Computational Geosciences

, Volume 16, Issue 2, pp 297–322 | Cite as

Open-source MATLAB implementation of consistent discretisations on complex grids

  • Knut–Andreas Lie
  • Stein Krogstad
  • Ingeborg Skjelkvåle Ligaarden
  • Jostein Roald Natvig
  • Halvor Møll Nilsen
  • Bård Skaflestad
Open Access
Original Paper

Abstract

Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab® toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains add-on modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.

Keywords

Mimetic schemes MPFA methods Consistent discretisations Unstructured grids Open-source implementation Multiscale methods Rate optimisation 

References

  1. 1.
    Aarnes, J.E., Gimse, T., Lie, K.A.: An introduction to the numerics of flow in porous media using Matlab. In: Hasle, G., Lie, K.A., Quak, E. (eds.) Geometrical modeling, numerical simulation and optimisation: industrial mathematics at SINTEF, pp. 265–306. Springer, Berlin (2007)CrossRefGoogle Scholar
  2. 2.
    Aarnes, J.E., Kippe, V., Lie, K.A.: Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Resour. 28(3), 257–271 (2005)CrossRefGoogle Scholar
  3. 3.
    Aarnes, J.E., Krogstad, S., Lie, K.A.: A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids. Multiscale model. Simul. 5(2), 337–363 (2006, electronic)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Aarnes, J.E., Krogstad, S., Lie, K.A.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12(3), 297–315 (2008). doi:10.1007/s10596-007-9072-8 MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405–432 (2002). doi:10.1023/A:1021291114475 MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on non-orthogonal, curvilinear grids for multi-phase flow. In: Proc. of the 4th European Conf. on the Mathematics of Oil Recovery (1994)Google Scholar
  7. 7.
    Aziz, K., Settari, A.: Petroleum reservoir simulation. Elsevier, London (1979)Google Scholar
  8. 8.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer series in computational mathematics, vol. 15. Springer, New York (1991)MATHCrossRefGoogle Scholar
  9. 9.
    Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonial and polyhedral meshes. Math. Models Methods Appl. Sci. 15, 1533–1553 (2005). doi:10.1142/S0218202505000832 MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chavent, G., Jaffre, J.: Mathematical models and finite elements for reservoir simulation. North Holland, Amsterdam (1982)Google Scholar
  11. 11.
    Chen, Z., Hou, T.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72, 541–576 (2003)MathSciNetMATHGoogle Scholar
  12. 12.
    Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide, 2nd edn. Oxford University Press, New York (1998)Google Scholar
  13. 13.
    Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. Presented at 8th International Forum on Reservoir Simulation Iles Borromees, Stresa, Italy, 20–24 June 2005Google Scholar
  14. 14.
    Edwards, M.G., Rogers, C.F.: A flux continuous scheme for the full tensor pressure equation. In: Proc. of the 4th European conference on the mathematics of oil recovery (1994)Google Scholar
  15. 15.
    Efendiev, Y., Hou, T.Y.: Multiscale finite element methods, surveys and tutorials in the applied mathematical sciences, vol. 4. Springer, Berlin (2009)Google Scholar
  16. 16.
    Farmer, C.L.: Upscaling: a review. Int. J. Numer. Methods Fluids 40(1–2), 63–78 (2002). doi:10.1002/fld.267 MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hou, T., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187, 47–67 (2003)MATHCrossRefGoogle Scholar
  19. 19.
    Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. In: International conference on parallel processing, pp. 113–122 (1995)Google Scholar
  20. 20.
    Kippe, V., Aarnes, J.E., Lie, K.A.: A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci. 12(3), 377–398 (2008). doi:10.1007/s10596-007-9074-6 MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Klausen, R.A., Stephansen, A.F.: Mimetic MPFA. In: Proc. 11th European conference on the mathematics of oil recovery, 8–11 Sept 2008, A12. EAGE, Bergen, Norway (2008)Google Scholar
  22. 22.
    Krogstad, S., Hauge, V.L., Gulbransen, A.F.: Adjoint multiscale mixed finite elements. SPE J. 16(1), 162–171 (2011). doi:10.2118/119112-PA Google Scholar
  23. 23.
    Ligaarden, I.S.: Well models for mimetic finite difference methods and improved representation of wells in multiscale methods. Master thesis, University of Oslo (2008)Google Scholar
  24. 24.
    Lipnikov, K., Shashkov, M., Yotov, I.: Local flux mimetic finite difference methods. Numer. Math. 112(1), 115–152 (2009). doi:10.1007/s00211-008-0203-5 MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Manzocchi, T.., et al.: Sensitivity of the impact of geological uncertainty on production from faulted and unfaulted shallow-marine oil reservoirs: objectives and methods. Pet. Geosci. 14(1), 3–15 (2008)CrossRefGoogle Scholar
  26. 26.
    MATLAB: The MATLAB reservoir simulation toolbox, version 2011a (2011). http://www.sintef.no/MRST/
  27. 27.
    Muskat, M.: The flow of homogeneous fluid through porous media. McGraw-Hill, New York (1937)Google Scholar
  28. 28.
    Natvig, J.R., Lie, K.A.: Fast computation of multiphase flow in porous media by implicit discontinuous Galerkin schemes with optimal ordering of elements. J. Comput. Phys. 227(24), 10108–10124 (2008). doi:10.1016/j.jcp.2008.08.024 MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Notay, Y.: An aggregation-based algebraic multigrid method. Electron. Trans. Numer. Anal. 37, 123–146 (2010)MathSciNetMATHGoogle Scholar
  30. 30.
    Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE, Trans. AIME 275, 10–22 (1983)Google Scholar
  31. 31.
    Skaflestad, B., Krogstad, S.: Multiscale/mimetic pressure solvers with near-well grid adaption. In: Proceedings of ECMOR XI–11th European Conference on the Mathematics of Oil Recovery, A36. EAGE, Bergen, Norway (2008)Google Scholar
  32. 32.
    Zhou, H., Tchelepi, H.: Operator-based multiscale method for compressible flow. SPE J. 13(2), 267–273 (2008)Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Knut–Andreas Lie
    • 1
    • 2
  • Stein Krogstad
    • 1
  • Ingeborg Skjelkvåle Ligaarden
    • 1
  • Jostein Roald Natvig
    • 1
  • Halvor Møll Nilsen
    • 1
  • Bård Skaflestad
    • 1
  1. 1.SINTEFOsloNorway
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations