CWP-730

Laser excitation of elastic waves at a fracture

Thomas Blum(1), Kapser van Wijk(1), Roel Snieder(2), and Mark Willis(3)

(1)Physical Acoustics Laboratory and Department of Geosciences, Boise State University, Boise, ID 83725(2) Center for Wave Phenomena, Colorado School of Mines, Golden CO 80401, email rsnieder@mines.edu(3) Formerly at ConocoPhillips Company, Houston, TX 77079, Now at Halliburton, Houston, TX 77032

ABSTRACTWe show that elastic waves can be excited at a fracture inside a transparentsample, by focusing laser light directly onto this fracture. The associated dis-placement field, measured by a laser interferometer, has pronounced waves thatare diffracted at the fracture tips. We confirm that these are tip diffractionsfrom direct excitation of the fracture by comparing them with tip diffractionsfrom scattered elastic waves excited on the exterior of the sample. Being ableto investigate fractures – in this case in an optically-transparent material – viadirect excitation opens the door to more detailed studies of fracture propertiesin general.

Key words: scattering, fracture, ultrasound

1 INTRODUCTION

Being able to remotely sense the properties of fractureswith elastic waves is of great importance in seismology,e.g. Nakahara et al., (2011) and non-destructive test-ing, e.g. Larose et al. (2010). For example, in geother-mal and hydrocarbon reservoirs, it is very commonto use hydraulic fracturing methods to attempt to in-crease the native permeability of the rocks above whatis present in any naturally occurring fractures. The mi-croseismic events associated with the fracturing processtypically radiate seismic energy, which is recorded innearby wells or at the surface. Much is left to be un-derstood about the nature of such fractures and theirrelationship to elastic waves, but the scaling issues in-volved make numerical modeling a challenge. On theother hand, laboratory studies of fractures or faults areused to investigate their mechanical properties, such asstiffness (Pyrak-Nolte and Nolte, 1992), fracture slip-rate, stress drop, or rupture propagation (Nielsen et al.,2008). Typically, fractures under laboratory investiga-tion are either on the surface of samples, or the resultof new or growing fractures from an applied stress toinduce fracture stick-slip creep (Thompson et al., 2009;Gross et al., 1993). Recently, Blum et al. (2010) usednoncontacting techniques to probe a fracture inside aclear sample to recover the fracture compliance. A high-powered laser excites the surface of the sample creatingultrasonic waves. These waves scatter from the fractureand are recorded at the surface of the sample with a

laser interferometer (Scruby and Drain, 1990). Here, in-stead of only exciting the ultrasonic waves at the sam-ple surface, we focus a pulsed infrared (IR) laser beamat the fracture location, turning it into an ultrasonicsource. This technique makes it possible to measure thefracture response as a function of source energy, stresson the sample, or the laser beam size and location. Byscanning the fracture with a focused IR laser beam itmay be possible to measure spatial variations in thefracture properties and delineate barriers and asperi-ties (Scholz, 1990), concepts that are of great impor-tance in earthquake dynamics for example. A localizedexcitation, along the fracture, could also be used to ex-cite interface waves traveling along the fracture (Royand Pyrak-Nolte, 1997; Gu et al., 1996) to probe forproperties such as fault gouge or the fluids filling thefracture. Here, we illustrate the use of direct excitationof a fracture to investigate the elastic effective size of thefracture by means of tip diffractions. Until now, theseare most commonly studied on surface cracks for crackcharacterization (Masserey and Mazza, 2005).

2 EXPERIMENT

We create a single disk-shaped fracture by focusing ahigh power Q-switched Nd:YAG laser in a cylinder madeof extruded Poly(methyl methacrylate, PMMA), with adiameter of 50.8 mm and a height of 150 mm. The lasergenerates a short pulse (∼ 20 ns) of infrared (IR) light

304 Blum, van Wijk, Snieder & Willis

Figure 1. Photograph of the laboratory sample and zoom

around the disk-shaped fracture, with ruler units in cm. Thesample is cut in half longitudinally to display the fracture

without optical deformation by the curvature of the sample.

The diameter of the fracture is ∼ 7 mm, and the diameter ofthe cylinder is 50.8 mm.

that is absorbed by the sample material at the focalpoint and converted into heat. The sudden thermal ex-pansion generates sufficient stress to form a fracture in-side the plastic material (Zadler and Scales, 2008; Blumet al., 2011). Anisotropy in the elastic moduli, causedby the extrusion process, results in a fracture with anorientation parallel to the cylindrical axis. The fracturestudied here is approximately circular with a diameterof ∼7 mm (Figure 1).

