American Institute of Mathematical Sciences

July  2013, 18(5): 1217-1251. doi: 10.3934/dcdsb.2013.18.1217

The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types

 1 Laboratoire de Mathématiques et de leurs Applications, CNRS-UMR 5142, Université de Pau, Av. de l’Université, 64000 Pau 2 Department of Mathematics, B.Verkin Institute for Low Temperature Physics and Engineering, 47, av. Lenin, 61103, Kharkov 3 Narvik University College, Postbox 385, Narvik, 8505

Received  September 2011 Revised  December 2012 Published  March 2013

We study a model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media taking into account capillary and gravity effects. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. This process can be formulated as a coupled system of partial differential equations which includes a nonlinear parabolic pressure equation and a nonlinear degenerate diffusion-convection saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. There are two kinds of degeneracy in the studied system: the first one is the degeneracy of the capillary diffusion term in the saturation equation, and the second one appears in the evolution term of the pressure equation. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result in order to pass to the limit in nonlinear terms. This passage to the limit is nontrivial due to the degeneracy of the system.
Citation: Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217
References:
 [1] H. W. Alt and E. Di Benedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scu. Norm. Sup. Pisa Cl. Sci., 12 (1985), 335-392. [2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 3 (1983), 311-341. doi: 10.1007/BF01176474. [3] B. Amaziane, S. Antontsev, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in porous media: Application to gas migration in a nuclear waste repository, SIAM J. Multiscale Model. Simul., 8 (2010), 2023-2047. doi: 10.1137/100790215. [4] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media, C. R. Mécanique, 336 (2008), 600-605. [5] B. Amaziane, M. Jurak and A. Žgaljić-Keko, Modeling and numerical simulations of immiscible compressible two-phase flow in porous media by the concept of global pressure, Transport in Porous Media, 84 (2010), 133-152. doi: 10.1007/s11242-009-9489-8. [6] B. Amaziane, M. Jurak and A. Žgaljić-Keko, An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media, J. Differential Equations, 250 (2011), 1685-1718. doi: 10.1016/j.jde.2010.09.008. [7] Y. Amirat, K. Hamdache and A. Ziani, Mathematical analysis for compressible miscible displacement models in porous media, Math. Models Methods Appl. Sci., 6 (1996), 729-747. doi: 10.1142/S0218202596000316. [8] Y. Amirat and M. Moussaoui, Analysis of a one-dimensional model for compressible miscible displacement in porous media, SIAM J. Math. Anal., 26 (1995), 659-674. doi: 10.1137/S003614109223297X. [9] Y. Amirat and V. Shelukhin, Global weak solutions to equations of compressible miscible flows in porous media, SIAM J. Math. Anal., 38 (2007), 1825-1846. doi: 10.1137/050640321. [10] ANDRA, Couplex-Gas Benchmark, 2006. Available online at: http://www.gdrmomas.org/ex_qualifications.html. [11] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids," North-Holland, Amsterdam, 1990. [12] T. Arbogast, The existence of weak solutions to single-porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19 (1992), 1009-1031. doi: 10.1016/0362-546X(92)90121-T. [13] J. Bear and Y. Bachmat, "Introduction to Modeling of Transport Phenomena in Porous Media," Kluwer Academic Publishers, London, 1991. [14] A. Bourgeat and A. Hidani, A result of existence for a model of two-phase flow in a porous medium made of different rock types, Appl. Anal., 56 (1995), 381-399. doi: 