Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids

Journal of Computational Physics

Volumn 229, Issue 3, 2010, Pages 594-625

  Edwards, Michael G ^a   Zheng, Hongwen ^a  

^a SWANSEA UNIVERSITY   (United Kingdom)

Author keywords

Anisotropy and pressure equation; Control volume distributed (CVD); Double family; Finite volume; Flux continuous schemes; Full pressure continuity; Locally conservative; M matrix; Maximum principle; Monotonicity; Multi point flux approximation (MPFA); Positivity

Indexed keywords

ANISOTROPIC MEDIA; MAXIMUM PRINCIPLE; PETROLEUM RESERVOIR ENGINEERING; PETROLEUM RESERVOIRS; TENSORS;

ANISOTROPY AND PRESSURE EQUATION; CONTROL VOLUMES; CONTROL-VOLUME DISTRIBUTED; DOUBLE-FAMILY; FINITE-VOLUME; FLUX-CONTINUOUS SCHEME; FULL-PRESSURE CONTINUITY; LOCALLY CONSERVATIVE; M-MATRICES; M-MATRIX; MONOTONICITY; MULTI-POINT FLUX APPROXIMATION; MULTI-POINTS; POSITIVITY; PRESSURE EQUATIONS;

ANISOTROPY;

EID: 70449732080     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.09.037     Document Type: Article

References (34)

