# American Institute of Mathematical Sciences

March  2017, 12(1): 147-171. doi: 10.3934/nhm.2017006

## An improved homogenization result for immiscible compressible two-phase flow in porous media

 1 CNRS / UNIV PAU & PAYS ADOUR, Laboratoire de Mathématiques et de leurs Applications-IPRA, UMR 5142, Av. de l'Université, 64000 Pau, France 2 Laboratory of Fluid Dynamics and Seismics, 9 Institutskiy per., Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700, Russia 3 CNRS / UNIV PAU & PAYS ADOUR, Laboratoire de Mathématiques et de leurs Applications-IPRA, UMR 5142, Av. de l'Université, 64000 Pau, France 4 University of Tromsø, Campus in Narvik, Postbox 385, Narvik, 8505, Norway 5 Institute for Information Transmission Problems of RAS, Bolshoy Karetny per., 19, Moscow, 127051, Russia

Received  November 2015 Revised  June 2016 Published  February 2017

The paper deals with a degenerate model of immiscible compressible two-phase flow in heterogeneous porous media. We consider liquid and gas phases (water and hydrogen) flow in a porous reservoir, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. The gas phase is supposed compressible and obeying the ideal gas law. The flow is then described by the conservation of the mass for each phase. The model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. The aim of this paper is to extend our previous results to the case of an ideal gas. In this case a new degeneracy appears in the pressure equation. With the help of an appropriate regularization we show the existence of a weak solution to the studied system. We also consider the corresponding nonlinear homogenization problem and provide a rigorous mathematical derivation of the upscaled model by means of the two-scale convergence.

