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 ${L}^{2}\left(0,T;{H}^{1}\left(Ø\right)\right)$ 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.

DOI : https://doi.org/10.1051/m2an:2003062
Classification:  35K65,  76S05,  65M12
Keywords: multiphase flow, Darcy's law, porous media, finite volume scheme
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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 = {http://www.numdam.org/item/M2AN_2003__37_6_937_0}
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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. http://www.numdam.org/item/M2AN_2003__37_6_937_0/`

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