Multi-component multiphase flow through a poroelastic medium. (English) Zbl 1415.76615
Summary: 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.
MSC:
| 76S05 | Flows in porous media; filtration; seepage |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 80A17 | Thermodynamics of continua |
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 · Zbl 0608.76082 |
| [2] | Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1972) · Zbl 1191.76001 |
| [3] | Biot, M.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155-164 (1941) · JFM 67.0837.01 · doi:10.1063/1.1712886 |
| [4] | Biot, M.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182-185 (1955) · Zbl 0067.23603 · doi:10.1063/1.1721956 |
| [5] | Biot, M.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 78, 91-96 (1956) · Zbl 0074.19101 |
| [6] | Biot, M., Willis, D.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594-601 (1957) |
| [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 · Zbl 0991.35047 · doi:10.1006/jdeq.2000.3848 |
| [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 · Zbl 1097.76064 · doi:10.1007/s002110100291 |
| [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 · Zbl 1092.76001 · doi:10.1137/1.9780898718942 |
| [10] | Cheng, A.H.: Poroelasticity. Theory and Applications of Transport in Porous Media, vol. 27. Springer, Berlin (2016) · Zbl 1402.74004 |
| [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 · Zbl 1338.82034 · doi:10.3390/e16031756 |
| [12] | Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167-178 (1963) · Zbl 0113.17802 · doi:10.1007/BF01262690 |
| [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 · Zbl 0477.47037 · doi:10.1137/0512062 |
| [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 · Zbl 1392.76103 · doi:10.1007/s10596-015-9527-2 |
| [16] | Ewing, R.: The Mathematics of Reservoir Simulation. Frontiers in Applied Mathematics. SIAM, Philadelphia (1983). https://books.google.com/books?id=k_4Y-RBOK9oC · Zbl 0533.00031 · doi:10.1137/1.9781611971071 |
| [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 · Zbl 1312.76027 · doi:10.1137/140952922 |
| [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 · Zbl 1310.35051 · doi:10.1016/j.jmaa.2013.07.023 |
| [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 · Zbl 0509.35048 · doi:10.1016/0022-0396(84)90084-6 |
| [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 · doi:10.1016/j.advwatres.2011.04.021 |
| [21] | Noll, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Ration. Mech. Anal. 52, 62-92 (1973) · Zbl 0284.73002 · doi:10.1007/BF00249093 |
| [22] | Peszyńska, M.; Lu, Q.; Wheeler, M. F., Coupling different numerical algorithms for two phase fluid flow, MAFELAP 1999, Uxbridge, Oxford · Zbl 0990.76044 · doi: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 · Zbl 1338.47126 · doi:10.1002/mma.3401 |
| [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 · doi:10.1029/RG014i002p00227 |
| [25] | Sanchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin (1980) · Zbl 0432.70002 |
| [26] | Seguin, B., Walkington, N.J.: Multi-component multiphase flow. Arch. Ration. Mech. Anal. (2017, submitted) · Zbl 1434.35119 |
| [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 · Zbl 0979.74018 · doi:10.1006/jmaa.2000.7048 |
| [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 · Zbl 1284.82001 · doi:10.1093/acprof:oso/9780199646944.001.0001 |
| [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 · Zbl 0598.73002 |
| [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 · Zbl 0882.35004 · doi:10.1007/978-1-4612-4078-5 |
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