Comptes Rendus
An existence result for the equations describing a gas–liquid two-phase flow
Comptes Rendus. Mécanique, Volume 337 (2009) no. 4, pp. 226-232.

We consider the immiscible two-phase mixture of water and hydrogen in a porous medium. The water phase is incompressible and the hydrogen phase is compressible. The hydrogen dissolves in the water. The flow is described by the system of non-linear evolution equations for the water saturation and the hydrogen pressure. Under non-degeneracy and slow oscillation assumptions on the diagonal coefficients and with small data for the hydrogen, we establish the existence of a weak solution.

Nous considérons le mélange diphasique immiscible de l'eau et de l'hydrogène dans un milieu poreux. L'eau est incompressible et l'hydrogène est compressible. L'hydrogène se dissout dans l'eau. L'écoulement est décrit par le système des équations non linéaires d'évolution pour la saturation de l'eau et la pression d'hydrogène. Sous les conditions de la non-dégénérescence et des petites oscillations des coefficients diagonaux et avec de petites données pour l'hydrogène, nous établissons l'existence d'une solution faible.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2009.04.007
Keywords: Porous media, Two-phase flow, Compressible gas phase, Existence of a solution
Mot clés : Milieux poreux, Écoulement diphasique, Phase gazeuse compressible, Existence d'une solution

Andro Mikelić 1, 2

1 Université de Lyon, 69003 Lyon, France
2 Université Lyon 1, Institut Camille-Jordan, UMR 5208, UFR Mathématiques, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex 07, France
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Andro Mikelić. An existence result for the equations describing a gas–liquid two-phase flow. Comptes Rendus. Mécanique, Volume 337 (2009) no. 4, pp. 226-232. doi : 10.1016/j.crme.2009.04.007. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.04.007/

[1] S.N. Antontsev; A.V. Kazhikhov; V.N. Monakhov Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and its Applications, vol. 22, North-Holland Publishing Co., Amsterdam, 1990

[2] G. Chavent; J. Jaffré Mathematical Models and Finite Elements for Reservoir Simulation. Single Phase, Multiphase and Multicomponent Flows through Porous Media, North-Holland, 1986

[3] G. Gagneux; M. Madaune-Tort Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière, Mathématiques et Applications, vol. 22, Springer-Verlag, Berlin, 1996

[4] D. Kröner; S. Luckhaus Flow of oil and water in a porous medium, J. Differential Equations, Volume 55 (1984) no. 2, pp. 276-288

[5] H.W. Alt; E. DiBenedetto Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 12 (1985) no. 3, pp. 335-392

[6] C. Galusinski; M. Saad On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media, Adv. Differential Equations, Volume 9 (2004) no. 11–12, pp. 1235-1278

[7] R.B. Bird; W.E. Stewart; E.N. Lightfoot Transport Phenomena, Wiley, New York, 1960

[8] M.B. Allen Numerical modeling of multiphase flow in porous media, Adv. Water Resources, Volume 8 (1985), pp. 162-187

[9] J. Talandier Présentation d'un benchmark sur la simulation des écoulements biphasiques en milieux poreux : application au transfert des gaz autour du stockage de déchets radioactifs http://www.andra.fr/interne.php3?id_article=913&id_rubrique=76 (Benchmark presented on the web page)

[10] A. Bourgeat, M. Jurak, Models for two-phase partially miscible flow, published online in Comput. Geosci. on August 30, 2008

[11] H.W. Alt; S. Luckhaus Quasilinear elliptic–parabolic differential equations, Math. Z., Volume 183 (1983) no. 3, pp. 311-341

[12] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam–New York, 1978

[13] O.A. Ladyženskaja; V.A. Solonnikov; N.N. Ural'ceva Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1967

[14] A. Mikelić, An existence result for the equations describing a gas–liquid two-phase flow, article in preparation

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