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et al.( 1998 ) derived a modified Voigt bound, with a
curve drawn from the mineral point at zero porosity
to the point where the critical porosity intersects the
Reuss bound (the so-called critical porosity model).
Other possibilities include the Hashin
6000
Voigt average
5000
Shtrikman
(HS) upper bound modified to include the critical
porosity (Avseth et al., 2005 ) and the contact cement
model (Dvorkin and Nur, 1996 )( Chapter 8 ). Within
the envelope defined by the elastic bounds, sandstones
tend to fall along trends either sub-parallel to the
Reuss bound (with porosity increase related to
improving sorting) or steeper near linear (modified
Voigt or critical porosity) trends related to diagenesis
(i.e. cementation).
A good example of these trends is shown in
Fig. 5.19 . The data are from an 80 m section in which
unconsolidated sands and sandy shales (sometimes
referred to as heterolithics; Avseth et al., 2003 ) overlie
cemented sands. The low angle sorting trend is clear
in the cluster associated with the uncemented sands as
is the high angle trend associated with the change
from uncemented to cemented sands (these sands
have roughly 2% cement). Presumably, with add-
itional chemical diagenesis the clean sands would
follow the modified HS bound to the mineral point
at about 6000 m/s. Figure 5.19b shows that the
cemented sands have higher values of AI and lower
values of Poisson
-
Rock stiffening
through compaction
and diagenesis
4000
3000
Decrease in sorting
Critical porosity
for sand
2000
Reuss average
1000
0
20
40
60
80
100
Porosity %
Suspensions
Sand-clay mixtures
Sand
Clay-free sandstone
Clay-bearing sandstone
Figure 5.18 Velocity-porosity characteristics of brine-bearing
siliciclastic sediments (data from Han, 1986 ; Yin, 1992 ; Hamilton 1956 ;
modified after Marion et al., 1992 Avseth et al., 2005 ).
Variations in stiffness of sandstones can be related to
a number of factors such as the number of grain
contacts, the amount and type of cement, clay/shale
content and the distribution of pore shapes through-
out the rock.
The broader context for the effects of pore geom-
etry and rock fabric on elastic properties is illustrated
in Fig. 5.18 . On this plot of compressional velocity vs
porosity, sand/shale rocks fall within an envelope
defined by two bounds, the Reuss ( 1929 ) and Voigt
( 1910 ) bounds. These bounds are a useful type of rock
physics model, essentially representing different ways
of mixing rock and fluid; the softest possible mix is
the Reuss bound (harmonic average) and the stiffest is
the Voigt bound (arithmetic average) (see Chapter 8
for more discussion on Voigt and Reuss bounds). It is
evident from Fig. 5.18 that the Reuss bound (also
referred to as Wood
s ratio. The rock fabric differences
shown here have a significant impact on the AVO
response. Taking average values for the shales above
the sand, two single interface AVO crossplots have
been generated for each of the sand types. The unce-
mented sands show Class II behaviour in which oil
sands would be expected to give a brightening of the
negative amplitude response at top sand. In contrast,
the cemented sands show Class I behaviour and the
presence of oil would be evident as a dimming of the
positive amplitude.
The inference from Fig. 5.19a is that the initial
compaction phase is characterised by a steep slope
on the velocity vs porosity crossplot. Figure 5.20
illustrates some data from 12 wells offshore West
Africa and Gulf of Mexico for arenites (sands and
sandstones with 2%
'
s( 1955 ) relation) effectively
describes the behaviour of suspensions. Note that
the yellow points refer to marine ooze data acquired
at or close to the seabed.
It is also clear from Fig. 5.18 that the Voigt bound
in this context is not especially useful to describe
trends of compressional velocity change with
changing porosity. It is possible to draw a more
effective upper bound in a number of ways. Nur
'
12% clay content). The figure
clearlyshowsthechangeinslopeonthevelocityvs
porosity plot marking the transition from domin-
antly mechanical compaction and initial pressure
solution in unconsolidated sands (high rate of vel-
ocity increase with decreasing porosity) to the zone
of lower porosities in consolidated sands within
-
70
 
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