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Successful application of multiscale methods in a real reservoir simulator environment

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Abstract

For the past 10 years or so, a number of so-called multiscale methods have been developed as an alternative approach to upscaling and to accelerate reservoir simulation. The key idea of all these methods is to construct a set of prolongation operators that map between unknowns associated with cells in a fine grid holding the petrophysical properties of the geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as for flow-based upscaling methods, and can be used to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, the multiscale basis functions have subscale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner. Among all multiscale formulations discussed in the literature, the multiscale restriction-smoothed basis (MsRSB) method has proved to be particularly promising. This method has been implemented in a commercially available simulator and has three main advantages. First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and robust when used as an approximate solver and converges relatively fast when used as an iterative fine-scale solver. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. We discuss numerical challenges posed by contemporary geomodels and report a number of validation cases showing that the MsRSB method is an efficient, robust, and versatile method for simulating complex models of real reservoirs.

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References

  1. Aarnes, J.: 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)

    Article  Google Scholar 

  2. Aarnes, J. E., Kippe, V., Lie, K.: Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Resour. 28(3), 257–271 (2005)

    Article  Google Scholar 

  3. Aarnes, J. E., Krogstad, S., Lie, K.: 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)

  4. Aarnes, J. E., Krogstad, S., Lie, K.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12(3), 297–315 (2008)

    Article  Google Scholar 

  5. Alpak, F. O., Pal, M., Lie, K.: A multiscale method for modeling flow in stratigraphically complex reservoirs. SPE J. 17(4), 1056–1070 (2012)

    Article  Google Scholar 

  6. Arbogast, T: Numerical Subgrid Upscaling of Two-Phase Flow in Porous Media. In: Numerical Treatment of Multiphase Flows in Porous Media (Beijing, 1999), Lecture Notes in Phys., vol. 552, pp 35–49. Springer, Berlin (2000)

  7. Arbogast, T: Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6(3-4), 453–481 (2002)

    Article  Google Scholar 

  8. Arbogast, T: Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal. 42(2), 576–598 (2004)

    Article  Google Scholar 

  9. Arbogast, T., Bryant, S.: A two-scale numerical subgrid technique for waterflood simulations. SPE J. 7 (4), 446–457 (2002)

    Article  Google Scholar 

  10. Arbogast, T., Pencheva, G., Wheeler, M. F., Yotov, I: A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6(1), 319–346 (2007)

    Article  Google Scholar 

  11. Arnold, D., Demyanov, V., Tatum, D., Christie, M., Rojas, T., Geiger, S., Corbett, P: Hierarchical benchmark case study for history matching, uncertainty quantification and reservoir characterisation. Comput. Geosci. 50, 4–15 (2013)

    Article  Google Scholar 

  12. Audigane, P., Blunt, M.: Dual mesh method for upscaling in waterflood simulation. Transp. Porous Media 55(1), 71–89 (2004)

    Article  Google Scholar 

  13. Babaei, M., King, P.: A modified nested-gridding for upscaling-downscaling in reservoir simulation. Transp. Porous Media 93(3), 753–775 (2012)

    Article  Google Scholar 

  14. Babaei, M., King, P.: An upscaling-static-downscaling scheme for simulation of enhanced oil recovery processes. Transp. Porous Media 98(2), 465–484 (2013)

    Article  Google Scholar 

  15. Babuska, I., Caloz, G., Osborn, J: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31(4), 945–981 (1994)

    Article  Google Scholar 

  16. Babuska, I., Osborn, J.: Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Num. Anal. 20(3), 510–536 (1983)

    Article  Google Scholar 

  17. Barker, J., Thibeau, S: A critical review of the use of pseudorelative permeabilities for upscaling. SPE Res. Engrg. 12(2), 138–143 (1997)

    Article  Google Scholar 

  18. Bergamashi, L., Mantica, S., Manzini, G: A mixed finite element-finite volume formulation of the black-oil model. SIAM J. Sci. Comput. 3(20), 970–997 (1998)

    Article  Google Scholar 

  19. Bonfigli, G., Jenny, P.: Recent developments in the multi-scale-finite-volume procedure. In: Lirkov, I., Margenov, S., Wasniewski, J. (eds.) Large-Scale Scientific Computing, Lecture Notes in Computer Science. doi:10.1007/978-3-642-12535-5_13, vol. 5910, pp 124–131. Springer, Berlin / Heidelberg (2010)

  20. Bush, L., Ginting, V., Presho, M.: Application of a conservative, generalized multiscale finite element method to flow models. J. Comput. Appl. Math. 260, 395–409 (2014)

    Article  Google Scholar 

  21. Castelletto, N., Hajibeygi, H., Tchelepi, H.: Hybrid Multiscale Formulation for Coupled Flow and Geomechanics. In: ECMOR XV – 15Th European Conference on the Mathematics of Oil Recovery. EAGE, Amsterdam, The Netherlands (2016)

  22. Castelletto, N., Hajibeygi, H., Tchelepi, H.: Multiscale finite-element method for linear elastic geomechanics. J. Comput. Phys. 331, 337–356 (2017)

    Article  Google Scholar 

  23. Chen, T., Gerritsen, M.G., Durlofsky, L.J., Lambers, J.V.: Adaptive local-global VCMP methods for coarse-scale reservoir modeling. In: 2009 SPE Reservoir Simulation Symposium, 2-4 February, The Woodlands, Texas. SPE 118994-MS (2009)

  24. Chen, T., Gerritsen, M. G., Lambers, J. V., Durlofsky, L.: Global variable compact multipoint methods for accurate upscaling with full-tensor effects. Comput. Geosci. 14(1), 65–81 (2010)

    Article  Google Scholar 

  25. Chen, Y., Durlofsky, L.: Adaptive local-global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media 62, 157–182 (2006)

    Article  Google Scholar 

  26. Chen, Y., Durlofsky, L. J., Gerritsen, M., Wen, X.: A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26, 1041–1060 (2003)

    Article  Google Scholar 

  27. Chen, Z.: Formulations and numerical methods of the black oil model in porous media. SIAM J. Numer. Anal. 38(2), 489–514 (2000)

    Article  Google Scholar 

  28. Chen, Z., Hou, T.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72, 541–576 (2003)

    Article  Google Scholar 

  29. Christie, M.: Upscaling for reservoir simulation. J. Pet. Tech. 48(11), 1004–1010 (1996)

    Article  Google Scholar 

  30. Chung, E. T., Efendiev, Y., Li, G: An adaptive GMsFEM for high-contrast flow problems. J. Comput. Phys. 273, 54–76 (2014)

    Article  Google Scholar 

  31. Cortinovis, D., Jenny, P: Iterative Galerkin-enriched multiscale finite-volume method. J. Comput. Phys. 277, 248–267 (2014)

    Article  Google Scholar 

  32. Cusini, M., Lukyanov, A. A., Natvig, J. R., Hajibeygi, H: Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous media. J. Comput. Phys. 299, 472–486 (2015)

    Article  Google Scholar 

  33. DeBaun, D., Byer, T., Childs, P., Chen, J., Saaf, F., Wells, M., Liu, J., Cao, H., Pianelo, L., Tilakraj, V., Crumpton, P., Walsh, D., Yardumian, H., Zorzynski, R., Lim, K.T., Schrader, M., Zapata, V., Nolen, J., Tchelepi, H.A.: An extensible architecture for next generation scalable parallel reservoir simulation. In: SPE Reservoir Simulation Symposium, 31 January–2 Febuary, The Woodlands, Texas, USA. SPE 93274-MS (2005)

  34. Dolean, V., Jolivet, P., Nataf, F., Spillane, N., Xiang, H: Two-level domain decomposition methods for highly heterogeneous Darcy equations. Connections with multiscale methods. Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles 69(4), 731–752 (2014)

    Article  Google Scholar 

  35. Efendiev, Y., Galvis, J., Hou, T.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)

    Article  Google Scholar 

  36. Efendiev, Y., Galvis, J., Wu, X.: Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230(4), 937–955 (2011)

    Article  Google Scholar 

  37. Farmer, C.: Upscaling: a review. Int. J. Numer. Meth. Fluids 40(1–2), 63–78 (2002)

    Article  Google Scholar 

  38. Fjerstad, P.A., Dasie, W.J., Sikandar, A.S., Cao, H., Liu, J.: Next generation parallel computing for large-scale reservoir simulation. In: SPE International Improved Oil Recovery Conference in Asia Pacific, 5-6 December, Kuala Lumpur, Malaysia. SPE 97358-MS (2007)

  39. Fossen, H., Hesthammer, J: Structural Geology of the Gullfaks Field. In: Coward, M. P., Johnson, H., Daltaban, T. (eds.) Structural Geology in Reservoir Characterization, 127, Geological Society Special Publication, pp 231–261 (1998)

