ELSEVIER Physica A 233 (1996) 835-847

Fluid flow across mass fractals and self-affine surfaces

Xiaodong Zhang a, Mark A. Knackstedt a,,, Muhammad Sahirni b a Department of Applied Mathematics, Research School of Physical Sciences and Enoineerino,

Australian National University, Canberra ACT 0200, Australia b Department of Chemical Enoineerin9, University of Southern California, Los Angeles,

CA 90089-1211, USA

Abstract

We use a lattice-gas method to simulate the slow flow of a fluid in systems with fractal surfaces and volumes. Two systems are studied. One is flow in a single three-dimensional fracture with self-affine surfaces. The other is flow across a three-dimensional diffusion-limited aggregate. In both cases, significant deviations from classical results are observed.

I. Introduction

Flow and transport in systems with fractal and self-affine surfaces and boundaries

are relevant to a wide variety of scientific and industrial problems. For example, nat-

ural porous media and rock contain a wide variety of pores and fractures with broad distributions of sizes and shapes (for recent reviews see, for example, [1,2]). There is

now experimental evidence [1,2] that the internal surface of the pores and the frac-

tures is very rough, and that the roughness obeys fractal statistics. Conventionally a fracture surface is modelled as perfectly smooth, and thus flow in a single fracture was

modelled as flow between parallel flat plates. Volumetric flow rate Q is then given by, Q = (w6 3 AP)/(12qL), where w is the width of the fracture and 6 its aperture, AP/L is the pressure gradient along fracture, and t/ is the viscosity of the fluid. According to this flat-plate model Q depends on the third power of 6, and the permeability K varies as 6 2 . However, there is ample experimental evidence [3 -5 ] for deviations from the cubic law, Q ~ 33; this has been attributed to the roughness of the fracture surface. Attempts to quantitatively verify this prediction is problematic. Flow in a single fracture with a realistic shape cannot be solved analytically, and only numerical solutions of the

* Corresponding author.

0378-4371/96/$15.00 Copyright @ 1996 Published by Elsevier Science B.V. All rights reserved PII S0378-4371 (96)00203-8

836 X.. Zhan 9 et al./Physica A 233 (1996) 835~847

problem can be obtained. The numerical methods used so far have been mostly based on discretizing the governing equations, the usual continuity and the Stokes' equations, by a finite-difference or finite-element method and solving the resulting equations. If the effect of surface roughness of a fracture is to be taken into account, the grid has to be very refined near the surface and this requires prohibitive computations.

The problem of hydrodynamics of particles or aggregates dispersed or suspended in a fluid is a second example of a flow problem of considerable importance to a wide variety of natural or man-made systems including colloids, polymers, granular and com- posite materials and ceramics. Colloidal particles for example, flocculate irreversibly and form aggregates which exhibit a fractal structure. The aggregates have a very low density when compared with bulk matter, yet they are very effective at screening hydrodynamic forces. The quantitative nature of the force depletion around this mass fractal structure is important as these materials are used for these screening properties

(e.g., as thickeners in fluids). The most important theoretical (and practical) problem is to predict the effective equilibrium and transport properties of these materials, from a knowledge of their microstructural mechanics. The microstructural mechanics entails the distribution of various forces acting on the particles, such as the hydrodynamic, Brownian, and interparticle forces, as well as the morphology or the microstructure of the particles. From the knowledge of microstructural mechanics, the physical proper- ties of the particles and the fluid that surrounds them, and the location and motion of the boundaries of the system, one can, in principle, predict the macroscopic proper- ties of the system with the given configuration, such as the diffusion coefficient and the thermal conductivity. Although this is a well-defined boundary value problem, it is difficult, since one has to deal with a many-body problem. To tackle this problem, various schemes have been developed in the past [6]. The problem is particularly diffi- cult when the particles or aggregates have a complex shape. Colloidal aggregates [7-9] are fractal objects with very complex structures, and moreover their properties scale with their linear size or mass, so that their true fractal behaviour does not manifest itself, unless the size of the aggregate becomes large. For these reasons, calculating various properties of aggregates that are suspended in a flowing fluid, such as the dis- tribution of the hydrodynamic forces that act on the aggregates, and the dependence of the friction factor on the Reynolds number, is a computationally difficult problem. Even sophisticated approaches that have been developed in the past [6] cannot be used with aggregates of more than a few hundred particles, unless one is willing to use a prohibitive amount of computer time. Moreover, the distribution of hydrodynamic forces exerted by the fluid on the particles of the aggregate becomes very broad, since (relatively) large forces will be exerted on the most exposed parts of the aggregate, whereas the particles deep inside the aggregate will feel small forces. These measures require the consideration of large aggregates.

