\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract / Introduction Related Papers Cited by
  • 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.
    Mathematics Subject Classification: 65M08, 76S05.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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 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.

    [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).

    [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.

    [5]

    T. Arbogast, Two-phase incompressible flow in a porous medium with various non homogeneous boundary conditions, IMA Preprint Series 606, (1990).

    [6]

    J. Bear, "Dynamics of Fluids in Porous Media," Dover, 1986.

    [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).

    [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.

    [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.

    [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.

    [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.

    [12]

    Z. Chen and R. E. Ewing, Mathematical analysis for reservoirs models, SIAM J. Math. Anal., 30 (1999), 431-452.doi: 10.1137/S0036141097319152.

    [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.

    [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.

    [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.

    [16]

    L. Evans, "Partial Differential Equations," Second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [30]

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

    [31]

    D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation," Elsevier Scientific Publishing, 1977.

    [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.

    [33]

    B. Saad, "Modélisation et Simulation Numérique d'Écoulements Multi-Composants en Milieu Poreux," Thèse de doctorat de l'Ecole Centrale de Nantes, 2011.

    [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(110) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return