The failure of rock in compression is a complex process that involves microscopic failures manifest as the creation of small tensile cracks and frictional sliding on grain boundaries (Brace, Paulding et al. 1966). Eventually, as illustrated in Figure 4.2a, there is a coalescence of these microscopic failures into a through-going shear plane (Lockner, Byerlee et al. 1991). In a brittle rock (with stress–strain curves like that shown in Figure 3.2) this loss occurs catastrophically, with the material essentially losing all of its strength when a through-going shear fault forms. In more ductile materials (such as poorly cemented sands) failure is more gradual. The strength is defined as the peak stress level reached during a deformation test after which the sample is said to strain soften, which simply means that it weakens (i.e. deforms at lower stresses) as it continues to deform. Simply put, rock failure in compression occurs when the stresses acting on a rock mass exceed its compressive strength. Compressive rock failure involves all of the stresses acting on the rock (including, as discussed below, the pore pressure). By rock strength we typically mean the value of the maximum principal stress at which a sample loses its ability to support applied stress.
The strength of rock depends on how it is confined. For the time being we will restrict discussion of rock strength to triaxial compression tests with non-zero pore pressure (with effective stressesσ1 > σ2=σ3). It is universally observed in such tests that sample strength is seen to increase monotonically with effective confining pressure (e.g. Jaeger and Cook 1979). Because of this, it is common to present strength test results using Mohr circles and Mohr failure envelopes (Figure 4.2b,c).
The basis for the Mohr circle construction is that it is possible to evaluate graphically the shear stress,τf, and effective normal stress (σn =Sn−Pp) on the fault that forms
s
3b
t
s
nt = f(sn)
Failure occurs when:
sn= Normal stress
t = Shear stress
s3 s3 = 0
s3 s3 = 0
s1 = UCS (C0)
s1 = UCS (C0) Mohr envelope
sn tf
s1
s1
b.
2b
mi (coefficient of internal friction)
s t
Linearized Mohr envelope
S0
c .
t
s
Figure 4.2. (a) In triaxial strength tests, at a finite effective confining pressureσ3(S3–P0), samples typically fail in compression when a through-going fault develops. The angle at which the fault develops is described byβ, the angle between the fault normal and the maximum compressive stress,σ1. (b) A series of triaxial strength tests at different effective confining pressures defines the Mohr failure envelope which typically flattens as confining pressure increases. (c) The linear simplification of the Mohr failure envelope is usually referred to as Mohr–Coulomb failure.
during the failure process in terms of the applied effective principal stressesσ1andσ3,
τf=0.5(σ1−σ3) sin 2β (4.1)
σn =0.5(σ1+σ3)+0.5(σ1−σ3) cos 2β (4.2) whereβis the angle between the fault normal andσ1(Figure 4.2a).
Conducting a series of triaxial tests defines an empirical Mohr–Coulomb failure envelope that describes failure of the rock at different confining pressures (Figure 4.2b).
Allowable stress states (as described by Mohr circles) are those that do not intersect the Mohr–Coulomb failure envelope. Stress states that describe a rock just at the failure point “touch” the failure envelope. Stress states corresponding to Mohr circles which exceed the failure line are not allowed because failure of the rock would have occurred prior to the rock having achieved such a stress state.
The slope of the Mohr failure envelopes for most rocks decreases as confining pres- sure increases, as shown schematically in Figure 4.2b and for a sandstone in Figure 4.3a.
