Mathematical study of a petroleum-engineering scheme
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, p. 937-972

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 L 2 (0,T;H 1 (Ø)) 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.

Classification:  35K65,  76S05,  65M12
Keywords: multiphase flow, Darcy's law, porous media, finite volume scheme
     author = {Eymard, Robert and Herbin, Rapha\`ele and Michel, Anthony},
     title = {Mathematical study of a petroleum-engineering scheme},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {6},
     year = {2003},
     pages = {937-972},
     doi = {10.1051/m2an:2003062},
     zbl = {1118.76355},
     mrnumber = {2026403},
     language = {en},
     url = {}
Eymard, Robert; Herbin, Raphaèle; Michel, Anthony. Mathematical study of a petroleum-engineering scheme. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, pp. 937-972. doi : 10.1051/m2an:2003062.

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