As alluded to above, many geomechanical problems associated with drilling must be addressed when core samples are unavailable for laboratory testing. In fact, core samples
a .
b.
60
30 90
350
250
150
0 200
100 300
50
3 5
5 2 0 200
100
0 300
400 200
100
0 300
Normal stress (MPa) Shear stress (MPa) Strength, s1 (MPa)
M u - B i G n e i s s
s
0c
0m
ib
b
b a.
b.
Figure 4.13. Fit of compressive strength tests to the theory illustrated in Figure4.12and defined by equation (4.33). Modified from Vernik, Lockner et al. (1992).
of overburden formations (where many wellbore instability problems are encountered) are almost never available for testing. To address this, numerous relations have been proposed that relate rock strength to parameters measurable with geophysical well logs.
The basis for these relations is the fact that many of the same factors that affect rock strength also affect elastic moduli and other parameters, such as porosity.
Nearly all proposed formulae for determination of rock strength from geophysical logs utilize either:
r P-wave velocity, Vp, or equivalently, the travel time of compressional waves along the wellbore wall,t (t=Vp−1), expressed as slowness, typicallyµs/ft,
r Young’s modulus, E (usually derived from Vpand density data as illustrated in Table 3.1), or
r Porosity,φ(or density) data.
The justification for proposed relations is illustrated by the dependence of uniaxial compressive strength on these parameters as shown in Figures 4.14, 4.15 and 4.16 for sandstones, shales, and limestone and dolomite, respectively utilizing data from Lama and Vutukuri (1978), Carmichael (1982), Jizba (1991), Wong, David et al. (1997), Horsrud (2001) and Kwasniewski (1989). In each of the figures, the origin of the laboratory data is indicated by the symbol used. Despite the considerable scatter in the data, for each rock type, there is marked increase of strength with Vpand E and a marked decrease in strength with increased porosity.
Table 4.1 presents a number of relations for different sandstones from different geo- logical settings for predicting rock strength from log data studied (Chang, Zoback et al. 2006). Equations (1)–(3) use P-wave velocity, Vp (or equivalently ast) mea- surements obtained from well logs. Equations (5)–(7) utilize both density and Vpdata, and equation (4) utilizes Vp, density, Poisson’s ratio (requiring Vsmeasurements) and clay volume (from gamma ray logs). Equation (8) utilizes Young’s modulus, E, derived from Vpand Vs, and equations (9) and (10) utilize log-derived porosity measurements to estimate rock strength. Because of the considerable scatter in Figure 4.14, it is obvious that it would be impossible for any of the relations in Table 4.1 to fit all of the data shown. It also needs to be remembered that P-wave velocity in the lab is measured at ultrasonic frequencies (typically∼1 MHz) in a direction that is frequently orthogonal to bedding and typically on the most intact samples available that may not be representa- tive of weaker rocks responsibility for wellbore failure. As discussed in Chapter 3, sonic velocities used in geophysical well logs operate at much lower frequencies. Moreover, such measurements are made parallel to the wellbore axis, which frequently is not per- pendicular to bedding. Hence, there are significant experimental differences between field measurements and laboratory calibrations that need to be taken into consideration.
For the relations that are based on Vp(Figure 4.14a), it is noteworthy that except for equations (1) and (6) (derived for relatively strong rocks), all of the relations predict extremely low strengths for very slow velocities, or high travel times (t≥100µs/ft), and appear to badly underpredict the data. Such velocities are characteristic of weak sands such as found in the Gulf of Mexico, and a number of the relations in Table 4.1 were derived for these data. However, while equation (6) appears to fit the reported values better than the other equations, one needs to keep in mind that there are essen- tially no very weak GOM sands represented in the strength data available in the three studies represented in this compilation. Similarly, for fast, high-strength rocks, equation
0 100 200 300
50 100 150 200 250
Lama and Vutukuri (1978) Carmichael (1982) Jizba (1991)
UCS (MPa)
3
1 2
4
5 6
7
S A N D S TO N E
b.
c . 0 100 200 300 400
0 10 20 30 40 50 60 70 80
Lama and Vutukuri (1978) t (µs/ft)
Carmichael (1982) Jizba (1991) Wong et al. (1997)
UCS (MPa)
E (GPa)
8 S A N D S TO N E
0 100 200 300 400
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Lama and Vutukuri (1978) Jizba (1991)
Kwasniewski (1989) Wong et al. (1997)
UCS (MPa)
f 9
1 0
S A N D S TO N E
Figure 4.14. Comparison between different empirical laws (Table4.1) for the dependence of the strength of sandstones on (a) compressional wave velocity, (b) Young’s modulus and (c) porosity.
After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier.
0 50 100 150 200
0 100 200 300 400 500 600
Lama and Vutukuri (1978) Jizba (1991)
Carmichael (1982)
UCS (MPa)
∆t (µs/ft)
1 2 1 3
1 4 11 1 5
0 100 200 300
0 20 40 60 80 100
Lama and Vutukuri (1978) Carmichael (1982) Jizba (1991)
E (GPa) 1 6
0 50 100 150 200
0 0.1 0.2 0.3 0.4 0.5
Lama and Vutukuri (1978) Jizba (1991)
f 1 7 1 8
1 9
S H A L E
c . b.
S H A L E
UCS (MPa)UCS (MPa)
S H A L E
Figure 4.15. Comparison between different empirical laws (Table4.2) for the dependence of the strength of shale on (a) compressional wave velocity, (b) Young’s modulus and (c) porosity. After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier.
0 50 100 150 200 250 300
40 60 80 100 120 140 160 180 200
Lama and Vutukuri (1978) Carmichael (1982)
UCS (MPa)
∆t (µs/ft) 2 1
2 0
0 50 100 150 200 250 300
0 20 40 60 80 100
Lama and Vutukuri (1978) Carmichael (1982)
E (GPa)
2 2 2 3
0 50 100 150 200 250 300
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Lama and Vutukuri (1978)
f 2 4
2 6 2 5
L I M E S TO N E & D O L O M I T E
b.
c .
L I M E S TO N E & D O L O M I T E
L I M E S TO N E & D O L O M I T E
UCS (MPa)UCS (MPa)
Figure 4.16. Comparison between different empirical laws (Table4.3) for the dependence of the strength of carbonate rocks on (a) compressional wave velocity, (b) Young’s modulus and (c) porosity. After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier.
Equation
No. UCS, MPa Region where developed General comments Reference
1 0.035 Vp−31.5 Thuringia, Germany – (Freyburg 1972)
2 1200 exp(−0.036t) Bowen Basin, Australia Fine grained, both consolidated and unconsolidated sandstones with wide porosity range
(McNally 1987)
3 1.4138×107t−3 Gulf Coast Weak and unconsolidated sandstones Unpublished
4 3.3×10−20ρ2Vp2[(1+ν)/(1−ν)]2(1−2ν) [1+0.78Vclay]
Gulf Coast Applicable to sandstones with UCS
>30 MPa
(Fjaer, Holt et al. 1992)
5 1.745×10−9ρVp2−21 Cook Inlet, Alaska Coarse grained sands and
conglomerates
(Moos, Zoback et al. 1999)
6 42.1 exp(1.9×10−11ρVp2) Australia Consolidated sandstones with 0.05
< φ <0.12 and UCS>80MPa
Unpublished
7 3.87 exp(1.14×10−10ρVp2) Gulf of Mexico – Unpublished
8 46.2 exp(0.000027E) – – Unpublished
9 A (1−Bφ)2 Sedimentary basins
worldwide
Very clean, well consolidated sandstones withφ <0.30
(Vernik, Bruno et al. 1993)
10 277 exp(−10φ) – Sandstones with 2<UCS<
360 MPa and 0.002< φ <0.33
Unpublished
Units used: Vp(m/s),t (µs/ft),ρ(kg/m3), Vclay(fraction), E (MPa),φ(fraction)
Table 4.2. Empirical relationships between UCS and other physical properties in shale.
After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier
UCS, MPa
Region where
developed General comments Reference
11 0.77 (304.8/t)2.93 North Sea Mostly high porosity Tertiary shales
(Horsrud2001) 12 0.43 (304.8/t)3.2 Gulf of Mexico Pliocene and younger Unpublished
13 1.35 (304.8/t)2.6 Globally – Unpublished
14 0.5 (304.8/t)3 Gulf of Mexico – Unpublished
15 10 (304.8/t−1) North Sea Mostly high porosity Tertiary shales
(Lal1999)
16 0.0528E0.712 – Strong and compacted shales Unpublished
17 1.001φ−1.143 – Low porosity (φ <0.1), high strength shales
(Lashkaripour and Dusseault1993) 18 2.922φ−0.96 North Sea Mostly high porosity
Tertiary shales
(Horsrud2001)
19 0.286φ−1.762 – High porosity (φ >0.27)
shales
Unpublished
Units used:t (µs/ft), E (MPa),φ(fraction)
(3) (derived for low-strength rocks) does a particularly poor job of fitting the data. The single equation derived using Young’s modulus, (8), fits the available data reasonably well, but there is considerable scatter at any given value of E. Both of the porosity relations in Table 4.1 seem to generally overestimate strength, except for the very lowest porosities.
Overall, it is reasonable to conclude that none of the equations in Table 4.1 seem to do a very good job of fitting the data in Figure 4.14. That said, it is important to keep in mind that the validity of any of these relations is best judged in terms of how well it would work for the rocks for which it was originally derived. Thus, calibration is extremely important before utilizing any of the relations shown. Equation (5), for example, seems to systematically underpredict all the data in Figure 4.14a, yet worked very well for the clean, coarse-grained sands and conglomerates for which it was derived (Moos, Zoback et al. 1999). It is also important to emphasize that relations that accurately capture the lower bound of the strength data (such as equations 2–5”) can be used to take a conservative approach toward wellbore stability. While the strength may be larger than predicted (and thus the wellbore more stable) it is not likely to be lower.
Considering now the empirical relations describing the strength of shales (Table 4.2), equations (11)–(15) seem to provide a lower bound for the data in Figure 4.15a. While it might be prudent to underestimate strength, the difference between these relations and the measured strengths is quite marked, as much as 50 MPa for high-velocity rocks (t<150µs/ft). For low-velocity rocks (t>200µs/ft), the relations under-predict strengths by 10–20 MPa. However, the porosity relations (equations 17–19) seem to
Table 4.3. Empirical relationships between UCS and other physical properties in limestone and dolomite. After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier
UCS, MPa
Region where
developed General comments Reference
20 (7682/t)1.82/145 – – (Militzer1973)
21 10(2.44+109.14/(t)/145 – – (Golubev and
Rabinovich1976)
22 0.4067 E0.51 – Limestone with 10<UCS<
300 MPa
Unpublished
23 2.4 E0.34 – Dolomite with 60<UCS<100
MPa
Unpublished 24 C (1−Dφ)2 Korobcheyev
deposit, Russia
C is reference strength for zero porosity (250<C<300 MPa). D ranges between 2 and 5 depending on pore shape
(Rzhevsky and Novick1971)
25 143.8 exp(−6.95φ) Middle East Low to moderate porosity (0.05
< φ <0.2) and high UCS (30
<UCS<150 MPa)
Unpublished
26 135.9 exp(−4.8φ) – Representing low to moderate
porosity (0< φ <0.2) and high UCS (10<UCS<300 MPa)
Unpublished
Units used:t (µs/ft), E (MPa),φ(fraction)
fit the available data for shales quite well, especially high-porosity Tertiary shales. It should be noted that in the context of these equations, porosity is defined as the effective porosity that one would derive from well logs.
Another type of correlation is sometimes useful for predicting shale strength is its dependence on shaliness. This arises in a case study considered in Chapter 10, where the stability of wells drilled through shales is highly variable because of strong variations of rock strength. In very relatively low gamma shales (<80 API), the UCS is quite high (˜125 MPa) whereas in shalier formations (<100 API) the UCS is only about 50 MPa.
