Advertisement

Effective Buckley-Leverett Equations by Homogenization

  • C. J. van Duijn
  • I. S. Pop
  • A. Mikelic
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 1)

Abstract

In this paper we consider water-drive to recover oil from a strongly heterogeneous porous column The two-phase model uses Corey relative permeabilities and Brooks-Corey capillary pressure. The heterogeneities are perpendicular to flow and have a periodic structure. This results in one-dimensional flow and a space periodic absolute permeability, reflecting alternating coarse and fine layers. Assuming many — or thin — layers, we use homogenization techniques to derive the effective transport equations. The form of these equations depend critically on the capillary number. The analysis is confirmed by numerical experiments. This paper summarises the results obtained in [10]

Keywords

Porous Medium Capillary Pressure Relative Permeability Capillary Number Auxiliary Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alt, H.W. and DiBenedetto, E. (1985) Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 12, 335–392.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alt, H.W. and Luckhaus, S. (1983) Quasilinear elliptic—parabolic differential equations. Math. Z. 183, 311–341.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bear, J. (1972) Dynamics of Fluids in Porous Media. American Elsevier, New York.zbMATHGoogle Scholar
  4. 4.
    Benilan, P. and Toure, H. (1995) Sur l’equation generale u t = a dans L1. II: Le probleme d’evolution. Ann. Inst. Henri Poincare, Anal. Non Lineaire, 12, 727–761.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bensoussan, A., Lions, J.L. and Papanicolaou, G. (1978) Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam New York Oxford.zbMATHGoogle Scholar
  6. 6.
    Bertsch, M., Dal Passo, R. and van Duijn, C.J. Analysis of oil trapping in porous media flow. (to appear).Google Scholar
  7. 7.
    Brooks, R.H. and Corey, A.T. (1964) Hydraulic Properties of Porous Media. Colorado State University, Fort Collins, Colorado.Google Scholar
  8. 8.
    Corey, A.T. (1954) The interrelation between gas and oil relative permeabilities. Producer’s Monthly, 19, 38–41.Google Scholar
  9. 9.
    Dale, M., Ekrann, S., Mykkeltveit, J. and Virnovsky, G. (1997) Effective relative permeabilities and capillary pressure for one-dimensional heterogeneous media. Transport in Porous Media, 26, 229–260.CrossRefGoogle Scholar
  10. 10.
    van Duijn, C.J., Mikelic, A. and Pop, I.S. Effective equations for two-Phase flow with trapping on the micro scale. (to appear).Google Scholar
  11. 11.
    van Duijn, C.J., Molenaar J. and de Neef, M.J. (1995) Effects of capillary forces on immiscible two-phase flow in heterogeneous porous media. Transport in Porous Media, 21, 71–93.CrossRefGoogle Scholar
  12. 12.
    Gilding, B.H. (1991) Qualitative mathematical analysis of the Richards equation. Transport in Porous Media, 5, 651–666.Google Scholar
  13. 13.
    Hornung, U., ed. (1997) Homogenization and Porous Media. Springer-Verlag, New York.zbMATHGoogle Scholar
  14. 14.
    Kortekaas, T.F.M. (1985) Water/oil displacement characteristics in crossbedded reservoir zones. Soc. Pet. Eng. J., 917–926.Google Scholar
  15. 15.
    Leverett, M.C. (1941) Capillary behavior in porous solids. Trans AIME Petr. Eng. Div.5, 142, 152–169.Google Scholar
  16. 16.
    van Lingen, P. (1998) Quantification and reduction of capillary entrapment in cross-laminated oil reservoirs. Sub-Faculty of Applied Earth Sciences, Delft University of Technology.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • C. J. van Duijn
    • 1
  • I. S. Pop
    • 1
  • A. Mikelic
    • 2
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Laboratoire d’Analyse NumériqueUniversité Lyon 1Villeurbanne CedexFrance

Personalised recommendations