Computational Geosciences

, Volume 12, Issue 2, pp 193–208 | Cite as

Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations

Original paper

Abstract

We propose a methodology, called multilevel local–global (MLLG) upscaling, for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local–global (LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of LG upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore, the number of outer iterations is bounded above, and hence, the computational costs of this approach are low. We design a new flow-based weighting of transmissibility values in LG upscaling that significantly improves the accuracy of LG and MLLG over traditional local transmissibility calculations. For highly heterogeneous (e.g., channelized) systems, the integration of grid adaptivity and LG upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is, the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic boundary conditions used to compute the upscaled parameters. The method is demonstrated on Cartesian cell-based anisotropic refinement (CCAR) grids, but it can be applied to other adaptation strategies for structured grids and extended to unstructured grids.

Keywords

Scale up Subsurface Heterogeneity Flow simulation Channelized Permeability Transmissibility Adaptivity Enhanced oil recovery Gas injection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aavatsmark, I., Barkve, T., Mannseth, T.: Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127, 2–14 (1996)MATHCrossRefGoogle Scholar
  2. 2.
    Aavatsmark, I., Eigestad, G.T., Nordbotten, J.M.: A compact MPFA method with improved robustness. In: Proceedings of the 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, 4–7 September 2006 (2006)Google Scholar
  3. 3.
    Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1989)MATHCrossRefGoogle Scholar
  4. 4.
    Berger, M.J., Oliger, J.E.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512 (1984)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bourgeat, A.: Homogenized behavior of two-phase flows in naturally fractured reservoirs with uniform fractures distribution. Comput. Methods Appl. Mech. Eng. 47, 205–16 (1984)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, Y., Durlofsky, L.J.: Adaptive local–global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media 62, 157–185 (2006)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chen, Y., Durlofsky, L.J., Gerritsen, M.G., Wen, X.H.: A coupled local–global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26, 1041–1060 (2003)CrossRefGoogle Scholar
  8. 8.
    Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Evalu. Eng. 4, 308–17 (2001)Google Scholar
  9. 9.
    Deutsch, C., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide, 2nd edition. Oxford Press, London (1998)Google Scholar
  10. 10.
    Durlofsky, L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27, 699–708 (1991)CrossRefGoogle Scholar
  11. 11.
    Edwards, M.G.: Elimination of adaptive grid interface errors in the discrete cell centered pressure equation. J. Comput. Phys. 126, 356–372 (1996)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259–290 (1998)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gautier, Y., Blunt, M.J., Christie, M.A.: Nested gridding and streamline-based simulation for fast reservoir performance prediction. In: SPE Reservoir Simulation Symposium, vol. 51931. SPE, Richardson (1999)Google Scholar
  14. 14.
    Gerritsen, M.G., Jessen, K., Mallison, B.T., Lambers, J.V.: A fully adaptive streamline framework for the challenging simulation of gas-injection processes. In: SPE ATC, vol. 97270. SPE, Richardson (2005)Google Scholar
  15. 15.
    Gerritsen, M.G., Lambers, J.V.: Solving elliptic equations in heterogeneous media using Cartesian grid methods with anisotropic adaptation. J. Comput. Phys. (in preparation)Google Scholar
  16. 16.
    Ham, F.E., Lien, F.S., Strong, A.B.: A Cartesian grid method with transient anisotropic adaptation. J. Comput. Phys. 179, 469–494 (2002)MATHCrossRefGoogle Scholar
  17. 17.
    He, C.: Structured flow-based gridding and upscaling for reservoir simulation. Ph.D. Thesis, Department of Petroleum Engineering, Stanford University (2004)Google Scholar
  18. 18.
    Henson, V.E., Yang, U.M.: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41, 155–77 (2002)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Holden, L., Nielsen, B.F.: Global upscaling of permeability in heterogeneous reservoirs: the output least squares (OTL) method. Transp. Porous Media 40, 115–43 (2000)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Hornung, R., Trangenstein, J.: Adaptive mesh refinement and multilevel iteration for flow in porous media. J. Comput. Phys. 136, 522–545 (1997)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kippe, V., Aarnes, J.E., Lie, K.A.: A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci. (2008, in press)Google Scholar
  22. 22.
    Lambers, J.V., Gerritsen, M.G., Mallison, B.T.: Accurate local upscaling with variable compact multi-point transmissibility calculations. Comput. Geosci. (2008). doi:10.1007/s10596-007-9068-4
  23. 23.
    Lee, S.H., Tchelepi, H.A., Jenny, P., DeChant, L.J.: Implementation of a flux-continuous finite-difference method for stratigraphic, hexahedron grids. SPE J. 7, 267–277 (2002)Google Scholar
  24. 24.
    Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Nilsson, J., Gerritsen, M.G., Younis, R.M.: A novel adaptive anisotropic grid framework for efficient reservoir simulation. In: Proc. of the SPE Reservoir Simulation Symposium, vol. 93243. SPE, Richardson (2005)Google Scholar
  26. 26.
    Nordbotten, J.M., Aavatsmark, I., Eigestad, G.T.: Mono tonicity of control volume methods. Numer. Math. 106, 255–288 (2007)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Pickup, G.E., Ringrose, P.S., Jensen, J.L., Sorbie, K.S.: Permeability tensors for sedimentary structures. Math. Geol. 26, 227–250 (1994)CrossRefGoogle Scholar
  28. 28.
    Sammon, PH.: Dynamic Grid Refinement and amalgamation for compositional simulation. In: SPE RSS, vol. 79683. SPE, Richardson (2003)Google Scholar
  29. 29.
    Trangenstein, J., Bi, Z.: Large multi-scale iterative techniques and adaptive mesh refinement for miscible displacement simulation. In: SPE/DOE Improved Oil Recovery Symposium, vol. 75232. SPE, Richardson (2002)Google Scholar
  30. 30.
    Watson, D.F.: Contouring: A Guide to the Analysis and Display of Spacial Data. Pergamon, New York (1994)Google Scholar
  31. 31.
    Wen, X.H., Durlofsky, L.J., Edwards, M.G.: Use of border regions for improved permeability upscaling. Math. Geol. 35, 521–547 (2003)MATHCrossRefGoogle Scholar
  32. 32.
    Younis, R.M., Caers, J.: A method for static-based upgridding. In: Proc. of the 8th European Conference on the Mathematics of Oil Recovery, Freiberg, 3–6 September 2002Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

Personalised recommendations