Computational Geosciences

, Volume 15, Issue 3, pp 399–419 | Cite as

Multidimensional upstream weighting for multiphase transport in porous media

  • Jeremy Edward Kozdon
  • Bradley T. Mallison
  • Margot G. Gerritsen
Original Paper

Abstract

Truly multidimensional methods for hyperbolic equations use flow-based information to determine the computational stencil, as opposed to applying one-dimensional methods dimension by dimension. By doing this, the numerical errors are less correlated with the underlying computational grid. This can be important for reducing bias in flow problems that are inherently unstable at simulation scale, such as in certain porous media problems. In this work, a monotone, multi-D framework for multiphase flow and transport in porous media is developed. A local coupling of the fluxes is introduced through the use of interaction regions, resulting in a compact stencil. A relaxed volume formulation of the coupled hyperbolic–elliptic system is used that allows for nonzero residuals in the pressure equation to be handled robustly. This formulation ensures nonnegative masses and saturations (volume fractions) that sum to one (Acs et al., SPE J 25(4):543–553, 1985). Though the focus of the paper is on immiscible flow, an extension of the methods to a class of more general scalar hyperbolic equations is also presented. Several test problems demonstrate that the truly multi-D schemes reduce biasing due to the computational grid.

Keywords

Multi-D transport Porous media Phase-based upwinding Two phase Interaction regions Monotonicity Hyperbolic equations Finite volume Finite difference Immiscible Partially miscible 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3–4), 405–432 (2002)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Abgrall, R., Marpeau, F.: Residual distribution schemes on quadrilateral meshes. J. Sci. Comput. 30(1), 131–175 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Abreu, E., Pereira, F., Ribeiro, S.: Central schemes for porous media flows. Comput. Appl. Math. 28(1), 87–110 (2009)MathSciNetMATHGoogle Scholar
  4. 4.
    Acs, G., Doleschall, S., Farkas, E.: General purpose compositional model. SPE J. 25(4), 543–553 (1985)Google Scholar
  5. 5.
    Arbogast, T., Huang, C.: A fully mass and volume conserving implementation of a characteristic method for transport problems. SIAM J. Sci. Comput. 28(6), 2001–2022 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science, New York (1979)Google Scholar
  7. 7.
    Blunt, M.J., Rubin, B.: Implicit flux-limiting schemes for petroleum reservoir simulation. In: 2nd European Conference on the Mathematics of Oil Recovery, pp. 131–138 (1990)Google Scholar
  8. 8.
    Brand, C.W., Heinemann, J.E., Aziz, K.: The grid orientation effect in reservoir simulation. In: SPE Paper 21228 presented at the SPE Reservoir Simulation Symposium, 17–20 February. Anaheim, CA (1991)Google Scholar
  9. 9.
    Brenier, Y., Jaffré, J.: Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28(3), 685–696 (1991)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Colella, P.: Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87(1), 171–200 (1990)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Corey, A.T.: The interrelation between gas and oil relative permeabilities. Prod. Mon. 19(1), 38–41 (1954)Google Scholar
  13. 13.
    Deconinck, H., Ricchiuto, M., Sermeus, K.: Introduction to residual distribution schemes and comparison with stabilized finite elements. In: Deconinck, H. (ed.) 33rd Computational Fluid Dynamics—Novels Methods for Solving Convection Dominated Systems. von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode (2003)Google Scholar
  14. 14.
    Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and Users Guide. Oxford University Press, Oxford (1998)Google Scholar
  15. 15.
    Edwards, M.G.: Multi-dimensional wave-oriented upwind schemes with minimal cross-wind diffusion. In: SPE Paper 79689 Presented at the SPE Reservoir Simulation Symposium, 3–5 February. Houston, TX (2004)Google Scholar
  16. 16.
    Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2(4), 259–290 (1998)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Gerritsen, M., Durlofsky, L.: Modeling fluid flow in oil reservoirs. Annu. Rev. Fluid Mech. 37, 211–238 (2005)CrossRefGoogle Scholar
