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April  2014, 7(2): 191-205. doi: 10.3934/dcdss.2014.7.191

Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository

Florian Caro 1, , Bilal Saad 2, and  Mazen Saad 3,

1. 

CEA Saclay, DEN/DANS/DM2S/SFME/LSET5, 91191 Gif Sur Yvette, France
2. 

King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering, 23955-6900, Thuwal, Saudi Arabia
3. 

Ecole Centrale de Nantes, Département d' Informatique et Mathématiques, Laboratoire de Mathématiques Jean Leray (UMR 6629 CNRS), 1, rue de la Noé, BP 92101, France

Received  April 2013 Revised  July 2013 Published  September 2013

Our goal is the mathematical analysis of a two phase (liquid and gas) two components (water and hydrogen) system modeling the hydrogen displacement in a storage site for radioactive waste. We suppose that the water is only in the liquid phase and is incompressible. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water with the Henry law. The flow is described by the conservation of the mass of each components. The model is treated without simplified assumptions on the gas density. This model is degenerated due to vanishing terms. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.
Keywords: porous media., compressible flow, nonlinear parabolic system, Degenerate system.
Mathematics Subject Classification: 35K65, 35K55, 76S0.
Citation: Florian Caro, Bilal Saad, Mazen Saad. Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 191-205. doi: 10.3934/dcdss.2014.7.191
References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 3 (1983), 311-341. doi: 10.1007/BF01176474.

[2]

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, Multiscale Modeling and Simulation, 8 (2010), 2023-2047. doi: 10.1137/100790215.

[3]

B. Amaziane and M. Jurak, Formulation of immiscible compressible two-phase flow in porous media, Comptes Rendus Mécanique, 7 (2008), 600-605. doi: 10.1016/j.crme.2008.04.008.

[4]

B. Andreianov, M. Bendahmane, K. H. Karlsen and S. Ouaro, Well-posedness results for triply nonlinear degenerate parabolic equations, Journal of Differential Equations, 247 (2009), 277-302. doi: 10.1016/j.jde.2009.03.001.

[5]

M. Bendahmane and K. H. Karlsen, Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), 405-422. doi: 10.1137/S0036141003428937.

[6]

A. Bourgeat, M. Jurak and F. Smai, Two-phase, partially miscible flow and transport modeling in porous media; application to gaz migration in a nuclear waste repository, Computational Geosciences, 4 (2009), 309-325.

[7]

F. Caro, B. Saad and M. Saad, Two-component two-compressible flow in a porous medium, Acta Applicandae Mathematicae, 117 (2012), 15-46. doi: 10.1007/s10440-011-9648-0.

[8]

G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows through Porous Media," Studies in Mathematics and its Applications, Elsevier, 1986.

[9]

Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution, Journal of Differential Equations, 171 (2001), 203-232. doi: 10.1006/jdeq.2000.3848.

[10]

Z. Chen, Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization, Journal of Differential Equations, 186 (2002), 345-376. doi: 10.1016/S0022-0396(02)00027-X.

[11]

C. Choquet, Asymptotic analysis of a nonlinear parabolic problem modelling miscible compressible displacement in porous media, Nonlinear Differential Equations and Appl., 15 (2008), 757-782. doi: 10.1007/s00030-008-8010-3.

[12]

C. Choquet, On a fully nonlinear parabolic problem modelling miscible compressible displacement in porous media, Journal of Mathematical Analysis and Applications, 339 (2008), 1112-1133. doi: 10.1016/j.jmaa.2007.07.037.

[13]

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 pressure, Journal of Mathematical Analysis and Applications, 326 (2007), 332-351. doi: 10.1016/j.jmaa.2006.02.082.

[14]

X. Feng, On existence and uniqueness results for a coupled systems modelling miscible displacement in porous media, J. Math. Anal. Appl., 194 (1995), 883-910. doi: 10.1006/jmaa.1995.1334.

[15]

G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingéniere Pétrolière," Mathématiques & Applications (Berlin), Vol. 22, Springer-Verlag, Berlin, 1996.

[16]

C. Galusinski and M. Saad, A nonlinear degenerate system modelling water-gas flows in porous media, Discrete and Continuous Dynamical System Ser. B, 9 (2008), 281-308.

[17]

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.

[18]

C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media, C. R. Acad. Sci. Paris, 347 (2009), 249-254. doi: 10.1016/j.crma.2009.01.023.

