# American Institute of Mathematical Sciences

April  2014, 7(2): 317-346. doi: 10.3934/dcdss.2014.7.317

## Numerical analysis of a non equilibrium two-component two-compressible flow in porous media

 1 King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering, 23955-6900, Thuwal 2 Ecole Centrale de nantes, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 1, rue de la Noé, 44321 Nantes

Received  May 2013 Revised  June 2013 Published  September 2013

We propose and analyze a finite volume scheme to simulate a non equilibrium two components (water and hydrogen) two phase flow (liquid and gas) model. In this model, the assumption of local mass non equilibrium is ensured and thus the velocity of the mass exchange between dissolved hydrogen and hydrogen in the gas phase is supposed finite.
The proposed finite volume scheme is fully implicit in time together with a phase-by-phase upwind approach in space and it is discretize the equations in their general form with gravity and capillary terms We show that the proposed scheme satisfies the maximum principle for the saturation and the concentration of the dissolved hydrogen. We establish stability results on the velocity of each phase and on the discrete gradient of the concentration. We show the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. At our knowledge, this is the first convergence result of finite volume scheme in the case of two component two phase compressible flow in several space dimensions.
Citation: 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
##### References:
 [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 3 (1983), 311-341. doi: 10.1007/BF01176474.  Google Scholar [2] B. Amaziane and M. El Ossmani, Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods, Numer. Methods Partial Differential Equations, 24 (2008), 799-832. doi: 10.1002/num.20291.  Google Scholar [3] Y. Amirat, D. Bates and A. Ziani, Convergence of a mixed finite element-finite volume scheme for a parabolic-hyperbolic system modeling a compressible miscible flow in porous media, Numer. Math., (2005). Google Scholar [4] B. Andreianov, M. Bendahmane and R. Ruiz Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics, M3AS Math. Models Meth. Appl. Sci., 21 (2011), 307-344. doi: 10.1142/S0218202511005064.  Google Scholar [5] T. Arbogast, Two-phase incompressible flow in a porous medium with various non homogeneous boundary conditions, IMA Preprint Series 606, (1990). Google Scholar [6] J. Bear, "Dynamics of Fluids in Porous Media," Dover, 1986. Google Scholar [7] M. Bendahmane, Z. Khalil and M. Saad, Convergence of a finite volume scheme for gas water flow in a multi-dimensional porous media, arXiv:1102.0582, (2011). Google Scholar [8] Y. Brenier and J. Jaffré, Upstream differencing for multiphase flow in reservoir simulation, SIAM J. Numer. Anal., 28 (1991), 685-696. doi: 10.1137/0728036.  Google Scholar [9] A. Bourgeat, M. Jurak and F. Smai, Two partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository, Comput. Geosci., 6 (2009), 309-325. doi: 10.1007/s10596-008-9102-1.  Google Scholar [10] 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.  Google Scholar [11] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows through Porous Media," North Holland, 1986. Google Scholar [12] Z. Chen and R. E. Ewing, Mathematical analysis for reservoirs models, SIAM J. Math. Anal., 30 (1999), 431-452. doi: 10.1137/S0036141097319152.  