Transport in Porous Media

, Volume 113, Issue 1, pp 111–135 | Cite as

Multiscale Simulation of Polymer Flooding with Shear Effects

  • Sindre T. Hilden
  • Olav MøynerEmail author
  • Knut-Andreas Lie
  • Kai Bao


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 petrophysical properties and unknowns on a coarser grid used for dynamic simulation. Herein, we extend one such method—the multiscale restricted-smoothed basis (MsRSB) method—to polymer flooding including shear-thinning (and thickening) effects, which gives highly nonlinear fluid models that are challenging to simulate. To this end, we first formulate a sequentially implicit solution procedure for polymer models with non-Newtonian rheology. By treating the implicit velocity dependence of the viscosities in an inner iteration loop, we obtain a formulation that appears to be more robust and stable than the standard fully-implicit approach. We then use a general algebraic multiscale framework to formulate an efficient and versatile multiscale solver. The unique feature of the MsRSB method is how the prolongation operators are constructed. By using restricted smoothing much in the same way as in smoothed aggregation multigrid methods, one gets a robust and flexible method that enables coarse partitions and prolongation operators to be constructed in an semi-automated manner even for highly complex geo-cellular models with high media contrasts and unstructured cell connections. By setting iterative tolerances appropriately, the resulting iterative multiscale solver can be set to compute mass-conservative approximations to the sequential or fully-implicit solution to arbitrary accuracy and hence be used to trade accuracy for efficiency. We first verify the sequential solution procedure and multiscale solver against a well-established commercial simulator on a test case with simple geometry, highly heterogeneous media properties, and strongly nonlinear fluid behavior. Next, the sequential fine-scale and multiscale solvers are validated on a synthetic simulation model of a shallow-marine reservoir. Here, the computational time is dominated by the pressure solves, and 5–8 times speedup is observed when replacing the fine-scale pressure solver by the iterative multiscale method. We also demonstrate the flexibility of the method by applying it to model with unstructured polyhedral cells that adapt to well positions and faults.


Multiscale simulation Porous media Nonlinear solver Polymer flooding Enhanced oil recovery 



The research was partly funded by VISTA, which is a basic research programme funded by Statoil and conducted in close collaboration with The Norwegian Academy of Science and Letters. The authors would also like to acknowledge funding by Schlumberger Information Solutions and by the Research Council of Norway under grant no. 226035.


