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Geophysical Journal InternationalGeophys. J. Int. (2013) doi: 10.1093/gji/ggs130

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Shear wave anisotropy from aligned inclusions: ultrasonic frequencydependence of velocity and attenuation

J. J. S. de Figueiredo,1,2,∗ J. Schleicher,2,3 R. R. Stewart,4 N. Dayur,4 B. Omoboya,4

R. Wiley4 and A. William4

1Faculty of Geophysics, Federal University of Para (UFPA), Belem, PA, Brazil, 66075-110. E-mail: [email protected] Institute of Petroleum Geophysics (INCT-GP), Salvador, BA, Brazil, 40170-1153Department of Applied Mathematics, IMECC/University of Campinas (UNICAMP), Campinas, SP, Brazil, 13083-9704Allied Geophysical Laboratories, University of Houston, Houston, Texas, USA, TX-77204

Accepted 2012 December 30. Received 2012 December 24; in original form 2011 December 13

S U M M A R YTo understand their influence on elastic wave propagation, anisotropic cracked media havebeen widely investigated in many theoretical and experimental studies. In this work, wereport on laboratory ultrasound measurements carried out to investigate the effect of sourcefrequency on the elastic parameters (wave velocities and the Thomsen parameter γ ) and shearwave attenuation) of fractured anisotropic media. Under controlled conditions, we preparedanisotropic model samples containing penny-shaped rubber inclusions in a solid epoxy resinmatrix with crack densities ranging from 0 to 6.2 per cent. Two of the three cracked sampleshave 10 layers and one has 17 layers. The number of uniform rubber inclusions per layerranges from 0 to 100. S-wave splitting measurements have shown that scattering effects aremore prominent in samples where the seismic wavelength to crack aperture ratio ranges from1.6 to 1.64 than in others where the ratio varied from 2.72 to 2.85. The sample with thelargest cracks showed a magnitude of scattering attenuation three times higher comparedwith another sample that had small inclusions. Our S-wave ultrasound results demonstratethat elastic scattering, scattering and anelastic attenuation, velocity dispersion and crack sizeinterfere directly in shear wave splitting in a source-frequency dependent manner, resulting inan increase of scattering attenuation and a reduction of shear wave anisotropy with increasingfrequency.

Key words: Body waves; Coda waves; Seismic anisotropy; Seismic attenuation; Wave scat-tering and diffraction; Wave propagation.

1 I N T RO D U C T I O N

Cracks and fractures in subsurface rocks are strong indicators oflithologic stress. Moreover, many hydrocarbon reservoirs are sit-uated in anisotropically cracked and fractured media. Thus, un-derstanding wave propagation in such media is of utmost impor-tance to be able to extract a maximum of information from seismicdata, which has motivated many studies in earthquake seismologyand seismic exploration of hydrocarbon reservoirs (Crampin 1981;Thomsen 1986, 1995; Crampin et al. 1999; Crampin & Peacock2005; Crampin & Gao 2010). Because of the geological complex-ities often attended by natural fracture structures, reliable conclu-sions about elastic properties are usually difficult to achieve withsufficient accuracy from field data. Thus, numerical and physi-

∗ Formerly at: CEP/UNICAMP, Department of Petroleum Engineering,Campinas (SP), Brazil.

cal simulations of elastic wave propagation in anisotropic mediabased on some previous knowledge are generally used as a toolto enhance the understanding of this complexity structures. How-ever, numerical simulation of wave propagation in cracked mediacan be computationally and mathematically expensive and intense(Hudson 1981; Crampin 1981; Hudson et al. 2001). When scatter-ing effects are taken into account, these costs become even moresignificant (Willis 1964; Mal 1970; Yang & Turner 2003, 2005).Moreover, such a computational approach can only study phenom-ena that are sufficiently well described by the underlying equations.Unfortunately, in many situations some of the assumptions made innumerical modelling may be oversimplified, constraining, or evenquestionable.

Some difficulties with the understanding and interpretation ofdata from both field acquisition and anisotropic modelling can beovercome using experimentally scaled physical modelling. Labora-tory measurements have been shown to be a useful tool for mod-elling conditions present in the field, helping to reduce uncertainties

C© The Authors 2013. Published by Oxford University Press on behalf of The Royal Astronomical Society. 1

Geophysical Journal International Advance Access published February 5, 2013

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2 J. J. S. de Figueiredo et al.

about elastic parameters in numerical methods. Assad et al. (1992,1996), Wei (2004) and Wei et al. (2007) established an experimen-tal relationship between crack density and shear velocity based ontheoretical predictions by Hudson (1981). Melia & Carison (1984)investigated the effect of layered media on compressional wavepropagation in a series of experiments in anisotropic samples. Theynoted that P-wave dispersion in anisotropic layered media is a func-tion of the concentration of different layered materials as well as thethickness of the layers. In a similar approach, Marion et al. (1994)and Rio et al. (1996) studied dispersion and multiple scattering ofshort and long wavelengths in stratified media. The main proposalof these latter works was to establish a description of wave prop-agation in the transition zone between ray theory and equivalentmedium theory.

Other sets of experimental work performed by Rathore et al.(1995) and Peacock et al. (1994) demonstrated the feasibility ofultrasonic measurements to study wave propagation in artificiallycracked porous media. Using experimental data obtained by Rathoreet al. (1995), the theoretical predictions of Thomsen (1995) foraligned cracks in porous rock received strong support. More re-cently, experiments by Tillotson et al. (2011) have suggested thepossible use of shear wave data to discriminate fluids on the basisof viscosity variations.

