Comptes Rendus
no. 9-10
Numerical Analysis
Accurate velocity reconstruction for Discontinuous Galerkin approximations of two-phase porous media flows
[Reconstruction précise de la vitesse pour des approximations par la méthode de Galerkine discontinue d'écoulements diphasiques en milieu poreux]

Présenté par : Olivier Pironneau

Alexandre Ern1 ; Igor Mozolevski1, 2 ; L. Schuh3
1 Université Paris-Est, CERMICS, École des ponts, 6 & 8, avenue B. Pascal, 77455 Marne-la-Vallée cedex 2, France
2 Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis-SC, Brasil
3 IME, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo, SP, Brasil
Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 551-554.

Nous considérons une méthode de Galerkine discontinue pour approcher les écoulements diphasiques non-miscibles en milieu poreux dans la formulation en pression globale. Une approche séquentielle est utilisée avec un schéma d'Euler implicite pour l'équation de la saturation, le même ordre polynomial pour la pression et la saturation, et en l'absence de limiteurs. Nous montrons comment reconstruire à partir de l'équation en pression une vitesse totale précise pour l'équation de la saturation. Des exemples numériques illustrent les avantages de l'approche proposée.

We consider Discontinuous Galerkin approximations of two-phase, immiscible porous media flows in the global pressure/fractional flow formulation with capillary pressure. A sequential approach is used with a backward Euler step for the saturation equation, equal-order interpolation for the pressure and the saturation, and without any limiters. An accurate total velocity field is recovered from the global pressure equation to be used in the saturation equation. Numerical experiments show the advantages of the proposed reconstruction.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.02.011
Alexandre Ern 1 ; Igor Mozolevski 1, 2 ; L. Schuh 3

1 Université Paris-Est, CERMICS, École des ponts, 6 & 8, avenue B. Pascal, 77455 Marne-la-Vallée cedex 2, France
2 Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis-SC, Brasil
3 IME, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo, SP, Brasil 
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     author = {Alexandre Ern and Igor Mozolevski and L. Schuh},
     title = {Accurate velocity reconstruction for {Discontinuous} {Galerkin} approximations of two-phase porous media flows},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {551--554},
     publisher = {Elsevier},
     volume = {347},
     number = {9-10},
     year = {2009},
     doi = {10.1016/j.crma.2009.02.011},
     language = {en},
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Alexandre Ern; Igor Mozolevski; L. Schuh. Accurate velocity reconstruction for Discontinuous Galerkin approximations of two-phase porous media flows. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 551-554. doi : 10.1016/j.crma.2009.02.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.011/

[1] P. Bastian; B. Rivière Superconvergence and H(div) projection for discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids, Volume 42 (2003) no. 10, pp. 1043-1057

[2] P. Bastian, B. Rivière, Discontinuous Galerkin methods for two-phase flow in porous media, Technical Report 2004-28, 2004

[3] G. Chavent; J. Jaffré Mathematical Models and Finite Elements for Reservoir Simulation, Elsevier, North-Holland, 1986

[4] Z. Chen; R.E. Ewing Degenerate two-phase incompressible flow, Numer. Math., Volume 90 (2001), pp. 215-240

[5] Y. Epshteyn; B. Rivière Fully implicit discontinuous finite element methods for two-phase flow, Appl. Numer. Math., Volume 57 (2007), pp. 383-401

[6] A. Ern; S. Nicaise; M. Vohralík An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 12, pp. 709-712

[7] R. Eymard; R. Herbin; A. Michel Mathematical study of a petroleum-engineering scheme, Math. Modell. Numerical Anal., Volume 37 (2003) no. 6, pp. 937-972

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