S T RO N G RO C K
W E A K RO C K
Effective normal stress Pm > P0
Pm = P0
srr = s3 sqq = s1
Shear stress
Figure 4.5. Schematic illustration of how raising mud weight helps stabilize a wellbore. The Mohr circle is drawn for a point around the wellbore. For weak rocks (low cohesion), when mud weight and pore pressure are equal, the wellbore wall fails in compression as the radial stress,σrris equal to 0. Raising mud weight increasesσrrand decreasesσθθ, the hoop stress acting around the wellbore. This stabilizes the wellbore by reducing the size of the Mohr circle all around the circumference of the well.
the mud weight, Pm, and the pore pressure, P0. The maximum principal stress driving failure of the wellbore wall isσθθ, the hoop stress acting parallel to the wellbore wall in a circumferential manner (Figure 6.1). Note that if the cohesive strength of the rock is quite low, when the mud weight is exactly equal to the pore pressure (i.e. the mud weight is exactly balanced with the pore pressure),σθθdoes not have to be very large to exceed the strength of the rock at the wellbore wall and cause wellbore failure because σrr =0. However, if the mud weight exceeds the pore pressure,σrrincreases andσθθ
decreases, thus resulting in a more stable wellbore. This is discussed more thoroughly in Chapter 6. Of course, drillers learned this lesson empirically a century ago as the use of mud weight to stabilize wellbores is one of a number of considerations which are discussed at some length in Chapter 10.
these criteria are intended to better utilize laboratory strength data in actual case stud- ies. It is obvious that the loading conditions common to laboratory tests are not very indicative of rock failure in cases of practical importance (such as wellbore stability).
However, while it is possible in principle to utilize relatively complex failure criteria, it is often impractical to do so because core is so rarely available for comprehensive laboratory testing (most particularly in overburden rocks where many wellbore sta- bility problems are encountered). Moreover, because the stresses acting in the earth at depth are strongly concentrated around wellbores (as discussed in Chapter 6), it is usually more important to estimate the magnitudes of in situ stresses correctly than to have a precise value of rock strength (which would require exhuming core samples for extensive rock strength tests) in order to address practical problems (as demonstrated in Chapter 10).
In this section, we will consider five different criteria that have been proposed to describe the value of the maximum stress,σ1, at the point of rock failure as a function of the other two principal stresses, σ2 and σ3. Two commonly used rock strength criteria (the Mohr–Coulomb and the Hoek–Brown criteria), ignore the influence of the intermediate principal stress,σ2, and are thus derivable from conventional triaxial test data (σ1 > σ2=σ3). We also consider three true triaxial, or polyaxial criteria (modified Wiebols–Cook, modified Lade, and Drucker–Prager), which consider the influence of the intermediate principal stress in polyaxial strength tests (σ1 > σ2> σ3). We illustrate below how well these criteria describe the strength of five rocks: amphibolite from the KTB site, Dunham dolomite, Solenhofen limestone, Shirahama sandstone and Yuubari shale as discussed in more detail by Colmenares and Zoback (2002).
Linearized Mohr–Coulomb
The linearized form of the Mohr failure criterion may be generally written as
σ1=C0+qσ3 (4.6)
where C0is solved-for as a fitting parameter, q =
µ2i +1 1/2+µi
2
=tan2(π/4+φ/2) (4.7)
and
φ=tan−1(µi) (4.8)
This failure criterion assumes that the intermediate principal stress has no influence on failure.
As viewed inσ1,σ2,σ3 space, the yield surface of the linearized Mohr–Coulomb criterion is a right hexagonal pyramid equally inclined to the principal stress axes. The intersection of this yield surface with theπ-plane is a hexagon. Theπ-plane is the plane
−150 −100 −50 0 50 100 150
−100
−50 0 50 100 150
A X I S H Y D RO S TAT I C I N S C R I B E D
D RU C K E R – P R AG E R
C I R C U M S C R I B E D D RU C K E R – P R AG E R
M O D I F I E D W I E B O L S – C O O K
M O H R – C O U L O M B
H O E K – B ROW N
s2 s1
s3
sy
sx( M Pa )
(MPa)
Figure 4.6. Yield envelopes projected in theπ-plane for the Mohr–Coulomb criterion, the Hoek–Brown criterion, the modified Wiebols–Cook criterion and the circumscribed and inscribed Drucker–Prager criteria. After Colmenares and Zoback (2002). Reprinted with permission of Elsevier.
perpendicular to the straight lineσ1 = σ2 = σ3. Figure 4.6 shows the yield surface of the linearized Mohr–Coulomb criterion is hexagonal in the π-plane. Figure 4.7a shows the representation of the linearized Mohr–Coulomb criterion inσ1−σ2space for C0 = 60 MPa and µi = 0.6. In this figure (and Figures 4.8, 4.9 and 4.10b below), σ1 at failure is shown as a function of σ2 for experiments done at different values ofσ3.
Hoek– Brown criterion
This empirical criterion uses the uniaxial compressive strength of the intact rock mat- erial as a scaling parameter, and introduces two dimensionless strength parameters,
0 50 100 150 200 250 300 350 400 450 500 0
50 100 150 200 250 300 350 400 450 500
Mohr Coulomb criterion
s2 (MPa) s1 (MPa)
C0 = 60 MPa mi = 0.6 s3 = 0 MPa
s3 = 90 MPa
s3 = 60 MPa
s3 = 30 MPa s1 = s2
0 50 100 150 200 250 300 350 400 450 500 0
50 100 150 200 250 300 350 400 450 500
Hoek and Brown criterion
s2 (MPa) s1 (MPa)
m = 16 s = 1 C0 = 60 MPa s3 = 0 MPa
s3 = 90 MPa
s3 = 60 MPa
s3 = 30 MPa s1 =s2
0 50 100 150 200 250 300 350 400 450 500 0
50 100 150 200 250 300 350 400 450 500
Modified Lade criterion
s2 (MPa) s1 (MPa)
C0 = 60 MPa mi = 0.6 s3 = 0 MPa
s3 = 60 MPa
s3 = 30 MPa s1 =s2
0 50 100 150 200 250 300 350 400 450 500 0
50 100 150 200 250 300 350 400 450 500
s2 (MPa) s1 (MPa)
Modified Wiebols and Cook criterion C0 = 60 MPa
mi = 0.6 s3 = 0 MPa
s3 = 90 MPa
s3 = 60 MPa
s3 = 30 MPa s1 =s2
0 50 100 150 200 250 300 350 400 450 500 0
50 100 150 200 250 300 350 400 450 500
s2 (MPa) s1 (MPa)
C0 = 60 MPa mi = 0.6 s3 = 0 MPa
s3 = 0 MPa s3 = 30 MPa
s1 =s2
Circumscribed Drucker–Prager criterion Inscribed Drucker–Prager criterion
a . b .
e .
c . d .
s3 = 30 MPa s3 = 90 MPa
Figure 4.7. To observe how different compressive failure criteria define the importance of the intermediate principal stress,σ2, on rock strength, forσ3=0,30,60 and 90 MPa and C0= 60 MPa andµi=0.6, we show the curves corresponding to (a) linearized Mohr–Coulomb criterion; (b) Hoek–Brown criterion (m=16 and s=1); (c) modified Lade criteria; (d) modified Wiebols–Cook criterion; (e) inscribed and circumscribed Drucker–Prager criteria (shown for only forσ3=0 and 30 MPa for simplicity). After Colmenares and Zoback (2002). Reprinted with permission of Elsevier.
