# American Institute of Mathematical Sciences

March  2018, 23(2): 629-665. doi: 10.3934/dcdsb.2018037

## Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media

 1 Laboratoire de Mathématiques et de leurs Applications-IPRA, CNRS/UNIV PAU & PAYS ADOUR, UMR 5142, Av. de l'Université, 64000 Pau, France 2 Faculty of Science, University of Zagreb, Bijenička 30,10000 Zagreb, Croatia 3 Laboratory of Fluid Dynamics and Seismic (RAEP 5 Top 100), Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation 4 Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6,10000 Zagreb, Croatia

Received  July 2016 Revised  October 2017 Published  December 2017

This paper presents a study of immiscible incompressible two-phase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by ${\varepsilon }^θ$, where $\varepsilon$ is the size of a typical porous block and $θ>0$ is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.

Citation: 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
##### References:
 [1] E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percival, An extension theorem from connected sets, and homogenization in general periodic domains, J. Nonlinear Analysis, 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7. Google Scholar [2] L. Ait Mahiout, B. Amaziane, A. Mokrane and L. Pankratov, Homogenization of immiscible compressible two-phase flow in double porosity media, Electron. J. Differential Equations, 2016 (2016), 1-28. Google Scholar [3] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar [4] G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (eds. A. Bourgeat et al. ), World Scientific Pub., Singapore, (1996), 15–25.Google Scholar [5] 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 MMS, 8 (2010), 2023-2047. doi: 10.1137/100790215. Google Scholar [6] B. Amaziane and L. Pankratov, Homogenization of a model for water-gas flow through double-porosity media, Math. Methods Appl. Sci., 39 (2016), 425-451. doi: 10.1002/mma.3493. Google Scholar [7] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media, Proc. Roy. Soc. Edinburgh, 136 (2006), 1131-1155. doi: 10.1017/S0308210500004911. Google Scholar [8] B. Amaziane, L. Pankratov and A. Piatnitski, Nonlinear flow through double porosity media in variable exponent Sobolev spaces, Nonlinear Anal. Real World Appl., 10 (2009), 2521-2530. doi: 10.1016/j.nonrwa.2008.05.008. Google Scholar [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 Contin. Dyn. Syst. Ser. B, 18 (2013), 1217-1251. doi: 10.3934/dcdsb.2013.18.1217. Google Scholar [10] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures, Math. Models Methods Appl. Sci., 24 (2014), 1421-1451. doi: 10.1142/S0218202514500055. Google Scholar [11] B. Amaziane, L. Pankratov and V. Rybalko, On the homogenization of some double porosity models with periodic thin structures, Appl. Anal., 88 (2009), 1469-1492. doi: 10.1080/00036810903114817. Google Scholar [12] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. Google Scholar [13] T. Arbogast, J. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-826. doi: 10.1137/0521046. Google Scholar [14] G. I. Barenblatt, Yu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303. doi: 10.1016/0021-8928(60)90107-6. Google Scholar [15] J. Bear, C. F. Tsang and G. de Marsily, Flow and Contaminant Transport in Fractured Rock, Academic Press Inc, London, 1993.Google Scholar [16] A. Bourgeat, G. Chechkin and A. Piatnitski, Singular double porosity model, Appl. Anal., 82 (2003), 103-116. doi: 10.1080/0003681031000063739. Google Scholar [17] A. Bourgeat, M. Goncharenko, M. Panfilov and L. Pankratov, A general double porosity model, C. R. Acad. Sci. Paris, Série IIb, 327 (1999), 1245-1250. Google Scholar [18] A. Bourgeat, S. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immicible two-phase flow, SIAM J. Math. Anal., 27 (1996), 1520-1543. doi: 10.1137/S0036141094276457. Google Scholar [19] A. Bourgeat, A. Mikelic and A. Piatnitski, Modèle de double porosité aléatoire, C. R. Acad. Sci. Paris, Sér. 1, 327 (1998), 99-104. Google Scholar [20] A. Braides, V. Chiadò Piat and A. Piatnitski, Homogenization of discrete high-contrast energies, SIAM J. Math. Anal., 47 (2015), 3064-3091. doi: 10.1137/140975668. Google Scholar [21] G. Chavent, J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986.Google Scholar [22] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. Google Scholar [23] C. Choquet, Derivation of the double porosity model of a compressible miscible displacement in naturally fractured reservoirs, Appl. Anal., 83 (2004), 477-499. doi: 10.1080/00036810310001643194. Google Scholar [24] C. Choquet and L. Pankratov, Homogenization of a class of quasilinear elliptic equations with non-standard growth in high-contrast media, Proc. Roy. Soc. Edinburgh, 140 (2010), 495-539. doi: 10.1017/S0308210509000985. Google Scholar [25] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9. Google Scholar [26] G. W. Clark and R. E. Showalter, Two-scale convergence of a model for flow in a partially fissured medium, Electron. J. Differential Equations, 1999 (1999), 1-20. Google Scholar [27] H. I. Ene and D. Polisevski, Model of diffusion in partially fissured media, Z. Angew. Math. Phys., 53 (2002), 1052-1059. doi: 10.1007/PL00013849. Google Scholar [28] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer, Berlin, 1997.Google Scholar [29] 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. Google Scholar [30] U. Hornung, Homogenization and Porous Media, Springer-Verlag, New York, 1997.Google Scholar [31] M. Jurak, L. Pankratov and A. Vrbaški, A fully homogenized model for incompressible two-phase flow in double porosity media, Appl. Anal., 95 (2016), 2280-2299. doi: 10.1080/00036811.2015.1031221. Google Scholar [32] V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations Boston, Birkhäuser, 2006. Google Scholar [33] M. Panfilov, Macroscale Models of Flow Through Highly Heterogeneous Porous Media, Kluwer Academic Publishers, Dordrecht-Boston-London, 2000. doi: 10.1007/978-94-015-9582-7. Google Scholar [34] L. Pankratov and V. Rybalko, Asymptotic analysis of a double porosity model with thin fissures, Mat. Sb., 194 (2003), 121-146. Google Scholar [35] G. Sandrakov, Averaging of parabolic equations with contrasting coefficients, Izv. Math., 63 (1999), 1015-1061. Google Scholar [36] R. P. Shaw, Gas Generation and Migration in Deep Geological Radioactive Waste Repositories. Geological Society, 2015.Google Scholar [37] J. Simon, Compact sets in the space $L^p(0,t; B)$, Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. Google Scholar [38] T. D. van Golf-Racht, Fundamentals of Fractured Reservoir Engineering, Elsevier Scientific Pulishing Company, Amsterdam, 1982.Google Scholar [39] J. L. Vázquez, The Porous Medium Equation, Oxford University Press Inc., New York, 2007. Google Scholar [40] L. M. Yeh, Homogenization of two-phase flow in fractured media, Math. Models Methods Appl. Sci., 16 (2006), 1627-1651. doi: 10.1142/S0218202506001650. Google Scholar