Citation: Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks and Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006
##### References:
 [1] E. Ahusborde, B. Amaziane and M. Jurak, Three-dimensional numerical simulation by upscaling of gas migration through engineered and geological barriers for a deep repository for radioactive waste, J. of the Geological Society, 294 (2014), 1-19. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [3] H. W. Alt and E. di Benedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 335-392. [4] 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. [5] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media, C. R. Mecanique, 336 (2008), 600-605.  doi: 10.1016/j.crme.2008.04.008. [6] B. Amaziane, M. Jurak and A. Vrbaški, Homogenization results for a coupled system modeling immiscible compressible two-phase flow in porous media by the concept of global pressure, Appl. Anal., 92 (2013), 1417-1433.  doi: 10.1080/00036811.2012.682059. [7] 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, Transp. Porous Media, 84 (2010), 133-152.  doi: 10.1007/s11242-009-9489-8. [8] 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. [9] B. Amaziane, L. Pankratov and A. 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 and Dynamical Systems, Ser. B, 18 (2013), 1217-1251.  doi: 10.3934/dcdsb.2013.18.1217. [10] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures, M3AS, 24 (2014), 1421-1451.  doi: 10.1142/S0218202514500055. [11] ANDRA, Dossier 2005 Argile, les Recherches de l'Andra sur le Stockage Géologique des Déchets Radioactifs á Haute Activité et á Vie Longue, Collection les Rapports, Andra, Châtenay-Malabry, 2005. [12] S.~N. Antontsev, On the solvability of boundary value problems for degenerating equations of two-phase flow, Solid-State Dynamics, 10 (1972), 28-53. [13] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Non-Homogeneous Fluids, (in Russian), Nauka, Novosibirsk, 1983. [14] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. [15] T. J. 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. [16] A. Bourgeat, O. Gipouloux and E. Marušić-Paloka, Mathematical modeling of an under-ground waste disposal site by upscaling, Math. Methods Appl. Sci., 27 (2004), 381-403.  doi: 10.1002/mma.459. [17] A. Bourgeat, O. Gipouloux and F. Smai, Scaling up of source terms with random behavior for modelling transport migration of contaminants in aquifers, Nonlinear Anal. Real World Appl., 11 (2010), 4513-4523.  doi: 10.1016/j.nonrwa.2008.10.062. [18] A. Bourgeat and E. Marušić-Paloka, A homogenized model of an underground waste repository including a disturbed zone, Multiscale Model. Simul., 3 (2005), 918-939.  doi: 10.1137/040605424. [19] A. Bourgeat, E. Marušić-Paloka and A. Piatnitski, Scaling up of an underground nuclear waste repository including a possibly damaged zone, Asymptot. Anal., 67 (2010), 147-165. [20] A. Bourgeat and A. Piatnitski, Averaging of a singular random source term in a diffusion convection equation, SIAM J. Math. Anal., 42 (2010), 2626-2651.  doi: 10.1137/080736077. [21] C. Cancés and P. Michel, An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field, SIAM J. Math. Anal., 44 (2012), 966-992.  doi: 10.1137/11082943X. [22] F. Caro, B. Saad and M. Saad, Study of degenerate parabolic system modellingthe hydrogen displacement in a nuclear waste repository, Discrete and Continuous Dynamical Systems, Ser. S, 7 (2014), 191-205. [23] G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986. [24] 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. [25] Z. Chen, Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization, J. Differential Equations, 186 (2002), 345-376.  doi: 10.1016/S0022-0396(02)00027-X. [26] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. doi: 10.1137/1.9780898718942. [27] J. Croisé, G. Mayer, J. Talandier and J. Wendling, Impact of water consumption and saturation-dependent corrosion rate on hydrogen generation and migration from an intermediate-level radioactive waste repository, Transp. Porous Media, 90 (2011), 59-75. [28] [29] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modéles Non-Linéaires de L'ingénierie Pétroliére, Springer-Verlag, Berlin, 1996. [30] 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. [31] C. Galusinski and M. Saad, Water-gas flow in porous media, Discrete Contin. Dyn. Syst., Ser. B, 9 (2008), 281-308. [32] 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. [33] 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. [34] O. Gipouloux and F. Smai, Scaling up of an underground waste disposal model with random source terms, Internat. J. Multiscale Comput. Engin., 6 (2008), 309-325. [35] A. Gloria, T. Goudon and S. Krell, Numerical homogenization of a nonlinearly coupled elliptic-parabolic system, reduced basis method, and application to nuclear waste storage, Math. Models Methods Appl. Sci., 23 (2013), 2523-2560.  doi: 10.1142/S0218202513500395. [36] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer, Berlin, 1997. [37] P. Henning, M. Ohlberger and B. Schweizer, Homogenization of the degenerate two-phase flow equations, Math. Models Methods Appl. Sci., 23 (2013), 2323-2352.  doi: 10.1142/S0218202513500334. [38] U. Hornung, Homogenization and Porous Media, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1920-0. [39] 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. [40] 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. [41] D. Kroener 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. [42] R. Senger, J. Ewing, K. Zhang, J. Avis, P. Marschall and I. Gauss, Modeling approaches for investigating gas migration from a deep low/intermediate level waste repository (Switzerland), Transp. Porous Media, 90 (2011), 113-133.  doi: 10.1007/s11242-010-9709-2. [43] R. P. Shaw, Gas Generation and Migration in Deep Geological Radioactive Waste Repositories, Geological Society, 2015. [44] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford University Press, Oxford, 2007.

