Chemxol Engineering Science. Vol. 46. No. 3, pp. 2225-2233, 1991. am!-2509/91 S3.03 + 0.00 Pnnted in Greal Britain 0 1991 Pergamon Press plc EFFICIENT SIMULATION OF FLOW AND TRANSPORT IN POROUS MEDIA MUHAMMAD SAHIMI + and DIETRICH STAUFFER* Supercomputer Center HLRZ, c/o KFA, D-5170 Jiilich 1, Germany (First receiued 10 Auyusf 1990; accepted in revised form 20 November 1990) Abstract-We describe two efficient computer simulation methods for investigating flow and transport processes in porous media. The first method, which used random-walk techniques, is used for estimating effective diffusion coefficients in disordered porous media. The second method, which is based on lattice gases and celIular automata models, is used for calculating the effective permeability of a porous medium. We describe in some detail the computer algorithms that employ these methods. Etticlent use of vectonz- ation enables us to simulate diffusion in disordered porous media, represented by two- or three-dimensional lattices, with up to IO9 sites, and a speed of nearly four steps per microsecond and per Gray-YMP processor. Vectorization of the algorithm based on cellular automata also enables us to simulate flow in a two- dimensional model porous medium with more than a million sites. As examples, we study dilksion and flow in stratified and macroscopically heterogeneous porous media and Row through periodic arrays of obstacles. The relation between the effective diffusivity and permeability of such porous media is-also briefly discussed. 1. INTRODUCTION The problems of flow and transport through porous media are of great theoretical and practical interest (see, e.g., Scheidegger, 1974; Mason and Malinauskas, 1983; and references therein). For example, in the petroleum industry, the reservoir engineers have been dealing with such problems since oil production be- gan. Problems such as drainage and imbibition in soil, multiphase flow through trickle-bed reactors, mer- cury porosimetry for determining the pore size distri- bution of a porous catalyst, channeling in packed columns and ground water flow are but a few pro- cesses of interest to hydrologists, soil scientists, and chemical and petroleum engineers. Historically, flow and transport processes in porous media have been simulated starting from continuum equations in which the relevant variables are related to conserved quantities, such as local density, local momentum flux, etc. Macroscopic properties, such as the effective transport coefficients, are then defined as averages of the corresponding microscopic or local quantities (see, e.g., Whitaker, 1986; and references therein). The averages are taken over a volume which is small compared to the volume of the system, but large enough for the transport equations to hold when applied to the volume. At every point of a porous medium one uses the smallest such volume and, there- by, generates macroscopic field variables obeying equations such as Fick s law of diffusion and Darcy s law of flow in porous media. This approach has been useful for macroscopic modelling and has been used Author to whom all correspondence should be ad- dressed. Present and permanent address: Department of Chemical Engineering, University of Southern California, Los Angeles, CA 90089-1211, U.S.A. z Present address: Institute for Theoretical Physics, Cologne University, D-5CMl Cologne 41, Germany. extensively in the past. However, this approach is purely phenomenological and provides little insight into how flow and transport processes depend on the morphology of the pore space, i.e. its topology or pore connectivity, and geometry which is usually repres- ented by a pore size distribution. In fact, in order to use such averaged transport equations, transport co- efficients must a priori be known. Therefore, if the morphology of the pore space is strongly chaotic or disordered, a continuum approach may not be a quantitative method of studying transport processes in porous media. But when applicable, one usually uses numerical simulations to solve the continuum equations and obtains a few of the quantities of inter- est. Such numerical methods often involve some kind of discretization of the continuum equations. The second method of studying flow and transport processes in porous media is based on network or lattice models of porous media. One models the me- dium as a network of bonds, which represent the pore throats of the actual medium, and sites, which repres- ent the pore bodies of the real system. The chaotic morphology of the medium can be incorporated into the network model by assigning randomly-selected effective sizes to the bonds and sites, to mimic the disordered geometry of the medium, and by using networks of random local coordination number, i.e. the number of bonds connected to the same site, to represent the chaotic topology of the pore space. It is often true