Scaling Concepts in Porous Media

  • Christophe Baudet
  • Elixabeth Charlaix
  • Eric Clément
  • Etienne Gyron
  • Fean-Pierre Hulin
  • Christophe Leroy


Scaling concepts in disordered matter should be of use to the physics of porous media (POM)1. The applications of such media are numerous, ranging from hydrogeology, oil industry, chromatography and filtration, and justify the lasting interest for this physics2. We will focus our attention towards the effect of disorder in POM and, more precisely, that of multiple scales in the geometry of the fluid(s) penetrating them; but we will also recall some basic properties of POM. Geometrical disorder in POM has several origins:
  • There is a strong contrast between the material properties of the solid phase and the fluid one(s) penetrating it.

  • The pore space can range from homogeneous to very heterogeneous (e.g. from sintered materials obtained with a relatively uniform particle size distribution to fractured rocks presenting an irregular distribution of cracks of variable size, aperture...). Heterogeneous POM often present a large range of geometrical scales.

  • In multiple phase flows, the distribution of the fluid phases contained in the POM introduces additional heterogeneities and multiplicity of scales.

  • Local heterogeneities like those due to roughness or of chemical nature (e.g. presence of clays in sandstones) control the wetting properties in polyphasic flows. We will not consider this last class of parameters which, nevertheless, are of dramatic practical importance. Chemical surface effects can also model the properties of walls in random field problems: the advancing front of a wetting fluid on an heterogeneous surface (or, possibly, in a porous material made of wettable and non wettable properties) displays some characteristic features (rough interfaces, hysteresis) of R.F. systems4 discussed in the article by Villain.


