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Multi-Component Multiphase Flow Through a Poroelastic Medium

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Abstract

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.

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Notes

  1. In porous flow contexts \(\{m_{c}\}_{c=1}^{N_{c}}\) denote mass densities of components and \(\{\rho _{\pi }\}_{\pi =1}^{N_{p}}\) are the mass densities of the phases.

  2. Here we assume that the stresses are symmetric. In more general mixture theory, it is possible for stresses to be nonsymmetric. In this case one must postulate a torque balance for each phase. However, if for each phase there is no external supply of torques, then torque balance implies that the stresses are symmetric. See Truesdell [29] for details.

  3. One could also consider small changes in \(e_{R}\), but here this is not done so we can compare the result with the traditional Biot theory, which is isothermal.

References

  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). http://www.numdam.org/item?id=ASNSP_1985_4_12_3_335_0

    MathSciNet  MATH  Google Scholar 

  2. Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1972)

    MATH  Google Scholar 

  3. Biot, M.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  ADS  MATH  Google Scholar 

  4. Biot, M.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185 (1955)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Biot, M.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 78, 91–96 (1956)

    MathSciNet  MATH  Google Scholar 

  6. Biot, M., Willis, D.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601 (1957)

    MathSciNet  Google Scholar 

  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). https://doi.org/10.1006/jdeq.2000.3848

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Chen, Z., Ewing, R.E.: Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90(2), 215–240 (2001). https://doi.org/10.1007/s002110100291

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Computational Science & Engineering SIAM, Philadelphia (2006). https://doi.org/10.1137/1.9780898718942

    Book  MATH  Google Scholar 

  10. Cheng, A.H.: Poroelasticity. Theory and Applications of Transport in Porous Media, vol. 27. Springer, Berlin (2016)

    MATH  Google Scholar 

  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). https://doi.org/10.3390/e16031756

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  13. Darcy, H.: Recherches expérimentales relatives au mouvement de l’eau dans les tuyaux. Mallet-Bachelier, Paris (1857)

    Google Scholar 

  14. DiBenedetto, E., Showalter, R.E.: Implicit degenerate evolution equations and applications. SIAM J. Math. Anal. 12(5), 731–751 (1981). https://doi.org/10.1137/0512062

    Article  MathSciNet  MATH  Google Scholar 

  15. Dinariev, O., Evseev, N.: Multiphase flow modeling with density functional method. Comput. Geosci. 20(4), 835–856 (2016). https://doi.org/10.1007/s10596-015-9527-2

    Article  MathSciNet  MATH  Google Scholar 

  16. Ewing, R.: The Mathematics of Reservoir Simulation. Frontiers in Applied Mathematics. SIAM, Philadelphia (1983). https://books.google.com/books?id=k_4Y-RBOK9oC

    Book  MATH  Google Scholar 

  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). https://doi.org/10.1137/140952922

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.jmaa.2013.07.023

    Article  MathSciNet  MATH  Google Scholar 

  19. Kroener, D., Luckhaus, S.: Flow of oil and water in a porous medium. J. Differ. Equ. 55(2), 276–288 (1984). https://doi.org/10.1016/0022-0396(84)90084-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.advwatres.2011.04.021

    Article  ADS  Google Scholar 

  21. Noll, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Ration. Mech. Anal. 52, 62–92 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/B978-008043568-8/50012-2

    Google Scholar 

  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). https://doi.org/10.1002/mma.3401

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1029/RG014i002p00227

    Article  ADS  Google Scholar 

  25. Sanchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin (1980)

    MATH  Google Scholar 

  26. Seguin, B., Walkington, N.J.: Multi-component multiphase flow. Arch. Ration. Mech. Anal. (2017, submitted)

  27. Showalter, R.E.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251(1), 310–340 (2000). https://doi.org/10.1006/jmaa.2000.7048

    Article  MathSciNet  MATH  Google Scholar 

  28. Swendsen, R.: An Introduction to Statistical Mechanics and Thermodynamics. Oxford Graduate Texts. Oxford University Press, Oxford (2012). https://books.google.com/books?id=hbW609uEobsC

    Book  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  30. Verruijt, A.: An Introduction to Soil Mechanics. Springer, Berlin (2010)

    Google Scholar 

  31. Visintin, A.: Models of Phase Transitions. Progress in Nonlinear Differential Equations and Their Applications, vol. 28. Birkhäuser, Boston (1996). https://doi.org/10.1007/978-1-4612-4078-5

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, 60660, USA

    Brian Seguin

  2. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Noel J. Walkington

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  1. Brian Seguin
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Correspondence to Brian Seguin.

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In memory of Walter Noll

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N.J. Walkington was supported in part by National Science Foundation Grants DMS-1418991 and DMREF-1729478. This work was also supported by the NSF through the Center for Nonlinear Analysis.

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Seguin, B., Walkington, N.J. Multi-Component Multiphase Flow Through a Poroelastic Medium. J Elast 135, 485–507 (2019). https://doi.org/10.1007/s10659-018-09721-9

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