10.1080/00036819508840332. [15] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation," North-Holland, Amsterdam, 1986. [16] Z. Chen, Degenerate two-phase incompressible flow I: Existence, uniqueness and regularity of a weak solution, J. Differential Equations, 171 (2001), 203-232. doi: 10.1006/jdeq.2000.3848. [17] Z. Chen and G. Huan and Y. Ma, "Computational Methods for Multiphase Flows in Porous Media," SIAM, Philadelphia, 2006. doi: 10.1137/1.9780898718942. [18] C. Choquet, On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media, J. Math. Anal. Appl., 339 (2008), 1112-1133. doi: 10.1016/j.jmaa.2007.07.037. [19] F. Z. Daïm, R. Eymard and D. Hilhorst, Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure, J. Math. Anal. Appl., 326 (2007), 332-351. doi: 10.1016/j.jmaa.2006.02.082. [20] B. K. Fadimba, On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium, J. Math. Anal. Appl., 328 (2007), 1034-1056. doi: 10.1016/j.jmaa.2006.06.012. [21] G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingénierie Pétrolière," Springer-Verlag, Berlin, 1996. [22] X. Feng, Strong solutions to a nonlinear parabolic system modeling compressible miscible displacement in porous media, Nonlinear Anal., 23 (1994), 1515-1531. doi: 10.1016/0362-546X(94)90202-X. [23] C. Galusinski and M. Saad, On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media, Adv. Differential Equations, 9 (2004), 1235-1278. [24] C. Galusinski and M. Saad, Water-gas flow in porous media, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 281-308. [25] C. Galusinski and M. Saad, Two compressible immiscible fluids in porous media, J. Differential Equations, 244 (2008), 1741-1783. doi: 10.1016/j.jde.2008.01.013. [26] C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media, C. R. Acad. Sci. Paris, Sér. I, 347 (2009), 249-254. doi: 10.1016/j.crma.2009.01.023. [27] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, 1983. [28] R. Helmig, "Multiphase Flow and Transport Trocesses in the Subsurface," Springer, Berlin, 1997. [29] Z. Khalil and M. Saad, Solutions to a model for compressible immiscible two phase flow in porous media, Electronic Journal of Differential Equations, 122 (2010), 1-33. [30] Z. Khalil and M. Saad, On a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media, Nonlinear Analysis: Real World Applications, 12 (2011), 1591-1615. doi: 10.1016/j.nonrwa.2010.10.015. [31] D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium, J. Differential Equations, 55 (1984), 276-288. doi: 10.1016/0022-0396(84)90084-6. [32] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow, C. R. Mécanique, 337 (2009), 226-232. [33] OECD/NEA, Safety of geological disposal of high-level and long-lived radioactive waste in France, An International Peer Review of the "Dossier 2005 Argile'' Concerning Disposal in the Callovo-Oxfordian Formation. OECD Publishing (2006). Available online at: http://www.nea.fr/html/rwm/reports/2006/nea6178-argile.pdf. [34] J. Simon, Compact sets in the space $L^p(0,t; B)$, Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. doi: 10.1007/BF01762360. [35] F. Smaï, A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository, C. R. Math. Acad. Sci. Paris, 347 (2009), 527-532. doi: 10.1016/j.crma.2009.03.011. [36] F. Smaï, Existence of solutions for a model of multiphase flow in porous media applied to gas migration in underground nuclear waste repository, Appl. Anal., 88 (2009), 1609-1616. doi: 10.1080/00036810902942226. [37] L. M. Yeh, On two-phase flow in fractured media, Math. Models Methods Appl. Sci., 12 (2002), 1075-1107. doi: 10.1142/S0218202502002045. [38] L. M. Yeh, Hölder continuity for two-phase flows in porous media, Math. Methods Appl. Sci., 29 (2006), 1261-1289. doi: 10.1002/mma.724.

