Page 51 - Applied Petroleum Geomechanics

42 Applied Petroleum Geomechanics

of the waves (Barton, 2007). Therefore, dynamic shear modulus controls
the shear velocity of propagation:
s ffiffiffiffiffiffi s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G d E d 1
V s ¼ ¼ (2.21)
r b r 2ð1 þ n d Þ
b
For a fluid the shear modulus is zero; from the above equation, S-wave
velocity is zero. Therefore, the S-wave is unable to travel through a fluid.
2.3.2 Sonic transit time

In well log measurements the sonic transit time (travel time or sonic
slowness) is used to represent the acoustic wave velocity. The compressional
transit time is the reciprocal of the compressional velocity and has the
following form in the English unit:
10 6
Dt p ¼ (2.22)
V p
where Dt p is the compressional transit time of the formation, in ms/ft; V p is
the compressional velocity, in ft/s.
Time average equation of Wyllie et al. (1956) can be written in the
following form in terms of the compressional transit time and porosity in
the rock:

Dt p ¼ fDt f þð1 fÞDt m (2.23)
where Dt m and Dt f are the sonic transit time in the rock matrix and the pore
fluid, respectively.

2.3.3 Relationship of V p and V s
For a gas-bearing formation, V p is slowed down by gas, and the relations of
V p and V s can be used to correct the measured V p (e.g., Zhang and
Wieseneck, 2011). The ratio of compressional to shear wave velocity
(V p /V s ) depends on dynamic Poisson’s ratio (n d ) according to the following
theoretical equation:
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V p 2ð1 n d Þ
¼ (2.24)
V s 1 2n d
p ffiffiffi
The ratio V p /V s is about 3 for hard rocks, for which n d is 0.25.
However, in the case of unconsolidated sediments, the ratio V p /V s can even