Advertisement

Transport in Porous Media

, Volume 2, Issue 1, pp 31–43 | Cite as

Permeability of a random array of fractures of widely varying apertures

  • Elisabeth Charlaix
  • Etienne Guyon
  • Stephane Roux
Article

Abstract

We modelize a fractured rock by a random array of plane cracks of finite extent having a very broad distribution of apertures (or of hydraulic conductances). If the rock is permeable, the flow will essentially take place along a ‘subnetwork’ made of the less resistant cracks. Using an analogy with the treatment of variable range transport in semiconductors, we evaluate the homogenization length and the permeability of this disordered network. This evaluation makes use of the notion of the critical bonds which are the weakest cracks among the good ones necessary for percolation; the remaining weaker bonds make a negligible contribution to the permeability. The method is applicable to other examples of transport in very heterogeneous macroscopic random materials.

Key words

Homogenization permeability percolation 

Notations

a

diameter of the disks; or typical pore (fracture) size

c

numerical prefactor of the order of unity

c′

numerical prefactor of the order of unity

d

space dimensionality

f(r)

normalized distribution of resistances

F(r)

cumulative distribution of resistances

K

medium permeability

Km, Km

lower and upper bounds for the medium permeability

k(r)

permeability of the subnetwork constituted of all resistances ⩾ r

k1(r)

lower bound of k(r)

k′(r)

permeability of the subnetwork constituted of all the resistances ⩽ r, the other ones being set to 0

ku(r)

upper bound of k′(r)

N

dimensionless number representing the density of cracks

Nc

critical value of N which separate the zero permeability region N < Nc from the permeable one N > Nc

nv

number of disk centers per unit volume

p

density of conductors on a lattice

r

resistance

rinfasupb
$$ = \int_a^b {rf(r)dr,} $$
{r}

‘representative’ value of r

s

critical exponent for the resistivity of a percolation medium constituted of normal conductors and superconductors

t

critical exponent for the conductivity of a percolation medium

δ

aperture of a crack; or pore entry radius

ξ

correlation length of a geometrical structure or of the velocity field

v

critical exponent for the correlation length

σ

conductivity

δ0, δ1

percentage of resistors between rc and r0, rc and r1

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ambegaokar, V., Halperin, B. I. and Langer, J. S., 1971, Phys. Rev. B4 2612.Google Scholar
  2. Broadbent, S. R. and Hammersley, J. S., 1957, Proc. Camb. Phil. Soc. 53, 629.Google Scholar
  3. Bourbié, T. and Zinszner, B., 1985, J. Géophys. Res. 90, 11524–11532.Google Scholar
  4. Charlaix, E., Hulin, J. P., and Plona, T., to be published in Phys. Fluids.Google Scholar
  5. Charlaix, E., Guyon, E., and Rivier, N., 1984, Solid State Comm. 50, 999.Google Scholar
  6. Charlaix, E., 1986, J. Phys. A. Lett., 19, L533-L536.Google Scholar
  7. Dullien, F. A. L., 1975, AIChE J. 21, 23.Google Scholar
  8. Groupe poreux PC, 1985, in R. Pynn (ed.), Scaling Concept in Porous Media, Plenum Press.Google Scholar
  9. Guyon, E., 1985, in E. Stanley and N. Ostrowski (eds.), On Growth and Form: a Modern View, Martinus Nijhoff, 163.Google Scholar
  10. Halperin, B., Feng, S., and Sen, P. M., 1985, Phys. Rev. Lett. 54, 2391.Google Scholar
  11. Kogut, P. M. and Straley, J. P., 1979, J. Phys. C12, 2151.Google Scholar
  12. Long, J. C. S., Remer, J. S., Wilson, C. R., and Witherspoon, P. A. 1982, Water Resour. Res. 18, 645.Google Scholar
  13. Long, J. C. S., Gilmour, P., and Witherspoon, P. A., 1985, Water Resour. Res. 21, 1105.Google Scholar
  14. De Marsily, G., 1981, Hydrogéologie quantitative, Masson, Paris.Google Scholar
  15. Rivier, N., Charlaix, E., and Guyon, E., 1985, Geol. Mag. 122, 157.Google Scholar
  16. Robinson, P. C., 1983, J. Phys. A16, 605 and thesis Oxford (1984).Google Scholar
  17. Rouleau, A., 1984, Thesis U. of Waterloo (Ontario).Google Scholar
  18. Seager, C. H. and Pike, G. E., 1983, Phys. Rev. B10, 1435.Google Scholar
  19. Shklovskii, B. I. and Efrös, A. L., 1984, Electronic Properties of Doped Semiconductors, Springer-Verlag, Berlin, p. 129.Google Scholar
  20. Stanley, H. E. and Coniglio, A. 1983, in G. Deutscher, R. Zaller, and J. Adler (eds.), Percolation Structures and Processes. Annals Isr. Phys. Soc. Vol. 5 (1983).Google Scholar
  21. Wilke, S., Guyon, E., and De Marsily, G., 1985, J. Math. Geol. 17, 17.Google Scholar

Copyright information

© D. Reidel Publishing Company 1987

Authors and Affiliations

  • Elisabeth Charlaix
    • 1
  • Etienne Guyon
    • 1
  • Stephane Roux
    • 1
  1. 1.LHMP ESPCIParis CédexFrance

Personalised recommendations