Page 29 - Applied Petroleum Geomechanics

Stresses and strains 19

fracture height is fixed and independent to the fracture length growth (or
length [ height).
1.4.5 Isotropic porous rocks

For the fluid-saturated rocks, the effect of pore pressure and Biot’s coef-
ficient (a) needs to be considered. Poroelasticity can be applied to consider
this effect (Detournay and Cheng, 1993). In porous rocks, the strain and
stress relations previously introduced should consider the effective stresses
instead of the total stresses (i.e., replacing each total stress s by effective
0
stress s using s ¼ s ap p ). For instance, the strain and stress relations in
0
isotropic porous rocks can be expressed as follows by replacing the total
stresses in Eq. (1.20) by the corresponding effective stresses (Eq. 1.9):
1
ε x ¼ ½s x nðs y þ s z Þ að1 2nÞp p Š
E
1
ε y ¼ ½s y nðs x þ s z Þ að1 2nÞp p Š
E
1
ε z ¼ ½s z nðs x þ s y Þ að1 2nÞp p Š
E
(1.33)
1
ε xy ¼ s xy
2G
1
ε xz ¼ s xz
2G
1
ε yz ¼ s yz
2G
The in situ stresses in the uniaxial strain condition (refer to Eq. 1.22)
with consideration of pore pressures can be solved from Eq. (1.33),
assuming ε x ¼ ε y ¼ 0, s x ¼ s h , s y ¼ s H , and s z ¼ s V , i.e.,
n
s h ¼ s H ¼ ðs V apÞþ ap p (1.34)
1 n

1.5 Stressestrain relations in anisotropic elastic rocks
Most rocks are anisotropic materials and have a characteristic orientation.
For example, in a shale formation, the clay minerals are oriented in the