  40. Gautier, Y., Blunt, M. J., Christie, M.: Nested gridding and streamline-based simulation for fast reservoir performance prediction. Comput. Geosci. 3, 295–320 (1999)

    Article  Google Scholar 

  41. Gerritsen, M. G., Durlofsky, L.: Modeling fluid flow in oil reservoirs. Annual Rev. Fluid Mech. 37(1), 211–238 (2005)

    Article  Google Scholar 

  42. Hajibeygi, H., Bonfigli, G., Hesse, M. A., Jenny, P: Iterative multiscale finite-volume method. J. Comput. Phys. 227(19), 8604–8621 (2008)

    Article  Google Scholar 

  43. Hajibeygi, H., Deb, R., Jenny, P.: Multiscale finite volume method for non-conformal coarse grids arising from faulted porous media. In: SPE Reservoir Simulation Symposium, The Woodlands, TX, USA, 21–23 February 2011. SPE 142205-MS (2011)

  44. Hajibeygi, H., Jenny, P.: Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media. J. Comput. Phys. 228(14), 5129–5147 (2009)

    Article  Google Scholar 

  45. Hajibeygi, H., Jenny, P.: Adaptive iterative multiscale finite volume method. J. Comput. Phys. 230(3), 628–643 (2011)

    Article  Google Scholar 

  46. Hajibeygi, H., Tchelepi, H.: Compositional multiscale finite-volume formulation. SPE J. 19(2), 316–326 (2014)

    Article  Google Scholar 

  47. Hesse, M. A., Mallison, B. T., Tchelepi, H.: Compact multiscale finite volume method for heterogeneous anisotropic elliptic equations. Multiscale Model. Simul. 7(2), 934–962 (2008)

    Article  Google Scholar 

  48. Hilden, S. T., Møyner, O., Lie, K.A., Bao, K.: Multiscale simulation of polymer flooding with shear effects. Transp. Porous Media 113(1), 111–135 (2016)

    Article  Google Scholar 

  49. Hou, T. Y., Wu, X.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)

    Article  Google Scholar 

  50. Hughes, T., Feijoo, G., Mazzei, L., Quincy, J.: The variational multiscale method – a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg 166, 3–24 (1998)

    Article  Google Scholar 

  51. Jenny, P., Lee, S. H., Tchelepi, H.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187, 47–67 (2003)

    Article  Google Scholar 

  52. Jenny, P., Lee, S. H., Tchelepi, H.: Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model. Simul. 3(1), 50–64 (2004)

    Article  Google Scholar 

  53. Jenny, P., Lee, S. H., Tchelepi, H.: 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 (2006)

    Article  Google Scholar 

  54. Jenny, P., Lunati, I: Modeling complex wells with the multi-scale finite-volume method. J. Comput. Phys. 228(3), 687–702 (2009)

    Article  Google Scholar 

  55. Juanes, R., Dub, F.: A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms. Comput. Geosci. 12(3), 273–295 (2008)

    Article  Google Scholar 

  56. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comp. 20(1), 359–392 (1998)

    Article  Google Scholar 

  57. Kippe, V., Aarnes, J. E., Lie, K.: A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci. 12(3), 377–398 (2008)

    Article  Google Scholar 

  58. Kozlova, A., Li, Z., Natvig, J.R., Watanabe, S., Zhou, Y., Bratvedt, K., Lee, S.H.: A real-field multiscale black-oil reservoir simulator. In: SPE Reservoir Simulation Symposium, 23-25 February, Houston, Texas, USA. SPE 173226-MS (2015)

  59. Kozlova, A., Walsh, D., Chittireddy, S., Li, Z., Natvig, J., Watanabe, S., Bratvedt, K.: A Hybrid Approach to Parallel Multiscale Reservoir Simulator. EAGE, Amsterdam, The Netherlands (2016)

    Book  Google Scholar 

  60. Krogstad, S.: A sparse basis POD for model reduction of multiphase compressible flow. In: 2011 SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 21-23 February 2011. SPE 141973-MS (2011)

  61. Krogstad, S., Lie, K.A., Møyner, O., Nilsen, H.M., Raynaud, X., Skaflestad, B.: MRST-AD – an open-source framework for rapid prototyping and evaluation of reservoir simulation problems. In: SPE Reservoir Simulation Symposium, 23–25 February, Houston, Texas. SPE 173317-MS (2015)