In this paper, we use an alternative method based on the lattice-gas (LG) simula- tion to study flow in these complex systems. We consider flow in a three-dimensional fracture with self-affine fractal surfaces, and evaluate the distribution of the hydrody- namic forces that are exerted on a colloidal fractal aggregate that is suspended in a

x. Zhang et al. I Physica A 233 (1996) 835-847 837

slowly moving fluid. Models based on LG are discrete in time and space, require only local rules for updating and are performed by a series of simple logical operations. By virtue of their construction, LG models of hydrodynamics are amenable to the study of both very large and very complex systems. Recent advances in parallel supercom- puter technology has made the LG method an extremely efficient computational tool and particularly appropriate for studying hydrodynamics in environments with complex boundaries.

The plan of this paper is as follows. In the next section, we briefly discuss the lattice-gas method that we use in this paper. In Section 3 we describe our simulation of flow in a fracture with rough surfaces, while Section 4 presents our results for the hydrodynamics of flow over fractal aggregates. The paper is summarized in Section 5, where we also point out future directions in this research field.

2. Lattice-gas simulation

LG models are [1,2, 10-12] are discrete analogues of molecular dynamics, in which particles with discrete velocities populate the links of a fixed array of lattice sites. The variables describing the state of the system are Boolean indicating the presence or absence of the particles in the bonds of the lattice. For example, the basic two- dimensional LG model consists of identical particles residing at sites on a triangular lattice. There are six different momentum states at each lattice site associated with the lattice vectors

ea = [COS ( ~ - ~ ) , sin ( ~ - ~ ) ] , a = l . . . . ,6.

An exclusion principle is imposed so that no more than one particle of a given site can have a given momentum state. The configuration of sites evolves in a sequence of discrete time steps. There are two microscopic updating processes at each step - advection and collision. In the advection step every particle moves from its present site to a nearest neighbour site in the direction ea. In the collision process the particles at each site are rearranged subject to the conservation of local mass and momentum. The important feature of the LG method is that all operations are purely discrete, local and logical - ideal for high speed simulation on supercomputers. A second feature of the method is its flexibility. Boundary conditions are very easy to implement. For example, total reflection of particles at a solid boundary simulates the macroscopic no-slip boundary condition.

The extension of LG models to three dimensions is more complex, since no regular three-dimensional lattice is isotropic, and thus in the continuum limit one has spurious terms, in addition to those in the Navier-Stokes equation, which are caused by the anisotropy of the lattice. There are several methods of circumventing this difficulty [14, 15]. For example, one can use [14, 15] the FHP model on a four -d imens iona l face- centred hypercubic (FCHC) lattice. The nodes (Xl,X2,Xa,X4) of the lattice satisfy the

838 X. Zhan 9 et al./Physica A 233 (1996) 835-847

condition that x~ +x2-I-x3-~-x4 is even, where Xi'S a r e integer numbers. For this lattice, which has a coordination number of 24, all pairwise symmetric fourth-order tensors are isotropic, and therefore one can simulate the Navier-Stokes equation on such a lattice. One may then make the observation that any solution of the four-dimensional model which does not depend on the fourth dimension is a solution of the three-dimensional model. This suggests the use of a FCHC lattice which wraps around periodically in the fourth direction. One actually uses a lattice which is only one lattice unit long in the fourth dimension, and therefore has an effectively three-dimensional structure. This is the approach we use in this paper. The only disadvantage of this model is that, although the fourth dimension is very thin, the discrete velocities still have components in all directions; therefore the model is bit intensive (24 or 25 bits per site as compared with 6 in two dimensions).

At the beginning of the simulation, one constructs a transition table by which the present states, determined by the velocity of the incoming particles, is transformed into the next state, given by outgoing velocities. The table contains in principle all of the possible states (for example, in two dimensions there are 28 possible states). In the case of the FCHC lattice the size of the table 224 is problematic. Symmetries are used to reduce the collision table to a size that fits into local memory on the 16K CM2 [16]. One iteration in the simulation consists of updating all lattice sites according to the reduced transition table. The number of required iterations for reaching a steady-state solution depends on the microstructure of the system. If the system has a complex configuration, then many iterations may be required. An issue of particular importance is the mean free path of the particles and its relation to the height of the roughness on the surface of the fracture. As Rothman [17] showed, in order for the LGA results to approach the continuum limit, the mean size of the void area in the lattice must be at least twice the mean free path of the particles, which is about 9 lattice bonds. This condition is obeyed in the simulations, insuring the result represents a true macroscopic continuum.

The LG model that we described has been used by various authors to investigate flow phenomena in porous media. Rothman [17] studied single-phase flow in porous media. Succi et al. [ 18] studied the same problem in three dimensions. Knackstedt et al. [ 19] (1993) used the LG method to study dynamic (frequency-dependent) permeability of a porous medium, a notoriously difficult problem, because one has to obtain the frequency-dependent permeability by Fourier transforming the numerical results. LG results are subject to relatively large noise and fluctuations, and therefore their Fourier transform is very difficult to obtain. Despite this, Knackstedt et al. [19] showed, by reproducing the exact results for the dynamic permeability of channel-like systems, that if a large number of realizations are used, the LG results are reliable. This indicates the reliability of LGA for simulating flow problems. Brosa and Stauffer [20], Kohring [21,22], and Sahimi and Stauffer [23] looked at flow in two-dimensional porous media with various obstacle shapes and arrangements (random versus regular and periodic), and paid particular attention to the efficiency of the simulation. Vollmar and Duarte [24] studied flow through a porous membrane, and investigated the effect of various

X. Zhan# et al. IPhysica A 233 (1996) 835-847 839

boundary conditions. Chen et al. [25, 26] used the LG and the related lattice-Boltzmann

methods to study a variety of flow problems.

3. Flow in a single fracture with fractal surfaces

The morphology of fracture surfaces is strongly dependent on the material, its fracture

mechanism and the scale of observation. There has been much progress recently in the statistical characterization of the topography of fracture surfaces. In particular, the

observation of scale invariance has been made in many experimental cases leading to

a more reliable description of the statistics of fracture roughness. It has been shown in

many instances that fracture surfaces exhibit statistically self-affine scaling properties.

A self-affine surface z(x,y), isotropic in the (x, y) plane, is invariant under the scale

transformation (2x, 2y, 2rtz) where H is the roughness exponent. For simulating flow in a single fracture, we model the fracture by a three-dimensional channel whose surface

is rough and obeys fractal statistics. The roughness of the surface was generated by a

fractional Brownian motion ( iBm) [27], which is a stationary stochastic process BH(r) with the following properties

( B . ( r ) - ~H(ro)) = O, (1)

( [B.( r ) - B.(r0)] 2) ~ ]r - rol 2H , (2)

where r=(x, y,z) and r0=(xo, yo,zo) are two arbitrary points, and H is the roughness or Hurst exponent. A remarkable property of IBm is that it generates correlations whose

extent is infinite. For example, if we define a one-dimensional correlation function C(r)

by

C(r) = ( -BH(-r )BH(r) ) (BH(r)2) , (3)

then one finds that C(r) = 22H-I - l, independent of r. Moreover, the type of cor-

relations can be tuned by varying H. If H > 1/2, then C(r) > 0 and IBm displays persistence, i.e., a trend (for example, a high or low value) at r is likely to be fol-

lowed by a similar trend at r + Ar. If H < 1/2, then C(r )< 0 and IBm generates antipersistence, i.e., a trend at r is not likely to be followed by a similar trend at

r + Ar. For H = 1/2 the trace of BH(r) is similar to that of a random walk, while H = - 1 / 2 represents the white noise limit. It has been shown by Schmittbuhl et al.

