A higher-order Godunov scheme coupled with dynamic local grid refinement for flow in a porous medium

Abstract Today's reservoir simulators are generally inefficient due to the use of low-order accurate schemes in space and time on regular grids. The numerical diffusion inherent in the single-point upstream weighting scheme can cause large errors in the numerical solutions, which can only be removed by the use of very fine uniform grids with many grid blocks. The aim of this work is to develop a method which can produce physically reliable and accurate results efficiently. A higher-order Godunov scheme is coupled with dynamic grid adaptivity, where grid blocks are inserted in highly active regions of the flow field and removed from regions of inactivity, thereby concentrating the computational effort where it is most needed. The quality of results computed by the adaptive higher-order scheme are comparable with those computed by the higher-order scheme on a uniform grid, globally refined to the level of the finest adaptive grid zones, while great savings in computer time are obtained. The adaptive high-order scheme is vastly superior compared to the first-order scheme on a uniform or adaptive grid. The results presented demonstrate the benefits of the method in reservoir simulation.

[1]  J. Tinsley Oden,et al.  An h-r-Adaptive Approximate Riemann Solver for the Euler Equations in Two Dimensions , 1993, SIAM J. Sci. Comput..

[2]  M. L. Wasserman Local Grid Refinement for Three-Dimensional Simulators , 1987 .

[3]  P. Quandalle,et al.  Reduction of Grid Effects Due to Local Sub-Gridding in Simulations Using a Composite Grid , 1985 .

[4]  Khalid Aziz,et al.  Efficient Use of Domain Decomposition and Local Grid Refinement in Reservoir Simulation , 1990 .

[5]  S. Osher,et al.  Upwind difference schemes for hyperbolic systems of conservation laws , 1982 .

[6]  Gary A. Pope,et al.  The Application of Fractional Flow Theory to Enhanced Oil Recovery , 1980 .

[7]  I. Yoshiaki The Evaluation of Interblock Mobility Using a Modified Midpoint Weighting Scheme , 1982 .

[8]  G. Karami Lecture Notes in Engineering , 1989 .

[9]  H. Morel‐Seytoux Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium , 1966 .

[10]  Michael J. King,et al.  High-resolution monotonic schemes for reservoir fluid flow simulation , 1991 .

[11]  Andrew B. White,et al.  Supra-convergent schemes on irregular grids , 1986 .

[12]  P. Sweby High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

[13]  Martin J. Blunt,et al.  Implicit Flux Limiting Schemes for Petroleum Reservoir Simulation , 1990 .

[14]  D. U. Rosenberg,et al.  Local Mesh Refinement for Finite Difference Methods , 1982 .

[15]  Z. E. Heinemann,et al.  Using local grid refinement in a multiple-application reservoir simulator , 1983 .

[16]  J. A. Trangenstein,et al.  The Use of Second-Order Godunov-Type Methods for Simulating EOR Processes in Realistic Reservoir Models , 1990 .

[17]  D. K. Han,et al.  A More Flexible Approach of Dynamic Local Grid Refinement for Reservoir Modeling , 1987 .

[18]  I. J. Taggart,et al.  The Use of Higher-Order Differencing Techniques in Reservoir Simulation , 1987 .

[19]  G. Strang On the Construction and Comparison of Difference Schemes , 1968 .

[20]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[21]  J. Brackbill,et al.  Adaptive zoning for singular problems in two dimensions , 1982 .

[22]  F. J. Jacobs,et al.  Adaptive local grid refinement and multi-grid in numerical reservoir simulation , 1988 .

[23]  Wim A. Mulder,et al.  Numerical Simulation of Two-Phase Flow Using Locally Refined Grids in Three Space Dimensions , 1993 .

[24]  Michael G. Edwards A Dynamically Adaptive Higher Order Godunov Scheme in Two Dimensions for Reservoir Simulation , 1992 .

[25]  T. F. Russell,et al.  Mixed Methods, Operator Splitting, and Local Refinement Techniques for Simulation on Irregular Grids , 1990 .

[26]  Phillip Colella,et al.  Higher order Godunov methods for general systems of hyperbolic conservation laws , 1989 .

[27]  P. Colella Multidimensional upwind methods for hyperbolic conservation laws , 1990 .

[28]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[29]  M. R. Todd,et al.  Methods for Increased Accuracy in Numerical Reservoir Simulators , 1972 .

[30]  John A. Trangenstein,et al.  Mathematical structure of the black-oil model for petroleum reservoir simulation , 1989 .

[31]  Peter A. Forsyth,et al.  Local Mesh Refinement and Modeling of Faults and Pinchouts , 1986 .

[32]  John A. Trangenstein,et al.  Numerical Analysis of Reservoir Fluid Flow , 1988 .