Page 593 - Petrophysics

560 PETROPHYSICS: RESERVOIR ROCK PROPERTIES

The maximum shear strain, ymm, occurs on a plane whose normal
makes an angle of 45" with the ~1 and ~3 directions, and is calculated from:

Ymax = E1 - E3 (9.9)
where: ~1 = maximum principal normal strain, and
~3 = minimum principal normal strain.
Ep = [" E2 ::I
The principal strain tensor is represented as

0
O
0 0 E3 (9.10)

The sum of the principal strains is the volumetric strain or dilatation, AV

AV = (~1 + ~2 + ~3)v (9.1 1)

where V is the initial volume of the rock. Inasmuch as strain is a ratio of
volumes or lengths, it is dimensionless.

TWO-DIMENSIONAL STRESS-STRAIN SYSTEMS

Many situations in rock mechanics can be treated as two-dimensional
problems in which only the stresses or strains in a single plane need be
considered. Poulos and Davis showed that the normal and shear stresses
on a plane making an angle 8 with the zdirection, as shown in Figure 9.5,
are [7] :
1 1
00 = - (ox + 02) + (ox - 02) cos 28 + Tx2sin 28 (9.12)
2
1
20 = T:,~COS 28 + - (ox - oZ) sin 28 (9.13)
2
and the principal stresses are:

O1 = - (Ox + (32) + - [(ox - O2I2 - 42 3 (9.14a)
1
1
2 0.5
2 2

The principal planes are inclined at an angle (81 + 90) to the z axis, where:

(9.15)