# 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 & 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. Google Scholar [2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 3 (1983), 311. doi: 10.1007/BF01176474. Google Scholar [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. doi: 10.1137/100790215. Google Scholar [4] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media,, C. R. Mécanique, 336 (2008), 600. Google Scholar [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. doi: 10.1007/s11242-009-9489-8. Google Scholar [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. doi: 10.1016/j.jde.2010.09.008. Google Scholar [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. doi: 10.1142/S0218202596000316. Google Scholar [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. doi: 10.1137/S003614109223297X. Google Scholar [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. doi: 10.1137/050640321. Google Scholar [10] ANDRA, Couplex-Gas Benchmark, 2006., Available online at: , (). Google Scholar [11] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,", North-Holland, (1990). Google Scholar [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. doi: 10.1016/0362-546X(92)90121-T. Google Scholar [13] J. Bear and Y. Bachmat, "Introduction to Modeling of Transport Phenomena in Porous Media,", Kluwer Academic Publishers, (1991). Google Scholar [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. doi: 10.1080/00036819508840332. Google Scholar [15] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation,", North-Holland, (1986). Google Scholar [16] Z. Chen, Degenerate two-phase incompressible flow I: Existence, uniqueness and regularity of a weak solution,, J. Differential Equations, 171 (2001), 203. doi: 10.1006/jdeq.2000.3848. Google Scholar [17] Z. Chen and G. Huan and Y. Ma, "Computational Methods for Multiphase Flows in Porous Media,", SIAM, (2006). doi: 10.1137/1.9780898718942. Google Scholar [18] C. Choquet, On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media,, J. Math. Anal. Appl., 339 (2008), 1112. doi: 10.1016/j.jmaa.2007.07.037. Google Scholar [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. doi: 10.1016/j.jmaa.2006.02.082. Google Scholar [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. doi: 10.1016/j.jmaa.2006.06.012. Google Scholar [21] G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingénierie Pétrolière,", Springer-Verlag, (1996). Google Scholar [22] X. Feng, Strong solutions to a nonlinear parabolic system modeling compressible miscible displacement in porous media,, Nonlinear Anal., 23 (1994), 1515. doi: 10.1016/0362-546X(94)90202-X. Google Scholar [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. Google Scholar [24] C. Galusinski and M. Saad, Water-gas flow in porous media,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 281. Google Scholar [25] C. Galusinski and M. Saad, Two compressible immiscible fluids in porous media,, J. Differential Equations, 244 (2008), 1741. doi: 10.1016/j.jde.2008.01.013. Google Scholar [26] C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media,, C. R. Acad. Sci. Paris, 347 (2009), 249. doi: 10.1016/j.crma.2009.01.023. Google Scholar [27] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983). Google Scholar [28] R. Helmig, "Multiphase Flow and Transport Trocesses in the Subsurface,", Springer, (1997). Google Scholar [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. Google Scholar [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. doi: 10.1016/j.nonrwa.2010.10.015. Google Scholar [31] D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium,, J. Differential Equations, 55 (1984), 276. doi: 10.1016/0022-0396(84)90084-6. Google Scholar [32] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow,, C. R. Mécanique, 337 (2009), 226. Google Scholar [33] OECD/NEA, Safety of geological disposal of high-level and long-lived radioactive waste in France,, An International Peer Review of the, (2006). Google Scholar [34] J. Simon, Compact sets in the space $L^p(0,t; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [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. doi: 10.1016/j.crma.2009.03.011. Google Scholar [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. doi: 10.1080/00036810902942226. Google Scholar [37] L. M. Yeh, On two-phase flow in fractured media,, Math. Models Methods Appl. Sci., 12 (2002), 1075. doi: 10.1142/S0218202502002045. Google Scholar [38] L. M. Yeh, Hölder continuity for two-phase flows in porous media,, Math. Methods Appl. Sci., 29 (2006), 1261. doi: 10.1002/mma.724. Google Scholar

