Mathematical study of a petroleum-engineering scheme
Université de Marne-la-Vallée, 5 Bld Descartes, Bat. Lavoisier, 77454 Marne-la-Vallée, France.
2 Université de Aix-Marseille 1, 39 rue Joliot Curie, 13453 Marseille, France. firstname.lastname@example.org.
3 Institut Français du Pétrole, 1 et 4 avenue Bois Préau, 92000 Rueil-Malmaison, France. email@example.com.
Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete L2(0,T;H1(Ω)) estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.
Mathematics Subject Classification: 35K65 / 76S05 / 65M12
Key words: Multiphase flow / Darcy's law / porous media / finite volume scheme.
© EDP Sciences, SMAI, 2003