Computing the effective permeability of log-normal permeability fields using renormalization methods

We consider the computation of the equivalent permeability K-eff of a heterogeneous porous medium. The logarithm of local permeabilities is a stationary isotropic gaussian random function, characterized by its covariance function rho>(*) over bar * (r). Using a renormalization group analysis, we compute the equivalent permeability K-eff(Gamma) as a function of the wave vector cut-off Gamma. Finally, using a mean-field approximation, we obtain a very simple differential equation linking the equivalent permeability to the power spectrum of the permeability fluctuations. After integration over the whole wave vectors, K-eff is expressed as a power averaging formulae, which was conjectured by Landau and Lifshitz. (C) 2000 Academie des sciences/Editions scientifiques et medicates Elsevier SAS.