Advertisement

Computational Geosciences

, Volume 19, Issue 1, pp 99–114 | Cite as

Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media

  • Patrick Henning
  • Mario Ohlberger
  • Ben Schweizer
ORIGINAL PAPER

Abstract

In this contribution, we present the first formulation of a heterogeneous multiscale method for an incompressible immiscible two-phase flow system with degenerate permeabilities. The method is in a general formulation, which includes oversampling. We do not specify the discretization of the derived macroscopic equation, but we give two examples of possible realizations, suggesting a finite element solver for the fine scale and a vertex-centered finite volume method for the effective coarse scale equations. Assuming periodicity, we show that the method is equivalent to a discretization of the homogenized equation. We provide an a posteriori estimate for the error between the homogenized solutions of the pressure and saturation equations and the corresponding HMM approximations. The error estimate is based on the results recently achieved as reported by Cancès et al. (Math. Comp. 83(285):153–188, 2014). An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow.

Keywords

Adaptivity HMM Multiscale problem Two-phase flow Porous media 

Mathematics Subject Classification (2010)

76S05 35B27 65G99 65N08 65N30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aarnes, J.E., Efendiev, Y.: An adaptive multiscale method for simulation of fluid flow in heterogeneous porous media. Multiscale Model. Simul. 5(3), 918–939 (2006)CrossRefGoogle Scholar
  2. 2.
    Aarnes, J.E., Krogstad, S., Lie, K.-A.: 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)CrossRefGoogle Scholar
  3. 3.
    Aarnes, J.E., Lie, K.-A., Kippe, V., Krogstad, S.: Multiscale methods for subsurface flow. In: Multiscale modeling and simulation in science, Vol. 66, p 3–48. Springer, Berlin (2009)Google Scholar
  4. 4.
    Abdulle, A.: On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model.Simul. 4(2), 447–459 (2005)CrossRefGoogle Scholar
  5. 5.
    Abdulle, A.: The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. In: Multiple scales problems in biomathematics, mechanics, physics and numerics, of GAKUTO International Series Mathematics Science Applications, Vol. 31, pp 133–181 (2009)Google Scholar
  6. 6.
    Abdulle, A., Engquist, W.E.B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer 21, 1–87 (2012)CrossRefGoogle Scholar
  7. 7.
    Abdulle, A., Nonnenmacher, A.: Adaptive finite element heterogeneous multiscale method for homogenization problems. Mech. Engrg Comput. Methods Appl. 200(37-40), 2710–2726 (2011)CrossRefGoogle Scholar
  8. 8.
    Alt, H.W., DiBenedetto, E.: Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Scuola Norm. Sup. Pisa Cl Sci. (4) 12(3), 335–392 (1985)Google Scholar
  9. 9.
    Allaire, G.: Homogenization, two-scale, convergence. SIAM J. Math. Anal 23(6), 1482–1518 (1992)CrossRefGoogle Scholar
  10. 10.
    Amaziane, B., Antontsev, S., Pankratov, L., Piatnitski, A.: Homogenization of immiscible compressible two-phase flow inporous media: application to gas migration in a nuclear waste repository. Multiscale Model. Simul. 8(5), 2023–2047 (2010)CrossRefGoogle Scholar
  11. 11.
    Arbogast, T.: The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow. Nonlinear Anal. 19(11), 1009–1031 (1992)CrossRefGoogle Scholar
  12. 12.
    Antontsev, S., Kazhikhov, A., Monakhov, V.: Boundary value problems in mechanics of nonhomogeneous fluids.Transl. from the Russian. In: Studies in mathematics and its applications 22, p. 309, Amsterdam etc. North- Holland. xii (1990)Google Scholar
  13. 13.
    Bear, J.: Dynamics of fluids in porous media. American Elsevier, New York (1972)Google Scholar
  14. 14.
    Bourgeat, A., Hidani, A.: Effective model of two-phase flow in a porous medium made of different rock types. Appl. Anal. 58(1-2), 1–29 (1995)CrossRefGoogle Scholar
  15. 15.
    Bourgeat, A., Luckhaus, S., Mikelić, A.: Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow. SIAM J. Math. Anal. 27(6), 1520–1543 (1996)CrossRefGoogle Scholar
  16. 16.
    Brezzi, F., Franca, L.P., Hughes, T.J.R., Russo, A.: b = ∫ g. Comput. Methods Appl. Mech. Engrg. 145(3-4), 329–339 (1997)CrossRefGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Cancès, C., Pop, I.S., Vohralík, M.: An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Math. Comp. 83(285), 153–188 (2014)CrossRefGoogle Scholar
  19. 19.
    Chavent, G., Jaffré, J.: Mathematical models and finite elements for reservoir simulation: single phase, multiphase, and multicomponent flows through porous media. Elsevier Science Pub. Co., North-Holland New York (1986)Google Scholar
  20. 20.
    Chen, Y., Li, Y.: Local-global two-phase upscaling of flow and transport in heterogeneous formations. Multiscale Model.Simul. 8(1), 125–153 (2009)CrossRefGoogle Scholar
  21. 21.
    Chen, Z.: Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Diff. Equat. 171(2), 203–232 (2001)CrossRefGoogle Scholar
  22. 22.
    Chen, Z.: Degenerate two-phase incompressible flow. II. Regularity stability and stabilization. J. Differ. Equ. 186(2), 345–376 (2002)CrossRefGoogle Scholar
  23. 23.
    Chen, Z., Ewing, R.E.: Degenerate two-phase incompressible flow. III. Sharp error estimates . Numer. Math. 90(2), 215–240 (2001)CrossRefGoogle Scholar
