On the Steady-State Flow of an Incompressible Fluid through a Randomly Perforated Porous Medium

Abstract

An incompressible fluid is assumed to satisfy the time-independent Stokes equations in a porous medium. The porous medium is modeled by a bounded domain inRnthat is perforated for eachε>0 byε-dilations of a subset ofRnarising from a family of stochastic processes which generalize the homogeneous random fields. The solution of the Stokes equations on these perforated domains is homogenized asε→0 by means of stochastic two-scale convergence in the mean and the homogenized limit is shown to satisfy a two-pressure Stokes system containing both deterministic and stochastic derivatives and a Darcy-type law which generalizes the Darcy law obtained for fluid flow in periodically perforated porous media.