Page 228 - Applied Petroleum Geomechanics

In situ stress estimate 223

In transversely isotropic formations, the maximum horizontal stress
can be expressed in the following form (refer to Chapter 1, Section 1.5.2
for derivations):
E h n V E h
s H ¼ ðs V a V p p Þþ a h p p þ 2 ðε H þ n h ε h Þ (6.72)
E V ð1 n h Þ 1 n h
where subscripts h and V represent the rock properties in the horizontal and
vertical directions, respectively.
The major barrier for applying these equations is the difficulty in
accurately determining the horizontal strains. Similar to Section 6.3, the
horizontal strains can be back-calculated from stress measurements using the
following relation:

E h
max
s tect ¼ ðε H þ n h ε h Þ (6.73)
1 n 2 h
The maximum horizontal stress may also be estimated from the
following empirical equation:
n
s H ¼ ðs V ap p Þþ ap p þ cs V (6.74)
1 n
where c is a calibration constant.

6.4.6 Maximum horizontal stress from equilibrium of in situ
stresses and pore pressure

To keep the stressestrain equilibrium, the three in situ stresses in elastic
formations should satisfy Hooke’s law. According to Hooke’s law, the
maximum effective horizontal stress (s H ) can be written as the following
0
form:
0
s Eε h
0 h 0
s ¼ s V (6.75)
H
n
where ε h is the strain in the minimum horizontal stress direction.
Replacing the effective stresses by total stresses, the in situ stresses have
the following relation:
ðs h ap p Þ Eε h
s H ¼ s V þ 2ap p (6.76)
n
Eq. (6.76) shows that if the minimum horizontal stress and horizontal
strain are available, the maximum horizontal stress can be calculated.