Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 8 (2017), 1845-1876.

Incompressible immiscible multiphase flows in porous media: a variational approach

Clément Cancès, Thomas O. Gallouët, and Léonard Monsaingeon

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Abstract

We describe the competitive motion of N+1 incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of nonnegative measures with prescribed masses, endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization scheme á la R. Jordan, D. Kinderlehrer and F. Otto (SIAM J. Math. Anal. 29:1 (1998) 1–17). This allows us to obtain a new existence result for a physically well-established system of PDEs consisting of the Darcy–Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure.

Article information

Source
Anal. PDE, Volume 10, Number 8 (2017), 1845-1876.

Dates
Received: 13 July 2016
Revised: 23 May 2017
Accepted: 29 June 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843584

Digital Object Identifier
doi:10.2140/apde.2017.10.1845

Mathematical Reviews number (MathSciNet)
MR3694008

Zentralblatt MATH identifier
1370.35230

Subjects
Primary: 35K65: Degenerate parabolic equations 35A15: Variational methods 49K20: Problems involving partial differential equations 76S05: Flows in porous media; filtration; seepage

Keywords
multiphase porous media flows Wasserstein gradient flows constrained parabolic system minimizing movement scheme

Citation

Cancès, Clément; Gallouët, Thomas O.; Monsaingeon, Léonard. Incompressible immiscible multiphase flows in porous media: a variational approach. Anal. PDE 10 (2017), no. 8, 1845--1876. doi:10.2140/apde.2017.10.1845. https://projecteuclid.org/euclid.apde/1510843584