0 500 1000 1500 0
200 400 600 800 1000 1200 1400 1600 1800 2000
KTB amphibolite C0 = 300 MPa mi = 1.2
Mean misfit = 77.9 MPa s3 = 150 s3 = 0 s3 = 60 s3 = 100 s3 = 30 s1 = s2
s3 = 150
s3 = 0
s3 = 60 s3 = 100
s3 = 30 s3 (MPa)
(MPa) (MPa)
0 200 400 600 800 1000 1200
0 250 500 750 1000 1250
s2 (MPa) s1 (MPa)
C0 = 450 MPa mi = 0.65
Mean misfit = 56.0 MPa Dunham dolomite
s3 = 125 s3 = 105 s3 = 25 s3 = 65 s3 = 85 s3 = 45 s1 = s2
s3 = 25 s3 = 45
s3 = 65 s3 = 85
s3 = 105 s3 = 125
s3 = 145
s3
s3 = 145
0 100 200 300 400 500 600 700 800
0 100 200 300 400 500 600 700 800
C0 = 375 MPa = 0.55
Mean misfit = 37.1 MPa
Solenhofen limestone s3 = 20 s3 = 60 s3 = 80 s3 = 40 s1 = s2
s3 (MPa) s3 = 20
s3 = 40 s3 = 60
s3 = 80
0 50 100 150 200 250 300
0 50 100 150 200 250 300 350
C0 = 120 MPa mi = 0.5
Mean misfit = 13.5 MPa Yuubari shal e
s3 = 50 s3 = 25 s3 s1 = s2
s3 = 25
s3 = 50
Mean misfit = 9.6 MPa mi = 0.8
C0 = 95 MPa Shirahama sandstone
s3 = 5
s3 = 40 s3 = 8 s3 = 20 s3 = 30 s3 = 15 s3 = 8
s3 = 15
s3 = 5
s3 = 20
s3 = 40
s3 = 30
s1 = s2
s3 (MPa)
0 50 100 150 200 250 300
0 50 100 150 200 250 300 350
a . b.
e.
c . d .
s1 (MPa)
s2 (MPa)
s1 (MPa)
s2 (MPa) s1 (MPa)
s2 (MPa)
s1 (MPa)
s2 (MPa)
(MPa)
mi
Figure 4.8. Best-fitting solution for all the rocks using the Mohr–Coulomb criterion. (a) Dunham dolomite. (b) Solenhofen limestone. (c) Shirahama sandstone. (d) Yuubari shale. (e) KTB amphibolite. After Colmenares and Zoback (2002). Reprinted with permission of Elsevier.
a . b.
e.
c . d .
0 500 1000 1500
0 250 500 750 1000 1250 1500 1750 2000
KTB amphibolite
s2 (MPa) Mean misfit = 91.3 MPa C0 = 250 MPa mi = 0.85
s3 = 150 s3 = 0 s3 = 60 s3 = 100 s3 = 30 s1 = s2
s3 = 150
s3 = 0
s3 = 60 s3 = 100
s3 = 30 s3 (MPa)
0 200 400 600 800 1000 1200
0 250 500 750 1000 1250
Dunham dolomite
s2 (MPa) s1 (MPa)
C0 = 380 MPa mi = 0.5
Mean misfit = 27.8 MPa
s3 = 125 s3 = 105 s3 = 25 s3 = 65 s3 = 85 s3 = 45 s1 = s2
s3 = 145 s3 = 25
s3 = 45 s3 = 65
s3 = 85 s3 = 105
s3 = 125 s3 = 145
s3 (MPa)
0 100 200 300 400 500 600 700 800
0 100 200 300 400 500 600 700 800
Solenhofen limestone
s2 (MPa) s1 (MPa)
C0 = 335 MPa mi = 0.4
Mean misfit = 23.3 MPa
s3 = 20 s3 = 60 s3 = 80 s3 = 40 s1 = s2
s3 = 20 s3 = 60
s3 = 80 s3 = 40
s3 (MPa)
0 50 100 150 200 250 300
0 50 100 150 200 250 300 350
s2 (MPa) s1 (MPa)
C0 = 110 MPa mi = 0.4
Mean misfit = 13.7 MPa
s3 = 25 s3 = 50 s3 (MPa) s3 = 25
s3 = 50
Yuubari shale s1 = s2
0 50 100 150 200 250 300
0 50 100 150 200 250 300 350
s2 (MPa)
s1 s1
(MPa) (MPa)
s3 = 5
s3 = 40 s3 = 8 s3 = 20 s3 = 30 s3 = 15 s3 (MPa)
Shirahama sandstone C0 = 55 MPa mi = 0.7
Mean misfit = 11.9 MPa s3 = 8
s3 = 15
s3 = 5
s3 = 20
s3 = 40
s3 = 30
s1 = s2
Figure 4.9. Best-fitting solution for all the rocks using the modified Lade criterion; (a) Dunham dolomite; (b) Solenhofen limestone; (c) Shirahama sandstone; (d) Yuubari shale; (e) KTB amphibolite. After Colmenares and Zoback (2002). Reprinted with permission of Elsevier.
a .
b.
0 100 200 300 400 500
0 50 100 150 200 250 300
(MPa)
(MPa) (MPa)
(MPa)
(MPa)
0 100 200 300 400 500 600 700 800
0 100 200 300 400 500 600 700 800
Modified Lade criterion C0 = 325 MPa
Mean misfit = 9.5 MPa Mohr–Coulomb criterion C
mi
mi 0 = 325 MPa = 0.4
t
s
s
s2
s3 s1
s2 s3
s3s3s
3
s3
s3
s3 s3 s3 s3 s3 s3
= 0
= 6
= 1 5
= 2 4
= 4 6
= 7 2
Figure 4.10. Best-fitting solution for the Solenhofen limestone using the triaxial test data;
(a) Mohr–Coulomb criterion; (b) modified Lade criterion. After Colmenares and Zoback (2002).
Reprinted with permission of Elsevier.
m and s. After studying a wide range of experimental data, Hoek and Brown (1980) proposed that the maximum principal stress at failure is given by
σ1 =σ3+C0
mσ3
C0 +s (4.9)
where m and s are constants that depend on the properties of the rock and on the extent to which it had been broken before being tested. Note that this form of the failure law
results in a curved (parabolic) Mohr envelope, similar to that shown in Figures 4.2b and 4.3a. The Hoek–Brown failure criterion was originally developed for estimating the strength of rock masses for application to excavation design.
According to Hoek and Brown (1997), m depends on rock type and s depends on the characteristics of the rock mass such that:
r 5<m<8: carbonate rocks with well-developed crystal cleavage (dolomite, lime- stone, marble).
r 4<m<10: lithified argillaceous rocks (mudstone, siltstone, shale, slate).
r 15<m<24: arenaceous rocks with strong crystals and poorly developed crystal cleavage (sandstone, quartzite).
r 16<m<19: fine-grained polyminerallic igneous crystalline rocks (andesite, dolerite, diabase, rhyolite).
r 22<m<33: coarse-grained polyminerallic igneous and metamorphic rocks (amphi- bolite, gabbro, gneiss, granite, norite, quartz-diorite).
While these values of m obtained from lab tests on intact rock are intended to represent a good estimate when laboratory tests are not available, we will compare them with the values obtained for the five rocks studied. For intact rock materials, s is equal to one.
For a completely granulated specimen or a rock aggregate, s is equal to zero.
Figure 4.6 shows that the intersection of the Hoek–Brown yield surface with the π-plane is approximately a hexagon. The sides of the Hoek–Brown failure cone are not planar, as is the case for the Mohr–Coulomb criterion but, in the example shown, the curvature is so small that the sides look like straight lines. Figure 4.7b shows this criterion inσ1−σ2space for C0=60 MPa, m=16 and s=1. The Hoek–Brown criterion is independent ofσ2, like the Mohr–Coulomb criterion. One practical disadvantage of the Hoek–Brown criterion, discussed later, is that correlations are not readily available in the published literature to relate m to commonly measured with geophysical well logs, nor are there relationships to relate m to the more commonly defined angle of internal friction.
Modified Lade criterion
The Lade criterion (Lade 1977) is a three-dimensional failure criterion that was orig- inally developed for frictional materials without effective cohesion (such as granular soils). It was developed for soils with curved failure envelopes. This criterion is given by
I13 I3
−27 I1 pa
m
=η1 (4.10)
where I1and I3are the first and third invariants of the stress tensor
I1 =S1+S2+S3 (4.11)
I3 =S1·S2·S3 (4.12)
pa is atmospheric pressure expressed in the same units as the stresses, and mandη1
are material constants.
In the modified Lade criterion developed by Ewy (1999) m was set equal to zero in order to obtain a criterion which is able to predict a linear shear strength increase with increasing mean stress, I1/3. For considering materials with cohesion, Ewy (1999) included pore pressure as a necessary parameter and introduced the parameters S and ηas material constants. The parameter S is related to the cohesion of the rock, while the parameterηrepresents the internal friction.
Doing all the modifications and defining appropriate stress invariants, the following failure criterion was obtained:
(I1)3
I3 =27+η (4.13)
where
I1 =(σ1+S)+(σ2+S)+(σ3+S) (4.14)
and
I3 =(σ1+S)(σ2+S)(σ3+S) (4.15)
S and η can be derived directly from the Mohr–Coulomb cohesion S0 and internal friction angleφby
S= S0
tanφ (4.16)
η= 4(tanφ)2(9−7sinφ)
(1−sinφ) (4.17)
where tanφ=µiand S0=C0/(2 q1/2) with q as defined in equation (4.7).
