Page 145 - Introduction to Petroleum Engineering

132 RESERVOIR GEOPHYSICS
Seismic velocities can be estimated from correlations that depend on rock type.
For example, Castagna et al. (1985) presented a correlation for seismic velocities in
sandstone. The velocities depend on porosity and clay content C:

−
.
.
.
V = 5819 42φ − 221 C (7.25)
P
and
−
.
.
.
V = 3897 07φ − 204 C (7.26)
S
where the seismic velocities are in km/s, C is expressed as the fraction of clay volume,
and porosity ϕ is a fraction.
7.5.2 IFM Moduli
Gassmann’s equation (Gassmann, 1951) was introduced to provide an estimate of
saturated bulk modulus, which is the bulk modulus of a rock saturated with fluids.
Gassmann’s equation is widely used in rock physics because it is relatively simple
and does not require much data. Saturated bulk modulus K* in the IFM is approxi-
mated as a form of Gassmann’s equation. It is given by

1 − K ( / K ) 2
K * = K +  IFM m  (7.27)
IFM K ) +(1 − / K ) −( K / K )
φ
2
( /φ f m IFM m
where K IFM is dry frame bulk modulus, K is rock matrix grain bulk modulus, and K
m
f
is fluid bulk modulus. The term “dry” means no fluid is present in the rock pore
space. The rock would not be considered “dry” if the pore space was filled with gas,
including air; instead, the rock would be considered gas saturated. The variables in
Equation 7.27 can change as reservoir conditions change.
Fluid bulk modulus is the inverse of fluid compressibility c :
f
1 1
K = = (7.28)
f c f ( cS + cS + cS )
o o w w g g
where fluid compressibility is the saturation weighted average of phase compress-
ibility. The bulk modulus of rock saturated with pore fluid can be calculated in terms
of seismic velocities by rearranging Equations 7.19 and 7.20:

K * = ρ  V − 4 V 2   (7.29)
2

B  P 3 S 
Similarly, effective shear modulus is
V
G * = ρ BS 2 (7.30)
Saturated bulk modulus K* equals grain modulus K when porosity goes to zero.
m