Geology Reference
In-Depth Information
be output from the Kuster
Toksöz model but owing
to the isolated nature of the pores within the solid this
represents a high-frequency solution.
The effect of pore aspect ratio using the Xu-White
model is shown in
Fig. 8.37
. For reference, a curve
representing Wyllie
-
Constant
cement
Cementing (diagenesis)
trend
s equation is shown, with a good
fit for pores with an aspect ratio of about 0.09. This is
consistent with the knowledge of consolidated sands
in which most of the crack-like pores have been closed
during compaction and diagenesis. Whilst direct
measurement of pore aspect ratios is possible, for
example from thin sections,
'
V
p
Variable
cement
Sorting
trend
in practice the pore
-
Porosity
aspect ratios input to the Xu
White model are effect-
ively fitting parameters, tuned to give results in agree-
ment with observed velocity logs. It should be noted
that the mineral modulus of clay is also a variable in
the fitting procedure.
Xu found typical values of 0.02
Figure 8.35
Schematic illustration of sorting and cementing
trends in clean sandstones.
8.2.7 Inclusion models
Inclusion models approximate the rock as an
elastic solid containing inclusions (i.e. pores). The
Xu
0.05 for clay-
related pores, 0.12 for sand-related pores, and 0.1 for
carbonate-related pores. Typically, single best-fit
values for the sand aspect ratio and the clay aspect
ratio are determined over a logged section of a few
hundred metres.
Figure 8.38
shows an example, where
there is a good fit to the observed P wave velocity log.
In this case there is no logged shear data available, but
the Xu
-
White (
1995
,
1996
) model is an excellent example
of an inclusion model that is readius applied to log
data. The model depends on theory developed by Kus-
ter and Toksöz (
1974
) for a two-phase medium, relat-
ing porosity and pore aspect ratio to P and S velocities.
The reader is recommended to consult the work of
Mavko et al.(
1998
) for a detailed discussion of the
Kuster
-
White prediction for the shear log is higher
than the Greenberg
-
Castagna prediction (black curve
in shear velocity track) which has been proven to
work in this area. After obtaining the required esti-
mates of the aspect ratios from the log fit, the model
can be used to predict the effect of varying clay or
porosity; it could also be used to replace defective
sections of the sonic logs (
Section 8.4.3
).
As a velocity model, the Xu
-
Toksöz model and its context within the broad
range of rock physics models. The key aspect of the
Xu
-
White pore model is that pores are split into clay-
related and sand-related pores and each has a different
pore aspect ratio (the ratio of shortest axis to longest
axis). Clay particles are expected to create pores with
low aspect ratios, and these crack-like pores will have
limited stiffness, whereas sand grains form pores with
larger, stiffer aspect ratios, and these sub-spherical
pores will have high stiffness.
The original Kuster
-
White (
1995
) model
is reasonably straightforward and provides insight
into the reasons for velocity change. For example it
predicts the expected behaviour with increasing shale
content for dispersed shale. Yan et al.(
2007
) have
suggested improvements to the estimation of sand
pore aspect ratios by using a relationship with poros-
ity and shale content. Keys and Xu (
2002
) have shown
that it is possible to make an accurate approximation
to the Xu
-
Toksöz theory required the
pore concentration to be dilute, in the sense that the
porosity had to be much less than the aspect ratio. In
order to avoid this restriction, Xu and White use an
effective medium approach, in which the properties of
the medium are calculated in stages; at each step a
proportion of the porosity is introduced, small
enough for the dilute porosity condition to be satis-
fied, and the output medium properties are used as the
input to the next stage of porosity introduction. The
outputs from Kuster
-
White method which is computationally
less intensive, by making certain assumptions about
the dry rock properties. The Xu
-
White (
1995
) model
is, however, purely volumetric, taking no account of
where the clay or shale is located within the rock. The
reader is referred to Sams and Andrea (
2001
) for a
discussion of the effects of clay distribution in the
context of this type of velocity modelling.
-
Toksöz modelling are the dry
rock moduli which can be saturated using Gassmann
-
s
equations (
Fig. 8.36
). The saturated moduli can also
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