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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. Google Scholar [2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 3 (1983), 311. doi: 10.1007/BF01176474. Google Scholar [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. doi: 10.1137/100790215. Google Scholar [4] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media,, C. R. Mécanique, 336 (2008), 600. Google Scholar [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. doi: 10.1007/s11242-009-9489-8. Google Scholar [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. doi: 10.1016/j.jde.2010.09.008. Google Scholar [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. doi: 10.1142/S0218202596000316. Google Scholar [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. doi: 10.1137/S003614109223297X. Google Scholar [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. doi: 10.1137/050640321. Google Scholar [10] ANDRA, Couplex-Gas Benchmark, 2006., Available online at: , (). Google Scholar [11] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,", North-Holland, (1990). Google Scholar [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. doi: 10.1016/0362-546X(92)90121-T. Google Scholar [13] J. Bear and Y. Bachmat, "Introduction to Modeling of Transport Phenomena in Porous Media,", Kluwer Academic Publishers, (1991). Google Scholar [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. doi: 10.1080/00036819508840332. Google Scholar [15] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation,", North-Holland, (1986). Google Scholar [16] Z. Chen, Degenerate two-phase incompressible flow I: Existence, uniqueness and regularity of a weak solution,, J. Differential Equations, 171 (2001), 203. doi: 10.1006/jdeq.2000.3848. Google Scholar [17] Z. Chen and G. Huan and Y. Ma, "Computational Methods for Multiphase Flows in Porous Media,", SIAM, (2006). doi: 10.1137/1.9780898718942. Google Scholar [18] C. Choquet, On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media,, J. Math. Anal. Appl., 339 (2008), 1112. doi: 10.1016/j.jmaa.2007.07.037. Google Scholar [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. doi: 10.1016/j.jmaa.2006.02.082. Google Scholar [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. doi: 10.1016/j.jmaa.2006.06.012. Google Scholar [21] G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingénierie Pétrolière,", Springer-Verlag, (1996). Google Scholar [22] X. Feng, Strong solutions to a nonlinear parabolic system modeling compressible miscible displacement in porous media,, Nonlinear Anal., 23 (1994), 1515. doi: 10.1016/0362-546X(94)90202-X. Google Scholar [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. Google Scholar [24] C. Galusinski and M. Saad, Water-gas flow in porous media,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 281. Google Scholar [25] C. Galusinski and M. Saad, Two compressible immiscible fluids in porous media,, J. Differential Equations, 244 (2008), 1741. doi: 10.1016/j.jde.2008.01.013. Google Scholar [26] C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media,, C. R. Acad. Sci. Paris, 347 (2009), 249. doi: 10.1016/j.crma.2009.01.023. Google Scholar [27] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983). Google Scholar [28] R. Helmig, "Multiphase Flow and Transport Trocesses in the Subsurface,", Springer, (1997). Google Scholar [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. Google Scholar [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. doi: 10.1016/j.nonrwa.2010.10.015. Google Scholar [31] D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium,, J. Differential Equations, 55 (1984), 276. doi: 10.1016/0022-0396(84)90084-6. Google Scholar [32] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow,, C. R. Mécanique, 337 (2009), 226. Google Scholar [33] OECD/NEA, Safety of geological disposal of high-level and long-lived radioactive waste in France,, An International Peer Review of the, (2006). Google Scholar [34] J. Simon, Compact sets in the space $L^p(0,t; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [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. doi: 10.1016/j.crma.2009.03.011. Google Scholar [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. doi: 10.1080/00036810902942226. Google Scholar [37] L. M. Yeh, On two-phase flow in fractured media,, Math. Models Methods Appl. Sci., 12 (2002), 1075. doi: 10.1142/S0218202502002045. Google Scholar [38] L. M. Yeh, Hölder continuity for two-phase flows in porous media,, Math. Methods Appl. Sci., 29 (2006), 1261. doi: 10.1002/mma.724. Google Scholar
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