Convergence of an MPFA finite volume scheme for a two-phase porous media flow model with dynamic capillarity

We discuss an O-type multi-point flux approximation finite volume scheme for the discretization of a system modelling two-phase flow in porous media. The particular feature in this model is that dynamic effects are taken into account in the capillary pressure. This leads to a nonlinear system of three evolution equations, written in terms of the nonwetting-phase saturation and of the two pressures. Based on a priori estimates and compactness arguments, we prove the convergence of the numerical approximation to the weak solution. In the final part, we present numerical results that confirm the convergence analysis. These results show that the method is first-order convergent for the flux, and second-order convergent for the saturation and the pressures.