Page 174 - Reservoir Geomechanics

157 Faults and fractures at depth

where

S r = R 3 S f R 3 (5.16)

and

 
cos(rake) sin(rake) 0
R 3 =  −sin(rake) cos(rake) 0  (5.17)


0 0 1
Here rake of the slip vector is given by

S f (3, 2)
rake = arctan (5.18a)
S f (3, 1)
if S f (3,2) > 0 and S f (3,1) > 0or S f (3,2) > 0 and S f (3,1) < 0; alternatively

S f (3, 2)

◦
rake = 180 − arctan (5.18b)
−S f (3, 1)
if S f (3,2) < 0 and S f (3,1) > 0; or

−S f (3, 2)

rake = arctan − 180 ◦ (5.18c)
−S f (3, 1)
if S f (3,2) < 0 and S f (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, S v .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.