Two-scale analytical homogenization of Richards' equation for flows through block inclusions

In this paper we propose an analytical form for the upscaled coefficient applicable to the nonlinear Richards' equation. Block inclusions are symmetrically centered in a grid cell, and flow is dominated by capillary forces on the small scale. The nonlinear boundary value problems can be defined in a three-dimensional bounded domain, with a periodic and rapidly oscillating saturated hydraulic conductivity coefficient. The new result is derived by applying a corrector to an analytical approximation of the well-known cell problem obtained by the two-scale asymptotic expansion to the original heterogeneous nonlinear problem. The previously known analytical results for the upscaled coefficient, including the geometric average for the checkerboard geometry, can be regarded as particular cases of this new form. We perform numerical simulations to obtain the error between the fine-scale solution and the upscaled solution, respectively, as well as convergence properties of the approximation, corroborating results in the literature. There is no limitation regarding either the ratio of permeability between the matrix and inclusion or its shape for both computing the upscaled coefficient and obtaining the upscaled equation. This is illustrated by the comparison with known results and by numerically demonstrating convergence properties with a medium having square and circular inclusions in a main matrix with heterogeneity ratio of 1:100 and 1:10. Even though the derivation and applications are presented for single phase and steady state, the results are valid for multiphase and transient flow conditions. We also show that the approximation is independent of the relative saturated hydraulic relationship.