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References:
 [1] H. W. Alt and E. Di Benedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scu. Norm. Sup. Pisa Cl. Sci., 12 (1985), 335-392. [2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 3 (1983), 311-341. doi: 10.1007/BF01176474. [3] B. Amaziane, S. Antontsev, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in porous media: Application to gas migration in a nuclear waste repository, SIAM J. Multiscale Model. Simul., 8 (2010), 2023-2047. doi: 10.1137/100790215. [4] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media, C. R. Mécanique, 336 (2008), 600-605. [5] B. Amaziane, M. Jurak and A. Žgaljić-Keko, Modeling and numerical simulations of immiscible compressible two-phase flow in porous media by the concept of global pressure, Transport in Porous Media, 84 (2010), 133-152. doi: 10.1007/s11242-009-9489-8. [6] B. Amaziane, M. Jurak and A. Žgaljić-Keko, An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media, J. Differential Equations, 250 (2011), 1685-1718. doi: 10.1016/j.jde.2010.09.008. [7] Y. Amirat, K. Hamdache and A. Ziani, Mathematical analysis for compressible miscible displacement models in porous media, Math. Models Methods Appl. Sci., 6 (1996), 729-747. doi: 10.1142/S0218202596000316. [8] Y. Amirat and M. Moussaoui, Analysis of a one-dimensional model for compressible miscible displacement in porous media, SIAM J. Math. Anal., 26 (1995), 659-674. doi: 10.1137/S003614109223297X. [9] Y. Amirat and V. Shelukhin, Global weak solutions to equations of compressible miscible flows in porous media, SIAM J. Math. Anal., 38 (2007), 1825-1846. doi: 10.1137/050640321. [10] ANDRA, Couplex-Gas Benchmark, 2006. Available online at: http://www.gdrmomas.org/ex_qualifications.html. [11] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids," North-Holland, Amsterdam, 1990. [12] T. Arbogast, The existence of weak solutions to single-porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19 (1992), 1009-1031. doi: 10.1016/0362-546X(92)90121-T. [13] J. Bear and Y. Bachmat, "Introduction to Modeling of Transport Phenomena in Porous Media," Kluwer Academic Publishers, London, 1991. [14] A. Bourgeat and A. Hidani, A result of existence for a model of two-phase flow in a porous medium made of different rock types, Appl. Anal., 56 (1995), 381-399. doi: 10.1080/00036819508840332. [15] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation," North-Holland, Amsterdam, 1986. [16] Z. Chen, Degenerate two-phase incompressible flow I: Existence, uniqueness and regularity of a weak solution, J. Differential Equations, 171 (2001), 203-232. doi: 10.1006/jdeq.2000.3848. [17] Z. Chen and G. Huan and Y. Ma, "Computational Methods for Multiphase Flows in Porous Media," SIAM, Philadelphia, 2006. doi: 10.1137/1.9780898718942. [18] C. Choquet, On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media, J. Math. Anal. Appl., 339 (2008), 1112-1133. doi: 10.1016/j.jmaa.2007.07.037. [19] F. Z. Daïm, R. Eymard and D. Hilhorst, Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure, J. Math. Anal. Appl., 326 (2007), 332-351. doi: 10.1016/j.jmaa.2006.02.082. [20] B. K. Fadimba, On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium, J. Math. Anal. Appl., 328 (2007), 1034-1056. doi: 10.1016/j.jmaa.2006.06.012. [21] G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingénierie Pétrolière," Springer-Verlag, Berlin, 1996. [22] X. Feng, Strong solutions to a nonlinear parabolic system modeling compressible miscible displacement in porous media, Nonlinear Anal., 23 (1994), 1515-1531. doi: 10.1016/0362-546X(94)90202-X. [23] C. Galusinski and M. Saad, On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media, Adv. Differential Equations, 9 (2004), 1235-1278. [24] C. Galusinski and M. Saad, Water-gas flow in porous media, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 281-308. [25] C. Galusinski and M. Saad, Two compressible immiscible fluids in porous media, J. Differential Equations, 244 (2008), 1741-1783. doi: 10.1016/j.jde.2008.01.013. [26] C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media, C. R. Acad. Sci. Paris, Sér. I, 347 (2009), 249-254. doi: 10.1016/j.crma.2009.01.023. [27] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, 1983. [28] R. Helmig, "Multiphase Flow and Transport Trocesses in the Subsurface," Springer, Berlin, 1997. [29] Z. Khalil and M. Saad, Solutions to a model for compressible immiscible two phase flow in porous media, Electronic Journal of Differential Equations, 122 (2010), 1-33. [30] Z. Khalil and M. Saad, On a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media, Nonlinear Analysis: Real World Applications, 12 (2011), 1591-1615. doi: 10.1016/j.nonrwa.2010.10.015. [31] D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium, J. Differential Equations, 55 (1984), 276-288. doi: 10.1016/0022-0396(84)90084-6. [32] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow, C. R. Mécanique, 337 (2009), 226-232. [33] OECD/NEA, Safety of geological disposal of high-level and long-lived radioactive waste in France, An International Peer Review of the "Dossier 2005 Argile'' Concerning Disposal in the Callovo-Oxfordian Formation. OECD Publishing (2006). Available online at: http://www.nea.fr/html/rwm/reports/2006/nea6178-argile.pdf. [34] J. Simon, Compact sets in the space $L^p(0,t; B)$, Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. doi: 10.1007/BF01762360. [35] F. Smaï, A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository, C. R. Math. Acad. Sci. Paris, 347 (2009), 527-532. doi: 10.1016/j.crma.2009.03.011. [36] F. Smaï, Existence of solutions for a model of multiphase flow in porous media applied to gas migration in underground nuclear waste repository, Appl. Anal., 88 (2009), 1609-1616. doi: 10.1080/00036810902942226. [37] L. M. Yeh, On two-phase flow in fractured media, Math. Models Methods Appl. Sci., 12 (2002), 1075-1107. doi: 10.1142/S0218202502002045. [38] L. M. Yeh, Hölder continuity for two-phase flows in porous media, Math. Methods Appl. Sci., 29 (2006), 1261-1289. doi: 10.1002/mma.724.

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