Abstract
We propose and analyze a combined finite volume–nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on its own pressure.
Similar content being viewed by others
Notes
This means that there exists a positive constant \(c\) such that for all \(a, b \in [0,\mathcal {B}(1)],\) one has \(|\mathcal {B}^{-1}(a)-\mathcal {B}^{-1}(b)|\le c|a - b|^{\theta }\).
References
Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers LTD, London (1979)
Barrett, J.W., Knabner, P.: Finite element approximation of the transport of reactive solutes in porous media. ii. error estimates for equilibrium adsorption processes. SIAM J. Numer. Anal. 34(2), 455–479 (1997)
Bendahmane, M., Khalil, Z., Saad, M.: Convergence of a finite volume scheme for gas water flow in a multi-dimensional porous media. Math. Models Methods Appl. Sci. 24(1), 145–185 (2014)
Brenier, Y., Jaffré, J.: Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28, 685–696 (1991)
Brezis, H.: Analyse fonctionnelle. Collection of Applied Mathematics for the Master’s Degree, Theory and applications. Masson, Paris (1983)
Caro, F., Saad, B., Saad, M.: Two-component two-compressible flow in a porous medium. Acta Appl. Math. 117(1), 15–46 (2012)
Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows Through Porous Media. Elsevier Science Publishers B. V., North Holland (1986)
Chen, Z., Ewing, R.E.: Degenerate two-phase incompressible flow. iii. Sharp error estimates. Numer. Math. 90(2), 215–240 (2001)
Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, North-Holland (1991)
Coudiére, Y., Vila, J.P., Villedieu, P.: Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. M2AN. Math. Model. Numer. Anal. 33(3), 493–516 (1999)
Dawson, C.: Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations. SIAM J. Numer. Anal. 35(5), 1709–1724 (1998)
Debiez, C., Dervieux, A., Mer, K., Nkonga, B.: Computation of unsteady flows with mixed finite volume/finite element upwind methods. Internat. J. Numer. Methods Fluids 27(1–4, Special Issue), 193–206 (1998)
Evans, L.: Partial Differential Equations. American Mathematical Society, Washington, D.C (2010)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, North-Holland (2000)
Eymard, R., Gallouët, T., Herbin, R.: A finite volume scheme for anisotropic diffusion problems. C. R. Math. Acad. Sci. Paris 339(4), 299–302 (2004)
Eymard, R., Herbin, R., Michel, A.: Mathematical study of a petroleum-engineering scheme. Math. Model. Numer. Anal. 37(6), 937–972 (2003)
Eymard, R., Hilhorst, D., Vohralik, M.: A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105, 73–131 (2006)
Faille, I.: A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Comput. Methods Appl. Mech. Eng. 100(2), 275–290 (1992)
Feistauer, M., Felcman, J., Lukáčová, M.: On the convergence of a combined finite volume-finite element method for nonlinear convection. Numer. Methods Partial Differ. Equ. 13(2), 163–190 (1997)
Galusinski, C., Saad, M.: Two compressible immiscible fluids in porous media. J. Differ. Equ. 244, 1741–1783 (2008)
Khalil, Z., Saad, M.: Solutions to a model for compressible immiscible two phase flow in porous media. Electron. J. Differ. Equ. 2010(122), 1–33 (2010)
Michel, A.: A finite volume scheme for the simulation of two-phase incompressible flow in porous media. SIAM J. Numer. Anal. 41, 1301–1317 (2003)
Nochetto, R.H., Schmidt, A., Verdi, C.: A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comput. 69(229), 1–24 (2000)
Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing, Amsterdam (1977)
Rulla, J., Walkington, N.J.: Optimal rates of convergence for degenerate parabolic problems in two dimensions. SIAM J. Numer. Anal. 33(1), 56–67 (1996)
Saad, B., Saad, M.: Study of full implicit petroleum engineering finite volume scheme for compressible two phase flow in porous media. SIAM J. Numer. Anal. 51(1), 716–741 (2013)
Vohralik, M.: On the discrete poincaré-friedrichs inequalities for nonconforming approximations of the sobolev space h1. Numer. Funct. Anal. Optim. 26(78), 925–952 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST) and this work is partially supported by GDR MOMAS.
Rights and permissions
About this article
Cite this article
Saad, B., Saad, M. A combined finite volume–nonconforming finite element scheme for compressible two phase flow in porous media. Numer. Math. 129, 691–722 (2015). https://doi.org/10.1007/s00211-014-0651-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-014-0651-z