Mixed finite element and related control volume discretizations for reservoir simulation on three-dimensional unstructured grids

SPE Reservoir Simulation Symposium Proceedings

Volumn , Issue , 2007, Pages 201-213

  Matringe, S F ^a   Juanes, R ^b   Tchelepi, H A ^a  

^a STANFORD UNIVERSITY   (United States)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ANISOTROPY; APPROXIMATION THEORY; COMPUTER SIMULATION; DEGREES OF FREEDOM (MECHANICS); FINITE ELEMENT METHOD; FINITE VOLUME METHOD;

MULTIPOINT FLUX APPROXIMATION (MPFA); PERMEABILITY ANISOTROPY; RESERVOIR FLOW EQUATIONS; TWO-POINT FLUX APPROXIMATION (TPFA); UNSTRUCTURED GRIDS;

PETROLEUM RESERVOIRS;

EID: 34249081135     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.2118/106117-ms     Document Type: Conference Paper

References (36)

1. L. J. Durlofsky. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res., 27(5):699-708, 1991.
2. Y. Chen, L. J. Durlofsky, M. Gerritsen, and X. H. Wen. A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour., 26(10):1041-1060, 2003.
3. K. Aziz and A. Settari. Petroleum Reservoir Simulation. Elsevier, London, 1979.
4. G. Chavent and J. Jaffré. Mathematical Models and Finite Elements for Reservoir Simulation, volume 17 of Studies in Mathematics and its Applications. Elsevier, North-Holland, 1986.
5. R. A. Klausen and T. F. Russell. Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci., 8(4):341-377, 2004.
6. I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth. Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I; Derivation of the methods. SIAM J. Sci. Comput., 19(5):1700-1716, 1998.
7. I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth. Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results. SIAM J. Sci. Comput., 19(5):1717-1736, 1998.
8. I. Aavatsmark, T. Barkve, and T. Mannseth. Control-volume discretization methods for 3D quadrilateral grids in inhomogeneous, anisotropic reservoirs. Soc. Pet. Eng. J., 3(2):146-154, June 1998.
9. I. Aavatsmark. An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci., 6:405-432, 2002.
10. M. G. Edwards and C. F. Rogers. Finite-volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci., 2(4):259-290, 1998.
11. S. H. Lee, H. Tchelepi, P. Jenny, and L. F. DeChant. Implementation of a flux-continuous finite-difference method for stratigraphic, hexahedron grids. Soc. Pet. Eng. J., 7(3):267-277, September 2002.
12. S. H. Lee, P. Jenny, and H. A. Tchelepi. A finite-volume method with hexahedral multiblock grids for modeling flow in porous media. Comput. Geosci., 6:353-379, 2002.
13. J. M. Nordbotten and I. Aavatsmark. Monotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous media. Comput. Geosci., 9(1):61-72, 2005.
14. J. M. Nordbotten and G. T. Eigestad. Discretization on quadrilateral grids with improved monotonicity properties. J. Comput. Phys., 203(2):744-760, 2005.
15. M. J. Mlacnik and L. J. Durlofsky. Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. J. Comput. Phys., 216:337-361, 2006.
16. I. Aavatsmark, G. T. Eigestad, J. M. Nordbotten, and B. T. Mallison. A compact MPFA method with improved monotonicity. Numer. Meth. Part Differ. Equ., 2007. (Submitted).
17. T. F. Russell and M. F. Wheeler. Finite element and finite difference methods for continuous flows in porous media. In R. E. Ewing, editor. The Mathematics of Reservoir Simulation, pages 35-106. SIAM, Philadelphia, PA, 1983.
18. A. Weiser and M. F. Wheeler. On convergence of blockcentered finite differences for elliptic problems. SIAM J. Numer. Anal., 25(2):351-375, 1988.
19. T. Arbogast, M. F. Wheeler, and I. Yotov. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal., 34:828-852, 1997.
20. T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov. Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput., 19:404-425, 1998.
21. M. F. Wheeler and I. Yotov. A multipoint flux mixed finite element method. SIAM J. Numer. Anal., 44(5):2082-2106, 2006.
22. S. F. Matringe, R. Juanes, and H. A. Tchelepi. Robust streamline tracing for the simulation of porous media flow on general triangular and quadrilateral grids. J. Comput. Phys., 219:992-1012, doi:10.1016/j.jcp.2006.07.004, 2006.
23. S. F. Matringe, R. Juanes, and H. A. Tchelepi. Streamline tracing on general triangular or quadrilateral grids. Soc. Pet. Eng. J., 2007. (Accepted, in press).
24. S. F. Matringe, R. Juanes, and H. A. Tchelepi. Robust discretization method for tensor permeabilities and unstructured grids. In European Conference on the Mathematics of Oil Recovery, ECMOR X, page Paper B025, Amsterdam, The Netherlands, September 4-7 2006.
25. F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods, volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.
26. M. Fortin and R. Glowinski. Augmented Lagrangian Methods: Applications to the Numerical Solution of BoundaryValue Problems. North-Holland, The Netherlands, 1983.
27. S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, New York, 1994.
28. I. Babuska. The finite element method with Lagrangian multipliers. Numer. Math., 20:179-192, 1973.
29. F. Brezzi. On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers. KAIRO Anal. Numér., 8:129-151, 1974.
30. J. E. Marsden and T. J. R. Hughes. Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs, NJ, 1983. Reprinted with corrections, Dover, New York, 1994.
31. P. A. Raviart and J. M. Thomas. A mixed finite element method for second order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical Aspects of the Finite Element Method, volume 606 of Lecture Notes in Mathematics, pages 292-315, New York, 1977. Springer-Verlag.
32. 3. Numer. Math., 50:57-81, 1986.
33. F. Brezzi, J. Douglas, Jr., and L. D. Marini. Two families of mixed finite elements for second order elliptic problems. Numer. Math., 47:217-235, 1985.
34. F. Brezzi, J. Douglas, Jr., R. Duran, and M. Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer. Math., 51:237-250, 1987.
35. T. J. R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1987. Reprinted with corrections, Dover, New York, 2000.
36. R. L. Naff, T. F. Russell, and J. D. Wilson. Shape functions for velocity interpolation in general hexahedral cells. Comput. Geosci., 6(3-4):285-314, 2002.