However, for most rocks it is possible to consider the change of strength with confin- ing pressure in terms of a linearized Mohr–Coulomb failure envelope (Figures 4.2c and 4.3a) defined by two parameters:µi, the slope of the failure line, termed the coef- ficient of internal friction, and the unconfined compressive strength (termed the UCS or C0). One could also describe the linear Mohr failure line in terms of its intercept whenσ3=0 which is called the cohesive strength (or cohesion), S0, as is common in soil mechanics. In this case, the linearized Mohr failure line can be written as
τ =S0+σnµi (4.3)
As cohesion is not a physically measurable parameter, it is more common to express rock strength in terms of C0. The relationship between S0 and C0is:
C0=2S0
µ2i +1 1/2+µi
(4.4) While uniaxial tests are obviously the easiest way to measure C0, it is preferable to determine C0by conducting a series of triaxial tests to avoid the axial splitting of the samples that frequently occurs during uniaxial tests and the test results are sensitive to the presence of pre-existing flaws in the samples. Once a Mohr envelope has been obtained through a series of tests, one can find C0by either fitting the envelope with a linear Mohr failure line and determining the uniaxial compressive strength graphically, or simply by measuring strength at many pressures and plotting the data as shown in Figure 4.3b, for Darley Dale sandstone (after Murrell 1965). As shown, C0 is the intercept in the resultant plot (94.3 MPa) andµiis found to be 0.83 from the relationship µi = n−1
2√
n (4.5)
50 100 150 200 250 300 350 400 450
0 10 20 30 40 50 60 70
S1(MPa)
S3 (MPa) DA R L E Y DA L E S A N D S TO N E
94.3 MPa
n = 4.55
−20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 20
40 60 80 100 120 140 160 180
Normal stress (MPa)
Shear stress (MPa)
Tr u e ve r t i c a l d e p t h : 3 0 6 5 m e t e r s
Pconf = 15 MPa
C0 = 105 MPa T0 = 6.3 MPa
Mohr F ailure en
velope
Pconf = 10 MPa
Pconf Linear Mohr en
velope
S0 = 24 MPa
mi = 0.9 a .
b.
= 0
Figure 4.3. (a) Laboratory strength tests at 0, 10 and 15 MPa confining pressure defines the Mohr envelope for a sandstone sample recovered from a depth of 3065 m in southeast Asia. A linearized Mohr–Coulomb failure envelope and resultant values of S0, C0andµiare shown. (b) Illustration of how strength values determined from a series of triaxial tests can be used to extrapolate a value of C0. As explained in the text, the slope of this line can be used to determineµi.
40 30 20 10 0
lithic tuff andesites basalts granitic rocks graywackes limestone conglomerates mudstones sandstones phyllites schists hornfels gneiss
I g n e o u s S e d i m e n t a r y M e t a m o r p h i c
Cohesive strength S0 (MPa) lithic tuff andesites basalts granitic rocks graywackes limestone conglomerates mudstones sandstones phyllites schists hornfels gneiss
I g n e o u s S e d i m e n t a r y M e t a m o r p h i c
2.0 1.5 1.0 0.5 0
2 . 9
m
Figure 4.4. Cohesion and internal friction data for a variety of rocks (data replotted from the compilation of Carmichael1982). Note that weak rocks with low cohesive strength still have a significant coefficient of internal friction.
where n is the slope of failure line when the stress at failure, S1, is plotted as a function of the confining pressure, S3, as shown in Figure 4.3b.
The fact that the test data can be fairly well fitted by a straight line in Figure 4.3b illustrates that using a linearized Mohr failure envelope for these rocks is a reasonable approximation. An important concept to keep in mind when considering rock strength is that while strong rocks have high cohesion and weak rocks have low cohesion, nearly all rocks have relatively high coefficients of internal friction. In other words, the rocks with low cohesion (or low compressive strength) are weak at low mean stresses but increase in strength as the mean stress increases. This is shown in the compilation shown in Figures 4.4a,b (data from Carmichael 1982). For sedimentary rocks, cohesive strengths are as low as 1 MPa and as high as several tens of MPa. Regardless, coefficients of internal friction range from about 0.5 to 2.0 with a median value of about 1.2. One exception to this is shales, which tend to have a somewhat lower value ofµi. This is discussed below in the section discussing how rock strength is derived from geophysical logs.
A simple, but very important illustration of the importance of cohesion on wellbore stability is illustrated in Figure 4.5. Linearized Mohr envelopes are shown schematically for a strong rock (high cohesive strength) and weak rock (low cohesive strength) with the sameµi. As discussed in detail in Chapter 6, when one considers the stresses at the wall of a vertical wellbore that might cause compressive rock failure, the least principal stress,σ3, is usually the radial stress,σrr, which is equal to the difference between
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.