In anisotropic cracked media, the influence of frequency on prop-erties of wave propagation is determined by the size of the hetero-geneities (Assad et al. 1992, 1996; Wei 2004; Wei et al. 2007).However, quantification of this influence is still desirable. To better

understand the influence of cracks and fractures on the frequencyresponse in such media, we conducted a series of experiments aimedat extending the work of the cited authors. We used a shear wavesource with different frequencies: low frequency (LF = 90 kHz),intermediate frequency (IF = 431 kHz) and high frequency (HF =840 kHz) to carry out experiments on a reference sample withoutinclusions and three other samples with different inclusion sizes,thereby simulating different crack densities. In this arrangement,shear wave splitting was observed with different magnitudes as afunction of frequency. In the same set of experiments, we also quan-tified attenuation using the frequency shift method (Quan & Harris1997). Our results show that S-wave attenuation, both intrinsic anddue to scattering (Toksoz & Johnston 1981; Gorich & Muller 1987;Tselentis 1998), correlates directly with shear wave splitting, whichin turn is related to crack density. Furthermore, we observe that theanisotropy parameter γ (Thomsen 1986) varies with frequency andcrack size.

2 E X P E R I M E N TA L P RO C E D U R E

The construction of the cracked samples as well as the ultrasonicmeasurements were carried out at the Allied Geophysical Labora-tories (AGL) at the University of Houston, Texas. Under controlledconditions, we constructed three cracked samples (M2, M3 and M4)with different crack densities and one uncracked sample (M1) forreference. Pictures of all samples are shown in Fig. 1.

Figure 1. (a) From right to left: Photograph of the reference sample M1 (uncracked) and cracked samples M2, M3; (b) sample M4. Also shown are theorientations of the coordinate systems. All ultrasonic measurements were made in the Y direction. The sample size in the Y direction is quantified in Table 1.


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Shear wave anisotropy from aligned inclusions 3

Table 1. Physical parameters of samples M1, M2, M3 and M4. Precision of length measurements is about 0.02 cm.

Sample Dimensions of fracture zone Measure Number Crack Crack Cracks Aspect CrackX Y Z Volume length of diameter aperture per ratio density

(cm) (cm) (cm) (cm3) (cm) layers (cm) (cm) layer (per cent)

M1 – – – – 7.31 0 – – 0 0 0M2 7.08 7.30 4.90 253.25 7.29 10 0.70 0.091 36 0.13 5.0M3 7.02 7.16 4.92 247.29 7.32 17 0.40 0.051 90 0.12 .0M4-1 4.98 5.18 4.85 125.11 7.64 10 0.70 0.091 30 0.13 8.4M4-3 4.73 5.18 4.85 118.83 7.74 10 0.44 0.091 80 0.20 9.3M4-5 4.28 5.18 4.85 107.53 7.74 10 0.32 0.091 100 0.28 6.8

Figure 2. (a) Device developed for S-wave polarization rotation and velocity measurements. (b) Sketch of experiment used for seismogram records.

2.1 Sample preparation

The isotropic sample M1 consists of a single cast of epoxy resin.Samples M2 and M3 contain cracks aligned along the Y and Xdirections, respectively. Sample M4 has three differently crackedregions, but five different positions used for measurements, labelled1–5 in Fig. 1. Positions 2 and 4 are at the boundaries betweenthe three different regions. The cracked samples were constitutedone layer at a time, alternating with the introduction of rubbercracks. To reduce possible boundary effects to a minimum, the timeinterval between the creation of separate layers was kept as shortas possible. Constant layer thickness (0.5 cm for M2 and M4 and0.25 cm for M3) was ensured by using the same volume of epoxyresin poured for each layer. After each layer with inclusions wasadded to the sample, air was extracted using a vacuum pump toavoid inhomogeneities in the epoxy resin (material of the matrix).The solid rubber material used to simulate the cracks in samplesM2 and M4 was neoprene rubber, while in sample M3 we usedsilicone rubber. The compressional wave-velocity ratio was around1.5 between solid epoxy and neoprene and about 2.25 betweensolid epoxy and silicone rubber. Note that these values are onlyrough estimates, because the S-wave velocity in rubber was difficultto determine due to the low shear modulus of this material.

The physical parameters of the included rubber cracks in eachsample are displayed in Table 1. The crack density ε in the crackedsamples was determined by

ε = Nπr 2h

V, (1)

where N is total number of inclusions, r is their radius, h is theinclusion’s thickness (crack aperture) and, finally, V is the volumeof the cracked region for each sample (see Fig. 1). Eq. (1) is amodification of the relation of Hudson (1981) for crack densityestimation.

