Journal of Elasticity

, Volume 135, Issue 1–2, pp 485–507 | Cite as

Multi-Component Multiphase Flow Through a Poroelastic Medium

  • Brian SeguinEmail author
  • Noel J. Walkington


An axiomatic development for a continuum description of a multi-component multiphase porous flow in an elastic medium is developed. The Coleman–Noll procedure is used to derive constitutive restrictions which guarantee that the resulting model satisfies an appropriate statement of the second law of thermodynamics and a corresponding dissipation inequality. Many of the models and formulations appearing in the engineering literature are shown to be special cases of the model developed here.


Poroelasticity Multiphase flow Biot theory Thermodynamics 

Mathematics Subject Classification

76S05 74F10 80A17 



  1. 1.
    Alt, H.W., DiBenedetto, E.: Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 12(3), 335–392 (1985). MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1972) zbMATHGoogle Scholar
  3. 3.
    Biot, M.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941) ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Biot, M.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185 (1955) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biot, M.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 78, 91–96 (1956) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Biot, M., Willis, D.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601 (1957) MathSciNetGoogle Scholar
  7. 7.
    Chen, Z.: Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differ. Equ. 171(2), 203–232 (2001). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z., Ewing, R.E.: Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90(2), 215–240 (2001). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Computational Science & Engineering SIAM, Philadelphia (2006). CrossRefzbMATHGoogle Scholar
  10. 10.
    Cheng, A.H.: Poroelasticity. Theory and Applications of Transport in Porous Media, vol. 27. Springer, Berlin (2016) zbMATHGoogle Scholar
  11. 11.
    Cimmelli, V.A., Jou, D., Ruggeri, T., Ván, P.: Entropy principle and recent results in non-equilibrium theories. Entropy 16(3), 1756–1807 (2014). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Darcy, H.: Recherches expérimentales relatives au mouvement de l’eau dans les tuyaux. Mallet-Bachelier, Paris (1857) Google Scholar
  14. 14.
    DiBenedetto, E., Showalter, R.E.: Implicit degenerate evolution equations and applications. SIAM J. Math. Anal. 12(5), 731–751 (1981). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dinariev, O., Evseev, N.: Multiphase flow modeling with density functional method. Comput. Geosci. 20(4), 835–856 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ewing, R.: The Mathematics of Reservoir Simulation. Frontiers in Applied Mathematics. SIAM, Philadelphia (1983). CrossRefzbMATHGoogle Scholar
  17. 17.
    Ganis, B., Kumar, K., Pencheva, G., Wheeler, M.F., Yotov, I.: A global Jacobian method for mortar discretizations of a fully implicit two-phase flow model. Multiscale Model. Simul. 12(4), 1401–1423 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gibson, N.L., Medina, F.P., Peszynska, M., Showalter, R.E.: Evolution of phase transitions in methane hydrate. J. Math. Anal. Appl. 409(2), 816–833 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kroener, D., Luckhaus, S.: Flow of oil and water in a porous medium. J. Differ. Equ. 55(2), 276–288 (1984). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lauser, A., Hager, C., Helmig, R., Wohlmuth, B.: A new approach for phase transitions in miscible multi-phase flow in porous media. Adv. Water Resour. 34, 957–966 (2011). ADSCrossRefGoogle Scholar
  21. 21.
    Noll, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Ration. Mech. Anal. 52, 62–92 (1973) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Peszyńska, M., Lu, Q., Wheeler, M.F.: Coupling different numerical algorithms for two phase fluid flow. In: The Mathematics of Finite Elements and Applications, X, MAFELAP 1999, Uxbridge, pp. 205–214. Elsevier, Oxford (2000). Google Scholar
  23. 23.
    Peszynska, M., Showalter, R.E., Webster, J.T.: Advection of methane in the hydrate zone: model, analysis and examples. Math. Methods Appl. Sci. 38(18), 4613–4629 (2015). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rice, J.R., Cleary Michael, P.: Some basic stress diffusion solutions for fluid saturated elastic porous media with compressible constituents. Rev. Geophys. 14(2), 227–241 (1976). ADSCrossRefGoogle Scholar
  25. 25.
    Sanchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin (1980) zbMATHGoogle Scholar
  26. 26.
    Seguin, B., Walkington, N.J.: Multi-component multiphase flow. Arch. Ration. Mech. Anal. (2017, submitted) Google Scholar
  27. 27.
    Showalter, R.E.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251(1), 310–340 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Swendsen, R.: An Introduction to Statistical Mechanics and Thermodynamics. Oxford Graduate Texts. Oxford University Press, Oxford (2012). CrossRefzbMATHGoogle Scholar
  29. 29.
    Truesdell, C.: Rational Thermodynamics. McGraw-Hill, New York (1969). A course of lectures on selected topics, with an appendix on the symmetry of the heat-conduction tensor by C.C. Wang zbMATHGoogle Scholar
  30. 30.
    Verruijt, A.: An Introduction to Soil Mechanics. Springer, Berlin (2010) Google Scholar
  31. 31.
    Visintin, A.: Models of Phase Transitions. Progress in Nonlinear Differential Equations and Their Applications, vol. 28. Birkhäuser, Boston (1996). CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

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

Personalised recommendations