Page 105 - Reservoir Geomechanics

89 Rock failure in compression, tension and shear

during the failure process in terms of the applied effective principal stresses σ 1 and σ 3 ,

τ f = 0.5(σ 1 − σ 3 ) sin 2β (4.1)
σ n = 0.5(σ 1 + σ 3 ) + 0.5(σ 1 − σ 3 ) cos 2β (4.2)

where β is the angle between the fault normal and σ 1 (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: µ i , the slope of the failure line, termed the coef-
ficient of internal friction, and the unconfined compressive strength (termed the UCS
or C 0 ). One could also describe the linear Mohr failure line in terms of its intercept
when σ 3 = 0 which is called the cohesive strength (or cohesion), S 0 ,asis common in
soil mechanics. In this case, the linearized Mohr failure line can be written as

τ = S 0 + σ n µ i (4.3)

As cohesion is not a physically measurable parameter, it is more common to express
rock strength in terms of C 0 . The relationship between S 0 and C 0 is:

2 1/2
C 0 = 2S 0 µ + 1 + µ i (4.4)
i
While uniaxial tests are obviously the easiest way to measure C 0 ,itis preferable to
determine C 0 by 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 C 0 by 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, C 0 is the
intercept in the resultant plot (94.3 MPa) and µ i is found to be 0.83 from the relationship

n − 1
µ i = √ (4.5)
2 n