Slow wave propagation in air-filled permeable solids
Peter B. Nagy
Departmentof WeldingEngineering,The Ohio State University,Columbus,Ohio 43210
(Received 15 October 1992; acceptedfor publication16 February 1993)
The propagationof slowcompressional
wavesin air-saturatedpermeablesolidswasstudiedby
experimentalmeansbetween10 and 500 kHz. The velocityand attenuationcoefficientwere
measuredas functionsof frequencyfrom the insertiondelay and loss of airborne ultrasonic
wavestransmittedthroughthin slabsof 1-5 mm in thickness.Porousceramicsof 2-70 Darcy
and natural rocksof 200-700 mDarcy permeabilitywere tested.In the low-frequency(diffuse)
regime,the experimentalresultsare consistent
with theoreticalpredictions;
the phasevelocity
and attenuationcoefficientare essentiallydeterminedby the permeabilityof the specimenand
both increaseproportionallyto the square root of frequency. In the high-frequency
(propagating)regime,the experimentalresultsare consistent
with the theoreticalpredictionsfor
the phase velocity but not for the attenuationcoefficient.The phasevelocity asymptotically
approachesa maximum value determinedby the tortuosity of the specimenwhile the
attenuationcoefficientbecomeslinearly proportionalto frequencyinsteadof the expected
square-rootrelationship.It is suggested
that the observeddiscrepancyis due to the irregular
pore geometry that significantlyreducesthe high-frequencydynamic permeabilityof the
specimens.
PACS numbers: 43.35.Cg
INTRODUCTION
The claycontentgreatlyincreases
viscous
dragbetween
The most interestingfeatureof acousticwave propagationin fluid-saturatedporousmediais the appearance
of
a secondcompressional
wave,the so-calledslowwave.The
existence
of a slowcompressional
wavein an isotropicand
macroscopically
homogeneous
fluid-saturatedporousme-
the fluid and solid frame, which results in excessiveatten-
uation and usually completedisappearance
of the slow
wave. Recently,Gist suggested
a simplemodelto demonstratethat, due to inherentpore-wallroughness,
the attenuation of the slow compressionalwave in rocks can be
thanthepredictions
oftheBlottheory.
5
diumwaspredicted
by Biotin 1956.
]'2Themaincharac- manytimeshigher
teristicof this modeis that its velocityis alwayslower than
both the compressional
wave velocityin the fluid and the
longitudinal velocity in the solid frame. Below a critical
frequency,which dependson the poresizein the frame and
the viscosityof the fluid, the slow compressional
wave is
highly dispersiveand strongly attenuatedover a single
wavelength.Above this critical frequency,it becomesa
dispersion-free
propagatingwavewith increasingbut fairly
low attenuation.The slowcompressional
waverepresents
a
relative motion between the fluid and the solid frame. This
motion is very sensitiveto the viscosityof the fluid and the
dynamic permeabilityof the porousformation. Naturally,
low-viscosityliquidssuchas water are the fluidsmostoften
used in experimentalstudiesof slow wave propagation.
However,it will be shownin this paperthat it alsomakes
good senseto use gaseousfluids suchas air to saturatethe
The excessviscousdragexplainsthe difficultyto detectthe
slow wave in fluid-saturated
rocks as well as the observed
correlation betweenultrasonicattenuationand clay content in sandstones.One way to reducethe excessiveattenuation of slow wavesin porousmaterialsis to usespecial
fluids of very low viscosityto saturatethe specimen.For
instance,
superfluid
4Hebelow1.1K hasbeenshown
to
workverywellinfused
glass
beadsamples,
6superleak
materialsconsisting
of compacted
powders,
7-9andin sandstones.10
The questionof whether or not excessiveattenuation
in viscous fluid-saturated
natural
rocks renders the detec-
tion of slowwavesimpossiblearises.Not necessarily!
Even
a very weak slow wave, attenuated by as much as 50-60
dB, couldeasilybe detectedbut for the presenceof much
strongerbackground"noise" causedby the direct arrivals
and scatteredcomponentsof the fast compressionaland/or
porous specimens.
shearwaves.If we could generatethe slow wave only and
Since 1980, when Plona was able to observeslow wave
propagation
inwater-saturated
porous
ceramics,
3theques- nothingelse,it would be much easierto detectin spite of
the substantialattenuation.Comparedto the solid frame,
tion of why slow wavescannotbe detectedin real rockshas
liquidslike water usuallyhavea lower, but still comparable
been one of the major issuesin the acousticsof fluidsaturated materials.
Klimenatos
and McCann
showed that
this lack of perceivableslowwave propagationis probably
dueto inherentinternalimpurities,suchassubmicronclay
density
pf andbulkmodulus
Bf. Althoughtheirviscosity
bt is also relatively high, which makes saturationof the
poroussamplesomewhattroublesome,
their kinematicvis-
particles,
foundin all types
of natural
rocks.
4 These
clay cosityr/=p/pf is fairlylow.On theotherhand,gaseous
particlesare depositedwithin the porethroatsand on the
surfaces
of the rockgrainsaswell assuspended
in thefluid.
3224
J. Acoust.Soc.Am.93 (6),June1993
fluids like air have very low density,bulk modulus, and
viscosity,while their kinematicviscosityis usuallyrather
0001-4966/93/063224-11506.00 ¸ 1993Acoustical
Societyof America
3224
o
fast
slow
-20
TABLE I. Physicalparametersof water and air at 20 øC.
shear
•
Type
Water
-40
Pf3
(kg/m)
1000
Air
1.3
vf
(m/s)
•
(mm2/s)
fmin
fmax
(kHz)
(kHz)
1480
I
5
810
332
15
75
180
-60
-80
slow880m/s
-lOO
5
,o
10
(a)
15
20
25
30
35
40
45
Angleof Incidence
[deg]
slow
-20
-40
-60
fast
-80
-100
velocities:
fluid 332m/s
fast
s--•
0
(b)
s_.•
5
10
2437 m/s
slow
225
m/s
shear 1380 m/s
15
20
25
30
35
40
45
Pmgle
ofIncidence
[deg]
FIG. 1. Slow, fast,and sheartransmission
coefficients
through (a) waterand (b) air-saturatedporousglassbeadplates.
nately, this doesnot happenin most natural rocks where
the shearvelocityis rather low). On the other hand, in the
caseof air saturation,the slow compressional
wave is at
least70 dB strongerthan all other modesand, due to the
very low sound velocity in air, the shear critical angle
dropsbelow 15ø,abovewhich only the slow wave is transmitted through the sample.This meansthat the highly
attenuated slow wave is submergedin electrical noise
ratherthan spurioussignalssoit canbe easilyrecoveredby
simple time-averaging.
In spiteof the excellentcouplingbetweenthe incident
compressional
waveand the transmittedslowwaveand the
obviousadvantageof saturatingthe specimenwith lowviscosityair rather than high-viscositywater, slow wave
propagationin air-filledporoussampleshas neverbeen
extensivelystudiedabovethe critical frequencywhere it
becomes
a propagating
mode.It shouldbe mentionedthat
considerablework has been done on air-filled porousmaterialsin the so-calleddiffuseregime,i.e., at relativelylow
frequencies
between
50Hz and4 kHz?-17Above
thecritical frequency,slowwavesare not expectedto propagatein
air-saturatedporoussamplesas well as in water-saturated
ones?Sincethekinematic
viscosity
of air is solargeand
the velocityof soundso small, there is but a very narrow
high. Therefore,it is very simpleto saturatea poroussample by air, but the slow wave is expectedto be highly
dispersive
andstronglyattenuated.In spiteof theseadverse
effects,slow wavescan be readily observedin air-filled poroussamples,includingcertainnaturalrocks,by usingairborneultrasonicwaves.•1
In the case of air saturation, because of the tremen-
dous acousticalmismatch between the incident compressionalwaveand the poroussolid,all the incidentenergyis
either reflected or transmitted
via the slow wave without
generatingappreciablefast compressional
or sheartransmitted waves. In order to demonstrate this crucial effect,
Fig. I showsthe slow, fast, and shear wave transmission
coefficientsthrough water- and air-saturatedglassbead
specimens.
The physicalparametersof the glassbeadspecimen and details of the calculation are given in Ref. 12. In
the caseof water saturation,the slow compressionalwave
is usually 5-10 dB weaker than the fast compressionalor
shear modes and it is much more attenuated. Also, because
of its lower velocity, it arrives later than the other modes
and it is often overshadowedby multiple reflectionsand
scatteredcomponents
of thesestrongersignals.Maybe the
only exception is when the shear velocity is sufficiently
high so that we can work above the secondcritical angle
where the slow compressionalwave becomesthe only
propagatingmode in the fluid-saturatedsample (unfortu3225
J. Acoust. Soc. Am., Vol. 93, No. 6, June 1993
frequencywindowwherethe attenuationcoefficient
is sufficiently low to observea more or less dispersion-free,
scattering-free
slow wave. This "window" is set by the
conditions
that the viscous
skindepth5=(2•1/to)1/2,
where to denotesthe angularfrequency,be lessthan the
poresizeapand,simultaneously,
the wavelength
)[ be
largerthanthegrainsizeag.TableI summarizes
therelevantphysicalparameters
of waterand air as well asfmm
and fmax, i.e., the limits of the frequencywindowwhere
slowwavepropagation
is expected.