  24. 24.
    Chen, Z., Hou, T.Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp 72(242), 541–576 (2003)CrossRefGoogle Scholar
  25. 25.
    E, W., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)CrossRefGoogle Scholar
  26. 26.
    Efendiev, Y., Ginting, V., Hou, T., Ewing, R.: Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220(1), 155–174 (2006)CrossRefGoogle Scholar
  27. 27.
    Epshteyn, Y., Rivière, B.: Analysis of hp discontinuous Galerkin methods for incompressible two-phase flow. Comput, J. Appl. Math 225(2), 487–509 (2009)CrossRefGoogle Scholar
  28. 28.
    Eymard, R., Herbin, R., Michel, A.: Mathematical study of a petroleum-engineering scheme. M2AN Math. Model. Numer. Anal. 37(6), 937–972 (2003)CrossRefGoogle Scholar
  29. 29.
    Ginting, V.: Analysis of two-scale finite volume element method for elliptic problem. J. Numer. Math. 12(2), 119–141 (2004)CrossRefGoogle Scholar
  30. 30.
    Gloria, A.: An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. Multiscale Model. Simul. 5(3), 996–1043 (2006)CrossRefGoogle Scholar
  31. 31.
    Gloria, A.: An analytical framework for numerical homogenization. II. Windowing and oversampling . Multiscale Model. Simul. 7(1), 274–293 (2008)CrossRefGoogle Scholar
  32. 32.
    Henning, P., Ohlberger, M.: The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. Numer. Math. 113(4), 601–629 (2009)CrossRefGoogle Scholar
  33. 33.
    Henning, P., Ohlberger, M.: Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems. Discrete Contin. Dyn. Syst. Ser. S 8(1), 119–150 (2015)CrossRefGoogle Scholar
  34. 34.
    Henning, P., Ohlberger, M., Schweizer, B.: Homogenization of the degenerate two-phase flow equations. Math. Models and Methods in Appl. Sciences 23(12), 2323–2352 (2013)CrossRefGoogle Scholar
  35. 35.
    Hou, T.Y., Wu, X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)CrossRefGoogle Scholar
  36. 36.
    Huber, R., Helmig, R.: Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media. Comput. Geosci. 4(2), 141–164 (2000)CrossRefGoogle Scholar
  37. 37.
    Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187(1), 47–67 (2003)CrossRefGoogle Scholar
  38. 38.
    Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model. Simul. 3(1), 50–64 (2004)CrossRefGoogle Scholar
  39. 39.
    Jiang, L., Aarnes, J.E., Efendiev, Y.: Some multiscale results using limited global information for two-phase flow simulations. Int. J. Numer. Anal. Model. 9(1), 115–131 (2012)Google Scholar
  40. 40.
    Jiang, L., Mishev, I.D.: Mixed multiscale finite volume methods for elliptic problems in two-phase flow simulations. Commun. Comput. Phys. 11(1), 19–47 (2012)Google Scholar
  41. 41.
    Jikov, V.V., Kozlov, S.M., Oleı̆nik, O.A.: Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994). (translated from Russian by G.A. Yosifian)CrossRefGoogle Scholar
  42. 42.
    Chen, Z., Huan, G., Ma., Y.: Computational methods for multiphase flows in porous media. Society for Industrial and Applied Mathematics (SIAM). PA, Philadelphia (2006)CrossRefGoogle Scholar
  43. 43.
    Kröner, D., Luckhaus, S.: Flow of oil and water in a porous medium. J. Differ. Equ. 55(2), 276–288 (1984)CrossRefGoogle Scholar
  44. 44.
    Leverett, M.C., Lewis, W.B.: Steady flow of gas-oil-water mixtures through unconsolidated sands. Trans. of the AIME 142(1), 107–116 (1941)CrossRefGoogle Scholar
  45. 45.
    Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2(1), 35–86 (2002)Google Scholar
  46. 46.
    Lunati, I., Jenny, P.: Multiscale finite-volume method for density-driven flow in porous media. Comput. Geosci. 12(3), 337–350 (2008)CrossRefGoogle Scholar
  47. 47.
    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)CrossRefGoogle Scholar
  48. 48.
    Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comp. 83(290), 2583–2603 (2014)CrossRefGoogle Scholar
  49. 49.
    Michel, A.: A finite volume scheme for two-phase immiscible flow in porous media. SIAM J. Numer. Anal. 41(4), 1301–1317 (2003)CrossRefGoogle Scholar
  50. 50.
    Murat, F., Tartar, L.: H-convergence, topics in the mathematical modeling of composite materials. Progr. Nonlinear Differ. Equ. Appl. 31, 21–43 (1997)Google Scholar
  51. 51.
    Nordbotten, J.M.: Adaptive variational multiscale methods for multiphase flow in porous media. Multiscale Model Simul 7(3), 1455–1473 (2008)CrossRefGoogle Scholar
  52. 52.
    Ohlberger, M.: Convergence of a mixed finite elements–finite volume method for the two phase flow in porous media. East-West J. Numer. Math. 5(3), 183–210 (1997)Google Scholar
  53. 53.
    Ohlberger, M.: A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems. Multiscale Model. Simul. 4(1), 88–114 (2005)CrossRefGoogle Scholar
  54. 54.
    Owhadi, H., Zhang, L., Berlyand, L.: Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. M2AN Math. Model. Numer. Anal. 48(2), 517–552 (2014)CrossRefGoogle Scholar
  55. 55.
    Yeh, L.-M.: Homogenization of two-phase flow in fractured media. Math. Models Methods Appl. Sci. 16(10), 1627–1651 (2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Patrick Henning
    • 1
  • Mario Ohlberger
    • 1
  • Ben Schweizer
    • 2
  1. 1.Institut für Numerische und Angewandte MathematikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

Personalised recommendations