2.2 Ultrasonic measurements

We carried out ultrasonic measurements using the Ultrasonic Re-search System at AGL with the pulse transmission technique. Thesampling rate per channel for all experiments was 0.1 µs. Fig. 2(a)shows the device developed for recording S-wave seismograms withrotating polarization. The source and receiver transducers were ar-ranged on opposing sides of the samples, separated by the measuringlength (see Table 1). To ensure the wave propagation to take placeinside the desired region of the samples, the transducers on eitherside were placed at the centre of each region. The initial shear wavepolarization was parallel to the cracks. Changes in polarization wereachieved by rotating both transducers by 10◦ degrees at a time untilpolarization was again parallel (i.e. 0–180◦) to the XZ-plane (seeFig. 2b). In total, 19 traces were recorded in each seismic sectionwith 20-fold stack to eliminate ambient noise. The polarizations of0◦ and 180◦ correspond to the fast S wave (S1) and 90◦ correspondsto the slow S wave (S2).

The reproducibility of the ultrasonic recordings for all sampleswas ensured by preserving the same physical condition of the com-plete electronic apparatus. Furthermore, the same coupling betweentransducers and samples was guaranteed by a holder with a springattached (see Fig. 2). In order to establish good contact betweentransducer and sample, a very slim layer of natural honey was placedat the surface of the samples.

The source wavelet functions generated by the employed trans-ducers as well as their physical specifications are shown in Table 2.More information about these transducers can be found at the web-site of the manufacturer.1 The field condition for these transducer

1www.olympus-ims.com/en/ultrasonic-transducers


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Table 2. Physical parameters of transducers used to record S wave seismo-grams in the samples depicted in Fig. 1.

Transducer Catalogue Transducer Ricker Near fieldfrequency number diameter wavelet distance

90 kHz V1548 25 mm (−t3 + 3t)et22 ∼1.0 cm

431 kHz V151-RB 25 mm (t2 − t − 1)et22 ∼5.7 cm

840 kHz V153-RB 13 mm (t2 − t − 1)et22 ∼3.2 cm

was verified based on the relationship of Thompson & Chimenti(1995),

N = D2 − λ2

4λ, (2)

where D is the transducer’s diameter and λ is the dominant wave-length in the sample. The values of the ratio between the measuringlength of the samples (see Table 1) and the near-field distance shownin Table 2 is greater than 1 for all transducers. Therefore, the far-fieldcondition is satisfied for all recordings.

2.3 Attenuation estimation procedure

We start by analysing the S-wave source signatures and Fourieramplitude spectra of the three source transducers used to obtainthe data later on (see Figs 3a and b). The small secondary peak inthe spectrum of the signature of the 90-kHz transducer (blue line inFig. 3b) can be attributed to some artefact in the piezoelectric crystalused for this transducer. Since the amplitude of this secondary peakis rather small, we can neglect its presence in the further analysis. Tosimplify the interpretation of the spectra, we performed a Gaussiannon-linear fit to each amplitude spectrum (see Fig. 3c). The time

Figure 3. The time domain, S-wave source signatures of the three transduc-ers: LF = 90 kHz, IF = 431 kHz and HF = 840 kHz. (b) Fourier transformof each signature trace. (c) Fourier transform after Gaussian nonlinear fit.Here the dominant frequencies have become 89, 386 and 805 kHz.

windows used to evaluate the Fourier spectra of the signature traceswere 10 µs for the IF and HF sources and 28 µs for the LF transducer.We used this Gaussian fit to determine the centroid frequency as wellas the variance of frequency content. This information is requiredlater on for the attenuation estimation using the frequency shiftmethod (Quan & Harris 1997).

The frequency-shift method determines the quality factor Q ofa seismic event from a comparison of the centroid frequency be-fore and after a transmission experiment. We applied this methodusing a source-signature trace and the S-wave arrival after trans-mission through our samples. The experimental setup is depictedschematically in Fig. 4. According to Matsushima et al. (2011), thefrequency shift between the two events determines the Q factor as

Q = σ 2s π�t

� f, (3)

where �t is the traveltime difference between two different record-ings, �f = (fi − fo) is the difference in centroid frequency be-tween the source (fi) and the sample-trace pulse fo after Gaussiannon-linear fit and σ 2

s is the corresponding variance of the source fre-quency (as depicted in Fig. 4c). It should be noted that the frequency-shift method measures total attenuation without distinction of thephysical effects that might be causing it. Note that, even thoughthe frequency shift method is a stable method to estimate attenua-tion, the associated determination of the variance is very sensitiveto the noise (Nunes et al. 2011). Thus, to avoid overestimation ofthe attenuation, the noise in the transmission seismograms of ourexperiments was significantly reduced by 20-fold stack for eachrecorded trace.

We see in Fig. 3(a) that all transducers have an intrinsic timedelay. For all S-wave transducers used in this work, we estimated adelay time of 2.7 µs. For the velocity calculations, the delay timewas subtracted from the observed arrival time. The accuracy of timepicking was ±0.1 µs, which allows to determine the wave velocitieswith an accuracy of ±0.3 per cent.

3 E X P E R I M E N TA L R E S U LT S

In this section, we present our experimental results for S-wave split-ting in three cracked samples and one uncracked sample. It alsoincludes a frequency-domain attenuation analysis in the three dif-ferent frequency ranges (LF, IF and HF).

3.1 Shear wave seismograms

We start with the analysis of the transmission seismograms for thelow, intermediate and high-frequency sources in samples M2 andM3. They are depicted in Fig. 5, which also includes the corre-sponding seismograms for the isotropic sample M1 for reference.All seismograms in Fig. 5 are scaled to the maximum of the respec-tive section. We observe shear wave splitting for all frequenciesin both samples M2 and M3. The magnitude of this birefringencedepends on the source frequency. As expected, the isotropic sam-ple M1 shows uniform S-wave arrivals for all polarizations and allrecording frequencies, not separating fast (S1, 0◦ and 180◦) andslow (S2, 90◦) S waves.