show all references

##### References:
 [1] E. Ahusborde, B. Amaziane and M. Jurak, Three-dimensional numerical simulation by upscaling of gas migration through engineered and geological barriers for a deep repository for radioactive waste, J. of the Geological Society, 294 (2014), 1-19. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [3] H. W. Alt and E. di Benedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 335-392. [4] 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. [5] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media, C. R. Mecanique, 336 (2008), 600-605.  doi: 10.1016/j.crme.2008.04.008. [6] B. Amaziane, M. Jurak and A. Vrbaški, Homogenization results for a coupled system modeling immiscible compressible two-phase flow in porous media by the concept of global pressure, Appl. Anal., 92 (2013), 1417-1433.  doi: 10.1080/00036811.2012.682059. [7] 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, Transp. Porous Media, 84 (2010), 133-152.  doi: 10.1007/s11242-009-9489-8. [8] 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. [9] B. Amaziane, L. Pankratov and A. 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 and Dynamical Systems, Ser. B, 18 (2013), 1217-1251.  doi: 10.3934/dcdsb.2013.18.1217. [10] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures, M3AS, 24 (2014), 1421-1451.  doi: 10.1142/S0218202514500055. [11] ANDRA, Dossier 2005 Argile, les Recherches de l'Andra sur le Stockage Géologique des Déchets Radioactifs á Haute Activité et á Vie Longue, Collection les Rapports, Andra, Châtenay-Malabry, 2005. [12] S.~N. Antontsev, On the solvability of boundary value problems for degenerating equations of two-phase flow, Solid-State Dynamics, 10 (1972), 28-53. [13] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Non-Homogeneous Fluids, (in Russian), Nauka, Novosibirsk, 1983. [14] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. [15] T. J. 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. [16] A. Bourgeat, O. Gipouloux and E. Marušić-Paloka, Mathematical modeling of an under-ground waste disposal site by upscaling, Math. Methods Appl. Sci., 27 (2004), 381-403.  doi: 10.1002/mma.459. [17] A. Bourgeat, O. Gipouloux and F. Smai, Scaling up of source terms with random behavior for modelling transport migration of contaminants in aquifers, Nonlinear Anal. Real World Appl., 11 (2010), 4513-4523.  doi: 10.1016/j.nonrwa.2008.10.062. [18] A. Bourgeat and E. Marušić-Paloka, A homogenized model of an underground waste repository including a disturbed zone, Multiscale Model. Simul., 3 (2005), 918-939.  doi: 10.1137/040605424. [19] A. Bourgeat, E. Marušić-Paloka and A. Piatnitski, Scaling up of an underground nuclear waste repository including a possibly damaged zone, Asymptot. Anal., 67 (2010), 147-165. [20] A. Bourgeat and A. Piatnitski, Averaging of a singular random source term in a diffusion convection equation, SIAM J. Math. Anal., 42 (2010), 2626-2651.  doi: 10.1137/080736077. [21] C. Cancés and P. Michel, An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field, SIAM J. Math. Anal., 44 (2012), 966-992.  doi: 10.1137/11082943X. [22] F. Caro, B. Saad and M. Saad, Study of degenerate parabolic system modellingthe hydrogen displacement in a nuclear waste repository, Discrete and Continuous Dynamical Systems, Ser. S, 7 (2014), 191-205. [23] G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986. [24] 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. [25] Z. Chen, Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization, J. Differential Equations, 186 (2002), 345-376.  doi: 10.1016/S0022-0396(02)00027-X. [26] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. doi: 10.1137/1.9780898718942. [27] J. Croisé, G. Mayer, J. Talandier and J. Wendling, Impact of water consumption and saturation-dependent corrosion rate on hydrogen generation and migration from an intermediate-level radioactive waste repository, Transp. Porous Media, 90 (2011), 59-75. [28] [29] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modéles Non-Linéaires de L'ingénierie Pétroliére, Springer-Verlag, Berlin, 1996. [30] 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. [31] C. Galusinski and M. Saad, Water-gas flow in porous media, Discrete Contin. Dyn. Syst., Ser. B, 9 (2008), 281-308. [32] 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. [33] 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. [34] O. Gipouloux and F. Smai, Scaling up of an underground waste disposal model with random source terms, Internat. J. Multiscale Comput. Engin., 6 (2008), 309-325. [35] A. Gloria, T. Goudon and S. Krell, Numerical homogenization of a nonlinearly coupled elliptic-parabolic system, reduced basis method, and application to nuclear waste storage, Math. Models Methods Appl. Sci., 23 (2013), 2523-2560.  doi: 10.1142/S0218202513500395. [36] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer, Berlin, 1997. [37] P. Henning, M. Ohlberger and B. Schweizer, Homogenization of the degenerate two-phase flow equations, Math. Models Methods Appl. Sci., 23 (2013), 2323-2352.  doi: 10.1142/S0218202513500334. [38] U. Hornung, Homogenization and Porous Media, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1920-0. [39] 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. [40] 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. [41] D. Kroener 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. [42] R. Senger, J. Ewing, K. Zhang, J. Avis, P. Marschall and I. Gauss, Modeling approaches for investigating gas migration from a deep low/intermediate level waste repository (Switzerland), Transp. Porous Media, 90 (2011), 113-133.  doi: 10.1007/s11242-010-9709-2. [43] R. P. Shaw, Gas Generation and Migration in Deep Geological Radioactive Waste Repositories, Geological Society, 2015. [44] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford University Press, Oxford, 2007.
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