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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
[email protected](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
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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
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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
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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
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and laboratory experiments of elasticwave scattering by dry planar
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Blum, T., K. van Wijk, B. Pouet, and A. Wartelle,2010,
Multicomponent wavefield characterizationwith a novel scanning
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Gu, B., K. Nihei, L. Myer, and L. Pyrak-Nolte, 1996,Fracture
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Keller, J., 1978, Rays, Waves and Asymptotics: Bull.Am. Math.
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Locating a small change in a multiple scatteringenvironment: Appl.
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Laser excitation of elastic waves at a fracture 307
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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).
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308 Blum, van Wijk, Snieder & Willis