Page 130 - Applied Petroleum Geomechanics

122 Applied Petroleum Geomechanics

3.4.6 HoekeBrown failure criterion
By studying experimental results of a wide variety of rocks, Hoek and
Brown (1980) presented the following empirical failure criterion for jointed
rock masses:
s ¼ s þ UCS ms UCS þ s a (3.62)

0
0
0
1 3 3
where s 1 and s 3 are the maximum and minimum effective principal
0
0
stresses, respectively; m and s are the constants depending on the properties
of the rock and on the extent to which it has been broken before being
subject to the stresses; a is a constant depended on the rock mass character-
istics; m ranges from 0.001 (extremely weak rock) to 25 (extremely strong
rock) and s ¼ 1 for intact rock; s < 1 for previously broken rock.
For intact rocks that make up the rock mass, Eq. (3.62) simplifies to:
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 0 mUCSs þ sUCS 2
0
s ¼ s þ (3.63)
1 3 3
For clastic sediments, Hoek and Brown (1997) suggested using the
following values for m:
m ¼ 22 for conglomerate; m ¼ 19 for sandstone; m ¼ 9 for siltstone; and
m ¼ 4 for claystone.
3.4.7 True triaxial failure criterion

Polyaxial compression (true triaxial) tests demonstrate that rock strength is a
function of the major principal stress (s 1 ) and the minor principal stress (s 3 )
as well as the intermediate stress (s 2 ). Therefore, rock failure characteristic
depends on the effects of all three principal stresses. For ductile materials,
the von Mises criterion is defined as the following form to consider all three
principal stresses:
1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2
2
s oct ¼ ðs 1 s 2 Þ þðs 1 s 3 Þ þðs 3 s 1 Þ ¼ d (3.64)
3
where s oct is the octahedral shear stress; d is a material-dependent constant,
p ffiffi
2
d ¼ s y ; s y is the yield strength of the material (UCS for the rock). Eq.
3
(3.64) states that the yield point is reached when the distortional energy,
represented by the octahedral shear stress increases to a constant d. Nadai
(1950) recommended that the von Mises yield criterion for ductile metals
can be adapted to rocks by replacing the constant d with a monotonically
rising function f N of the octahedral normal stress s oct or the mean stress s m :
s oct ¼ f N ðs oct Þ (3.65)