Page 96 - Reservoir Geomechanics

80 Reservoir geomechanics

of the high-frequency limit) matches the compaction observed in the field. Assuming
complete depletion of the producing reservoir over its ∼30 year history (prior to water
flooding and pressure support) results in a predicted total vertical compaction of 1.5%.
This value matches closely with well-casing shortening data from the reservoir, which
indicates a total vertical compaction of 2% (Kosloff and Scott 1980).
The power-law model that fits the viscous deformation data best has the form

P c 0.1
ε(P c , t) = ε 0 1 + t (3.12)
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where ε 0 is the instantaneous volumetric strain, P c is the confining pressure, and t is the
time in hours. In order to complete the constitutive law, we need to combine our model
for the time-dependent deformation with a model for the instantaneous deformation.
Hagin and Zoback (2004) show that the instantaneous volumetric strain is also a power-
law function of confining pressure, and can be described empirically with the following
equation:

ε 0 = 0.0083P 0.54 (3.13)
c
Combining the two equations results in a constitutive equation for Wilmington sand in
which the volumetric strain depends on both pressure and time:

ε(P c , t) = 0.0083P 0.54 1 + P c 0.1 (3.14)
t
c
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Hence, this is a dual power-law constitutive law. Strain is a function of both confining
pressure raised to an empirically determined exponent (0.54 for Wilmington sand) and
time raised to another empirically determined exponent (0.1 for Wilmington sand).
Other workers have also reached the conclusion that a power-law constitutive law best
describes the deformation behavior of these types of materials (de Waal and Smits
1988; Dudley, Meyers et al. 1994).
By ignoring the quasi-static and higher frequency data, these dual power-law consti-
tutive laws can be simplified, and the terms in the model needed to model the quasi-static
data in Figure 3.13a can be eliminated (Hagin and Zoback 2007). In fact, by focusing
on long-term depletion, the seven-parameter best-fitting model for Wilmington sand
proposed by Hagin and Zoback (2004c) can be simplified to a three-parameter model
without any loss of accuracy when considering long-term effects.
The model assumes that the total deformation of unconsolidated sands can be decou-
pled in terms of time. Thus, the instantaneous (time-independent) elastic–plastic com-
ponent of deformation is described by a power-law function of pressure, and the viscous
(time-dependent) component is described by a power-law function of time. The pro-
posed model has the following form (written in terms of porosity for simplicity):

φ(P c , t) = φ i − (P c /A) t b (3.15)