Transport in Porous Media

, Volume 9, Issue 3, pp 275–286 | Cite as

Percolation approach to the problem of hydraulic conductivity in porous media

  • B. Berkowitz
  • I. Balberg


While percolation theory has been studied extensively in the field of physics, and the literature devoted to the subject is vast, little use of its results has been made to date in the field of hydrology. In the present study, we carry out Monte Carlo computer simulations on a percolating model representative of a porous medium. The model considers intersecting conducting permeable spheres (or circles, in two dimensions), which are randomly distributed in space. Three cases are considered: (1) All intersections have the same hydraulic conductivity, (2) The individual hydraulic conductivities are drawn from a lognormal distribution, and (3) The hydraulic conductivities are determined by the degree of overlap of the intersecting spheres. It is found that the critical behaviour of the hydraulic conductivity of the system,K, follows a power-law dependence defined byK ∞ (N/Nc−1)x, whereN is the total number of spheres in the domain,Nc is the critical number of spheres for the onset of percolation, andx is an exponent which depends on the dimensionality and the case. All three cases yield a value ofx≈1.2±0.1 in the two-dimensional system, whilex≈1.9±0.1 is found in the three-dimensional system for only the first two cases. In the third case,x≈2.3±0.1. These results are in agreement with the most recent predictions of the theory of percolation in the continuum. We can thus see, that percolation theory provides useful predictions as to the structural parameters which determine hydrological transport processes.

Key words

Percolation theory hydraulic conductivity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aharony, A., 1986, Percolation, in G. Grinstein and G. Mazenko (eds)Directions in Condensed Matter Physics, World Scientific, Singapore, pp, 1–50.Google Scholar
  2. Aitchison, J. and Brown, J. A. C., 1957,The Lognormal Distribution, Cambridge University Press, Cambridge.Google Scholar
  3. Balberg, I., 1986, Excluded-volume explanation of Archie's law,Phys. Rev. B 33(5), 3618–3620.Google Scholar
  4. Balberg, I., 1987, Recent developments in continuum percolation,Phil. Mag. 56(6), 991–1003.Google Scholar
  5. Balberg, I. and Binenbaum, N., 1985, Cluster structure and conductivity of three-dimensional continuum systems,Phys. Rev. A 31(2), 1222–1225.Google Scholar
  6. Balberg, I. and Binenbaum, N., 1986, Direct determination of the conductivity exponent in directed percolation,Phys. Rev. B 33(3), 2017–2019.Google Scholar
  7. Balberg, I., Wagner, N., Hearn, D. W., and Ventura, J. A., 1988, Critical behavior of the electrical resistance and its noise in inverted random-void systems,Phys. Rev. Lett. 60(19), 1887–1890.Google Scholar
  8. Batchelor, G. K., 1967,An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge.Google Scholar
  9. Bear, J., 1972,Dynamics of Fluids in Porous Media, American Elsevier, New York.Google Scholar
  10. Bear, J., Braester, C., and Menier, P. C., 1987, Effective and relative permeabilities of anisotropic porous media,Transport in Porous Media 2, 301–316.Google Scholar
  11. Broadbent, S. R. and Hammersely, J. M., 1957, Percolation processes: 1. Crystals and mazes.Proc. Camb. Phil. Soc. 53, 629–641.Google Scholar
  12. Carman, P. C., 1956,Flow of Gases Through Porous Media, Butterworths, London.Google Scholar
  13. Doyen, P. M., 1988, Permeability, conductivity, and pore geometry of sandstone,J. Geophys. Res. 93(B7), 7729–7740.Google Scholar
  14. Dullien, F. A. L., 1979,Porous Media: Fluid Transport and Pore Structure, Academic Press, New York.Google Scholar
  15. Fair, G. M. and Hatch, L. P., 1933, Fundamental factors governing the streamline flow of water through sand,J. Amer. Water Works Accos. 25, 1551–1565.Google Scholar
  16. Feng, S., Halperin, B. I., and Sen, P. N., 1987, Transport properties of continuum systems near the percolation threshold,Phys. Rev. B 35(1), 197–214.Google Scholar
  17. Gueguen, Y. and Dienes, J., 1989, Transport properties of rocks from statistics and percolation,Math. Geol. 21(1), 1–13.Google Scholar
  18. Halperin, B. I., Feng, S., and Sen, P. N., 1985, Differences between lattice and continuum percolation transport exponents,Phys. Rev. Lett. 54(22), 2391–2394.Google Scholar
  19. Johnson N. L., 1949a, Systems of frequency curves generated by the method translation,Biometrika 36, 149–176.Google Scholar
  20. Johnson N. L., 1949b, Bivariate distributions based on simple translated systems,Biometrika 36, 297–315.Google Scholar
  21. Kim, D. Y., Hermann, H. J., and Landau, D. P., 1987, Percolation on a random lattice,Phys. Rev. B 35(7), 3661–3662.Google Scholar
  22. Kogut, P. M. and Straley, J., 1979, Distribution-induced non-universality of the percolation conductivity exponents,J. Phys. C 12, 2151–2159.Google Scholar
  23. Kozeny, J., 1927, Uber kapillare Leitung des Wassers im Boden,Sitzungsber. Akad. Wiss. Wien 136, 271–306.Google Scholar
  24. Meyer, P. D., Valocchi, A. J., Ashby, S. F., and Saylor, P. E., 1989, A numerical investigation of the conjugate-gradient method as applied to three-dimensional groundwater flow problems in randomly heterogeneous porous media,Water Resour. Res. 25(6), 1440–1446.Google Scholar
  25. Sen, P. N., Roberts, J. N., and Halperin, B. I., 1985, Nonuniversal critical exponents for transport in percolating systems with a distribution of bond strengths,Phys. Rev. B 32(5), 3306–3308.Google Scholar
  26. Stauffer, D., 1985,Introduction to Percolation Theory, Taylor and Francis, New York.Google Scholar
  27. Thompson, A. H., Katz, A. J., and Krohn, C. E., 1987, The microgenity and transport properties of sedimentary rock,Adv. Phys. 36(5), 625–694.Google Scholar
  28. Wagner, N. and Balberg, I., 1987, Anomalous diffusion and continuum percolation,J. Stat. Phys. 49(1/2), 369–382.Google Scholar
  29. Wong, P., 1988, The statistical physics of sedimentary rock,Physics Today, December, 24–32.Google Scholar
  30. Wong, P., Koplik, J., and Tomanic, J. C., 1984, Conductivity and permeability of rocks,Phys. Rev. B 30(11), 6606–6614.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • B. Berkowitz
    • 1
  • I. Balberg
    • 2
  1. 1.Hydrological Service, Water CommissionMinistry of AgricultureJerusalemIsrael
  2. 2.Racah Institute of PhysicsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations