Page 17 - Applied Petroleum Geomechanics

Stresses and strains 7

where K dry is the bulk modulus of the dry porous rock; K m is the bulk
modulus of the matrix mineral in the rock; a is restricted to the range of
f < a 1(f is porosity). In Terzaghi’s effective stress law, a ¼ 1.
In the three-dimensional condition the relation between changes in
0
total stress (s ij ) and effective stress (s ij ) can be expressed in the following
equation (Biot, 1941):
0
s ¼ s ij ap p d ij (1.11)
ij
where s ij is the index notation of the total stress tensor, as shown in Eq. (1.3);
d ij is Kronecker’s delta, d ij ¼ 1, when i ¼ j;and d ij ¼ 0, when i s j.

1.1.6 In situ stresses, far-field and near-field stresses

In situ stress state is the original stress status in the rock before excavations
or other perturbations. In situ stresses are also called far-field stresses. For
example, the stress state before a borehole is drilled shown in Fig. 1.6 is the
in situ stress state (Zhang, 2013). As a first approximation, one can assume
that the three principal stresses of a natural in situ stress field are acting
vertically (one component, s V ) and horizontally (two components, s H and
s h ). More details about in situ stresses can be found in Chapters 5 and 6.

Figure 1.6 Schematic representation of in situ stresses (far-field stresses) and near-
field stresses (2-D case) in a borehole.