Page 97 - Reservoir Geomechanics

81 Basic constitutive laws

where the second term describes creep compaction normalized by the pressure and the
first term describes the instantaneous compaction:

φ i = φ 0 P c d (3.16)

which leaves three unknown constants, A, b, and d where φ 0 is the initial porosity.
Apart from mathematical simplicity, this model holds other advantages over the
model proposed by Hagin and Zoback (2004c). While the previous model required
data from an extensive set of laboratory experiments in order to determine all of the
unknown parameters, the proposed model requires data from only two experiments.
The instantaneous parameters can be solved for by conducting a single constant strain-
rate test, while the viscous parameters can be derived from a single creep strain test
conducted at a pressure that exceeds the maximum in situ stress in the field (Hagin and
Zoback 2007).
Predicting the long-term compaction of the reservoir from which the samples were
taken can now be accomplished using the following equation:

φ(P c , t) = 0.272P −0.046 − P c t 0.164 (3.17)
c
5410
The first term of equation (3.17) represents the instantaneous porosity as a function
of effective pressure, with φ 0 equal to 0.27107 and the parameter d equal to −0.046.
The second term describes creep compaction normalized by effective pressure, with
the parameters A equal to 5410 and b equal to 0.164.
Hagin and Zoback (2007) tested an uncemented sand from the Gulf of Mexico using
experimental conditions very similar to those for the Wilmington sample, except that
in this case the effective pressure was increased to 30 MPa to reflect the maximum in
situ effective stress in the reservoir. They found that the GOM sand constant strain-rate
data could be fit with the following function:

−0.1518
φ i = 0.2456P c (3.18)
where φ i is the instantaneous porosity, P c is effective pressure, and the intercept of the
equation is taken to be the initial porosity (measured gravimetrically).
The creep compaction experimental procedure used to determine the time-dependent
model parameters for the GOM sand sample was also similar to that used for the
Wilmington sand. Hagin and Zoback (2007) found that the creep compaction data are
described quite well by the following equation:

φ(P c , t) = 0.0045105t 0.2318 (3.19)
where φ is the porosity, P c is the effective pressure, and t is the time in days. For details
on how to determine the appropriate length of time for observing creep compaction,
see Hagin and Zoback (2007).