - •]/(e - •2) (2b)

where u = [[u[[. Notice that ( • - •2 ) in eq. 2b equals var(Z) since Z2(x) = Z(x). These measurements were made in a single, but otherwise arbitrary, plane

since Fontainebleau sandstones are known to be isotropic. They were made by image analysis. The image must first be binarized as indicated in Fig. 1. Some experimental correlation functions are shown in Fig. 2; note that these functions do not depend very much upon porosity, since • varies by a factor 3 in Fig. 2.

, - / . g "~ lp<, 'IIF'.- It

v'÷ 4"2 N-

d

Fig. 1. Thin sections of the Fonta ineb leau sandstone. The porc space appears in bie, cf , , ia), (b) and (d) are the real pictures of thin sections, while in (c) the contras t between the solid and lhe " l iqu id" phases has been enhanced by various t reatments , mostly manual . The scale is indicated on each picture by a t~ar which corresponds to 0.5 mm. The names and the rneasured surface porosity of each picture are as follows: (a) 2 A 3 , • = 0.31; (b) GF2. • -- 0.25: (c) ( 'J . ~ = 0.21: a n d ( d ) 1 2 A 1 3 , ~ - 0.11.

RECONSTRUCTED POROUS MEDIA AND FLUID FLOW AND SOLUTE TRANSPORT 7

I

R~

0.5

I I I I

10 20 30 u ~0

Fig. 2. Experimental correlation functions R~ as functions of the translation u which is graduated in pixels. The length scale a is always equal to ~t = 3.8/~m/pixel. Data are for: image 2A3 (+); image GF2 (×); image CJ (v); andimage 12A13 (0).

Additional details on image analysis, size of the samples, the checking of the statistical homogeneity, etc., can be found in P.M. Adler et al. (1990).

2.2. Generation of random discrete variables with given average and correlation function

Let us now briefly sketch the reconstruction of three-dimensional random media. We want to generate a three-dimensional random porous medium with a given porosity e and a given correlation function; the medium is homogeneous and isotropic - - but this last property is not essential. It should be emphasized that the correlation function of isotropic media only depends on the norm u of the vector u (see R.J. Adler, 1981).

Similarity, we want to generate a random function of space Z(x) which is equal to 0 in the solid phase and to 1 in the liquid phase. Z(x) has to verify the two average properties eqs. 2a and 2b (Quiblier, 1984). It should be emphasized that the point of view is quite different here; E is a given positive number < 1; R~ (u) is a given function of u which verifies the general properties of a correlation (see R.J. Adler, 1981) but is otherwise arbitrary.

For practical purposes only, the porous medium is constructed in a discrete manner. It is considered to composed of N~ 3 small cubes, each of the same size a. These elementary cubes are filled either with liquid, or with solid. Examples of such porous media have already been given elsewhere (P.M. Adler, 1989; Lemaitre and Adler, 1990). Hence the position vector x and the translation vector u will only take discrete values; the corresponding trios of integers are denoted by:

v' =-x ,a = [i,j,k) (3a)

u' = u/a =: (r,s,l) (3bJ

An additional condition is imposed by the fact that the sample of generated porous medium has a finite size aN~. This is equivalent to the covering of the whole porous medium by an infinite number of identical unit cells; this covering has been used in asymptotic analysis (Bensoussan et al., 1978) and molecular dynamics (Hansen and McDonald, 1976). Since the medium is assumed to be uniformly correlated, the random field Z(x) has to verify:

= Z(x) (4a)

R: (u) = R: (u) = [Z(x ) - E][Z(x t ) - E]/(~ - ~2 ) (4b)

where the translated vector x, is defined mod aNt for each of its components:

x t = x + u (modaN c)

This equality means that, for instance,

i, = i + r (modN~,)

(5a)

(5b)

Because of this spatial periodicity, all the physical quantities are independent of the choice of origin and of the faces of unit cells.

There are several methods to generate discrete random variables which verify eqs. 4a and 4b. Here we adapt to isotropic media an algorithm due to Quiblier (1984) for general three-dimensional porous media. This algorithm is itself an extension of the two-dimensional scheme devised by Joshi (1974).

For the sake of clarity, we briefly present this algorithm in the present section and recall the major properties of the corresponding random functions. It can be shown that a random and discrete field Z(x) can be devised from a Gaussian field X(x) when the latter is successively passed through a linear and a nonlinear filter. Let us summarize the influence of these filters and relate their properties to the statistical properties of the resulting fields. A detailed presentation can be found in P.M. Adler et al. (1990).

