Page 48 - Origin and Prediction of Abnormal Formation Pressures

ORIGIN OF ABNORMAL FORMATION PRESSURES 31

Hydrostatic stress state

The component attributable to the interstitial fluid is the hydrostatic stress (pressure),
O-w, which can be regarded as being continuous throughout the medium. The normal and
shear stress components are given by:

O-wx 72wxy "Cwxz

P -- "Cwyx O-wy rwyz (2-22)
rwzx "gwzy Crwz
where P is the hydrostatic tensor. It can be assumed that under hydrostatic conditions
no shearing stresses exist in the interstitial fluid. By definition, a fluid is a substance
that cannot sustain tangential or shear forces when in static equilibrium. This may not
hold true for adsorbed water because of its probable quasi-crystalline nature. Hubbert
and Rubey (1959, p. 138) noted that if a viscous fluid occupies the pore space, there
are then microscopic shear stresses, which are expended locally against the fluid-solid
boundaries. Thus, their only macroscopic effect is to transmit to the solid skeleton by
viscous coupling whatever net impelling force may be applied to the interstitial fluid.
In any stress system with the principal stresses, O-~, O-y, and cr z, one can determine the
local mean value for the hydrostatic stress, 6w, as:
1
6w -- 5(o-wx .qt_ O-wy + O-wz) (2-23)
Now, the hydrostatic stress tensor, P, can be represented by

-iw 0 0
P ---- 6w 0 (2-24)
0 ~w
and
P--5 1 (36w) -- 6w (2-25)
The above expression represents the hydrostatic pressure of a fluid whether it is
flowing or is stationary in the porous system of the shale. Note that O-wx -- O-wy =
O-wz - 6w, and that the hydrostatic portion of the total stress system causes only volume
changes in the deformed material.

Deviatoric stress state

The second component is known as the stress deviator from the hydrostatic state. It is
expressed as the difference between total stress and the hydrostatic stress, which resists
deformation:
(O-x -- O-wx ) 75xy "Cxz
D -- ryx (o-y -- O-wy) ryz (2-26)

rzx rzy (O-z -- O-wz)