Elastic waves are excited at the surface of the sam-ple by using the same high-power Q-switched Nd:YAGlaser, operated at a much lower power, and with apartially focused beam. When an energy pulse fromthe laser hits an optically absorbing surface, part ofthat energy is absorbed and converted into heat. Theresulting localized heating causes thermal expansion,which in turn results in elastic waves in the ultrasonicrange (Scruby and Drain, 1990).

Typically, the laser is focused on the outside of thesample – but as we explore in this Letter – the lasercan also be focused inside the sample. In this case, theplanar fracture has a visible contrast with the rest ofthe sample, seen as a darker region in Figure 1. TheNd:YAG pulsed laser generates energy at a wavelengthof 1064 nm, in the near infrared. Therefore, we assumethat the optical contrast due to the fracture is alsopresent at the IR wavelength, leading to energy absorp-tion and thermoelastic expansion at the fracture loca-tion.

We measure elastic displacement with a laser inter-ferometer, based on a doubled Nd:YAG laser, generat-ing a Constant Wave 250 mW beam at a wavelengthof 532 nm. The light is split between a beam reflectingoff the sample and one following a reference track insidethe sensor. Two-wave mixing of the reflected and refer-

sourceLaser

y

x

Sample rotation

Receiver

S1

θ

δ

Fracture

Figure 2. Top view of the experimental setup for direct frac-

ture excitation. The laser source beam (red) excited elastic

waves (blue) at S1.

ence beams in a photo-refractive crystal delivers a pointmeasurement of the out-of-plane displacement field atthe sample surface. The output is calibrated to givethe absolute displacement in nanometers (Blum et al.,2010). The frequency response is flat between 20 kHzand 20 MHz, and accurately detects displacements ofthe order of parts of Ångstroms. Since the PMMA sam-ple is transparent for green light, we apply a reflectivetape to the surface to reflect light back to the laser re-ceiver.

The location of the non-contacting ultrasonicsource and receiver are fixed in the laboratory frame ofreference, but the PMMA sample is mounted on a rota-tional stage. The source-receiver angle δ (defined in Fig-ure 2) is therefore constant, here δ = 20◦, and only theorientation of the fracture with respect to the frame ofreference, characterized by the angle θ, changes. More-over, the source and receiver are focused on the samplein an x − y plane normal to the cylinder axis (z axis,Figure 2). While anisotropic, as mentioned above, theextruded PMMA is transversely isotropic, and its elas-tic properties are therefore invariant with respect to thedefined angles of interest.

By computer-controlled rotation of the stage, wemeasure the elastic field in the (x, y)-plane for values ofθ in increments of 1 degree. The signal is digitized with16-bit precision and a sampling rate of 100 MS/s (megasamples per second) and recorded on a computer acqui-sition board. For each receiver location, 256 waveformsare acquired and averaged after digitization.

Figure 3 shows the ultrasonic displacement field forthe source S1 at the fracture for all recorded azimuths,after applying a 1-5 MHz band-pass filter. As definedin Figure 2, the horizontal axis represents the angle θbetween the normal to the fracture and the source di-rection. Electromagnetic interferences are generated bythe high-power source laser when the light pulse is emit-ted, and leads to noise being recorded for short arrivaltimes (0 – 3 µs, highlighted in Figure 2). The arrival

Laser excitation of elastic waves at a fracture 305

θ (°)

Tim

e (

µs

)

−90 0 90 180 270

0

10

20

30

40

Source

laser

noise

fP

PfP

fPP

PP

Figure 3. Displacement field generated by excitation of the

fracture. fP is the P-wave generated at S1 and traveling di-rectly to the receiver. PfP is the P-wave generated at S2 andscattered by the fracture before reaching the receiver. fPP isthe P-wave generated at S1, traveling away from the receiver

before bouncing back to the sample surface. Finally, PP isthe P-wave generated at S2, traveling across the sample andbouncing back to the receiver.

at approximately 10 µs denoted fP corresponds to thewavefield excited at the fracture. The fPP wave is ex-cited at the fracture and reflects off the backside of thesample.

Next, we apply reflective tape where the sourcelaser beam hits the sample surface at S2, increasingthe IR light absorption at the surface and lowering theamount of energy reaching the fracture (Figure 4). Werepeat with this configuration the acquisition procedureused in the first experiment (Figure 5). The PfP wave isgenerated at the surface of the sample, and then scat-tered by the fracture, while PP is scattered from thebackside of the sample. PfP and PP phases are strongerthan fP and fPP in Figure 3, confirming that more ofthe thermoelastic expansion takes place at the surfaceof the cylinder.

