Geophys. 1. Inl. (1990) 101, 169-180
Attenuation of elastic waves in a cracked solid
S. Xu and M. S. King
Deparhent of Geology and Mineral Resources Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BP, V K
Accepted 1989 October 17. Received 1989 October 10; in original form 1988 December 7
SUMMARY
The spectral ratios technique is used to measure the attenuation and phase
dispersion of the compressional wave and two shear waves polarized parallel and
perpendicular to the cleavage of a slate, before and after cracks had been induced in
the cleavage plane. The experimental results show that the quality factor Q of the
rock sample is affected significantly by the presence of cracks, and that Q is more
sensitive to crack parameters than the corresponding wave velocity.
The frequency dependence of the attenuation coefficient a is generally assumed to
be proportional to the nth power of frequency, con, where it is suggested that n has a
value lying between 0.5 and 4,depending on the mechanism of the attenuation. The
measured frequency dependence of a in this study, where a is influenced strongly by
presence of aligned cracks, is demonstrably non-linear for both P- and Sz- (polarized
parallel to the cleavage) waves, implying that Q is frequency dependent. The convex
shape of the attenuation coefficient curves for P-waves indicates a value of n of
rather less than unity (0.5 f 0.2), and the concave shape of the curves for &-waves
suggests a value of n of' more than unity (1.8 f0.2), indicating that the frequency
dependence of the attenuation also depends on wave mode. The attenuation
coefficient for &-waves (polarized perpendicular to the cleavage of the slate) has
also been observed to behave in a non-linear manner ( n = 3.5 f 0.3) at low confining
pressures. This is probably due to attenuation by scattering, where the frequency
dependence in the long-wavelength limit is predicted to be to the fourth power of
frequency. At high confining pressures, the scattering factor is found to be negligible
and a linear relationship (constant Q) is observed, probably as a result of friction
between crack surfaces.
The crack-induced phase dispersion for P-, S1- and &-waves has been observed to
maintain causality. Although the P- and &-wave velocities are not found to be
sensitive to cracking (as predicted by Hudson's theory), the corresponding Q values
have been found to vary significantly with confining pressure. This behaviour can be
explained as due to the closure of low aspect ratio cracks at high confining pressure.
The &-wave phase dispersion is the only one which can be demonstrated to obey the
Kramers-Kronig relation. This is made possible because the &-waves can be
recorded over a sufficiently wide frequency band with high enough signal-to-noise
ratios.
Key words: anisotropy , attenuation, cracks, dispersion, fractures.
INTRODUCTION
Elastic wave attenuation and phase dispersion are two
important phenomena of wave propagation, showing
significant characteristics of rock properties. In a theoretical
study, Crampin (1981) shows that wave attenuation has a
much greater anisotropy induced by aligned cracks than
does the corresponding velocity. This implies that the wave
attenuation is much more sensitive to stress-induced cracks.
Increasing interest in seismic wave attenuation has created
a need for a further understanding of the mechanism of
anelastic wave absorption. It is well known that wave
attenuation is due to the following two main effects.
(a) Intrinsic attenuation, which converts part of the
elastic energy into heat. The proposed mechanism for
169
170
S.
Xu and M. S. King
intrinsic attenuation include friction at the grain boundaries
and across the surfaces of microcracks, microscopic fluid
flow (Biot 1956a, b), intercrack squirt flow (Mavko & Nur
1975; O’Connell & Budiansky 1977), intracrack flow
(Mavko & Nur 1979), anelasticity, viscosity, and the
relaxation of pore fluids.
(b) Apparent attenuation, which includes reflection,
scattering and geometric spreading, and involves no energy
loss.
Biot (1956a. b) developed a comprehensive theory of
wave propagation in a fluid-saturated two-phase medium.
The model consists of a matrix containing parallel cylindrical
pores saturated with a liquid and a plane P-wave is assumed
to travel in the direction of these cylindrical pores. Biot
(1956a, b) shows that shear stresses are generated within a
skin depth which depends on the frequency of the
compressional wave and the viscosity of the fluid. The
amplitude of these stresses decreases rapidly away from the
wall of the pores, but it is large enough to influence the
overall elastic behaviour within a skin depth. Biot theory
assumes that the anelasticity arises from the viscous
interaction between the fluid and the framework of the solid
particles, and predicts a frequency dependence of
attenuation coefficient of f at low frequencies and f ‘I2 at
high frequencies.
One of the assumptions made in Biot’s (1956a, b) theory
is that the pore sizes are closely grouped around a mean
value. McCann & McCann (1985) extended Biot’s theory to
the case of a distribution of pore sizes and in so doing
obtained a better agreement with experimental results.
Based on the modified Biot theory, these authors predict
a linear (f’) variation of attenuation coefficient with
frequency in the frequency range 10KHz to 2.25MHz for
water-saturated porous rocks.
Frictional sliding has been suggested as the main effect of
cracks on wave attenuation (Murphy 1982). A considerable
amount of data indicates that the quality factor Q is
essentially independent of frequency from low to high
frequencies (Wuenschel 1965; Hovem & Ingram 1979). The
attenuation coefficient LY is proportional to w e - ’ , so a
constant Q is equivalent to the linear frequency dependence
of LY as proposed by McCann & McCann (1985). Hamilton
(1972) concluded that the attenuation coefficient is a linear
function of frequency in the range 0.01-1000KHz for the
majority of sediments. A frequency-independent Q has been
interpreted by many authors in terms of frictional sliding at
grain boundaries or across crack faces (e.g. Johnston,
Toksoz & Timur 1979; Johnston & Toksoz 1980). Stoll
(1974, 1977, 1980) attempted to account for the linear
variation of attenuation coefficient with frequency by
introducing model complex moduli which are assumed to
arise from frictional losses generated by relative movement
of solid particles. McCann & McCann (1985) comment that
although this theory correctly predicts a linear relation of
attenuation coefficient with frequency, the attenuation is
amplitude dependent and there is no velocity dispersion.
This certainly does not conform to the principle of causality
(cf. Aki & Richards 1980). Savage (1969) points out that for
typical strain amplitudes of seismic waves and for reasonable
microcrack dimensions, the computed slip across crack faces
is less than the interatomic spacing. This small interaction
probably cannot be described with conventional macrofrictional models.
The attenuation of elastic waves in a cracked solid in the
long wavelength limit (ka << 1; where k is the wavenumber
and a is the mean crack radius) has been studied
theoretically by, amongst others, Chatterjee, Knopoff &
Hudson (1980) and Hudson (1981). In the case of thin
penny-shaped cracks filled with a viscoelastic fluid,
Chatterjee et al. (1980) suggest that elastic waves travelling
through such a medium are attenuated by loss of energy due
to anelasticity of the material filling the cracks, as well as by
scattering of the waves by the cracks. In their model, the
crack concentration is supposed to be dilute, and no
interaction of the scattering waves between cracks is taken
into account. The Chatterjee el al. (1980) derivation shows
that the attenuation coefficient caused by anelasticity of the
filling material is proportional to frequency, while the
attenuation coefficient caused by scattering is proportional
to the crack density and to the fourth power of frequency
for a weakly viscous fluid contained in the cracks. They also
conclude that the effect of the fluid or filling material is
significantly more important at low frequencies than that of
scattering, since the scattering term varies as the fourth
power of frequency. Hudson’s (1981) theory only takes the
scattering effects into account and shows the same frequency
dependence of the attenuation due to scattering effects as
that derived by Chatterjee el al. (1980). Hudson (1981) also
demonstrates that the attenuation coefficient of a cracked
solid is proportional to crack density and to the third power
of ratio of mean crack radius to wavelength.
The main limitation of the theories of Chatterjee et al.
(1980) and Hudson (1981) in application to the real Earth is
that they are based on long-wavelength scattering (Rayleigh
scattering). Wu & Aki (1985) point out that in the
intermediate- to high-frequency range (ka = l), the frequency dependence of attenuation due to scattering is
dependent on the inhomogeneity spectrum of the medium.
This makes the frequency dependence of the attenuation
coefficient due to scattering even more complicated.
It is generally found that wave attenuation is accompanied
by phase dispersion in an anelastic solid. This implies that
any wave travelling through an anelastic solid must show a
frequency-dependent shift of phase. Futterman (1962)
assumes that the attenuation and phase dispersion of an
anelastic wave obey the Kramers-Kronig relation, which
states that the attenuation coefficient and the corresponding
slowness are a Hilbert transform pair. This indicates that
phase dispersion can be used as an independent means for
measuring wave attenuation and as a method for
distinguishing the frequency dependence of velocity and Q
(Jacobson 1987). However, problems still exist. As Jacobson
(1987) points out, if the Hilbert transform is to be used to
determine attenuation, the phase dispersion for all
frequencies must be known, not just over the limited
bandwidth inherent in digital data.
It seems probable that the Kramers-Kronig relation is
valid for intrinsic attenuation. Apparent attenuation should
produce velocity dispersion. Jacobson (1987) points out that
the notion that the apparent attenuation function and
velocity dispersion obey the Kramers-Kronig relation needs
further clarification.
Liu, Anderson & Kanamori (1976) proposed a concept of
Wave attenuation in cracked solid
a spectrum of relaxation mechanisms to represent the
absorption and dispersion of a linear viscoelastic solid and
showed that the Q value for a single relaxation mechanism is
not necessarily frequency independent, and hence the
attenuation coefficient could be a non-linear function of
frequency. However, after solving the Boltzmann aftereffect equation for a standard linear solid with both a finite
number or a continuous distribution of relaxation times, Liu
et al. (1976) found that Q is approximately constant. Both
the attenuation coefficient and phase velocity were found to
be linear functions of frequency over a limited frequency
range.
It is well established that microcracks in rock have a
significant effect on wave attenuation. In fact, most
proposed attenuation mechanisms, such as friction, microscopic flow, intracrack flow, intercrack squirt flow, and
scattering, are associated with cracks. In this paper we
report experimental measurements of wave attenuation and
velocity dispersion of a slate specimen before and after
cracking, obtained using a spectral ratios technique. Some
of the proposed mechanisms are eliminated by ensuring that
the rock was tested in a dry condition.
BASIC B A C K G R O U N D
Assuming there is a plane wave travelling through a linear
viscoelastic solid in the x-direction, the amplitude is given
by
A ( o ) = AO(w)ei(m'-KX)
(1)
where w is the angular frequency and K is the complex
wavenumber defined by
in which C ( w ) is the phase velocity and a ( w ) is the
attenuation coefficient. Considering both the intrinsic and
apparent attenuation, the attenuation coefficient can be
expressed as
.(w) = yw"
171
Kramers-Kronig relation. Azimi, Kalinin & Pivovarov
(1968) suggest the attenuation coefficient is weakly
frequency dependent through the relationship
(.
0 )=
ff0"
-
1 + (Y1w
where mo and a1are constants, and crlw << 1 for all seismic
frequencies. It can be seen that ( ~ ( 0
aw
) is not apparent
until w is very large. In the case where (alw)' is vanishing
small, the phase velocity is found to be
1
-=-
1
2cYO
(7)
in which In denotes the natural logarithm.
EXPERIMENT
One of the most frequently used methods for measuring the
elastic wave attenuation and quality factor Q in the
laboratory has been described by Toksoz, Johnston & Timur
(1979). In this method, the attenuation is measured in
relation to a reference sample with very low attenuation,
using the spectral ratios technique. Toksoz et al. (1979) state
that the sample to be studied and the reference sample
should have the same shape and geometry, and the two
measurements should be made using identical procedures.
The Fourier transforms of plane elastic waves for the
reference and the sample are of following forms
A,(w) = GI(x)ei(m'-K1x),
(8)
A,(w) = G2(x)ei(W'-KzX)
(9)
where A is the Fourier transform, w is angular frequency, x
is distance, K is the complex wavenumber and G(x) is a
geometric factor which includes the effects of spreading,
reflection, coupling etc. Dividing A,(w) by A2(w) and
taking natural logarithms, we obtain
Introducing (2) into (lo), we obtain
or
Q-1
= ylgn-l
(3)
where y and y' are constants. The exponent n lies in the
range 0.5-4, depending on the dominant mechanism of the
wave attenuation.
For a causal wave propagating through an anelastic solid,
Futterman (1962) suggests that the relationship between
attenuation and dispersion is given by a Hilbert transform
pair
(4)
or
(5)
where H denotes the Hilbert transform, and C , denotes the
highest phase velocity. This relation is known as the
where LU, and a1 are the attenuation coefficients of the
specimen to be studied and the reference respectively, and
C , and C , are the corresponding phase velocities. If the
specimen and the reference sample have the same geometry,
GI = G2 and the second term on the left may be ignored. In
this case, the amplitude of In (A,/A2) is used to estimate the
wave attenuation and the phase of In(A,/A,) is used to
estimate phase dispersion.
If the Q value of the reference sample is large enough, the
attenuation coefficient a1 can be ignored; the attenuation
coefficient a, can then be determined. In these experiments,
aluminium was selected as a standard reference. The Q
value of aluminium is greater than 150 0o0, so that the error
introduced by ignoring
is negligible.
The system employed for the experiments reported here is
172
S. Xu and
M. S. King
P
Controller
Figure 1. Simplified testing system
shown in Fig. 1. It consists of a PUNDIT pulser, a digital
oscilloscope, an oscilloscope controller, a preamplifier, two
switch boxes and two stacks of transducers. A voltage signal
generated by the pulser is fed to one of the three
piezo-electric transducers mounted in the transmitter, which
converts it into elastic energy. The pulse of the mechanical
energy travels through the transducer holder face, the rock
sample and finally the face of the receiving ,transducer
holder. The corresponding piezo-electric transducer
mounted in the receiver converts it back to an electrical
signal. This signal is amplified before being digitized by a
programmable digitizer in the oscilloscope, and displayed on
the screen. The controller is used to transfer the digitized
waveform from the oscilloscope to a microcomputer or store
it on a magnetic tape within the controller.
The P-wave and S-wave transmitters and receivers are
broadband, with a frequency range from 300 to 800KHz.
One of the three piezo-electric transducers mounted in the
transmitter and receiver respectively is used for generating
or receiving P-waves and the others for two orthogonally
polarized S-waves. The rock sample is a right cylinder
approximately 60 mm in length and 55 mm in diameter, with
its axis parallel to the cleavage. The rock sample is mounted
between the two transducer holders so that one of the shear
waves is polarized perpendicular to the cleavage. This shear
wave is referred to as S, and the other, polarized parallel to
the cleavage, is referred to as S,.
An axial loading stress on the specimen is applied through
the transducer holders by means of a compression testing
machine, where a confining pressure is applied by mineral
oil under pressure between the sample and a Hoek cell
jacket. A load cell and a pressure transducer are employed
to monitor the axial stress and the confining pressure,
respectively. The attenuation measurements are made under
a constant axial stress of lOMPa, while the confining
pressure ranged from 1.4 to 20.7 MPa.
Cracks were formed in the slate specimen prior to testing
by subjecting it to axial stress condition to failure, which is
defined as a point after which the stress-strain curve has a
negative slope, in a servo-controlled compression testing
machine. The specimen was loaded under constant
strain-rate control until failure was reached. It was then
unloaded to less than 50 per cent failure load, and reloaded
to failure a number of times until the desired crack density
had been achieved. Measurements of P, S, and S, velocities
before, during and after the loading cycles indicate that the
cracking lies completely within the cleavage plane, as did
observations of the specimen made after removal from the
test cell.
Figure 2 shows waveforms before and after cracking,
where (a) refers to the P-wave, (b) to the &-wave and (c) to
the &-wave. These waveforms provided an indication of the
variation in velocities and attenuation with confining
pressure. It can be seen that the P- and &-wave velocities
are virtually unchanged before and after cracking, while S,
shows strong variation with confining pressure.
Figure 3 shows the attenuation coefficient of the specimen
during loading but before cracking for P-, S,- and $-waves,
respectively. It can be seen that the attenuation coefficients
for P- and &-waves remain identical with increasing
confining pressure. A slight change of the frequency
dependence of the attenuation coefficient for S,-wave has
been observed. The changing magnitude for the S,-wave is
probably due to variation in coupling between the
transducer holders and the rock samples during the
measurements.
In cases of non-linear frequency dependence of (Y it is
necessary to estimate mean Q values over a small frequency
range, assuming constant Q within that range, using a leastsquares fitting procedure. Selection of this frequency range
was found to be crucial. On the one hand, the energy of the
selected frequencies must be high enough to make the
measurement reliable, and on the other hand, the frequency
range should be chosen as wide as possible to obtain the
frequency dependence accurately enough. Fig. 4 shows the
amplitude spectra of P-, S,-and S,- waves for the reference,
and the slate sample before and after cracking, respectively.
In these experiments, the selected frequency bands were
500-750 KHz for the P-wave, 210-410 KHz for the S,-wave
and 300-600KHz for the &-wave. The reasons for these
different frequency bands are that the P - and S-wave
transducer characteristics are different and that in the
transducer stack the S, and S, transducers have different
frequency responses. The wavenumber corresponding to
these frequencies and to the velocities for the slate after
cracking lies in the range 600-800m-'. After cracking, the
crack size is of the order of 1 mm. This leads to a value of ka
of approximately 0.6-0.8. The calculated Q values are listed
in Table 1.
Table 1 shows that the Q of the specimen before cracking
varies very slightly with increasing confining pressure, well
within the estimates of errors involved in the measuring
technique. The variations for &-waves, however, are
unpredictable, and are most likely caused by noise. The
dramatic changes in the quality factor caused by fracturing
indicate that the variation in Q values is more sensitive to
induced cracks than variation in the corresponding wave
velocity. In this particular example, the relative changes in
Q values at a confining pressure of 1.4MPa are 63, 78 and
65 per cent for P-, S,- and &-waves, respectively, while the
relative changes in the corresponding wave velocities are 5,
8 and 3 per cent. The larger changes in both Q value and
velocity due to fracturing for the S,-wave indicate that the
slower shear wave is the best one to be used to monitor
crack parameters. The significant amount of change in Q
values and high signal-to-noise ratios for P- and &-waves
demonstrate that P- and &-waves should also be used to
monitor the newly produced cracks.
In order to investigate the effects of the newly produced
W A V E F 0R M S.
Ssmple: SL2.
Oste: 5-15-88.
Wsve mode:
p
Conflnlng Pressure
13.8
10
.20 .
,
Travel-tlme
. 30 .
. 40
(In micro-sl
( a )
W A V E F 0R M S.
SamDle: sL2.
. 20
10
Wave mode: si.
D a t e : 5-15-88
. 30
40
Jravel-tlme ( l n micro-s).
( b )
W A V E F 0R M S.
Samolc: sL2.
Confining P r e s s u r e [
D a t e : 5-15-88
I
wave mode: s 2
1
I
2 0 . 7 (HPal
13.0
z
6.9
r cracking
1.4
I
20.7
I
\
A
"
I
. 10
I
II
I
I
,
20
. 30
II
,
40
Travel-tlme (in micro-s).
( c )
Figure 2. Ultrasonic waveforms of the sample S12 at confining pressures 1.4, 6.9, 13.8 and 20.7 MPa before and after cracking: (a) refers to
P-waves; (b) &-waves; (c) &waves.
16.0 -
18.0
14.0
12.0
-
-
0.07
5
,
,
,
I
I
I
5.8
5.4
I
,
6.2
I
I
6.6
I
I
I
I
7.4
7
J
7.8
FREQUENCY (*lo0 KHZI
(a)
30,
I
ij
-
c--f
0-
22
Pen 6.9 MPa
Pea13.8 MPa
Pen20.7 MPa
Pa: Conflnlng Prssaura
20
18
0
;
2.1
,
,
2.3
,
,
2.5
,
,
2.7
,
,
2.9
,
I
3.1
FREOUENCY I100
-
I
I
3.3
I
I
3.5
I
I
,
3.7
3.9
I
4.1
KHr)
6.9 MPa
Pe=13.8 MPa
P ~ 2 0 . 7MPa
ct Pe=
*-•
12
Po: Conflnlng Pressure
10
6
:
0
-2
-10
;
2
I
I
I
I
3
4
I
I
5
I
I
6
FREQUENCY (*lo0 KHz)
( c )
Figure 3. Illustration of attenuation coe5icient of P-waves (a), &-waves (b), and $-waves (c) before cracking at confining pressures of 1.4,6.9,
13.8 and 20.7 MPa.
0.14
.
-
0.13
-
0.15
0.12
-
0.11
-
Sanple b e f o r e cracking
0.08 0.07 0.06 0.1
0.09
0.05
-
0.03
-
0.02
-
0.04
0
2
6
8
10
12
14
FREOUENCY (100 KHr)
( a )
0.13
0.12
Sample before cracking
0.11
0.1
0.09
0.08
W
0
0.07
c
(
-I
4
0.06
0.05
0.04
0.03
0.02
0.01
0
0
2
4
6
8
10
12
10
12
14
FREOUWCY I100 KHZ)
0.07
0.06 -
+
3
0.05
-
0.04
-
M
-I
a
5
0.03-
0
2
4
6
8
FREOUENCY [n 100 KHZI
( c l
Figure 4. Illustration of the amplitude spectra of P-waves (a), S,-waves (b), and $-waves (c) for aluminium and the slate sample, before and
after cracking.
176
S.
Xu
and
M. S. King
Table 1. Tabulation of Q values and the corresponding velocities of the slate
specimen for P-waves (a), &-waves (b), and &-waves (c). The selected frequency
range is 500-750 KHz for P-waves, 210-410 for &-waves and 300-600 KHz for
&-waves.
Confining
Pressure
(ma)
1.4
6.9
13.8
20.1
Before
Q-value
21.8
22.4
21.8
21.2
Cracking
Velocity
(ds)
After
Q-value
8.2
9.3
11.0
12.0
6789
6789
6789
6189
Cracking
Velocity
(ds)
6465
6510
6510
6555
(a)
Confining
Pressure
(MPa)
1.4
6.9
13.8
20.1
Conf ininq
Pressure
(MPa)
1.4
6.9
13.8
20.1
Before
Q-value
39.8
32.9
30.9
36.2
Before
Q-value
Cracking
Velocity
(m/s)
After
Q-value
2655
2715
2723
2731
8.6
16.3
13.8
17.5
Cracking
Velocity
After
Q-value
(m/s)
46.8
46.3
45.3
47.0
Cracking
Velocity
(rn/s)
2431
2514
2662
2700
Cracking
Velocity
( d s )
4168
4168
4168
4168
16.1
19.0
24.9
28.6
4027
4062
4079
4079
(C)
cracks on wave attenuation, we follow the Chatterjee et al.
(1980) approach and suppose that the attenuation is
composed of two parts:
(a) that due to all non-crack influences, which is assumed
to be a linear function of frequency; and
(b) that due to the presence of cracks, which includes
scattering, friction of the crack surfaces and attenuation due
to anelasticity of the material filling the cracks.
where
Ignoring the geometric term In (Gb/Ga),we obtain
Thus, it is assumed that
+ ac(w)
d w )=
(12)
where (Y is the total attenuation coefficient of the specimen
after cracking, a, is the attenuation resulting from the
non-crack effects, equivalent to the intrinsic attenuation of
the specimen before cracking, and a, is the attenuation
coefficient caused by newly produced cracks, which includes
the effects of frictional sliding across the crack surfaces,
anelasticity of the filling material and scattering of the elastic
waves.
In this case, the amplitudes of the waves transmitted
through the sample before cracking (Ab) and after cracking
(A,) are
If we take the sample before cracking as a reference, the
crack-induced attenuation coefficient can be estimated using
the spectral ratios technique,
In
(2) (2)
- In
=i(K,
- Kb)X
which shows that the amplitude of In (&/A,) can be used to
estimate the crack-induced attenuation coefficient a=,and
the phase spectrum of ln(Ab/A,) can be used to estimate
phase dispersion
CRACK-INDUCED WAVE ATTENUATION
Figures 5(a) and (c) show the attenuation coefficients of Pand &-waves due to cracking at confining pressures of 1.4,
6.9, 13.8 and 20.7MPa. In this particular case, the P- and
&wave attenuation due to scattering is predicted to be
vanishingly small by Hudson (1981). All the energy loss of
the waves must be caused by intrinsic attenuation, most
likely due to frictional sliding in this case where the sample
is tested in its dry condition. The results reported here show
that the attenuation coefficient curves of the P-waves are
convex down (Fig. 5a) and the curves for $-waves are
concave down (Fig. 5c). The least-square fitting technique
has been used to the exponent n, and it has been found that
n = 0.5 f 0.2 for P-waves and n = 1.8 f 0.2 for &-waves.
This indicates that the frequency dependence of wave
attenuation also depends on the wave mode. The results
also show that the frequency dependence of LY generally
24
I-
Gu
L
LL
W
8
z
0
5z
-
"t.."
2220
-
12
/
10
-
--+
W
5
*-•
Pe= 6.9 MPa
Pe=13.8 MPa
Pe=20.7 MPo
Pa: Confining Pressure
4-
20
I
I
I
I
I
I
1
I
I
I
t
4!
0
LL
L
W
0
0
z
0
I-
b:
3
z
W
6
( b )
-
*-)
t-
22
Go_
20
w
0
16
LL
LL
0
z
0
tb:
3
z
E
Pee 6.9 MPa
Pe-13.8 MPa
Pe-20.7 MPa
14
12
10
8
6
4
2
0 1
2
I
I
I
I
3
4
I
1
5
I
1
6
FAEQUWCY (a100 Mrl
( c )
Fngure 5. Illustration of the crack-induced attenuation coefficients for P-waves (a), &-waves (b), and &-waves (c) at confining pressures of 1.4,
L
n
1% Q ,.A
3n 7 MP-
20.00
19.00
18.00
17.00
16.00
15.00
14.00
13.00
12.00
11.00
10.00
9.00
-
--
-
Peg 6 . 9 MPa
Pe-13.0 MPa
Pe-20.7 MPa
0-
-
5.00
4.00
::L-----1 .oo
0.00 5
6.a
5.4
63
6.6
7.4
7
7.8
FREOVENCY (1100 KHz)
( a )
--
150
-
:7
*-
110
Pen 1 . 4
Pea 6 . 9
Pe-13.8
Pem20.7
Pressure
Pe: Confining
100
MPa
MPa
MPa
MPa
90
2
10
0
o
!
2.1
l
2.3
~
~
l
2.7
2.5
~
i
l
2.9
i
3.1
l
t
3.3
l
3.5
l
l
3.7
l
I l
3.9
I
i
I
l
4,l
FREQUENCY (mi00 Mrl
( b l
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
2
3
4
5
6
FREQVENCY ( ~ 1 0 0KHz)
( c )
Fire 6. Illustration of the crack-induced phase dispersions for P-waves (a), S,-waves (b), and &-waves (c) at confining pressures of 1.4,6.9,
13.8 and 20.7 MPa.
Wave attenuation in cracked solid
decreases with confining pressure (as shown as Fig. 5). The
vertical shift of the attenuation coefficient curves demonstrates a certain energy loss due to coupling.
Experimental results for the attenuation coefficients at
different confining pressures for the &-wave are shown in
Fig. 5(b). The effect of scattering on wave attenuation
reaches its maximum according to Hudson’s prediction for
shear waves polarized perpendicular to the plane of the
cracks. A strong non-linear relationship between the
attenuation coefficient and frequency is observed at a
confining pressure of 1.4 MPa and the estimated n is around
3.5. When the confining pressure is increased to 6.9 MPa or
above, a very good linear relation is observed. This is
probably because the mean radius of the cracks decreases
very quickly with increasing confining pressure. At low
confining pressures, nearly all the cracks are open, and the
attenuation is dominated by scattering. A non-linear
relation is therefore observed. As the confining pressure is
increased, cracks are closed and the attenuation due to
scattering becomes negligible. The effects of friction and
anelasticity of the filling material then become dominant. In
the experiments reported here, the crack condition is dry
and the most probable mechanism for wave attenuation at
high confining pressures is frictional sliding of the crack
surfaces.
PHASE DISPERSION
Tribolet (1977) and Stoffa, Buhl & Bryan (1974) present a
method for establishing the phase by first determining its
derivative with respect to angular frequency, and then
integrating the derivative. Jacobson (1987) points out that
the increment of frequency during the numerical integration
procedure should be very small to obtain high accuracy in
establishing the phase. We compared the phase spectra of
wavetrains with a total number of either 1024 or 2048
points. No difference was found for P- and S,-waves, but a
significant difference was found for &-waves. In this case,
we assumed that one with 2048 points is correct.
We still use the slate sample before cracking as a
reference and investigate the phase dispersion caused by
newly produced cracks. In this particular case the difference
between the slownesses of the sample before and after
cracking was calculated for each wave mode, instead of the
velocities. The experimental results are shown in Fig. 6,
where (a) refers to P-waves, (b) to Sl-waves and (c) to
&waves. We found that the crack-induced phase dispersion
for the &-wave is the most likely to obey the natural
logarithm law predicted by Futterman (1962) from the
Kramers-Kronig relation. The slowness of P-waves caused
by cracking at different confining pressures appear to be a
linear function of frequency in this limited frequency range,
as shown in Fig. 6(a). This is possibly because the &wave
signals have a much wider frequency range than P-wave
signals, and it is therefore difficult to establish the frequency
dependence of phase dispersion in such a narrow frequency
band.
The crack-induced phase dispersion for the S,-waves, as
shown in Fig. 6(b), is much more complicated. This is
because the mechanisms of wave attenuation for &-waves
include both intrinsic and apparent attenuation, and the
signal-to-noise ratio is much lower than that for P- and
179
&-waves. It is still obvious, however, that the crack-induced
phase change for S,-waves is greater than that for P- and
&-waves, which indicates that strong wave attenuation is
accompanied by severe phase dispersion. However the
results cannot establish whether or not the velocity
dispersion and attenuation obey the Kramers-Kronig
relation in this case.
CONCLUSIONS
The experimental results show that the quality factor Q of
rock samples is affected significantly by the presence of
cracks, and that Q is many times more sensitive to crack
parameters than the corresponding wave velocity. Hudson’s
theory predicts that the crack-induced attenuation due to
scattering for P- and &-waves should be vanishing small and
negligible. The significant changes in Q values for both wave
modes observed in the present work indicate that intrinsic
attenuation (most likely due to frictional sliding) is an
important mechanism of wave attenuation in dry slate.
As predicted by Hudson, the effect of S,-wave scattering
is found to be significant. It has been observed that at low
confining pressures the wave attenuation due to scattering is
dominant, and the attenuation coefficient versus frequency
relationship then behaves in a non-linear manner. At high
confining pressures the scattering factor is negligible and a
linear relation is observed, probably as a result of friction
between the crack surfaces.
Crack-induced phase dispersion for P-,Sl-and &-waves
have been observed. It seems that only the &-wave
dispersion can be demonstrated to obey the KramersKronig relation using the present experimental configuration, while the frequency dependence of the phase
dispersion for P-waves can be treated as linear within the
limited frequency band of the observation. The variation in
phase dispersion for Sl-waves is less well resolved, possibly
due to the complicated mechanisms of the wave attenuation
(including both intrinsic and apparent attenuation) and the
low signal-to-noise ratio of the &-wave.
ACKNOWLEDGMENTS
An equipment grant from Shell Expro in support of the
research is gratefully acknowledged. The British Council is
acknowledged for sponsoring S. X. and the Government of
the People’s Republic of China for his support. We thank an
anonymous referee for suggestions leading to major
improvements in this contribution.
REFERENCES
Aki, K. & Richards, P., 1980. Quontitotive Seismology, Theory and
Methodr I, W. H. Freeman, San Francisco.
Azimi, S. A . , Kalinin, A . V. & Pivovarov, B. L., 1968. Impulse
and transient characteristics of media with linear and quadratic
absorption laws, Izv. Earth Phys., 2, 42-54.
Biot, M. A , , 1956a. Theory of propagation of elastic waves in a
fluid saturated, porous solid. I. Low-frequency range, J .
Acowt. SOC. Am., 28, 168-178.
Biot, M. A . , 1956b. Theory of propagation of elastic waves in a
fluid saturated, porous solid. 11. Higher frequency range, J .
Acowt. SOC. Am., 28, 179-191.
Chatterjee, A. K., Knopoff, M. L. & Hudson, J. A., 1980.
180
S. Xu and
M. S. King
Attenuation of elastic waves in a cracked, fluid-saturated solid,
Math. Proc. Camb. Phil. SOC.,88, 547-561.
Crampin, S., 1981. A review of wave motion in anisotropic and
cracked elastic media, Waue Motion, 3, 341-391.
Futterman, W. I., 1%2. Dispersive body waves, 1. geophys. Res.,
67,5279-5291.
Hamilton, E. L.,1972. Compressional-wave attenuation in marine
sediments, Geophysics, 37, 620-646.
Hovem, J. M. & Ingram, G. D., 1979. Viscous attenuation of sound
in saturated sand, J. Acoust. SOC.A m . , 66, 1807-1812,
Hudson, J. A., 1981. Wave speed and attenuation of elastic waves
in material containing cracks, Geophys. 1. R. astr. SOC., 64,
133-150.
Jacobson, R. S., 1987. An investigation into the fundamental
relationship between attenuation, phase dispersion, and
frequency using seismic refraction profiles over sedimentary
structures, Geophysics, 52, 72-87.
Johnston, D. H . & Toksoz, M. N., 1980. Ultrasonic P- and S-wave
attenuation in dry and saturated rocks under pressure, J .
geophys. Res., 85,925-936.
Johnston, D. H., Toksiiz, M. N. & Timur, A., 1979. Attenuation of
seismic waves in dry and saturated rocks: 11. Mechanisms,
Geophysics, 44, 691-711.
Liu, H. P., Anderson, D. L. & Kanamori, H., 1976. Velocity
dispersion due to anelasticity: implications for seismology and
mantle composition, Ceophys. I. R. astr. SOC.,47, 41-58.
Mavko, G. & Nur, A., 1975. Melt squirt in the asthenosphere, J .
geophys. Res., 80, 1444-1448.
Mavko, G. & Nur, A., 1979. Wave attenuation in partially
saturated rocks, Geophysics, 44, 161-178.
McCann, C. & McCann, D. M., 1985. A theory of compressional
wave attenuation in non-cohesive sediments, Geophysics, 50,
1311-1317.
Murphy, W. F., 1982. Effects of partial water saturation on
attenuation in Massilon sandstone and Vycor porous glass, J.
Acoust. SOC.A m . , 71, 1458-1468.
O’Connell, R. J. & Budiansky, B., 1977. Viscoelastic properties of
fluid-saturated cracked solids, J. geophys. Res., 82, 5719-5736.
Savage, J. C., 1969. Comments on ‘velocity and attenuation of
seismic waves in imperfectly elastic rock’, 1. geophys. Res., 74,
726-728.
Stoffa, P. L., Buhl, P. & Bryan, G. M., 1974. The application of
homomorphic deconvolution to shallow-water marine seismology, part I: models, Geophysics, 39, 401-416.
Stoll, R. D., 1974. Acoustic waves in saturated sediments, in
Physics of Sound in Marine Sediments, Ed. Hampton, L. D.,
Plenum Press, New York.
Stoll, R. D., 1977. Acoustic waves in ocean sediments, Geophysics,
42,715-725.
Stoll, R. D., 1980. Theoretical aspects of sound transmission in
sediments, J . Acowt. SOC.A m . , 68, 1341-1350.
Toksoz, M. N., Johnston, D. H.& Timur, A., 1979. Attenuation of
seismic waves in dry and saturated rocks, I: Laboratory
measurements, Geophysics, 44,681-690.
Tribolet, 1. M., 1977. A new phase unwrapping algorithm, Inst.
Electr. Electron. Eng., Trans. Acoust. Speech Signal Proc., 25,
170-177.
Wu, R. S. & Aki, A., 1985. The fractal nature of the
inhomogeneities in the lithosphere evidenced from seismic
wave scattering, Pageoph., l23, 805-818.
Wuenschel, P. C., 1%5. Dispersive body waves-an experimental
study, Geophysics, 30, 539-551.