Geology Reference
In-Depth Information
10.1 Stoneley Waves in Permeable Wells - Background
Earlier in this topic, we had demonstrated the importance and role of
“dispersion relations” in wave propagation. For instance, the sinusoidal
propagating wave assumption “sin (kx - t)” in the non-dissipative wave
equation u
tt
- T u
xx
= 0 l e ads to a /k = (T/ ) relating the real frequency to
its wavenumber k in one-dimensional applications. We also addressed lateral
vibrations on beams, gravity waves in the ocean and other problems, showing
how different physical insights can be drawn from our kinematic wave
approach, key to which is the availability of a dispersion relation. It is important
to remember that, while a dispersion relation can be obtained from mathematical
analysis, it can also be deduced from experimental data when the mathematics is
intractable or not convenient. Not all dispersion relations are as simple as those
given previously; in fact, the great majority in engineering are not. Authors
Chang, Liu and Johnson (1988) in “Low-Frequency Tube Waves in Permeable
Rocks” demonstrated importantly that the dispersion relation
2 (1 -
i
)
(
/(2C
D
)) K
1
[(1 -
i
) r
b
(
/(2C
D
))]
1/V
T
2
=
bf
{1/K
bf
+ 1/N -
}
i
r
b
K
0
[(1 -
i
) r
b
( /(2C
D
))] (10.1)
applies to Stoneley wave propagation in
concentric
holes. This is a direct
consequence of the “Biot model” in the low-frequency limit. It is also consistent
with earlier results of White in
Underground Sound Application of Seismic
Waves
in the limit of high frame rigidity, published by Elsevier Science in 1983.
In Equation 10.1, the following nomenclature is assumed:
V
T
. . . Complex phase velocity
bf
. . . Density of borehole fluid
K
bf
. . . Bulk modulus of borehole fluid
N . . . Solid shear modulus
r
b
. . . Borehole radius
. . . Angular frequency
. . . Pore fluid viscosity
. . . Formation permeability
C
D
. . . Frame rigidity factor
K
0
. . . Modified Bessel function of order “0” with complex argument
K
1
. . . Modified Bessel function of order “1” with complex argument
Our purpose is
not
to analyze Equation 10.1, which shows that the complex
phase velocity depends on seven independent terms through complicated Bessel
functions with complex arguments. It is quite correct - but it goes without
saying that any data processing using it is likely to be numerically intensive and
unlikely to provide any significant physical insight. Is there a better way?
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