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Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids

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Abstract

Improved and enhanced oil recovery methods require sophisticated simulation tools to predict the injected flow pass together with the chemical reactions inside it. One approach is application of higher-order numerical schemes to avoid excessive numerical diffusion that is very typical for transport processes. In this work, we provide a first step towards higher-order schemes applicable on general polyhedral and corner-point grids typically used in reservoir simulation. We compare three possible approaches of linear reconstruction and slope limiting techniques on a variety of different meshes in two and three spatial dimensions and discuss advantages and disadvantages.

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References

  1. Aziz, K.: Reservoir simulation grids: opportunities and problems. J. Petrol. Technol. 45 (1993). doi:10.2118/25233-PA

  2. Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE. Computing 82(2–3), 121–138 (2008). doi:10.1007/s00607-008-0004-9

    Article  Google Scholar 

  3. Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework. Computing 82(2–3), 103–119 (2008). doi:10.1007/s00607-008-0003-x

    Article  Google Scholar 

  4. Blatt, M., Burchardt, A., Dedner, A., Engwer, C., Fahlke, J., Flemisch, B., Gersbacher, C., Gräser, C., Gruber, F., Grüninger, C., Kempf, D., Klöfkorn, R., Malkmus, T., Müthing, S., Nolte, M., Piatkowski, M., Sander, O.: The distributed and unified numerics environment, version 2.4. Arch. Numer. Softw. 4(100), 13–29 (2016). doi:10.11588/ans.2016.100.26526

    Google Scholar 

  5. Buffard, T., Clain, S.: Monoslope and multislope MUSCL methods for unstructured meshes. J. Comput. Phys. 229(10), 3745–3776 (2010). doi:10.1016/j.jcp.2010.01.026

    Article  Google Scholar 

  6. Calgaro, C., Chane-Kane, E., Creuse, E., Goudon, T.: \(L^{\infty }\)-stability of vertex-based MUSCL finite volume schemes on unstructured grids: simulation of incompressible flows with high density ratios. J. Comput. Phys. 229(17), 6027–6046 (2010). doi:10.1016/j.jcp.2010.04.034

    Article  Google Scholar 

  7. Chen, L., Li, R.: An integrated linear reconstruction for finite volume scheme on unstructured grids. J. Sci. Comput. 68(3), 1172–1197 (2016). doi:10.1007/s10915-016-0173-1

    Article  Google Scholar 

  8. Dedner, A., Klöfkorn, R.: A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems. J. Sci. Comput. 47(3), 365–388 (2011). doi:10.1007/s10915-010-9448-0

    Article  Google Scholar 

  9. Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptive scientific computing: abstraction principles and the DUNE-Fem module. Computing 90(3–4), 165–196 (2010). doi:10.1007/s00607-010-0110-3

    Article  Google Scholar 

  10. Durlofsky, L., Engquist, B., Osher, S.: Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comput. Phys. 98(1), 64–73 (1992)

    Article  Google Scholar 

  11. Eymard, R., Henry, G., Herbin, R., Hubert, F., Klöfkorn, R., Manzini, G.: 3D benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Fort, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds.) Finite Volumes for Complex Applications VI Problems & Perspectives, Springer Proceedings in Mathematics, vol. 4, pp 895–930. Springer, Berlin Heidelberg (2011), doi:10.1007/978-3-642-20671-9_89

  12. Feistauer, M., Felcman, J., Straskraba, I.: Mathematical and Computational Methods for Compressible Flow. Oxford University Press (2003)

  13. Heinemann, Z., Brand, C., Munka, M., Chen, Y.: Modeling Reservoir Geometry With Irregular Grids. SPE Res. Eval. Eng. 225–232 (1991). doi:10.2118/18412-PA

  14. Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids Finite Volumes for Complex Applications V, pp 659–692. ISTE, London (2008)

    Google Scholar 

  15. Hou, J., Liang, Q., Zhang, H., Hinkelmann, R.: An efficient unstructured MUSCL scheme for solving the 2D shallow water equations. Environ. Modell. Softw. 66, 131–152 (2015). doi:10.1016/j.envsoft.2014.12.007

