Geology Reference
In-Depth Information
including simple interference models. Importantly,
rock physics data can be used in conjunction with
relevant geological knowledge to generate modelled
scenarios, often termed
The Voigt and Reuss bounds are defined, for
example with a mix of quartz and water, as follows.
Voigt modulus:
+ K
w
Vol
w
, which can
contribute to the understanding of the relationship
between seismic amplitude attributes and reservoir
properties (
Chapter 10
) as well as the attendant
uncertainty in predictions.
Fundamental to each of these steps are good qual-
ity velocity and density log data, together with an
accurate petrophysical interpretation (i.e. mineral
content, porosity and water saturation determined
from various log suites). Another major component
is the use of rock physics models, algorithms or
empirical relationships, which can be fit to measured
velocity and density log responses and then used
predictively. In this way rock physics models can be
used to predict missing log sections or to highlight
potentially erroneous log sections. Perhaps the most
important rock physics model is Gassmann
'
pseudo-wells
'
¼
K
qtz
Vol
qtz
ð
Þ
,
ð
8
:
1
Þ
K
sat
voigt
where
K
qtz
¼
modulus of quartz,
K
w
¼
modulus of water,
Vol
qtz
¼
volume fraction of quartz,
volume fraction of water.
Reuss modulus (describing a lower bound for
mineral/fluid suspensions):
Vol
w
¼
1
¼
:
ð
:
Þ
K
sat Reuss
8
2
Vol
qtz
K
qtz
+
Vol
w
K
w
The modified Voigt or critical porosity model (Nur
et al.,
1998
) provides a more realistic upper bound for
sandstones and is defined by:
K
qtz
+K
s(
1951
)
equation, used for determining the effect of substitut-
ing fluids on the velocities of rocks with intergranular
porosity.
'
ϕ
ϕ
c
K
sat mod voigt
¼
1
K
ϕ
c
c
,
ð
8
:
3
Þ
ϕ
where K
ϕ
c
is the Reuss modulus at critical porosity.
Another commonly used set of bounds was pub-
lished by Hashin and Shtrikman (
1963
). These
bounds in general provide a slightly tighter constraint
than the Voigt
8.2 Rock physics models and relations
There are a large number of rock physics models and
relations that provide tools for data QC, characterisa-
tion and generation of model scenarios. Not all are
covered here for the sake of brevity; the reader is
referred to Mavko et al.(
1998
) and Avseth et al.
(
2005
) for a thorough treatment.
The most commonly employed models can be
categorised as follows:
Reuss values, although the equations
are more complex. For a mixture of two materials the
Hashin
-
Shtrikman upper (HS
+
) and lower (HS
)
bounds are defined by:
-
f
2
K
HS
¼
K
1
+
1
Þ
1
+ f
1
K
1
+
3
μ
1
ð
K
2
K
1
f
2
HS
μ
¼ μ
1
+
ð
8
:
4
Þ
1
+
2f
1
K
1
+2
μ
1
ð
Þ
theoretical bounds,
μ
2
μ
1
μ
1
K
1
+
4
3
μ
1
Þ
:
5
empirical models,
where K,
and f represent the bulk moduli, shear
moduli and volume fractions of the individual phases
1 and 2. If the stiffest material is denoted by 1 and the
softest material is denoted by 2 the equations calculate
the upper bound. To calculate the lower bound requires
that the stiffest material is denoted by 2 and the softest
material is denoted by 1. Berryman (
1995
)hasprovided
a general form that can applied to more than 2 phases
(see also Mavko et al.,
1998
).
Figure 8.2
illustrates the
bounds in terms of a mix of quartz and water. Note that
the lower HS bound is identical to the Reuss bound
because the shear modulus of brine is zero.
Whilst theoretical bounds are not especially pre-
dictive for the purposes of describing the velocities of
μ
Gassmann
'
s equation,
contact models,
inclusion models.
8.2.1 Theoretical bounds
Theoretical bounds establish the physical limits of the
properties of mixtures of minerals and fluids. The
idea of bounds was introduced in
Chapter 5
in terms
of the effective limits of porosity
velocity behaviour
of sandstones, with the lower bound determined by
the Reuss average (describing a suspension of mineral
and fluid) and the upper bound by a modified Voigt
bound.
-
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