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 (∼104 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.
5 10 15 20 25 30
00.0050.010.0150.020.0250.030.0350.04
0 10 20 30 40
Confining pressure (MPa) Axial strain (in/in)
Time (hour)
C O N F I N I N G P R E S S U R E I N S TA N TA N E O U S
S T R A I N
C R E E P S T R A I N
0 0.0005 0.001 0.0015 0.002 0.0025
0 5000 1 × 104 1.5 × 104 2 × 104 2.5 × 104 3 × 104
Volumetric strain
Time (seconds)
G O M – L e n t i c Wi l m i n g t o n –
U p p e r Te r m i n a l
O t t awa S a n d – 1 0 % M o n t m o r i l l o n i t e
O t t awa S a n d – 5 % M o n t m o r i l l o n i t e O t t awa S a n d
b.
Figure 3.8. (a) Incremental instantaneous and creep strains corresponding to 5 MPa incremental increases in pressure. The data plotted at each pressure reflect the increases in strain that occurred during each increase in pressure. Note that above 15 MPa the incremental creep strain is the same magnitude as the incremental instantaneous strain from Hagin and Zoback (2004b). (b) Creep strain is dependent upon the presence of clay minerals and mica. In the synthetic samples, the amount of creep is seen to increase with clay content. The clay content of the Wilmington sand samples is approximately 15% (modified from Chang, Moos et al.1997). Reprinted with permission of Elsevier.
0.015
a . b.
I n s t a n t a n e o u s s t ra i n
10 15 20
0.02
5 0.01
0.005
0
C r e e p s t ra i n Axial strain (in/in) for MPa increasing pressure increments
25 30 35
Confining pressure (MPa) Instantaneous vs. time-dependent strain
dry Wilmington sand
10 15 20
5 25 30 35
Confining pressure (MPa) 0.015
0.02
0.01 0.005 0 0.035
0.04
0.03 0.025
Cumulative axial strain (in/in)
Cumulative instantaneous and creep strain dry Wilmington sand
To t a l s t ra i n
C r e e p s t ra i n
Figure 3.9. (a) Incremental instantaneous and creep strains corresponding to 5 MPa incremental increases in pressure. The data plotted at each pressure reflect the increases in strain that occurred during each increase in pressure. Note that above 15 MPa the incremental creep strain is the same magnitude as the incremental instantaneous strain (from Hagin and Zoback2004b). The cumulative instantaneous and total (instantaneous plus creep) volumetric strain as a function of pressure.
Note that above 10 MPa, both increase by the same amount with each increment of pressure application.
stress that is greater than the sample experienced in situ. However, as will be shown below, once the previous highest load experienced has been exceeded, viscoplastic compaction can be quite appreciable. One would dramatically underpredict reservoir compaction from laboratory experiments on uncemented sands if one were to neglect viscoplastic effects.
In an attempt to understand the physical mechanism responsible for the creep in these samples, Figure 3.8b illustrates an experiment by Chang, Moos et al. (1997) that compares the time-dependent strain of Wilmington sand (both dry and saturated) with Ottawa sand, a commercially available laboratory standard that consists of pure, well-rounded quartz grains. Note that in both the dry and saturated Wilmington sand samples, a 5 MPa pressure step at 30 MPa confining pressure results in a creep strain of 2% after 2×104sec (5.5 hours). In the pure Ottawa samples, very little creep strain is observed, but when 5% and 10% montmorillonite was added, respectively, appreciably more creep strain occurred. Thus, the fact that the grains in the sample were uncemented to each other allowed the creep to occur. The presence of montmorillonite clay enhances this behavior. The Wilmington samples are composed of∼20% quartz,∼20% feldspar, 20% crushed metamorphic rocks, 20% mica and 10% clay. Presumably both the clay and mica contribute to the creep in the Wilmington sand.
Stress - strain S t ra i n Creep strain
Stress - strain
Modulus - 1/Q Stress
Strain Frequency
Time Time
S t r e s s
S t ra i n
S t r e s s
m o d u l u s Q−1
Increasing strain rate
Stress relaxation
Modulus dispersion and attenuation
Rate hardening
a . b.
c . d .
Figure 3.10. Time-dependent deformation in a viscoelastic material is most commonly observed as (a) creep strain or (b) stress relaxation. (c) Linear viscoelasticity theory predicts frequency- and rate-dependence for materials that exhibit time-dependence. Specifically, elastic modulus and attenuation (Q−1) should vary with loading frequency, and (d) stiffness should increase with strain rate. From Hagin and Zoback (2004b).
Figure 3.9 (after Hagin and Zoback 2004b) illustrates a set of experiments that illustrate just how important creep strain is in this type of reservoir sand. Note that after the initial loading step to 10 MPa, the creep strain that follows each loading step is comparable in magnitude to the strain that occurs instantaneously (Figure 3.9a). The cumulative strain (Figure 3.10b) demonstrates that the creep strain accumulates linearly with pressure.
Figure 3.10 summarizes four different ways in which viscoelastic deformation man- ifests itself in laboratory testing, and presumably in nature. As already noted, a viscous material strains as a function of time in response to an applied stress (Figure 3.10a), and differential stress relaxes at constant strain (Figure 3.10b). In addition, the elastic moduli are frequency dependent (the seismic velocity of the formation is said to be dispersive) and there is marked inelastic attenuation. Q is defined as the seismic quality factor such that inelastic attenuation is defined as Q−1(Figure 3.10c). Finally, a stress–
strain test (such as illustrated in Figure 3.2) is dependent on strain rate (Figure 3.10d)
such that the material seems to be both stiffer and stronger when deformed at higher rates.
The type of behavior schematically illustrated in Figure 3.10a is 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 stiffer and stronger at a strain rate of 10−5sec−1, than at 10−7sec−1.
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 dramatic dependence of the normalized bulk modulus with frequency. At the frequencies of seismic waves (10–100 Hz) and higher sonic logging frequencies of 104 Hz and ultrasonic lab frequencies of∼106Hz, a constant stiffness is observed. However, when deformed at very low frequencies (especially at<10−3Hz), 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
5 H Y D RO S TAT I C 25
20
15
10
00 5 10 15 20 25
Time (hours)
Differential stress (MPa)
10
S t r e s s r e l a x a t i o n a t c o n s t a n t a x i a l s t ra i n ( e11 = ~ 0 . 7 )
50
40
30
20 60
0
0.016
0 0.032 0.048 0.064 1 2 3 4 5
Axial strain (in/in) Time (hours)
Differential stress (MPa)
P c = 5 0 M Pa
P c = 1 5 M Pa S t ra i n ra t e = 1 0−5/ s
S t ra i n ra t e = 1 0−6/ s S t ra i n ra t e = 1 0−7/ s
b.
c .
0 0.2 0.4 0.6 0.8 1 1.2 1.4
1.00E−05 1.00E−03 1.00E−01 1.00E +01 1.00E +03 1.00E +05 1.00E +07
Frequency (Hz)
We l l - l o g d a t a U l t ra s o n i c d a t a P r e s s u r e - c y c l i n g d a t a
G a s s m a n n s t a t i c l i m i t
B i o t S q u i r t t h e o r y ( o i l s a t u ra t i o n ) L ow - f r e q u e n c y d i s p e r s i o n
S e i s m i c d a t a
S t a t i c " m o d u l u s "
Normalized bulk modulus
Figure 3.11. Experiments on dry Wilmington sand that illustrate different kinds of viscous deformation (modified from Hagin and Zoback2004b). (a) Relaxation of stress after application of a constant strain step (similar to what is shown schematically in Figure3.10b). (b) The strain rate dependence of sample stiffness (as illustrated in Figure3.10d). (c) Normalized bulk modulus of dry Wilmington sand as a function of frequency spanning 10 decades (as shown schematically in Figure3.10c).
Mechanical model Creep response Modulus disperson Attenuation response (at constant stress)
StrainStrainStrainStrainStrain ModulusModulusModulusModulusModulus
Time
Time
Time
Time
Time
Log frequency Log frequency
Log frequency Log frequency
Log frequency
Log frequency
Log frequency
Log frequency
Log frequency
Log frequency Maxwell solid
Voight solid
Standard linear solid
Burber’s solid
Power law E(t) = Eo + Ctn
E1
E2
E1
E2
E2
E1
h1
s/E1
s/E2
s/E1
s/E1 s(E1 + E1)/(E1E2)
s/h1
s/h1 h2
h2
h2 h1
h = a
h = a
h = a
h = a
h = a h = b
h = b
h = b
h = b
h = b h = b h = a a>b
a>b
a>b
a>b
a>b a>b
1/Q1/Q1/Q1/Q1/Q
Figure 3.12. Conceptual relationships between creep, elastic stiffness, and attenuation for different idealized viscoelastic materials. Note that the creep strain curves are all similar functions of time, but the attenuation and elastic stiffness curves vary considerably as functions of frequency. From Hagin and Zoback (2004b).
are illustrated in Figure 3.12. It is important to note that if one were simply trying to fit the creep behavior of an unconsolidated sand such as shown in Figure 3.8b, four of the constitutive laws shown in Figure 3.12 have the same general behavior and could be adjusted to fit the data.
Hagin and Zoback (2004b) independently measured dispersion and attenuation and thus showed that a power-law constitutive law (the last idealized model illustrated in Figure 3.12) appears to be most appropriate. Figure 3.13a shows their dispersion mea- surements for unconsolidated Wilmington sand (shown previously in Figure 3.11c) as fit by three different constitutive laws. All three models fit the dispersion data at inter- mediate frequencies, although the Burger’s model implies zero stiffness under static conditions, which is not physically plausible. Figure 3.13b shows the fit of various con- stitutive laws to the measured attenuation data. Note that attenuation is∼0.1 (Q∼10) over almost three orders of frequency and only the power-law rheology fits the essen- tially constant attenuation over the frequency range measured. More importantly, the power-law constitituve law fits the dispersion data, and its static value (about 40%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.00E−09 1.00E−07 1.00E−05 1.00E−03 1.00E−01 1.00E+01 1.00E+03 1.00E+05 Frequency (Hz)
Data S.L.Solid Burger’s Power/Maxwell M e a n p r e s s u r e = 2 2 . 5 M Pa
P r e s s u r e a m p l i t u d e = 5 M Pa
We l l - l o g d a t a
E x t ra p o l a t e t o 3 0 - ye a r s i mu l a t e d r e s e r vo i r l i fe s p a n
Q u a s i - s t a t i c l o a d c y c l i n g d a t a
1.00E−04 1.00E−03 1.00E−02 1.00E−01 1.00E+00 1.00E+01 1.00E+02
1.00E−06 1.00E−05 1.00E−04 1.00E−03 1.00E−02 1.00E−01
data S.L.Solid Burger’s Power/Maxwell
M e a n p r e s s u r e = 2 7 . 5 M Pa P r e s s u r e a m p l i t u d e = 5 M Pa
Normalized bulk modulus
U l t ra s o n i c d a t a
Frequency (Hz)
Attenuation (1/Q)
a .