In sample M2, the time delay observed between the arrivals as-sociated with S1 and S2, that is, the fast and slow shear waves, was6.9 µs for LF data (Fig. 5a) and 1.7 µs for IF data (Fig. 5b). Despitethe slightly larger measuring length in sample M3 (with smallercracks), the time delays were smaller here. We found 3.9 and 1.5 µs


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Figure 4. (a) Schematic representation of two recording with different source–receiver spacing. (b) Signature source trace (top) and pulse-transmission trace(bottom). (c) The corresponding Fourier spectra with central frequency at fi (input source signature) and fo (output pulse-transmission signature).

Figure 5. S-wave seismograms as a function of change in polarization from 0◦ to 180◦ for samples M1 (isotropic), M2 and M3 in the (a) low, (b) intermediateand (c) high frequency range.

for LF and IF data, respectively (see Figs 5a and b). In the case ofthe high frequency measurement, sample M2 (see Fig. 5c) showsfast and slow shear wave arrivals that are hard to interpret, whichprobably can be attributed to the pulse wavelength being of the same

order as the size of the cracks. We will elaborate on this in the nextsection. Similarly, due to the small ratio between wavelength andcrack size (see next section), the sample M3 for HF source presentsa time delay of 0.8 µs.


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Table 3. Source centroid frequencies fs and respective variances σ s, as well as sample-trace centroid frequenciesfm of samples M1, M2 and M3 for polarizations S1 and S2.

Source frequency (kHz) 90 431 840

Centroid frequency fs (kHz) 88.4 386 805Variance σ s (kHz) 38.5 193 271Sample-trace Sample S1 S2 S1 S2 S11 S21 S12 S22

centroid M1 87.5 87.5 317 317 552 552 – –frequency fm M2 84.3 83.2 137 106 172 206.5 829 750(kHz) M3 85.4 83.5 249 176 386 266 – –

3.2 Frequency analysis

The next step consists of analysing the Fourier spectra of the aboveseismograms and their respective Gaussian non-linear fit spectra.The purpose is to analyse the shear wave scattering and attenuationin the samples. The results are explained below and summarized inTable 3.

3.2.1 Sample M1

We start with the data for sample M1 (left column of Fig. 5). Thered box in the seismograms mark the time window used to performthe FFT operation over the traces corresponding to the 0◦ (S1) and90◦ (S2) polarizations. We observe from the resulting normalizedamplitude spectra (Fig. 6) that in this isotropic epoxy resin sampleM1, the peak for the HF waves is the most strongly shifted one. Thedominant frequency is shifted from 840 kHz (source frequency) to552 kHz (frequency response), while the shift for IF is from 431kHz to 317 kHz and the one for LF is 90 kHz to 87.5 kHz. This verysmall shift, which means that the LF waves are almost unattenuated,can be attributed to the fact that in this frequency range, the wave-length is of about the order of the sample size (deviating by only afactor 1.5).

3.2.2 Sample M2

The corresponding spectra for sample M2 are depicted in Fig. 7. Inthis sample, the ratio of wavelength to crack size ranges from 0.37(HF) to 3.57 (LF) (see Table 4) and hence effects associated withscattering or diffraction as well as effective media are expected tobe seen at the same time (Marion et al. 1994; Gibson et al. 2000;Matsushima et al. 2011).

In the HF spectra (see Fig. 7c) obtained from the seismograms inthe centre of Fig. 5c, we observe two independent peaks for both theS1 and S2 waves. The reason becomes evident from Table 4, whichpresents the ratio between crack size and seismic wavelength for S1and S2 waves in samples M2 and M3 in the LF, IF and HF range.The effective crack diameter d in Table 4 is that of an equivalentsphere with the same volume Vc as the cracks, that is, d = 3

√3Vc4π .

The contributions of the wavefield at the highest frequenciestravel practically unaffected in the nearly homogeneous mediumbetween the cracks, giving rise to an unperturbed S-wave arrivalof the observed wavefield at almost the same traveltime as in theisotropic sample M1 (see left seismograms in Fig. 5c). Contribu-tions at the lowermost frequencies propagate as if in an effectivemedium, almost unperturbed from the individual cracks, becausethe crack size is much smaller than the wavelength. On the other

Figure 6. Fourier spectra for sample M1 using S-wave sources in the (a) LW, (b) IF and (c) HF range.


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Figure 7. Fourier spectra for sample M2 using S-wave sources in the (a) LW, (b) IF and (c) HF range.

Table 4. Seismic wavelength λ to effective crack diameter d ratio for polarizations S1 and S2. Note that for sample M2 at HF,there are two peaks in the spectrum of S1 and S2, resulting in two values in the respective lines of this table.