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##### References:
 [1] E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percival, An extension theorem from connected sets, and homogenization in general periodic domains, J. Nonlinear Analysis, 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7. Google Scholar [2] L. Ait Mahiout, B. Amaziane, A. Mokrane and L. Pankratov, Homogenization of immiscible compressible two-phase flow in double porosity media, Electron. J. Differential Equations, 2016 (2016), 1-28. Google Scholar [3] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar [4] G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (eds. A. Bourgeat et al. ), World Scientific Pub., Singapore, (1996), 15–25.Google Scholar [5] 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 MMS, 8 (2010), 2023-2047. doi: 10.1137/100790215. Google Scholar [6] B. Amaziane and L. Pankratov, Homogenization of a model for water-gas flow through double-porosity media, Math. Methods Appl. Sci., 39 (2016), 425-451. doi: 10.1002/mma.3493. Google Scholar [7] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media, Proc. Roy. Soc. Edinburgh, 136 (2006), 1131-1155. doi: 10.1017/S0308210500004911. Google Scholar [8] B. Amaziane, L. Pankratov and A. Piatnitski, Nonlinear flow through double porosity media in variable exponent Sobolev spaces, Nonlinear Anal. Real World Appl., 10 (2009), 2521-2530. doi: 10.1016/j.nonrwa.2008.05.008. Google Scholar [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 Contin. Dyn. Syst. Ser. B, 18 (2013), 1217-1251. doi: 10.3934/dcdsb.2013.18.1217. Google Scholar [10] B. Amaziane, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures, Math. Models Methods Appl. Sci., 24 (2014), 1421-1451. doi: 10.1142/S0218202514500055. Google Scholar [11] B. Amaziane, L. Pankratov and V. Rybalko, On the homogenization of some double porosity models with periodic thin structures, Appl. Anal., 88 (2009), 1469-1492. doi: 10.1080/00036810903114817. Google Scholar [12] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. Google Scholar [13] T. Arbogast, J. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-826. doi: 10.1137/0521046. Google Scholar [14] G. I. Barenblatt, Yu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303. doi: 10.1016/0021-8928(60)90107-6. Google Scholar [15] J. Bear, C. F. Tsang and G. de Marsily, Flow and Contaminant Transport in Fractured Rock, Academic Press Inc, London, 1993.Google Scholar [16] A. Bourgeat, G. Chechkin and A. Piatnitski, Singular double porosity model, Appl. Anal., 82 (2003), 103-116. doi: 10.1080/0003681031000063739. Google Scholar [17] A. Bourgeat, M. Goncharenko, M. Panfilov and L. Pankratov, A general double porosity model, C. R. Acad. Sci. Paris, Série IIb, 327 (1999), 1245-1250. Google Scholar [18] A. Bourgeat, S. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immicible two-phase flow, SIAM J. Math. Anal., 27 (1996), 1520-1543. doi: 10.1137/S0036141094276457. Google Scholar [19] A. Bourgeat, A. Mikelic and A. Piatnitski, Modèle de double porosité aléatoire, C. R. Acad. Sci. Paris, Sér. 1, 327 (1998), 99-104. Google Scholar [20] A. Braides, V. Chiadò Piat and A. Piatnitski, Homogenization of discrete high-contrast energies, SIAM J. Math. Anal., 47 (2015), 3064-3091. doi: 10.1137/140975668. Google Scholar [21] G. Chavent, J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986.Google Scholar [22] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. Google Scholar [23] C. Choquet, Derivation of the double porosity model of a compressible miscible displacement in naturally fractured reservoirs, Appl. Anal., 83 (2004), 477-499. doi: 10.1080/00036810310001643194. Google Scholar [24] C. Choquet and L. Pankratov, Homogenization of a class of quasilinear elliptic equations with non-standard growth in high-contrast media, Proc. Roy. Soc. Edinburgh, 140 (2010), 495-539. doi: 10.1017/S0308210509000985. Google Scholar [25] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9. Google Scholar [26] G. W. Clark and R. E. Showalter, Two-scale convergence of a model for flow in a partially fissured medium, Electron. J. Differential Equations, 1999 (1999), 1-20. Google Scholar [27] H. I. Ene and D. Polisevski, Model of diffusion in partially fissured media, Z. Angew. Math. Phys., 53 (2002), 1052-1059. doi: 10.1007/PL00013849. Google Scholar [28] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer, Berlin, 1997.Google Scholar [29] 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. Google Scholar [30] U. Hornung, Homogenization and Porous Media, Springer-Verlag, New York, 1997.Google Scholar [31] M. Jurak, L. Pankratov and A. Vrbaški, A fully homogenized model for incompressible two-phase flow in double porosity media, Appl. Anal., 95 (2016), 2280-2299. doi: 10.1080/00036811.2015.1031221. Google Scholar [32] V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations Boston, Birkhäuser, 2006. Google Scholar [33] M. Panfilov, Macroscale Models of Flow Through Highly Heterogeneous Porous Media, Kluwer Academic Publishers, Dordrecht-Boston-London, 2000. doi: 10.1007/978-94-015-9582-7. Google Scholar [34] L. Pankratov and V. Rybalko, Asymptotic analysis of a double porosity model with thin fissures, Mat. Sb., 194 (2003), 121-146. Google Scholar [35] G. Sandrakov, Averaging of parabolic equations with contrasting coefficients, Izv. Math., 63 (1999), 1015-1061. Google Scholar [36] R. P. Shaw, Gas Generation and Migration in Deep Geological Radioactive Waste Repositories. Geological Society, 2015.Google Scholar [37] J. Simon, Compact sets in the space $L^p(0,t; B)$, Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. Google Scholar [38] T. D. van Golf-Racht, Fundamentals of Fractured Reservoir Engineering, Elsevier Scientific Pulishing Company, Amsterdam, 1982.Google Scholar [39] J. L. Vázquez, The Porous Medium Equation, Oxford University Press Inc., New York, 2007. Google Scholar [40] L. M. Yeh, Homogenization of two-phase flow in fractured media, Math. Models Methods Appl. Sci., 16 (2006), 1627-1651. doi: 10.1142/S0218202506001650. Google Scholar
(a) The domain $\Omega$. (b) The reference cell $Y$
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