Skip to main content
SpringerLink
Log in

Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-conditioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: (i) a sequential formulation that decouples flow and transport into separate subproblems, and (ii) a highly efficient Gauss–Seidel solver for the transport problems. This solver uses intercell fluxes to reorder the grid cells according to their upstream neighbors, and groups cells that are mutually dependent because of counter-current flow into local clusters. The cells and local clusters can then be solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process. This method has been successfully applied to real-field problems using the standard first-order finite volume discretization. Here, we extend the idea to first-order dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Appleyard, J.R., Cheshire, I.M.: The cascade method for accelerated convergence in implicit simulators. In: European Petroleum Conference, pp 25—28. Society of Petroleum Engineers, London (1982), https://doi.org/10.2118/12804-MS

  2. Bell, J.B., Trangenstein, J.A., Shubin, G.R.: Conservation laws of mixed type describing three-phase flow in porous media. SIAM J. Appl Math. 46(6), 1000–1017 (1986). https://doi.org/10.1137/0146059

    Article  Google Scholar 

  3. Berge, R.L., Klemetsdal, Ø.S., Lie, K.-A.: Unstructured voronoi grids conforming to lower dimensional objects. Comput. Geosci 23(1), 169–188 (2019). https://doi.org/10.1007/s10596-018-9790-0

    Article  Google Scholar 

  4. Bratvedt, F., Gimse, T., Tegnander, C.: Streamline computations for porous media flow including gravity. Transp. Porous Media 25(1), 63–78 (1996). https://doi.org/10.1007/BF00141262

    Article  Google Scholar 

  5. Brenier, Y., Jaffré, J.: Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer Anal. 28(3), 685–696 (1991). https://doi.org/10.1137/0728036

    Article  Google Scholar 

  6. Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: A comparison of upscaling techniques. SPE Reservoir Eval. Eng. 4, 308–317 (2001). https://doi.org/10.2118/72469-PA

    Article  Google Scholar 

  7. Cockburn, B., Shu, C.-W.: TVB runge-kutta local projection discontinuous galerkin finite element method for conservation laws ii: General. Math. Comp. 52(186), 411–435 (1989). https://doi.org/10.1090/S0025-5718-1989-0983311-4

    Google Scholar 

  8. Cockburn, B., Shu, C.-W.: The Runge-Kutta local projection p1- discontinuous-galerkin finite element method for scalar conservation laws. ESAIM–Math. Model. Num. 25(3), 337–361 (1991)

    Article  Google Scholar 

  9. Datta-Gupta, A., King, M.J.: Streamline simulation: Theory and practice, volume 11 of SPE Textbook Series Society of Petroleum Engineers (2007)

  10. Eikemo, B., Lie, K.-A., Dahle, H.K., Eigestad, G.T.: Discontinuous Galerkin methods for transport in advective transport in single-continuum models of fractured media. Adv Water Resour. 32(4), 493–506 (2009). https://doi.org/10.1016/j.advwatres.2008.12.010

    Article  Google Scholar 

  11. Gries, S., Stüben, K., Brown, G.L., Chen, D., Collins, D.A.: Preconditioning for efficiently applying algebraic multigrid in fully implicit reservoir simulations. SPE J. 19(04), 726–736 (2014). https://doi.org/10.2118/163608-PA

    Article  Google Scholar 

  12. Jenny, P., Lee, S.H., Tchelepi, H.A.: 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). https://doi.org/10.1016/j.jcp.2006.01.028

    Article  Google Scholar 

  13. Klausen, R.A., Rasmussen, A.F., Stephansen, A.: Velocity interpolation and streamline tracing on irregular geometries. Comput. Geosci. 16, 261–276 (2012). https://doi.org/10.1007/s10596-011-9256-0

    Article  Google Scholar 

  14. Klemetsdal, Ø.S., Berge, R.L., Lie, K.-A., Nilsen, H.M., Møyner, O.: Unstructured gridding and consistent discretizations for reservoirs with faults and complex wells. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers. https://doi.org/10.2118/182666-MS (2017)

  15. Klemetsdal, Ø.S., Møyner, O., Lie, K.-A.: Robust nonlinear newton solver with adaptive interface-localized trust regions. SPE J. https://doi.org/10.2118/195682-PA (2019)

  16. Klemetsdal, Ø.S., Møyner, O., Lie, K.-A.: Implicit high-resolution compositional simulation with optimal ordering of unknowns and adaptive spatial refinement. In: SPE Reservoir Simulation Conference, pp 10–11. Society of Petroleum Engineers, Galveston (2019), https://doi.org/10.2118/193934-MS

  17. Kwok, F., Tchelepi, H.: Potential-based reduced newton algorithm for nonlinear multiphase flow in porous media. J. Comput. Phys. 227(1), 706–727 (2007). https://doi.org/10.1016/j.jcp.2007.08.012

