Geology Reference
In-Depth Information
a)
b)
Figure 2.12
Volume and shape changes
in rocks under stress; (a) change of
volume with stress applied equally from
all directions, (b) change in shape
associated with shear stress (e.g. stresses
applied parallel to bedding boundaries).
Normal stress (S) =
Force (kg
.
m/sec
2
) / Area (m
2
)
no change
in volume
V
Shear stress = stress
parallel to a surface
w
=
w
Transverse strain
Longitudinal strain
ðσÞ¼
Δ
Poisson ratio
l
¼
l
=
Δ
3K
2
μ
¼
μÞ
:
ð
2
:
9
Þ
2
ð
3K+
The contrast in Poisson
s ratio across an interface can
have a large control on the rate of change of ampli-
tude with offset.
Elastic moduli are not generally measured directly,
for example with downhole logging tools, but they
can be calculated from velocity and density measure-
ments. Some key equations relating velocities and
densities to elastic properties are as follows:
'
l
V
s
K+4
s
λ
μ=
3
+2
μ
V
p
¼
or V
p
¼
ð
2
:
10
Þ
ρ
ρ
and
w
r
,
μ
ρ
V
s
¼
ð
2
:
11
Þ
ρ
where
is the density of the material.
It is evident from these equations that the com-
pressional velocity is a more complicated quantity
than the shear wave velocity, involving both bulk
and shear moduli. Useful equations that illustrate
the relationship between P and S velocities and Pois-
son
Figure 2.13
Poisson's ratio.
λ
and
μ
) in prefer-
'
s ratio (
σ
) are shown below:
s
2
ence to (K and
) believing that they offer greater
physical insight. The Lamé constant
μ
V
p
V
s
¼
ð
σÞ
λ
1
is given by
ð
2
:
12
Þ
ð
1
2
σÞ
2
3
:
λ ¼
K
ð
2
:
8
Þ
2
σ ¼
γ
2
V
p
V
s
2
σ
2
2
, where
γ ¼
and
γ ¼
1
:
ð
An important elastic parameter in AVO is Poisson
'
s
2
γ
2
σ
ratio. Poisson
s ratio is the ratio of the fractional
change in width to the fractional change in length
under uni-axial compression (
Fig. 2.13
). It can be
shown that it is given by:
'
2
:
13
Þ
'
The general relationship between V
p
/V
s
and Poisson
s
ratio is shown in
Fig. 2.14
together with typical ranges
13
