in Figure 5.2b was taken just west of well C in Figure 5.8. Numerous earthquakes occur in this region (shown by small dots), and are associated with slip on numerous reverse and strike-slip faults throughout the area. The trends of active faults and fold axes are shown on the map. The distribution of fractures and faults from analysis of ultrasonic borehole televiewer data in wells A–D are presented in the four stereonets shown. The direction of maximum horizontal compression in each well was determined from analy- sis of wellbore breakouts (explained in Chapter 6) and is northeast–southwest compres- sion (as indicated by the arrows on stereonets). Note that while the direction of maximum horizontal compression is relatively uniform (and consistent with both the trend of the active geologic structures and earthquake focal mechanisms in the region as discussed later in this chapter), the distribution of faults and fractures in each of these wells is quite different. The reason for this has to do with the geologic history of each site. The highly idealized relationships between stress directions and Mode I fractures and conjugate normal, reverse and strike-slip faults (illustrated in Figure 5.1) are not observed. Rather, complex fault and fracture distributions are often seen which reflect not only the cur- rent stress field, but deformational episodes that have occurred throughout the geologic history of the formation. Thus, the ability to actually map the distribution of fractures and faults using image logs is essential to actually knowing what features are present in situ and thus which fracture and faults are important in controlling fluid flow at depth (Chapter 11).
t
m = 0 . 6
2b1
s3 s2 s1 Sn
M o d e 1
m = 1 . 0
A c t i v e s h e a r p l a n e s M o d e 2 a n d M o d e 3
2b3
P
S3
S2 S1
P
b3
b1 FAU LT
a . b .
Figure 5.9. Representation of the shear and effective normal stress on an arbitrarily oriented fault can be accomplished with a three-dimensional Mohr circle (a). Although the exact position of point P can be determined with anglesβ1andβ3measured between the fault normal and S1
and S3directions (b) utilizing the graphical reconstruction shown, it is typical to calculate this mathematically (see text) and utilize the three-dimensional Mohr diagram for representation of the data.
location of P graphically, one utilizes the angles 2β1and 2β3 to find points on the two small circles, and by constructing arcs drawn from the center of the other Mohr circle, P is determined as the intersection of the two arcs. It is obvious that a Mode I plane (normal to the least principal stress) plots at the position ofσ3in the Mohr diagram.
Of course, it is not necessary these days to use graphical techniques alone for deter- mining the shear and effective normal stress on arbitrarily oriented planes, but three- dimensional Mohr diagrams remain quite useful for representing fault data, as will be illustrated in the chapters that follow.
There are two common methods for calculating the magnitude of shear and normal stress on an arbitrarily oriented plane. The first technique defines the shear and normal stress in terms of the effective principal stresses and the orientation of the fault plane to the stress field. The shear and effective normal stresses are given by
τ =a11a12σ1+a12a22σ2+a13a23σ3 (5.7)
σn =a211σ1+a122 σ2+a213σ3 (5.8)
where ai jare the direction cosines (Jaeger and Cook 1971):
A=
cosγ cosλ cosγ sinλ −sinγ
−sinλ cosλ 0 sinγ cosλ sinγsinλ cosγ
(5.9)
whereγ is the angle between the fault normal and S3, andλis the angle between the projection of the fault strike direction and S1in the S1–S2plane.
Alternatively, one can determine the shear and normal stress via tensor transforma- tion. If principal stresses at depth are represented by
S=
S1 0 0
0 S2 0
0 0 S3
we can express stress in a geographical coordinate system with the transform
Sg= R1SR1 (5.10)
where R1 =
cos a cos b sin a cos b −sin b
cos a sin b sin c −sin a cos c sin a sin b sin c +cos a cos c cos b sin c cos a sin b cos c +sin a sin c sin a sin b cos c −cos a sin c cos b cos c
(5.11) and the Euler (rotation) angles that define the stress coordinate system in terms of geographic coordinates are as follows:
a=trend of S1 b= −plunge of S1
c=rake S2.
However, if S1is vertical (normal faulting), these angles are defined as a=trend of SHmax−π/2
b= −trend of S1 c=0.
Using the geographical coordinate system, it is possible to project the stress tensor on to an arbitrarily oriented fault plane. To calculate the stress tensor in a fault plane coor- dinate system, Sf, we once again use the principles of tensor transformation such that
Sf= R2SgR2 (5.12)
where R2 =
cos(str) sin(str) 0
sin(str) cos(dip) −cos(str) cos(dip) −sin(dip)
−sin(str) sin(dip) cos(str) sin(dip) −cos(dip)
(5.13)
where str is the fault strike and dip is the fault dip (positive dip if fault dips to the right when the fault is viewed in the direction of the strike). The shear stress,τ, which acts in the direction of fault slip in the fault plane, and normal stress, Sn, are given by
τ =Sr(3,1) (5.14)
Sn =Sf(3,3) (5.15)
where
Sr= R3SfR3 (5.16)
and
R3 =
cos(rake) sin(rake) 0
−sin(rake) cos(rake) 0
0 0 1
(5.17)
Here rake of the slip vector is given by rake=arctan
Sf(3,2) Sf(3,1)
(5.18a) if Sf(3,2)>0 and Sf(3,1)>0 or Sf(3,2)>0 and Sf(3,1)<0; alternatively
rake=180◦−arctan
Sf(3,2)
−Sf(3,1)
(5.18b) if Sf(3,2)<0 and Sf(3,1) > 0; or
rake=arctan
−Sf(3,2)
−Sf(3,1)
−180◦ (5.18c)
if Sf(3,2)<0 and Sf(3,1)<0.
To illustrate these principles for a real data set, Figure 5.10 shows a stereonet repre- sentation of 1688 faults imaged with a borehole televiewer in crystalline rock from the Cajon Pass research borehole over a range of depths from 1750 to 3500 m depth (after Barton and Zoback 1992). Shear and normal stress were calculated using equations (5.14) and (5.15) with the magnitude and orientation of the stress tensor from Zoback and Healy (1992). We can then represent the shear and normal stress on each plane with a three-dimensional Mohr circle in the manner of Figure 5.9a. Because of the variation of stress magnitudes over this depth range, we have normalized the Mohr diagram by the vertical stress, Sv. As illustrated, most of the faults appear to be inactive in the current stress field. As this is Cretaceous age granite located only 4 km from the San Andreas fault, numerous faults have been introduced into this rock mass over tens of millions of years. However, a number of faults are oriented such that the ratio of shear to normal stress is in the range 0.6–0.9. These are active faults, which, in the context of the model shown in Figures 4.24c,d, are critically stressed and hence limit principal stress magnitudes. In Chapter 12 we show that whether a fault is active or inactive in the current stress field determines whether it is hydraulically conductive (permeable) at depth.
0 0.2 0.4
0 0.2 0.4 0.6 0.8 1
(sn - Pp)/Sv t/Sv
N a .
b .
m = 0 . 6 m = 1 . 0
Figure 5.10. (a) Stereographic representation of fault data detected through wellbore image analysis in highly fractured granitic rock encountered in the Cajon Pass research well from 1750 to 3500 m depth (after Barton and Zoback1992). (b) Representation of the same data utilizing a three-dimensional Mohr diagram normalized by the vertical stress. While many fractures appear to be critically stressed, most are not and thus reflect the rock’s geologic history (after Barton, Zoback et al.1995).