Page 163 - Applied Petroleum Geomechanics

156 Applied Petroleum Geomechanics

(A) σ n (B) σ σ n Intact Rock with
rock fracture
y

k
1

u
0
V
σ n
Figure 4.15 (A) A discontinuity loaded in compression; (B) normal stress and normal
displacement relationship.

The normal and shear stiffnesses are two basic parameters used to
describe mechanical behaviors of a discontinuity. The normal stiffness is
defined as the slope in the normal stress and displacement curve when a
discontinuity is loaded under a normal stress perpendicular to the direction
of the discontinuity (Fig. 4.15), i.e.,

vs n
k n ¼ (4.50)
vu
where k n is the normal stiffness of the discontinuity; s n is the normal stress
applied perpendicularly to the discontinuity; u is the normal displacement
of the discontinuity.
The shear stiffness (k s ) is the slope in the curve of the applied shear stress
(s) and resulted shear displacement (v) and can be expressed in the following
(refer to Fig. 4.16):
vs
k s ¼ (4.51)
vv
Laboratory test results show that the shear stiffness in the discontinuity is
much smaller than the normal stiffness. In other words, the discontinuity is
much weaker to resist shear stress.
For an interlocked joint (discontinuity) when a normal compressive
stress is exerted on the discontinuity as shown in Fig. 4.15, the normal stress
and discontinuity displacement have a hyperbolic (nonlinear) relation
(Goodman, 1976):

DV j
s n ¼ s i þ s i (4.52)
V m DV j