Page 92 - Reservoir Geomechanics

76 Reservoir geomechanics

such that the material seems to be both stiffer and stronger when deformed at higher
rates.
The type of behavior schematically illustrated in Figure 3.10ais shown for
Wilmington sand in Figure 3.8a,b and the type of behavior schematically illustrated
in Figure 3.10b (stress relaxation at constant strain) is shown for Wilmington sand in
Figure 3.11a. A sample was loaded hydrostatically to 3 MPa before an additional axial
stress of 27 MPa was applied to the sample in a conventional triaxial apparatus (see
Chapter 4). After loading, the length of the sample (the axial strain) was kept constant.
Note that as a result of creep, the axial stress relaxed from 30 MPa to 10 MPa over a
period of ∼10 hours. An implication of this behavior for unconsolidated sand reservoirs
in situ is that very small differences between principal stresses are likely to exist. Even
in an area of active tectonic activity, applied horizontal forces will dissipate due to creep
in unconsolidated formations.
The type of viscous behavior is illustrated schematically in Figure 3.10d, and the rate
dependence of the stress–strain behavior is illustrated in Figure 3.11b for Wilmington
sand (after Hagin and Zoback 2004b). As expected, the sample is stiffer at a confining
pressure of 50 MPa than it is at 15 MPa and at each confining pressure, the samples are
−1
−1
stiffer and stronger at a strain rate of 10 −5 sec , than at 10 −7 sec .
The dispersive behavior illustrated in Figure 3.10c can be seen for dry Wilmington
sand in Figure 3.11c (Hagin and Zoback 2004b). The data shown comes from a test
run at 22.5 MPa hydrostatic pressure with a 5 MPa pressure oscillation. Note the
dramaticdependenceofthenormalizedbulkmoduluswithfrequency.Atthefrequencies
4
of seismic waves (10–100 Hz) and higher sonic logging frequencies of 10 Hz and
6
ultrasonic lab frequencies of ∼10 Hz, a constant stiffness is observed. However, when
deformed at very low frequencies (especially at <10 −3 Hz), the stiffness is dramatically
lower. Had there been fluids present in the sample, the bulk modulus at ultrasonic
frequencies would have been even higher than that at seismic frequencies. The bulk
modulus increases to the Gassmann static limit at approximately 0.1 Hz and then
stays constant as frequency is increased through 1 MHz. The Gassmann static limit is
explained by Mavko, Mukerjii et al.(1998). While our experiments were conducted on
dry samples, we have included the effects of poroelasticity in this diagram by including
the predicted behavior of oil-saturated samples according to SQRT theory (Dvorkin,
Mavko et al. 1995).
Because viscous deformation manifests itself in many ways, and because it is impor-
tant to be able to predict the behavior of an unconsolidated reservoir sand over decades
of depletion utilizing laboratory measurements made over periods of hours to days,
it would be extremely useful to have a constitutive law that accurately describes the
long-term formation behavior. Hagin and Zoback (2004c) discuss a variety of idealized
viscous constitutive laws in terms of their respective creep responses at constant stress,
the modulus dispersion and attenuation. The types of idealized models they considered