Page 22 - Reservoir Geomechanics

7 The tectonic stress field

significant tensile stress cannot be supported in the earth. Second, because there is
alwaysafluidphasesaturatingtheporespaceofrockatdepth(exceptatdepthsshallower
than the water table), the pore pressure resulting from this fluid phase would cause the
rock to hydraulically fracture should the least compressive stress reach a value close to
the value of the pore pressure (Chapter 4).
Once a stress tensor is known in one coordinate system, it is possible to evaluate
stresses in any other coordinate system via tensor transformation. To accomplish this
transformation, we need to specify the direction cosines (a ij ,as illustrated in Figure 1.1)
that describe the rotation of the coordinate axes between the old and new coordinate
systems. Mathematically, the equation which accomplishes this is
T

S = A SA (1.3)
where,
 
a 11 a 12 a 13
 
A =  a 21 a 22 a 23 
a 31 a 32 a 33
There are two reasons why the ability to transform coordinate systems is of fundamental
importance here. First, once we know an in situ stress field in some coordinate system,
we can compute stresses in any other. For example, if we know the stress state in a
geographic coordinate system, we will show how it is possible to derive the stress
field surrounding a wellbore of arbitrary orientation (Chapter 8)to address problems
of stability (Chapter 10), or along a fault plane (Chapter 5)togauge its proximity to
frictional failure and slip (Chapter 11). Another reason why tensor transformation is
important is because we can choose to describe the state of stress at depth in terms of the
principal stresses (i.e. those acting in the principal coordinate system), making the issue
of describing the stress state in situ appreciably easier. The principal coordinate system
is the one in which shear stresses vanish and three principal stresses, S 1 ≥ S 2 ≥ S 3 fully
describe the stress field (as illustrated in the right side of Figure 1.1). In the principal
coordinate system we have diagonalized the stress tensor such that the principal stresses
correspond to the eigenvalues of the stress tensor and the principal stress directions
correspond to its eigenvectors:
 
0 0
S 1

S =  0 S 2 0  (1.4)


0 0 S 3
The reason this concept is so important is that because the earth’s surface is in contact
with a fluid (either air or water) which cannot support shear tractions, it is a principal
stress plane. Thus, one principal stress is generally normal to the earth’s surface with
the other two principal stresses acting in an approximately horizontal plane. While it
is clear that this must be true close to the earth’s surface, compilation of earthquake
focal mechanism data and other stress indicators (described below) suggest that it is