Variational calculation of the effective fluid permeability of heterogeneous media

Abstract

We evaluate the effective permeability of heterogeneous media with Gaussian local permeability disorder using the replica-variational approach. We obtain integral equations that determine the effective permeability kernel, and we study specific cases that admit analytical solutions. Specifically, in the case of homogeneous disorder we obtain a variational estimate for the uniform effective permeability. We compare the results of our analytical calculations with experimental and numerical data. Finally, we model the behavior of the effective permeability in the preasymptotic regime by means of momentum filters. Explicit finite-size expressions are obtained in terms of a support function that increases monotonically with the ratio of the support scale over the correlation length of the disorder. It is found that the asymptotic effective permeability is approached at a slower rate than expected.