Page 88 - Reservoir Geomechanics

72 Reservoir geomechanics

in velocity) associated with the amount of pressure “carried” by pore fluids as seismic
waves pass through rock. At very high (ultrasonic) frequencies, there is insufficient
time for localized fluid flow to dissipate local pressure increases. Hence, the rock
appears quite stiff (corresponding to the undrained modulus and fast ultrasonic P-wave
velocities as measured in the lab) because the pore fluid pressure is contributing to the
stiffness of the rock. Conversely, at relatively low (seismic or well logging) frequencies,
the rock deforms with a “drained” modulus. Hence, the rock is relatively compliant
(relatively slow P-wave velocities would be measured in situ). It is intuitively clear why
the permeability of the rock and the viscosity of the fluid affect the transition frequency
from drained to undrained behavior. This is illustrated in Figure 3.6b. SQRT theory
predicts the observed dispersion for a viscosity of 1 cp, which is appropriate for the
water filling the pores of this rock. Had there been a more viscous fluid in the pores
(or if the permeability of the rock was lower), the transition frequency would shift to
lower frequencies, potentially affecting velocities measured with sonic logging tools
4
(∼10 Hz). This type of phenomenon, along with related issues of the effect of pore
fluid on seismic velocity (the so-called fluid substitution effect), are discussed at length
by Mavko, Mukerjii et al.(1998), Bourbie, Coussy et al.(1987) and other authors.

Viscous deformation in uncemented sands

Although cemented sedimentary rocks tend to behave elastically over a range of applied
stresses (depending on their strength), uncemented sands and immature shales tend to
behave viscously. In other words, they deform in a time-dependent manner, or creep,
as illustrated schematically in Figure 3.1d. Such behavior has been described in several
studies (e.g. de Waal and Smits 1988; Dudley, Meyers et al. 1994; Ostermeier 1995;
Tutuncu, Podio et al. 1998a;Tutuncu, Podio et al. 1998b). In this section we will
concentrate on the behavior of the uncemented turbidite sand from a reservoir at a depth
of about 1 km in the Wilmington field in Long Beach, California. Chang, Moos et al.
(1997) discussed viscoelastic deformation of the Wilmington sand utilizing a laboratory
test illustrated in Figure 3.8a (from Hagin and Zoback 2004c). Dry samples (traces of
residual hydrocarbons were removed prior to testing) were subjected to discrete steps of
hydrostatic confining pressure. After several pressure steps were applied, loading was
stopped and the sample was allowed to creep. After this creeping period, the sample was
partially unloaded, then reloaded to a higher pressure level and allowed to creep again.
Note that after each loading step, the sample continued to strain at constant confining
pressure and there was appreciable permanent deformation at the conclusion of an
experiment. One point of note that will be important when we consider viscoplastic
compaction of reservoirs in Chapter 12 is that viscous effects are not seen until the
pressure exceeds the highest pressure previously experienced by a sample. In other
words, viscous compaction will only be important when depletion results in an effective