Cellular Automaton Model for Fluid Flow in Porous Media

A cellular automaton model for the simulation of fluid flow in porous media is presented. A lattice and a set of rules are int roduced, such th at the flow equations in the continuum limit are formally the same as the equations for one-phase liquid flow in porous media. The model is valid in two as well as th ree dimensions. Numerical calculatio ns of some simple problems are presented and compared with known analytical results. Agreement is within estim ated error s. 1. Introdu ction It is well known that the Navier-Stokes equation can be deri ved by two alternative methods, which may be called macroscopic and microscopic. The macroscopic me thod builds on the hypothesis that fluid s are structureless continua and uses conservation laws of general validity [6]. The viscosity coefficients are introduced as constants to be determined by experiments. T he microscopic approach, on the other hand, is based on the molecular structure of fluids and uses the framework of statistical physics [8]. The viscosity is here calcu lable in terms of the intermolecular potent ial. T here is an analogous division in the theory of flow in porous media and the Darcy equation . The macroscopic point of view st arts with Darcy 's law, which states that the rate of one-dimensional flow is proportional to the pressure gradient, and generalizes this law by introducing a permeabili ty te nsor. T he components of this tensor are to be determined by experiments. T he microscopic me thod is represented by an extensive literature (see for example [9] and refere nces given there) where it is shown that the Darcy equat ion ari ses from an averaging ou t of Navier-Stokes flow by the pores and that the permeability te nsor is in principle deducible from the pore geome try. The length scale is here microscopic only as far as the porous me dium is concerned: the pores are visible but the fluid is struct ureless . The recent paper by Rothman [7] on the cellular automaton simulation of flow in po rous media is in t he microscopic tradition. Using the FHP [3] triangular-lattice gas it shows how to estimate the permeability of almost © 1989 Comp lex Systems Publications , Inc.