Stable Hybrid Upwinding VAG scheme for the incompressible diphasic model with discontinuous capillary  pressure

In this work, we propose an improved discretization, in terms of stability and accuracy, for the incompressible two-phase Darcy flows in a heterogeneous porous medium with discontinuous capillary forces. For this purpose, the total velocity formulation of the model is used. The coupled system is composed of a degenerate parabolic equation for the non-wetting phase and a pressure equation for the total velocity. We combine a positive Vertex Approximation Gradient (VAG) type scheme for the gradient fluxes with a hybrid upwinding of the mobilities. This approach entails a maximum principle on the saturations, which remain in their physical ranges. Energy estimates are obtained by selecting key approximations of the fluxes. These stability results allow to prove the existence of discrete solutions. Numerical experiments on complex test-cases show the robustness of the new approach in terms of the accuracy as well as the nonlinear convergence. Comparison to the usual phase potential upwinding approach and to a previous hybrid upwinding scheme are also provided.