    Article  Google Scholar 

  18. Lie, K.-A.: An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User guide for the MATLAB Reservoir Simulation Toolbox (MRST). Cambridge University Press (2017)

  19. Lie, K.-A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012). https://doi.org/10.1007/s10596-011-9244-4. ISSN 1420-0597

    Article  Google Scholar 

  20. Lie, K.A., Natvig, J.R., Nilsen, H.M.: Discussion of dynamics and operator splitting techniques for two-phase flow with gravity. Int. J. Numer. Anal. Mod. 9(3), 684–700 (2012)

    Google Scholar 

  21. Lie, K.-A., Nilsen, H.M., Rasmussen, A.F., Raynaud, X.: Fast simulation of polymer injection in heavy-oil reservoirs on the basis of topological sorting and sequential splitting. SPE J. 19(06), 0991–1004 (2014). https://doi.org/10.2118/163599-PA

    Article  Google Scholar 

  22. Lie, K.-A., Møyner, O., Natvig, J.R., Kozlova, A., Bratvedt, K., Watanabe, S., Li, Z.: Successful application of multiscale methods in a real reservoir simulator environment. Comput. Geosci. 21(5), 981–998 (2017). https://doi.org/10.1007/s10596-017-9627-2

    Article  Google Scholar 

  23. Møyner, O: Nonlinear solver for three-phase transport problems based on approximate trust regions. Comput. Geosci. 21(5-6), 999–1021 (2017). https://doi.org/10.1007/s10596-017-9660-1

    Article  Google Scholar 

  24. Müller, B., Kummer, F., Oberlack, M.: Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int. J. Numer. Meth. Eng. 96(8), 512–528 (2013). https://doi.org/10.1002/nme.4569

    Article  Google Scholar 

  25. 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). https://doi.org/10.1016/j.jcp.2008.08.024

    Article  Google Scholar 

  26. Natvig, J.R., Lie, K.-A., Eikemo, B., Berre, I.: An efficient discontinuous Galerkin method for advective transport in porous media. Adv. Water Resour. 30(12), 2424–2438 (2007). https://doi.org/10.1016/j.advwatres.2007.05.015

    Article  Google Scholar 

  27. OPM: The open porous media (OPM) initative. https://opm-project.org/ (2019)

  28. Rasmussen, A.F., Lie, K.-A.: Discretization of flow diagnostics on stratigraphic and unstructured grids. In: 16th European Conference on the Mathematics of Oil Recovery. European Association of Geoscientits and Engineers, Catalania (2014), https://doi.org/10.3997/2214-4609.20141844

  29. Shahvali, M., Tchelepi, H.A.: Efficient coupling for non-linear multiphase flow with strong gravity. In: SPE Reservoir Simulation Symposium, 18-20 February. Society of Petroleum Engineers, The Woodlands (2013), https://doi.org/10.2118/163659-MS

  30. Sheth, S.M., Younis, R.M.: Localized solvers for general full-resolution implicit reservoir simulation. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers (2017), https://doi.org/10.2118/182691-MS

  31. The Open Porous Media (OPM) Initative: The Norne dataset. https://github.com/OPM/opm-data (2019)

  32. Trangenstein, J.A., Bell, J.B.: Mathematical structure of the black-oil model for petroleum reservoir simulation. SIAM J. Appl Math. 49(3), 749–783 (1989). https://doi.org/10.1137/0149044

    Article  Google Scholar 

  33. Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic Press (2000)

  34. Watts, J.: A compositional formulation of the pressure and saturation equations. SPE Reservoir Eng. 1(03), 243–252 (1986). https://doi.org/10.2118/12244-PA

    Article  Google Scholar 

Download references

Funding

The work of Klemetsdal, Rasmussen, and Lie is funded by the Research Council of Norway through grant no. 244361. Møyner is funded by VISTA, a basic research program funded by Equinor and conducted in close collaboration with the Norwegian Academy of Science and Letters.

Author information

Authors and Affiliations

  1. Norwegian University of Science and Technology, Trondheim, Norway

    Øystein S. Klemetsdal & Olav Møyner

  2. SINTEF Digital, Oslo, Norway

    Øystein S. Klemetsdal, Atgeirr F. Rasmussen, Olav Møyner & Knut-Andreas Lie

Authors
  1. Øystein S. Klemetsdal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Atgeirr F. Rasmussen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Olav Møyner
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Knut-Andreas Lie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Øystein S. Klemetsdal.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Klemetsdal, Ø.S., Rasmussen, A.F., Møyner, O. et al. Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models. Comput Geosci 24, 593–607 (2020). https://doi.org/10.1007/s10596-019-09844-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-019-09844-5

Keywords

Search

Navigation