Page 213 - Applied Petroleum Geomechanics

208 Applied Petroleum Geomechanics

where subscripts h and V represent rock properties in horizontal and
vertical directions, respectively. For example, if rock properties have the
following relations: E h /E V ¼ 2.0, n h /n V ¼ 1.4, n V ¼ 0.2, n h ¼ 0.28, and
a V ¼ a h ¼ a, the minimum horizontal stress in anisotropic and isotropic
cases can be calculated as follows:

Anisotropic case, from Eq. (6.36): s h VTI ¼ 0:56 s V ap p þ ap p ,

Isotropic case, from Eq. (6.26): s h ¼ 0:25 s V ap p þ ap p
The results in the above calculations indicate that the effective minimum
horizontal stress in the transversely isotropic case is more than two times of
that in the isotropic case. The difference in this example is significant;
therefore, the anisotropy of the rock needs to be considered.
Because the uniaxial strain model cannot describe the real state of in situ
stresses in most cases, the minimum stress model in Eq. (6.36) can be
modified by considering the tectonic strains. The equation considering the
horizontal strains derived in Chapter 1, Section 1.5.2 (Eq. 1.48) can be used
for the transversely isotropic formations:

E h n V E h
s h VTI ¼ ðs V a V p p Þþ a h p p þ 2 ðε h þ n h ε H Þ (6.37)
E V ð1 n h Þ 1 n h
The difficulty in applying this equation is to determine the minimum
and maximum horizontal strains ε H and ε h . To solve this issue, two methods
can be used: one is to use local or regional empirical equations to estimate
the minimum horizontal stress; the other is to back-calculate the horizontal
strains or tectonic stresses from stress measurements using the following
equation and Eq. (6.73):
E h
min
s tect ¼ ðε h þ n h ε H Þ (6.38)
1 n 2 h

6.3.4 Minimum horizontal stress from empirical equations
When measured data in the study area are not available, regional empirical
equations can be used as a first estimate. Empirical equations in several pe-
troleum basins are presented in the following. For application of any of these
empirical equations, calibrations from local data in the studied area are needed
because these equations might be obtained with some limitations, e.g., for a
certain formation or rock type without considering lithology effects.
Using more than 300 reliable data from leak-off tests and hydraulic
fracture tests from the US Gulf Coast region, Breckels and van Eekelen
(1982) proposed several relationships between the minimum horizontal