1. Edwards M.G., and Rogers C.F. Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geo. 2 (1998) 259-290
2. Edwards M.G. Unstructured, control-volume distributed, full-tensor finite volume schemes with flow based grids. Comput. Geo. 6 (2002) 433-452
3. Edwards M.G. Higher-resolution hyperbolic - coupled - elliptic flux-continuous CVD schemes on structured and unstructured grids in 3D. Int. J. Numer. Methods Fluids 51 (2006) 1079-1095
4. M. Pal, M.G. Edwards, Quasi-Monotonic Continuous Darcy-Flux Approximation for General 3D Grids of any Element Type Paper SPE 106486, SPE Reservoir Simulation Symposium Houston, Texas, USA, 26-28 February 2007, 14p, doi:10.2118/106486-MS.
5. Edwards M.G., and Pal M. Positive definite q-families of continuous subcell Darcy-flux CVD(MPFA) finite-volume schemes and the mixed finite element method. Int. J. Numer. Methods Fluids 57 (2008) 355-387
6. Aavatsmark I. Introduction to multipoint flux approximation for quadrilateral grids. Comput. Geo. 6 (2002) 405-432
7. Lee S.H., Jenny P., and Tchelepi H.A. A finite-volume method with hexahedral multiblock grids for modeling flow in porous media. Comput. Geo. 6 (2002) 353-379
8. S. Verma, Flexible Grids for Reservoir Simulation, Ph.D. Thesis, Stanford University, June 1996.
9. Wheeler M.F., and Yotov I. A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44 (2006) 2082-2106
10. R.A. Raviart, J.M. Thomas, A Mixed Finite Element method for Second Order Problems, Lecture Notes in Mathematics, vol. 606, Springer-Verlag, New York, 1977, pp. 292-315.
11. Russel T.F., and Wheeler M.F. Finite element and finite difference methods for continuous flows in porous media. In: Ewing R.E. (Ed). Chapter 2, in the Mathematics of Reservoir Simulation. Frontiers in Applied Mathematics (1983), SIAM 35-106
12. C.L. Farmer, D.E. Heath, R.O. Moody, A global optimization approach to grid generation, in: Proceedings of the 11th SPE Reservoir Simulation Symposium, Anaheim CA, USA, 17-20 February, 1991, pp. 341-350.
13. Durlofsky L.J. A triangle based mixed finite element finite volume technique for modeling two-phase flow through porous media. J. Comput. Phys (1993) 252-266
14. Arbogast T., Wheeler M.F., and Yotov I. Mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences. SIAM J. Numer. Anal. 34 2 (1997) 828
15. Russell T.F. Relationships among some conservative discretization methods. In: Chen Z., et al. (Ed). Numerical Treatment of Multiphase Flows in Porous Media. Lecture Notes in Physics vol. 552 (2000), Springer, Heidelberg 267-282
16. Hyman J.M., Shashkov M., and Steinberg S. The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130-148
17. B. Riviere, Discontinuous Galerkin Method for Solving the Miscible Displacement Problem in Porous Media, Ph.D. Thesis, The University of Texas at Austin, 2000.
18. Riviere B., Wheeler M.F., and Banas K. Discontinuous Glaerkin method applied to a single phase flow in porous media. Comput. Geo. 4 (2000) 337-349
19. M.G. Edwards, Symmetric, flux continuous, positive definite approximation of the elliptic full-tensor pressure equation in local conservation form, in: Proceedings of the 13th SPE Reservoir Simulation Symposium, San Antonio, Texas, USA, February 1995, pp. 553-562.
20. Nordbotten J.M., Aavatsmark I., and Eigestad G.T. Monotonicity of control-volume methods. Numerisch 106 (2007) 255-288
21. Mlacnik M.J., and Durlofsky L.J. Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. J. Comput. Phys. 216 1 (2006) 337-361 20 July
22. Edwards M.G., and Zheng H. A quasi-positive family of continuous Darcy-flux finite volume schemes with full pressure support. J. Comput. Phys. 227 (2008) 9333-9364
23. Aavatsmark I., Eigestad G.T., Mallison B.T., and Nordbotten J.M. A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Diff. Eqs. 24 (2008) 1329-1360
24. M. Pal, M.G. Edwards, Flux-splitting schemes for improved monotonicity of discrete solution of elliptic equation with highly anisotropic coefficients, in: Paper 384, Proceedings, ECCOMAS CFD-2006 Conference, Egmond aan Zee, The Netherlands, 5-8th September, 2006, ISBN:90-9020970-0.
25. M. Pal, M.G. Edwards, Family of flux-continuous finite-volume schemes with improved monotonicity, in: Paper B009, Proceedings of the 10th European Conference on the Mathematics of Oil Recovery, 4-7th September 2006, ISBN:90-73781-47-7.
26. Le Potier C. Schema volume finis pour des operator operateurs de diffusion fortement anisotropes sur des maillages de triangle nonstructures. C.R., Math., Acad. Sci. Paris, Ser. I. 340 12 (2005) 921-926
27. Lipnikov K., Shashkov M., Svyatskiy D., and Vassilevski Yu. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 227 (2008) 492-512
28. Edwards M.G. M-matrix flux splitting for general full-tensor discretization operators on structured and unstructured grids. J. Comput. Phys. 160 (2000) 1-28
29. M.G. Edwards, Symmetric positive definite general tensor discretization operator on unstructured and flow based grids, in: C.L. Farmer, Z.E. Heinemann (Eds.), Paper E04, Proceedings of the Eighth European Conference on Mathematics of Oil Recovery, Freiberg, Germany, 3-6th September, 2002, pp. 22-32, ISBN:90-73781-24-8.
30. E. Godlewski, P. Raviart, Numerical approximation of hyperbolic systems of conservation laws, App. Math. Sci., vol. 118, Springer-Verlag, New York, 1996.
31. S. Lamine, M.G. Edwards, Higher order multidimensional wave oriented upwind schemes for flow in porous media on structured and unstructured grids, in: Paper SPE 119187, Proceedings of the Reservoir Simulation Symposium, Houston, Texas, USA, February 2009.
32. Edwards M.G. Cross flow, tensors and finite volume approximation with deferred correction. Comput. Methods Appl. Mech. Eng. 151 (1998) 143-161
33. M.G. Edwards, Superconvergent renormalization and tensor approximation, in: Proceedings of the Fifth European Conference on the Mathematics of Oil Recovery, Leoben, Austria, 3-6 September 1996, pp. 445-454, ISBN:3-9500542-0-0.
34. Aziz K., and Settari A. Petroleum Reservoir Simulation (1979), Applied Science Publishers, London