[28] that, if we consider a two-dimensional roughness profile on a fracture surface, then the average height h of the profile is related to its length L by

k ~ L H , (4)

where H is the roughness or Hurst exponent defined above. A value H _~ 0.85 was found by Schmittbuhl et al. for granitic faults, indicating strong positive and long-range correlations. In the present paper, we use a range of values for H in order to assess its effect on the permeability of the fracture and its relation with the fracture aperture.

840 X. Zhano et al./Physica A 233 (1996) 835~47

Fig. 1. Example of two rough surfaces generated by fBm (H = 0.8) and subsequently used in the fracture flow simulations.

Fig. 1 shows an example of a two-dimensional rough surface generated by the fBm. Once the roughness of the surface is generated, simulation of the flow prob-

lem is started. We used an Lx x Ly x Lz system, where z denotes the direction of flow,

and in our simulations Lx =Ly = 64 and Lz = 128 (all distances are measured in units of lattice gas bonds). The simulations start by introducing a flat velocity profile (piston

flow) at the entrance to the fracture. We then allow the system to relax and reach a steady state. Typically, this is achieved after about 10000 time steps, after which we measured the desired quantities for another 20000 time steps. The pressure gradient

AP/Lz in the system is measured by calculating the momentum transfer across imag-

inary planes perpendicular to the flow direction. A fit to the permeability K of the fracture is then calculated by using Darcy's law:

K AP v - ( 5 )

~/Lz '

where v is the average fluid velocity. The simulations were carried out at low enough Reynolds number to ensure that Darcy's law is applicable. The results were then av- eraged over a minimum of 20 different realization of the fracture with a given H, the Hurst exponent. We implement the LG code on a 16K Connection Machine 2.

Of prime interest is the dependence of the permeability of the fracture on the fracture aperture. We generate self-affine surfaces of root mean square thickness ¢. The aperture

X. Zhano et al. IPhysica A 233 (1996) 835--847 841

5.0

4.5

4.0

3.5

3.0

2.5

2.0 2.0

I I

/ A

..~ ~" ./.~'"

/ ' . & - "

A "

, i I a L i I , i

2.5 3.0 In(Dis tance be tween Mean Fracture Surfaces)

i

3.5

Fig. 2. Dependence of permeability K on the aperture distance 6 for H = 0 . 8 . A fit to the data with f l=2.67 is shown: ( A ) d = 10; (*) f = 18. The crossover to f l=2.0 is evident in the case E= t8.

6 is initially defined as the distance between the two mean surfaces parallel to the flow

direction; 6 = Ly - 2ft. Aperture is varied by pushing the two surfaces of constant E

together. By varying d one may probe the effect of a range of dimensionless aperture 6/d on the results. Fig. 2 shows the dependence of K on the aperture for H = 0.8, a

value close to what Schmittbuhl et al. [28] obtained for rough fracture surfaces, and

two values of Y. As can be seen, in both cases one has

K, .~6 ~ , (6)

with fl ~_ 2.67, except that for f = 18 one has an apparent crossover to fl ~ 2. This

crossover is associated with the significant overlap of the rough fracture surfaces at small 6 - in this case the system reverts to behaving like a simple porous medium.

From Darcy's law we expect fl=2.0. At larger separations the results for both f = 10 and f = 18 differ from the classical result for fracture with smooth surfaces.

Fig. 3 shows the results for H = 0.3. No crossover is noted for both values of E. In both cases we find fl-~ 3.15, significantly different from the classical results. If we further decrease H, we find that fl increases significantly. For example, for H = - 1 / 2 we find fl > 4. Clearly, the classical result for flow between fractures (K ~ 62) does not hold across rough fracture surfaces. Our study shows quantitatively that the experimentally observed deviation from the cubic law can be attributed to surface roughness.

