Page 136 - Percolation Models for Transport in Porous Media With

130 CHAPTER 7. PARAMETERS OF FRACTURED ROCKS

7.1 Concentration and Average Length of Frac-
tures Determined from the Core

Consider a medium with identical disk fractures distributed chaotically in space
and oriented isotropically. Assume that the traces of these fractures on an ar-
bitrary cross-section are identical; let them be line segments of length 2d., with
concentration of centers nd. Calculate v', the average number of the traces of
length 2d 8 intersecting the butt surface of the core of radius R'. The surface of
the core can be intersected not only by the segments whose centers lie in the circle
of radius R~, but also by the segments whose centers are at a distance of no more
than (R' +d.) (see fig. 49) from the center 0 of the circle. Here the probability of
a fracture trace intersecting the circle depends on the orientation of the fracture
and is determined by the angle 8. Thus the number of intersections for the traces
of length 2d 8 and orientation 8 equals

(7.1)

After averaging both sides over the lengths of the traces and over the angles of
their orientation and taking account of the fact that the length and angle distri-
bution functions of the traces are normalized, we conclude that in the actual case
of arbitrary length distribution of segments the formula (7.1) defines the quantity
v'( <d. > ), where
00
<d. >= J d.f(d.) dd.

0
is the average value of lengths of the fracture traces distributed with the density
f(d.).
If we define v'( < d .. >) for circles of different radii R~, then from the system
of equations
(7.2)
it is possible to find the quantities we are seeking, namely the concentration nd of
the fracture traces on the cross-section and the average length < d 8 > of a fracture
trace on the cross-section. Knowing the quantity nd, we can find the concentration
of disk fracture centers n• and the average radius of a disk fracture < rt >. The
latter is easy to relate to the average length of a fracture trace on the cross-section,
by taking into account the fact that the distribution of fractures is homogeneous.
The probability of the length of the trace left on the core surface by a fracture to
belong to the interval 2d8 + 2(d8 + l::J..d8 ) does not depend on the distance x from
the center of the fracture to its intersection with the cross-section in the fracture
plane. This probability is the same for all distances x and equal to dx/ < rt >.
Using the correlation between d 8 and x we obtain the following