Healy (1984) based on stress measurements from much shallower depths. It should be noted that Townend (2003) pointed out that the uncertainty estimates in Figure 4.26 are likely significantly smaller than those shown.
There are two implications of the data shown in Figure 4.26. First, Byerlee’s law (equation 4.41), defined on the basis of hundreds of laboratory experiments, appears to correspond to faults in situ. This is a rather amazing result when one considers the huge difference between the size of samples used for friction experiments in the lab and the size of real faults in situ, the variability of roughness of the sliding surface, the idealized conditions under which laboratory experiments are conducted, etc. Second, everywhere that stress magnitudes have been measured at appreciable depth, they indicate that they are controlled by the frictional strength of pre-existing faults in the crust. In other words, the earth’s crust appears to be in a state of failure equilibrium and the law that describes that state is simple Coulomb friction, or Amontons’ law as defined in equation (4.39).
In fact, we will find that this is the case in many sedimentary basins around the world (Chapters 9–12).
In shaley rocks, it is widely suspected that the coefficient of friction may be sig- nificantly lower than 0.6, especially at low effective pressure. In fact, Byerlee pointed out that due to water layers within its crystallographic structure, montmorillonite has unusually lower frictional strength because intracrystalline pore pressure develops as it is being deformed. This manifests itself as a low friction. In recent drained labora- tory tests Ewy, Stankowich et al. (2003) tested a deep clay and three shale samples and found coefficients of friction that range between 0.2 and 0.3. The subject of the frictional strength of shaley rocks is complicated, not only by the issue of pore pressure but by the fact that many tests reveal that clays that have low frictional strength at low effective pressure have higher frictional strength at higher effective pressures (Morrow, Radney et al. 1992; Moore and Lockner 2006).
a .
S
1S
1b.
b
S
1c .
s3 s1 sn
S0
t m = 0 . 6
b = 6 0o m
2b 1
1
2
3
3 2
Figure 4.27. (a) Frictional sliding on an optimally oriented fault in two dimensions. (b) One can consider the Earth’s crust as containing many faults at various orientations, only some of which are optimally oriented for frictional sliding. (c) Mohr diagram corresponding to faults of different orientations. The faults shown by black lines in (b) are optimally oriented for failure (labeled 1 in b and c), those shown in light gray in (b) (and labeled 2 in b and c) in (b) trend more perpendicular to SHmax, and have appreciable normal stress and little shear stress. The faults shown by heavy gray lines and labeled 3 in (b) are more parallel to SHmaxhave significantly less shear stress and less normal stress than optimally oriented faults as shown in (c).
simply assume that stresses in the Earth cannot be such that they exceed the frictional strength of pre-existing faults. This concept is schematically illustrated in Figure 4.27.
We first consider a single fault in two dimensions (Figure 4.27a) and ignore the magnitude of the intermediate principal effective stress because it is in the plane of the fault. The shear and normal stresses acting on a fault whose normal makes and an angle β with respect to the direction of maximum horizontal compression, S1, was given by equations (4.1) and (4.2). Hence, the shear and normal stresses acting on the fault depend on the magnitudes of the principal stresses, pore pressure and the orientation of the fault with respect to the principal stresses.
It is clear in the Mohr diagram shown in Figure 4.27c that for any given value ofσ3
there is a maximum value ofσ1established by the frictional strength of the pre-existing
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,
β =π/2−tan−1µ (4.42)
Combining this relation with the principles of Anderson’s classification scheme (see also Figure 5.1) it is straightforward to see (assumingµ≈0.6):
r Normal faults are expected to form in conjugate pairs that dip∼60◦and strike parallel to the direction of SHmax.
r Strike-slip faults are expected to be vertical and form in conjugate pairs that strike
∼30◦from the direction of SHmax.
r Reverse faults are expected to dip∼30◦and form in conjugate pairs that strike normal to the direction of SHmax.
Jaeger and Cook (1979) showed that the values ofσ1andσ3(and hence S1and S3) 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:
σ1
σ3
= S1−Pp
S3−Pp =[(µ2+1)1/2+µ]2 (4.43)
such that forµ=0.6 (see Figure 4.26), σ1
σ3
=3.1 (4.44)
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 whichσ2=σvis in the plane of the faults (although this certainly need not be the case).
In this case, the difference betweenσHmax(defined as SHmax−Pp) andσhmin(defined as Shmin−Pp), 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 SHmaxincreases with respect to Shmin, 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 SHmaxwith respect to Shmincannot 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 SHmaxhave 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 SHmaxhave 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. SHmax, Shmin, or Sv) corresponds to S1, S2and S3,
respectively. This depends, of course, on whether it is a normal, strike-slip, or reverse faulting environment. In other words:
Normal faultingσ1
σ3
= Sv−Pp
Shmin−Pp ≤[(µ2+1)1/2+µ]2 (4.45) Strik-slip faultingσ1
σ3
= SHmax−Pp Shmin−Pp
≤[(µ2+1)1/2+µ]2 (4.46) Reverse faultingσ1
σ3
= SH max−Pp
Sv−Pp ≤[(µ2+1)1/2+µ]2 (4.47) As referred to above, the limiting ratio of principal effective stress magnitudes defined 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 Sv 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 Shmin∼0.6Sv, 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 SHmax(as given by equation 4.46) depends on the magnitude of the minimum horizontal stress, Shmin. 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 SHmax. The position of the heavy dashed line in Figure 4.28b shows the maximum value of SHmaxfor the Shminvalues 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, Sv, it is clear that the limiting value for SHmax(heavy dashed line) is very high. In fact, the limiting case for the value of SHmaxis∼2.2Svfor hydrostatic pore pressure andµ=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 Shminis near its lower limit (∼0.6Sv) and SHmax near its upper limit Sv(such that S1≈S2), 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
Depth (meters)
Normal faulting hydrostatic
10000 8000
12000 (feet) 4000
6000 1000
2000
3000
4000 0
psi 200
0 4000 6000 8000 10000 12000
SHmax Shmin
Sv Pp
Sv≥ SHmax ≥ Shmin
(~23 MPa/km) (~1 psi/ft)
(~10 MPa/km) (~0.44 psi/ft)
Stress or pressure
Depth (meters) 8000
12000 (feet) 4000
6000 1000
2000
3000
4000
20 40 60 80
0 MPa
2000 4000 6000 8000 10000 12000 0
10000 b.
Strike-slip faulting hydrostatic SHmax≥ Sv≥ Shmin
Sv SHmax Shmin
Pp
Stress or pressure
Depth (meters)
20 40 60 80
0 MPa
psi 2000 4000 6000 8000 10000 12000
10000 8000
12000 (feet) 4000
6000 1000
2000
3000
4000 0 c .
Reverse faulting hydrostatic
Sv SHmax Shmin
Pp
SHmax ≥ Shmin ≥ Sv
Figure 4.28. Limits on stress magnitudes defined by frictional faulting theory in normal (a), strike-slip and (b) reverse faulting (c) regimes assuming hydrostatic pore pressure. The heavy line in (a) shows the minimum value of the least principal stress, Shmin, in normal faulting environments, in (b) the maximum value of SHmaxfor the values of Shminshown by the ticks and (c) the maximum value of SHmaxfor reverse faulting regimes where the least principal stress is the vertical stress Sv.
Depth (meters)
psi 2000 4000 6000 8000 10000 12000
8000
12000 (feet) 4000
6000 1000
2000
3000
4000 0
10000
Normal faulting overpressure at depth
SHmax Shmin
Sv Pp
Sv≥ SHmax≥ Shmin
b. Stress or pressure
Depth (meters)
20 40 60 80
0 MPa
psi
2000 4000 6000 8000 12000
10000 8000
12000 (feet) 4000
6000 1000
2000
3000
4000 0
10000
Strike-slip faulting overpressure at depth SHmax ≥ Sv ≥ Shmin
Sv SHmax Shmin
Pp
c . Stress or pressure
Depth (meters)
20 40 60 80
0 MPa
psi 2000 4000 6000 8000 10000 12000
10000 8000
12000 (feet) 4000
6000 1000
2000
3000
4000 0
Reverse faulting overpressure at depth SHmax ≥ Shmin ≥ Sv
Sv
SHmax Shmin Pp
Figure 4.29. Same as Figure4.28when overpressure develops at depth as shown. Note that in all three stress states, when pore pressure is nearly lithostatic, all three principal stresses are also close to Sv.
Effective normal stress
Shear stress
15 30 45 60 75
H y d r o s t a t i c Pp = 3 0 M Pa mi
( Sv a t 3 k m ) Pp = 5 0 M Pa
Pp = 8 1 M Pa
Figure 4.30. In terms of frictional faulting theory, as pore pressure increases (and effective stress decreases), the difference between the maximum and minimum effective principal stress (which defines the size of the Mohr circle) decreases with increasing pore pressure at the same depth.
compressional environment, if Shminis approximately equal Sv(such that S2≈S3), and SHmaxmuch larger, it would correspond to a state in which both strike-slip and reverse faults were potentially active.
Figures 4.29a–c illustrate the limiting values of stress magnitudes when pore pres- sure increases markedly with depth in a manner similar to cases like that illustrated in Figure 1.4. As pore pressure approaches Sv at great depth, as is the case in some sedimentary basins, the limiting stress magnitudes (heavy dashed lines) are not signifi- cantly different from the vertical stress, regardless of whether it is a normal, strike-slip or reverse faulting environment. While this might seem counter-intuitive, when pore pressure is extremely high, fault slip will occur on well-oriented faults when there is only a small difference between the maximum and minimum principal stresses.
The way in which the difference in principal stresses is affected by pore pressure is illustrated by the Mohr circles in Figure 4.30. When stress magnitudes are controlled by the frictional strength of faults, as pore pressure increases, the maximum size of the Mohr circle decreases. In other words, as pore pressure gets higher and higher, faulting occurs with smaller and smaller differences between the maximum and mini- mum effective principal stress. Hence, the Mohr circle gets smaller as pore pressures increases.
While Figure 4.30 may seem obvious, there are two issues to draw attention to because it is most commonly assumed that for given values of S1, S2and S3, changing pore pressure simply shifts the position of a Mohr circle along the abscissa. The first point worth emphasizing is that when the state of stress at depth is limited by the frictional strength of pre-existing faults, the ratio of effective stresses remains the same
(in accord with equations 4.45, 4.46 and 4.47) as pore pressure changes (as illustrated).
But this is not true of the ratios (or differences) in the absolute stress magnitudes (as shown in Figure 4.29) such that the higher the pore pressure, the lower the principal stress differences. At extremely high pore pressure, relatively small stress perturbations are sufficient to change the style of faulting from one stress regime to the other (for example, to go from normal faulting to reverse faulting). This is dramatically different from the case in which pore pressure is hydrostatic. The second point to note is that perturbations of pore pressure associated with depletion (or injection) will also affect stress magnitudes through the types of poroelastic effects discussed in Chapter 3. Hence, the size and position of the Mohr circle is affected by the change in pore pressure. This can have an important influence on reservoir behavior, especially in normal faulting regions (Chapter 12).