Centroid wavelength (M2) λ (cm) 1.47 1.33 0.91 1.12 0.74 0.60 0.15 0.17Centroid wavelength (M3) λ (cm) 1.43 1.37 0.49 0.68 0.32 0.46 — —

Sample Crack volume Vc (mm3) Eff. Diameter d (cm) λS1d

λS2d

λS1d

λS2d

λS11d

λS21d

λS12d

λS22d

M2 35.02 0.41 3.57 1.71 2.22 2.73 1.80 1.46 0.37 0.41M3 6.41 0.23 6.21 5.97 2.15 2.97 1.39 1.98 – –

hand, intermediate frequency contributions with wavelengths of theorder of the effective size of the scatterers suffer from the strongestattenuation from scattering at the cracks. Thus, these contributionsare almost completely missing in the receiver wavefield, resultingin two peaks at either side of the resulting amplitude spectrum.Note that Figs 7(a) and (b) do not exhibit a second peak, indicatingthat the high frequencies that suffer very little attenuation are notpresent in these wavefields. This is corroborated by the seismogramsin Figs 5(a) and (b), in which the S-waves arrivals are recorded at asignificantly later time than in the isotropic sample M1.

Note that in the HF range (see Fig. 7c), the spectra of the S1 andS2 polarization exhibit two visible differences. (1) The shift of dom-inant frequency is stronger for the S1 polarization (from 840 kHz to172 kHz) than for S2 (from 840 kHz to 206.5 kHz). (2) The secondpeak at higher frequencies is much more pronounced relative tothe first one for the S2 polarization than for S1. In the LF and IFranges, the S1 and S2-wave polarizations exhibit a nearly identicalbehaviour. While for LF, almost no frequency shift is observed (seeFig. 7a), the IF range exhibits a pronounced frequency shift (seeFig. 7b).

The rather strong frequency shift for both polarizations for IFand HF may be explained by the fact that in the strongly attenu-ated frequency ranges, the wavelengths are of the size of the inclu-

sions lengths, which increases the scattering-related attenuation (seeTable 4). The difference in the frequency shift between the polar-izations and the fact that the second peak is much stronger for theS1 than for the S2 polarization indicates that scattering attenuationis dominant when the polarization is parallel to the crack, but thatintrinsic attenuation becomes more important when the polarizationis perpendicular to the cracks.

For a better understanding of the two separate peaks, we applieda band-pass filter of 10–50–350–400 kHz (perfect pass between 50and 350 kHz, with linear cut-off ramps in the ranges 10–50 kHzand 350–400 kHz) to the HF data of sample M2 (centre panel ofFig. 5c, replotted in Fig. 8a). The Cut-off frequency of 400 kHzapproximately coincides with the end of the first peak in Fig. 7(c).The result is depicted in Fig. 8(b). The HF part, obtained as thedifference between the original and filtered seismograms, is shownin Fig. 8(c). Note that after filtering, shear wave splitting with amagnitude of 1.4 µs becomes visible (Fig. 8b), which could notbe observed before. This corroborates our interpretation that thelow-frequency part of the wavefield behaves as if travelling in aneffective anisotropic medium. On the other hand, the seismic sec-tion in Fig. 8(c) corresponds to the combined frequency content ofthe two high-frequency peaks at 750 kHz (S2) and 820 kHz (S1) ofFig. 7(c). No shear wave splitting is visible in Fig. 8(c), indicating


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Figure 8. (a) S-wave seismogram for sample M2 (b) The same data after application of band-pass filter 10–50–350–400 kHz (S-wave time delay is 1.4 µs). (c)High-frequency section after subtraction of (b) from (a).

Figure 9. Fourier transform spectra for sample M3 using S-wave sources in the (a) LW, (b) IF and (c) HF range.

that the high-frequency part of the wavefield behaves as if travel-ling in an isotropic medium with the velocity of uncracked epoxyresin.

3.2.3 Sample M3

In sample M3, none of the frequency ranges produces a secondpeak (see Fig. 9), because the cracks are too small and too sparselydistributed to allow for perceptible scattering attenuation. All wavepropagation is in the effective-medium regime. However, like insample M2, also in this sample the shift in frequency associated withthe perpendicular polarization (S2) is more prominent than the onefor S1. As mentioned before (see Table 4), the pulse-wavelength-to-crack-size ratio for M3 ranges from 1.39 to 6.21, that is, thewavelengths are not much smaller than the crack size. This explainswhy there is less unscattered wave propagation and less unperturbedenergy as compared to sample M2.

3.3 Velocity results

3.3.1 Samples M2 and M3

We determine the S-wave velocities from the picked traveltimesof the first breaks. Fig. 10 depicts the resulting velocities of thefast (VS1) and slow (VS2) shear waves as functions of sample wave-propagation frequency according to Table 3. Fig. 10(a) shows thatthe isotropic medium suffers of very little dispersion. The dispersioneffect because of the cracks is more prominent for sample M2(Fig. 10b) than for sample M3 (Fig. 10b). We also note that in allcracked samples, the S2 wave is more dispersive.

From these velocity values, we calculate Thomsen’s anisotropyparameter γ using the relationship

γ = 1

2

(V 2

S1

V 2S2

− 1

). (4)


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Figure 10. Velocity plots for samples M1 (a), M2 (b) and M3 (c) as a function of effective dominant frequency. The dispersion curves show that in theinvestigated frequency range, the polarization S2 is more influenced by frequency than S1 for both cracked samples M2 and M3.

Figure 11. Anisotropy parameter γ calculated from the S-wave velocities of Fig. 10 using eq. (4).