[19]

Z. Khalil and M. Saad, Solutions to a model for compressible immiscible two phase flow in porous media, Electronic Journal of Differential Equations, 2010 (2010), 33 pp.

[20]

Z. Khalil and M. Saad, On a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media, Nonlinear Analysis, 12 (2011), 1591-1615. doi: 10.1016/j.nonrwa.2010.10.015.

[21]

A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow, Comptes rendus Mécanique, 337 (2009), 226-232.

[22]

F. Smaï, A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository, C. R. Acad. Sci. Paris, 347 (2009), 527-532. doi: 10.1016/j.crma.2009.03.011.

[23]

, J. Talandier. Available from: http://www.andra.fr.

[24]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New-York, 1993.

show all references

References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 3 (1983), 311-341. doi: 10.1007/BF01176474.

[2]

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, Multiscale Modeling and Simulation, 8 (2010), 2023-2047. doi: 10.1137/100790215.

[3]

B. Amaziane and M. Jurak, Formulation of immiscible compressible two-phase flow in porous media, Comptes Rendus Mécanique, 7 (2008), 600-605. doi: 10.1016/j.crme.2008.04.008.

[4]

B. Andreianov, M. Bendahmane, K. H. Karlsen and S. Ouaro, Well-posedness results for triply nonlinear degenerate parabolic equations, Journal of Differential Equations, 247 (2009), 277-302. doi: 10.1016/j.jde.2009.03.001.

[5]

M. Bendahmane and K. H. Karlsen, Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), 405-422. doi: 10.1137/S0036141003428937.

[6]

A. Bourgeat, M. Jurak and F. Smai, Two-phase, partially miscible flow and transport modeling in porous media; application to gaz migration in a nuclear waste repository, Computational Geosciences, 4 (2009), 309-325.

[7]

F. Caro, B. Saad and M. Saad, Two-component two-compressible flow in a porous medium, Acta Applicandae Mathematicae, 117 (2012), 15-46. doi: 10.1007/s10440-011-9648-0.

[8]

G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows through Porous Media," Studies in Mathematics and its Applications, Elsevier, 1986.

[9]

Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution, Journal of Differential Equations, 171 (2001), 203-232. doi: 10.1006/jdeq.2000.3848.

[10]

Z. Chen, Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization, Journal of Differential Equations, 186 (2002), 345-376. doi: 10.1016/S0022-0396(02)00027-X.

[11]

C. Choquet, Asymptotic analysis of a nonlinear parabolic problem modelling miscible compressible displacement in porous media, Nonlinear Differential Equations and Appl., 15 (2008), 757-782. doi: 10.1007/s00030-008-8010-3.

[12]

C. Choquet, On a fully nonlinear parabolic problem modelling miscible compressible displacement in porous media, Journal of Mathematical Analysis and Applications, 339 (2008), 1112-1133. doi: 10.1016/j.jmaa.2007.07.037.

[13]

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 pressure, Journal of Mathematical Analysis and Applications, 326 (2007), 332-351. doi: 10.1016/j.jmaa.2006.02.082.

[14]

X. Feng, On existence and uniqueness results for a coupled systems modelling miscible displacement in porous media, J. Math. Anal. Appl., 194 (1995), 883-910. doi: 10.1006/jmaa.1995.1334.

[15]

G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingéniere Pétrolière," Mathématiques & Applications (Berlin), Vol. 22, Springer-Verlag, Berlin, 1996.

[16]

C. Galusinski and M. Saad, A nonlinear degenerate system modelling water-gas flows in porous media, Discrete and Continuous Dynamical System Ser. B, 9 (2008), 281-308.

[17]

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.

[18]

C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media, C. R. Acad. Sci. Paris, 347 (2009), 249-254. doi: 10.1016/j.crma.2009.01.023.

[19]

Z. Khalil and M. Saad, Solutions to a model for compressible immiscible two phase flow in porous media, Electronic Journal of Differential Equations, 2010 (2010), 33 pp.

[20]

Z. Khalil and M. Saad, On a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media, Nonlinear Analysis, 12 (2011), 1591-1615. doi: 10.1016/j.nonrwa.2010.10.015.

[21]

A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow, Comptes rendus Mécanique, 337 (2009), 226-232.

[22]

F. Smaï, A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository, C. R. Acad. Sci. Paris, 347 (2009), 527-532. doi: 10.1016/j.crma.2009.03.011.

[23]

, J. Talandier. Available from: http://www.andra.fr.

[24]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New-York, 1993.

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