Page 64 - Applied Petroleum Geomechanics

Rock physical and mechanical properties 55

Jones (1975) provided an empirical relation for the fracture permeability
in carbonate rocks as follows:
0 i 3
h
s 0
k f ¼ k 0 log (2.38)
s 0
0
where k 0 is the initial permeability; s is the effective stress; s 0 is the effec-
0
tive stress when k f ¼ k 0 .
Louis (1974) suggested an alternative relationship based on well
pumping tests at different depths:
0
k f ¼ k 0 expð As Þ (2.39)
0
0
where s is the effective stress and can be expressed as s ¼ gH p p in
which g is the specific gravity of the rocks; H is the depth; p p is the pore
pressure; and A is a coefficient.
Walsh (1981) offered the following empirical relation derived from
laboratory test data:

h p ffiffiffi i 3
0 0
k f ¼ k 0 1 2 x lnðs =s 0 Þ (2.40)
where s 0 is the initial effective stress, and x is a constant related to the frac-
0
ture geometry.
Bai and Elsworth (1994) presented the following equation to describe
strain and permeability change (Dk):
3
" #
k n b b
1
Dk ¼ k 0 1 þ Dε þ (2.41)
E s
where Dε is the strain change; k n is the normal stiffness of the fracture; and
E is Young’s modulus.
Based on Eq. (2.37) and large synthetic rock tests (Zhang et al., 2007),
the permeability change in one direction (such as, z-direction) due to the
aperture change can be directly related to the stress resultant deformations.
For two mutually orthogonal sets of fractures, as shown in Fig. 2.15, the
permeability change due to the aperture changes in the z-direction can be
obtained as follows (Zhang et al., 2007):
3

Db x Db y
k z ¼ k 0z 1 (2.42)
b 0x b 0y