Page 137 - Percolation Models for Transport in Porous Media With

7.1 CONCENTRATION AND LENGTH OF FRACTURES 131

Figure 49: Refers to the determination of the probability of a circle with an
arbitrary radius intersecting with a fracture trace on the surface of the core

<r•>
< da >= J ~h- (xj < Tt >} dx (7.3}
2
0
From (7.3} we find that< ds >= 1r /4 < rt >. The correlation between the surface
concentration nd of the fracture traces and the volumetric concentration n* of the
disk fractures can be found as follows.
Consider the cross-section M (see fig. 49}. Intersection of an arbitrary disk
fracture with the cross-section is possible only when the distance from the frac-
ture center to the plane M does not exceed the quantity ft. The probability of
intersection is determined by the value of the solid angle n. If a fracture lies
inside this angle, then it intersects with the plane. It can be easily shown that
n = 21r(1 - cos8t}, where cos81 = xfrt. In this case, when we take into account
the symmetry of the problem, we obtain an expression for P(x}, the probability
of the plane intersecting a fracture with the average radius < Tt >
P(x) = 1 - xf < rt > (7.4}

The number of the fracture traces on a unit surface of the cross-section is
<r•>
nd = 2n* j P(x)dx (7.5}
0

Using (7.4} and (7.5} we find nd = n* < rt >. Recall that the density of fractures,
which characterizes the average distance between the fracture traces on the plane,
is defined by the relationship

f = Vn* < Tt >
The investigation just carried out shows that the concentration and the average
length of the fracture traces on the surface of the core can be found. Knowing