Geology Reference
In-Depth Information
(4) Fluid substitution
density
ρ
b2
¼ ρ
b1
ϕ
ρ
fl1
-
In
Chapter 5
it was shown that sandstones with the
same porosity but different stiffness can show different
magnitudes of fluid substitution effect on the compres-
sional velocity. This is explained in terms of the effect
of pore space compressibility in
Fig. 8.19
. Data from
two (unrelated) sands with 30% porosity are shown for
both water and gas fill. The change in compressional
velocity in the stiffer cemented sand is far less than in
the softer uncemented sand (
Figs. 8.19a
,
b
).
The effect of stiffness on fluid substitution is
elegantly explained by considering the crossplot of
porosity vs normalised bulk modulus (K
sat
/K
0
)
(Mavko and Mukerji,
1995
; Mavko et al.,
1998
; Avseth
et al.,
2005
). Given the Gassmann relation:
1
K
d
¼
ϕ
ρ
fl2
+1
ρ
fl2
¼ ρ
w
S
w
ð
S
w
Þρ
h
+1
ρ
b2
¼ ρ
fl2
ϕ
ð
ϕ
Þρ
0
:
(5) Fluid substitution
-
shear velocity
r
μ
ρ
b2
V
s2
¼
:
-
(6) Fluid substitution
compressional velocity
1
K
0
+
K
ϕ
ð
8
:
26
Þ
t
ρ
b1
N
2
N
1
2
V
p2
¼
V
p1
2
+
β
,
it is possible to draw lines of equal normalised pore
space modulus (K
ϕ
/K
0
) on the plot. Wide spacing of
the lines indicates relatively soft rock, whereas closely
spaced lines indicate relatively stiff rocks. Given that:
1
K
sat
≈
ρ
b1
+
ρ
fl2
ρ
fl1
ρ
b1
+
ρ
fl2
ρ
fl1
ϕ
ϕ
where
K
d
K
0
1
K
fl1
+
β
ϕ
1
K
fl2
+
β
ϕ
β ¼
1
N
1
¼
N
2
¼
:
1
K
0
ϕ
K
ϕ
+K
fl
+
ð
8
:
27
Þ
K
0
K
0
ð
8
:
21
Þ
the new saturated bulk modulus (K
sat
) can be esti-
mated by using the change in normalised fluid modu-
lus to move between the normalised pore modulus
lines. Thus, if the change in normalised fluid modulus
is 0.07 (as in
Fig. 8.19
) and the normalised pore
modulus lines have 0.02 spacing, the new K
sat
can be
read directly by moving 3.5 lines from the K
sat
starting value. For a given change in the fluid modu-
lus there will be a relatively large change in K
sat
for
soft rocks but a small change for stiff rocks.
The model that is fit to the dry rock data can take
various forms depending on the dominant trends in
the data. The following are examples of functions that
might be used:
Linear fits
for porosity variations controlled by
diagenesis (i.e the critical porosity model of Nur,
1992
):
-
K
d
K
0
¼
ϕ
+1
:
ð
8
:
22
Þ
ϕ
c
Iso-stiffness fits representing porosity variations con-
trolled by sorting:
8.2.4 Minerals, fluids and porosity
The
s equation
requires values for mineral and fluid moduli as well
as densities and porosity. In the application of Gass-
mann to well log data an accurate petrophysical
analysis of mineral fractions, porosity and fluid satur-
ations is a necessary starting point.
parameterisation of Gassmann
'
K
d
K
0
¼
1
1+c
ϕ
:
ð
8
:
23
Þ
Exponential fits
-
for intermediate trends:
K
d
K
0
¼
exp
c
ϕ
:
ð
:
Þ
8
24
8.2.4.1 Mineral parameters
Values for the matrix mineral density (
Models such as that proposed by Krief et al.(
1990
)
might also be used:
ρ
0
) and bulk
modulus (K
0
) are generally based on published tabu-
lations (
Table 8.4
). Information is therefore required
on the various minerals present in the rock. Often the
petrophysicist
K
d
K
0
¼
x
ð
ϕ
Þ
ð
:
Þ
1
ϕ
,
8
25
1
163
where x is a fitting coefficient.
'
s shale volume calculation is used as
