1887

Abstract

Summary

Multiscale methods for solving strongly heterogenous systems in reservoirs have a long history from the early ideas used on incompressible flow to the newly released version in commercial simulation. Much effort has been put into making the MsFV method work for fully unstructured multiphase problems. The MsRSB version is a newly developed version, which tackles most of the "real" world problems. It is to our knowledge, the only multiscale method that has been released in a commercial simulator. You can alternatively see the method as a variant of smoothed aggregation or as an iterative approach to AMG with energy minimizing basis functions. This will be discussed in detail.

So far, most work on comparing MsRSB with AMG methods has been on qualitative performance measures like iteration number rather than on pure runtime on fair code implementation. We discuss the theoretical performance and show the practical performance for our implementation. Here, we compare performance of pure AMG, standard two-level MsRSB with pure AMG as coarse solver, as well as a new truly multilevel MsRSB scheme. Our implementation uses the DUNE-ISTL framework. To limit the scope of the discussion we restrict our assessment to AMG with aggregation and smoothed aggregation and the MsRSB method. These three methods are closely related and are primarily distinguished in a preconditioner setting by the coarsening factors used, and the degree of smoothing applied to the basis. We also compare with other state-of-the-art AMG implementations, but do not investigate combinations of them with the MSRB method. For the MsRSB method, we also discuss practical considerations in different parallelization regimes including domain decomposition using MPI, shared memory using OpenMP, and GPU acceleration with CUDA.

All comparisons will focus on the setting in which many similar systems should be solved, e.g. during a large-scale, multiphase flow simulation. That is, our emphasis is on the performance of updating a preconditioner and on the apply time for the preconditioner relative to the convergence rate. Performance of the solvers will be tested for pure parabolic/elliptic problems that either arise as part of a sequential splitting procedure or as a pseudo-elliptic preconditioner/solver as a part of a CPR preconditioner for a multiphase system, for which block ILU0 is used as the outer smoother.

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2020-09-14
2021-06-20
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References

  1. Acs, G., Doleschall, S. and Farkas, E.
    [1985] General Purpose Compositional Model. Soc. Pet. Eng. J., 25(04), 543–553.
    [Google Scholar]
  2. Blatt, M. and Bastian, P.
    [2006] The iterative solver template library. In: Kågström, B., Elmroth, E., Dongarra, J. and Waśniewski, J. (Eds.) International Workshop on Applied Parallel Computing. Springer, 666–675.
    [Google Scholar]
  3. [2008] On the generic parallelisation of iterative solvers for the finite element method. Int. J. Comp. Sci. Eng., 4(1), 56–69.
    [Google Scholar]
  4. Blatt, M., Ippisch, O. and Bastian, P.
    [2012] A massively parallel algebraic multigrid preconditioner based on aggregation for elliptic problems with heterogeneous coefficients. arXiv preprint arXiv:1209.0960.
    [Google Scholar]
  5. Bröker, O. and Grote, M.J.
    [2002] Sparse approximate inverse smoothers for geometric and algebraic multigrid. Applied numerical mathematics, 41(1), 61–80.
    [Google Scholar]
  6. Bui, Q.M., Elman, H.C. and Moulton, J.D.
    [2017] Algebraic multigrid preconditioners for multiphase flow in porous media. SIAM Journal on Scientific Computing, 39(5), S662–S680.
    [Google Scholar]
  7. Christie, M.A. and Blunt, M.J.
    [2001] Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE Res. Eval. Eng., 4(04), 308–317.
    [Google Scholar]
  8. Coats, K.H.
    [1980] An Equation of State Compositional Model. Soc. Pet. Eng. J., 20(05), 363–376.
    [Google Scholar]
  9. [2000] A note on IMPES and some IMPES-based simulation models. SPE J., 5(03), 245–251.
    [Google Scholar]
  10. Demidov, D.
    [2019] AMGCL: an Efficient, Flexible, and Extensible Algebraic Multigrid Implementation. Lobachevskii Journal of Mathematics, 40(5), 535–546.
    [Google Scholar]
  11. Ehrmann, S., Gries, S. and Schweitzer, M.A.
    [2019] Generalization of algebraic multiscale to algebraic multigrid. Computational Geosciences, 1–14.
    [Google Scholar]
  12. Gries, S., Stüben, K., Brown, G.L., Chen, D. and Collins, D.A.
    [2014] Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations. SPE J., 19(04), 726–736.
    [Google Scholar]
  13. Hajibeygi, H., Bonfigli, G., Hesse, M.A. and Jenny, P.
    [2008] Iterative multiscale finite-volume method. J. Comput. Phys., 227(19), 8604–8621.
    [Google Scholar]
  14. Hajibeygi, H. and Tchelepi, H.A.
    [2014] Compositional multiscale finite-volume formulation. SPE J, 19(02), 316–326.
    [Google Scholar]
  15. Hesse, M.A., Mallison, B.T. and Tchelepi, H.A.
    [2008] Compact multiscale finite volume method for heterogeneous anisotropic elliptic equations. Multiscale Model. Simul., 7(2), 934–962.
    [Google Scholar]
  16. Hilden, S.T., Møyner, O., Lie, K.A. and Bao, K.
    [2016] Multiscale Simulation of Polymer Flooding with Shear Effects. Transp. Porous Media, 113(1), 111–135.
    [Google Scholar]
  17. Hou, T.Y. and Wu, X.H.
    [1997] A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media. J. Comput. Phys., 134(1), 169–189.
    [Google Scholar]
  18. Jenny, P., Lee, S.H. and Tchelepi, H.A.
    [2003] Multi-Scale Finite-Volume Method for Elliptic Problems in Subsurface Flow Simulation. J. Comput. Phys., 187, 47–67.
    [Google Scholar]
  19. [2006] Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. J. Comput. Phys., 217(2), 627–641.
    [Google Scholar]
  20. Kippe, V., Aarnes, J.E. and Lie, K.A.
    [2008] A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci., 12(3), 377–398.
    [Google Scholar]
  21. Klemetsdal, Ø.S., Møyner, O. and Lie, K.A.
    [2020] Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions. Comput. Geosci., 24, 459–476.
    [Google Scholar]
  22. Kozlova, A., Li, Z., Natvig, J.R., Watanabe, S., Zhou, Y., Bratvedt, K. and Lee, S.H.
    [2016a] A real-field multiscale black-oil reservoir simulator. SPE J., 21(6), 2049–2061.
    [Google Scholar]
  23. Kozlova, A., Walsh, D., Chittireddy, S., Li, Z., Natvig, J., Watanabe, S. and Bratvedt, K.
    [2016b] A hybrid approach to parallel multiscale reservoir simulator. In: ECMOR XV – 15th European Conference on the Mathematics of Oil Recovery.EAGE, Amsterdam, The Netherlands.
    [Google Scholar]
  24. Lacroix, S., Vassilevski, Y.V. and Wheeler, M.F.
    [2001] Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS). Numer. Linear Algebra Appl., 8(8), 537–549.
    [Google Scholar]
  25. Lie, K.A., Møyner, O. and Natvig, J.R.
    [2017a] Use of multiple multiscale operators to accelerate simulation of complex geomodels. SPE J., 22(6), 1929–1945.
    [Google Scholar]
  26. Lie, K.A., Møyner, O., Natvig, J.R., Kozlova, A., Bratvedt, K., Watanabe, S. and Li, Z.
    [2017b] Successful application of multiscale methods in a real reservoir simulator environment. Comput. Geosci., 21(5–6), 981–998.
    [Google Scholar]
  27. Lie, K.A., Nilsen, H.M., Rasmussen, A.F., Raynaud, X. et al
    . [2014] Fast simulation of polymer injection in heavy-oil reservoirs on the basis of topological sorting and sequential splitting. SPE J., 19(06), 991–1.
    [Google Scholar]
  28. Lunati, I. and Lee, S.H.
