Page 126 - Applied Petroleum Geomechanics

118 Applied Petroleum Geomechanics

friction in the plane of weakness, respectively; m w ¼ tan4 w ; and 4 w is the
angle of internal friction in the plane of weakness.
In terms of effective stresses, Eq. (3.49) can be expressed in the
following form:

2 c w þ m s 0
0 0 w 3
s s ¼ (3.50)
1 3
ð1 m cot bÞsin 2 b
w
The value of s 1 required to cause failure, as given by Eqs. (3.49) and
(3.50), trends to infinity as b /90 or b / 4 w (i.e., failure in the rock). In

other words, when 0 < b < 4 w and b ¼ 90 , the planes of weakness have
no impact on the rock strength. If 4 w < b < 90 , shear failure will occur in

the weak planes at a finite value of s 1 that varies with b, as shown in
Fig. 3.26B. The minimum strength occurs in the following condition (refer
to Fig. 3.26B):

b min ¼ 45 þ 4 =2 (3.51)
w
Compared to the experimental results in the outcrop of the Mancos
shale, the model of the planes of weakness Eq. (3.49) gives a good
prediction (Fig. 3.27). For multiple sets of parallel planes of weakness, su-
perposition principle can be used in Eq. (3.50) to solve the complex
problem.

Figure 3.27 Measured uniaxial compressive strengths versus b and the strength
predicted by the planes of weakness model in the Mancos shale (Eq. 3.49). (After Fjær,
E., Nes, O.M., 2013. with permission of ARMA).