Engineering Rock Mechanics

Collated worldwide in situ stress data 59

and so

V

6, =- 0”

1-v

where OH = oH1 = oHp This relation has been known for some time:
according to Turchaninov et al. (1979), it was first derived by Academician
Dinnik in 1925.
From this analysis, we find that the ratio between the horizontal stress
and the vertical stress of vl(1- v) is only a function of Poisson’s ratio. Hence,
knowing the extremes of Poisson’s ratio for rock-like materials, we can
find the theoretical upper and lower bounds for the induced horizontal
stress.
We have

v=o, OH=o

v = 0.25, OH = 0.330,

v = 0.5, OH = 0,

showing that the lower bound is for a value of Poisson‘s ratio of zero (i.e.
the application of a vertical stress does not induce any horizontal strain),
when there is no horizontal stress induced. At the other extreme, the upper
bound is given for a Poisson’s ratio of 0.5 (the value for a fluid) when the
induced horizontal stress equals the applied vertical stress. In between,
measured values of the Poisson’s ratio for intact rock are typically around
0.25, indicating that the induced horizontal stress might be approximately
one third of the applied vertical stress.
These calculations have indicated the likely values of the vertical and
horizontal natural in situ stress components based on the application of
elasticity theory to an isotropic rock. It is also implicit in the derivations that
gravity has been ’switched on’ instantaneously to produce the stresses: this
is manifestly unrealistic. Nevertheless, we can now compare these predic-
tions with measured data collated from stress determination programmes
worldwide.

4.7 Collated worldwide in situ stress data

Because of the need to know the in situ stress state for engineering
purposes, there have been many measurements made of the in situ stress
state over the last two or three decades. In some cases, the programmes
have been rather cursory and not all components of the stress tensor have
been determined; in other cases, the programmes have specifically
attempted to estimate all six independent components of the stress tensor.
Some of these data were collected by Hoek and Brown (1980) and are
presented in the two graphs shown in Figs 4.14 and 4.15.
In Fig. 4.14, the line representing one of the equations intimated in
Section 4.6.1, i.e. 0, = 0.0272, is also shown (here, the value of 0.027 has been