The Computation of Nonclassical Shock Waves in Porous Media with a Heterogeneous Multiscale Method: The Multidimensional Case

Abstract

We consider two-phase flow problems in porous media with overshooting waves. If higher-order effects are neglected, such problems are governed by a hyperbolic conservation law with a nonconvex flux function as the macroscale model. It is well known that there can be multiple weak solutions. To ensure uniqueness, we use on the microscale either a kinetic relation or in a second approach an extended system, taking rate-dependent capillary pressure effects into account. We present a new multidimensional mass-conserving numerical method to solve the macroscale model. The method belongs to the class of heterogeneous multiscale methods in the sense of E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132] and is $L^{\infty}$-stable. A key part of the approximation is a novel numerical flux function for the multidimensional setting, which captures undercompressive waves and generalizes the one-dimensional approach of Boutin et al. [Interfaces Free Bound., 10 (2008), pp. 399--421]. Furthermore, we improve the overall computational complexity, using a data-based approach. Finally, we validate the numerical method and test it on several infiltration problems.