ABSTRACT:

The cellular automata approximation of two-dimensional hydrodynamics is used to model viscous flow driven along a fracture channel by suction at porous media walls. Coupled fluid motions satisfying the Navier-Stokes equations in the free space and Darcy's law in the permeable walls are simulated indicating slip at the common boundaries. The slip velocity was found to be related to the permeability of the walls in the way proposed by Saffman. The numerical experiments show that the results are in reasonable agreement with the perturbation solutions based on the wall slip boundary condition proposed by Beavers and Joseph.

RÉSUMÉ:

La methode approximative des "cellular-automaton" a ete utilisee pour la modelisation bidimensionnelle hydrodynamique dun ecoulement visquex dans une fracture plane à parois poreuses. Le couplage des lois gouvernant les mouvements des fluids, satisfaisant à la fois les equations de Navier- Stokes pour lespace libre et la loi de Darcy pour les parois permeables, montre quÍl existe un glissement le long de la frontière commune. La vitesse de ce glissement est fonction de la permeabilite des parois, comme ce fut propose anterieurernent par Safran. Les resultats numeriques, quant à eux, sont raisonnablement en accord avec les solutions de perturbations proposees par Beavers et Joseph.

ABSTRACKT:

In diesern Aufsatz wird die "cellular automata" - Annaherungsmetnode fuer zweidimensiononle Hydrodynamik zur Modellierung von viskosen Fluβ entlang eines Riβkanales mit porösen Wanden angewandt. Zur Simulierung von gekoppelten Fluidebewegungen im freien Raum und in den porösen Wanden wurden die Navier-Stokes und die Darcy Gleichungen eingesetzt. Schlupferscheinungen am gemeinsamen Rand wurden dabei festgestellt. Die Saffman'sche Beziehung zwischen der Schlupfgeschwindigkeit und der Durchlassigkeit wird im Rahmen dieser Arbeit bestatigt. Die numerischen Untersuchungen zeigen, daβ die erzielten Ergebnissen mit der Perturbationslösung von Beaver und Jones, gut uebereinstimmen.

1 INTRODUCTION

The last several years have been seen convincing evidence that a certain class of cellular automata models originally introduced by Fisch, Hasslacher, and Pomeau (FHP) (Frisch, et al. 1986) can be used for modeling hydrodynamics fluid flow. One area in which these models should be particularly useful is that of fluid flow involving complicated geometries at low Reynolds number. In fact, Brosa has recently shown that for flows involving porous membranes. In this paper a related problem is studied for which there seems to be little progress via traditional methods, namely the flow of a viscous fluid driven along a fracture channel by suction at porous walls. Characterizing such flows is important to the hydraulic fracturing researcher in order to understand the evolution of viscous flow between parallel porous walls, as well as for the engineer attempting to estimate the oil and gas recovery potential after the stimulation treatment. The traditional approach to hydrodynamics is to start with the Navier Stokes Equation, and then, use a numerical approach such as finite element, to solve it with a given set of boundary conditions. The cellular automata approach reverses this process, i. e., one starts with a discrete model that simulates a simplified version of the microdynamics of the molecules of a gas, which, as it has been shown (Frisch et al., 1986), follows the Navier-Stokes equation at the macroscopic level. The model consists of a two-dimensional space discretized by an hexagonal lattice. Particles are placed on the links of the lattice. For each link, the particles can be traveling in one of the six possible directions, but there can be no more than one particle on a given link traveling in a given direction. The utility of the cellular automata approach lies on the ease in which computations are made, even in systems with complicated boundary conditions.

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