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Rock physical and mechanical properties 75

differential pressure (the confining pressure minus pore pressure) increases.
Laboratory bulk volume test results (Fatt, 1959) on the Boise sandstone
show that Biot’s coefficient values range from 1 (very low differential
pressure of 0.5 MPa), 0.82 (differential pressure of 20 MPa) to 0.77 (high
differential pressure of 60 MPa). Therefore, Biot’s coefficient values
decrease as the differential confining pressure increases.
The Bakken shale samples from the Williston Basin were tested to
obtain Biot’scoefficient (He et al., 2016) through measuring the varia-
tions of both confining pressure and pore pressure. The experimental
results show that Biot’scoefficients are dependent on rock permeability
and have anisotropic behavior. The vertical (perpendicular to the
bedding) samples have much lower Biot’scoefficients than those in the
horizontal samples, i.e.,

a V ¼ ca h (2.79)
where a V and a h are the vertical and horizontal Biot’s coefficients, respec-
tively; c is a parameter, c ¼ 0.79 to 1 from the test results.

2.7.2 Dynamic Biot’s coefficient
Theoretically, Biot’s coefficient cannot be calculated from the bulk
modulus obtained by dynamic methods (such as P-wave and S-wave
measurements) because the static modulus values are required by the the-
ory. However, dynamic bulk modulus, compressional velocity, and other
dynamic values can be used to obtain an empirical Biot’s coefficient. The
dynamic bulk modulus in a dry rock can be obtained from the following
theoretical equation:

K d ¼ r V 4V sd 2 3 (2.80)
2
d
pd
where K d , r d , V pd , and V sd are the dynamic bulk modulus, bulk density,
compressional velocity, and shear velocity of the dry rock, respectively.
The dynamic Biot’s coefficient may be obtained by replacing K dry by
Eq. (2.80) in Eq. (2.77), i.e.,

r V 4V sd 2 3
2
d
pd
a d ¼ 1 (2.81)
K m
where a d is the dynamic Biot’s coefficient.