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

Authors

  • Boris Andreianov,

    1. Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France
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  • Robert Eymard,

    Corresponding author
    1. Laboratoire d'Analyse et de Mathématiques Appliquées UMR CNRS 8050, Université Paris-Est, 5, Boulevard Descartes, Champs-sur-Marne, 77454 Marne La Vallée Cedex 2, France
    • Laboratoire d'Analyse et de Mathématiques Appliquées UMR CNRS 8050, Université Paris-Est, 5, Boulevard Descartes, Champs-sur-Marne, 77454 Marne La Vallée Cedex 2, France
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  • Mustapha Ghilani,

    1. Equipe EMMACS, ENSAM, BP 4024 Bni M'hamed, 50 000 Meknès, Morocco
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  • Nouzha Marhraoui

    1. Equipe EMMACS, Faculté des Sciences, BP 11201 Zitoune, 50 000 Meknès, Morocco
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Abstract

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 μ 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(Ω)) 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) 93–113), 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 μ, are finally given. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

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