# Mathematical study of a petroleum-engineering scheme

Robert Eymard; Raphaèle Herbin; Anthony Michel

- Volume: 37, Issue: 6, page 937-972
- ISSN: 0764-583X

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topEymard, Robert, Herbin, Raphaèle, and Michel, Anthony. "Mathematical study of a petroleum-engineering scheme." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.6 (2003): 937-972. <http://eudml.org/doc/245709>.

@article{Eymard2003,

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 $L^2(0,T;H^1(Ø))$ 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.},

author = {Eymard, Robert, Herbin, Raphaèle, Michel, Anthony},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {multiphase flow; Darcy’s law; porous media; finite volume scheme; Multiphase flow; Darcy's law},

language = {eng},

number = {6},

pages = {937-972},

publisher = {EDP-Sciences},

title = {Mathematical study of a petroleum-engineering scheme},

url = {http://eudml.org/doc/245709},

volume = {37},

year = {2003},

}

TY - JOUR

AU - Eymard, Robert

AU - Herbin, Raphaèle

AU - Michel, Anthony

TI - Mathematical study of a petroleum-engineering scheme

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2003

PB - EDP-Sciences

VL - 37

IS - 6

SP - 937

EP - 972

AB - 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.

LA - eng

KW - multiphase flow; Darcy’s law; porous media; finite volume scheme; Multiphase flow; Darcy's law

UR - http://eudml.org/doc/245709

ER -

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