  62. Krogstad, S., Lie, K.A., Nilsen, H.M., Natvig, J.R., Skaflestad, B., Aarnes, J.E.: A multiscale mixed finite-element solver for three-phase black-oil flow. In: SPE Reservoir Simulation Symposium, The Woodlands, TX, USA, 2–4 February 2009. SPE 118993-MS (2009)

  63. Krogstad, S., Lie, K. A., Skaflestad, B: Mixed Multiscale Methods for Compressible Flow. In: Proceedings of ECMOR XIII–13Th European Conference on the Mathematics of Oil Recovery EAGE. Biarritz, France (2012)

    Google Scholar 

  64. Larson, M.G., Målqvist, A.: Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations. Comp. Meth. Appl. Mech. Eng. 196(37–40), 3546–3561 (2007). Special Issue Honoring the 80th Birthday of Professor Ivo Babuska

    Article  Google Scholar 

  65. Lee, S. H., Efendiev, Y., Tchelepi, H.: Hybrid upwind discretization of nonlinear two-phase flow with gravity. Adv. Water Resour. 82, 27–38 (2015)

    Article  Google Scholar 

  66. Lee, S. H., Wolfsteiner, C., Tchelepi, H.: Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three phase flow with gravity. Comput. Geosci. 12(3), 351–366 (2008)

    Article  Google Scholar 

  67. Li, B., Tchelepi, H.: Unconditionally convergent nonlinear solver for multiphase flow in porous media under viscous force, buoyancy, and capillarity. Energy Procedia 59, 404–411 (2014)

    Article  Google Scholar 

  68. Lie, K.A.: An Introduction to Reservoir Simulation Using MATLAB: User guide for the Matlab Reservoir Simulation Toolbox (MRST). SINTEF ICT. http://www.sintef.no/Projectweb/MRST/publications, 3nd edn. (2016)

  69. Lie, K.A., Møyner, O., Natvig, J.R.: A Feature-Enriched Multiscale Method for Simulating Complex Geomodels. In: SPE Reservoir Simulation Conference, 20–22 February 2017, Montgomery, Texas, USA. SPE 182701-MS (2017)

  70. Lie, K. A., Natvig, J. R., Krogstad, S., Yang, Y., Wu, X.: Grid adaptation for the dirichlet–Neumann representation method and the multiscale mixed finite-element method. Comput. Geosci. 18(3), 357–372 (2014)

    Article  Google Scholar 

  71. Lunati, I., Jenny, P: Multiscale finite-volume method for compressible multiphase flow in porous media. J. Comput. Phys. 216(2), 616–636 (2006a)

  72. Lunati, I., Jenny, P: A Multiscale Finite-Volume Method for Three-Phase Flow Influenced by Gravity. In: Binning, P., Engesgaard, P., Dahle, H., Pinder, G., Gray, W (eds.) Proceedings of the XVI International Conference on Computational Methods in Water Resources. Copenhagen, Denmark (2006b)

  73. Lunati, I., Jenny, P.: Treating highly anisotropic subsurface flow with the multiscale finite-volume method. Multiscale Model. Simul. 6(1), 308–318 (2007)

    Article  Google Scholar 

  74. Lunati, I., Lee, S.: An operator formulation of the multiscale finite-volume method with correction function. Multiscale Model. Simul. 8(1), 96–109 (2009)

    Article  Google Scholar 

  75. Lunati, I., Tyagi, M., Lee, S.: An iterative multiscale finite volume algorithm converging to the exact solution. J. Comput. Phys. 230(5), 1849–1864 (2011)

    Article  Google Scholar 

  76. Magri, V. A.: Pré-Condicionador Multiescala AlgéBrico Aplicado à Simulação De Reservatórios De Petróleo. Master’s thesis. Universidade Federal de Santa Catarina - UFSC (2015)

  77. Manea, A., Hajibeygi, H., Vassilevski, P., Tchelepi, H.: Enriched Algebraic Multiscale Linear Solver. In: ECMOR XV – 15Th European Conference on the Mathematics of Oil Recovery. EAGE, Amsterdam, The Netherlands (2016)

  78. Manea, A.M., Sewall, J., Tchelepi, H.A.: Parallel Multiscale Linear Solver for Highly Detailed Reservoir Models. In: SPE Reservoir Simulation Symposium, 23-25 February, Houston, Texas, USA. SPE 173259-MS (2015)

  79. Montaron, B.A., Bradley, D.C., Cooke, A., Prouvost, L.P., Raffn, A.G., Vidal, A., Wilt, M.: Shapes of flood fronts in heterogeneous reservoirs and oil recovery strategies. In: SPE/EAGE Reservoir Characterization and Simulation Conference, 28-31 October, Abu Dhabi, UAE. SPE-111147-MS (2007)

