Page 63 - Applied Petroleum Geomechanics

54 Applied Petroleum Geomechanics

microcrack closing) commences from the origin until the crack-initiation
stress level (s ci ) is reached. This region corresponds to the closure of
existing microcracks pervading the rock sample. This causes the rock vol-
ume to contract as the load increases; therefore, permeability decreases
gradually. Region II (new crack growth) corresponds to the initiation of
new cracks, which induces a volumetric dilation and a dramatic increase in
permeability. It is obvious from Fig. 2.14 that changes of permeability and
volumetric strain are coherent in trend. This phenomenon illustrates that
the volumetric strain and permeability can be related by a certain function,
although it is difficult to determine a correlation between stress and
permeability, particularly after the peak strength (s c ) is reached.

2.4.4 Stress and permeability relations in fractured rocks
For a single fracture, the fracture permeability can be obtained from the
parallel plate model:
b 2
k f ¼ (2.36)
12
where k f is the fracture permeability; b is the fracture aperture.
The single fracture model can be extended to multiple fracture systems
by considering regular families of parallel fractures. The permeability
through a set of parallel fractures of equal aperture, oriented parallel to flow
direction, can be expressed in the following equation (the cubic law):
b 3
k f ¼ (2.37)
12s
where s is the mean fracture spacing.
Because natural fractures are neither smooth nor parallel, investigators
have questioned the accuracy of applying the cubic law to natural fractures.
Investigations show that the cubic law is valid when corrected by consid-
ering the fracture tortuosity, correction factor, or using effective fracture
aperture (Witherspoon et al., 1980). The cubic law may be also applicable
in hydraulic fractures for determining the stresseconductivity relation.
Decrease of formation pore pressure because of depletion will increase
effective stresses and cause the apertures (widths) of the hydraulic fractures
to reduce; therefore, the fracture conductivity decreases.
In fractured formations, permeability variations with the stresses have
been delineated through various laboratory and field tests, represented by
different empirical equations.