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Computational Geosciences

, Volume 22, Issue 2, pp 565–586 | Cite as

Monotone nonlinear finite-volume method for challenging grids

  • M. Schneider
  • B. Flemisch
  • R. Helmig
  • K. Terekhov
  • H. Tchelepi
Original Paper
  • 165 Downloads

Abstract

This article presents a new positivity-preserving finite-volume scheme with a nonlinear two-point flux approximation, which uses optimization techniques for the face stencil calculation. The gradient is reconstructed using harmonic averaging points with the constraint that the sum of the coefficients included in the face stencils must be positive. We compare the proposed scheme to a nonlinear two-point scheme available in literature and a few linear schemes. Using two test cases, taken from the FVCA6 benchmarks, the accuracy of the scheme is investigated. Furthermore, it is shown that the scheme is linearity-preserving on highly complex corner-point grids. Moreover, a two-phase flow problem on the Norne formation, a geological formation in the Norwegian Sea, is simulated. It is demonstrated that the proposed scheme is consistent in contrast to the linear Two-Point Flux Approximation scheme, which is industry standard for simulating subsurface flow on corner-point grids.

Keywords

Finite-volume method Monotone discretization Corner-point grid Challenging grids 

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Notes

Acknowledgements

The authors Bernd Flemisch, Rainer Helmig, and Martin Schneider would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.

