Page 15 - Applied Petroleum Geomechanics

Stresses and strains 5

Thus, in the direction q 1 , which identifies a principal axis, the normal
stress is s 1 and the shear stress is zero. In the direction q 2 , which identifies
the other principal axis, the normal stress is s 3 and the shear stress is also
zero. The principal axes are mutually orthogonal. The equations for
calculating principal stresses in three dimensions can be found in rock
mechanics textbooks (e.g., Jeager et al., 2007). All unsupported excavation
surfaces (including wellbores) are principal stress planes (Hudson and
Harrison, 1997). This is because all unsupported excavation surfaces have
no shear stresses acting on them and are therefore principal stress planes.
1.1.4 Principal stresses

It is possible to show that there is one set of axes with respect to which all
shear stresses are zero, and the three normal stresses have their extreme
values, as shown in Fig. 1.4. These three mutually perpendicular planes are
called principal planes, and the three normal stresses acting on these planes
are the principal stresses. It is convenient to specify the stress state using
these principal stresses because they provide direct information on the
maximum and minimum values of the normal stress components (Hudson
and Harrison, 1997). The values s 1 , s 2 , and s 3 in Fig. 1.4 are the principal
stresses, and s 1 > s 2 > s 3 , which are three principal stress components.
Therefore, the principal stress tensor can be expressed as follows:
2 3
0 0
s 1
s ¼ 4 0 s 2 0 5 (1.8)
6
7
0 0 s 3
1.1.5 Effective stresses

The effect of pore pressure on the mechanical properties of saturated rocks has
been extensively investigated by using the concept of the effective stress that

σ 1

σ
2
σ 3

Figure 1.4 Principal stress components in the principal planes.