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References

  • H. W. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential equations”, Math. Z. 183:3 (1983), 311–341.
  • B. Amaziane, M. Jurak, and A. Vrbaški, “Existence for a global pressure formulation of water-gas flow in porous media”, Electron. J. Differential Equations 2012:102 (2012), 1–22.
  • B. Amaziane, M. Jurak, and A. Žgaljić Keko, “Modeling compositional compressible two-phase flow in porous media by the concept of the global pressure”, Comput. Geosci. 18:3-4 (2014), 297–309.
  • L. Ambrosio and N. Gigli, “A user's guide to optimal transport”, pp. 1–155 in Modelling and optimisation of flows on networks, Lecture Notes in Math. 2062, Springer, 2013.
  • L. Ambrosio and S. Serfaty, “A gradient flow approach to an evolution problem arising in superconductivity”, Comm. Pure Appl. Math. 61:11 (2008), 1495–1539.
  • L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd ed., Birkhäuser, Basel, 2008.
  • L. Ambrosio, E. Mainini, and S. Serfaty, “Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices”, Ann. Inst. H. Poincaré Anal. Non Linéaire 28:2 (2011), 217–246.
  • B. Andreianov, C. Cancès, and A. Moussa, “A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs”, preprint, 2015, hook https://hal.archives-ouvertes.fr/hal-01142499/document \posturlhook.
  • S. N. Antoncev and V. N. Monahov, “Three-dimensional problems of transient two-phase filtration in inhomogeneous anisotropic porous media”, Dokl. Akad. Nauk SSSR 243:3 (1978), 553–556. In Russian; translated in Soviet Math., Dokl. 19 (1978), 1354–1358.
  • J. Bear and Y. Bachmat, Introduction to modeling of transport phenomena in porous media, Springer, 1990.
  • A. Blanchet, “A gradient flow approach to the Keller–Segel systems”, RIMS Kôkyûroku 1837 (2013), 52–73.
  • A. Blanchet, V. Calvez, and J. A. Carrillo, “Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model”, SIAM J. Numer. Anal. 46:2 (2008), 691–721.
  • F. Bolley, I. Gentil, and A. Guillin, “Uniform convergence to equilibrium for granular media”, Arch. Ration. Mech. Anal. 208:2 (2013), 429–445.
  • C. Cancès and T. Gallouët, “On the time continuity of entropy solutions”, J. Evol. Equ. 11:1 (2011), 43–55.
  • C. Cancès, T. O. Gallouët, and L. Monsaingeon, “The gradient flow structure for incompressible immiscible two-phase flows in porous media”, C. R. Math. Acad. Sci. Paris 353:11 (2015), 985–989.
  • G. Carlier and M. Laborde, “On systems of continuity equations with nonlinear diffusion and nonlocal drifts”, preprint, 2015.
  • J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, and D. Slepčev, “Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations”, Duke Math. J. 156:2 (2011), 229–271.
  • G. Chavent, “A new formulation of diphasic incompressible flows in porous media”, pp. 258–270 in Applications of methods of functional analysis to problems in mechanics (Marseille, 1975), Lecture Notes in Math. 503, Springer, 1976.
  • G. Chavent, “A fully equivalent global pressure formulation for three-phases compressible flows”, Appl. Anal. 88:10-11 (2009), 1527–1541.
  • G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation, Studies in Mathematics and its Applications 17, North Holland, Amsterdam, 1986.
  • G. Chavent and G. Salzano, “Un algorithme pour la détermination de perméabilités relatives triphasiques satisfaisant une condition de différentielle totale”, INRIA Technical Report 335, 1985, hook https://hal.inria.fr/inria-00076202v1 \posturlhook.
  • Z. Chen, “Degenerate two-phase incompressible flow, I: Existence, uniqueness and regularity of a weak solution”, J. Differential Equations 171:2 (2001), 203–232.
  • F. H. Clarke, Optimization and nonsmooth analysis, 2nd ed., Classics in Applied Mathematics 5, SIAM, Philadelphia, PA, 1990.
  • E. De Giorgi, “New problems on minimizing movements”, pp. 81–98 in Boundary value problems for partial differential equations and applications, edited by J.-L. Lions and C. Baiocchi, RMA Res. Notes Appl. Math. 29, Masson, Paris, 1993.
  • J. Dolbeault, B. Nazaret, and G. Savaré, “A new class of transport distances between measures”, Calc. Var. Partial Differential Equations 34:2 (2009), 193–231.
  • P. Fabrie and M. Saad, “Existence de solutions faibles pour un modèle d'écoulement triphasique en milieu poreux”, Ann. Fac. Sci. Toulouse Math. $(6)$ 2:3 (1993), 337–373.
  • G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière, Mathématiques & Applications (Berlin) 22, Springer, 1996.
  • N. Gigli and F. Otto, “Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric”, Calc. Var. Partial Differential Equations 47:1-2 (2013), 181–206.
  • H. Hanche-Olsen and H. Holden, “The Kolmogorov–Riesz compactness theorem”, Expo. Math. 28:4 (2010), 385–394.
  • R. Jordan, D. Kinderlehrer, and F. Otto, “The variational formulation of the Fokker–Planck equation”, SIAM J. Math. Anal. 29:1 (1998), 1–17.
  • D. Kinderlehrer, L. Monsaingeon, and X. Xu, “A Wasserstein gradient flow approach to Poisson–Nernst–Planck equations”, ESAIM Control Optim. Calc. Var. 23:1 (2017), 137–164.
  • M. Laborde, Systèmes de particules en interaction, approche par flot de gradient dans l'espace de Wasserstein, Ph.D. thesis, Université Paris-Dauphine, 2016, hook https://basepub.dauphine.fr/handle/123456789/16518 \posturlhook.
  • O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs 23, Amer. Math. Soc., Providence, RI, 1968.
  • P. Laurençot and B.-V. Matioc, “A gradient flow approach to a thin film approximation of the Muskat problem”, Calc. Var. Partial Differential Equations 47:1-2 (2013), 319–341.
  • E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics 14, Amer. Math. Soc., Providence, RI, 2001.
  • S. Lisini, “Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces”, ESAIM Control Optim. Calc. Var. 15:3 (2009), 712–740.
  • S. Lisini, D. Matthes, and G. Savaré, “Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics”, J. Differential Equations 253:2 (2012), 814–850.
  • D. Matthes, R. J. McCann, and G. Savaré, “A family of nonlinear fourth order equations of gradient flow type”, Comm. Partial Differential Equations 34:10-12 (2009), 1352–1397.
  • B. Maury, A. Roudneff-Chupin, and F. Santambrogio, “A macroscopic crowd motion model of gradient flow type”, Math. Models Methods Appl. Sci. 20:10 (2010), 1787–1821.
  • A. Moussa, “Some variants of the classical Aubin–Lions lemma”, J. Evol. Equ. 16:1 (2016), 65–93.
  • F. Otto, “Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory”, Arch. Rational Mech. Anal. 141:1 (1998), 63–103.
  • F. Otto, “The geometry of dissipative evolution equations: the porous medium equation”, Comm. Partial Differential Equations 26:1-2 (2001), 101–174.
  • E. Sandier and S. Serfaty, “Gamma-convergence of gradient flows with applications to Ginzburg–Landau”, Comm. Pure Appl. Math. 57:12 (2004), 1627–1672.
  • F. Santambrogio, Optimal transport for applied mathematicians; calculus of variations, PDEs, and modeling, Progress in Nonlinear Differential Equations and their Applications 87, Springer, 2015.
  • C. Villani, Optimal transport: old and new, Grundlehren der Mathematischen Wissenschaften 338, Springer, 2009.
  • J. Zinsl, “Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure”, Monatsh. Math. 174:4 (2014), 653–679.
  • J. Zinsl and D. Matthes, “Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis”, Anal. PDE 8:2 (2015), 425–466.
  • J. Zinsl and D. Matthes, “Transport distances and geodesic convexity for systems of degenerate diffusion equations”, Calc. Var. Partial Differential Equations 54:4 (2015), 3397–3438.