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Journal of Elasticity

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06-02-2019

Multi-Component Multiphase Flow Through a Poroelastic Medium

Authors: Brian Seguin, Noel J. Walkington

Published in: Journal of Elasticity | Issue 1-2/2019

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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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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.

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.

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.

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 MathSciNetMATH

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

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

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

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

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

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 ADSMathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

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 CrossRefMATH

10.

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

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 ADSMathSciNetCrossRefMATH

12.

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

13.

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

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 MathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

16.

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

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 MathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

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 ADSMathSciNetCrossRefMATH

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 ADSCrossRef

21.

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

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

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 ADSMathSciNetCrossRefMATH

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 ADSCrossRef

25.

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

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 MathSciNetCrossRefMATH

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 CrossRefMATH

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

30.

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

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 CrossRefMATH

Title: Multi-Component Multiphase Flow Through a Poroelastic Medium
Authors: Brian Seguin
Noel J. Walkington
Publication date: 06-02-2019
Publisher: Springer Netherlands
Published in: Journal of Elasticity / Issue 1-2/2019
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-018-09721-9

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