Page 47 - Origin and Prediction of Abnormal Formation Pressures

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

stresses"
AFt
Acr--
A (2-15)
AFtt
At--
A

Resolution of the total stress field

The stress tensor for a porous, homogeneous, isotropic shale body can be written in
the conventional way:

crx "gxy rxz
S ~ ry x Cry 12y z (2-16)
72zx Tzy Oz.
where S signifies the symmetrical tensor of the total stress; cri and rij represent the
normal and shear forces, respectively, acting on the faces of a unit of an argillaceous
sediment.
Next, one can take moments about point O (Fig. 1-2). The tangential stress, r~y,
multiplied by the area in which it acts, gives the force rxydzdy, and this times dx, gives
a clockwise moment about O. The stress, ryx, times the area gives r,.xdxdz, and the
latter times dy results in a counterclockwise moment ryxdxdzdy. At equilibrium, the
two moments balance each other:

rxydzdydx - ryxdxdzdy (2-17)
or
rxy -- ryx (2-18)
Then it follows that

rx: - r:x (2-19)
and
ryz - rzy (2-20)
The total stress array for a point in a cylindrical body under compaction can be
expressed in cylindrical coordinates r, 0, and z"

crr TrO Trz
S ~ "fOr crO gOz (2-21)
Tzr "gzO crz
The total stress tensor can be decomposed into two distinct parts for a body of
sediment in equilibrium: (1) hydrostatic stress; and (2) deviatoric stress.