Convergence Analysis of a Mixed Finite Volume Scheme for an Elliptic-Parabolic System Modeling Miscible Fluid Flows in Porous Media

We study a finite volume discretization of a strongly coupled elliptic-parabolic PDE system describing miscible displacement in a porous medium. We discretize each equation by a finite volume scheme which allows a wide variety of unstructured grids (in any space dimension) and gives strong enough convergence for handling the nonlinear coupling of the equations. We prove the convergence of the scheme as the time and space steps go to $0$. Finally, we provide numerical results to demonstrate the efficiency of the proposed numerical scheme.

[1]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[2]  Richard E. Ewing,et al.  Mathematical analysis for reservoir models , 1999 .

[3]  Jérôme Droniou,et al.  Finite volume schemes for fully non-linear elliptic equations in divergence form , 2006 .

[4]  Richard E. Ewing,et al.  A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media , 1983 .

[5]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[6]  Richard E. Ewing,et al.  Simulation of miscible displacement using mixed methods and a modified method of characteristics , 1983 .

[7]  A. Ziani,et al.  Asymptotic Behavior of the Solutions of an Elliptic-Parabolic System Arising in Flow in Porous Media , 2004 .

[8]  Richard E. Ewing,et al.  Efficient Time-Stepping Methods for Miscible Displacement Problems in Porous Media , 1982 .

[9]  Dong Liang,et al.  An Approximation to Miscible Fluid Flows in Porous Media with Point Sources and Sinks by an Eulerian-Lagrangian Localized Adjoint Method and Mixed Finite Element Methods , 2000, SIAM J. Sci. Comput..

[10]  Robert Eymard,et al.  A mixed finite volume scheme for anisotropic diffusion problems on any grid , 2006, Numerische Mathematik.

[11]  Ping Lin,et al.  An Iterative Perturbation Method for the Pressure Equation in the Simulation of Miscible Displacement in Porous Media , 1998, SIAM J. Sci. Comput..

[12]  Members Spe-Aime.  Simulation of Miscible Displacement Using Mixed Methods and a Modified Method of Characteristics , 1983 .

[13]  Richard E. Ewing,et al.  The approximation of the pressure by a mixed method in the simulation of miscible displacement , 1983 .

[14]  Dong Liang,et al.  An improved numerical simulator for different types of flows in porous media , 2003 .

[15]  X. B. Feng.  On Existence and Uniqueness Results for a Coupled System Modeling Miscible Displacement in Porous Media , 1995 .

[16]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[17]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

[18]  I. Faille,et al.  A control volume method to solve an elliptic equation on a two-dimensional irregular mesh , 1992 .

[19]  T. F. Russell,et al.  Finite Elements With Characteristics for Two-Component Incompressible Miscible Displacement , 1982 .

[20]  T. F. Russell,et al.  Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics , 1984 .