Page 41 - Percolation Models for Transport in Porous Media With

32 CHAPTER2. ONE-PHASE FLOW IN ROCKS

Figure 10: Geometry of intersection of fractures with the plane M

Here n~ = n° !(9, 1/J) d9 is the concentration of those fractures oriented at an
angle 9 from the interval 9 + 9 + d9; f(9, 1/J) is the angle distribution function for
the circular fractures. If the fractures are oriented isotropically, then their angu-
lar distribution is described in spherical coordinates by the distribution function
/(9,1/J) = (27r)- 1 (0 $ 9 $ 1rj2, 0 $ 1/J $ 21r). In this case we can average Yt
over the solid angle dO = sin 9 d9 d,P to obtain the following

y =I ytf(9, 1/J) dO. (2.21)

Taking account of the relationship (2.20), we find the average number of inter-
sections from (2.21) to equal
(2.22)

If the fractures have equal radii all, then y = 1rn°r:. In accordance with the
discussed above, the probability of a bond forming between two sites is
pb = yfz.

The threshold value of the probability in the considered bond problem can be
estimated using the invariant (1.1) for D = 3
P; = 1.5/z. (2.23}

In this case the permeability of the medium is determined according to (2.23},
i.e., by the following relationship

(2.24}
The coefficient 'Y can be found by comparing the relationship (2.24} to the
relationship for the coefficient of permeability of a medium pierced with infinitely
long fractures, oriented isotropically [47]

(2.25}