Page 119 - Reservoir Geomechanics

103 Rock failure in compression, tension and shear

and
√
3C 0 cos φ
k = √ (4.30)
q (3 − sin φ)
As equations (4.29) and (4.30) show, α only depends on φ which means that α has
an upper bound in both cases; 0.866 in the inscribed Drucker–Prager case and 1.732 in
the circumscribed Drucker–Prager case.
In Figure 4.7ewe show the Drucker–Prager criteria for C 0 = 60 MPa and µ i = 0.6
in comparison with other failure criteria. As shown in Figure 4.7e, for the same values
of C 0 and µ i , the inscribed Drucker–Prager criterion predicts failure at much lower
stresses as a function of σ 2 than the circumscribed Drucker–Prager criterion.
As mentioned above, Colmenares and Zoback (2002) considered these failure cri-
terion for five rock types: amphibolite from the KTB site in Germany (Chang and
Haimson 2000), Dunham dolomite and Solenhofen limestone (Mogi 1971) and Shira-
hama sandstone and Yuubari shale (Takahashi and Koide 1989).
Figure4.8presentsalltheresultsfortheMohr–Coulombcriterionwiththebest-fitting
parameters for each rock type. As the Mohr–Coulomb does not take into account the
influence of σ 2 , the best fit would be the horizontal line that goes through the middle of
the data for each σ 3 . The smallest misfits associated with the Mohr–Coulomb criterion
were obtained for the Shirahama sandstone and the Yuubari shale. The largest misfits
were for Dunham dolomite, Solenhofen limestone and KTB amphibolite, which are
rocks showing the greatest influence of the intermediate principal stress on failure.
The modified Lade criterion (Figure 4.9)works well for the rocks with a high σ 2 -
dependence of failure such as Dunham dolomite and Solenhofen limestone. For the
KTB amphibolite, this criterion reasonably reproduces the trend of the experimental
data but not as well as for the Dunham dolomite. We see a similar result for the Yuubari
shale. The fit to the Shirahama sandstone data does not reproduce the trends of the data
very well.
We now briefly explore the possibility of using triaxial test data to predict the σ 2 -
dependence using the modified Lade failure criterion. The reason for doing this is
to be able to characterize rock strength with relatively simple triaxial tests, but to
allow all three principal stresses to be considered when addressing problems such as
wellbore failure. We utilize only the triaxial test data for Solenhofen limestone (Figure
4.8b) which would not have detected the fact that the strength is moderately dependent
on α 2 .As shown by Colmenares and Zoback (2002), by using only triaxial test data
(shown in Figure 4.10a), we obtain a value of C 0 as a function of α 2 (Figure 4.10b) that
is within ±3% of that obtained had polyaxial test data been collected.
Because the subject of rock strength can appear to be quite complex, it might seem
quite difficult to know how to characterize the strength of a given rock and to utilize
this knowledge effectively. In practice, however, the size of the failure envelope (Figure
4.6)is ultimately more important than its exact shape. When applied to problems of