2.1 Fracture tip travel times

The waves fP and PfP in Figures 3 and 5 show a distinctlenticular pattern. For source angles θ = −10◦ and 170◦,the PfP phase is a specular reflection, and the amplitudeis a maximum. For intermediate angles, the scatteredamplitude decreases (Blum et al., 2011). Note splittingof the wave at intermediate angles into wavelets arrivingbefore and after the specular reflection (see Figure 6).These waves have the travel time and phase of wavesdiffracted by the crack tips. In particular, for θ = 70◦,the receiver is in the plane of the fracture, and there-fore the travel time difference between the tips of the

sourceLaser

^f

n̂

m̂

y

x

Receiver

Sample rotation

θ

δ

S2

Fracture

Figure 4. Top view of the experimental setup for elastic-

wave excitation at the sample surface. The laser source beam

(red) excited elastic waves (blue) at S2.

θ (°)

Tim

e (

µs)

−90 0 90 180 270

0

10

20

30

40

Source

laser

noise

PfP

PP

Figure 5. Displacement field generated by excitation at thesample interface. Signal for t < 3 µs corresponds to noise gen-

erated by the laser source, and to the direct P-wave traveling

directly from the source S1 to the receiver. Other arrivals aredefined in Figure 3.

fracture the closest and the farthest to the receiver islargest (Figure 4).

Equation 39 in Blum et al. (2011) shows that theP to P scattered amplitude for a planar fracture in alinear-slip model in the Born approximation can be writ-ten in the frequency domain as a product of a scalingfactor, a factor depending on the mechanical proper-ties of the fracture and the propagation medium, and aform factor that depends of the fracture shape and thewavenumber change from the fracture scattering. Onlythis last factor carries time information. We show in theAppendix that the corresponding travel times are

ttip-sc =R

α

(2± a

R(sin θ(1 + cos δ) + sin δ cos θ)

),

(1)where a is the radius of the fracture and R the ra-dius of the cylinder. The P-wave velocity is given by

306 Blum, van Wijk, Snieder & Willis

θ (°)

Tim

e (

µs)

−90 0 90 180

16

18

20

22

24

Figure 6. Detailed view of the scattered (PfP) arrival. Thesolid (orange) curves represent the tip arrival times computed

from equation (1).

θ (°)

Tim

e (

µs)

−90 0 90 180

6

8

10

12

14

Figure 7. Detailed view of the direct fracture excitation

arrival. The solid (purple) curves represent the tip arrival

times computed from equation (3).

α = 2600 m/s (Blum et al., 2011). Figure 6 shows thePfP arrival overlain by the computed travel times fromequation (1) with a fracture radius aPfP = 3.3 mm.

For the arrival time of the fP wave that is excitedat the fracture, we consider the geometry of rays orig-inating from the fracture tips and traveling directly tothe receiver. The raypaths are shown in Figure 2. Usingthis geometry the travel time can be expressed as

ttip-direct =

√a2 ± 2aR sin(θ) +R2

α. (2)

Due to the fact that the size of the fracture is smallcompared to the radius of the sample, this travel timeis to leading order in a/R given by

ttip-direct =R

α

(1± a

Rsin(θ)

). (3)

Figure 7 shows the fracture-source displacement fieldoverlain with the tip arrival time (in blue) computedfrom equation (3). Just as in Figure 6, the theoreticaltime for a radius afP = 3.3 mm agrees well with thearrival time of the fP wave, and the observed size inFigure 1. The good agreement with the visually esti-mated radius confirms that the whole visually fracturedarea is mechanically discontinuous and capable of beingexcited by elastic waves.

3 CONCLUSIONS

Laser-based ultrasonic techniques can not only exciteand detect elastic waves at the surface, but can also

be used to directly excite heterogeneities (such as frac-tures) inside an optically transparent sample. This re-sult opens up possibilities for diagnosing the mechanicalproperties of fractures by directly exciting them. Here,we estimate the effective elastic size of the excited frac-ture. By scanning the fracture with a focused IR laserbeam, it may be possible to measure spatial variationsin the fracture properties and delineate barriers andasperities. These concepts are of great importance inearthquake dynamics, although hard to investigate inthe field or numerically.

ACKNOWLEDGMENTS

We thank ConocoPhillips for funding this research, PaulMartin for his valuable suggestions, and Randy Nuxollfor his help with sample preparation.

REFERENCES

Blum, T., R. Snieder, K. van Wijk, and M. . Willis,2011, Theory and laboratory experiments of elasticwave scattering by dry planar fractures: J. Geophys.Res., 116, B08218.

Blum, T., K. van Wijk, B. Pouet, and A. Wartelle,2010, Multicomponent wavefield characterizationwith a novel scanning laser interferometer: Rev. Sci.Instrum., 81, 073101.

Gross, S., J. Fineberg, M. Marder, W. McCormick, andH. Swinney, 1993, Acoustic emissions from rapidlymoving cracks: Phys. Rev. Lett., 71, 3162–3165.

Gu, B., K. Nihei, L. Myer, and L. Pyrak-Nolte, 1996,Fracture interface waves: J. Geophys. Res., 101,PAGES 827–835.

