On Darcy- and Brinkman-type models for two-phase flow in asymptotically flat domains

We study two-phase flow for Darcy and Brinkman regimes. To reduce the computational complexity for flow in vertical equilibrium, various simplified models have been suggested. Examples are dimensional reduction by vertical integration, the multiscale model approach in Guo et al. (Water Resour. Res. 50(8), 6269-6284, 2014) or the asymptotic approach in Yortsos (Transport Porous Med. 18, 107-129, 1995). The latter approach uses a geometrical scaling. In this paper, we provide a comparative study on efficiency of the asymptotic approach and the relation to the other approaches. First, we prove that the asymptotic approach is equivalent to the multiscale model approach. Then, we demonstrate its accuracy and computational efficiency over the other approaches and with respect to the full two-phase flow model. We apply then asymptotic analysis to the two-phase flow model in Brinkman regimes. The limit model is a single nonlocal evolution law with a pseudo-parabolic extension. Its computational efficiency is demonstrated using numerical examples. Finally, we show that the new limit model exhibits overshoot behaviour as it has been observed for dynamical capillarity laws (Hassanizadeh and Gray, Water Resour. Res. 29, 3389-3406, 1993).