    [2009] An operator formulation of the multiscale finite-volume method with correction function. Multiscale Model. Simul., 8(1), 96–109.
    [Google Scholar]
  29. Lunati, I., Tyagi, M. and Lee, S.H.
    [2011] An iterative multiscale finite volume algorithm converging to the exact solution. J. Comput. Phys., 230(5), 1849–1864.
    [Google Scholar]
  30. Manea, A. and Almani, T.
    [2018] A multi-level algebraic multiscale solver (ML-AMS) for reservoir simulation. In: ECMOR XVI – 16th European Conference on the Mathematics of Oil Recovery, 1. European Association of Geoscientists & Engineers, 1–12.
    [Google Scholar]
  31. Manea, A.M., Hajibeygi, H., Vassilevski, P. and Tchelepi, H.A.
    [2017] Parallel enriched algebraic multiscale solver. In: SPE Reservoir Simulation Conference, 20–22 February, Montgomery, Texas, USA. Society of Petroleum Engineers.
    [Google Scholar]
  32. Manea, A.M., Sewall, J. and Tchelepi, H.A.
    [2016] Parallel multiscale linear solver for highly detailed reservoir models. SPE J., 21(06), 2062–2078.
    [Google Scholar]
  33. Moncorge, A., Møyner, O., Tchelepi, H.A. and Jenny, P.
    [2020] Consistent upwinding for sequential fully implicit multiscale compositional simulation. Comput. Geosci., 24, 533–550.
    [Google Scholar]
  34. Moncorgé, A., Tchelepi, H.A. and Jenny, P.
    [2017] Modified sequential fully implicit scheme for compositional flow simulation. J. Comput. Phys., 337, 98–115.
    [Google Scholar]
  35. [2018] Sequential Fully Implicit Formulation for Compositional Simulation using Natural Variables. J. Comput. Phys., 371, 690–711.
    [Google Scholar]
  36. Møyner, O. and Lie, K.A.
    [2014] The multiscale finite-volume method on stratigraphic grids. SPE J., 19(5), 816–831.
    [Google Scholar]
  37. [2016a] A Multiscale Restriction-Smoothed Basis Method for Compressible Black-Oil Models. SPE J., 21(06), 2079–2096.
    [Google Scholar]
  38. [2016b] A Multiscale Restriction-Smoothed Basis Method for High Contrast Porous Media Represented on Unstructured Grids. J. Comput. Phys., 304, 46–71.
    [Google Scholar]
  39. Møyner, O. and Moncorgé, A.
    [2020] Nonlinear domain decomposition scheme for sequential fully implicit formulation of compositional multiphase flow. Comp. Geosci., 24, 789–806.
    [Google Scholar]
  40. Møyner, O. and Tchelepi, H.A.
    [2018a] A Mass-Conservative Sequential Implicit Multiscale Method for Isothermal Equation-of-State Compositional Problems. SPE J., 23(6).
    [Google Scholar]
  41. [2018b] A Mass-Conservative Sequential Implicit Multiscale Method for Isothermal Equation-of-State Compositional Problems. SPE J., 23(06), 2376–2393.
    [Google Scholar]
  42. Nordbotten, J.M., Keilegavlen, E. and Sandvin, A.
    [2012] Mass conservative domain decomposition for porous media flow. In: Petrova, R. (Ed.) Finite Volume Method-Powerful Means of Engineering Design, InTech Europe, Rijeka, Croatia, 235–256.
    [Google Scholar]
  43. Notay, Y.
    [2010] An aggregation-based algebraic multigrid method. Electronic transactions on numerical analysis, 37(6), 123–146.
    [Google Scholar]
  44. Ruge, J.W. and Stüben, K.
    [1987] Algebraic multigrid. In: Multigrid methods, SIAM, 73–130.
    [Google Scholar]
  45. Stüben, K.
    [2001] A review of algebraic multigrid. In: Numerical Analysis: Historical Developments in the 20th Century, Elsevier, 331–359.
    [Google Scholar]
  46. Trangenstein, J.A. and Bell, J.B.
    [1989] Mathematical Structure of Compositional Reservoir Simulation. SIAM J. Sci. Stat. Comp., 10(5), 817–845.
    [Google Scholar]
  47. Vanek, P., Mandel, J. and Brezina, M.
    [1996] Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing, 56(3), 179–196.
    [Google Scholar]
  48. Wallis, J., Kendall, R. and Little, T.
    [1985] Constrained Residual Acceleration of Conjugate Residual Methods. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (SPE).
    [Google Scholar]
  49. Wang, Y., Hajibeygi, H. and Tchelepi, H.A.
    [2012] Algebraic multiscale linear solver for heterogeneous elliptic problems. In: ECMOR XIII – 13th European Conference on the Mathematics of Oil Recovery. EAGE, Biarritz, France.
    [Google Scholar]
  50. [2014] Algebraic multiscale solver for flow in heterogeneous porous media. J. Comput. Phys., 259, 284–303.
    [Google Scholar]
  51. [2016] Monotone multiscale finite volume method. Comput. Geosci., 20(3), 509–524.
    [Google Scholar]
  52. Watts, J.W.
    [1986] A Compositional Formulation of the Pressure and Saturation Equations. SPE Res. Eng., 1(03), 243–252.
    [Google Scholar]
  53. Xu, J. and Zikatanov, L.
    [2017] Algebraic multigrid methods. Acta Numerica, 26, 591–721.
    [Google Scholar]
  54. Zhou, H. and Tchelepi, H.A.
    [2008] Operator-based multiscale method for compressible flow. SPE J., 13(2), 267–273.
    [Google Scholar]
  55. [2012] Two-stage algebraic multiscale linear solver for highly heterogeneous reservoir models. SPE J., 17(2), 523–539.
    [Google Scholar]
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