Geology Reference
In-Depth Information
Table 8.1.
Coefficients for Gardner's relations. Note that, based
on practical experience, the coeficients initially published for
limestone have been amended from d ¼ 1.50,
6000
Wyllie
5500
f ¼ 0.225 to
d ¼ 1.55,
f ¼ 0.3 (H. Morris, personal communication).
5000
Lithology
d
f
4500
Ss/sh avg
1.741
0.25
4000
Shale
1.75
0.265
3500
Sandstone
1.66
0.261
3000
Limestone
1.55
0.3
2500
Reuss
Dolomite
1.74
0.252
2000
0
0.1
0.2
0.3
0.4
Anhydrite
2.19
0.16
Porosity
Figure 8.4
Crossplot of porosity vs compressional velocity for
various sandstone datasets: purple points - data from Han et al.
(
1986
), dark blue points - Tertiary sandstone dataset from North Sea,
dashed red line - Reuss mix of water and quartz, red and yellow
lines - trends from Oseberg high porosity data (Dvorkin and Nur,
1996
), blue lines - trends from Troll high porosity data (Dvorkin and
Nur,
1996
) with upper line 20MPa effective pressure and lower
line 5MPa effective pressure, red points
The Gardner relations may be used in transform-
ing sonic or density logs for the purposes of replacing
missing sections or in constraining the results of
inversions for P and S reflectivity (e.g. White,
2000
).
Owing to the lack of universal applicability it
is
advised that area-specific density
velocity relations
are derived from the available data. In some cases it
may be more appropriate to derive density from a
transform based on shear velocity rather than com-
pressional velocity (e.g. Potter and Stewart,
1998
).
-
-
selected unconsolidated
sand data.
carbonate data from Eberli et al.(
2003
). The best fit
line drawn on the graph is close to a prediction using
Wyllie
s equation. Intriguingly, high permeability car-
bonates with moldic and intra-frame porosity plot
above the trend and low permeability carbonates with
microporosity and interparticle porosity plot below
the trend (Eberli et al.,
2003
). In general, to quote
Weger et al., (
2009
),
'
8.2.2.2 Wyllie's('time average') equation
Wyllie et al.(
1958
) derived a relationship between
velocity and porosity that fits data from well consoli-
dated sandstones and limestones. The relation is
essentially intuitive rather than based on physical
principles. Wyllie
carbonates with a large amount
of microporosity, a complex pore structure (high
specific surface), and small pores generally show low
acoustic velocity at a given porosity. Samples with a
simple pore structure (low specific surface) and large
pores show high acoustic velocity for their porosity
'
s equation relates the sonic interval
transit time (i.e. the reciprocal of the seismic velocity,
usually measured in
'
s/ft in sonic logs) to the
weighted addition of interval transit times through
the pore fluid and the mineral matrix. It is often
referred to as the
μ
'
.
When traditional porosity logs are not available,
the Wyllie equation is sometimes used by petrophy-
sicists as a way to calculate porosity:
'
time-average
'
equation):
t
¼
ϕ
t
fl
+
ð
1
ϕ
Þ
t
0
,
ð
8
:
6
Þ
ϕ
¼
ð
t
t
0
Þ=
t
f
t
0
ð
8
:
7
Þ
where
is the porosity (as a fraction) and t, t
fl
and t
0
are the interval transit times in the rock, the fluid and
the mineral matrix.
Figure 8.4
illustrates the form of
Wyllie
ϕ
Clearly the porosity estimate will be erroneous for
rocks which fall below the Wyllie trend. An empirical
correction to this porosity estimate that extends it to
relatively unconsolidated rocks is to multiply it by
100/t
sh
, where t
sh
is the shale interval transit time at
or near the depth of interest, in
'
s equation in the context of a selection of
different (brine-bearing) sandstone data. It is evident
that Wyllie
s equation falls at the top of the data-
points, indicating that it is a model appropriate for
stiff well consolidated sands. It is often observed to fit
well with data from carbonate rocks (see
chapter 7
)
where the mineralogy has a key role in establishing a
relatively stiff rock frame.
Figure 8.5
illustrates some
'
s/ft.
An improved version of the Wyllie equation was
published by Raymer et al.(
1980
):
μ
153
2
V
0
+
V
¼
ð
1
ϕ
Þ
ϕ
V
fl
,
ð
8
:
8
Þ
