Approximating the vanishing capillarity limit of two‐phase flow in multi‐dimensional heterogeneous porous medium
Boris Andreïanov, Konstantin Brenner, Clément Cancès · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · 2013
Abstract Neglecting capillary pressure effects in two‐phase flow models for porous media may lead to non‐physical solutions: indeed, the physical solution is obtained as limit of the parabolic model with small but non‐zero capillarity. In this paper, we propose several numerical strategies designed specifically for approximating physically relevant solutions of the hyperbolic model with neglected capillarity, in the multi‐dimensional case. It has been shown in [Andreianov & Cancès, Comput. Geosci., DOI: 10.1007/s10596‐012‐9329‐8, 2013] that in the case of the one‐dimensional Buckley‐Leverett equation with distinct capillary pressure properties of adjacent rocks, the interface may impose an upper bound on the transmitted flux. This transmission condition may reflect the oil trapping phenomenon. We recall the theoretical results for the one‐dimensional case which are used to motivate the construction of multi‐dimensional finite volume schemes. We describe and discuss a coupled scheme resulting as the limit of the scheme constructed in [Brenner & Cancès & Hilhorst, HAL preprint no.00675681, 2012] and two IMplicit Pressure – Explicit Saturation (IMPES) schemes with different level of coupling. We finally provide numerical evidences of the good behavior of the fully decoupled version of the IMPES scheme.