Page 140 - Percolation Models for Transport in Porous Media With

134 CHAPTER 7. PARAMETERS OF FRACTURED ROCKS

goes up abruptly. Therefore reliable determination of the function nd(ds) is pos-
sible only for those fracture traces whose half-lengths are not too large compared
to the radius of the core (ds < R').
For further calculations, it is convenient to introduce a normalized distribution
function

(7.10)

Consider a disk fracture of radius Tt whose mouth is negligible. It is easy to
show that by reasons similar to those mentioned in deriving the relationship (7.3),
the probability of this fracture leaving on the cross-section a trace of length 2d 8 is
equal to
P(d)- ds
Tt rt - s
8 - J 2 d2
Let F(rt) be the normalized radius probability density function for disk fractures.
In this case the probability of the length of a trace on the surface of an arbitrary
cross-section to lie in the interval 2d 8 + 2{ d 8 + l:1d 8 ) is determined by the expression

(7.11)

In deriving the formula (7.11 ), it was taken into account that a fracture with radius
less than ds cannot leave a trace of length d 8 on a plane.
The integral equation of the first type {7.11) establishes the correlation between
the experimentally measured function f(ds) and the function F(rt) which is to
be determined. Since such equations represent examples of ill-posed problems,
their numerical solutions on computer are unstable. In the special case when
the function f(ds) and its derivative are continuous and bounded, the solution
of the equation {7.10) can be obtained in the explicit form. Aside from the fact
that the obtaining of an analytical solution of the equation (7.10) is of separate
mathematical interest, it is also very important in the methodological aspect of
the considered application. The exact analytical solution, unlike an approximate
numerical one, is stable. Since the outlined limitations on the function f(d 8 ) are
satisfied by the majority of functions obtained in experiment, this solution can be
used as a stable technique for the processing of a broad set of probability density
functions for traces of fractures on the core.
Make the following change of variables in (7.10), rt 1 = t. As a result, we
obtain
d-1
• 1
f( d ) = d I F(C ) dt (7.12)
8 8 vf1- fP..t2
0 8