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Simulation of multiphase porous media flows with minimising movement and finite volume schemes

Part of: Parabolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations Flows in porous media; filtration; seepage Numerical methods in calculus of variations and optimal control

Published online by Cambridge University Press:  31 October 2018

CLÉMENT CANCÈS Open the ORCID record for CLÉMENT CANCÈS [Opens in a new window] ,
THOMAS GALLOUËT ,
MAXIME LABORDE  and
LÉONARD MONSAINGEON
CLÉMENT CANCÈS*
Affiliation:
Inria, Univ. Lille, CNRS, UMR 8524 -Laboratoire Paul Painlevé, F-59000 Lille, France email: clement.cances@inria.fr
THOMAS GALLOUËT
Affiliation:
INRIA, Project team Mokaplan, Université Paris-Dauphine, PSL Research University, Ceremade, Paris, France Département de mathématiques, Universitè de Liège, Belgique email: thomas.gallouet@inria.fr
MAXIME LABORDE
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Canada email: maxime.laborde@mcgill.ca
LÉONARD MONSAINGEON
Affiliation:
IECL, Université de Lorraine, Nancy, France email: leonard.monsaingeon@univ-lorraine.fr GFM, Universidade de Lisboa, Lisbon, Portugal
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Abstract

The Wasserstein gradient flow structure of the partial differential equation system governing multiphase flows in porous media was recently highlighted in Cancès et al. [Anal. PDE10(8), 1845–1876]. The model can thus be approximated by means of the minimising movement (or JKO after Jordan, Kinderlehrer and Otto [SIAM J. Math. Anal.29(1), 1–17]) scheme that we solve thanks to the ALG2-JKO scheme proposed in Benamou et al. [ESAIM Proc. Surv.57, 1–17]. The numerical results are compared to a classical upstream mobility finite volume scheme, for which strong stability properties can be established.

Keywords

Multiphase porous media flowsWasserstein gradient flowminimising movement schemeaugmented Lagrangian methodfinite volumes

MSC classification

Primary: 35K65: Degenerate parabolic equations
Secondary: 35A15: Variational methods 49M29: Methods involving duality 65M08: Finite volume methods 76S05: Flows in porous media; filtration; seepage
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1123 - 1152
Copyright
© Cambridge University Press 2018 

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Footnotes

†

C. C. was supported by the French National Research Agency (ANR) through grant ANR-13-JS01-0007-01 (project GEOPOR) and ANR-11-LABX0007-01 (Labex CEMPI). L. M. was partially supported by the Portuguese Science Foundation through FCT grant PTDC/MAT-STA/0975/2014. T. O. G. was partially supported by the Fonds de la Recherche Scientifique – FNRS under grant MIS F.4539.16.

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