Consider first the initial random field X(ij,k); the random variables X(ij , k) are assumed to be normally distributed with a mean equal to 0 and a variance equal to 1; these variables are independent.

A linear operator can be defined by an array of coefficients ~(u') where u' belongs to a finite cube [0,Lc ]3 in Z 3 . Outside this cube, it is equal to 0. A new random field Y(x) can be expressed as a finear combination of the random variables X(x'):

r(x) = Y a(u')X(xO (6) u'~[0,G] 3

where the translated vector x[ is defined mod Nc for each of its components.


The definition (6) is identical to the definition used by Joshi (1974) and Quiblier (1984), except for the periodic character introduced by the condition mod Nc. Without any further requirements on the coefficients t~(u') of the linear filter, it can be shown that the random variables Y(x) are standard Gaussian if Nc > 2Lc. Let us further assume that the variance of Y(x) is equal to 1:

E{YZ(x)} = 1 (7)

Hence the random variables Y(x) have a standard normal distribution, although they are no longer statistically independent. Their correlation function Ry(u) is easily seen to be:

Ry(u) = E a., +r..2+s..3+, (8) r,s,t~[O,Lc]

where (u~+r), (u2+s) and (U3+t) are determined mod No, and u = + +

The random field Y(x) is correlated, but still not satisfactory since it takes its values in R, while the porous medium has to be represented by a discrete- valued field Z(x) (cf. Joshi, 1974). In order to extract such a field from Y(x), one applies a nonlinear filter G, i.e. the random variable Z is a deterministic function of Y:

Z = G(Y) (9)

If G is known, the statistical properties of the random field Z can be derived from those of Y. For the sake of completeness, this derivation, which can be found in Joshi (1974), is briefly repeated here.

Since the random variable Y(x) has a standard normal distribution (i.e. with a zero mean and a variance equal to 1), its distribution function P(y) is given by:

P(y) = (2n)- ' /2If e x p ( - y 2 /2)dy (10)

The deterministic function G is defined by the following condition: when the random variable Y is equal to y, Z takes the value z:

z = 1, if P(y)~<E ( l la)

z = 0, otherwise (1 lb)

It is, thus, fairly obvious that the average value of Z(x) is equal to e, and its variance to (e - e2).

The most difficult point is the determination of the correlation function Rz(u) of Z(x) as a function of Ry(u). One can start from the fact that the random variable (Y(x),Y(x+u)) is a bivariate Gaussian whose probability density is known (e.g., R.J. Adler, 1981); this density can be expanded in terms

of Hermite polynomials. After some tedious manipulations using classical identities (Gradshteyn and Ryshik, 1965), R~(u) can be expressed as a series in terms of R,(u):

R,(u)_ = ~ Cm R~ l ? l =: [)

where the coefficients C., are given by:

C m = (2rcm.t)-'/2f~fc(y)exp(-y2/2)Hm(y)dy

together with

c(y) = (E- 1)/[E(1 _~)],:2 if P(y) ~<e

c(y) = ~/[E(t -E)] '/2 if P(y) > e

and

Hm(Y ) = (__l)mexp(½y2) dm exp ( l y 2 ) dy ~

(12)

(13a)

(13b)

(13c)

(13d)

2.3. Simulation of real porous media

When the aim is to simulate a given porous medium, the first problem is the determination of the correlation function Ry(u) and of the set of coefficients ~: this is what we shall call the inverse problem. Once these coefficients have been calculated, porous media can be simulated and their general properties critically examined.

Let us first briefly sketch the inverse problem. When the porosity is given, the correlation function Ry(u) is easily derived from R~(u); this simply corresponds to the numerical inversion of eqs. 12 and 13 by any standard method such as a Newton iterative scheme.

When Ry(u) is known, one has to determine the coefficients ~ by solving the set of quadratic equations (8). This step can only be numerically performed by using standard optimization routines; it should be noticed that the solution is not unique and that it was sometimes difficult to determine h.

Again further details on the inverse problem can be found in P.M. Adler et al. (1990).

Once the coefficients ~ are obtained, arbitrary samples of porous media can be reconstructed. One starts from an arbitrary seed and then generates a set of independent Gaussian variables X(id, k); t l~n this set is successively passed through the linear filter (6) and the nonlinear filter (9)-(11).