  80. Møyner, O.: Multiscale Finite-Volume Methods on Unstructured Grids Master’s Thesis. Norwegian University of Science and Technology, Trondheim (2012)

    Google Scholar 

  81. Møyner, O.: Construction of multiscale preconditioners on stratigraphic grids. In: ECMOR XIV – 14th European Conference on the Mathematics of Oil Recovery, Catania, Sicily, Italy, 8-11 September 2014. EAGE (2014)

  82. Møyner, O.: Next Generation Multiscale Methods for Reservoir Simulation. Ph.D. thesis, Norwegian University of Science and Technology (2016a)

  83. Møyner, O.: Nonlinear Solver for Three-Phase Transport Problems Based on Approximate Trust Regions. In: ECMOR XV – 15Th European Conference on the Mathematics of Oil Recovery. EAGE, Amsterdam, The Netherlands (2016b)

  84. Møyner, O., Lie, K.A.: The multiscale finite volume method on unstructured grids. In: SPE Reservoir Simulation Symposium, The Woodlands, TX, USA, 18–20 February 2013. SPE 163649-MS (2013)

  85. Møyner, O., Lie, K.A: The multiscale finite-volume method on stratigraphic grids. SPE J. 19(5), 816–831 (2014a)

  86. Møyner, O., Lie, K.A: A multiscale two-point flux-approximation method. J. Comput. Phys. 275, 273–293 (2014b)

  87. Møyner, O., Lie, K.A.: A multiscale method based on restriction-smoothed basis functions suitable for general grids in high contrast media. In: SPE Reservoir Simulation Symposium held in Houston, Texas, USA, 23–25 February 2015. SPE 173265-MS (2015)

  88. Møyner, O., Lie, K.A: A multiscale restriction-smoothed basis method for compressible black-oil models. SPE J. 21(6), 2079–2096 (2016a)

  89. Møyner, O., Lie, K.A: A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids. J. Comput. Phys. 304, 46–71 (2016b)

  90. Møyner, O., Tchelepi, H.A.: A Multiscale Restriction-Smoothed Basis Method for Compositional Models. In: SPE Reservoir Simulation Conference, 20–22 February 2017, Montgomery, Texas, USA. SPE 182679-MS (2017)

  91. Natvig, J.R., Skaflestad, B., Bratvedt, F., Bratvedt, K., Lie, K.A., Laptev, V., Khataniar, S.K.: Multiscale mimetic solvers for efficient streamline simulation of fractured reservoirs. In: SPE Reservoir Simulation Symposium, The Woodlands, TX, USA, 2–4 February 2009. SPE 119132-MS (2009)

  92. Natvig, J. R., Skaflestad, B., Bratvedt, F., Bratvedt, K., Lie, K. A., Laptev, V., Khataniar, S.: Multiscale mimetic solvers for efficient streamline simulation of fractured reservoirs. SPE J. 16(4), 880–888 (2011)

    Article  Google Scholar 

  93. Nordbotten, J. M., Bjørstad, P.E.: On the relationship between the multiscale finite-volume method and domain decomposition preconditioners. Comput. Geosci. 12(3), 367–376 (2008)

    Article  Google Scholar 

  94. Nordbotten, J. M., Keilegavlen, E., Sandvin, A. Petrova, R. (ed.): Mass Conservative Domain Decomposition for Porous Media Flow. Intech Europe, Rijeka, Croatia (2012)

  95. Pal, M., Lamine, S., Lie, K. A., Krogstad, S.: Multiscale Method for Two and Three-Phase Flow Simulation in Subsurface Petroleum Reservoirs. In: Proceedings of ECMOR XIII–13Th European Conference on the Mathematics of Oil Recovery. EAGE, Biarritz, France (2012)

  96. Pal, M., Lamine, S., Lie, K. A., Krogstad, S: Validation of the multiscale mixed finite-element method. Int. J. Numer. Meth. Fluids 77(4), 206–223 (2015)

    Article  Google Scholar 

  97. Parramore, E., Edwards, M. G., Pal, M., Lamine, S: Multiscale finite-volume CVD-MPFA formulations on structured and unstructured grids. Multiscale Model. Simul. 14(2), 559–594 (2016)