ag=200-/.tm
graindiameterand •b=30% porositywere assumedin the calculations. By assuminga perfectlystiff frame, the highfrequencyasymptoticvalueof the slow wavevelocitycan
beeasily
calculated
asv=vf/r•/2,where
¾fdenotes
the
soundvelocityin the fluid and r is the tortuosityof the
permeableframe.For the purposes
of the examplegivenin
TableI, thetortuosity
wasestimated
fromtheporosity
as
r= 1/2(qb-1
+ 1) =2.17.19
Todetermine
fminandfmax,we
assumedthat the pore size is approximately six times
smallerthanthe grainsize(a•=ag/6) andat leastfour
timeslargerthanthe viscous
skindepth(ap•>45)to account for the smaller crosssectionsat the crucial pore
throats:
fmin
= 576'///rra2g
Peter B. Nagy: Slow-wave propagationin air-filledsolids
( 1)
3225
2.5
and
fmax
= ¾f/2Yrag
•'1/2'
(2)
Table I clearlydemonstrates
the greatlyreducedfrequencywindowwhere more or lessdispersion-free
and
attenuation-free
slowwavepropagation
canbe expected
in
typicalair-filledsamplesof approximately
200-/tin grain
size. On the other hand, these results do not exclude slow
Attenborough
1.5
ß
wave propagationover a much larger frequencyrange.
They simply mean that the slow wave becomesincreasinglydispersive
below100kHz andverystrongattenuation
can be expectedabove200 kHz.
0.5
I. THEORETICAL
BACKGROUND
3
4
5
6
7
8
Log {Frequency[Hz]}
For the specialcaseof air-saturatedpermeablesolids
of random
formation,
Attenborough's
theoretical
model
•6
canbe usedto determineboththe complexwavenumberk
and the complexacousticimpedanceZ:
k(co)=co[p(co)/K(co)
]1/2
FIG. 2. Comparisonbetweenthe slow wave propagationmodelsof Attenborough(Reft 16) and Johnsonet al. (Ref. 20).
(3)
sic parameters,namely porosity •b, high-frequency
tortuosityr•o, staticpermeability
K0, andporeshapefactor
and
Z(to)= [p(ro)K(ro)
]•/2,
(4)
wherep(to) andK(r0) denotethe complexdensityand the
complexmodulus,respectively,of the air-saturatedmaterial. The complexdensityincludesthe effectof viscosity
ratiosp,to describe
the porousformation.
This model
worksbetter for air-saturatedmaterialsthan the dynamic
permeability
modelof Johnson
etal.20because
the latter
oneis limitedto fluidsof negligible
thermalexpansion
coefficient.Comparisonof the two modelsrevealsthat the
actual difference between them is rather small. The best
],
(5)
while the complexmodulusincludesthe somewhatweaker
effectof heat conductionin air,
K(ro)
=•[1+(y--1)t(Pr
1/2
•)] '
(6)
Here, ?' denotesthe specificheat ratio, Pr is the Prandtl
number(y= 1.4 and Pr=0.74 for air), and t is a simple
functionof the normalizedporeradius•:
(7)
whereJ0 and Jl are the zero- and first-orderBesselfunctionsand i is the imaginaryunit. For cylindricaltubes,the
normalizedpore radiusis exactlyknown:
• =a( ro/*q
) •/2,
(8)
wherea denotesthe actual pore radius.For real porous
materialsof random pore geometry,there is no corre-
sponding
exactsolution.Attenborough
16suggested
that
the normalizedpore radius shouldbe calculatedas
•= (2r•oK0tO/•,/•)
1/2,
andhighfrequencies,
which
requires
that%=yl/2•0.423
andAm(5.6r©%/qfi)1/2,
where
A isa measure
ofthepore
sizein thenotationof Ref. 20. As an example,Fig. 2 shows
the calculatedattenuation
coefficient
of the slowcompressionalwavein a poroussolidof •b=0.3, r•o= 1.79,and
K0=2.2X10-12m2.Thelargest
discrepancy
between
the
two models occurs at the transition between the diffuse and
propagatingregimes,but it neverexceeds12%.
Sincethe staticpermeabilityK0 and the pore shape
factorratio % alwaysoccurin the samecombination
2J1(
t(•):x•S0(x/-•),
agreementcan be achievedby asymptoticmatchingat low
throughthe normalizedporeradius•, theycannotbe separatedby acousticalmeasurements.
