reservoir geomecanics

69 Basic constitutive laws
depend on the simple form of the effective stress law (equation 3.8) because there is no
stiff rock matrix to support externally applied stresses. Other types of effective stress
laws describe the dependence of rock permeability on external “confining” pressure
and internal pore pressure.
Poroelasticity and dispersion
As mentioned above, the stiffness (elastic moduli) of a poroelastic rock is rate dependent.
In regard to seismic wave propagation, this means that P-wave and S-wave velocities
will be frequency dependent. Figure 3.6a illustrates the difference between laboratory
bulk modulus measurements of an uncemented Gulf of Mexico sand determined
statically, and using ultrasonic (∼1 MHz) laboratory velocity measurements. Note that
at low confining pressure, there is about a factor of 2 difference between the moduli
determined the two different ways. As confining pressure increases the difference
increases significantly. Thus, there can be significant differences in velocity (or the
elastic modulus) depending on the frequency of seismic waves. Seismic-wave frequencies
typical of a reflection seismic measurement (∼10–50 Hz) are slower (yield lower
moduli) than sonic logs (typically ∼10 kHz), and sonic logs yield slower velocities than
ultrasonic laboratory measurements (typically ∼1 MHz). As illustrated in Figure 3.6b,
this effect is much more significant for P-wave velocity than S-wave velocity.
Figure 3.7a (after Zimmer 2004) clearly demonstrates the difference between static
and dynamic bulk modulus in an uncemented Gulf of Mexico sand. As shown by
the hydrostatic loading cycles, the static stiffness (corresponding to the slope of the
loading line) is much lower than the dynamic stiffness (indicated by the slope of the
short lines) determined from ultrasonic velocity measurements (see expanded scale in
Figure 3.7b). Upon loading, there is both elastic and inelastic deformation occurring
whereas upon unloading, the initial slope corresponds to mostly elastic deformation.
Hence, the unloading stiffnesses (as illustrated in Figure 3.2) are quite similar to the
dynamically measured stiffnesses during loading.
There are a number of different processes affecting seismic wave propagation that
contribute to the effects shown in Figure 3.6. First, the seismic waves associated with
seismic reflection profiling, well logging and laboratory studies sample very different
volumes of rock. Second, when comparing static measurements with ultrasonic
measurements, it is important to remember that the amount of strain to which the samples
are subjected is markedly different, which can affect the measurement stiffness
(Tutuncu, Podio et al. 1998a,b). Finally, pore fluid effects can contribute dramatically
to dispersion at high frequencies. SQRT (squirt, or local flow) is a theory used to
explain the dependence of wave velocity on frequency in a saturated poroelastic rock
at high frequency (see the review by Dvorkin, Mavko et al. 1995). Fundamentally,
SQRT (and theories like it) calculates the increase in rock stiffness (hence the increase