Multi-component Multiphase Porous Flow

  • Brian SeguinEmail author
  • Noel J. Walkington


An axiomatic foundation for models of multi-component multiphase porous flow appearing ubiquitously in the engineering literature is developed. This unifies and extends various disparate and empirical formulations appearing in the literature. Constitutive restrictions are derived from an appropriate statement of the second law of thermodynamics, and the corresponding dissipation inequalities establish stability of solutions. The convexity properties and variational structure of these models are elucidated.


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Allaire, G.: Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal. 2(3), 203–222, 1989MathSciNetzbMATHGoogle Scholar
  2. 2.
    Allaire, G.: Continuity of the Darcy’s law in the low-volume fraction limit. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)18(4), 475–499, 1991.
  3. 3.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518, 1992. MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ambrosio, L.: Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5)19, 191–246, 1995MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Mainini, E.: Infinite-dimensional porous media equations and optimal transportation. J. Evol. Equ. 10(1), 217–246, 2010. MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barenblatt, G., Zheltov, I., Kochina, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech. 24(5), 1286–1303, 1960. ADSCrossRefGoogle Scholar
  7. 7.
    Bear, J.: Dynamics of Fluids in Porous Media, Dover Civil and Mechanical Engineering Series. Dover, New York, 1988.
  8. 8.
    Beck, A.: Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with MATLAB. SIAM, Philadelphia 2014CrossRefGoogle Scholar
  9. 9.
    Bedford, A.: Theories of immiscible and structured mixtures. Int. J. Eng. Sci. 2(8), 863–960, 1983MathSciNetCrossRefGoogle Scholar
  10. 10.
    Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the monge-kantorovich mass transfer problem. Numer. Math. 84(3), 375–393, 2000. MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44, 375–417, 1991MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Computational Science and Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia 2006. CrossRefzbMATHGoogle Scholar
  13. 13.
    Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178, 1963MathSciNetCrossRefGoogle Scholar
  14. 14.
    Deseri, L., Zingales, M.: A mechanical picture of fractional-order Darcy equation. Commun. Nonlinear Sci. Numer. Simul. 20(3), 940–949, 2015. ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ewing, R.: The Mathematics of Reservoir Simulation. Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1983.
  16. 16.
    Flügge, S.: Principles of Classical Mechanics and Field Theory/Prinzipien der Klassischen Mechanik und Feldtheorie. Prinzipien der theoretischen Physik/Principles of Theoretical Physics. Springer, Berlin 1960CrossRefGoogle Scholar
  17. 17.
    Fomin, S., Chugunov, V., Hashida, T.: The effect of non-Fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2061), 2923–2939, 2005. ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Garibotti, C.R., Peszynska, M.: Upscaling non-Darcy flow. Transp. Porous Media80(3), 401, 2009. MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge 2010. CrossRefGoogle Scholar
  20. 20.
    Hassanizadeh, S.M., Gray, W.G.: High velocity flow in porous media. Transp. Porous Media2(6), 521–531, 1987. CrossRefGoogle Scholar
  21. 21.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17, 1998MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mielke, A.: Deriving effective models for multiscale systems via evolutionary \(\varGamma \)-convergence. In: Control of Self-Organizing Nonlinear Systems, pp. 235–251. Springer, Cham, 2016Google Scholar
  23. 23.
    Noll, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Ration. Mech. Anal. 52, 62–92, 1973MathSciNetCrossRefGoogle Scholar
  24. 24.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174, 2001MathSciNetCrossRefGoogle Scholar
  25. 25.
    Otto, F., Weinan, E.: Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys. 107(23), 10177–10184, 1997ADSCrossRefGoogle Scholar
  26. 26.
    Peszynska, M., Trykozko, A.: Pore-to-core simulations of flow with large velocities using continuum models and imaging data. Comput. Geosci. 17(4), 623–645, 2013. CrossRefzbMATHGoogle Scholar
  27. 27.
    Sanchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin 1980Google Scholar
  28. 28.
    Showalter, R.E.: Distributed microstructure models of porous media. In: Flow in Porous Media (Oberwolfach, 1992). International Series of Numerical Mathematics, vol. 114, pp. 155–163. Birkhäuser, Basel, 1993CrossRefGoogle Scholar
  29. 29.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence, 1997.
  30. 30.
    Showalter, R.E., Walkington, N.J.: Micro-structure models of diffusion in fissured media. J. Math. Anal. Appl. 155(1), 1–20, 1991. MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Swendsen, R.: An Introduction to Statistical Mechanics and Thermodynamics. Oxford Graduate Texts. Oxford University Press, Oxford, 2012.
  32. 32.
    Truesdell, C.: Rational Thermodyanics, 2nd edn. Springer, New York 1984CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLoyola University ChicagoChicagoUSA
  2. 2.Department of MathematicsCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations