45
Μ = Κ g
2 / (D - K m )
(2.17c)
D = K g · (1 + φ · ( K g /K f - 1))
(2.17d)
The apparent mass factor m = a’⋅ρ f /φ (a’ ≥ 1) in
Equation 2.16b considers that not all of the pore
fluid moves along the maximum pressure gradient
in case of tortuous, curvilinear capillaries. As a result the pore fluid seems to be more dense, with
higher inertia. (a’) is called structure factor and is
equal to 1 in case of uniform parallel capillaries.
In the low frequency limit H - 4/3µ m (Eq. 2.17a)
represents the bulk modulus computed by Gassmann (1951) for a ‘closed system’ with no pore
fluid flow. If the shear modulus (µ m ) of the frame is
additionally zero, the sediment is approximated by
a dilute suspension and Equation 2.17a reduces to
the reciprocal bulk modulus of Wood’s equation for
‘zero’ acoustic frequency, (K
-1
= φ/K f + (1-φ)/K g ;
Wood 1946).
Additionally, Biot (1956a, b) introduced a
complex correction function (F) which accounts
for a frequency-dependent viscous flow resistance (η/κ). In fact, while the assumption of an
ideal Poiseuille flow is valid for lower frequencies,
deviations of this law occur at higher frequencies.
For short wavelengths the influence of pore fluid
viscosity confines to a thin skin depth close to
the sediment frame, so that the pore fluid seems to
be less viscous. To take these effects into account
the complex function (F) modifies the viscous flow
resistance (η/κ) as a function of pore size, pore
fluid density, viscosity and frequency. A complete
definition of (F) can be found in Stoll (1989).
The equations of motions 2.16 are solved by a
plane wave approach which leads to a 2 x 2
determinant for P-waves
0
det
2
2
2
2
2
2
2
2
=
⎟
⎟
⎠
⎞
−
⋅
−
⋅
−
⎜
⎜
⎝
⎛
−
⋅
−
⋅
κ
η
ω
ω
ω
ρ
ω
ρ
ρω
F
i
k
M
m
k
C
k
C
k
H
f
f
(2.18)
and a similar determinant for S-waves (Stoll 1989).
The variable k(ω) = k r (ω) + ik i (ω) is the complex
wavenumber. Computations of the complex zeroes
of the determinant result in the phase velocity
c(ω) = ω/k r (ω) and attenuation coefficient α(ω) =
k i (ω) as real and imaginary parts. Generally, the
determinant for P-waves has two and that for Swaves one zero representing two P- and one Swave propagating in the porous medium. The first
P-wave (P-wave of first kind) and the S-wave are
well known from conventional seismic wave
propagation in homogeneous, isotropic media.
The second P-wave (P-wave of second kind) is
similar to a diffusion wave which is exponentially
attenuated and can only be detected by specially
arranged experiments (Plona 1980).
An example of such frequency-dependent phase
velocity and attenuation curves presents Figure 2.12
for P- and S-waves together with the slope (power (n))
of the attenuation curves (α = k ⋅ f
n
). Three sets of
physical properties representing typical sand, silt
and clay (Table 2.2) were used as model parameters. The attenuation coefficients show a significant change in their frequency dependence.
They follow an α∼f
2
power law for low and an
α∼√f law for high frequencies and indicate a
continuously decreasing power (n) (from 2 to 0.5)
near a characteristic frequency f c = (ηφ)/(2πκρ f ).
This characteristic frequency depends on the
microstructure (porosity (φ), permeability (κ)) and
pore fluid (viscosity (η), density (ρ f )) of the sediment and is shifted to higher frequencies if
porosity increases and permeability decreases.
Below the characteristic frequency the sediment
frame and pore fluid moves in phase and coupling
between solid and fluid components is at maximum. This behaviour is typical for clayey sediments in which low permeability and viscous
friction prevent any relative movement between
pore fluid and frame up to several MHz, in spite of
their high porosity. With increasing permeability
and decreasing porosity the characteristic
frequency diminishes so that in sandy sediments
movements in phase only occur up to about 1 kHz.
Above the characteristic frequency wavelengths
are short enough to cause relative motions
between pore fluid and frame.
Phase velocities are characterized by a lowand high-frequency plateau with constant values
and a continuous velocity increase near the characteristic frequency. This dispersion is difficult to
detect because it is confined to a small frequency
band. Here, dispersion could only be detected
from 1 - 10 kHz in sand, from 50 - 500 kHz in silt
and above 100 MHz in clay. Generally, velocities in
coarse-grained sands are higher than in finegrained clays.
S-waves principally exhibit the same attenuation and velocity characteristics as P-waves.
However, at the same frequency attenuation is
2.4
Acoustic and Elastic Properties
Μ = Κ g
2 / (D - K m )
(2.17c)
D = K g · (1 + φ · ( K g /K f - 1))
(2.17d)
The apparent mass factor m = a’⋅ρ f /φ (a’ ≥ 1) in
Equation 2.16b considers that not all of the pore
fluid moves along the maximum pressure gradient
in case of tortuous, curvilinear capillaries. As a result the pore fluid seems to be more dense, with
higher inertia. (a’) is called structure factor and is
equal to 1 in case of uniform parallel capillaries.
In the low frequency limit H - 4/3µ m (Eq. 2.17a)
represents the bulk modulus computed by Gassmann (1951) for a ‘closed system’ with no pore
fluid flow. If the shear modulus (µ m ) of the frame is
additionally zero, the sediment is approximated by
a dilute suspension and Equation 2.17a reduces to
the reciprocal bulk modulus of Wood’s equation for
‘zero’ acoustic frequency, (K
-1
= φ/K f + (1-φ)/K g ;
Wood 1946).
Additionally, Biot (1956a, b) introduced a
complex correction function (F) which accounts
for a frequency-dependent viscous flow resistance (η/κ). In fact, while the assumption of an
ideal Poiseuille flow is valid for lower frequencies,
deviations of this law occur at higher frequencies.
For short wavelengths the influence of pore fluid
viscosity confines to a thin skin depth close to
the sediment frame, so that the pore fluid seems to
be less viscous. To take these effects into account
the complex function (F) modifies the viscous flow
resistance (η/κ) as a function of pore size, pore
fluid density, viscosity and frequency. A complete
definition of (F) can be found in Stoll (1989).
The equations of motions 2.16 are solved by a
plane wave approach which leads to a 2 x 2
determinant for P-waves
0
det
2
2
2
2
2
2
2
2
=
⎟
⎟
⎠
⎞
−
⋅
−
⋅
−
⎜
⎜
⎝
⎛
−
⋅
−
⋅
κ
η
ω
ω
ω
ρ
ω
ρ
ρω
F
i
k
M
m
k
C
k
C
k
H
f
f
(2.18)
and a similar determinant for S-waves (Stoll 1989).
The variable k(ω) = k r (ω) + ik i (ω) is the complex
wavenumber. Computations of the complex zeroes
of the determinant result in the phase velocity
c(ω) = ω/k r (ω) and attenuation coefficient α(ω) =
k i (ω) as real and imaginary parts. Generally, the
determinant for P-waves has two and that for Swaves one zero representing two P- and one Swave propagating in the porous medium. The first
P-wave (P-wave of first kind) and the S-wave are
well known from conventional seismic wave
propagation in homogeneous, isotropic media.
The second P-wave (P-wave of second kind) is
similar to a diffusion wave which is exponentially
attenuated and can only be detected by specially
arranged experiments (Plona 1980).
An example of such frequency-dependent phase
velocity and attenuation curves presents Figure 2.12
for P- and S-waves together with the slope (power (n))
of the attenuation curves (α = k ⋅ f
n
). Three sets of
physical properties representing typical sand, silt
and clay (Table 2.2) were used as model parameters. The attenuation coefficients show a significant change in their frequency dependence.
They follow an α∼f
2
power law for low and an
α∼√f law for high frequencies and indicate a
continuously decreasing power (n) (from 2 to 0.5)
near a characteristic frequency f c = (ηφ)/(2πκρ f ).
This characteristic frequency depends on the
microstructure (porosity (φ), permeability (κ)) and
pore fluid (viscosity (η), density (ρ f )) of the sediment and is shifted to higher frequencies if
porosity increases and permeability decreases.
Below the characteristic frequency the sediment
frame and pore fluid moves in phase and coupling
between solid and fluid components is at maximum. This behaviour is typical for clayey sediments in which low permeability and viscous
friction prevent any relative movement between
pore fluid and frame up to several MHz, in spite of
their high porosity. With increasing permeability
and decreasing porosity the characteristic
frequency diminishes so that in sandy sediments
movements in phase only occur up to about 1 kHz.
Above the characteristic frequency wavelengths
are short enough to cause relative motions
between pore fluid and frame.
Phase velocities are characterized by a lowand high-frequency plateau with constant values
and a continuous velocity increase near the characteristic frequency. This dispersion is difficult to
detect because it is confined to a small frequency
band. Here, dispersion could only be detected
from 1 - 10 kHz in sand, from 50 - 500 kHz in silt
and above 100 MHz in clay. Generally, velocities in
coarse-grained sands are higher than in finegrained clays.
S-waves principally exhibit the same attenuation and velocity characteristics as P-waves.
However, at the same frequency attenuation is
2.4
Acoustic and Elastic Properties