Correlation structure dependence of the effective permeability of heterogeneous porous media

Abstract

A theory is given in which the effective permeability tensor Keff of heterogeneous porous media is derived by a perturbation expansion of Darcy’s law in the variance σ2 of the log-permeability ln[κ(ub;;-45rubx)]. The only assumption is that the spatially varying permeability κ(ub;;-45rubx) is a expressed in terms of the moments of the distribution of ln[κ(ub;;-45rubx)], i.e. Keff can formally be computed for any given distribution of the fluctuations of the log-permeability. The explicit dependence of Keff on multi-point statistics is given for non-gaussian log-permeability fluctuations up to order σ6. As a special case of the theory, we examine Keff for a normal distribution function for both isotropic and anisotropic media. In the case of three-dimensional isotropic porous media, a conjecture has been made in the past according to which the scalar effective permeability κeff=KGexp[σ2/6] where KG is the geometric mean of the log-permeability. It is shown here that this conjecture is incorrect as the σ6-order term of κeff contains additional terms than those corresponding to the development of the above formula. Moreover, these additional terms depend on the structure of the two-point correlation function of ln[κ]. The resulting κeff computed for both Gaussian and exponentially decaying covariances lies below the exponential formula. This result might suggest the exponential formula as being an upper bound for κeff. For anisotropic systems, Keff is given up to the σ4-order for the general case where the mean flow is arbitrarily oriented with regard to the axes of stratification.