Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Robust Multi-D Transport Schemes with Reduced Grid Orientation Effects

  • 188 Accesses

  • 15 Citations

Abstract

In this paper we investigate truly multi-D upwind schemes for simulating adverse mobility ratio displacements in porous media. Due to an underlying physical instability at the simulation scale, numerical results are highly sensitive to discretization errors and hence the orientation of the underlying computational grid. We use modified equations analysis to predict preferred flow angles on structured grids for several popular methods and present a conservative, multi-D framework for designing positive upwind schemes for general velocity fields. After placing the common schemes in this framework, we go on to develop a novel scheme with “minimal” constant transverse (cross-wind) diffusion. Results for miscible gas injection into homogeneous and heterogeneous media demonstrate that truly multi-D schemes, and in particular our new scheme, greatly reduce grid orientation effects and numerical biasing as compared to dimensional upwinding.

This is a preview of subscription content, log in to check access.

References

  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)

Download references

Author information

Correspondence to Jeremy Kozdon.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kozdon, J., Mallison, B. & Gerritsen, M. Robust Multi-D Transport Schemes with Reduced Grid Orientation Effects. Transp Porous Med 78, 47–75 (2009). https://doi.org/10.1007/s11242-008-9281-1

Download citation

Keywords

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