Page 42 - Percolation Models for Transport in Porous Media With

2.4 CONDUCTIVITY OF A FRACTURED MEDIUM 33

where r is the thickness of fractures.
Using the geometry of the problem we can relate the thickness of fractures to
their concentration. Consider the cross-section M (see fig. 10). Intersection of an
arbitrary circular fracture with M is possible only if the distance from the center
of the fracture toM does not exceed Tt. The intersection probability is determined
from the solid angle fh. If the fracture lies inside this angle, then it can intersect
with the plane M. It can be seen from fig. 10, that 0 1 = 211'(1- cos81), where
cos81 = xfrt (the segment x lies in the same plane as the fracture). Taking
account of symmetry of the problem, we obtain the formula for the probability of
the plane intersecting with the fracture
Pt(x) = 1- xfrt. (2.26)

The number of traces on a unit surface of the cross-section is
r,
n2 = 2n° / Pt(x)dx. (2.27)
0
Using (2.26) and {2.27), we find that n 2 = n°rt. The thickness of fractures,
which characterizes the average distance between the fracture traces on the plane,
is determined from the relationship

f= #r"t. (2.28)

When y ~ 1.5, virtually all fractures intersect with each other, and the perme-
ability of the medium can be described by formula (2.25), if r is found from (2.28).
After comparing the dependence {2.24) for 1rn°rr = 1.5 to the relationship {2.25),
we can find 'Y using (2.28}. The resultant expression for the coefficient of perme-
ability of the fractured medium is

Consider the case when the size distribution of the circular fractures in the
medium is described by the function /{rt)· Using this distribution and the for-
mula (2.23}, we obtain the following expression

y = 1rn° < r~ >< Tt >, {2.29}

where< Tt >= f 0 rtf(rt) drt, < r~ >= f 0 r~ /(rt) drt. Consequently the number
00
00
of adjacent sites is
z = 211'[(1 + 11'/2) < r~ >< Tt > +(2/3) < r~ >]- 1, {2.30)