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Mathematical study of a petroleum-engineering scheme

Published online by Cambridge University Press:  15 November 2003

Robert Eymard ,
Raphaèle Herbin  and
Anthony Michel
Robert Eymard
Affiliation:
Université de Marne-la-Vallée, 5 Bld Descartes, Bat. Lavoisier, 77454 Marne-la-Vallée, France. eymard@math.univ-mlv.fr.
Raphaèle Herbin
Affiliation:
Université de Aix-Marseille 1, 39 rue Joliot Curie, 13453 Marseille, France. herbin@cmi.univ-mrs.fr.
Anthony Michel
Affiliation:
Institut Français du Pétrole, 1 et 4 avenue Bois Préau, 92000 Rueil-Malmaison, France. anthony.michel@ifp.fr.
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Abstract

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.

Keywords

Multiphase flowDarcy's lawporous mediafinite volume scheme.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 6 , November 2003 , pp. 937 - 972
Copyright
© EDP Sciences, SMAI, 2003

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