Google Scholar [13] Z. Chen, Degenerate two-phase incompressible flow. Existence, uniqueness and regularity of a weak solution, Journal of Differential Equations, 171 (2001), 203-232. doi: 10.1006/jdeq.2000.3848.  Google Scholar [14] Z. Chen, Degenerate two-phase incompressible flow. Regularity, stability and stabilization, Journal of Differential Equations, 186 (2002), 345-376. doi: 10.1016/S0022-0396(02)00027-X.  Google Scholar [15] Y. Coudiére, J. P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem, M2AN Math. Model. Numer. Anal., 33 (1999), 493-516. doi: 10.1051/m2an:1999149.  Google Scholar [16] L. Evans, "Partial Differential Equations," Second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar [17] R. Eymard, T. Gallouët and R. Herbin, A finite volume scheme for anisotropic diffusion problems, C. R. Math. Acad. Sci. Paris, 339 (2004), 299-302. doi: 10.1016/j.crma.2004.05.023.  Google Scholar [18] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis. Vol. VII" (eds. P. Ciarlet and J.-L. Lions), North-Holland, Amsterdam, (2000), 713-1020.  Google Scholar [19] R. Eymard, R. Herbin and A. Michel, Mathematical study of a petroleum-engineering scheme, Mathematical Modelling and Numerical Analysis, 37 (2003), 937-972, doi: 10.1051/m2an:2003062.  Google Scholar [20] P. Fabrie, P. Le Thiez and P. Tardy, On a system of nonlinear elliptic and degenerate parabolic equations describing compositional water-oil flows in porous media, Nonlinear Anal., 28 (1997), 1565-1600. doi: 10.1016/S0362-546X(96)00002-8.  Google Scholar [21] I. Faille, A control volume method to solve an elliptic equation on a two-dimensional irregular mesh, Comput. Methods Appl. Mech. Engrg., 100 (1992), 275-290. doi: 10.1016/0045-7825(92)90186-N.  Google Scholar [22] G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière, Mathématiques & Applications (Berlin), 22, Springer-Verlag, Berlin, 1996.  Google Scholar [23] C. Galusinski and M. Saad, On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media, Advances in Diff. Eq., 9 (2004), 1235-1278.  Google Scholar [24] C. Galusinski and M. Saad, A nonlinear degenerate system modeling water-gas in reservoir flows, Discrete and Continuous Dynamical System, 9 (2008), 281-308.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] 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.  Google Scholar [28] 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.  Google Scholar [29] A. Michel, A finite volume scheme for the simulation of two-phase incompressible flow in porous media, SIAM J. Numer. Anal., 41 (2003), 1301-1317. doi: 10.1137/S0036142900382739.  Google Scholar [30] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow, Comptes rendus Mécanique, 337 (2009), 226-232. Google Scholar [31] D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation," Elsevier Scientific Publishing, 1977. Google Scholar [32] B. Saad and M. Saad, Study of full implicit petroleum engineering finite-volume scheme for compressible two-phase flow in porous media, SIAM J. Numer. Anal., 51 (2013), 716-741. doi: 10.1137/120869092.  Google Scholar [33] B. Saad, "Modélisation et Simulation Numérique d'Écoulements Multi-Composants en Milieu Poreux," Thèse de doctorat de l'Ecole Centrale de Nantes, 2011. Google Scholar [34] 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, Ser. I, 347 (2009), 527-532. doi: 10.1016/j.crma.2009.03.011.  Google Scholar