Accordingto the dynamicpermeabilitymodel,at low frequencies,
the complex
wave numberof the slow compressional
wave can be ap-
proximated
as2ø
k(løw)
• kf( i&*l/KotO
) 1/2.
(10)
Attenborough's
model
16yields
a similar
formof
k(•o,•)
y-,kf(iq•,l/Kato
)1/2,
( 11)
where% denotesthe "acoustic"permeability:
(9)
Ka= K0/4yS•p.
( 12)
wherego is the staticpermeability,%0 is the real-valued
Accordingto Attenborough'smodel,only three inde-
high-frequency
tortuosity,and st,is the so-called
pore
pendent parameterscan be determined from acousticmea-
shapefactor ratio that is usually between0.1 and 0.5.
Attenborough'sanalyticaltechniqueprovidesa unified
surementson air-filled porous solids: porosity •fi, highfrequencytortuositytoo, and the low-frequencyacoustic
permeabilitytq, which is a combinationof the static per-
model for both low-frequency (diffuse) and highfrequency (propagating) regimes of the slow compressionalwave in air-saturatedporoussolids.It usesfour ba3226
d. Acoust.Soc. Am., Vol. 93, No. 6, June 1993
meability
•0 andtheporeshape
factorratiost,.Thelatter
one is alwayslessthan 0.5 while y is higher than one for
Peter B. Nagy: Slow-wavepropagationin air-filledsolids
3226
I
TABLE II. Estimatedmaterialparameters
of cementedglassbeadspecimens (•=30%
and •-©= 1.79 for all grades).
K0
Grade
( 10-12m2)
sp
15
2.2
0.460
40
6.5
0.460
55
II
0.475
90
27
0.475
175
67
0.475
• 0.6
Grade 175
Grade 90
• 0.4
Grade 55
Grade 40
Grade 15
• 0.:2
0
gases.As we mentionedabove,the acousticpermeability
100
0
equals
thestaticonefors•,=0.423.
Forcylindrical
pores,
200
(a)
300
400
500
600
500
600
Frequency[KHz]
theporeshape
factorratioreaches
itsmaximum
of%=0.5
and the acousticpermeabilityis slightly lower than the
staticpermeability.This is becausethe specificheatratio is
40
Grade 1$
higherthanonein airandtheslowwavevelocity
depends
ontheisothermal
sound
velocity
Vr=Vf/•/1/2
rather
than
ontheadiabatic
velocity
vœ,
which
ismeasured
inaninfi-
Grade 40
• 30
nitemedium.For poreswith noncircularcrosssections,
the
Grade
55
Grade
90
Grade175
pore
shape
factor
ratio
s•,
can
bemuch
lower
than
0.5.
For
• 20
example,
in thecaseof equilateral
triangle
cross
section,
s•,
is aslowas0.158.16
We shallshowthat,in thecaseof the
measured
high-permeability
sandstones,
theporeshape
factorratiois between
0.2 and0.3,therefore
theacoustic
permeabilityis somewhathigherthan the staticone.
Recently, Champoux and Allard extended the dy-
0
0
namicpermeability
model
2øof Johnson
etal. for airsaturated
solids.
21'22
Thisnewmodelproperly
accounts
for
100
200
(b)
300
400
Frequency
[kHz]
the fact that viscouseffectsare dominatedby small pore
FIG. 3. (a) Normalizedvelocityand (b) attenuationcoefficientof the
crosssectionsand thermal effectsby large onesby introslow compressional
wave in air-filledcementedglassbeadspecimens
as
ducing an additional characteristiclength parameter. functionsof frequencyfor differentgrades.
However,the useof this more rigoroustheoryapparently
would not improvethe agreementwith our experimental
(13)
data at high frequencies
wherethe only major discrepancy
occurs.Therefore,we shall useAttenborough'ssimpleran-
alytical
technique
]5'•6to calculate
thevelocity
andattenuation coefficientof slow compressionalwavespropagating
in air-filledporousmaterialsfor comparisonto our experimental data.
In order to demonstrate
the main features of slow
wavepropagationin air-filledporoussolids,let uscalculate
the propagationparametersand the complexacousticimpedance in cementedglass bead specimensof different
grades.Table II liststhe material parametersusedin these
calculations.
The porosity(•6=0.3) andstaticpermeability
were estimated from the manufacturer's (Eaton Products
International, Inc.) specifications.
The high-frequencytortuosity was taken to be 1.79 as the most typical parameter
and
a.ow)
= (20nys/K0f)1/2.
