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www.elsevier.com/locate/jappgeo
Journal of Applied Geophysics 54 (2003) 319–333
Observation and analysis of frequency-dependent anisotropy from
a multicomponent VSP at Bluebell-Altamont Field, Utah$
E. Liua,*, J.H. Queenb, X.Y. Lia, M. Chapmana, S. Maultzscha,H.B. Lynnc, E.M. Chesnokovd
aBritish Geological Survey, Murchison House, West Mains Road, Edinburgh EH9 3LA, UKbConocoPhillips Inc., Houston, TX, USA
cLynn Inc., 14732 F Perthshire, Houston, TX 77079, USAdUniversity of Oklahoma, Norman, OK 73019, USA
Received 23 April 2002; received in revised form 12 November 2002; accepted 20 January 2003
Abstract
In this paper, we present results from the analysis of a multicomponent VSP from a fractured gas reservoir in the Bluebell-
Altamont Field, Utah. Our analysis is focused on frequency-dependent anisotropy. The four-component shear-wave data are
first band-pass filtered into different frequency bands and then rotated to the natural coordinates so that the fast and slow shear-
waves are effectively separated. We find that the polarisations of the fast shear-waves are almost constant over the whole depth
interval, and show no apparent variation with frequency. In contrast, the time delays between the split shear-waves decrease as
the frequency increases. A linear regression is then applied to fit the time-delay variations in the target and we find that the
gradients of linear fits to time delays show a decrease as frequency increases. Finally, we apply a time-frequency analysis
method based on the wavelet transform with a Morlet wavelet to the data. The variation of shear-wave time delays with
frequency is highlighted in the time-delay and frequency spectra. We also discuss two mechanisms giving rise to dispersion and
frequency-dependent anisotropy, which are likely to explain the observation. These are scattering of seismic waves by
preferentially aligned inhomogeneneities, such as fractures or fine layers, and fluid flow in porous rocks with micro-cracks and
macro-fractures.
D 2003 Elsevier B.V. All rights reserved.
Keywords: Anisotropy; Fracture characterisation; Velocity dispersion; Frequency-dependent anisotropy; Multicomponent seismic
0926-9851/$ - see front matter D 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.jappgeo.2003.01.004
$ Presented at the 10th International Workshop on Seismic
Anisotropy, Tulzing, Germany, 14–19, April 2002. Revised version
submitted to Journal of Applied Geophysics (Proceedings of
10IWSA) on 4 Oct. 2002.
* Corresponding author. Tel.: +44-131-650-0362; fax: +44-131-
667-1877.
E-mail addresses: [email protected] (E. Liu),
[email protected] (J.H. Queen),
[email protected] (H.B. Lynn),
[email protected] (E.M. Chesnokov).
1. Introduction
The presence of micro-scale (grain) cracks and
macro-scale (formation) fractures is considered to be
one of the dominant causes of observed anisotropy in
hydrocarbon reservoirs. One commonly used method
to examine fracture-induced azimuthal anisotropy is
shear-wave splitting. The two basic parameters are the
polarisation direction of fast shear-waves, interpreted
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333320
as the strike of aligned vertical fractures and the time-
delay between the fast and slow shear-waves, giving
the degree of anisotropy (i.e. fracture density). Re-
cently, observational evidence suggests that the mea-
sured seismic anisotropy as inferred from time delays
of split shear-waves does actually depend on frequen-
cy (Marson-Pidgeon and Savage, 1997; Chesnokov et
al., 2001; Liu et al., 2001), which we believe has an
important implication in our understanding of the
causes of seismic anisotropy.
The interpretation of anisotropic measurements
made from seismic data requires theoretical models
that relate measurable seismic parameters (such as
polarisation and time delays of split shear-waves) to
macroscopically determined rock properties (e.g. frac-
ture density and orientations). Based on the assump-
tion that the scale-lengths associated with the cracks
and fractures are considerably smaller than the seismic
wavelength, a description of the average properties of
a medium will be sufficient, i.e. an equivalent (or
effective) medium description. Various equivalent me-
dium theories have been proposed (e.g. Thomsen,
1995; Hudson et al., 1996, 2001; Hudson, 1981;
Bayuk and Chesnokov, 1998; Chesnokov et al.,
1998; Liu et al., 2000; Pointer et al., 2000). These
theories have provided the foundation of anisotropy
analysis, but most theories fail to provide an adequate
explanation of velocity dispersion at ‘low’ or seismic
frequency and hence variation of anisotropy with
frequency is not predicted. It is also not possible to
determine the size of fractures (thus fracture volume or
storability) from seismic data based on conventional
equivalent medium theories. In those theories, veloc-
ities are given in terms of crack density, a single
parameter that depends on both the crack radius and
the number density of the cracks.
Two mechanisms which can introduce velocity
dispersion and thus frequency-dependent anisotropy
are the scattering of seismic waves in media with
preferentially aligned heterogeneities, such as fractures
or fine layers, and fluid flow in fractured porous rock.
The former has been investigated by Werner and
Shapiro (1999); and Chesnokov et al. (2001), while
the latter has been the subject of Magnitsky and
Chesnokov (1986); Bayuk and Chesnokov (1998);
Parra (2000); Tod and Liu (2002). Chesnokov et al.
(2001) have attempted to model the observed varia-
tions of shear-wave anisotropy with frequency using a
multiple scattering correlation approximation, aiming
at estimating the size of the scatterers (fractures).
Bayuk and Chesnokov (1998), Chapman et al.
(2002); and Tod and Liu (2002) have presented theo-
retical models to allow vertically aligned meso-scale
fractures inserted in porous rock to include multi-scale
fluid flow. Their models predict velocity dispersion at
‘low’ or seismic frequencies and can explain the
observed frequency-dependent seismic anisotropy.
In this study, we analyse multicomponent VSP data
acquired from a fractured tight gas reservoir in the
Bluebell-Altamont Field in the Uinta Basin, Utah. Our
work may be regarded as a continuous and comple-
mentary study of early work by Lynn et al. (1995,
1996, 1999) who have used P- and S-wave azimuthal
anisotropy to characterise fractured reservoirs. How-
ever, we focus our analysis on frequency dependent
anisotropy. In Section 2, we use a conventional
technique to process the four-component shear-wave
data, that is, rotating the four-component data into the
natural coordinate system to separate the fast and slow
shear-waves. The rotation is performed after a series
of band-pass filters have been applied so that aniso-
tropic parameters can be extracted for each of the
frequency bands. Our results show that the polar-
isations of the fast shear-waves are almost constant,
and show no apparent variation with frequency (ex-
cept at very low frequency). However, the time delays
between split shear-waves decrease as the frequency
increases. We apply a linear regression to fit the time-
delay variations in the target formation and find that
the gradients of the linear fits show a decrease as
frequency increases, indicating that anisotropy
decreases as frequency increases. In Section 3, we
apply a time-frequency analysis method based on the
wavelet transform to the data in order to highlight the
variation of shear-wave time delays with frequency.
Finally, in Section 4, we discuss two possible causes
of observed variation of anisotropy with frequency:
seismic scattering and multi-scale fluid flow in frac-
tured porous rock.
2. Data and multicomponent shear-wave analysis
for anisotropy
The geological setting of the Bluebell-Altamont
field, Uinta Basin in northeastern Utah, has been
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333 321
described in detail by Lynn et al. (1995, 1996, 1999).
A brief summary of their description is given below.
Gas in the Bluebell-Altamont field is produced from a
tight sand/shale reservoir in the Tertiary upper Green
River Formation at depths of approximately 6500 ft
(1981 m) to 8500 ft (2590 m). Individual lenticular
sands are encased in tight shales and carbonates and
vary from 5 to 20 ft (2 to 6 m), but they often stack
vertically to thicknesses over 80 ft (24 m). The local
structure is subhorizontal sediments with no faulting
observed at target intervals, and it has a very gentle
anticlinal closure that is less than 50 ft (15 m). Matrix
porosity and permeability in the reservoir rocks are
generally low (porosity is less than 8% and perme-
ability is less than 1 mD), and fracturing yields
substantially higher production rates. Consequently,
seismic detection of fracturing is potentially an im-
portant aid to field development. Two joint and
fracture sets were identified in surface mapping, one
striking N22jW to N32jWand one striking N60jE to
N77jE. Maximum horizontal stress was estimated as
N40jW to N45jW from borehole elongation ob-
served in four-arm calliper logs in two adjacent wells
and from the N40jW orientation of recent, nearby,
gilsonite dikes. These are seen as a natural hydro-
fracturing event, when over-pressured fluids fractured
the overlying rocks upon expulsion from the lower
Green River formation.
Lynn et al. (1995, 1996, 1999) have analysed
extensive datasets as part of a US Department of
Energy funded project. They have used both surface
and borehole P-waves and shear-wave 3D-3C data to
characterize fractured reservoirs in the Bluebell-Alta-
mont field. The P-wave response is an azimuthal
variation of AVO signature, which depends on
source-receiver azimuths relative to fracture strike.
Shear-wave multicomponent seismic data are used to
obtain fracture orientations and fracture intensity in
the reservoirs using the concept of shear-wave split-
ting (Crampin, 1985). Our interest here is mainly in
the understanding of the variation of seismic anisot-
ropy with frequency, and for this purpose, we re-
analyse the multicomponent shear-wave VSP data.
Four vertical (P-wave) vibrators and four horizontal
(shear-wave) vibrators were used for the VSP acqui-
sition. The S-wave vibrators were Amoco S-vibrators
with rotatable baseplate. We analyse only the 550 ft
(168 m) offset VSP shear-wave data, and the receiver
depths are between 2800 ft (853 m) and 8650 ft (2636
m). A total of 118 receiver positions with an interval
of 50 ft (15 m) of VSP data were acquired. Direct
evidence of vertical fractures from cores, and fractures
mapped on outcrop, were used to orient the nine-
component seismic lines.
Fig. 1 displays the horizontal four-component data
matrix that has been rotated to the acquisition coor-
dinate, i.e. in-line (X) and cross-line (Y) directions.
The top rows marked with XX and XY are from in-
line sources, and the bottom rows marked with YX
and YY are from cross-line sources. The first letter
denotes the source orientation, and the second the
receiver orientation. The data are displayed with true
relative amplitudes so that a direct comparison be-
tween different components can be made. We can
immediately see that there is strongly coupled energy
in the cross-components, i.e. energy from the in-line
source is recorded in the cross-line receiver (XY) and
energy from the cross-line source is recorded in the in-
line receiver (YX). This feature is usually a direct
indication of the presence of azimuthal or fracture-
induced anisotropy (Crampin, 1985; Liu et al., 1993;
Queen and Rizer, 1990). Conventional analysis of
multicomponent shear-wave data involves rotation
of horizontal four-component shear-wave data into
the natural coordinate system through the use of
standard methods such as Alford rotation (Alford,
1986), or the linear transform (Li and Crampin,
1993) with the aim of separating the two split shear-
waves by minimising the off-diagonal energy. The
rotated four-component data are displayed in Fig. 2,
which shows clearly that the off-diagonal energy has
been significantly reduced. Once the shear-wave data
are rotated into their natural coordinate system, the
fast and slow shear-waves are separated. The rotation
angle then represents the polarisation angle of the fast
split shear-wave, which is commonly interpreted as
the fracture orientation. A cross-correlation can be
applied to the fast and slow components to obtain the
time-delays between the two split shear-waves. Note
that the sample rate is 2 ms, and the data have been re-
sampled to 0.5 ms using a sinc interpolation function.
Fig. 3 (top) shows the polarisation angles obtained
through successive rotations after the data were band-
pass filtered into five frequency bands. The band pass
filter that we use is a zero-phased, since-squired tapered
filter (‘sufilter’ in the Seismic Unix package of the
Fig. 1. Original four-component shear-wave data that have been rotated to the acquisition coordinates, i.e. in-line (X) and cross-line (Y)
directions. The top rows with XX and XYare from in-line sources, the bottom rows with YX and YYare from cross-line sources. The first letter
denotes the source orientation, and the second the receiver orientation.
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333322
Fig. 2. Four-component shear-wave data that have been rotated to the natural coordinates through the use of a standard method (Alford
rotation), i.e., by minimising the off-diagonal energy in the original data matrix. XX section is for fast shear-waves, and YY section is for slow
shear-waves.
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333 323
Fig. 3. Variation of polarisation of fast split shear-waves (top) and time-delays of split shear-waves (bottom) with depth after the data have been
band-pass filtered into five frequency bands. The angles are relative to the in-line component.
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333324
Fig. 4. Top: Same as Fig. 3 (bottom), but the only the time delays for the target formation between the depths of 6800 ft (2072 m) and 8650 ft
(2636 m) are displayed. The straight lines in each plot are from the linear regression algorithm to fit the variation of time delays for each
frequency. Bottom: Variation of anisotropy (in percentage) estimated from the gradient in Fig. 4a with frequency in the target interval between
the depth of 6800 ft (2072 m) and 8650 ft (2636 m). The error bars represent the uncertainty to the fit of time delays.
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333 325
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333326
Colorado School of Mines). Except for the very low
frequency band between 0 and 10 Hz, the polarisations
are generally constant over the whole depth interval at
40j to 45j from the in-line direction, which agrees with
the direction of predominant fracture orientations of
N43jW in the study area (Lynn et al., 1999), and there
is no apparent dependence of polarisation on frequen-
cy. Fig. 3 (bottom) shows the variations of time delays
between fast and slow shear-waves. We can identify
three distinct intervals: In Interval I (from 2800 ft (853
m) to about 4000 ft (1219 m)), as receiver depth
increases, time delays linearly increases, indicating this
interval is seismically anisotropic. In Interval II be-
tween the depth of 4000 ft (1219 m) and 6800 ft (2072
m), the time delays remain almost constant, implying
this interval is isotropic for the propagating waves as
there is no further shear-wave splitting in this interval.
Below a depth of about 6800 ft (2072 m), the time
delays begin to increase abruptly. This interval (Inter-
val III), which is also the target of the reservoir in the
Bluebell-Altamont field, thus, shows strong anisotropy
(about 3% to 4%), which is attributed to the presence of
intense fracturing in the reservoir. Note that a layer-
stripping procedure is not necessary as there is no
Fig. 5. Examples of spectral plots for all 4-component shear-w
obvious change of the polarisation with depth, imply-
ing there is no change of fracture orientation with depth
in at least the interval between 2800 ft (853 m) and
8650 ft (2636 m). In particular, we see the variation of
time delays with frequency shown in Fig. 3 (bottom).
The variation of shear-wave time delays with frequency
was noticed by one of the authors of this paper (Heloise
Lynn, Lynn) and the results were included in a report to
the US Department of Energy in 2000.
The shear-wave anisotropy is interpreted as due to
the presence of open and aligned vertical fractures,
striking northwest in the Upper Green River Forma-
tion. If we assume that the magnitude of shear-wave
anisotropy (time delays between split shear-waves) is
proportional to the fracture density, then the highest
density of open, gas-filled fractures is interpreted to be
in the interval between 6800 ft (2072 m) and 8650 ft
(2636 m). In Fig. 4 (top), we plot the time delays at
the target interval between 6800 ft (2072 m) and 8650
ft (2636 m) estimated for five different frequency
bands. In general, time-delays show linear an increase
with depth, and a decrease as frequency increases. A
linear regression is applied to fit the time delay
variations with depth for each frequency band. Except
aves recorded at the receiver depth of 6800 ft (2072 m).
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333 327
for the lowest frequency band (0–10 Hz), at which the
amplitudes are small (Fig. 5), we see a slight decrease
in the gradient of time delays, and this can be clearly
seen in Fig. 4 (bottom). Note that in Fig. 4 (bottom),
we have omitted the first point (0–10 Hz). However,
Tod and Liu (2002) have presented an alternative
model that predicts an increase of anisotropy with
frequency before it begins to decrease with frequency,
and the low anisotropy at 0–10 Hz supports their
theoretical prediction.
Fig. 5 shows the examples of spectral plots for all
four-component shear-waves recorded at the receiver
depth of 6800 ft (2072 m). The data have a frequency
band between 10 and 35 Hz. One may say that the
band-pass filtering technique used in Figs. 3 and 4
might be one of the factors of the observed variation
of anisotropy with frequency. According to the un-
certainty principle, we cannot have optimal resolution
in time and frequency at the same time. This restricts
to what we can do with our data using the band pass
filtering technique. However, we have performed a
series of tests on synthetic data generated using
Fig. 6. Variation of time-delays of split shear-waves with depth from a synth
filtered into five frequency bands. The synthetic data were computed usi
anisotropy.
frequency-independent elastic constants, and the syn-
thetic data are band-pass filtered in exactly the same
way as the real data, no frequency-dependent time-
delays are observed. An example of such test is given
in Fig. 6, demonstrating that the band-pass filtering
technique does not introduce frequency-dependency.
3. Time-frequency analysis
In this section, we shall present a technique to
process multicomponent shear-wave data based on a
continuous time-frequency analysis. One of the
advantages of such analysis is that frequency content
from multiple arrivals, such as the scattered waves and
split shear-waves, can be identified. In our analysis,
the continuous wavelet transform with a Morlet wave-
let used (Niitsuma et al., 1993).
Our analysis procedure is described below. Firstly,
the four-component data are rotated into the natural
coordinate system, i.e., fast and slow shear-waves are
separated into different components as discussed in
etic multicomponent VSP dataset after the data have been band-pass
ng Hudson’s (1981) model which predicts no frequency-dependent
Fig. 7. Time-frequency spectra of the fast and slow shear-waves recorded at the receiver depth of 6800 ft (2072 m). Top plot is for the fast shear-
wave and bottom plot is for the slow shear-wave at this depth.
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333328
Time-frequency analysis
Time-frequency analysis
Depth (m)
dt (ms)50
45
30
25
20
15
10
5
0
35
40
dt (ms)50
45
30
25
20
15
10
5
0
35
40
Fre
quen
cy (
Hz)
0
5
10
15
20
25
30
351000 1200 1400 1600 1800 2000 2200 2400 2600
Depth (m)
Fre
quen
cy (
Hz)
0
5
10
15
20
25
30
352000 2100 2200 2300 2400 2500 2600
Fig. 8. Time-frequency analysis of shear-wave splitting. It shows a colour-coded map of the time-delays and frequency spectra. The colour bar
shows the time-delay in milliseconds. The top plot displays the whole depth intervals and bottom plot shows the time-delays for the target
formation.
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333 329
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333330
the previous section. Secondly, we apply the wavelet
transform with a Morlet wavelet to the fast and slow
shear-wave components to obtain time-frequency
spectra (or local spectra) of recorded waves. The
number of samples used in the wavelet transform is
32 samples. Examples from trace number 80 (receiver
depth at 6800 ft (2072 m)) are shown in Fig. 7. We
can see a clear difference in travel times between the
fast and slow shear-waves, and also the difference in
frequency spectra between the fast and slow shear-
waves. Thirdly, we pick the maximum amplitudes and
their corresponding arrival times for each frequency in
the transformed fast and slow components [or their
power spectra]. Finally, the time-delays between split
shear-waves for each frequency are obtained by sub-
tracting the travel-times corresponding to the maxi-
mum amplitudes (or power spectra) of the transformed
fast and slow components. This procedure is repeated
trace by trace for all traces to build complete time-
delay and frequency spectra.
The results from the analysis procedure described
above are given in Fig. 8 (top). It shows a colour-
coded map of the variation of time-delays with
frequency for all depths. The colour bar is the time-
delays in milliseconds. The results confirm the in-
crease of time-delay with depth (colour changes from
light yellow to red) and decrease of time-delays as
frequency increases (colour changes from yellow-red
to light green-yellow). Fig. 8 (bottom) displays the
same colour map as Fig. 8 (top), but highlights the
target interval. Clearly, we see an increase in time-
delays with depths (colour changes from light yellow
to red), and a decrease of time-delays with frequency
(colour changes from red to yellow). This is basically
the confirmation of the result in Fig. 4, where a band-
pass filter was applied. The time-frequency analysis,
thus, can give a continuous variation of time-delays
with frequency.
4. Discussion: mechanisms of frequency-dependent
anisotropy
The phenomenon of seismic anisotropy is now in
widespread use as a tool for investigating fractured
rock, accessing sub-seismic scales of importance to
fluid migration. This advance has significantly in-
creased our ability to measure sub-seismic properties
of the rock-mass from seismic measurements. Esti-
mates of fracture intensity and orientation can be
found using existing equivalent medium theories cited
in Section 1. However, the application of seismic
anisotropy is limited by the necessary theoretical
constraints of dilute concentration and very small
scale-size to wavelength ratios, as heterogeneous
and fractured porous rock may be characterised by
observations in different critical wavelength ranges,
each reflecting different physical mechanisms. This
limitation has not yet been satisfactorily addressed.
We do not claim that frequency-dependent anisot-
ropy is unexpected or indeed surprising. However,
there are very few reports of frequency-dependent
anisotropy in the literature, especially in exploration
geophysics community. Evidence has been reported
before for earthquake data, e.g. by Marson-Pidgeon
and Savage (1997). They show a systematic increase
in time-delay with period, i.e. time-delay decreases
with frequency, which is very similar to the observa-
tion reported in this paper. Marson-Pidgeon and
Savage (1997) offer three possible explanations: (1)
shear-wave splitting resolution versus waveform peri-
od; (2) multiple layers of anisotropy; or (3) seismic
scattering by heterogeneities in the crust. If a proper
mechanism is understood, it may provide the means of
going beyond the concept of the conventional static
equivalent medium theories to potentially map the
size of meso-scale fractures whose lengths are much
larger than microcracks or grain boundary pores, but
less than seismic wavelength, and to extract informa-
tion about fluid flow (or transport) properties in
reservoir rocks. This is a subject in our companion
paper by Liu et al. (2003), where we have presented a
quantitative interpretation and also shown comparison
of synthetic results with the observed frequency-
dependent anisotropy using the multi-scale fracture
model presented by Chapman et al. (2002, this issue).
Here, we discuss the two most likely mechanisms
that can cause velocity dispersion and, hence, fre-
quency-dependent seismic anisotropy and some of the
implications from our study.
4.1. Seismic scattering
It has long been known by earthquake seismologists
that wave scattering is frequency-dependent (e.g.
Leary and Abercrombie, 1994; Wu 1981). It has also
E. Liu et al. / Journal of Applied G
been recognised that heterogeneities will produce ap-
parent anisotropy (Marson-Pidgeon and Savage,
1997). Frequency-dependent anisotropy is unarguably
related to the scattering of seismic waves in heteroge-
neous rocks. A typical example is the wave propaga-
tion in finely layered media as demonstrated by
Shapiro et al. (1994); Werner and Shapiro (1999),
who show that anisotropy differs for waves with
different frequencies. This leads to the conclusion that
heterogeneous materials with certain alignments or
ordered distributions must behave as an anisotropic
medium to elastic waves at some frequencies (e.g.
Chesnokov et al., 1998). The presence of a heteroge-
neous medium instead of a homogeneous isotropic
medium can produce frequency-dependent anisotropy,
in which the influence of heterogeneities decreases as
frequency increases. The scale of heterogeneities can
be constrained by the fact that they need to be small
enough to cause effective anisotropy instead of scat-
tering. This requires a quasi-homogeneous propagation
regime which can be expressed mathematically as ka
«1 (for example, say ka < 0.1, e.g. Sato and Fehler,
1997), where k is the wavenumber, and a is the scale-
length of the heterogeneities. For the data presented in
this paper, the dominant frequency is f = 25 Hz, and the
average shear-wave velocity is about V= 7000 ft/s
(2133 m/s). We expect that scale-lengths smaller than
about 10 m (V/fc 2133/25� 0.1) would cause effec-
tive anisotropy. Heterogeneities larger than 10 m will
cause scattering of seismic waves, and so reduce the
time delays due to effective anisotropy (the discussion
here follows that of Marson-Pidgeon and Savage,
1997).
Chesnokov et al. (2001) present a model based on
the correlation approach addressing frequency depen-
dent anisotropy when there is order in the orientation
of contrasting inhomogeneities. Anisotropy can only
be observed when the wavelengths of propagating
waves (P or S) are much greater than at least one (of
possibly many) scale lengths associated with the sizes
of inhomogeneities, i.e., in the case where the assump-
tions of the long wavelength approximation are satis-
fied. As frequency increases and wavelengths become
commensurate with these scale-lengths, anisotropy
becomes smaller and smaller. Beyond a certain tran-
sition frequency, it becomes minimal, with the prop-
agation being better characterised by discrete scatter-
ing effects.