RECONSTRUCTED POROUS MEDIA AND FLUID FLOW AND SOLUTE TRANSPORT 1 1

2.4. Results and discussion

'Examples of numerical thin sections inside the same cube are shown in Fig. 3. If 1 is the arbitrary length of the side of the cube, the horizontal cross- sections correspond to values of the vertical coordinate z equal to 0, 0.25, 0.5 and 0.75. The visual aspect of these sections is very different from the sections obtained by site percolation (see Lemaitre and Adler, 1990); the elementary cubes are gathered in larger pores because of the correlation function. It is believed that most of the isolated elementary cubes do not belong to the general pore system.

These reconstructed media can also be represented in three dimensions with adequate software and hardware. An example is shown in Fig. 4. The apparent realism of the porous medium is quite striking although some fine features do not exist in the reconstructed media. This is more apparent in a close comparison between Fig. 1 and Fig. 3; thin connections along grain boundaries are missing. However, these connections are not likely to play an important role in quantities such as the macroscopic permeability when porosity is > 10%.

It is important to note that the model uses a number of parameters, such as the size of the elementary cubes a, Nc, Lc, etc. For the sake of brevity, the detailed discussion done by P.M. Adler et al. (1990) is not repeated here. In it, the reader will find general conditions on these parameters. For instance, a is a critical parameter; if it is too large, adjacent cubes are not correlated and the model is identical to site percolation. Usually, a is chosen so as to be about one-tenth of the correlation length defined as the integral of R~(u). Other parameters such as the number M of terms in the series (12) are automatically controlled to achieve the calculations with a given precision.

Recently, some substantial progress has been made both in the generation process and in its properties. The coefficients h are not obtained by inverting eq. 8, but directly from the Fourier components of the correlation function Ry(u). This is much less demanding in terms of computer time.

Moreover, a fundamental question may be raised on the reconstruction process itself. When Gaussian variables are used, all the moments of the phase functions are determined when the two first moments (i.e. porosity and correlation function) are given. Hence, one may wonder if the moments of order n:

[ ]} Rz(u, , . . . , U,_l) = Z ( x ) - e ] I-[ ~ = 1

are the same in the real and in the reconstructed materials. Yao et al. (1993) recently showed that this is indeed the case on three-dimensional samples at

I ~ - '

~m J . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . t

. . . " ' ~ l k e . , l , , . , . , , .

~.T' . , ~ , - , t~

: - , - " 'It '4-~' ik

' ~ . r l i r ~ ," ¢~ .~

1 2

I I

. . . . ~11--- ~ . . - ~ , - - . - r . . . . . . . . ~, . . . . . . . . . . . . . . . . . r[ - - - ~ - - - : ~ : ,Ik" , . --

~,., ,Ir ~" ,4" t " ~ . . ~ ,

• I . ' f l ~

: ~ ~ .,. ~. qq~.i - - . • • I " i l l . : ? . . - -

iL . . . . .._~., ' I I . . . . . . . . . . . . . . . . . . ,':_ . . . . ~ . . ' ! . ,

3 4

Fig. 3. Cross-sections of a sample of reconstructed porous medium. The pores are black; the solid phase is White; the boundaries o f the sample are indicated by the broken lines. This sample has the same characteristics as the image displayed in Fig. ld. The bar corresponds to 250/~m. Data are for: Nc = 80, Lc = 16.

least up to the fourth-order moment of the phase function. This is a very important result which gives some confidence in the reconstruction method.

The comparison of, this method with more trad-aional methods such as packing of particles of a given shape is currently being made. In our opinion,

RECONSTRUCTED POROUS MEDIA AND FLUID FLOW AND SOLUTE TRANSPORT | 3

Fig. 4. A three-dimensional porous medium.

the present method is interesting only when the particles which compose a medium do not belong to a well-defined class of shapes; in this situation, it is more reliable to use only the general statistical features of the pore space.

3. TRANSPORT PROCESSES

In this section, we show how some of the macroscopic transport properties of these reconstructed media can be numerically determined by solving the local equations with the adequate boundary conditions. When possible, these properties are compared with experimental data.

3.1. Permeability

Once finite samples of porous media have been generated, the flow field of Newtonian fluids at low Reynolds number can be determined with the

program which was first described by Lemaitre and Adler (1990). The flow problem will be briefly recalled here together with a description of the program itself and of its possibilities. Then this program is applied to reconstructed porous media and compared with some experimental data due to Jacquin (1964).

