Partial differential equations/Mathematical problems in mechanics
The gradient flow structure for incompressible immiscible two-phase flows in porous media
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 985-989.

We show that the widely used model governing the motion of two incompressible immiscible fluids in a possibly heterogeneous porous medium has a formal gradient-flow structure. More precisely, the fluid composition is governed by the gradient flow of some non-smooth energy. Starting from this energy together with a dissipation potential, we recover the celebrated Darcy–Muskat law and the capillary pressure law governing the flow thanks to the steepest descent condition for the energy. Our interpretation does not require the introduction of any algebraic transformation like, e.g., the global pressure or the Kirchhoff transform, and can be transposed to the case of more phases.

Nous montrons qu'un modèle très couramment utilisé dans l'industrie pour décrire un écoulement diphasique incompressible et immiscible dans un milieux poreux possiblement hétérogène possède une structure de flot gradient. Plus précisément, la composition du fluide est gouvernée par le flot gradient d'une énergie singulière. En partant de cette énergie et d'un potentiel de dissipation, nous retrouvons les lois de Darcy–Muskat et de pression capillaire gouvernant l'écoulement à l'aide d'un principe de moindre dissipation de l'énergie. Notre interprétation ne nécessite pas l'introduction d'une transformation algébrique du type pression globale ou transformée de Kirchhoff, ce qui permet son extension à un nombre plus grand de phases.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.021
Cancès, Clément 1; Gallouët, Thomas O. 2; Monsaingeon, Léonard 3

1 Team RAPSODI, Inria Lille–Nord Europe, 40, avenue Halley, 59650 Villeneuve-d'Ascq, France
2 CMLS, UMR 7640, École polytechnique, 91128 Palaiseau cedex, France
3 CAMGSD, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
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Cancès, Clément; Gallouët, Thomas O.; Monsaingeon, Léonard. The gradient flow structure for incompressible immiscible two-phase flows in porous media. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 985-989. doi : 10.1016/j.crma.2015.09.021. http://www.numdam.org/articles/10.1016/j.crma.2015.09.021/

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