Keller, J., 1978, Rays, Waves and Asymptotics: Bull.Am. Math. Soc., 84, 727–750.

Larose, E., T. Planes, V. Rossetto, and L. Margerin,2010, Locating a small change in a multiple scatteringenvironment: Appl. Phys. Lett., 96, 204101.

Masserey, B., and E. Mazza, 2005, Analysis of thenear-field ultrasonic scattering at a surface crack: J.Acoust. Soc. Am., 118, 3585.

Nakahara, H., H. Sato, T. Nishimura, and H. Fujiwara,2011, Direct observation of rupture propagation dur-ing the 2011 off the Pacific coast of Tohoku earth-quake (mw 9.0) using a small seismic array: EarthPlanets Space, 63, 589.

Nielsen, S., J. Taddeucci, S. Vinciguerra, and A. Schub-nel, 2008, Tangled dynamic rupture propagation onlaboratory faults: Eos, 89, Abstract S44A–04.

Pyrak-Nolte, L., and D. Nolte, 1992, Frequency depen-dence of fracture stiffness: Geophys. Res. Lett., 19,325–328.

Roy, S., and L. Pyrak-Nolte, 1997, Observation of adistinct compressionalmode interface wave on a singlefracture: Geophys. Res. Lett., 24, PP. 173–176.

Laser excitation of elastic waves at a fracture 307

Scholz, C., 1990, The Mechanics of Earthquakes andFaulting: Cambridge Univ. Press.

Scruby, C., and L. Drain, 1990, Laser Ultrasonics Tech-niques and Applications, 1st ed.: Taylor & Francis.

Snieder, R., 2009, A Guided Tour of MathematicalMethods: For the Physical Sciences, 2nd ed.: Cam-bridge University Press.

Thompson, B., R. Young, and D. Lockner, 2009, Pre-monitory acoustic emissions and stick-slip in natu-ral and smooth-faulted Westerly granite: J. Geophys.Res., 114, 14 PP.

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APPENDIX A: TIP-DIFFRACTION TIMESFROM FORM FACTOR

Equation 39 of Blum et al. (2011) shows that the P to Pscattered amplitude for a planar fracture in a linear-slipmodel in the Born approximation is

fP,P (n̂; m̂) =ω2

4πρα4AF (kα(n̂− m̂))

×{λ2ηN + 2λµηN

((n̂ · f̂)2 + (m̂ · f̂)2

)+4µ2(ηN − ηT )(n̂ · f̂)2(m̂ · f̂)2

+4µ2ηT (n̂ · m̂)(n̂ · f̂)(m̂ · f̂)},

(A1)

where ω is the angular frequency, α the P-wave velocity,ρ the density of the material, λ and µ the Lamé param-eters, A the surface area of the fracture, and ηN and ηTthe normal and tangential compliances, respectively, forthe linear-slip model. The unit vectors n̂ and m̂ denotethe directions of incoming and outgoing waves, respec-tively, and f̂ is the unit vector normal to the fracture(see Figure 4).

The prefactor (ω2/4πρα4)A does not carry time in-formation. The factor in curly brackets contains the an-gular dependence of the scattering amplitude, and de-pends only on the mechanical properties of the fractureηN and ηT of the sample material, and on the direc-tions of the incoming and outgoing waves relative to thefracture orientation. The form factor F (kα(n̂− m̂)) de-pends on the fracture size and shape, and contains traveltime information. For the case of a circular fracture, theform factor is given by eq. (33) of Blum et al. (2011):

F (kα(n̂− m̂)) =2

k‖aJ1(k‖a) , (A2)

where a is the radius of the fracture, k‖ the projectionof the wavenumber change during the scattering onto

the fracture plane, and J1 the first order Bessel func-tion. According to equation (20.53) of Snieder (2009),the asymptotic form of the Bessel function is

Jm(x) =

√2

πxcos(x− (2m+ 1)π

4

)+O(x−3/2) ,

(A3)For the geometry described in Figure 4, the wavenumberchange can be expressed as

k‖ =ω

α(sin θ(1 + cos δ) + sin δ cos θ) . (A4)

Inserting equations (A3) and (A4) into expression (A2),and expanding the cosine in exponentials gives

F (k) ∝(eiπ/4eiωT + e−iπ/4e−iωT

), (A5)

where T = (a/α) (sin θ(1 + cos δ) + sin δ cos θ). T and−T quantify the delay time of the tip diffraction arrivalsrelative to the arrival time t = 2R/α for a ray reflect-ing at the center of the fracture. Therefore, the totaltip diffraction travel times for the scattered arrival aregiven by equation (1). Note that this expression pre-dicts a phase shift exp(±iπ/4) for these waves that ischaracteristic of edge-diffracted waves (Keller, 1978).

308 Blum, van Wijk, Snieder & Willis