American Institute of Mathematical Sciences

Advanced
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

Brahim Amaziane 1, , Leonid Pankratov 2, and  Andrey Piatnitski 3,

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.
Keywords: water-gas., nonlinear degenerate system, two-phase flow, Immiscible compressible, nuclear waste, porous media.
Mathematics Subject Classification: Primary: 76S05, 76T10; Secondary: 35D30, 35K65, 35Q3.
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

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.   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

[1]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006
[2]

Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281
[3]

Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020
[4]

Clément Cancès. On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Networks & Heterogeneous Media, 2010, 5 (3) : 635-647. doi: 10.3934/nhm.2010.5.635
[5]

Brahim Amaziane, Mladen Jurak, Leonid Pankratov, Anja Vrbaški. Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 629-665. doi: 10.3934/dcdsb.2018037
[6]

Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307
[7]

Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146
[8]

Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[9]

Florian Caro, Bilal Saad, Mazen Saad. Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 191-205. doi: 10.3934/dcdss.2014.7.191
[10]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
[11]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
[12]

Bilal Saad, Mazen Saad. Numerical analysis of a non equilibrium two-component two-compressible flow in porous media. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 317-346. doi: 10.3934/dcdss.2014.7.317
[13]

Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93
[14]

Shifeng Geng, Zhen Wang. Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant. Communications on Pure & Applied Analysis, 2012, 11 (2) : 475-500. doi: 10.3934/cpaa.2012.11.475
[15]

Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021
[16]

Yingshan Chen, Mei Zhang. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinetic & Related Models, 2016, 9 (3) : 429-441. doi: 10.3934/krm.2016001
[17]

Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
[18]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665
[19]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[20]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences