Page 117 - Reservoir Geomechanics

101 Rock failure in compression, tension and shear

with features similar to the Wiebols–Cook criterion which is an extension of the cir-
cumscribed Drucker–Prager criterion (described below).
The failure criterion proposed by Zhou predicts that a rock fails if
1/2 2
J = A + BJ 1 + CJ (4.18)
2 1
where
1
J 1 = (σ 1 + σ 2 + σ 3 ) (4.19)
3
and

1/2 1 2 2 2
J = (σ 1 − σ 2 ) + (σ 1 − σ 3 ) + (σ 2 − σ 3 ) (4.20)
2
6
1/2 1/2
J 1 is the mean effective confining stress and, for reference, J 2 is equal to (3/2) τ oct ,
where τ oct is the octahedral shear stress
1
2
2
τ oct = (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 2 − σ 1 ) 2 (4.21)
3
The parameters A, B, and C are determined such that equation (4.18)is constrained
by rock strengths under triaxial (σ 2 = σ 3 ) and triaxial extension (σ 1 = σ 2 ) conditions
(Figure 4.1). Substituting the given conditions plus the uniaxial rock strength (σ 1 = C 0 ,
σ 2 = σ 3 = 0) into equation (4.18), it is found that

√
27 C 1 + (q − 1)σ 3 − C 0 q − 1
C = − (4.22)
2C 1 + (q − 1)σ 3 − C 0 2C 1 + (2q + 1)σ 3 − C 0 q + 2
with C 1 = (1 + 0.6 µ i )C 0 and q given by equation (4.7),
√
3(q − 1) C
B = − [2C 0 + (q + 2)σ 3 ] (4.23)
q + 2 3
and

C 2
C 0 C 0 0
A = √ − B − C (4.24)
3 3 9
The rock strength predictions produced using equation (4.18) are similar to those of
Wiebols and Cook and thus it is referred to as the modified Wiebols–Cook criterion.
For polyaxial states of stress, the strength predictions made by this criterion are slightly
higher than those found using the linearized Mohr–Coulomb criterion. This can be
seen in Figure 4.6 because the failure cone of the modified Wiebols–Cook criterion just
coincides with the outer apices of the Mohr–Coulomb hexagon. This criterion is plotted
in σ 1 −σ 2 space in Figure 4.7d. Note its similarity to the modified Lade criterion.