    Article  Google Scholar 

  16. Hubbard, M.: Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155(1), 54–74 (1999). doi:10.1006/jcph.1999.6329

    Article  Google Scholar 

  17. Ju, L., Ringler, T., Gunzburger, M.: Numerical Techniques for Global Atmospheric Models. chap. Voronoi Tessellations and Their Application to Climate and Global Modeling, pp 313–342. Springer Berlin Heidelberg, Berlin Heidelberg (2011)

    Google Scholar 

  18. Kiris, C.C., Housman, J.A., Barad, M.F., Sozer, E., Brehm, C., Moini-Yekta, S.: Computational framework for launch, ascent, and vehicle aerodynamics (LAVA). Aerospace Science and Technology, pp– (2016). doi:10.1016/j.ast.2016.05.008

  19. Klöfkorn, R., Kröner, D., Ohlberger, M.: Local adaptive methods for convection dominated problems. Int. J. Numer. Methods Fluids 40(1-2), 79–91 (2002). doi:10.1002/fld.268

    Article  Google Scholar 

  20. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004). doi:10.1016/j.jcp.2003.08.010

    Article  Google Scholar 

  21. Kröner, D.: Numerical Schemes for Conservation Laws. Wiley-Teubner (1997)

  22. Lamine, S., Edwards, M.: Multidimensional upwind schemes and higher resolution methods for three-component two-phase systems including gravity driven flow in porous media on unstructured grids. Comput. Methods Appl. Mech. Eng. 292, 171–194 (2015). doi:10.1016/j.cma.2014.12.022

    Article  Google Scholar 

  23. van Leer, B.: Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14(4), 361–370 (1974). doi:10.1016/0021-9991(74)90019-9

  24. LeVeque, R.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1990)

    Book  Google Scholar 

  25. LeVeque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33(2), 627–665 (1996). doi:10.1137/0733033

    Article  Google Scholar 

  26. May, S., Berger, M.: Two-dimensional slope limiters for finite volume schemes on non-coordinate-aligned meshes. SIAM J. Sci. Comput. 35(5), A2163—A2187 (2013). doi:10.1137/120875624

    Article  Google Scholar 

  27. 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). doi:10.1016/j.jcp.2015.10.010

    Article  Google Scholar 

  28. Shu, C.-W.: High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments. J. Comput. Phys. 316, 598–613 (2016). doi:10.1016/j.jcp.2016.04.030

    Article  Google Scholar 

  29. Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 2nd edn. Springer, Berlin (1999)

    Book  Google Scholar 

  30. Wesenberg, M.: Efficient finite – volume schemes for magnetohydrodynamics simulations in solar physics. Ph.D. thesis. Albert–Ludwigs–Universität Freiburg, Mathematisches Institut (2003). http://www.freidok.uni-freiburg.de/volltexte/792/

    Google Scholar 

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Acknowledgements

Robert Klöfkorn and Anna Kvashchuk thank the Research Council of Norway and the industry partners—ConocoPhillips Skandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy AS Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall and Norge AS—of The National IOR Centre of Norway for the financial support.

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Authors and Affiliations

  1. International Research Institute of Stavanger, Thormøhlensgt. 55, 5008, Bergen, Norway

    Robert Klöfkorn

  2. University of Stavanger, P.O. Box 8600 Forus, 4036, Stavanger, Norway

    Anna Kvashchuk

  3. University of Freiburg, Hermann-Herder-Str. 10, 79104, Freiburg, Germany

    Martin Nolte

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  1. Robert Klöfkorn
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  2. Anna Kvashchuk
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  3. Martin Nolte
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Correspondence to Robert Klöfkorn.

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Klöfkorn, R., Kvashchuk, A. & Nolte, M. Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids. Comput Geosci 21, 909–919 (2017). https://doi.org/10.1007/s10596-017-9658-8

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  • DOI: https://doi.org/10.1007/s10596-017-9658-8

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