842 ~ Zhang et al./Physica A 233 (1996) 835447

4

' ' ' ' [ ' ' ' ' I ' ' ' ' I ' ' ' ' 1

2

o

2.0 I I t I ] t r I I I I I I I I L I I t I

2.5 3.0 3.5 4.0 In(Distance between Mean Fracture Surface)

i

4.5

Fig. 3. Dependence of permeability K on the aperture distance ~ for H = 0.3. A fit to the data with fl = 3.15 is shown: ( A ) f = 10; (*) f = 18.

4. Hydrodynamics of flow over a fractal aggregate

We next consider the problem of fluid flow over a fractal aggregate. This study represents the first step towards our ultimate goal of studying suspensions of colloidal

aggregates and calculating their dynamical properties, such as the viscosity of the

suspension. Since it is now well-established that many colloidal aggregates have a fractal structure, we studied flow over a three-dimensional DLA aggregate which has

been extensively invoked as a model stucture for colloidal aggregates [9]. DLA has

a fractal dimension D f "~ 2.5. A similar problem has already been studied by Meakin

[30,31], who used the Kirkwood-Riseman theory to investigate hydrodynamics of a fractal aggregate moving with a constant velocity in a quiescent fluid. However, due

to computational difficulties, this study was limited to the consideration of only very small aggregates with at most 400 elementary particles in them (Meakin used diffusion-

limited cluster-cluster aggregates which have a fractal dimension Df _~ 1.8 in three dimensions).

Using the LG method, we have studied the problem with aggregates with masses M which range up to 4507 particles. We first generated the DLA cluster and then placed it in the lattice system. For clusters with M < 800 we use 32 × 32 × 64 lattices, while for the larger aggregates we used 64 × 64 × 128 lattices. We implemented periodic boundary conditions to minimize the effect of the finite lattice size. No-slip boundary conditions are used at the interface between the fluid and solid phases. At the entrance to the system, a velocity profile is introduced, and the system is allowed to relax. In

X. Zhan9 et al. IPhysica A 233 (1996) 835447 843

q

o

z

1000

800

600

400

200

0

- 7

. . . . . . . . . i . . . . . . . . . i . . . . . . . . . , . . . . . . . . . , . . . . . . . . .

/

- 6 - 5 - 4 - 3 - 2 log(F,.....)

Fig. 4. Logarithmic force distributions for the components of the forces exerted by the particles on the fluid parallel to the direction of motion through the fluid. Results show for DLA aggregates of size M = 428-4037. (-), M=4037; (- - -), M=1056; ( . . . ) , M=646; ( - • - ) , M=514; ( . . . . . ), M=428.

this problem, the system reached equilibrium after about 7000 time steps. We then measured the quantities of interest over the next 3000 time steps. The results were then averaged over more than 100 realizations of the system.

An important property of the system is the normalized force distribution defined by

(FII)i (7) (F~)i-- M F '

where (F~)i is the normalized force parallel to the flow direction, and (FII)/ is the cor- responding unnormalized force, The logarithmic distributions are shown in Fig. 4 for various aggregate sizes, where N[ln(Fii )] is the number of aggregate particles per unit interval of ln(FiI ) with a force component Fll. These distributions seem to be skewed and similar to a log-normal distribution. Typical of fractal aggregates, these distributions can be collapsed onto a single scaling curve. Shown in Fig. 5 are ln(N[ln(F~ )])fin(M) versus ln(F~)/ln(M). Although we used large aggregates, due to the noise in the sim- ulations the collapse is not complete. This figure however does indicate that there is a single universal curve for the distribution of the hydrodynamic forces for various aggregate sizes. If true, we expect to have

ln(N[ln(F~)]) = ln(M)g[ln(F~)/ln(M)], (8)

where g(x) is the scaling function that describes the collapsed curve in Fig. 5. The force distribution shown in Fig. 4 is multifractal [32], i.e., each of its moment scales with a different power of M. From the knowledge of g(x) one can construct a spectrum of singularities f (~ ) [32], describing the multifractal properties of the distribution, given by

f (~ ) = D f g ( - D f l ~ ) . (9)

Fig. 6 shows the resulting spectrum.

