# 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  March 2018 Early access  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 and 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. [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. [3] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [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. [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. [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. [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. [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. [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. [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. [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. [12] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. [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. [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. [15] J. Bear, C. F. Tsang and G. de Marsily, Flow and Contaminant Transport in Fractured Rock, Academic Press Inc, London, 1993. [16] A. Bourgeat, G. Chechkin and A. Piatnitski, Singular double porosity model, Appl. Anal., 82 (2003), 103-116.  doi: 10.1080/0003681031000063739. [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. [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. [19] A. Bourgeat, A. Mikelic and A. Piatnitski, Modèle de double porosité aléatoire, C. R. Acad. Sci. Paris, Sér. 1, 327 (1998), 99-104. [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. [21] G. Chavent, J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986. [22] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. [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. [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. [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. [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. [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. [28] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer, Berlin, 1997. [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. [30] U. Hornung, Homogenization and Porous Media, Springer-Verlag, New York, 1997. [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. [32] V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations Boston, Birkhäuser, 2006. [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. [34] L. Pankratov and V. Rybalko, Asymptotic analysis of a double porosity model with thin fissures, Mat. Sb., 194 (2003), 121-146. [35] G. Sandrakov, Averaging of parabolic equations with contrasting coefficients, Izv. Math., 63 (1999), 1015-1061. [36] R. P. Shaw, Gas Generation and Migration in Deep Geological Radioactive Waste Repositories. Geological Society, 2015. [37] J. Simon, Compact sets in the space $L^p(0,t; B)$, Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. [38] T. D. van Golf-Racht, Fundamentals of Fractured Reservoir Engineering, Elsevier Scientific Pulishing Company, Amsterdam, 1982. [39] J. L. Vázquez, The Porous Medium Equation, Oxford University Press Inc., New York, 2007. [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.

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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. [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. [3] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [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. [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. [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. [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. [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. [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. [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. [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. [12] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. [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. [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. [15] J. Bear, C. F. Tsang and G. de Marsily, Flow and Contaminant Transport in Fractured Rock, Academic Press Inc, London, 1993. [16] A. Bourgeat, G. Chechkin and A. Piatnitski, Singular double porosity model, Appl. Anal., 82 (2003), 103-116.  doi: 10.1080/0003681031000063739. [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. [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. [19] A. Bourgeat, A. Mikelic and A. Piatnitski, Modèle de double porosité aléatoire, C. R. Acad. Sci. Paris, Sér. 1, 327 (1998), 99-104. [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. [21] G. Chavent, J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986. [22] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. [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. [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. [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. [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. [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. [28] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer, Berlin, 1997. [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. [30] U. Hornung, Homogenization and Porous Media, Springer-Verlag, New York, 1997. [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. [32] V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations Boston, Birkhäuser, 2006. [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. [34] L. Pankratov and V. Rybalko, Asymptotic analysis of a double porosity model with thin fissures, Mat. Sb., 194 (2003), 121-146. [35] G. Sandrakov, Averaging of parabolic equations with contrasting coefficients, Izv. Math., 63 (1999), 1015-1061. [36] R. P. Shaw, Gas Generation and Migration in Deep Geological Radioactive Waste Repositories. Geological Society, 2015. [37] J. Simon, Compact sets in the space $L^p(0,t; B)$, Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. [38] T. D. van Golf-Racht, Fundamentals of Fractured Reservoir Engineering, Elsevier Scientific Pulishing Company, Amsterdam, 1982. [39] J. L. Vázquez, The Porous Medium Equation, Oxford University Press Inc., New York, 2007. [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.
(a) The domain $\Omega$. (b) The reference cell $Y$
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