Degenerate two-phase incompressible flow III. Sharp error estimates

Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system.

[1]  Stephan Luckhaus,et al.  Flow of Oil and Water in a Porous Medium , 1984 .

[2]  D. W. Peaceman.  Fundamentals of numerical reservoir simulation , 1977 .

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  Zhangxin Chen,et al.  Degenerate two-phase incompressible flow II: regularity, stability and stabilization☆ , 2002 .

[5]  Ricardo H. Nochetto,et al.  Approximation of Degenerate Parabolic Problems Using Numerical Integration , 1988 .

[6]  Zhangxin Chen.  Degenerate Two-Phase Incompressible Flow: I. Existence, Uniqueness and Regularity of a Weak Solution , 2001 .

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

[8]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[9]  Todd Arbogast,et al.  A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media , 1996 .

[10]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[11]  S. Kružkov,et al.  BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF EQUATIONS OF TWO-PHASE POROUS FLOW TYPE; STATEMENT OF THE PROBLEMS, QUESTIONS OF SOLVABILITY, JUSTIFICATION OF APPROXIMATE METHODS , 1977 .

[12]  H. Alt,et al.  Nonsteady flow of water and oil through inhomogeneous porous media , 1985 .

[13]  Todd Arbogast.  The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow , 1992 .

[14]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[15]  J. Douglas,et al.  Prismatic mixed finite elements for second order elliptic problems , 1989 .

[16]  M. Fortin,et al.  E cient rectangular mixed fi-nite elements in two and three space variables , 1987 .

[17]  E. Shamir,et al.  Regularization of mixed second-order elliptic problems , 1968 .

[18]  M. Fortin,et al.  Mixed finite elements for second order elliptic problems in three variables , 1987 .

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

[20]  Magne S. Espedal,et al.  Continuous-time finite element analysis of multiphase flow in groundwater hydrology , 1994 .

[21]  Zhangxin Chen,et al.  Fully Discrete Finite Element Analysis of Multiphase Flow in Groundwater Hydrology , 1997 .