Page 132 - Applied Petroleum Geomechanics

124 Applied Petroleum Geomechanics

The linear Mogi criterion, Eq. (3.68), then reduces to (Al-Ajmi and
Zimmerman, 2005):
p ffiffiffi
2 b
ðs 1 s 3 Þ ¼ a þ ðs 1 þ s 3 Þ (3.70)
3 2
Comparison with the linear MohreCoulomb failure criterion shows
that for triaxial compression case, the linear Mogi criterion, Eq. (3.70),
coincides with the MohreCoulomb criterion, if the following relations are
satisfied:
p ffiffiffi
2 2
a ¼ c cos 4
3
p ffiffiffi (3.71)
2 2
b ¼ sin 4
3

Under this condition the linear Mogi criterion given by Eq. (3.68) is
exactly equivalent to the MohreCoulomb criterion.
The equations presented in this section are expressed in total stresses.
They need to be expressed in effective forms for porous rocks by consid-
ering pore pressure effect, i.e., replacing s by s .
0
3.4.8 Cam-Clay failure criterion

The Cam-Clay model was developed to describe soil yield by researchers at
the University of Cambridge (hence its name) (Roscoe et al., 1958, 1963).
Roscoe and Burland (1968) proposed the modified Cam-Clay model. It is a
very influential soil yield model and recently has been used to model the
compaction of sediments and rock failures. The primary assumptions of
the Cam-Clay and modified Cam-Clay models are characterized by the
following three parameters:
0
(1) Effective mean stress, p , can be calculated in terms of principal stresses
s 1 , s 2 , and s 3 as:
0
0
0
1
0 0 0 0
p ¼ s þ s þ s 3 (3.72)
1
2
3
(2) Deviatoric (shear stress) q:

1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 2
0 2
0 2
0
0
0
q ¼ p ffiffiffi ðs s Þ þðs s Þ þðs s Þ (3.73)
3
1
1
2
3
2
2