Robust multi-D transport schemes with reduced grid orientation effects

Transport in Porous Media

Volumn 78, Issue 1, 2009, Pages 47-75

  Kozdon, Jeremy ^a   Mallison, Brad ^b   Gerritsen, Margot ^c  

^a Stanford University   (United States)

^b CHEVRON ENERGY TECHNOLOGY COMPANY   (United States)

^c STANFORD UNIVERSITY   (United States)

Author keywords

Adverse mobility ratio; Finite difference; Finite volume; Grid orientation effect; Hyperbolic equations; Interaction regions; Miscible gas injection; Multi D transport; Physical instabilities; Positivity

Indexed keywords

ADVERSE MOBILITY RATIO; FINITE DIFFERENCE; FINITE VOLUME; GRID ORIENTATION EFFECT; HYPERBOLIC EQUATIONS; INTERACTION REGIONS; MISCIBLE GAS INJECTION; MULTI-D TRANSPORT; PHYSICAL INSTABILITIES; POSITIVITY;

POROUS MATERIALS;

DIFFERENCE EQUATIONS;

DIFFUSION; FINITE DIFFERENCE METHOD; FINITE VOLUME METHOD; HETEROGENEOUS MEDIUM; NUMERICAL MODEL; POROUS MEDIUM; SIMULATION;

EID: 64749105839     PISSN: 01693913     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11242-008-9281-1     Document Type: Article

References (25)

1. Aavatsmark I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3-4), 405-432 (2002)
2. Berger M.J., Helzel C., LeVeque R.J.: H-box methods for the approximation of hyperbolic conservation laws on irregular grids. SIAM J. Numer. Anal. 41(3), 893-918 (2003)
3. Chen W.H., Durlofsky L.J., Engquist B., Osher S.: Minimization of grid orientation effects through use of higher-order finite difference methods. SPE Adv. Technol. Ser. 1(2), 43-52 (1993)
4. Colella P.: Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87(1), 171-200 (1990)
5. Deconinck, H., Koren, B. (eds.): Euler and Navier-Stokes Solvers Using Multi-Dimensional Upwind Schemes and Multigrid Acceleration. Vieweg (1997)
6. Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and Users Guide. Oxford Univsity Press (1998)
7. Heinemann Z.E., Brand C.W., Munka M., Chen Y.M.: Modeling reservoir geometry with irregular grids. SPE Reserv. Eng. 6(2), 225-232 (1991)
8. Helzel C., Berger M.J., LeVeque R.J.: A high-resolution rotated grid method for conservation laws with embedded geometries. SIAM J. Sci. Comput. 26(3), 785-809 (2005)
9. Hurtado, F.S.V., Maliska, C.R., da Silva, A.F.C., Cordazzo, J.: A quadrilateral element-based finite-volume formulation for the simulation of complex reservoirs. In SPE paper 107444-MS presented at the SPE Latin American and Caribbean Petroleum Engineering Conference held in Buenos Aires, Argentina, 15-18 April 2007
10. Koren B.: Low-diffusion rotated upwind schemes, multigrid and defect corrections for steady, multi-dimensional euler flows. Int. Ser. Numer. Math. 98, 265-276 (1991)
11. Kozdon J., Gerritsen M., Christie M.: Grid orientation revisited: near-well, early-time effects and solution coupling methods. Trans. Porous Media 73, 255-277 (2008)
12. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. 1st edn. Cambridge Univsity Press (2002)
13. Liu, K., Subramanian, G., Dratler, D.I., Lebel, J., Yerian, J.: A general unstructured grid, EOS based, fully implicit thermal simulator for complex reservoir processes. In SPE paper 106073 presented at the SPE reservoir simulation symposium, Houston, Texas, 26-28 February 2007
14. Potempa, T.: An improved implementation of the McCracken and Yanosik nine point finite difference procedure. Technical Report 83-14, Rice University (1983)
15. Pruess, K., Bodvarsson, G.S.: A seven-point finite difference method for improved grid orientation performance in pattern steamfloods. In SPE Paper 12252 presented at the SPE reservoir simulation symposium, San Francisco, California, 15-18 November 1983
16. Riaz A., Meiburg E.: Linear stability of radial displacements in porous media: Influence of velocity-induced dispersion and concentration-dependent diffusion. Phys. Fluids 16(10), 3592 (2004)
17. Robertson G.E., Woo P.T.: Grid-orientation effects and the use of orthogonal curvilinear coordinates in reservoir simulation. SPE J. 18(1), 13-19 (1978)
18. Roe P.L., Sidilkover D.: Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal. 29(6), 1542-1568 (1992)
19. Schneider M.J.: A skewed, positive influence coefficient upwinding procedure for control-volume-based finite-element convection-diffusion computation. Numer. Heat Trans. Part A: Appl. 9(1), 1-26 (1986)
20. Shubin G.R., Bell J.B.: An analysis of the grid orientation effect in numerical simulation of miscible displacement. Comput. Meth. Appl. Mech. Eng. 47(1-2), 47-71 (1984)
21. Tan C.T., Homsy G.M.: Stability of miscible displacements in porous media: Radial source flow. Phys. Fluids 30(5), 1239-1245 (1997)
22. Todd M.R., ODell P.M., Hiraski G.J.: Methods for increased accuracy in numerical reservoir simulators. SPE J. 12(6), 515-530 (1972)
23. Van Ransbeeck, P., Hirsch, Ch.: A general analysis of 2d/3d multidimensional upwind convection schemes. In Deconinck, H., Koren, B. (eds.) Euler and Navier-Stokes Solvers Using Multi-Dimensional Upwind Schemes and Multigrid Acceleration. Vieweg (1997)
24. Wolcott, D.S., Kazemi, H., Dean, R.H.: A practical method for minimizing the grid orientation effect in reservoir simulation. In SPE paper 36723 presented at the SPE annual technical conference and exhibition, Denver, Colorado, 6-9 October 1996
25. Yanosik J.L., McCracken T.A.: A nine-point, finite difference reservoir simulator for realistic prediction of adverse mobility ratio displacements. SPE J. 19(4), 253-262 (1979)