Page 150 - Reservoir Geomechanics

133 Rock failure in compression, tension and shear

respectively. This depends, of course, on whether it is a normal, strike-slip, or reverse
faulting environment. In other words:

σ 1 S v − P p 2 1/2 2
Normal faulting = ≤ [(µ + 1) + µ] (4.45)
σ 3 S hmin − P p
σ 1 S Hmax − P p 2 1/2 2
Strik-slip faulting = ≤ [(µ + 1) + µ] (4.46)
σ 3 S hmin − P p
σ 1 S Hmax − P p 2 1/2 2
Reverse faulting = ≤ [(µ + 1) + µ] (4.47)
σ 3 S v − P p

Asreferredtoabove,thelimitingratioofprincipaleffectivestressmagnitudesdefined
in equations (4.45)–(4.47)is 3.1 for µ = 0.6, regardless of whether one considers
normal, strike-slip or reverse faulting regime. However, it should be obvious from these
equations that stress magnitudes will increase with depth (as S v increases with depth).
The magnitude of pore pressure will affect stress magnitudes as will whether one is in
a normal, strike-slip, or reverse faulting environment. This is illustrated in Figures 4.28
and 4.29, which are similar to Figure 1.4 except that we now include the limiting values
of in situ principal stress differences at depth for both hydrostatic and overpressure
conditions utilizing equations (4.45)–(4.47). In a normal faulting environment in which
pore pressure is hydrostatic (Figure 4.28a), equation (4.45) defines the lowest value of
the minimum principal stress with depth. It is straightforward to show that in an area
of critically stressed normal faults, when pore pressure is hydrostatic, the lower bound
value of the least principal stress S hmin ∼ 0.6S v ,as illustrated by the heavy dashed line
in Figure 4.28a. The magnitude of the least principal stress cannot be lower than this
value because well-oriented normal faults would slip. Or in other words, the inequality
in equation (4.45)would be violated. In the case of strike-slip faulting and hydrostatic
pore pressure (Figure 4.28b), the maximum value of S Hmax (as given by equation 4.46)
depends on the magnitude of the minimum horizontal stress, S hmin .If the value of the
minimum principal stress is known (from extended leak-off tests or hydraulic fracturing,
as discussed in Chapter 6), equation (4.46) can be used to put an upper bound on S Hmax .
The position of the heavy dashed line in Figure 4.28b shows the maximum value of
S Hmax for the S hmin values shown by the tick marks. Finally, for reverse faulting (equation
4.47 and Figure 4.27c), because the least principal stress is the vertical stress, S v ,it
is clear that the limiting value for S Hmax (heavy dashed line) is very high. In fact, the
limiting case for the value of S Hmax is ∼2.2S v for hydrostatic pore pressure and µ = 0.6.
Many regions around the world are characterized by a combination of normal and
strike-slip faulting (such as western Europe) and reverse and strike-slip faulting (such as
the coast ranges of western California). It is clear how these types of stress states come
about. In an extensional environment, if S hmin is near its lower limit (∼0.6S v ) and S Hmax
near its upper limit S v (such that S 1 ≈ S 2 ), the equalities in equations (4.45) and (4.46)
could both be met and both normal and strike-slip faults would be potentially active. In a