Page 85 - Applied Petroleum Geomechanics

76 Applied Petroleum Geomechanics

It is rarely reported on how to convert dynamic Biot’scoefficient to
the static one. However, dynamic bulk modulus can be related to static
bulk modulus by using empirical equations presented in Section 2.5.5.
As described in Section 2.6.3, the difference of dynamic and static
Poisson’s ratios is small; therefore, the correlations of dynamic and static
Young’s moduli can be used to obtain the correlations in dynamic and
static bulk moduli. For example, Eq. (2.72) can be expressed in the
following form:

K s ¼ kK d (2.82)
where K s and K d are the static and dynamic bulk moduli, respectively; k is a
correlation factor between 0.5 and 0.79 from Eq. (2.72).
Combining Eqs. (2.81) and (2.82), the static Biot’s coefficient, a, can be
obtained in the following form:

kr V 4V sd 2 3
2
d
pd
a ¼ 1 (2.83)
K m
where k is a correlation factor between 0.5 and 0.79 from Eq. (2.72).
Combining Eqs. (2.83) and (2.81), the relation of dynamic and static
Biot’s coefficients can be obtained, i.e.,

a ¼ð1 kÞþ ka d (2.84)
For example, if a d ¼ 0.7 and k ¼ 0.6, then from Eq. (2.84) a ¼ 0.82.
Fabricius et al. (2008) studied dynamic Biot’s coefficients in the North
Sea chalk samples, which varied considerably in depositional texture and
mineralogy. They calculated dynamic Biot’s coefficients from Eq. (2.81)
using the velocities of the dry chalk samples and the matrix bulk modulus of
calcite. Their results indicate that dynamic Biot’s coefficient tends to
decrease with porosity in clay-poor samples as a reflection of calcite
cementation, causing the mineral frame to stiffen as porosity is filled by
cement. Samples rich in smectite or chlorite tend to have relatively high
Biot’s coefficient, probably because the presence of clay prevents calcite
cement from bridging between particles.

2.7.3 Empirical methods for Biot’s coefficient
If no measured Biot’s coefficient is available, the empirical equations can be
used to estimate Biot’s coefficient. Experimental results show that Biot’s
coefficient is a function of porosity; therefore, most empirical equations for