Page 21 - Reservoir Geomechanics

6 Reservoir geomechanics

X 2
S A SA

S
X' 2 2
S 22
S = (new) X' S = (old)
S 11 2
X 2
S 12
S 21 −1
cos a
S 23 12
S 13
X' 1
S 31 S 1
S 32
X 1
X' 1
S
S 33 X 3 3
X' 3
X 3
X' 3
X 1

S 11 S 12 S 13 a 11 a 12 a 13 S 1 0 0
S S 21 S 22 S 23 A a a a S' 0 S 0
23
22
21

2
S 31 S 32 S 33 a 31 a 32 a 33 0 0 S 3

Figure 1.1. Definition of stress tensor in an arbitrary cartesian coordinate system (Engelder and
Leftwich 1997), rotation of stress coordinate systems through tensor transformation (center) and
principal stresses as defined in a coordinate system in which shear stresses vanish (right).
be defined in terms of any reference system. An arbitrarily oriented cartesian coordinate
system is shown. Because of equilibrium conditions
s 12 = s 21
(1.2)
s 13 = s 31
s 23 = s 32
so that the order of the subscripts is unimportant. In general, to fully describe the state of
stress at depth, one must define six stress magnitudes or three stress magnitudes and the
three angles that define the orientation of the stress coordinate system with respect to
a reference coordinate system (such as geographic coordinates, wellbore coordinates,
etc.).
In keeping with the majority of workers in rock mechanics, tectonophysics and
structural geology, I utilize the convention that compressive stress is positive because
in situ stresses at depths greater than a few tens of meters in the earth are always
compressive. Tensile stresses do not exist at depth in the earth for two fundamental
reasons. First, because the tensile strength of rock is generally quite low (see Chapter 4),