Page 135 - Reservoir Geomechanics

118 Reservoir geomechanics

velocity, gamma ray and density. The coefficient of internal friction was determined
using the gamma relation, equation (28) in Table 4.4. Although this interval is comprised
of almost 100% shale, the value of µ i obtained using equation (28) ranges between 0.7
and 0.84. Using the velocity data, the UCS was determined using equations (11) and
(12) (Table 4.2, Figure 4.17a, b). While the overall shape of the two strength logs is
approximately the same (as both are derived from the V p data), the mean vertically aver-
aged strength derived using equations (11) is 1484 ± 233 psi (Figure 4.18a) whereas
that derived with equations (12) has a strength of 1053 ± 182 psi (Figure 4.18b). Poros-
3
ity was derived from the density log assuming a matrix density of 2.65 g/cm and a fluid
3
density of 1.1 g/cm . The porosity-derived UCS shown in Figure 4.17c with equation
(18) indicates an overall strength of 1878 ± 191 psi (Figure 4.18c). It is noteworthy
in this single example that there is an almost factor of 2 variation in mean strength.
However, as equation (12) was derived for the Gulf of Mexico region, it is probably
more representative of actual rock strength at depth as it is was derived for formations
of that particular region.

Shear-enhanced compaction

Another form of compressional rock failure of particular interest in porous rocks is
sometimes referred to as shear-enhanced compaction. It refers to the fact that there will
be irreversible deformation (i.e. plasticity) characterized by the loss of porosity due
to pore collapse as confining pressure and/or shear stress increases beyond a limiting
value. To represent these ductile yielding behaviors of rocks, end-caps (or yield surfaces
of constant porosity) are used. These end-caps represent the locus of points that have
reached the same volumetric plastic strain and their position (and exact shape) depends
on the properties of the specific rock being considered.
A theoretical formalism known as the Cambridge Clay (or Cam-Clay) model is
useful for describing much laboratory end-cap data (Desai and Siriwardane 1984). In
this case, failure envelopes are determined by relatively simple laboratory experiments
and are commonly represented in p–q space where p is the mean effective stress and q
is the differential stress. Mathematically, the three principal stresses and p–q space are
related as follows:
1 1
p = J 1 = (σ 1 + σ 2 + σ 3 )
3 3
(4.35)
1
p = (S 1 + S 2 + S 3 ) − P P
3
√
q = 3J 2D
1 (4.36)
2 2 2 2
q = [(S 1 − S 2 ) + (S 2 − S 3 ) +(S 1 − S 3 ) ]
2