Second‐order computational homogenization of nonlinear fluid flow through porous media

Abstract

We present a second‐order computational multiscale model for heterogeneous porous media, which allows for the scale bridging of transient fluid‐flow processes through porous materials with non‐separated scales. The formulation connects a homogenized macroscopic scale described by a local theory of grade two with heterogeneous microstructures described by a local theory of grade one. At the macroscale, this leads to C1‐continuity requirements on the macroscopic solution field; at the microscale, we need to take account of constraints on fluctuation fields that require H(div)∩H(grad)‐conformity of microscopic solutions. Both these challenges are addressed through the development of mixed Hu-Washizu formulations that result in a variationally consistent homogenization framework with minimization structure across scales. We validate the second‐order multiscale model by means of fully resolved, direct numerical simulations and provide comparisons with results of first‐order FE 2 simulations. By considering linear Darcy and nonlinear Darcy-Forchheimer flow through two‐ and three‐dimensional porous microstructures, we provide further insights into the framework and associated length‐scale effects.