Page 87 - Applied Petroleum Geomechanics

78 Applied Petroleum Geomechanics

For unconsolidated sediments Biot’s coefficient is large and may
approach 1, and the following empirical equation was suggested by Lee
(2003):
184:05
a ¼ 0:99494 (2.89)
1 þ e ðfþ0:56468Þ=0:10817
where porosity f is in fractions.
Biot’s coefficient predicted from Eq. (2.89) is adequate for a differential
pressure about 20 MPa based on the calibration from the data presented by
Domenico (1976). Notice that these empirical equations did not consider
the effect of the differential pressure.

2.7.4 Biot’s coefficient estimate from well logs

Shear deformation does not produce a pore-volume change, and conse-
quently different fluids do not affect shear modulus. Therefore, dry rock
shear modulus (m d ) should be equal to the saturated formation shear
modulus (m fm ). Thus, Biot’s coefficient can be approximately obtained using
the formation shear modulus and matrix shear modulus (m ma ) as shown in
the following equation, particularly for a gas-filled formation (Krief et al.,
1990):
m fm
a ¼ 1 (2.90)
m ma
Therefore, dynamic Biot’s coefficient can be expressed as the following
form:
r V 2
b
a d ¼ 1 S (2.91)
r V Sma
2
m
where V S and V Sma are the shear velocities of the formation and the matrix,
respectively.
From Eq. (2.90), if the dynamic formation shear and matrix shear
moduli have the same constant (k) to convert to static ones, then Eq. (2.91)
is also the static Biot’s coefficient (a).
The formation bulk density can be expressed as a function of the matrix
density (r m ) and porosity, i.e., r b ¼ (1 f)r m þ fr f . Substituting it to
Eq. (2.91), the following equation can be obtained:
V 2
S
a ¼ 1 ½ð1 fÞþ fr =r Š V Sma (2.92)
f
m
2