Empirical relations relating the strength of carbonate rocks to geophysical parameters are presented in Table 4.3 and do a fairly poor job whether considering velocity, modulus or porosity data (Figure 4.16). One of the reasons for this is that there is a very large variation in strength of any given rock type. For example, strong carbonate rocks of low porosity, high velocity and high stiffness show strength values that vary by almost a factor of 4. The same is true for high porosity and low velocity, very large fluctuations in observed strength are observed. All of this emphasizes the importance of being able to calibrate strength relations in any particular case.
There have been relatively few attempts to find relationships between the angle of internal friction, and geophysical measurements, in part because of the fact that even weak rocks have relatively high(Figure 4.4), and there are relatively complex
Table 4.4. Empirical relationships betweenand other logged measurements. After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier
degree General comments Reference
27 sin−1((Vp−1000)/(Vp+1000)) Applicable to shale (Lal1999) 28 70−0.417GR Applicable to shaly sedimentary rocks
with 60<GR<120
Unpublished
Units used: Vp(m/s), GR (API)
10000 9800 9600 9400 9200 9000 8800 8600 8400 8200 8000
TVD below mud line (feet)
UCS (psi) UCS (psi) UCS (psi) mi
0 1000 2000 0 1000 2000 0 1000 2000 0 0.5 1
a . b. c . d .
Figure 4.17. Utilization of equations (11) (a), (12) (b) and (19) (c) from Table4.2, to predict rock strength for a shale section of a well drilled in the Gulf of Mexico. (d) The coefficient of internal friction is from equation (28) in Table4.4. After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier.
relationships betweenand micro-mechanical features of rock such as a rock’s stiff- ness, which largely depends on cementation and porosity. Nonetheless, some exper- imental evidence shows that shale with higher Young’s modulus generally tends to possess a higher(Lama and Vutukuri 1978). Two relationships relating to rock properties for shale and shaley sedimentary rocks are listed in Table 4.4. It is relatively straightforward to show that the importance ofin wellbore stability analysis is much less significant than UCS.
An example illustrating how rock strength is determined from geophysical logs using three of the empirical relations in Table 4.2 is illustrated in Figures 4.17 and 4.18 for a shale section in a vertical well in the Gulf of Mexico. We focus on the interval from 8000 to 10,000 ft where there are logging data available that include compressional wave
0 1000 2000 3000 4000 5000 0
100 200 300 400 500
UCS (psi)
Frequency (counts)
M E A N = 1 4 8 4 p s i
0 1000 2000 3000 4000 5000
0 100 200 300 400 500 600 700
UCS (psi)
Frequency (counts)
M E A N = 1 0 5 3 p s i
0 1000 2000 3000 4000 5000
0 500 1000 1500
UCS (psi)
Frequency (counts)
M E A N = 1 8 7 8 p s i c .
b.
Figure 4.18. Histogram of shale strengths for the log-derived values shown in Figure4.17: (a), (b) and (c) correspond to equations (11), (12) and (18) in Table4.2, respectively. Note that the mean strength varies considerably, depending on which empirical relation is chosen. After Chang, Zoback et al. (2006). Reprinted with permission of Elsevier.
velocity, gamma ray and density. The coefficient of internal friction was determined using the gamma relation, equation (28) in Table 4.4. Although this interval is comprised of almost 100% shale, the value ofµiobtained using equation (28) ranges between 0.7 and 0.84. Using the velocity data, the UCS was determined using equations (11) and (12) (Table 4.2, Figure 4.17a, b). While the overall shape of the two strength logs is approximately the same (as both are derived from the Vpdata), the mean vertically aver- aged strength derived using equations (11) is 1484±233 psi (Figure 4.18a) whereas that derived with equations (12) has a strength of 1053±182 psi (Figure 4.18b). Poros- ity was derived from the density log assuming a matrix density of 2.65 g/cm3and a fluid density of 1.1 g/cm3. The porosity-derived UCS shown in Figure 4.17c with equation (18) indicates an overall strength of 1878±191 psi (Figure 4.18c). It is noteworthy in this single example that there is an almost factor of 2 variation in mean strength.
However, as equation (12) was derived for the Gulf of Mexico region, it is probably more representative of actual rock strength at depth as it is was derived for formations of that particular region.