Finite volume discretization with imposed flux continuity for the general tensor pressure equation

We present a new family of flux continuous, locally conservative, finite volume schemes applicable to the diagonal and full tensor pressure equations with generally discontinuous coefficients. For a uniformly constant symmetric elliptic tensor field, the full tensor discretization is second order accurate with a symmetric positive definite matrix. For a full tensor, an M-matrix with diagonal dominance can be obtained subject to a sufficient condition for ellipticity. Positive definiteness of the discrete system is illustrated. Convergence rates for discontinuous coefficients are presented and the importance of modeling the full permeability tensor pressure equation is demonstrated.

[1]  L. Durlofsky.  Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media , 1991 .

[2]  M. Wheeler,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences , 1997 .

[3]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[4]  J. Oden.  Finite Elements: A Second Course , 1983 .

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

[6]  Y. M. Chen,et al.  Modeling reservoir geometry with irregular grids , 1991 .

[7]  M. Edwards.  Elimination of Adaptive Grid Interface Errors in the Discrete Cell Centered Pressure Equation , 1996 .

[8]  J. L. Yanosik,et al.  A Nine-Point, Finite-Difference Reservoir Simulator for Realistic Prediction of Adverse Mobility Ratio Displacements , 1979 .

[9]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[10]  R. Horne,et al.  Computing Absolute Transmissibility in the Presence of Fine-Scale Heterogeneity , 1987 .

[11]  Louis J. Durlofsky,et al.  Finite Difference Simulation of Geologically Complex Reservoirs With Tensor Permeabilities , 1998 .

[12]  I. Aavatsmark,et al.  Control-Volume Discretization Methods for 3D Quadrilateral Grids in Inhomogeneous, Anisotropic Reservoirs , 1998 .

[13]  M. G. Edwards.  Superconvergent Renormalization and Tensor Approximation , 1996 .

[14]  Todd Arbogast,et al.  Logically rectangular mixed methods for Darcy flow on general geometry , 1995 .

[15]  T. F. Russell,et al.  Finite element and finite difference methods for continuous flows in porous media. , 1800 .

[16]  M. Edwards,et al.  Multigrid and Renormalization for Reservoir Simulation , 1994 .

[17]  Michael G. Edwards.  Symmetric Flux Continuous Positive Definite Approximation of the Elliptic Full Tensor Pressure Equation in Local Conservation Form , 1995 .

[18]  G. Shiralkar.  Reservoir simulation of generally anisotropic systems , 1990 .

[19]  G. R. Shubin,et al.  An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions , 1988 .

[20]  Ivar Aavatsmark,et al.  Discretization on Non-Orthogonal, Quadrilateral Grids for Inhomogeneous, Anisotropic Media , 1996 .

[21]  Michael G. Edwards.  Cross flow tensors and finite volume approximation with by deferred correction , 1998 .

[22]  Michael G. Edwards,et al.  A Flux Continuous Scheme for the Full Tensor Pressure Equation , 1994 .

[23]  M. Shashkov,et al.  The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials , 1997 .

[24]  C. L. Farmer,et al.  A Global Optimization Approach to Grid Generation , 1991 .

[25]  I. Faille.  Control Volume Method to Model Fluid Flow on 2D Irregular Meshing , 1990 .

[26]  K. H. Coats,et al.  Effects of grid type and difference scheme on pattern steamflood simulation results , 1982 .

[27]  M. J. King.  Application and Analysis of Tensor Permeability to Crossbedded Reservoirs , 1993 .

[28]  K. Aziz,et al.  Petroleum Reservoir Simulation , 1979 .

[29]  Keith H. Coats.  A consistent method for calculating transmissibilities in nine-point difference equations , 1983 .

[30]  R. Aris.  Vectors, Tensors and the Basic Equations of Fluid Mechanics , 1962 .