The modified Lade criterion predicts a strengthening effect with increasing interme- diate principal stress, σ2, followed by a slight reduction in strength asσ2 increases.
It should be noted that the equations above allow one to employ this criterion using the two parameters most frequently obtained in laboratory strength tests, C0 andµi. This makes this criterion easy to use, and potentially more generally descriptive of rock failure, when considering problems such as wellbore stability. The modified Lade criterion can be observed in Figure 4.7c where it has been plotted inσ1−σ2space for C0=60 MPa andµi=0.6, the same parameters used for the Mohr–Coulomb criterion in Figure 4.7a.
Modified Wiebols–Cook criterion
Wiebols and Cook (1968) proposed an effective strain energy criterion for rock failure that depends on all three principal stresses. Zhou (1994) presented a failure criterion
with features similar to the Wiebols–Cook criterion which is an extension of the cir- cumscribed Drucker–Prager criterion (described below).
The failure criterion proposed by Zhou predicts that a rock fails if
J21/2= A+B J1+C J12 (4.18)
where J1 = 1
3(σ1+σ2+σ3) (4.19)
and J21/2=
1 6
(σ1−σ2)2+(σ1−σ3)2+(σ2−σ3)2
(4.20) J1is the mean effective confining stress and, for reference, J21/2is equal to (3/2)1/2τoct, whereτoctis the octahedral shear stress
τoct= 1 3
(σ1−σ2)2+(σ2−σ3)2+(σ2−σ1)2 (4.21)
The parameters A, B, and C are determined such that equation (4.18) is constrained by rock strengths under triaxial (σ2=σ3) and triaxial extension (σ1=σ2) conditions (Figure 4.1). Substituting the given conditions plus the uniaxial rock strength (σ1=C0, σ2=σ3=0) into equation (4.18), it is found that
C =
√27
2C1+(q−1)σ3−C0
C1+(q−1)σ3−C0
2C1+(2q+1)σ3−C0
− q−1 q+2
(4.22) with C1=(1+0.6µi)C0 and q given by equation (4.7),
B=
√3(q−1) q+2 − C
3[2C0+(q+2)σ3] (4.23)
and A= C0
√3− C0
3 B−C02
9 C (4.24)
The rock strength predictions produced using equation (4.18) are similar to those of Wiebols and Cook and thus it is referred to as the modified Wiebols–Cook criterion.
For polyaxial states of stress, the strength predictions made by this criterion are slightly higher than those found using the linearized Mohr–Coulomb criterion. This can be seen in Figure 4.6 because the failure cone of the modified Wiebols–Cook criterion just coincides with the outer apices of the Mohr–Coulomb hexagon. This criterion is plotted inσ1−σ2space in Figure 4.7d. Note its similarity to the modified Lade criterion.
Drucker–Prager criterion
The extended von Mises yield criterion, or Drucker–Prager criterion, was originally developed for soil mechanics (Drucker and Prager 1952). The von Mises criterion may be written in the following way
J2=k2 (4.25)
where k is an empirical constant. The yield surface of the modified von Mises criterion in principal stress space is a right circular cone equally inclined to the principal stress axes.
The intersection of theπ-plane with this yield surface is a circle. The yield function used by Drucker and Prager to describe the cone in applying the limit theorems to perfectly plastic soils has the form:
J21/2 =k+αJ1 (4.26)
whereαand k are material constants. The material parametersαand k can be determined from the slope and the intercept of the failure envelope plotted in the J1 and (J2)1/2 space. α is related to the internal friction of the material and k to the cohesion of the material. In this way, the Drucker–Prager criterion can be compared to the Mohr–
Coulomb criterion. Whenαis equal to zero, equation (4.26) reduces to the von Mises criterion.
The Drucker–Prager criteria can be divided into an outer bound criterion (or cir- cumscribed Drucker–Prager) and an inner bound criterion (or inscribed Drucker–
Prager). These two versions of the Drucker–Prager criterion come from comparing the Drucker–Prager criterion with the Mohr–Coulomb criterion. In Figure 4.6 the two Drucker–Prager criteria are plotted in the π-plane. The inner Drucker–Prager circle only touches the inside of the Mohr–Coulomb criterion and the outer Drucker-Prager circle coincides with the outer apices of the Mohr–Coulomb hexagon.