Consider an infinite medium made of identical unit cells of size aN~. The low Reynolds number flow of an incompressible Newtonian fluid is governed by the usual Stokes equations:

Vp = /~V 2 v (14a)

V.v - 0 (14bl

where v, p and # are the velocity, pressure and viscosity of the fluid, respectively In general, v satisfies the no-slip condition at the wall:

v - 0 on S tl5a)

where S denotes the surface of the wetted solid inside the unit cell. The volume t0 of this cell is equal to ( N , a ) 3. Because of the spatial periodicity of the medium, it can be shown (see Bensoussan et al.. 1978) that v possesses the following property:

v is spatially periodic, with period aN t. in the three directions of space (16a)

As in Section 2, one considers a finite sample of size Nc a (see Lemaitre and Adler, 1990). This system of equations and the conditions apply locally at each point R of the interstitial fluid. In addition, it is assumed that either the seepage velocity vector ~ is specified, i.e.:

- - l v = j R d s ' v - a prescribed constant vector t l6a)

or else that the macroscopic pressure gradient Vp is specified:

7p = a prescribed constant vector (16b)

Since the system (14)-(16) is linear, it can be shown that ~ is a linear function of Vp. These two quantities are related by the permeability tensor K such that:

-v - p- ~ K ' V p (17)

Here K is a symmetric tensor that is positive definite. It only depends on the geometry of the system and thus can be simplified when the porous medium possesses geometric symmetries. A good example is given by the regular fractals studied by Lemaitre and Adler (1990), which possessed cubic symmetry; hence K is a spherical tensor, i.e.:

K = K 1 (18)

where I is the unit tensor.


The same property holds for the average permeability K of the random medium since it is isotropic only in the average.

The numerical method which is used here is a finite-difference scheme identical to the one used by Lema~tre and Adler (1990). In order to cope with the continuity equation, the so-called artificial compressibility method was applied with a staggered marker-and-cell (MAC) mesh (Peyret and Taylor, 1985). In essence, the problem is replaced by an unsteady compressible one which is assumed to converge towards the steady incompressible situation of interest.

The number of iterations was minimized by an implicit scheme and the equations were solved successively along each direction; this is the so-called alternating-direction-implicit (ADI) scheme. Additional details can be found in P.M. Adler et al. (1990). Note that our formulation is consistent; for instance, if two solid cubes have only one edge in common, no fluid can flow through this edge.

In order to evaluate our methodology, the experimental data of Jacquin (1964) were used. The porosity E and the permeability K were measured on a large number of cylindrical samples with a diameter of 2.5 cm and a length of 3-4 cm. The global porosity was measured by comparison of three different weights: weight of the dry sample; weight of the sample saturated with water; and apparent weight of the saturated sample immersed in water. Permeability was measured with air and water on a few samples; except for the small permeabilities where the Klinkenberg effect occurs in air (Jacquin, 1964), no significant difference was observed. Hence all the other measurements were performed with air only. Note that the thin sections of Fig. 1 were taken from some of these samples.

The permeability data are shown in Fig. 5 and they are seen not to be too scattered for real data. They are also compared to the numerical results; it is important to note that this comparison does not involve any hidden adjusted parameter and that every quantity is measured or calculated. The calculated permeability differs by, at most, a factor 5 from the measured one. However, the general shape of the experimental curve is predicted in quite an accurate way as if a systematic "error" was incorporated in the measurement of the unit scale.

Since Yao et al. (1993) showed that the moments of the phase function are identical up to the fourth order in real and simulated media, the discrepancy is likely to be due to the variations of the sandstone properties at some scale of intermediate length of maybe some few mm.

1 6 i ~ , \ a l ( S I [ \

K (mdy) • . ' : '4" T • A 10 3

• "~' " . % v

10 2 "°" o °oo • •

°o

10

u ! |

0 0t 0.2 0.3

Fig. 5. Permeability (for air in mD) as a funct ionof the porosity E (Jacquin, 1964)~ The dots are the experimental data. The average numerical permeability g was calculated for the four previous samples plus an additional one; they are indicated by crosses; data are for N~ = 27, Lc = 8; the vertical bars J, indicate the interval of variation of the individual permeabilities.

