Page 139 - Percolation Models for Transport in Porous Media With

7.2 FRACTURE LENGTH DISTRIBUTION 133

which intersects with the boundary of the core. This is due to the fact that those
fracture traces that intersect with the boundary of the core from the inside and
from the outside are indistinguishable. Let Nd( d., 9) be the number ofthe fracture
traces whose half-lengths lie in the interval d. + d8 + 6.d8 , oriented at an angle
from the interval (} + (} + 6.9 and lying completely inside the core. This number is
equal to the difference between the total number of all such fractures in the circle
and the number of the fracture traces that intersect with the circumference. For
the fractures oriented at an angle 9 (see fig. 50) we have

(7.6)

where Sn is the unshaded part of the circle.
After integrating (7.6) with respect to the angle 9, we can find the number of
the fracture traces with half-lengths lying in the interval d. +d. +ad. and falling
completely inside the core

1r
Nd(ds) =I nd(d8 ,9)Snd9 (7.7)
0
The area Sn is equal to twice the area of the section

Sn = 2 [R' cos- (d./R')- dsVR' 2 - ~] (7.8)
1
2
After substituting (7.8) in (7.7) and taking account of the fact that

1r
I nd(d.,9)dfJ = nd(ds)
0
where nd(d.) is the surface density of the centers of those fracture traces, whose
half-lengths lie in the interval d.+ ds + 6.d., we obtain the following expression,

Upon calculating the number of the fracture traces with length 2d8 falling
completely inside the core of radius R', one can determine the length probability
density for fractures

The quantity Nd(d.), where ds lies in the interval d8 +ds+f:1d8 , has to be deter-
mined from experiment by gathering the amount of data necessary for statistical
analysis. As it can be seen from {7.9), as d. -+ R', the error in determining nd(d.)