Page 40 - Percolation Models for Transport in Porous Media With

2.4 CONDUCTIVITY OF A FRACTURED MEDIUM 31

Figure 9: For determining the average number of intersections for circular frac-
tures

Consider a set of randomly distributed circular fractures in an infinite medium.
Assume that the fractures are oriented isotropically and all have the same radius
Tt and opening b. Assume also that the centers of the fractures are sites, and
intersections of any two fractures form a bond between the sites. In this case the
probability of bonds having a non-zero conductivity is determined from the ratio
of two quantities, namely the number of intersections an arbitrary fracture has
with other fractures and the number of its nearest neighbors (sites). Consider (fig.
8) an arbitrary "reference" fracture. Among other fractures, only those intersect
with it, whose centers lie in the domain V. The total number of fractures in Vis

t
z1 =21rn°r~ /(1+2coscfo+cos cfo)coscfodcfo,
2
0
where n° is the concentration of the fracture centers. The greatest number of sites
nearest to the "reference" fracture is

z = Zt- 1 = 21rn°(l + 11"/2 + 2/3)r~- 1.

Calculate the average number y of intersections the "reference" fracture has
with its nearest neighbors. Obviously, such an intersection is possible only if
the center of the other fracture lies inside V. En route, consider a more general
problem of intersections for those circular fractures which lie in planes forming an
angle 8 between them (fig. 9). The radius of the "reference" fracture equals Tt
and the radii of the fractures intersecting with it are equal to R1 • Intersection of
fractures is possible if the center of the second one lies inside a slanted cylinder
whose base has an area equals 1rr~ and whose altitude is 2R1 cosO. Consequently
the number of intersections for such fractures (or, more exactly, the expectation
of this number) is
(2.20)