Analysis and numerical approximation of Brinkman regularization of two phase flows in porous media

We consider a hyperbolic-elliptic system of PDEs that arises in the modeling of two-phase flows in a porous medium. The phase velocities are modeled using a Brinkman regularization of the classical Darcy's law. We propose a notion of weak solutions for these equations and prove existence of these solutions. An efficient finite difference scheme is proposed and is shown to converge to the weak solutions of this system. The Darcy limit of the Brinkman regularization is studied numerically using the convergent finite difference scheme in two space dimensions as well as using both analytical and numerical tools in one space dimension. The results suggest that the Darcy limit of the Brinkmann regularization may not lead to the recovery of standard entropy solutions of the classical model for two-phase flows.