Page 22 - Applied Petroleum Geomechanics

12 Applied Petroleum Geomechanics

where ε x , ε y , ε z are the normal strains in x, y, and z directions, respectively;
u x , u y , u z are the displacements in x, y, and z directions, respectively.
The shear strains can be expressed as follows:
vu x vu y
g ¼ g ¼ vy þ vx ;
yx
xy
vu y vu z
g ¼ g ¼ þ ; (1.17)
zy
yz
vz vy
vu z vu x
g ¼ g ¼ vx þ vz
zx
xz
where g xy , g yz , g zx are the engineering shear strains in x, y, and z direc-
tions, respectively; and g xy ¼ g yx ¼ 2ε xy ¼ 2ε yx ; g yz ¼ g zy ¼ 2ε yz ¼ 2ε zy ;
g zx ¼ g xz ¼ 2ε zx ¼ 2ε xz
The tensorial normal and shear strain components of the infinitesimal
strain tensor can be expressed in the following matrix forms:

2 3 2 3
ε xx ε xy ε xz ε x g xy 2 g =2
xz

6 7 6 7
ε ¼ 4 ε yx ε yy ε yz 5 ¼ 4 g yx 2 ε y g yz 2 5 (1.18)

ε zx ε zy ε zz g =2 g zy 2 ε z
zx
Note that this matrix is symmetrical and hence has six independent
components.
1.4 Stressestrain relations in isotropic rocks
1.4.1 Stressestrain relations for different rocks

Rocks behave mechanically different under compression tests, and different
models can be used to describe the stressestain behaviors. Fig. 1.11 illustrates
some typical models to describe stressestrain constitutive relationships. The
commonly used model assumes that the rock has a linear elastic stressestrain
relationship in which elasticity can be applied, as shown in Fig. 1.11A.
The stress and strain in uniaxial compression in Fig. 1.11A follows a linear
relationship, and rock failure happens when the stress reaches the rock
strength. This behavior is mainly for brittle rocks.
Fig. 1.11B illustrates the elastic perfectly plastic behavior, i.e., the rock
has the elastic behavior before the stress reaches the peak strength. After
reaching the peak strength, a constant stress state is kept (residual strength is
the same as the peak strength, i.e., no stress drop), but straining continues,