Page 149 - Reservoir Geomechanics

132 Reservoir geomechanics

fault (the Mohr circle cannot exceed the maximum frictional strength). If the fault is
critically oriented, that is, at the optimal angle for frictional sliding,
−1
β = π/2 − tan µ (4.42)
Combining this relation with the principles of Anderson’s classification scheme (see
also Figure 5.1)itis straightforward to see (assuming µ ≈ 0.6):
Normal faults are expected to form in conjugate pairs that dip ∼60 and strike parallel
◦
to the direction of S Hmax .
Strike-slip faults are expected to be vertical and form in conjugate pairs that strike

∼30 from the direction of S Hmax .
◦
Reverse faults are expected to dip ∼30 and form in conjugate pairs that strike normal
◦
to the direction of S Hmax .
Jaeger and Cook (1979) showed that the values of σ 1 and σ 3 (and hence S 1 and S 3 )
that corresponds to the situation where a critically oriented fault is at the frictional limit
(i.e. equation 4.39 is satisfied) are given by:
σ 1 S 1 − P p 2 1/2 2
= = [(µ + 1) + µ] (4.43)
σ 3 S 3 − P p
such that for µ = 0.6 (see Figure 4.26),
σ 1
= 3.1 (4.44)
σ 3
In Figure 4.27c, we generalize this concept and illustrate the shear and normal stresses
acting on faults with three different orientations. As this is a two-dimensional illustra-
tion, it is easiest to consider this sketch as a map view of vertical strike-slip faults in
which σ 2 = σ v is in the plane of the faults (although this certainly need not be the case).
In this case, the difference between σ Hmax (defined as S Hmax − P p ) and σ hmin (defined
as S hmin − P p ), the maximum and minimum principal effective stresses for the case of
strike-slip faulting, is limited by the frictional strength of these pre-existing faults as
defined in equation (4.43). In other words, as S Hmax increases with respect to S hmin , the
most well-oriented pre-existing faults begin to slip as soon as their frictional strength is
reached (those shown by heavy black lines and labeled 1). As soon as these faults start
to slip, further stress increases of S Hmax with respect to S hmin cannot occur. We refer to
this subset of faults in situ (those subparallel to set 1) as critically stressed (i.e. to be just
on the verge of slipping), whereas faults of other orientations are not (Figure 4.27b,c).
The faults that are oriented almost orthogonally to S Hmax have too much normal stress
and not enough shear stress to slip (those shown by thin gray lines and labeled set 2)
whereas those striking sub parallel to S Hmax have low normal stress and low shear stress
(those shown by thick gray lines and labeled set 3).
We can use equation (4.43)to estimate an upper bound for the ratio of the maximum
and minimum effective stresses and use Anderson’s faulting theory (Chapter 1)to
determine which principal stress (i.e. S Hmax , S hmin ,or S v ) corresponds to S 1 , S 2 and S 3 ,