A phase-by-phase upstream scheme that converges to the vanishing capillarity solution for countercurrent two-phase flow in two-rock media

We discuss the convergence of the upstream phase-by-phase scheme (or upstream mobility scheme) towards the vanishing capillarity solution for immiscible incompressible two-phase flows in porous media made of several rock types. Troubles in the convergence were recently pointed out by Mishra and Jaffr, (Comput. Geosci. 14, 105-124, 2010) and Tveit and Aavatsmark (Comput. Geosci. 16, 809-825, 2012). In this paper, we clarify the notion of vanishing capillarity solution, stressing the fact that the physically relevant notion of solution differs from the one inferred from the results of Kaasschieter (Comput. Geosci. 3, 23-48, 1999). In particular, we point out that the vanishing capillarity solution depends on the formally neglected capillary pressure curves, as it was recently proven in by Andreianov and CancSs (Comput. Geosci. 17, 551-572, 2013). Then, we propose a numerical procedure based on the hybridization of the interfaces that converges towards the vanishing capillarity solution. Numerical illustrations are provided.