Page 165 - Applied Petroleum Geomechanics

158 Applied Petroleum Geomechanics

where k ni is the initial normal stiffness in MPa/mm; a j is the initial mechan-
ical aperture in mm; JCS is in MPa.
The above equations are for interlocked joints. However, for a dis-
located joint the normal stress and displacement have the following relation:

log s n ¼ k ni þ MDV j (4.57)
where M is a constant.
Based on laboratory tests on fractured concrete blocks, Zhang et al.
(1999) found that the applied stress and fracture displacement follow an
exponential relation. For example, for the uniaxial loading perpendic-
ular to the fracture plane, the fracture displacement (aperture) decreases
as the applied stress increases, which can be expressed in the following
form:
u ¼ b 0 e as n (4.58)

where b 0 is the initial fracture width (aperture); a is a testing constant
(a > 0).
Eq. (4.58) indicates that the normal stress, which is perpendicular to
the fracture plane, causes the fracture to close. However, for the
uniaxial loading parallel to the fracture plane, the fracture aperture
increases as the applied stress increases. This is because the tensile stress
is induced in the direction of perpendicular to the fracture plane. The
fracture displacement can be defined by the following relation (Zhang
et al., 1999):

u ¼ b 0 e bs x (4.59)

where s x is the applied stress parallel to fracture plane; b is a constant
(b > 0).
The shear stress and shear displacement of a rough discontinuity
depend on the normal stress and the discontinuity surface characteristics
because the shear deformation involves dilation and fracture asperities. A
hyperbolic function is frequently used to describe the shear stress and
shear displacement relationship for a discontinuity in the prepeak stress
regime:
v
s ¼ (4.60)
a þ bv
where v is the shear displacement of the discontinuity; s is the shear stress;
and c and d are constants.