3.2. Formation factor

The formation factor Fis usually defined as the inverse of the dimensionless electrical conductivity a/ao of a porous medium filled by a conducting liquid phase of conductivity a0:

F = ao/~ (19)

The electrical terminology is used here but the following developments are also valid for thermal conduction and for diffusion of particles whose size is small with respect to a typical size of the medium. At the macroscopic scale, an isotropic porous medium can be characterized by a macroscopic diffusion coefficient/3 such that:

5 = (20)

where D is the diffusion coefficient of the particles in an infinite liquid. In order to determine F or/3, one has to solve the field equation in samples

of reconstructed media with periodic boundary conditions at the surface of the unit cell. Electrical a n d thermal conductions are both governed by a Laplace equation:

V 2 T = 0 (21)

where T is the local field, together with the no-flux boundary condition at the wall Sp, when the solid phase is assumed to be insulating:


m ' V T = 0 on Sp (22)

where m is the unit normal vector to Sp. This hypothesis of insulating solid phase is well verified for Fontainebleau sandstones.

VT is assumed to be spatially periodic with a period aNt in the three directions of space. In addition, either the macroscopic temperature gradient VT or the average heat flux ~:

= (a c)-3fsR.dsq (23) 4

is specified. S is the surface of the unit cell. These two quantities are related by the symmetric positive definite conduc-

tivity tensor:

= - a . V T (24)

which depends only on the geometry of the medium. In the average, for an isotropic random medium, a is a spherical tensor

equal to ~ ' / . The Neumann problem (eqs. 21-23) is solved via a second-order finite-

difference formulation. A conjugate-gradient method turned out to be very effective for the problem at hand, primarily because it is better suited to vectorial programming than implicit relaxation schemes. The computations were run on various vectorial computers, with an acceleration factor ranging from 4 to 6 with respect to an optimized scalar execution.

Again the numerical results can be compared with the experimental data obtained by Jacquin (1964) on the same samples as before. The data and the comparison are displayed on Fig. 6. First, it should be noticed that the data are well correlated by the so-called Archie's law (Dullien, 1979):

Foc~-" (25)

with a cementation factor m = 1.64. The comparison between data and predictions is better than for the per-

meability since the ratio between them is always < 3. This improvement might be due to the fact that the electrical problem does not involve any length scale, while permeability has the dimension of the square of a length and therefore is likely to be more sensitive to any small change than the formation factor. Note that permeability was always underestimated while F is overestimated; this is consistent since permeability is, so to speak, a conductivity while the formation factor is a resistance.

Again in view of the fact that no external parameter whatsoever is fitted, the agreement between experimental data and numerical predictions is the best one at the moment, to our knowledge.

100

5O

10

" . . . . I

O.O5 0.1 0.2 E Fig. 6. The formation factor F as a function of the porosity E. The dots are the experimental data. The average numerical formation factor is indicated by a cross. Numerical data are for: Nc = 80: The vertical bars ~ indicate the interval of variation of the individual formation factors.

3.3. Dispersion of a passive solute

The physical situation can be summarized as follows: a neutrally buoyant, spherical Brownian particle is injected at some arbitrary interstitial position R' at time t = 0; this particle is convected by the interstitial fluid and simultaneously undergoes Brownian mot ion characterized by the diffusion coefficient D. Within the limit of long times, the moments of order m of the probability distribution are defined by (Brenner, I980):

M,~ = ~v~ ( R - R') m P(R,t/R')d 3 R (26)

where ( R - R ' ) m represents the rn-adic ( R - R ' ) . . . ( R - R ' ) . The probability density is denoted by P(R,t/R'). The two first moments verify (Brenner, 1980):

dMi -, (27a) lim - v , ~ dt

l d lim~w-(M2 - M I M I ) = /)* (27b)

where v* is the mean interstitial fired velocity vector in rE, the portion of the unit cell r0 occupied by the liquid phase:


v* = l I~ vd3 (28)

The macroscopic dispersion tensor/5* can be calculated in two different ways. The first one which is due to Brenner (1980), can be summarized as follows: the general expression of/~* is:

/~, = _D f~LVBt • VBd3R (29) •L

B is a vector field satisfying:

B(R) = I~(R)-R (30)

where/~ is the solution of:

v-~* = V'(v/~)-DV2/~ (31a)

n" V/~ = n on Sp (31 b)

/~ spatially periodic (3 lc)

The second classical manner to determine/5* is to perform a Monte Carlo calculation by simulating the displacement inside the fluid of a large number of particles. Let Rj be the position of particlej at time t. During the time step At, this particle is convected by the fluid and undergoes a displacement v(Rj)At; because of the Brownian motion, it will also undergo a random displacement ~j. The new position of particle j at time (t + At) is:

Rj + v(Rj)At + ~j (32)

This calculation can be repeated a large number of times on a large number of particles. The two first moments can be evaluated with eq. 26 and/~* is immediately derived from (27b); the derivative (27a) of the first moment can be used as a useful check since it yields a statistical estimation of ~*. Note that the value of the equivalent diffusion coefficient is proportional to II 6j II 2/At; the proportionality coefficient depends upon the precise random rule (see Sall6s et al., 1993).

