Page 109 - Reservoir Geomechanics

93 Rock failure in compression, tension and shear

these criteria are intended to better utilize laboratory strength data in actual case stud-
ies. It is obvious that the loading conditions common to laboratory tests are not very
indicative of rock failure in cases of practical importance (such as wellbore stability).
However, while it is possible in principle to utilize relatively complex failure criteria,
it is often impractical to do so because core is so rarely available for comprehensive
laboratory testing (most particularly in overburden rocks where many wellbore sta-
bility problems are encountered). Moreover, because the stresses acting in the earth
at depth are strongly concentrated around wellbores (as discussed in Chapter 6), it is
usually more important to estimate the magnitudes of in situ stresses correctly than to
have a precise value of rock strength (which would require exhuming core samples for
extensive rock strength tests) in order to address practical problems (as demonstrated
in Chapter 10).
In this section, we will consider five different criteria that have been proposed to
describe the value of the maximum stress, σ 1 ,at the point of rock failure as a function
of the other two principal stresses, σ 2 and σ 3 .Two commonly used rock strength
criteria (the Mohr–Coulomb and the Hoek–Brown criteria), ignore the influence of the
intermediate principal stress, σ 2 , and are thus derivable from conventional triaxial test
data (σ 1 >σ 2 = σ 3 ). We also consider three true triaxial, or polyaxial criteria (modified
Wiebols–Cook, modified Lade, and Drucker–Prager), which consider the influence of
the intermediate principal stress in polyaxial strength tests (σ 1 >σ 2 >σ 3 ). We illustrate
below how well these criteria describe the strength of five rocks: amphibolite from the
KTB site, Dunham dolomite, Solenhofen limestone, Shirahama sandstone and Yuubari
shale as discussed in more detail by Colmenares and Zoback (2002).

Linearized Mohr–Coulomb

The linearized form of the Mohr failure criterion may be generally written as

σ 1 = C 0 + qσ 3 (4.6)
where C 0 is solved-for as a fitting parameter,

2

2 1/2 2
q = µ + 1 = tan (π/4 + φ/2) (4.7)
i + µ i
and
φ = tan −1 (µ i ) (4.8)

This failure criterion assumes that the intermediate principal stress has no influence on
failure.
As viewed in σ 1 , σ 2 , σ 3 space, the yield surface of the linearized Mohr–Coulomb
criterion is a right hexagonal pyramid equally inclined to the principal stress axes. The
intersection of this yield surface with the π-plane is a hexagon. The π-plane is the plane