    Article  Google Scholar 

  98. Renard, P., De Marsily, G: Calculating equivalent permeability: a review. Adv. Water Resour. 20(5), 253–278 (1997)

    Article  Google Scholar 

  99. Sandve, T. H., Berre, I., Keilegavlen, E., Nordbotten, J.: Multiscale Simulation of Flow and Heat Transport in Fractured Geothermal Reservoirs: Inexact Solvers and Improved Transport Upscaling. Stanford, California, USA (2013)

    Google Scholar 

  100. Shah, S., Møyner, O., Tene, M., Lie, K.A., Hajibeygi, H.: The multiscale restriction smoothed basis method for fractured porous media. J. Comput. Phys. 318, 36–57 (2016)

    Article  Google Scholar 

  101. 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, vol. A36. EAGE, Bergen, Norway (2008)

  102. Smith, B. F., Bjørstad, P.E., Gropp, W.D: Domain decomposition: Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  103. Spillette, A.G., Hillestad, J.G., Stone, H.L.: A high-stability sequential solution approach to reservoir simulation. In: Fall Meeting of the Society of Petroleum Engineers of AIME, 30 September-3 October, Las Vegas, Nevada. SPE-4542-MS (1973)

  104. Tchelepi, H. A., Jenny, P., Lee, S. H., Wolfsteiner, C.: Adaptive multiscale finite volume framework for reservoir simulation. SPE J. 12, 188–195 (2007)

    Article  Google Scholar 

  105. Tene, M., Kobaisi, M. S. A., Hajibeygi, H.: Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS). J. Comput. Phys. 321, 819–845 (2016a)

  106. Tene, M., Kobaisi, M. S. A., Hajibeygi, H.: Multiscale Projection-Based Embedded Discrete Fracture Modeling Approach (F-AMS-PEDFM). In: ECMOR XV – 15Th European Conference on the Mathematics of Oil Recovery. EAGE, Amsterdam, The Netherlands (2016b)

  107. Wang, Y., Hajibeygi, H., Tchelepi, H.: Algebraic Multiscale Linear Solver for Heterogeneous Elliptic Problems. In: ECMOR XIII - 13Th European Conference on the Mathematics of Oil Recovery. EAGE, Biarritz, France (2012)

  108. Wang, Y., Hajibeygi, H., Tchelepi, H.: Algebraic multiscale solver for flow in heterogeneous porous media. J. Comput. Phys. 259, 284–303 (2014)

    Article  Google Scholar 

  109. Wang, Y., Hajibeygi, H., Tchelepi, H.: Monotone multiscale finite volume method. Comput. Geosci. 20(3), 509–524 (2016)

    Article  Google Scholar 

  110. Watanabe, S., Li, Z., Bratvedt, K., Lee, S.: A Stable Multi-Phase Nonlinear Transport Solver with Hybrid Upwind Discretization in Multiscale Reservoir Simulators. In: ECMOR XV – 15Th European Conference on the Mathematics of Oil Recovery. EAGE, Amsterdam, The Netherlands (2016)

  111. Wolfsteiner, C., Lee, S. H., Tchelepi, H.: Well modeling in the multiscale finite volume method for subsurface flow simulation. Multiscale Model. Simul. 5(3), 900–917 (2006)

    Article  Google Scholar 

  112. Zhou, H., Tchelepi, H.: Operator-based multiscale method for compressible flow. SPE J. 13(2), 267–273 (2008)

    Article  Google Scholar 

  113. Zhou, H., Tchelepi, H.: Two-stage algebraic multiscale linear solver for highly heterogeneous reservoir models. SPE J. 17(2), 523–539 (2012)

    Article  Google Scholar 

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Acknowledgments

The authors thank Schlumberger for the permission to publish, and the Research Council of Norway (grant no. 226035) for partial financial support.

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Authors and Affiliations

  1. Mathematics and Cybernetics, SINTEF Digital, Oslo, Norway

    K.-A. Lie & O. Møyner

  2. Schlumberger, Asker, Norway

    J. R. Natvig

  3. Schlumberger, Houston, USA

    A. Kozlova, K. Bratvedt, S. Watanabe & Z. Li

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  2. O. Møyner
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  3. J. R. Natvig
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  4. A. Kozlova
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  6. S. Watanabe
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  7. Z. Li
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Correspondence to O. Møyner.

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Lie, KA., Møyner, O., Natvig, J.R. et al. Successful application of multiscale methods in a real reservoir simulator environment. Comput Geosci 21, 981–998 (2017). https://doi.org/10.1007/s10596-017-9627-2

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  • DOI: https://doi.org/10.1007/s10596-017-9627-2

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