Porous Medium Representative Elementary Volume Percolation Threshold Fracture Rock Capillary Number 
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  1. 1.
    F. A. L. Dullien, Porous Media, Fluid Transport and Pore Structure, Acad. Press, New-York (1979).Google Scholar
  2. 2.
    The present review extends and complements the article by E. Guyon, P. P. Hulin, R. Lenormand (in french) in Ann. des Mines, special issue “Ecoulements dans les Milieux Fissurés”, 191, (5.6) p. 17 (1984) where a larger number of references can be found.Google Scholar
  3. 3.
    Physics and Chemistry of Porous Media, D.L. Johnson and P.M. Sen edit. A.I.P. Conference Proceed. n° 107, AIP, New-York (1984).Google Scholar
  4. 4.
    For example, see J.F. Joanny, Thèse Paris, p. 135 (1985).Google Scholar
  5. 5.
    H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris (1856).Google Scholar
  6. 6.
    L. Sander, in these proceedings. The Hele Shaw geometry involving a flow between two parallel plates distant of b can be described by eq. (1) with a constant K - b3/12.Google Scholar
  7. 7.
    L. A. Santaló, Integral Geometry and Geometric Probability, Encyclopedia of Mathematics, vol. I, Addison Wesley Publishing Cie, Reading Massachusetts.Google Scholar
  8. 8.
    P. Z. Wong, J. Koplik and J. P. Tomanic, to appear in Phys. Rev. B.Google Scholar
  9. 9.
    D. L. Johnson, Appl. Phys. Lett. 37: 1065 (1980).CrossRefGoogle Scholar
  10. 10.
    R. Lemaitre, thèse Université Rennes (F385).Google Scholar
  11. 11.
    R. Omnes, J. de Phys. 46: 139 (1985).MathSciNetCrossRefGoogle Scholar
  12. 12.
    A.J. Katz et A. H. Thompson, Phys. Rev. Lett. 54, 1325 (1985).CrossRefGoogle Scholar
  13. 13.
    R. Orbach, these lectures.Google Scholar
  14. 14.
    P. G. deGennes, Partial Filling of Fractal Structure by Wetting Fluid, to be published.Google Scholar
  15. 15.
    T. Madden, Geophysics, 31: 1104 (1976).CrossRefGoogle Scholar
  16. 16.
    C.J. Allègre„ J. L. Le Mouel, A. Provost, Nature 297, 5861, 47–49 (1982).CrossRefGoogle Scholar
  17. 17.
    S. R. Broadbent and J.M. Hammersley, Proc. Camb. Phil. Soc. 53: 629 (1957).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    I. Bernabe, W. C. Brace, Mech. of Materials 1: 173 (1982).CrossRefGoogle Scholar
  19. 19.
    P. Z. Wong, J. Koplik and J. P. Tomanic, to appear in Phys. Rev. B.Google Scholar
  20. 20.
    G. de Marsily, Hydrogeologie Quantitative, Masson, Paris (1981).Google Scholar
  21. 21.
    E. Charlaix, E. Guyon, N. Rivier, Sol. State Com. 50, 11:999 (1984). The article also shows that it is possible to derive these parameters form random plane cuts.Google Scholar
  22. 22.
    G. E. Pike and C. H. Seayer, Phys. Rev. B 10: 1421 (1974).CrossRefGoogle Scholar
  23. 23.
    The same dimensionless quantity describes within a numerical factor the onset of nematic ordering of percolation of arrays of rods (or disks) and that of ordering of calamitic (or discotic) nematics. Indeed both phases can coexist in molecular systems. See L. Onsager, Ann. N.Y. Acad. Sci. 51: 627 (1949).Google Scholar
  24. 24.
    P. C. Robinson, J. Phys. A, 16: 605 (1983).MathSciNetCrossRefGoogle Scholar
  25. 25.
    I. Balberg, C. H. Anderson, S. Alexander and N. Wagner, Phys. Rev. B 30: 3933 (1984).CrossRefGoogle Scholar
  26. 26.
    S. Wilke, E. Guyon and G. de Marsily, Math. Geol. 17: 17 (1985).CrossRefGoogle Scholar
  27. 27.
    B. Halperin, S. Fang, P.N. Sen, to be published.Google Scholar
  28. 28.
    A. Rouleau, PH.D Thesis, Waterloo, Ontario (1984).Google Scholar
  29. J. C. S. Long, PH.D Thesis, University of California (1983).Google Scholar
  30. 29.
    B. I. Shklovskii, A. L. Efros, Electronic Properties of Doped Semi-Conductor, Springer Verlag, Berlin (1984).CrossRefGoogle Scholar
  31. 30.
    J. Bear, Dynamics of Flow in Porous Media, Chapter 9, American Elsevier, New-York (1972).Google Scholar
  32. 31.
    Groupe Poreux P.C., Two components Properties in Heterogeneous Porous Media, Proc. “Physics offinely divided matter”, Les Houches, Ed. by Daoud (Springer ) (1985).Google Scholar
  33. 32.
    Ch. G. Jacquin, and P. M. Adler, to be published in S. Coll. Int. Sci. (1985).Google Scholar
  34. 33.
    P. G. de Gennes and E. Guyon, J. Meca. 17: 403 (1978).Google Scholar
  35. I. Chatzis and F. A. L. Dullien, Journal of Canadian Petroleum Technology, January-March (1977).Google Scholar
  36. R. G. Larson, L. E. Scriven and H. T. Davis, Chem. Eng. Sci. 36:57, Pergamon Press Ldt, Great Britain (1980).Google Scholar
  37. 34.
    R. Lenormand and S. Bories, C.R. Acad. Sc. Paris, 291 B: 279 (1980).Google Scholar
  38. 35.
    R. Lenormand and C. Zarcone, to be submitted to Phys. Rev. Lett. (1985).Google Scholar
  39. 36.
    D. Wilkinson, Phys. Rev. A 30: 520 (1984).CrossRefGoogle Scholar
  40. 37.
    R. Lenormand and C. Zarcone, J. Phys. Chem. Hydr., January (1985).Google Scholar
  41. 38.
    L. Paterson, Phys. Rev. Lett. 52–18: 1621 (1984).Google Scholar
  42. 39.
    P. G. Saffman and G. I. Taylor, Proc. R. Soc. Lond. A 245–311 (1985).Google Scholar
  43. S. B. Gorell and G. M. Homsy, S.I.AM, J. Appl. Math. 43–1:79 (1983). A general condition for marginal stability of the S.T. instability including permeability as well as density difference effect is where the subscripts 1,2 refer to the displacing and displaced fluid.Google Scholar
  44. 40.
    J. Bear, Dynamics of Fluids in Porous Media, chap. 10, American Elsevier, N.Y. We have benefited from several discussions with J. Koplik on chapter 4 1Google Scholar
  45. 41.
    G. I. Taylor in Low Reynolds Number Flows in Illustrated Experiments in Fluid Mechanics, MIT Press (1982) and film of the N.C.F.M. Films.Google Scholar
  46. 42.
    We thank J. Feder, U. Oxaal and their group for communication of their unpublished data.Google Scholar
  47. 43.
    P. G. Saffman, J. Fl. Mech. 6: 321 (1959).MathSciNetCrossRefGoogle Scholar
  48. 44.
    R. Aris, Proc. Roy. Soc. A. 235: 67 (1956).CrossRefGoogle Scholar
  49. The effect of velocity field is quite different from that of an externally applied field considered by Dhar and Barma for a biased ant walk. In this case the local probabilities are determined by the orientation of bonds with respect to the field (like in trickled bed flows in a gravitational field“). In the present problem, there is a continuum of particles which respond the local field (the local velocity field) which can be obtained only from a knowledge of the connectivity properties.Google Scholar
  50. 45.
    M. Crine, P. Marchot and G. L’Homme, Chem. Eng. Comm. 7: 377 (1980).CrossRefGoogle Scholar
  51. 46.
    C. D. Catalin and J. Roussenq, work in progress.Google Scholar
  52. 47.
    J. M. Hammersley and D.J.A. Welsh, Cont. Phys. 21: 593 (1980).CrossRefGoogle Scholar
  53. 48.
    A recent discussion of the so called “Richardson Pair Diffusion” can be found in S. Grossmann and I. Prococcia, Phys. Rev. A 29:1358 (1984).Google Scholar
  54. 49.
    This correlation function can be measured by forced Rayleigh Scattering M. Cloitre and E. Gyron, to appear in Jour. F1. Mech.Google Scholar
  55. 50.
    G. Matheron, unpublished.Google Scholar
  56. 51.
    P. G. de Gennes, J. F1. Mech. 136: 189 (1983).zbMATHCrossRefGoogle Scholar
  57. 52.
    M. Sahimi, H. T. Davis and L. E. Scriven, Chem. Eng. Com. 23: 329 (1983).CrossRefGoogle Scholar
  58. 53.
    J. P. Gaudet, Thèse Grenoble (1978).Google Scholar
  59. 54.
    L. de Arcangelis, S. Redner and A. Coniglio, preprint.Google Scholar
  60. 55.
    A. Dieulin, G. Matheron, G. de Marsily, The Science of Total Environment, 21: 319 (1981).CrossRefGoogle Scholar
  61. 56.
    D. L. Koch, J. P. Brady, to be published.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Christophe Baudet
    • 1
  • Elixabeth Charlaix
    • 1
  • Eric Clément
    • 1
  • Etienne Gyron
    • 1
  • Fean-Pierre Hulin
    • 1
  • Christophe Leroy
    • 1
  1. 1.Laboratoire d’Hydrodynamique et de Mécanique Physique E.S.P.C.I.Groupe Poreux P.C.Paris Cedex 05France

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