In this work, we show an alternative way of handling the spatially discontinuous capillary pressure models in two-phase flow in porous media. This topic is very challenging and has also been studied theoretically and numerically in recent works [1,2,3]. We propose a new numerical formulation, by combining mixed hybrid finite element and finite volume discretization strategies along with novel coupling conditions at interfaces for numerically solving convection-diffusion problems with gravity and diffusive discontinuous capillary pressure. The novelties are twofold: 1) a reinterpretation of the Robin interface conditions between elements to accommodate the nontrivial effects of heterogeneities in capillary pressure and 2) the use of a conservative finite volume framework for approximation of the first-order hyperbolic flux in a robust fashion as a source term in the fully coupled formulation to preserve the delicate nonlinear balance with the diffusive operator. By following , we also present preliminary results and further directions that relates the new approach to multiscale formulations for porous media transport problems in the presence of high contrast geological properties.
 B. Andreianov, Clément Cancès, Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks’ medium, Computational Geosciences 17(3) (2013) 551–572.
 E. Abreu, Numerical modelling of three-phase immiscible flow in heterogeneous porous media with gravitational effects, Mathematics and Computers in Simulation 97 (2014) 234-259.
 P. Bastian, A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure, Computational Geosciences 18(5) (2014) 779-796.
 Arthur Santo, Conservative numerical formulations for approximating hyperbolic models with source terms and related transport models. University of Campinas, Institute of Mathematics, Statistics and Scientific Computing (2017).
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