Second-Order Computational Homogenization of Nonlinear Fluid Flow through Porous Media

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 FE2 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.