Page 43 - Percolation Models for Transport in Porous Media With

34 CHAPTER2. ONE-PHASE FLOW IN ROCKS

where < r~ >= f 0 r~ J(rt) drt.
00
Let the size distribution of fractures be represented by the following exponential
relation
(2.31)

where r 1 is the greatest radius of a circular fracture.
When r 0 «: r 1 , it follows from (2.29) and (2.30) in accordance with (2.31 ), that
y ~ 21rn°rg, z ~ 1811'n°rg - 1. After substituting the found relations into (2.24)
and taking account of (2.28) for Tt = r 0 , we obtain the resultant expression for
the permeability of the medium for the case of the exponential size distribution of
fractures

(2.32) implies that when the variance of the size distribution function of frac-
tures increases, a substantial drop of the percolation threshold follows. This phe-
nomenon is caused primarily by the increase of the number of the nearest neigh-
bors, and consequently by the increase of the probability of fractures intersecting
with each other. It is quite interesting to compare the sizes of the pores and the
fractures which correspond to the percolation threshold for a given concentration
n° in porous and fractured media. It follows from formulas (2.14) and (2.24) that
Rp ~ 0.55rt. That is, the presence of circular fractures in a medium is as effective
in encouraging formation of an IC, as is the presence of pores with radii equal to
exactly half the radius of a circular fracture. This fact implies that the conduc-
tivity of a medium is primarily affected not by the form of the conducting cut-ins,
but by their maximal size.

2.5 Conductivity As a Function of the Strained

State

Experimental data [48] show that the variation of the strained state of a medium
can notably affect its conductivity. Relationships (2.1) and (2.4) obtained in §2.1
allow to find the change of the conductivity in the medium caused by external
factors (pressure, temperature, etc.), if the nature of the effect which these pa-
rameters have on the distribution function f(r), is known. Thus the problem
of finding the correlation between the conductivity and the strained state of a
medium reduces to that of determining the dependence of the intrinsic conduc-
tivity distribution for channels on the strained state of the medium. That is, the
problem reduces to the study of the change in the effective capillary radii under
external pressure. The problem of the correlation between the capillary size and
the value of the strain tensor was discussed in [49] using the model of a nonlinearly