On the double porosity model of a single phase flow in random media

Abstract

We consider the linearized equations of slightly compressible single fluid flow through a highly heterogeneous random porous medium, consisting of two types of material. Due to the high heterogeneity of the two materials the ratio of their permeability coefficients is of order ε², where ε is the characteristic scale of heterogeneities. Supposing that the matrix blocks set of the porous medium consists of random stationary inclusions, and assuming positive definiteness of the effective permeability tensor associated to the corresponding Neumann problem for the random fractures system, we obtain the homogenized problem for a random version of the double porosity model used in geohydrology. It includes as a particular case the periodic setting, already studied by homogenization theory methods (see, for example, [1,7]). The homogenized problem is obtained by using the stochastic two scale convergence in the mean, and by means of convergence results specially adapted to our a priori estimates and to the random geometry, which do not require extension of solutions to the matrix part.