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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 |
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: , ().
|
[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. |
show all references
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: , ().
|
[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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