Page 49 - Origin and Prediction of Abnormal Formation Pressures

32 G.V. CHILINGAR, J.O. ROBERTSON JR. AND H.H. RIEKE III

where D is the deviatoric portion of the total stress tensor. The effect of the deviator
stress is to produce a distortion, which is elastic or plastic in nature and is introduced
into the shale body.

Total stress tensor
If the sediment body is not in equilibrium, the second component will not be a
symmetric tensor for "gxy ~ "gyx. Ramsay (1967, p. 282) subdivided the asymmetric
tensor into symmetric and skew-symmetric parts. The hydrostatic stress component is
the same as in Eq. 2-23. The second symmetrical part is the deviatoric stress component
which can be expressed as follows:
l(r~z + rz~)
(O'x --O'w) l "~ ( 72 x y -+- "C y x ) -~
1 1
D -- ~('gxy + "gyx) (Oy - ~w) ~("gyz 'Jl- "gzy) (2-27)
1 1
~ ( "g x z + "g z x ) "~ ( 72 y z + 7Jay) ( O'z --O'w)
The skew-symmetric part is termed the disequilibrium component, which causes the
shale to undergo a rotation in space and is expressed as:

0 l ~('rxy + ryx) l(r~ + r~)
R = 1 ! (r~,:. + r:..~,) (2-28)
1
1
~(r:.~ + r~:.) ~(r:.y + ry:.) 0
where R is the disequilibrium component. Such a stress state would be anticipated if
tectonic forces were acting on the shale mass in a basin within a geosyncline. The
total stress tensor for a shale body not in equilibrium is expressed as the sum of the
above-described parts:

S= P+D+R (2-29)
Namely, the total stress = hydrostatic stress + deviatoric stress + disequilibrium
component. Each one of the three components making up the state of stress is directly
related to the respective component of the strain tensor. The hydrostatic portion of
the stress system causes changes in volume, the deviatoric stress components cause
distortion, and the disequilibrium components cause the material to undergo rotation in
space (Ramsay, 1967).
Lo (1969) demonstrated that the pore pressure induced by shear may be expressed as
a sole function of the major principal strain. According to him, the only unambiguous
and correct principle of superposition of pore pressure is to consider an isotropic stress
system and a deviatoric stress system, namely,
m
ACrl 0 0 Act3 0 0 (ACrl -- Act3) 0 0
0 Act 2 0 -- 0 Act 3 0 + 0 (Ao 2 -- /ko 3) 0
(2-30)
0 0 AO- 3 0 0 AO" 3 0 0 0