The graphs of the anisotropy parameter γ for samples M2 and M3(Fig. 11) show that γ decreases with increasing dominant frequencyin the samples. For both samples containing cracks, splitting is morepronounced at the lowest frequency (90 kHz) than at intermediateand high frequencies (431 and 840 kHz). As expected from theanisotropic theories for cracked media (Hudson 1981; Crampin1984), at LF the value of γ = 12.2 per cent in sample M2 is higherthan γ = 7.2 per cent in sample M3, which has a lower crack densitythan sample M2. However, at the highest frequency, the value of γ =2.0 per cent in sample M2 is equal to the one for M3.

While these values of the velocities and anisotropy parameter γ

seem to be in conflict with standard anisotropic theories, a betterunderstanding can be obtained when plotting them as a function ofthe ratio between centroid wavelength and effective crack size (seeFig. 12).

From Fig. 12, we can infer a relationship between anisotropy andseismic frequency (or wavelength) relative to crack size. When plot-ted as a function of λ/d, the velocities in both samples exhibit anapproximately linear behaviour. The different polarizations with re-spect to the crack orientations give rise to different slopes (see Figs12a and b). Further investigations will be necessary to determinewether the variation of the slope as a function of S-wave polariza-tion could be used for an estimation of crack orientation. From thesevelocity values, we extracted the anisotropy parameter γ as a func-tion of λ/d by means of eq. (4), where at each frequency, we haveused a measured velocity value for one polarization and the linearlyinterpolated one for the other polarization. The resulting γ valuesare even better approximated by a straight line (Fig. 12c). At longwavelengths (LF), where effective-media theory is more realistic(see Table 4) the effective anisotropy parameter γ is higher than at


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Figure 12. Linear fit of velocity as function of ratio between wavelength and effective crack size for polarization S1 and S2. (a) sample M2 and (b) sampleM3. (c) Anisotropy parameter γ calculated from eq. (4).

short wavelengths (HF). These velocity results allow to conjecturethat the magnitude of shear wave splitting depends on dominantsource frequency, crack size and density. Further experiments willbe necessary to establish how the slope of γ over λ/d depends onthe physical crack parameters.

3.3.2 Sample M4

Sample M4 was devised to study the effect of different crack densityat constant crack aperture. In this sample, all cracks have the sameaperture (0.091 cm), but three different diameters (0.7, 0.44 and0.32 cm) and thus different aspect ratios (0.13, 0.20 and 0.28).Together with different numbers of cracks per layer, this led todifferent densities (8.4, 9.3 and 6.8 per cent) in the three regions ofsample M4. The physical information of this sample is containedalso in Table 1. To separately interpret the S1 and S2 waves in the HFseismograms, we applied again the 10–50–350–400 kHz bandpassfilter. The S-wave velocities of the fast and slow shear waves at thefive measurement points in sample M4 are summarized in Table 5and graphically represented in Fig. 13.

3.4 Size effect investigation

The above observations regarding the dependence of the anisotropyon the size parameters of the cracks are confirmed from the resultsin sample M4. Table 5 shows the velocity values of S1 and S2 wavesin samples M1, M2, M3 and M4 together with the relevant phys-ical crack parameters diameter, aperture and density. We see that

a simultaneous decrease in diameter, aperture, and density, fromsample M2 to M3, led to decreasing S1 and HF S2 velocities, whileonly LF and IF S2 velocities increased as expected. On the otherhand, from the measuring points M4-1, M4-3 and M4-5, we seethat the velocities are practically insensitive to the crack diameter.Slight velocity variations seem to be correlated with decreasingcrack density. Comparing the values for M2 with those for M4-1,where the crack size is the same, we see no sensitivity of LF andIF S1 velocities to crack density, while S2 and HF S1 velocitiesconsistently decrease with increasing density. Generalizing theseobservations to the assumption that a crack density increase shouldnever result in increased velocities, we can study the influence of thecrack aperture by comparing the values for M3 with those for M4-3,where the crack size is very similar. We see that an increase in crackaperture causes a measurable increase in the S1 velocities for allfrequencies and in the HF S2 velocities. The strong reduction in theLF and IF velocities between samples M3 and M4-3 can probablybe attributed to the difference in crack density. From the observeddependencies of the shear wave velocities on the physical crackparameters, we infer that crack aperture is the most important pa-rameter for shear wave splitting. The crack density is somewhat lessimportant, and the crack diameter seems to have the least influence.While the effective crack size determines scattering attenuation, itdoes not seem to be a determining parameter for the S-wave veloc-ity. Further experiments varying only one crack parameter at a timewill be needed for a more conclusive study.

As shown in Fig. 13, S-wave splitting does not show a strongdependency on the physical crack parameters that were varied insample M4. In Figs 13(a) and (b), where long wavelengths are


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Table 5. Velocity values for samples M1, M2, M3 and M4 for LF, IF and Hf ranges together with crack diameters and crackapertures. The velocity of the higher-frequency event in M2 is the same as for S1 of the lower-frequency event for both polarizations,that is, V(S12) = V(S22) = 1267 m s−1.


Sample Crack parameters Shear wave velocitiesDiameter Aperture Eff. Size Density V(S1) V(S2) V(S1) V(S2) V(S1) V(S2)

(cm) (cm) (cm) (per cent) (m s−1) (m s−1) (m s−1) (m s−1) (m s−1) (m s−1)

M1 – – – – 1254 1254 1264 1264 1271 1271M2 0.7 0.091 0.41 5.0 1235 1106 1245 1186 1267 1237M3 0.4 0.051 0.23 4.0 1220 1146 1230 1204 1232 1214M4-1 0.7 0.091 0.41 8.4 1237 1082 1243 1129 1253 1217M4-3 0.44 0.091 0.30 9.3 1232 1088 1247 1139 1257 1221M4-5 0.32 0.091 0.24 6.8 1233 1086 1247 1139 1261 1221

Figure 13. Velocities for five different points in sample M4 and the respective anisotropic parameter γ associated with these velocities for S-wave sourcetransducers: (a) LF, (b) IF and (c) HF ranges.

dominant, the anisotropy parameter γ slightly decreases with re-duced crack density and individual crack length. On the other hand,for high frequency (see Fig. 13c), a decrease in crack density orcrack size leads to a slight increase in magnitude of γ . However,the variations are very small.

3.5 Shear wave attenuation measurement

To estimate the shear wave attenuation, we applied the frequency-shift method of Quan & Harris (1997) using a source-signaturetrace and the S-wave arrivals from the pulse-transmission exper-iment. The details of the procedure are described in Section 2.3.The centroid frequencies and respective variances of sources in thedifferent frequency ranges, as well as the centroid frequencies ofsamples M1, M2 and M3 for polarizations S1 and S2 after Gaussiannon-linear fit are presented in Table 3.

Using the values from Table 3 and the first-break arrival travel-times of the S1 and S2 waves in the seismic profiles shown in Fig. 5,we calculate the quality factor from eq. (3). Note again that the re-

Table 6. Quality factor estimates for samples M1, M2 and M3.

Source frequency 90 kHz 431 kHz 840 kHz

Sample Q (S1) Q (S2) Q (S1) Q (S2) Q (S1) Q(S2)

M1 294.93 294.93 99.94 99.94 53.94 53.94M2 67.52 59.63 27.69 27.14 21.20 23.20M3 93.57 61.08 51.71 35.81 32.97 27.35

sulting values for Q, presented in Table 6, refer to total attenuation,including intrinsic and scattering attenuation.

Fig. 14 shows a graphical representation of the correspondingattenuation values (Q−1), together with error bars estimated fromthe neighboring traces. We see that in all samples the attenuationincreases with increasing frequency, and the increase in the crackedsamples M2 and M3 is stronger than in the isotropic sample M1.In sample M2, the behaviour of the two polarizations is not signif-icantly different, but differs distinctly from that in sample M1. Insample M3, the S2 wave is significantly stronger attenuated thanthe S1 wave.


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Figure 14. Total attenuation Q−1total as a function of effective dominant frequency. (a) Comparison of sample M2 to M1. (b) Comparison of sample M3 to M1.

Figure 15. Scattering attenuation Q−1scattering for the shear wave polarizations S1 and S2 in samples (a) M2 and (b) M3.

Since the epoxy resin that constitutes the background medium ofsamples M2 and M3 is very similar to sample M1, we can assumethat the intrinsic attenuation in the resin of samples M2 and M3 isapproximately equal to the total attenuation in sample M1, that is,Q−1

resin ≈ Q−1total(M1). Thus, we can calculate the attenuation effects

due to the presence of the rubber inclusions using the approach ofBrown & Seifert (1997) and Tselentis (1998). For this purpose, welinearly interpolate the value of the intrinsic resin attenuation Q−1

resin

in model M1 (model without inclusions) as a function of frequency,and subtract the result from the total attenuation Q−1

total for samplesM2 and M3 at the same frequencies, that is,

Q−1inclusions = Q−1

total − Q−1resin . (5)

Since the rubber inclusions are much smaller than the involvedwavelengths, we assume that the intrinsic attenuation inside therubber cracks is much smaller than the attenuation effect of scatter-ing. Thus, from now on we will briefly refer to Q−1

inclusions ≈ Q−1scattering

as scattering attenuation.Fig. 15 shows that the so-obtained scattering attenuation Q−1

scattering

for both (fast and slow) polarizations increases with increasingsource frequency from LF to HF in both samples M2 and M3. Notethat at LF, we observe little scattering attenuation with Q−1

s ≈ 0.01in both samples. In sample M2, both S1 and S2 polarizations ex-hibit a very similar behaviour of the scattering behaviour, whilein sample M3, the S2 wave is clearly more attenuated than the S1wave. Another difference between the samples regards the increaseof scattering attenuation with frequency. While in sample M3 thescattering attenuation increases linearly from LF to HF for S2, itsslope rises between IF and HF for S1. On the other hand, the slope

decreases significantly for both S1 and S2 in sample M2. This lat-ter behaviour is consistent with our previous interpretation that forhigher frequencies, there are more and more waves that can prop-agate in the space between the cracks in the isotropic backgroundmedium in sample M2.

4 D I S C U S S I O N

All our above observations indicate that the S2 wave is more stronglyinfluenced by cracks in the medium when the propagation is closer tothe effective-medium condition, that is, for low and intermediate fre-quencies. However, the main difference between the behaviours ofQ−1

s in the samples is noted in the intermediate and high-frequencyregimes. At these frequencies, the S1 attenuation is increased withrespect to S2. We see in Fig. 15 for equivalent high frequency thescattering attenuation of the S1 wave in sample M2 is 66.66 per centstronger than that of S1 in sample M3 while for the S2 polarizationthis difference is only 33.33 per cent.