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##### References:
 [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 3 (1983), 311-341. doi: 10.1007/BF01176474.  Google Scholar [2] B. Amaziane and M. El Ossmani, Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods, Numer. Methods Partial Differential Equations, 24 (2008), 799-832. doi: 10.1002/num.20291.  Google Scholar [3] Y. Amirat, D. Bates and A. Ziani, Convergence of a mixed finite element-finite volume scheme for a parabolic-hyperbolic system modeling a compressible miscible flow in porous media, Numer. Math., (2005). Google Scholar [4] B. Andreianov, M. Bendahmane and R. Ruiz Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics, M3AS Math. Models Meth. Appl. Sci., 21 (2011), 307-344. doi: 10.1142/S0218202511005064.  Google Scholar [5] T. Arbogast, Two-phase incompressible flow in a porous medium with various non homogeneous boundary conditions, IMA Preprint Series 606, (1990). Google Scholar [6] J. Bear, "Dynamics of Fluids in Porous Media," Dover, 1986. Google Scholar [7] M. Bendahmane, Z. Khalil and M. Saad, Convergence of a finite volume scheme for gas water flow in a multi-dimensional porous media, arXiv:1102.0582, (2011). Google Scholar [8] Y. Brenier and J. Jaffré, Upstream differencing for multiphase flow in reservoir simulation, SIAM J. Numer. Anal., 28 (1991), 685-696. doi: 10.1137/0728036.  Google Scholar [9] A. Bourgeat, M. Jurak and F. Smai, Two partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository, Comput. Geosci., 6 (2009), 309-325. doi: 10.1007/s10596-008-9102-1.  Google Scholar [10] 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.  Google Scholar [11] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows through Porous Media," North Holland, 1986. Google Scholar [12] Z. Chen and R. E. Ewing, Mathematical analysis for reservoirs models, SIAM J. Math. Anal., 30 (1999), 431-452. doi: 10.1137/S0036141097319152.  Google Scholar [13] Z. Chen, Degenerate two-phase incompressible flow. Existence, uniqueness and regularity of a weak solution, Journal of Differential Equations, 171 (2001), 203-232. doi: 10.1006/jdeq.2000.3848.  Google Scholar [14] Z. Chen, Degenerate two-phase incompressible flow. Regularity, stability and stabilization, Journal of Differential Equations, 186 (2002), 345-376. doi: 10.1016/S0022-0396(02)00027-X.  Google Scholar [15] Y. Coudiére, J. P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem, M2AN Math. Model. Numer. Anal., 33 (1999), 493-516. doi: 10.1051/m2an:1999149.  Google Scholar [16] L. Evans, "Partial Differential Equations," Second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar [17] R. Eymard, T. Gallouët and R. Herbin, A finite volume scheme for anisotropic diffusion problems, C. R. Math. Acad. Sci. Paris, 339 (2004), 299-302. doi: 10.1016/j.crma.2004.05.023.  Google Scholar [18] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis. Vol. VII" (eds. P. Ciarlet and J.-L. Lions), North-Holland, Amsterdam, (2000), 713-1020.  Google Scholar [19] R. Eymard, R. Herbin and A. Michel, Mathematical study of a petroleum-engineering scheme, Mathematical Modelling and Numerical Analysis, 37 (2003), 937-972, doi: 10.1051/m2an:2003062.  Google Scholar [20] P. Fabrie, P. Le Thiez and P. Tardy, On a system of nonlinear elliptic and degenerate parabolic equations describing compositional water-oil flows in porous media, Nonlinear Anal., 28 (1997), 1565-1600. doi: 10.1016/S0362-546X(96)00002-8.  Google Scholar [21] I. Faille, A control volume method to solve an elliptic equation on a two-dimensional irregular mesh, Comput. Methods Appl. Mech. Engrg., 100 (1992), 275-290. doi: 10.1016/0045-7825(92)90186-N.  Google Scholar [22] G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière, Mathématiques & Applications (Berlin), 22, Springer-Verlag, Berlin, 1996.  Google Scholar [23] C. Galusinski and M. Saad, On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media, Advances in Diff. Eq., 9 (2004), 1235-1278.  Google Scholar [24] C. Galusinski and M. Saad, A nonlinear degenerate system modeling water-gas in reservoir flows, Discrete and Continuous Dynamical System, 9 (2008), 281-308.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] 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.  Google Scholar [28] 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.  Google Scholar [29] A. Michel, A finite volume scheme for the simulation of two-phase incompressible flow in porous media, SIAM J. Numer. Anal., 41 (2003), 1301-1317. doi: 10.1137/S0036142900382739.  Google Scholar [30] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow, Comptes rendus Mécanique, 337 (2009), 226-232. Google Scholar [31] D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation," Elsevier Scientific Publishing, 1977. Google Scholar [32] B. Saad and M. Saad, Study of full implicit petroleum engineering finite-volume scheme for compressible two-phase flow in porous media, SIAM J. Numer. Anal., 51 (2013), 716-741. doi: 10.1137/120869092.  Google Scholar [33] B. Saad, "Modélisation et Simulation Numérique d'Écoulements Multi-Composants en Milieu Poreux," Thèse de doctorat de l'Ecole Centrale de Nantes, 2011. Google Scholar [34] 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, Ser. I, 347 (2009), 527-532. doi: 10.1016/j.crma.2009.03.011.  Google Scholar
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