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44 Applied Petroleum Geomechanics

60

Vp/Vs =1.7 1.8 1.9 2.0 2.1
70

1/Vp (μs/ft) 80

90
shale
shaly sand
sandstone
100
100 110 120 130 140 150 160 170 180
1/Vs (μs/ft)
Figure 2.6 V p /V s plot at the true vertical depth of 27,000e30,000 ft in an ultradeep
well for shale, shaly sandstone, and sandstone in the Green Canyon, the Gulf of Mexico
(Zhang et al., 2009).

shaly sands, from 4% to 39% porosity, with clay volume fractions spanning
0%e55%, and over a confining pressure range of 0e40 MPa, equivalent to
depths up to roughly 3 km.

2.3.4 Velocity and porosity relationship
The velocity of a wave is controlled by the elastic properties and density of
material. However, porosity has a significant impact on the P-wave and
S-wave velocities in the porous rock. In general, an approximate inverse
proportionality is found between velocity and porosity, and time average
equation proposed by Wyllie et al. (1956) can be rewritten to link the
compressional velocity to porosity of the rock:
1 f ð1 fÞ
¼ þ (2.27)
V p V f V m
where V p , V m , and V f are the compressional velocities for the formation,
matrix, and pore fluid, respectively.
Analyzing laboratory-measured results in sandstones (Fig. 2.7) from
literature, the following simple relation of V p /V s and porosity was given
(Zhang et al., 2009):
A
V p =V s ¼ B (2.28)
ð1 fÞ