These two methods were used indifferently in a number of situations. The flow field was first determined by the routine which yields permeability. The B-equation was discretized by a second-order finite-difference formulation and solved by a conjugate-gradient iterative scheme. These two methods are in good agreement with each other; the resolution of the B-equation is much faster than the Monte Carlo scheme, but the latter is much more versatile.

The dispersion tensor/~* in an isotropic medium can generally be written as:

[ /3 0 0 ]

/)* = 0 /3 0 133~

Lo o j The z-axis is assumed to be parallel to the interstitial velocity ~*, so that the x- and ),-axes play an equivalent role.

In the rest of this subsection, we shall only deal with the longitudinal component I7)* of/5*. It is customary to represent its variations as a function of the P6ctet number:

Pe = hS*/D (34)

where D is again the diffusion coefficient of the solute particles in an infinite fluid; and l is some characteristic length of the medium. Pe represents the ratio between the convective and the diffusive effects. At low P+clet number, diffusion is predominant and one has:

z3 = E,0" (35)

Our routines were also checked with respect to existing results such as the ones by Edwards et al. (1991) for square arrays of cylinders and the classical experimental data by Gunn and Pryce (1969) for cubic arrays of spheres. They were also systematically used for various structures such as fractals and random media derived from site percolation. This material will be given in a forthcoming publication (Sail,s et al., 1993).

Results relative to reconstructed media are shown in Fig. 7. Although porosity is multiplied by a factor 3, the data are quite similar: this may be explained by the fact that the dispersion tensor is an interstitial quantity like the interstitial velocity ~*. This figure only gives trends since it is based on a single sample for each porosity.

At large P6clet numbers, i.e. when convection is predominant, the dispersion coefficient can be represented by a power law:

/3* /D~Pe ~ (36)

where ~ ~ 1.6. This intermediate value of the exponent has been confirmed by the extensive calculations by Salt+s et al. (1993) in this range of P6clet numbers. This might be due to the limited value of P6clet numbers which were used here, but it was difficult to investigate higher values with an acceptable accuracy.

4. CONCLUDING REMARKS

Realistic representations of real porous media are obtained by means of the method of reconstructed media based on the reproduction of porosity and

R E C O N S T R U C T E D POROUS M E D I A A N D F L U I D FLOW A N D SOLUTE TRANSPORT 21

D~/D I0 s

10 4

10 3

10 2

10

1

10 -~ '-b

10 -~ 10 -2

+

o

v +

o

÷

Pe , •

10 ~ 1 10 10 ~ 10 ~

Fig. 7. Dimensionless longitudinal dispersion coefficient ~*/D for reconstructed media as a function of the P6clet number Pe = f~*L/D. L is the correlation length of the medium. Data are for: 12A13 (E = 0.11, O); CJ (E = 0.21, v); and 2A3 (E = 0.31, +).

correlation function. This method is very useful when the structure of the medium is not well defined.

Transport processes can be systematically studied in these reconstructed media. This was made possible by the development of a series of numerical programs which are able to solve the local field equations in any geometry made of elementary cubes. So far, the permeability, the formation factor and the dispersion tensor have been investigated.

Many extensions of this method can be envisioned. At the moment, deposition in reconstructed media is actively studied by

means of Monte Carlo calculations. A large number of particles is injected into a medium; they are convected, they diffuse and they interact with the wall; from time to time, the geometry of the medium is updated.

A second important class of extensions is colloidal particles when they are small with respect to the pores. The specific interactions of these particles with the wall can be easily taken into account; the colloidal forces are the super- position of van der Waals and double-layer forces. Many of the previous calculations can be done with these forces.

A last class of extensions is multiphase flow in reconstructed media.

ACK NOWLEDGEM ENTS

O u r sincere g ra t i tude goes to A N D R A (Agence N a t i o n a l e p o u r la Gest iol l

des D6chets Radioac t i f s ) and I F P ( Ins t i tu t Franc;ais du P6trole) which par t ly

s u p p o r t this work . The help o f an a n o n y m o u s referee is gra teful ly acknowl - edged.

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reconstructed porous media and their application to fluid flow and solute transport

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