Finite volume approximation of degenerate two-phase flow model with unlimited air mobility

被引:5
作者
Andreianov, Boris [1 ]
Eymard, Robert [2 ]
Ghilani, Mustapha [3 ]
Marhraoui, Nouzha [4 ]
机构
[1] Univ Franche Comte, Lab Math Besancon, UMR 6623, CNRS, F-25030 Besancon, France
[2] Univ Paris Est, Lab Anal & Math Appl, UMR 8050, CNRS, F-77454 Champs Sur Marne 2, Marne La Vallee, France
[3] ENSAM, Equipe EMMACS, Meknes 50000, Morocco
[4] Fac Sci, Equipe EMMACS, Meknes 50000, Morocco
关键词
convergence of approximate solutions; discrete a priori estimates; finite volume method; flow in porous media; infinite mobility limit; Richards model; two-phase flow model; CONVERGENCE; SCHEME;
D O I
10.1002/num.21715
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Models of two-phase flows in porous media, used in petroleum engineering, lead to a coupled system of two equations, one elliptic and the other degenerate parabolic, with two unknowns: the saturation and the pressure. In view of applications in hydrogeology, we construct a robust finite volume scheme allowing for convergent simulations, as the ratio mu of air/liquid mobility goes to infinity. This scheme is shown to satisfy a priori estimates (the saturation is shown to remain in a fixed interval, and a discrete L2(0,T;H1(Omega)) estimate is proved for both the pressure and a function of the saturation), which are sufficient to derive the convergence of a subsequence to a weak solution of the continuous equations, as the size of the discretization tends to zero. We then show that the scheme converges to a two-phase flow model whose limit, as the mobility of the air phase tends to infinity, is the quasi-Richards equation (Eymard et al., Convergence of two phase flow to Richards model, F. Benkhaldoun, editor, Finite Volumes for Complex Applications IV, ISTE, London, 2005; Eymard et al., Discrete Cont Dynam Syst, 5 (2012) 93113), which remains available even if the gas phase is not connected with the atmospheric pressure. Numerical examples, which show that the scheme remains robust for high values of mu, are finally given. (C) 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013
引用
收藏
页码:441 / 474
页数:34
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