References

  1. 1.
    Aarnes, J.E., Krogstad, S., Lie, K.A.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12(3), 297–315 (2008)CrossRefGoogle Scholar
  2. 2.
    Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3-4), 405–432 (2002)CrossRefGoogle Scholar
  3. 3.
    Aavatsmark, I.: Multipoint flux approximation methods for quadrilateral grids. In: 9th International Forum on Reservoir Simulation, Abu Dhabi (2007)Google Scholar
  4. 4.
    Aavatsmark, I.: Comparison of monotonicity for some multipoint flux approximation methods. R. Eymard et JM hérard (rédacteurs). Finite Volumes for Complex Applications, tome 5, 19–34 (2008)Google Scholar
  5. 5.
    Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127(1), 2–14 (1996)CrossRefGoogle Scholar
  6. 6.
    Agélas, L., Eymard, R., Herbin, R.: A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. Comptes Rendus Mathématique 347(11), 673–676 (2009)CrossRefGoogle Scholar
  7. 7.
    Alkämper, M., Dedner, A., Klöfkorn, R., Nolte, M.: The DUNE-ALUGrid module. arXiv:1407.6954 (2014)
  8. 8.
    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO-Modé,lisation Mathématique et Analyse Numérique 19(1), 7–32 (1985)Google Scholar
  9. 9.
    Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization, vol. 6. MA, Athena Scientific Belmont (1997)Google Scholar
  10. 10.
    Blatt, M., Burchardt, A., Dedner, A., Engwer, C., Fahlke, J., Flemisch, B., Gersbacher, C., Gräser, C., Gruber, F., Grüninger, C., et al.: The distributed and unified numerics environment, version 2.4. Archive of Numerical Software 4(100), 13–29 (2016)Google Scholar
  11. 11.
    Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005)CrossRefGoogle Scholar
  12. 12.
    Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005)CrossRefGoogle Scholar
  13. 13.
    Cancès, C., Cathala, M., Le Potier, C.: Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125(3), 387–417 (2013)CrossRefGoogle Scholar
  14. 14.
    Cao, Y., Helmig, R., Wohlmuth, B.: The influence of the boundary discretization on the multipoint flux approximation l-method. Finite Volumes for Complex Applications V, ISTE, London, pp. 257–263 (2008)Google Scholar
  15. 15.
    Cao, Y., Helmig, R., Wohlmuth, B.: Geometrical interpretation of the multi-point flux approximation L-method. Int. J. Numer. Methods Fluids 60(11), 1173–1199 (2009)CrossRefGoogle Scholar
  16. 16.
    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media, vol. 2. SIAM (2006)Google Scholar
  17. 17.
    Danilov, A., Vassilevski, Y.V.: A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russ. J. Numer. Anal. Math. Model. 24(3), 207–227 (2009)CrossRefGoogle Scholar
  18. 18.
    Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20(3), 720–755 (1999)CrossRefGoogle Scholar
  19. 19.
    Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2(4), 259–290 (1998)CrossRefGoogle Scholar
  20. 20.
    Edwards, M.G., Zheng, H.: Quasi M-matrix multifamily continuous Darcy-flux approximations with full pressure support on structured and unstructured grids in three dimensions. SIAM J. Sci. Comput. 33(2), 455–487 (2011)CrossRefGoogle Scholar
  21. 21.
    Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3d schemes for diffusive flows in porous media. ESAIM: Mathematical Modelling and Numerical Analysis 46(2), 265–290 (2012)CrossRefGoogle Scholar
  22. 22.
    Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., et al.: DuMux: DUNE for multi-{phase, component, scale, physics,...} flow and transport in porous media. Adv. Water Resour. 34 (9), 1102–1112 (2011)CrossRefGoogle Scholar
  23. 23.
    Fort, J., Fürst, J., Halama, J., Herbin, R., Hubert, F.: Finite volumes for complex applications. VI. Problems & Perspectives, vol. 1, 2. Springer Proceedings in Mathematics (20011)Google Scholar
  24. 24.
    Friis, H.A., Edwards, M.G., Mykkeltveit, J.: Symmetric positive definite flux-continuous full-tensor finite-volume schemes on unstructured cell-centered triangular grids. SIAM J. Sci. Comput. 31(2), 1192–1220 (2008)CrossRefGoogle Scholar
  25. 25.
    Gao, Z., Wu, J.: A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes. Int. J. Numer. Methods Fluids 67(12), 2157–2183 (2011)CrossRefGoogle Scholar
  26. 26.
    Gao, Z., Wu, J.: A second-order positivity-preserving finite volume scheme for diffusion equations on general meshes. SIAM J. Sci. Comput. 37(1), A420–A438 (2015)CrossRefGoogle Scholar
  27. 27.
    Hoteit, H., Mosé, R., Philippe, B., Ackerer, P., Erhel, J.: The maximum principle violations of the mixed-hybrid finite-element method applied to diffusion equations. Int. J. Numer. Methods Eng. 55(12), 1373–1390 (2002)CrossRefGoogle Scholar
  28. 28.
    Krogstad, S., Lie, K.A., Møyner, O., Nilsen, H.M., Raynaud, X., Skaflestad, B., et al.: Mrst-ad–an open-source framework for rapid prototyping and evaluation of reservoir simulation problems. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2015)Google Scholar
  29. 29.
    Le Potier, C.: Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés. Comptes Rendus Mathématique 341(12), 787–792 (2005)CrossRefGoogle Scholar
  30. 30.
    Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163–1227 (2014)CrossRefGoogle Scholar
  31. 31.
    Lipnikov, K., Manzini, G., Svyatskiy, D.: Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230(7), 2620–2642 (2011)CrossRefGoogle Scholar
  32. 32.
    Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes. J. Comput. Phys. 228(3), 703–716 (2009)CrossRefGoogle Scholar
  33. 33.
    Nikitin, K., Terekhov, K., Vassilevski, Y.: A monotone nonlinear finite volume method for diffusion equations and multiphase flows. Comput. Geosci. 18(3-4), 311–324 (2014)CrossRefGoogle Scholar
  34. 34.
    Nordbotten, J., Aavatsmark, I., Eigestad, G.: Monotonicity of control volume methods. Numer. Math. 106(2), 255–288 (2007)CrossRefGoogle Scholar
  35. 35.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, pp. 292–315. Springer (1977)Google Scholar
  36. 36.
    Schneider, M., Agélas, L., Enchéry, G., Flemisch, B.: Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. J. Comput. Phys. 351, 80–107 (2017).  https://doi.org/10.1016/j.jcp.2017.09.003 CrossRefGoogle Scholar
  37. 37.
    Schneider, M., Flemisch, B., Helmig, R.: Monotone nonlinear finite-volume method for nonisothermal two-phase two-component flow in porous media. Int. J. Numer. Methods Fluids 84(6), 352–381 (2016)CrossRefGoogle Scholar
  38. 38.
    Schneider, M., Gläser, D., Flemisch, B., Helmig, R.: Nonlinear finite-volume scheme for complex flow processes on corner-point grids. In: Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems: FVCA 8, Lille, France, June 2017, pp. 417–425. Springer International Publishing (2017)Google Scholar
  39. 39.
    Sheng, Z., Yuan, G.: The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes. J. Comput. Phys. 230(7), 2588–2604 (2011)CrossRefGoogle Scholar
  40. 40.
    Sun, W., Wu, J., Zhang, X.: A family of linearity-preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids. Comput. Methods Appl. Mech. Eng. 267, 418–433 (2013)CrossRefGoogle Scholar
  41. 41.
    Terekhov, K., Vassilevski, Y.: Two-phase water flooding simulations on dynamic adaptive octree grids with two-point nonlinear fluxes. Russ. J. Numer. Anal. Math. Model. 28(3), 267–288 (2013)CrossRefGoogle Scholar
  42. 42.
    Terekhov, K.M., Mallison, B.T., Tchelepi, H.A.: Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem. J. Comput. Phys. 330, 245–267 (2017)CrossRefGoogle Scholar
  43. 43.
    Vidović, D., Dotlić, M., Dimkić, M., Pušić, M., Pokorni, B.: Convex combinations for diffusion schemes. J. Comput. Phys. 246, 11–27 (2013)CrossRefGoogle Scholar
  44. 44.
    Wolff, M., Cao, Y., Flemisch, B., Helmig, R., Wohlmuth, B.: Multi-point flux approximation L-method in 3d: numerical convergence and application to two-phase flow through porous media. Radon Ser. Comput. Appl. Math., De Gruyter 12, 39–80 (2013)Google Scholar
  45. 45.
    Wu, J., Gao, Z.: Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids. J. Comput. Phys. 275, 569–588 (2014)CrossRefGoogle Scholar
  46. 46.
    Yuan, G., Sheng, Z.: Monotone finite volume schemes for diffusion equations on polygonal meshes. J. Comput. Phys. 227(12), 6288–6312 (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Modelling Hydraulic and Environmental Systems (IWS)University StuttgartStuttgartGermany
  2. 2.Energy Resources EngineeringStanford UniversityStanfordUSA

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