Advertisement

Computational Geosciences

, Volume 12, Issue 3, pp 399–416 | Cite as

Accurate local upscaling with variable compact multipoint transmissibility calculations

  • James V. Lambers
  • Margot G. Gerritsen
  • Bradley T. Mallison
Original paper

Abstract

We propose a new single-phase local upscaling method that uses spatially varying multipoint transmissibility calculations. The method is demonstrated on two-dimensional Cartesian and adaptive Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve two generic local flow problems. The multipoint stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to ensure that the flux approximation is as close as possible to being two-point. The resulting multipoint flux approximations are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to two-point expressions in cases without full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local–global upscaling method.

Keywords

Scale up Subsurface Heterogeneity Flow simulation Channelized Permeability Transmissibility Multiscale Adaptivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aarnes, J.E.: On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul. 2(3), 421–439 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publisher, Essex (1979)Google Scholar
  3. 3.
    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
  4. 4.
    Arbogast, T.: An overview of subgrid upscaling for elliptic problems in mixed form. In: Chen, Z., Glowinski, R., Li, K. (eds.) Current Trends in Scientific Computing. Contemporary Mathematics, pp. 21–32. AMS, Providence (2003)Google Scholar
  5. 5.
    Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1989)MATHCrossRefGoogle Scholar
  6. 6.
    Berger, M.J., Oliger, J.E.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512 (1984)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    Caers, J.: Petroleum Geostatistics. SPE, Richardson 2005Google Scholar
  9. 9.
    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
  10. 10.
    Chen, Y., Mallison, B.M., Durlofsky, L.J.: Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations. Comput. Geosci. (2008). doi:10.1007/s10596-007-9067-5
  11. 11.
    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
  12. 12.
    Chen, Z., Hou, T.Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72(242), 541–576 (2003)MATHMathSciNetGoogle Scholar
  13. 13.
    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
  14. 14.
    Durlofsky, L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27, 699–708 (1991)CrossRefGoogle Scholar
  15. 15.
    Durlofsky, L.J., Efendiev, Y., Ginting, V.: An adaptive local–global multiscale finite volume element method for two-phase flow simulations. Adv. Water Res. 30, 576–588 (2006)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    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
  18. 18.
    Gautier, Y., Blunt, M.J., Christie, M.A.: Nested gridding and streamline-based simulation for fast reservoir performance prediction. In: SPE Reservoir Simulation Symposium, SPE51931. Houston, TX, 14–17 February 1999Google Scholar
  19. 19.
    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 ATCE, SPE 97270. Dallas, TX, 9–12 October 2005Google Scholar
  20. 20.
    Gerritsen, M.G., Lambers, J.V.: Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations. Comput. Geosci. (2008). doi:10.1007/s10596-007-9078-2
  21. 21.
    Gerritsen, M.G., Lambers, J.V., Mallison, B.T.: A variable and compact MPFA for transmissibility upscaling with guaranteed monotonicity. In: Proceedings of the 10th European Conference on the Mathematics of Oil Recovery. Amsterdam, 4–7 September 2006Google Scholar
  22. 22.
    Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic, London (1981)MATHGoogle Scholar
  23. 23.
    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
  24. 24.
    He, C.: Structured flow-based gridding and upscaling for reservoir simulation. Ph.D. thesis, Department of Petroleum Engineering, Stanford University (2004)Google Scholar
  25. 25.
    Henson, V.E., Yang, U.M.: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41, 155–77 (2002)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    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
  27. 27.
    Hornung, R., Trangenstein, J.: Adaptive mesh refinement and multilevel iteration for flow in porous media. J. Comput. Phys. 136, 522–545 (1997)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Jenny, P., Lee, S.H., Tchelepi, H.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187(1), 47–67 (2003)MATHCrossRefGoogle Scholar
  30. 30.
    Kippe, V., Aarnes, J.E., Lie, K.-A.: A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci. (2008). doi:10.1007/s10596-007-9074-6
  31. 31.
    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
  32. 32.
    Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–61 (2005)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Nilsson, J., Gerritsen, M.G., Younis, R.M.: A novel adaptive anisotropic grid framework for efficient reservoir simulation. In: Proceedings of the SPE Reservoir Simulation Symposium, SPE 93243, Houston, TX, 31 January–2 February 2005Google Scholar
  34. 34.
    Nordbotten, J.M., Aavatsmark, I., Eigestad, G.T.: Monoton icity of control volume methods. Numer. Math. 106, 255–288 (2006)CrossRefMathSciNetGoogle Scholar
  35. 35.
    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
  36. 36.
    Pollock, D.: Semianalytical computation of path lines for finite difference models. Ground Water 26, 743–750 (1988)CrossRefGoogle Scholar
  37. 37.
    Sammon, P.H.: Dynamic grid refinement and amalgamation for compositional simulation. In: SPE RSS, SPE 79683. SPE, Richardson (2003)Google Scholar
  38. 38.
    Trangenstein, J., Bi, Z.: Large multi-scale iterative techniques and adaptive mesh refinement for miscible displacement simulation. In: SPE/DOE Improved Oil Recovery Symposium, SPE75232. Tulsa, OK, 13–17 April 2002Google Scholar
  39. 39.
    Watson, D.F.: Contouring: a guide to the analysis and display of spacial data. Pergamon, Oxford (1994)Google Scholar
  40. 40.
    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
  41. 41.
    Younis, R.M., Caers, J.: A method for static-based upgridding. In: Proceedings 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. 2007

Authors and Affiliations

  • James V. Lambers
    • 1
  • Margot G. Gerritsen
    • 2
  • Bradley T. Mallison
    • 3
  1. 1.Stanford UniversityStanfordUSA
  2. 2.Stanford UniversityStanfordUSA
  3. 3.Chevron ETCSan RamonUSA

Personalised recommendations