844 X. Zhang et al./Physica A 233 (1996) 835-847

O

! v

Q

Z v

O

1 . 0

0 .5

0 . 0

- 0 . 5 . . . .

- 1 , 4 - 1 . 2

• • " ..'..*' . i . ' " "" " "- - ' • ' ' '

!,

i i ,

- 1 . 0

l o g ( F L , ~ ) / l o g ( M ) - 0 . 8 - 0 . 6

Fig. 5. Scaling of the normalized force components parallel to the direction of motion through the fluid for cluster of size M = 428-4037.

2.5

2.0

1.5

3

1 . 0 -

0 . 5

0 . 0

2 . 0

. . . . . . " " . , . L J . . . . . . . . I . . . .

.~ . .

x" x \ , " \ ",,

] " \ ' \ \ . ,

x~'\,\ " \

~ \ ' \ \

\

i i i , I l i , , I i l l i ] , l , i I l l i i

2 . 5 3 . 0 3 . 5 4 . 0 4 . 5

Fig. 6. Shapes obtained for the function f (~) for clusters of three different sizes. The fact that distinctly different f(~) curves are obtained indicates that clusters are still far from the asymptotic (M ~ oc) limit.

Another important quantity is the friction factor Cf and its dependence on the

Reynolds number Re. Cf is defined by

"gs c j - ½p( )2, (10)

where p is the density o f the fluid particles, Zs is the shear stress on the sol id surface,

and (v) is the mean f low velocity. For a spherical particle in low Reynolds number

X. Zhan# et al. IPhysica A 233 (1996) 835847 845

2.5

E.O

1.5

ttO 0 ,- 1.0

0.5

0.0

. . . . . . . . I . . . .

Q.

......... ili?".o.i ..........

I , , , , I , , , ,

.0 - 0 . 5 0.0 0.5 log,0(R.)

Fig. 7. Dependence of Cf o n Re for a sphere and for DLA clusters of various hydrodynamic radius. (D),

sphere, R = 6; (o), aggregate, R = 21,Rh ---- 6.3; (x), aggregate, R = 23,Rh = 8.2; (A), aggregate,

R = 24,Rh = 8.4.

fluid flow, one has the classical result [33]

Cf = 24/Re. (11 )

We implemented LG simulations to measure Cf across spherical particles and con- finned this equation. Fig. 7 shows Cf across a spherical particle of radius R -- 6 (lat-

tice gas units) as well as results for the DLA aggregates with three different masses.

In one case the hydrodynamic radius (Rh) of the aggregate is close to that of the

sphere (cluster radius = 21, Rh=6.3). Comparing the Cf for these two systems we

find that for Re < 20 the aggregate has a smaller Cf than the sphere - at higher Re a crossover seems evident. For over two orders of magnitude variations in Re in the range 0.1 < Re < 10 we have

C f = a/Re ~ , (12)

where ~ ~_ 0.85. ~ is found to be independent of cluster mass. We have no simple theoretical explanation for this value of ft.

5. Summary and conclusions

The study of flow in systems with rough surfaces involves complex phenomenon. No analytic solutions to these hydrodynamic problems exist. While several models have been developed for studying these systems, they are not completely satisfactory and cannot provide quantitative, and in some cases even qualitative, predictions for the quantities of interest. We have studied slow fluid flow across fractal surfaces. We show quantitatively that self-affine fracture roughness accounts for strong deviations from the

846 X, Zhan9 et al./Physica A 233 (1996) 835~47

simple fracture flow model based on a parallel plate geometry. We also find that the friction factor across fractal aggregates has a anomalous Reynolds number dependence. In our opinion, the systems studied in this paper represent definitive examples of the problems for which lattice-gas models are definitely more appropriate than the contin- uum, or other discrete models. This method is the ideal tool for vector computers and parallel computations, which are the way the present massive computations are carried out. As such models become more sophisticated and realistic, more traditional numer- ical methods of simulating fluid flow in complex systems, such as the finite-difference methods, lose their competitive edge, and may be phased out in the future.