Note that the different material of the inclusions (neoprene insample M2, silicone in sample M3) did not seem to have a stronginfluence on the attenuation in our experiments. Intuitively, oneshould expect that the significantly lower value of the shear wavemodulus for silicone (μsil ≈ 1.0 MPa) than for neoprene (μneo ≈3.6 MPa) would lead to much stronger attenuation. However, weobserved stronger attenuation in sample M2, where the cracks aremade of neoprene rubber. This indicates that the influence of cracksize and aperture is much more prominent than that of the crackfilling. Moreover, it is consistent with our assumption that the atten-uation due to the presence of cracks in the samples is determined


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by scattering and that intrinsic attenuation within the cracks can beneglected.

There are many difficulties that are encountered in the labora-tory and field to accurately measure an attenuation value (intrinsic,scattering or apparent). Effects related to the near-field, sphericaldivergence, boundaries, reflectors, coupling and scattering are fac-tors that change the amplitude of a seismic trace. To avoid theseeffects, we used a method that basically depends on the frequencyshift observed in the direct-arrival measurements at two differentspacings. This method, which does not require any amplitude ra-tio approach (like, e.g. the spectral ratio), was established by Quan& Harris (1997). As shown above, the application of this methodrequires two wave traces recorded at two different positions. Dueto the characteristics of the experimental setup in this work, westrongly believe that the application of the frequency shift-methoddid not lead to overestimation of attenuation.

The results obtained from our three cracked samples have shownevidence of the frequency-dependent behaviour of wave propaga-tion in anisotropic elastic media. The rubber inclusions used in thiswork simulated ideal cylindrical cracks consisting of solid materialshowing a low shear modulus as compared to the surrounding ma-trix. Our experiments used an idealized fracture system exhibitingaligned crack distributions with different fracture parameters. Thesize of the individual cracks was much below the seismic wave-lengths. Our results indicate that the attenuation in such a systemdepends stronger on the geometric properties of the cracks thanon the filling material. Regarding the geometric parameters, crackaperture and crack density were more important than crack diam-eter. Waves in different frequency ranges react slightly different tothese parameters.

5 C O N C LU S I O N S

This experimental study aimed to investigate the influence of sourcefrequency on elastic wave propagation in anisotropic media contain-ing aligned penny-shaped cracks. The results show that the mag-nitude of S-wave birefringence in cracked media directly dependson the source frequency as well as crack size and density. In thelow-frequency range, splitting was more conspicuous in all crackedsamples than for intermediate and higher frequencies. For increasingfrequencies, the magnitude of S-wave splitting (measured by meansof the S-wave Thomson anisotropy parameter) decreases drastically.Low-pass filtering of high-frequency data turned out to be helpful tomake a small shear wave splitting visible. This splitting was higherfor larger cracks with smaller density.

We observed the dispersive effect of cracked media to be higherfor the (slow) S2 than for the (fast) S1 polarization. Furthermore thisdispersion is predominant when the crack length is smaller or of thesame order as the wavelengths used in the investigation. Moreover,the lower the source frequency was, the more pronounced were theobserved dispersive effects.

Contrary to the typical behaviour of shear wave splitting, theS1 wave seems to be more influenced by scattering than S2 whenthe crack size is larger than the wavelength. If this statement canbe confirmed by future experiments, the crack aperture may be lessrelevant than the individual crack size in the HF range. An additionalexperiment in the high-frequency range with the same crack aperturebut varying crack size and density showed an almost constant butslightly increasing anisotropy parameter with decreasing crack size.

From our experiments, we can establish an order of importancefor the influence of different physical crack parameters on shear

wave birefringence in anisotropic cracked media. Our results showthat the crack aperture is the most relevant parameter, followed bycrack density. Of the geometric crack parameters, their diameterseems to have the least influence on shear wave velocities. Par-ticularly in the low-frequency case, where the S-wave propagationbehaves like in an effective medium, the anisotropic parameter γ

does not strongly depend on the crack size, but much more onthe crack density. Because of our strict adherence to scalability ofthe experiments (except, of course, regarding the physical dimen-sions of the sources and receivers), we expect the results of ourlaboratory measurements to apply correspondingly in the seismicfrequency range.

As discussed by Shaw (2005), the knowledge about fracture pa-rameters such as density, size, spacing and aperture can help toreduce uncertainties in seismic hydrocarbon exploration in frac-tured media. We believe that the main contribution of this work is toprovide more data to better understand the dependence of seismicwave propagation on the properties of fractured reservoirs, withspecial regard to source frequency and fracture parameters. More-over, the data set supplied by these laboratory measurements canbe used for theoretical model validation. However, further modelexperiments under variation of a single parameter will be necessaryto further corroborate our findings.

A C K N OW L E D G M E N T S

This work was made possible by the financial and facility sup-port at Allied Geophysics Laboratories, University of Houston.The authors are grateful to Dr. Leon Thomsen and Dr. Evgeny Ches-nokov (AGL) for their expertise and advice, and to two anonymousreferees whose remarks greatly helped to improve the manuscript.The first author wishes to thank CAPES and CNPq from Brazil forhis scholarship (contract # 201461/2009-9). Also, we are grateful toCNPq, Petrobras and the sponsors of the Wave Inversion Technology(WIT) Consortium (www.wit-consortium.de) for their support.

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