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


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 



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


numerical prefactor of the order of unity


numerical prefactor of the order of unity


space dimensionality


normalized distribution of resistances


cumulative distribution of resistances


medium permeability

Km, Km

lower and upper bounds for the medium permeability


permeability of the subnetwork constituted of all resistances ⩾ r


lower bound of k(r)


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


upper bound of k′(r)


dimensionless number representing the density of cracks


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


number of disk centers per unit volume


density of conductors on a lattice



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

‘representative’ value of r


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


critical exponent for the conductivity of a percolation medium


aperture of a crack; or pore entry radius


correlation length of a geometrical structure or of the velocity field


critical exponent for the correlation length



δ0, δ1

percentage of resistors between rc and r0, rc and r1


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Copyright information

© D. Reidel Publishing Company 1987

Authors and Affiliations

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

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