4.2. Fluid flow in fractured porous rock
The other likely cause of frequency-dependent an-
isotropy is related to the fluid flow in fractured porous
rock. There have been several studies aimed at predict-
ing the velocity dispersion in porous rock and, hence,
variation of anisotropy with frequency. Bayuk and
Chesnokov (1998) investigate the correlation of elastic
wave anisotropy of P- and shear-waves and transport
properties of cracked porous rock and conclude that
seismic waves can be used to distinguish fluid proper-
ties (oil and gas) of reservoir rocks. Chapman et al.
(2002) have developed a model by placing a set of
aligned fractures into porous media, based on a squirt
flow model. Coupled fluid flow motion occurs on two
scales: the grain scale and the scale of the fractures. A
consequence of this is that the predicted anisotropy
becomes frequency-dependent, with the form of the
dependence related directly to the fracture scale. Pre-
vious estimates of the squirt flow frequency have given
high values, typically between the sonic and ultrasonic
bands. This has led to the suggestion, implicit in
Thomsen’s (1995) equant porosity model, that at seis-
mic frequencies, there should be little dispersion.
Chapman et al. (2002) demonstrated that in the pres-
ence of larger scale fractures, substantial frequency
dependence may be expected in the seismic frequency
ranges and, therefore, it is not safe to treat seismic
frequency as a low frequency limit. Tod and Liu (2002)
have presented an alternative model that describes a
fluid flow mechanism between elliptical cracks (bed-
limited cracks) that produces a frequency dependence
in the resulting effective material parameters, and
hence in the shear-wave splitting. They demonstrate
that the effects of fluid flow (either gas or liquid) are
non-negligible and, thus, an important factor contrib-
uting to observed frequency-dependent anisotropy.
eophysics 54 (2003) 319–333 331
5. Summary
We have provided observational evidence showing
the variation of shear-wave anisotropy with frequen-
cy from analysis of multicomponent VSP data in a
fractured gas reservoir. There is no apparent varia-
tion of polarisation with frequency, which is not
surprising as we do not expect that stress pattern
and, thus, spatial distribution of fracture orientation
E. Liu et al. / Journal of Applied Geophysics 54 (2003) 319–333332
would change with frequency. Time-delays between
split shear-waves, which are proportional to fracture
density, show a decrease as frequency increases. We
have presented a processing technique based on the
time-frequency representation of transient signals to
extract time-delays and frequency information. Time-
frequency analysis of shear-wave splitting enables us
to detect multiple arrivals in time, and variations in
shear-wave splitting with frequency. The method we
have proposed in this paper can be used to analyse
anisotropy and shear-wave splitting.
Acknowledgements
We thank John A. Hudson, Simon Tod (Cambridge
Univ.), Kurt Nihei, Seiji Nakagawa, Larry Myer, Ernie
Majer, Tom Daley (Lawrence Berkeley National Lab.),
and our colleagues Alexander Druzinin, David Booth
and Russ Evans for their comments on this manuscript.
We thank Professor Dirk Gajewski for many insight
discussions and for reminding us of the limitation of
band-pass filtering technique used to produce Figs. 3
and 4, and Claudia Vanelle and an anonymous reviewer
for their constructive comments that have significantly
improved the readability of our paper. The VSP data
presented here were acquired as part of the US DOE
project (Contract number DE-AC21-92MC28135).
The Seismic Unix code ‘sufilter’ of the Colorado
School of Mines was used to perform band-pass
filtering, and the BGS/EAP SWAP package was used
for anisotropic parameter estimation from shear-wave
data. This work was supported by the Natural En-
vironment Research Council (UK) through project
GST22305 as part of the thematic programme Under-
standing the micro-to-Macro Behaviour of Rock Fluid
Systems, and Edinburgh Anisotropy Project (EAP),
and is published with the approval of the Executive
Director of the British Geological Survey and the EAP
sponsors.
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