Page 153 - Applied Petroleum Geomechanics

146 Applied Petroleum Geomechanics

4.2.6.2 Fracture process zone at the fracture tip in rock
A fracture in rock propagates generally in a brittle manner than in plastic
yielding; therefore, the plastic zone at the crack front of the rock is very
different from metallic materials. Fracture propagation in rock is charac-
terized by the generation of microcracks around the crack tip and inter-
locking in a portion of the crack where displacement has not reached a
critical value. This zone of inelastic behavior is called the fracture process
zone (FPZ), analogous to the plastic zone in metals (Labuz et al., 1985).
However, there are no sound theoretical models available to fully describe
the shape and size of the crack tip FPZ, and it is often described by the
approximate models developed to describe the plastic zone in metals
(Whittaker et al., 1992).
Schmidt (1980) suggested a maximum normal stress criterion to describe
the shape of the crack tip FPZ in rock. The criterion is that when the local
maximum principal stress in the vicinity of the crack tip reaches the uniaxial
tensile strength of the rock (T 0 ), the FPZ is generated, i.e.,

(4.35)
s 1 ¼ T 0
Here the tensile stress is positive and compressive stress is negative to be
consistent with the fracture mechanics notation.
Substituting Eq. (4.13) into the above equation yields the shape of the
FPZ in Mode I fracture as follows:
2 2
1 K I 2 q q
rðqÞ¼ cos 1 þ sin (4.36)

2p T 0 2 2
The length of the FPZ can be obtained from the above equation by
substituting q ¼ 0:

2
1 K I
r p ¼ (4.37)
2p T 0
This is the same form as that of metallic materials for plane stress from
the Von Mises criterion (Eq. 4.33) if the yield strength (s 0 ) is equal to the
uniaxial tensile strength (T 0 ). The shape and size of the FPZ is independent
of whether the crack is under plane stress or plane strain condition because
the out-of-plane stress does not enter onto the expression for r(q) as given
by Eq. (4.36). Because the maximum normal stress criterion does not
consider the stress redistribution outside the FPZ, the actual size of the FPZ
should be much larger than the one predicted from Eq. (4.36).