Page 116 - Reservoir Geomechanics

100 Reservoir geomechanics

p a is atmospheric pressure expressed in the same units as the stresses, and m and η 1
are material constants.

In the modified Lade criterion developed by Ewy (1999) m was set equal to zero
in order to obtain a criterion which is able to predict a linear shear strength increase
with increasing mean stress, I 1 /3. For considering materials with cohesion, Ewy (1999)
included pore pressure as a necessary parameter and introduced the parameters S and
η as material constants. The parameter S is related to the cohesion of the rock, while
the parameter η represents the internal friction.
Doing all the modifications and defining appropriate stress invariants, the following
failure criterion was obtained:
3
(I )
1 = 27 + η (4.13)
I
3
where
I = (σ 1 + S) + (σ 2 + S) + (σ 3 + S) (4.14)

1
and

I = (σ 1 + S)(σ 2 + S)(σ 3 + S) (4.15)
3
S and η can be derived directly from the Mohr–Coulomb cohesion S 0 and internal
friction angle φ by

S 0
S = (4.16)
tan φ
2
4(tan φ) (9 − 7sin φ)
η = (4.17)
(1 − sin φ)
where tan φ = µ i and S 0 = C 0 /(2 q 1/2 ) with q as defined in equation (4.7).
The modified Lade criterion predicts a strengthening effect with increasing interme-
diate principal stress, σ 2 , followed by a slight reduction in strength as σ 2 increases.
It should be noted that the equations above allow one to employ this criterion using
the two parameters most frequently obtained in laboratory strength tests, C 0 and µ i .
This makes this criterion easy to use, and potentially more generally descriptive of
rock failure, when considering problems such as wellbore stability. The modified Lade
criterion can be observed in Figure 4.7c where it has been plotted in σ 1 −σ 2 space for
C 0 = 60 MPa and µ i = 0.6, the same parameters used for the Mohr–Coulomb criterion
in Figure 4.7a.

Modified Wiebols–Cook criterion
Wiebols and Cook (1968) proposed an effective strain energy criterion for rock failure
that depends on all three principal stresses. Zhou (1994) presented a failure criterion