  1. Aarnes, J.E., Krogstad, S.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12(3), 297–315 (2008). doi: 10.1007/s10596-007-9072-8 CrossRefGoogle Scholar
  2. Alpak, F.O., Pal, M., Lie, K.A.: A multiscale method for modeling flow in stratigraphically complex reservoirs. SPE J. 17(4), 1056–1070 (2012). doi: 10.2118/140403-PA CrossRefGoogle Scholar
  3. AlSofi, A.M., Blunt, M.J.: Streamline-based simulation of non-Newtonian polymer flooding. SPE J 15(4), 895–905 (2010). doi: 10.2118/123971-PA CrossRefGoogle Scholar
  4. Arbogast, T.: Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6(3–4), 453–481 (2002). doi: 10.1023/A:1021295215383 CrossRefGoogle Scholar
  5. Babaei, M., King, P.R.: A modified nested-gridding for upscaling-downscaling in reservoir simulation. Transp. Porous Media 93(3), 753–775 (2012). doi: 10.1007/s11242-012-9981-4 CrossRefGoogle Scholar
  6. Babaei, M., King, P.R.: An upscaling-static-downscaling scheme for simulation of enhanced oil recovery processes. Transp. Porous Media 98(2), 465–484 (2013). doi: 10.1007/s11242-013-0154-x CrossRefGoogle Scholar
  7. Brenier, Y.: Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal 28(3), 685–696 (1991). doi: 10.1137/0728036 CrossRefGoogle Scholar
  8. Chen, Z., Hou, T.Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp 72, 541–576 (2003). doi: 10.1090/S0025-5718-02-01441-2 CrossRefGoogle Scholar
  9. Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4(4), 308–317 (2001). doi: 10.2118/66599-MS CrossRefGoogle Scholar
  10. Clemens, T., Abdev, J., Thiele, M.R.: Improved polymer-flood management using streamlines. SPE J. 14(2), 171–181 (2011). doi: 10.2118/132774-PA Google Scholar
  11. Cusini, M., Lukyanov, A.A., Natvig, J., 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). doi: 10.1016/ CrossRefGoogle Scholar
  12. Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys 251, 116–135 (2013). doi: 10.1016/ CrossRefGoogle Scholar
  13. Gautier, Y., Blunt, M.J., Christie, M.A.: Nested gridding and streamline-based simulation for fast reservoir performance prediction. Comput. Geosci. 3(3), 295–320 (1999). doi: 10.1023/A:1011535210857 CrossRefGoogle Scholar
  14. Hajibeygi, H., Bonfigli, G., Hesse, M.A., Jenny, P.: Iterative multiscale finite-volume method. J. Comput. Phys. 227(19), 8604–8621 (2008). doi: 10.1016/ CrossRefGoogle Scholar
  15. Hajibeygi, H., Jenny, P.: Adaptive iterative multiscale finite volume method. J. Comput. Phys. 230(3), 628–643 (2011). doi: 10.1016/ CrossRefGoogle Scholar
  16. Hoteit, H., Chawathé, A.: Making field-scale chemical EOR simulations a practical reality using dynamic gridding. In: SPE EOR Conference at Oil and Gas West Asia, 31 March–2 April, Muscat, Oman (2014). doi: 10.2118/169688-MS
  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). doi: 10.1006/jcph.1997.5682 CrossRefGoogle Scholar
  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). doi: 10.1016/S0021-9991(03)00075-5 CrossRefGoogle Scholar
  19. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998). doi: 10.1137/S1064827595287997 CrossRefGoogle Scholar
  20. 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 (2015). doi: 10.2118/173226-MS
  21. 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 (2015). doi: 10.2118/173317-MS
  22. 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 (2009). doi: 10.2118/118993-MS
  23. Lake, L.W.: Enhanced oil recovery, reprint 2010 edn. Society of Petroleum Engineers (1989)Google Scholar
  24. Lie, K.A., Nilsen, H.M., Rasmussen, A.F., Raynaud, X.: An unconditionally stable splitting method using reordering for simulating polymer injection. In: ECMOR XIII - 13th European Conference on the Mathematics of Oil Recovery, Biarritz, France, 10-13 September 2012 (2012). B25Google Scholar
  25. Lie, K.A.: An introduction to reservoir simulation using MATLAB: user guide for the Matlab Reservoir Simulation Toolbox (MRST). SINTEF ICT, 2nd edn. (2015)
  26. Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H., Skaflestad, B.: Open-source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012). doi: 10.1007/s10596-011-9244-4 CrossRefGoogle Scholar
  27. Lie, K.A., Natvig, J.R., Krogstad, S., Yang, Y., Wu, X.H.: Grid adaptation for the Dirichlet–Neumann representation method and the multiscale mixed finite-element method. Comput. Geosci. 18(3), 357–372 (2014). doi: 10.1007/s10596-013-9397-4 CrossRefGoogle Scholar
  28. Lie, K.A., Nilsen, H.M., Rasmussen, A.F., Raynaud, X.: Fast simulation of polymer injection in heavy-oil reservoirs based on topological sorting and sequential splitting. SPE J. 19(6), 991–1004 (2014). doi: 10.2118/163599-PA CrossRefGoogle Scholar
  29. Lipnikov, K., Moulton, J.D., Svyatskiy, D.: A multilevel multiscale mimetic (m\(^3\)) method for two-phase flows in porous media. J. Comput. Phys. 227(14), 6727–6753 (2008). doi: 10.1016/ CrossRefGoogle Scholar
  30. Lunati, I., Lee, S.H.: An operator formulation of the multiscale finite-volume method with correction function. Multiscale Model. Simul. 8(1), 96–109 (2009). doi: 10.1137/080742117 CrossRefGoogle Scholar
  31. Lunati, I., Tyagi, M., Lee, S.H.: An iterative multiscale finite volume algorithm converging to the exact solution. J. Comput. Phys. 230(5), 1849–1864 (2011). doi: 10.1016/ CrossRefGoogle Scholar
  32. Manzocchi, T., et al.: Sensitivity of the impact of geological uncertainty on production from faulted and unfaulted shallow-marine oil reservoirs: objectives and methods. Petrol. Geosci. 14(1), 3–15 (2008). doi: 10.1144/1354-079307-790 CrossRefGoogle Scholar
  33. 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 (2015). doi: 10.2118/173265-MS
  34. Møyner, O., Lie, K.A.: A multiscale restriction-smoothed basis method for compressible black-oil models. To be appear in SPE J. (2016)Google Scholar
  35. Møyner, O., Lie, K.A.: The multiscale finite-volume method on stratigraphic grids. SPE J. 19(5), 816–831 (2014). doi: 10.2118/163649-PA CrossRefGoogle Scholar
  36. Møyner, O., Lie, K.A.: A multiscale two-point flux-approximation method. J. Comput. Phys. 275, 273–293 (2014). doi: 10.1016/ CrossRefGoogle Scholar
  37. 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 (2016)CrossRefGoogle Scholar
  38. 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. SPE J. 16(4), 880–888 (2011). doi: 10.2018/119132-PA CrossRefGoogle Scholar
  39. 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), 10,108–10,124 (2008). doi: 10.1016/ CrossRefGoogle Scholar
  40. Notay, Y.: An aggregation-based algebraic multigrid method. Electron. Trans. Numer. Anal. 37, 123–140 (2010)Google Scholar
  41. 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). doi: 10.1002/fld.3978 CrossRefGoogle Scholar
  42. Schlumberger: ECLIPSE 2013.2 Technical Description (2013)Google Scholar
  43. The MATLAB Reservoir Simulation Toolbox, MRST 2015a (2015).
  44. Todd, M., Longstaff, W.: The development, testing, and application of a numerical simulator for predicting miscible flood performance. J. Pet. Technol. 24(7), 874–882 (1972)CrossRefGoogle Scholar
  45. Wang, Y., Hajibeygi, H., Tchelepi, H.A.: Algebraic multiscale solver for flow in heterogeneous porous media. J. Comput. Phys. 259, 284–303 (2014). doi: 10.1016/ CrossRefGoogle Scholar
  46. Zhou, H., Tchelepi, H.A.: Operator-based multiscale method for compressible flow. SPE J. 13(2), 267–273 (2008). doi: 10.2118/106254-PA CrossRefGoogle Scholar
  47. Zhou, H., Tchelepi, H.A.: Two-stage algebraic multiscale linear solver for highly heterogeneous reservoir models. SPE J. 17(2), 523–539 (2012). doi: 10.2118/141473-PA CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Sindre T. Hilden
    • 1
    • 2
  • Olav Møyner
    • 2
    Email author
  • Knut-Andreas Lie
    • 2
  • Kai Bao
    • 2
  1. 1.Department of Mathematical SciencesNTNUTrondheimNorway
  2. 2.Department of Applied MathematicsSINTEFOsloNorway

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