  18. 18.
    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)Google Scholar
  19. 19.
    Jessen, K., Gerritsen, M.G., Mallison, B.T.: High-resolution prediction of enhanced condensate recovery processes. SPE J. 13(2), 257–266 (2008)Google Scholar
  20. 20.
    Jiang, G., Tadmor, E.: Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19(6), 1892–1917 (1998)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Keilegavlen, E.: Robust control volume methods for reservoir simulation on challenging grids. Ph.D. thesis, The University of Bergen (2009)Google Scholar
  22. 22.
    Keilegavlen, E., Kozdon, J.E., Mallison, B.T.: Monotone multi-dimensional upstream weighting on general grids. In: 12th European Conference on the Mathematics of Oil Recovery, 6–9 September. EAGE, Amsterdam (2010)Google Scholar
  23. 23.
    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)MathSciNetGoogle Scholar
  24. 24.
    Koval, E.J.: A method for predicting the performance of unstable miscible displacements in heterogeneous media. SPE J. 3, 145–154 (1963)Google Scholar
  25. 25.
    Kozdon, J., Mallison, B., Gerritsen, M.: Robust multi-D transport schemes with reduced grid orientation effects. Transp. Porous Media 78(1), 47–75 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kozdon, J., Mallison, B., Gerritsen, M., Chen, W.: Multi-D upwinding for multi phase transport in porous media. SPE J. (2010, in press)Google Scholar
  27. 27.
    Krishnamurthy, S.: Relaxation schemes for multiphase, multicomponent for in gas injection processes. Ph.D. thesis, Stanford University, Stanford, CA (2008)Google Scholar
  28. 28.
    Kwok, F., Tchelepi, H.: Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media. J. Comput. Phys. 227(1), 706–727 (2007)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Lamine, S., Edwards, M.G.: Higher order multidimensional upwind convection schemes for flow in porous media on structured and unstructured quadrilateral grids. SIAM J. Sci. Comp. 32(3), 1119–1139 (2010)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    LeVeque, R.J.: Wave propagation algorithms for multidimensional hyperbolic systems. J. Comput. Phys. 131(2), 327–353 (1997)MATHCrossRefGoogle Scholar
  31. 31.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, 1st edn. Cambridge University Press, Cambridge (2003)Google Scholar
  32. 32.
    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)CrossRefGoogle Scholar
  33. 33.
    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)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Sammon, P.H.: An analysis of upstream differencing. SPE Reserv. Eng. 3(3), 1053–1056 (1988)Google Scholar
  35. 35.
    Schneider, G.E., Raw, M.J.: A skewed, positive influence coefficient upwinding procedure for control-volume-based finite-element convection–diffusion computation. Numer. Heat Transf., A Appl. 9(1), 1–26 (1986)CrossRefGoogle Scholar
  36. 36.
    Shubin, G.R., Bell, J.B.: An analysis of the grid orientation effect in numerical simulation of miscible displacement. Comput. Methods Appl. Mech. Eng. 47(1–2), 47–71 (1984)MATHCrossRefGoogle Scholar
  37. 37.
    Tadmor, E.: Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43(168), 369–381 (1984)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Tan, C.T., Homsy, G.M.: Stability of miscible displacements in porous media: radial source flow. Phys. Fluids 30(5), 1239–1245 (1997)CrossRefGoogle Scholar
  39. 39.
    Todd, M.R., Longstaff, W.J.: The development, testing and application of a numerical simulator for predicting miscible flood performance. Trans. AIME. 253, 874–882 (1972)Google Scholar
  40. 40.
    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, Wiesbaden (1997)Google Scholar
  41. 41.
    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)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Jeremy Edward Kozdon
    • 1
  • Bradley T. Mallison
    • 2
  • Margot G. Gerritsen
    • 3
  1. 1.Geophysics DepartmentStanford UniversityStanfordUSA
  2. 2.Reservoir and Production EngineeringChevron Energy Technology CompanySan RamonUSA
  3. 3.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

Personalised recommendations