Page 59 - Applied Petroleum Geomechanics

50 Applied Petroleum Geomechanics

where c 11 , c 33 , c 44 , and c 66 are the elastic constants (refer to Eq. (1.39)). For
VTI rocks (see Fig. 1.14), g determines variation of horizontally polarized
shear velocity with angle from vertical, and ε relates the horizontal P veloc-
ity, V PH , to the vertical P velocity, V PV :
2
V 2 ¼ V ð1 2εÞ
PH
PV
(2.30)
V PV z V PH ð1 εÞ
Wang (2002) measured velocities and anisotropies of many shales and
reservoir rocks from oil and gas fields around the world. The results show
that anisotropy in shales ranges from 6% to 33% for P-waves (ε) and
2%e55% for S-waves (g). The coal sample is extremely anisotropic,
showing over 40% anisotropy for both P- and S-waves. The magnitude of
anisotropy decreases exponentially in shales as porosity increases.

2.4 Permeability
2.4.1 Permeability and hydraulic conductivity
Permeability is one of the most important physical properties of a porous
medium. It measures quantitatively the ability of a porous medium to
conduct fluid flow. Permeability of a rock depends largely on the
connectedness of the void spaces, the grain size of the rock, and the
cementation between rock grains. A rock could be extremely porous, but if
each pore is isolated from the others, the rock would be impermeable. If
rock grain size is small, then void spaces of the rock are small; therefore, the
surface film of the fluid can actually choke the movement of additional
fluids through the small spaces.
Permeability is a tensor in a manner similar to the stress tensor. More
often the permeability is isotropic in the direction of the bedding plane but
anisotropic in the direction of perpendicular to the bedding plane.
Therefore, if two coordinates are in the plane of the bedding, the two
permeabilities having equal values and denoted as the horizontal perme-
ability, k h , and the other coordinate in the direction perpendicular to the
bedding denoted as the vertical permeability, k V , then the permeability
tensor can be expressed in the following form:

2 3
k h 0 0
k ¼ 4 0 k h 0 5 (2.31)
7
6
0 0 k V