The inscribed Drucker–Prager criterion is obtained when (Veeken, Walters et al.
1989; McLean and Addis 1990) α= 3sinφ
9+3sin2φ (4.27)
and
k = 3C0cosφ 2√
q
9+3sin2φ (4.28)
whereφis the angle of internal friction, as defined above.
The circumscribed Drucker–Prager criterion (McLean and Addis 1990; Zhou 1994) is obtained when
α= 6 sinφ
√3 (3−sinφ) (4.29)
and k=
√3C0cosφ
√q (3−sinφ) (4.30)
As equations (4.29) and (4.30) show,αonly depends onφwhich means thatαhas an upper bound in both cases; 0.866 in the inscribed Drucker–Prager case and 1.732 in the circumscribed Drucker–Prager case.
In Figure 4.7e we show the Drucker–Prager criteria for C0 =60 MPa andµi=0.6 in comparison with other failure criteria. As shown in Figure 4.7e, for the same values of C0 and µi, the inscribed Drucker–Prager criterion predicts failure at much lower stresses as a function ofσ2than the circumscribed Drucker–Prager criterion.
As mentioned above, Colmenares and Zoback (2002) considered these failure cri- terion for five rock types: amphibolite from the KTB site in Germany (Chang and Haimson 2000), Dunham dolomite and Solenhofen limestone (Mogi 1971) and Shira- hama sandstone and Yuubari shale (Takahashi and Koide 1989).
Figure 4.8 presents all the results for the Mohr–Coulomb criterion with the best-fitting parameters for each rock type. As the Mohr–Coulomb does not take into account the influence ofσ2, the best fit would be the horizontal line that goes through the middle of the data for eachσ3. The smallest misfits associated with the Mohr–Coulomb criterion were obtained for the Shirahama sandstone and the Yuubari shale. The largest misfits were for Dunham dolomite, Solenhofen limestone and KTB amphibolite, which are rocks showing the greatest influence of the intermediate principal stress on failure.
The modified Lade criterion (Figure 4.9) works well for the rocks with a highσ2- dependence of failure such as Dunham dolomite and Solenhofen limestone. For the KTB amphibolite, this criterion reasonably reproduces the trend of the experimental data but not as well as for the Dunham dolomite. We see a similar result for the Yuubari shale. The fit to the Shirahama sandstone data does not reproduce the trends of the data very well.
We now briefly explore the possibility of using triaxial test data to predict theσ2- dependence using the modified Lade failure criterion. The reason for doing this is to be able to characterize rock strength with relatively simple triaxial tests, but to allow all three principal stresses to be considered when addressing problems such as wellbore failure. We utilize only the triaxial test data for Solenhofen limestone (Figure 4.8b) which would not have detected the fact that the strength is moderately dependent onα2. As shown by Colmenares and Zoback (2002), by using only triaxial test data (shown in Figure 4.10a), we obtain a value of C0as a function ofα2(Figure 4.10b) that is within ±3% of that obtained had polyaxial test data been collected.
Because the subject of rock strength can appear to be quite complex, it might seem quite difficult to know how to characterize the strength of a given rock and to utilize this knowledge effectively. In practice, however, the size of the failure envelope (Figure 4.6) is ultimately more important than its exact shape. When applied to problems of
wellbore stability (Chapter 10), for example, practical experience has shown that for relatively strong rocks, either the Mohr–Coulomb criterion or the Hoek–Brown criterion yield reliable results. In fact, when using these data to fit the polyaxial strength data shown in Figure 4.8, the two criteria worked equally well (Colmenares and Zoback 2002). However, because the value for the parameter m in the Hoek–Brown criterion is rarely measured, it is usually most practical to use the Mohr–Coulomb criterion when considering the strength of relatively strong rocks. Similarly, in weaker rocks, both the modified Lade and the modified Wiebols–Cook criteria, both polyaxial criteria, seem to work well and yield very similar fits to the data shown in Figure 4.8 (Colmenares and Zoback 2002). The modified Lade criterion is easily implemented in practice as it is used with the two parameters most commonly measured in laboratory tests, µi
and C0.