Theory and high-resolution finite element analysis of 2-D and 3-D effective permeabilities in strongly heterogeneous porous media

Starting from the premise that Darcy's law applies with a random hydraulic conductivity field K(x) defined on a support ω, we consider steady state flow in a larger domain Ω driven by statistically independent random source and boundary functions. Writing K(x) as the sum of a slowly varying unbiased estimate κ(x) and a zero mean estimation error K′(x), the conditional ensemble moments 〈h(x)〉_κ≡〈h(x)|κ(x)〉 and 〈q(x)〉_κ≡〈q(x)|κ(x)〉 become conditionally unbiased predictors of the hydraulic head h(x) and Darcy flux q(x), respectively; 〈q(x)〉_κ satisfies a standard continuity equation driven by ensemble mean source and boundary functions; and 〈q(x)〉_κ = -κ(x)▽〈h(x)〉_κ+r_κ(x) where r_κ(x) is a residual flux. We present an exact formal expression for 〈r(x)〉_κ which demonstrates that 〈q(x)〉_κ is nonlocal (depends on head gradients at points other than x) with a well-defined kernel, and non-Darcian (there is no effective or equivalent conductivity valid for arbitrary directions of conditional mean flow), except in special cases. In some of these cases the effective conductivity is κ(x), in some it is a symmetric or a nonsymmetric second-rank tensor, and in yet other situations it exists only as a set of directional scalars but not as a tensor. We describe a weak approximation for 〈r(x)〉_κ which improves with the quality of the estimate κ(x), and an ancillary approximation for the effective hydraulic conductivity tensor in an infinite domain. We then use high-resolution finite element Monte Carlo simulation to verify aspects of these approximations under uniform 3-D and radial 2-D flows in strongly heterogenous, statistically homogeneous and isotropic porous media.