Geology Reference
In-Depth Information
Undisturbed rock
x
z
Direction of propagation
Rarefaction
P wave
x
Compression
Particle
motion
z
Direction of propagation
S wave
x
Figure 2.11 Density components of a rock with intergranular
porosity.
z
necessary to perform the calculation in terms of elas-
tic moduli before the effect on velocities and density
can be appreciated (see Chapter 8 ).
There are a variety of elastic parameters that can
be used to describe the isotropic and elastic behaviour
of rocks. Table 2.1 provides a useful reference (from
Smidt, 2009 ) , illustrating that two independent meas-
urements can be used to calculate any other elastic
parameter. Elastic moduli describe rock responses to
different types of stress (i.e. force applied over a unit
area). The bulk modulus, for example, is the rock
response to normal stress applied in all directions on
a unit of rock ( Fig. 2.12 ) and relates fractional volume
change
Direction of propagation
Figure 2.10 Schematic illustration of P and S waves.
can indicate the presence and orientation of fractures
in the subsurface (Lynn, 2004 ) ( Chapters 5 and 7 ).
Bulk density
ρ b is a relatively simple parameter,
being calculated as the weighted average of the dens-
ities of the components ( Fig. 2.11 ):
ϕ ¼ðρ 0 ρ b Þ=ðρ 0 ρ fe Þ
,
ð
2
:
4
Þ
where
ρ 0 parameters
are the densities of the fluid in the pore space and of
the rock matrix respectively. If more than one fluid or
mineral is present the effective densities are calculated
simply by weighting the various component densities
according to their proportions. Equation (2.4) can be
re-written to calculate porosity from bulk density:
ϕ ¼ðρ b ρ 0 Þ=ρ 0 ρ fl Þ:
ϕ
is the porosity and the
ρ fl and
V/V to the uniform compressive stress S:
Δ
S
K
¼
V :
ð
2
:
6
Þ
Δ
V
=
As such the bulk modulus is an indicator of the
extent to which the rock can be squashed. It is some-
times encountered as its reciprocal, 1/K, called the
compressibility. The shear modulus (
ð
:
Þ
2
5
) is the response
to a tangential or shearing stress and is defined by
μ
shear stress
shear strain ,
2.3.2.2 Isotropic and elastic moduli
Unfortunately it is insufficient simply to focus on
velocity and bulk density. Understanding the isotropic
and elastic context of velocities and density is neces-
sary in order to appreciate the rock physics tools at the
disposal of the seismic modeller. For example, when
calculating the effect of varying fluid fill on sandstone
velocities and densities (i.e. fluid substitution) it is
μ ¼
ð
2
:
7
Þ
where the shear strain is measured through the shear
angle ( Fig. 2.12 ). As such the shear modulus indicates
the rigidity of the rock or the resistance of the rock
to a shaking motion. Most fluids are not able to resist
a shear deformation, so it is usually assumed that the
shear modulus of fluids is zero.
11
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