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 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.
Mots-clés : multiphase flow, Darcy's law, porous media, finite volume scheme
@article{M2AN_2003__37_6_937_0,
author = {Eymard, Robert and Herbin, Rapha\`ele and Michel, Anthony},
title = {Mathematical study of a petroleum-engineering scheme},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {937--972},
publisher = {EDP-Sciences},
volume = {37},
number = {6},
year = {2003},
doi = {10.1051/m2an:2003062},
mrnumber = {2026403},
zbl = {1118.76355},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003062/}
}
TY - JOUR AU - Eymard, Robert AU - Herbin, Raphaèle AU - Michel, Anthony TI - Mathematical study of a petroleum-engineering scheme JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 937 EP - 972 VL - 37 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003062/ DO - 10.1051/m2an:2003062 LA - en ID - M2AN_2003__37_6_937_0 ER -
%0 Journal Article %A Eymard, Robert %A Herbin, Raphaèle %A Michel, Anthony %T Mathematical study of a petroleum-engineering scheme %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 937-972 %V 37 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003062/ %R 10.1051/m2an:2003062 %G en %F M2AN_2003__37_6_937_0
Eymard, Robert; Herbin, Raphaèle; Michel, Anthony. Mathematical study of a petroleum-engineering scheme. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 937-972. doi : 10.1051/m2an:2003062. http://www.numdam.org/articles/10.1051/m2an:2003062/
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