Page 141 - Percolation Models for Transport in Porous Media With

7.2 FRACTURE LENGTH DISTRIBUTION 135

Another change of variables, t =sin f/J/d., in the equation (7.12) reduces it to

w/2
/(d.)= IF (si~f/J) df/J (7.13)
0
The equation (7.13) is one of the SchlOmilch integral equations. The solu-
tion of this equation can be obtained as follows. After introducing the function
F 1(sin¢J/d.) = F(d./sinf/J) into the equation (7.13) and then differentiating it
with respect to d., we obtain
w/2

d~f'(ds) =-IF{ (sinf/J/d.)sinf/Jdf/J (7.14)
0
By substituting xf sin 1/J fords in (7.14), integrating it with respect to 1/J from 0
to 1rj2, and changing the order of integration in the right side, the equation (7.14)
can be reduced to
w/2 w/2 w/2

I _j:_ !' (~) d,P = 2 I df/J IF,' (sin,Psinf/J) sinf/Jd'I/J (7.15)
SID 2 1/J
SID 1/J
l
X
0 0 0
Having introduced a new variable sin A = sin f/Jsin 1/J, after changing the order
of integration in the right side, we obtain
w/2 w/2 w/2
.x: !' (~) d,P =-IF{ (sin A) cosAdA I
sinf/Jdf/J
I SID 1/J SID 1/J x J cos2 A - cos2 ¢J
0 0 .\
Taking account of the fact that
w/2
sin ¢J df/J 1r
I Jcos2A- cos2¢J = 2'
.\
after integrating with respect to A, we obtain the resultant expression

w/2
F(rt) = F(oo)- ~ J -4:- f' ( .rt.,,) d,P (7.16)
7r SID 1/J SID 'I'
0
Formula (7.16) permits to find the length distribution function F(rt) of disk
fractures if the length distribution function /(d.) of fracture traces is known.
In particular, if the distribution f(d 8 ) is defined as an exponential relation

f(d.) = aexp(-cd.)