b.
Figure 3.13. (a) Low-frequency bulk modulus dispersion predicted using parameters derived from fitting the creep strain curves compared with experimental results. The standard linear solid and Burger’s solid models provide the best fit to the quasi-static data, but fail to predict reasonable results at time-periods associated with the life of a reservoir. The power law–Maxwell mixed model reproduces the overall trend of the data, and predicts a physically reasonable value of stiffness at long time-periods. The error bars represent the inherent uncertainty of the measurements (from Hagin and Zoback (2004c). (b) Low-frequency attenuation predicted using the parameters derived from the creep strain data compared with experimental results.
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 ε(Pc,t)=ε0
1+ Pc
15t0.1
(3.12) whereε0is the instantaneous volumetric strain, Pcis 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:
ε0=0.0083Pc0.54 (3.13)
Combining the two equations results in a constitutive equation for Wilmington sand in which the volumetric strain depends on both pressure and time:
ε(Pc,t)=0.0083Pc0.54
1+ Pc
15t0.1
(3.14) 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(Pc,t )=φi−(Pc/A) tb (3.15)
where the second term describes creep compaction normalized by the pressure and the first term describes the instantaneous compaction:
φi=φ0Pcd (3.16)
which leaves three unknown constants, A, b, and d whereφ0is 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(Pc,t )=0.272Pc−0.046− Pc
5410t0.164
(3.17) The first term of equation (3.17) represents the instantaneous porosity as a function of effective pressure, withφ0 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:
φi=0.2456Pc−0.1518 (3.18)
whereφiis the instantaneous porosity, Pcis 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(Pc,t )=0.0045105t0.2318 (3.19)
whereφis the porosity, Pcis 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).
The total deformation for this Gulf of Mexico sample can now be described in terms of porosity, effective pressure and time by combining equations (3.18) and (3.19) into the following:
φ(Pc,t )=0.2456Pc−0.1518−(Pc/6666.7)t0.2318 (3.20) As before, the first term of equation (3.20) represents the instantaneous component of deformation, withφ0equal to 0.2456 and the parameter d equal to−0.1518. The second term describes the time-dependent component of deformation, with the parameters A equal to 6667 and b equal to 0.2318. Assuming complete drawdown of the producing reservoir and an approximately 30 year history results in a predicted total vertical compaction of nearly 10%.
Table 3.2 summarizes the fitting parameters obtained from the creep strain tests and constant strain rate tests described earlier. The parameters to make note of are the exponent parameters, b and d. The apparent viscosity of a reservoir sand is captured in Table 3.2. Creep parameters for two uncemented sands
Reservoir sand A (creep)
b (creep)
φ0
(instant) d
(instant) Notes
Wilmington 5410.3 0.1644 0.271 −0.046 Stiffer and more viscous
GOM – Field X 6666.7 0.2318 0.246 −0.152 Softer and less viscous
1.2 × 10−5
1.0 × 10−5
8.0 × 10−6
6.0 × 10−6
4.0 × 10−6
2.0 × 10−6 1.0 × 10−7
−
0 20 40 60 80 100
Percent of silica Coefficient of thermal expansion (oC)
Cherts, quartzites Sandstones Granitoid rocks Slates Andesites
Gabbros, basalts, diabase
Figure 3.14. Measurements of the coefficient of linear thermal expansion for a variety of rocks as a function of the percentage of silica (data from Griffith1936). As the coefficient of thermal expansion of silica (∼10−5◦C−1) is an order of magnitude higher than that of most other rock forming minerals (∼10−6◦C−1), the coefficient of thermal expansion ranges between those two amounts, depending on the percentage of silica.
the b parameter, and smaller values of b represent greater viscosities. Thus, Wilmington sand is more viscous than the sand from the Gulf of Mexico. The effective pressure exponent d represents compliance, with smaller values being stiffer. Thus, Wilmington sand is stiffer than the GOM sand. Note that stiffness is not related to d in a linear way, because strain and stress are related via a power law.
We will revisit the subject of viscoplastic compaction in weak sand reservoirs in Chapter 12 and relate this phenomenon to the porosity change accompanying com- paction of the Wilmington reservoir in southern California and a field in the Gulf of Mexico.