Acknowledgements

MAK was supported by the Australian Research Council and MS was supported in part by the Petroleum Research Fund, administered by the American Chemical Soci- ety. We thank the PCRF at ANU and the National Resource Information Council for

generous allocations of computer time.

References

[1] M. Sahimi, Rev. Mod. Phys. 65 (1993) 1393. [2] M. Sahimi, Flow and Transport in Porous Media and Fractured Rock (VCH, Weinheim, Germany,

1995). [3] S.R. Brown, Geophys. Res. Lett. 14 (1987) 1095. [4] S.R. Brown, J. Geophys. Res. B92 (1987) 1337. [5] Y.W. Tsang and P.A. Witherspoon, J. Geophys. Res. 86 (1981) 9287. [6] J.F. Brady and G. Bossis, Annu. Rev. Fluid Mech. 20 (1988) 111. [7] W.B. Russel, D.A. Saville and W.R. Schowalter, Colloidal Dispersions (Cambridge University Press,

Cambridge, 1989). [8] D.W. Shaefer, J.E. Martin, J.E. Wiltzius and D.S. Cannell, Phys. Rev. Lett. 52 (1984) 2371. [9] P. Meakin, in: Phase Transitions and Critical Phenomena, Vol. 12, eds. C. Domb and J.L. Lebowitz

(Academic, London, 1988), p. 335. [10] S. Wolfram, Theory and Applications of Cellular Automata (World Scientific, Singapore, 1986). [11] R. Benzi, S. Succi and M. Vergassola, Phys. Rep. 222 (1992) 145. [12] D.H. Rothman and S. Zaleski, Rev. Mod. Phys. 66 (1994) 1417. [13] U. Frisch, B. Hasslacher and Y. Pomeau, Phys. Rev. Lett. 56 (1986) 1505. [14] U. Frisch, D. d'Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau and J.-P. Rivet, Complex Syst. 1

(1987) 649. [15] D. d'Humieres, P. Lallemand and U. Frisch, Europhys. Lett. 2 (1986) 291. [16] J.A. Somers and P.C. Rem, Appl. Sci. Res. 48 (1991) 391. [17] D.H. Rothman, Geophysics 53 (1988) 509. [18] S. Succi, E. Foti and F. Higuera, Europhys. Lett. 10 (1989) 433. [19] M.A. Knackstedt, M. Sahimi and D.Y.C. Chan, Phys. Rev. E 47 (1993) 2593. [20] U. Brosa and D. Stauffer, J. Stat. Phys. 57 (1989) 63; 63 (1991) 405. [21] G.A. Kohring, J. Physique II 1 (1991) 593. [22] G.A. Kohring, J. Stat. Phys. 63 (1991) 411. [23] M. Sahimi and D. Stauffer, Chem. Eng. Sci. 46 (1991) 2225. [24] S. Vollmar and J.A.M.S. Duarte, J. Physique II 2 (1992) 1565. [25] S. Chen, G. Doolen, K. Eggert, D.G. Grunau and E.Y. Loh, Phys. Rev. A 43 (1991) 245. [26] S. Chen, G. Doolen and W.H. Matthaues, J. Stat. Phys. 64 (1991) 1133.

X. Zhan9 et al. IPhysica A 233 (1996) 835-847 847

[27] B.B. Mandelbrot and J.W. van Ness, SIAM Rev. 10 (1968) 422. [28] J. Schmittbuhl, S. Gentier and S. Roux, Geophys. Res. Lett. 20 (1993) 639. [29] X. Zhang, M.A. Knackstedt and M. Sahimi, Water Resour. Res. (to be published). [30] P. Meakin and J.M. Deuteh, J. Chem. Phys. 86 (1987) 4648. [31] P. Meakin, J. Chem. Phys. 88 (1988) 2042. [32] T.C. Halsey, M.H. Jensen, LP. Kadanoff, 1. Procaccia and B. Shraiman, Phys. Rev. A 33, (1986) 1141. [33] R.B. Bird, W.E. Stewart and L.N. Lightfoot, Transport Phenomena (Wiley, New York, 1960). [34] X. Zhang, M.A. Knackstedt and M. Sahimi, Chem. Eng. Sci. (to be published).