Dual homogenization of immiscible steady two-phase flows in random porous media

Abstract

We develop a dual conductivity-resistivity homogenization for immiscible steady two-phase flow in heterogeneous, spatially correlated, random porous media. The objective is to obtain macroscale equations for an equivalent homogeneous medium. We assume that the flow is governed locally by the generalized Darcy equations. Two formulations are developed, one in terms of capillary pressure pc and the other in terms of saturation s. In each case the perturbation approach is used to express average and fluctuation equations, and the closure problems are solved using the Wiener-Kinchine spectral theory. The macroscale permeability Kea and resistivity R ea tensors are determined for each fluid a, which also yield the relative permeabilities. These dual results provide a physically consistent correction of the perturbation-based tensors. The corrected macroscale tensors are then always positive, symmetric, and reciprocal to each other. We also develop, similarly, the homogenization of the capillary pressure curve pc(X, θ) and of its reciprocal θ(x, pc), where θ is the wetting fluid content. Finally, these general analytical results are applied to the special case where 〈∇pc〉 ≈ 0 and/or 〈∇θ〉 ≈ 0, which can be interpreted as a mean capillary equilibrium.