Page 151 - Applied Petroleum Geomechanics

144 Applied Petroleum Geomechanics

r ffiffiffiffiffiffi r ffiffiffiffiffiffi
K I r q 1 2 q K II r q 1 2 q
u x ¼ cos ðk 1Þþ sin þ sin ðk þ1Þþ cos
G 2p 2 2 2 G 2p 2 2 2
ð1 JÞs
þ fr½cosðq þ 2aÞþ k cosðq 2aÞ
8G

2 sin q sin 2 aŠþðk þ 1Þa cos 2 ag
r ffiffiffiffiffiffi r ffiffiffiffiffiffi
K I r q 1 2 q K II r q 1 2 q
u y ¼ sin ðk þ 1Þ cos þ cos ð1 kÞþ sin
G 2p 2 2 2 G 2p 2 2 2
ð1 JÞs
þ fr½sinð2a qÞþ k sinð2a þ qÞ 2 sin q cos 2 aŠ
8G
þðk þ 1Þa sin 2 ag
(4.23)
where s is the far-field stress in y-direction (see Fig. 4.6A); J is the ratio of
the far-field stress in x-direction to the far-field stress in y-direction.
The crack intensity factors K I and K II can be obtained from the
following equations:
p
K I ¼ s n pa (4.24)
ffiffiffiffiffi
p ffiffiffiffiffi
K II ¼ s n pa (4.25)
where s n is the normal stress perpendicular to the fracture surface; and s n is
the shear stress parallel to the fracture surface. For the loading conditions
shown in Fig. 4.6A, the intensity factors are specified by the following
equations (Eftis and Subramonian, 1978):
p
s pa
ffiffiffiffiffi
K I ¼ ½ð1 þ JÞ ð1 JÞcos 2 aŠ (4.26)
2
p
s pa
ffiffiffiffiffi
K II ¼ ð1 JÞsin 2 a (4.27)
2
The equations proposed by Eftis and Subramonian (1978) indicate that
presence of the horizontal load for the inclined crack shows up both in the
terms involving K I and K II , as well as in Eqs. (4.22) and (4.23). Both are
necessary to give full account of load biaxiality.
Consider a pressurized crack subjected to an internal pressure, p 0 , and
the far-field compressive stresses s h and s H (see Fig. 4.6B), the stress