# Mathematical study of a petroleum-engineering scheme

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

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## Abstract

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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.

## How to cite

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Eymard, 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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