Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-12T19:45:33.903Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: [email protected]
THOMAS GALLOUËT
Affiliation:
INRIA, Project team Mokaplan, Université Paris-Dauphine, PSL Research University, Ceremade, Paris, France Département de mathématiques, Universitè de Liège, Belgique email: [email protected]
MAXIME LABORDE
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Canada email: [email protected]
LÉONARD MONSAINGEON
Affiliation:
IECL, Université de Lorraine, Nancy, France email: [email protected] GFM, Universidade de Lisboa, Lisbon, Portugal
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Ait Hammou Oulhaj, A. (2018) Numerical analysis of a finite volume scheme for a seawater intrusion model with cross-diffusion in an unconfined aquifer. Numer. Methods Partial Differ. Equ. 34(3), 857–880.CrossRefGoogle Scholar
Ambrosio, L., Gigli, N. & Savaré, G. (2008) Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, Basel.Google Scholar
Bear, J. & Bachmat, Y. (1990) Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht, The Netherlands.CrossRefGoogle Scholar
Benamou, J.-D. & Brenier, Y. (2000) A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393.CrossRefGoogle Scholar
Benamou, J.-D., Brenier, Y. & Guittet, K. (2004) Numerical analysis of a multi-phasic mass transport problem. In: Recent Advances in the Theory and Applications of Mass Transport. Contemporary Mathematics, Vol. 353. American Mathematical Society, Providence, RI, pp. 1–17.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G. & Laborde, M. (2016) An augmented Lagrangian approach to Wasserstein gradient flows and applications. In: Gradient Flows: From Theory to Application. ESAIM Proceedings and Surveys, Vol. 54, EDP Sciences, Les Ulis, pp. 1–17.Google Scholar
Brenier, Y. & Jaffré, J. (1991) Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28(3), 685–696.CrossRefGoogle Scholar
Brenier, Y. & Puel, M. (2002) Optimal multiphase transportation with prescribed momentum. ESAIM Control Optim. Calc. Var. 8, 287–343.CrossRefGoogle Scholar
Cancès, C. (2018) Energy stable numerical methods for porous media flow type problems. https://hal.archives-ouvertes.fr/hal-01719502/document.CrossRefGoogle Scholar
Cancès, C., Gallouët, T. O. & Monsaingeon, L. (2015) The gradient flow structure of immiscible incompressible two-phase flows in porous media. C. R. Acad. Sci. Paris Sér. I Math. 353, 985–989.CrossRefGoogle Scholar
Cancès, C., Gallouët, T. O. & Monsaingeon, L. (2017) Incompressible immiscible multiphase flows in porous media: a variational approach. Anal. PDE 10(8), 1845–1876.CrossRefGoogle Scholar
Cancès, C., Matthes, D. & Nabet, F. (2017) A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow. https://hal.archives-ouvertes.fr/hal-01665338/document.Google Scholar
Cancès, C. & Nabet, F. (2017) Finite volume approximation of a degenerate immiscible two-phase flow model of Cahn–Hilliard type. In: Cancès, C., and Omnes, P. (editors), Finite Volumes for Complex Applications VIII – Methods and Theoretical Aspects: FVCA 8, Lille, France, June 2017. Proceedings in Mathematics and Statistics, vol. 199, Springer International Publishing, Cham, pp. 431–438.CrossRefGoogle Scholar
Darcy, H. (1856) Les fontaines publiques de la ville de Dijon, Dalmont, Paris.Google Scholar
De Giorgi, E. (1993) New problems on minimising movements. In: Boundary Value Problems for Partial Differential Equations and Applications. RMA Research Notes in Applied Mathematics, Vol. 29, Masson, Paris, pp. 81–98.Google Scholar
Deimling, K. (1985) Nonlinear Functional Analysis, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Dolbeault, J., Nazaret, B. & Savaré, G. (2009) A new class of transport distances between measures. Calc. Var. Partial Differ. Equ. 34(2), 193–231.CrossRefGoogle Scholar
Droniou, J., Eymard, R., Gallouët, T., Guichard, C. & Herbin, R. (2016) The gradient discretisation method. A framework for the discretisation and numerical analysis of linear and non-linear elliptic and parabolic problems, Springer International Publishing, Cham, Switzerland.Google Scholar
Erbar, M., Kuwada, K. & Sturm, K.-T. (2015) On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. Math. 201(3), 993–1071.CrossRefGoogle Scholar
Eymard, R., Gallouët, T. & Herbin, R. (2000) Finite volume methods. In: Ciarlet, P. G., et al. (editors), Handbook of Numerical Analysis, North-Holland, Amsterdam, pp. 713–1020.Google Scholar
Eymard, R., Herbin, R. & Michel, A. (2003) Mathematical study of a petroleum-engineering scheme. M2AN Math. Model. Numer. Anal. 37(6), 937–972.CrossRefGoogle Scholar
Fortin, M. & Glowinski, R. (1983) Augmented Lagrangian Methods. Studies in Mathematics and its Applications, Vol. 15, North-Holland Publishing Co., Amsterdam. Applications to the numerical solution of boundary value problems, Translated from the French by B. Hunt and D. C. Spicer.Google Scholar
Gallouët, T., Laborde, M. & Monsaingeon, L. (2017) An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems. ESAIM: COCV.Google Scholar
Hecht, F. (2012) New development in FreeFEM++. J. Numer. Math. 20(3-4), 251–265.CrossRefGoogle Scholar
Jordan, R., Kinderlehrer, D. & Otto, F. (1998) The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17.CrossRefGoogle Scholar
Legendre, G. & Turinici, G. (2017) Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces. C. R. Acad. Sci. Paris Sér. I Math. 353(3), 345–353.CrossRefGoogle Scholar
Lisini, S. (2009) Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM Control Optim. Calc. Var. 15(3), 712–740.CrossRefGoogle Scholar
Loeschcke, C. (2012) On the Relaxation of a Variational Principle for the Motion of a Vortex Sheet in Perfect Fluid. PhD thesis, University of Bonn.Google Scholar
Matthes, D., McCann, R. J. & Savaré, G. (2009) A family of nonlinear fourth order equations of gradient flow type. Comm. Partial Differ Equ. 34(11), 1352–1397.CrossRefGoogle Scholar
Matthes, D. & Plazotta, S. (2017) A Variational Formulation of the BDF2 Method for Metric Gradient Flows. https://arxiv.org/pdf/1711.02935.pdf.Google Scholar
McCann, R. J. (1997) A convexity principle for interacting gases. Adv. Math. 128(1), 153–179.CrossRefGoogle Scholar
Otto, F. & E, W. (1997) Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys. 107(23), 10177–10184.CrossRefGoogle Scholar
Papadakis, N., Peyré, G. & Oudet, E. (2014) Optimal transport with proximal splitting. SIAM J. Imaging Sci. 7(1), 212–238.CrossRefGoogle Scholar
Peaceman, D. W. (1977) Fundamentals of Numerical Reservoir Simulation. Developments in Petroleum Science, Vol. 6, Elsevier.CrossRefGoogle Scholar
Santambrogio, F. (2015) Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, 1st ed. Progress in Nonlinear Differential Equations and Their Applications, Vol. 87, Birkhäuser, Basel.Google Scholar
Santambrogio, F. (2017) Euclidean, metric, and Wasserstein gradient flows: an overview. Bull. Math. Sci. 7(1), 87–154.CrossRefGoogle Scholar
Villani, C. (2009) Optimal Transport. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin. Old and new.CrossRefGoogle Scholar