Page 115 - Reservoir Geomechanics

99 Rock failure in compression, tension and shear

results in a curved (parabolic) Mohr envelope, similar to that shown in Figures 4.2b
and 4.3a. The Hoek–Brown failure criterion was originally developed for estimating
the strength of rock masses for application to excavation design.
According to Hoek and Brown (1997), m depends on rock type and s depends on the
characteristics of the rock mass such that:
5 < m < 8: carbonate rocks with well-developed crystal cleavage (dolomite, lime-

stone, marble).
4 < m < 10: lithified argillaceous rocks (mudstone, siltstone, shale, slate).

15 < m < 24: arenaceous rocks with strong crystals and poorly developed crystal

cleavage (sandstone, quartzite).
16<m<19:fine-grainedpolyminerallicigneouscrystallinerocks(andesite,dolerite,

diabase, rhyolite).
22 < m < 33: coarse-grained polyminerallic igneous and metamorphic rocks (amphi-

bolite, gabbro, gneiss, granite, norite, quartz-diorite).
Whilethesevaluesofmobtainedfromlabtestsonintactrockareintendedtorepresent
a good estimate when laboratory tests are not available, we will compare them with the
values obtained for the five rocks studied. For intact rock materials, s is equal to one.
Fora completely granulated specimen or a rock aggregate, s is equal to zero.
Figure 4.6 shows that the intersection of the Hoek–Brown yield surface with the
π-plane is approximately a hexagon. The sides of the Hoek–Brown failure cone are
not planar, as is the case for the Mohr–Coulomb criterion but, in the example shown,
the curvature is so small that the sides look like straight lines. Figure 4.7b shows this
criterion in σ 1 −σ 2 space for C 0 = 60 MPa, m = 16 and s = 1. The Hoek–Brown criterion
is independent of σ 2 , like the Mohr–Coulomb criterion. One practical disadvantage of
the Hoek–Brown criterion, discussed later, is that correlations are not readily available
in the published literature to relate m to commonly measured with geophysical well
logs, nor are there relationships to relate m to the more commonly defined angle of
internal friction.

Modified Lade criterion
The Lade criterion (Lade 1977)isa three-dimensional failure criterion that was orig-
inally developed for frictional materials without effective cohesion (such as granular
soils). It was developed for soils with curved failure envelopes. This criterion is given by

I I 1

3
m
1
− 27 = η 1 (4.10)
I 3 p a
where I 1 and I 3 are the first and third invariants of the stress tensor
I 1 = S 1 + S 2 + S 3 (4.11)
(4.12)
I 3 = S 1 · S 2 · S 3