A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities

• Published in: IMA journal of numerical analysis 2018-04, Vol.38 (2), p.884-920
• Main Authors: Radu, Florin A, Kumar, Kundan, Nordbotten, Jan M, Pop, Iuliu S
• Format: Article
• Language: English
• Subjects: Matematik; Mathematics
• ISSN: 0272-4979; 1464-3642; 1464-3642
• DOI: 10.1093/imanum/drx032

Abstract In this work, we present a mass conservative numerical scheme for two-phase flow in porous media. The model for flow consists of two fully coupled, nonlinear equations: a degenerate parabolic equation and an elliptic one. The proposed numerical scheme is based on backward Euler for the temporal discretization and mixed finite element method for the spatial one. A priori stability and error estimates are presented to prove the convergence of the scheme. A monotone increasing, Hölder continuous saturation is considered. The convergence of the scheme is naturally dependant on the Hölder exponent. The nonlinear systems within each time step are solved by a robust linearization method, called the $L$-scheme. This iterative method does not involve any regularization step. The convergence of the $L$-scheme is rigorously proved under the assumption of a Lipschitz continuous saturation. For the Hölder continuous case, a numerical convergence is established. Numerical results (two-dimensional and three-dimensional) are presented to sustain the theoretical findings.