(14)
The normalizedattenuationcoefficienta•, i.e., the total
attenuationover a singlewavelength,is constant,
cz(•
•øw)
=ct(løw)v/f=2;r(Neper)
•55 riB.
(15)
In the propagatingregime, i.e., at high frequencies,the
velocityapproaches
a constantvaluewhile the attenuation
coefficientis againproportionalto the square-rootof frequency,althoughthe proportionalitycoefficientis slightly
different from the low-frequencyvalue (see Fig. 2):
obtained from acoustical and electrical measurements on
similar
materials.
6'18'23'24
Finally,theporeshape
factorratioswerechosen
bymatching
theanalytical
results
toour
experimentaldata to be presentedin the next section.
Figure3 shows
thenormalized
velocityv/vf andthe
v(high)/¾f=y•
1/2
(16)
and
cz(high)=
(c0'/7d}'ro•S2/4K04
.)1/2[1-1(y--1)Pr•/2],(17)
attenuationcoefficient
a of the slowcompressional
wavein
the air-filled poroussampleslistedin Table II as functions
of frequency.In the diffuseregime,i.e., at low frequencies,
both the velocityand the attenuationcoefficientare pro-
where the secondterm of Eq. (17) is approximately1.46
for air. It is interestingto note that the ratio betweenthe
high- and low-frequencyasymptoticvaluesof the attenua-
portional to the square-rootof frequency:
tion coefficients
3227
d. Acoust.Soc. Am., Vol. 93, No. 6, June 1993
is
Peter B. Nagy: Slow-wave propagationin air-filledsolids
3227
5O
coefficientof the slow compressional
wave. Even then, but
especiallyin the transitionregionbetweenthe diffuseand
propagatingregimes,we have to take into accountthe total
(fluid/fluid-saturatedporoussolid/fluid) transmission
loss
40t/
Grade 15
Grade 40
Grade 55
3o
Grade 90
T o causedby the significantacousticalimpedancemismatchbetweenthe air and the air-filledspecimen:
20
4
To(w)
=2+Z(ca)/Zf+
Zf/Z(ca)
'
IO
0
10
20
30
40
50
6O
(20)
The simplestthing to do is to approximatethe actual
acousticimpedanceby its real-valuedhigh-frequency
asymptote. From Eqs. (19) and (20),
Frequency[kHz]
4
TO
(high)
---
(21)
, s/ 1/2'
2+ 7.1/2•s
• /•q-•/•-•
0
Forexample,
T0(hig•l
•_.--4.5dBfor•=0.3 andr• = 1.79.
In this way, we inevitably underestimatethe total transmissionlossand, consequently,
overestimatethe attenua-
• -30-[I
Grade00
•
Grade
40
II
tion coefficient.
Also, because
of the phaseshiftcausedby
the complexnature of To(ca), we slightlyunderestimate
the slow wave velocityin samplesof small thickness.In
order to get better agreementbetweenthe experimental
measurements
and theoreticalcalculations,we can easily
correctour analyticalresultsfor the differencebetweenthe
actualtransmission
lossTo(ca)and its real valuedasymp-
z
-50
,
,
,
0
10
20
(b)
toteT(0
high)
. Themeasured
transmission
coefficient,
30
40
50
60
Tm(ca) = To( ca)exp( icad/v)exp( -ad),
Frequency
[kHz]
(22)
can be expressedas
FIG. 4. (a) Normalizedrealand (b) imaginarycomponents
of the acoustic impedancein air-filled cementedglassbeadspecimensas functionsof
frequencyfor differentgrades.
Tm(ca)
= T?ig•lexp(icad/va)exp(--a•),
(23)
whered isthethickness
of thespecimen
andVaandaa are
•(high)/•(low)
•0.43•%1/2,
(18)
i.e., fairly closeto onefor mostporoussolidsof interestto
us. Of course,in the propagatingregime,where the Mow
wavevelocityasymptoticallyapproaches
a constantvalue,
the normalized
attenuation
coefficient decreases with fre-
quency,at least to a point where other attenuationmechanismssuchas scatteringare still negligible.
Figure 4 showsthe real and imaginarycomponentsof
the acousticimpedancein air-filled cementedglassbead
specimens as functions of frequencyfor five different
grades.Both components
were normalizedto the acoustic
impedance
of the saturating
air, Zœ=v/p/. At low frequencies,
the moduliof bothcomponents
are inverselyproportionalto the squareroot of frequency.At highfrequencies, the real part approachesa finite asymptotic value of
Z(high)/zf:
Tloo/2/q•,
( 19)
while the imaginary part diminishes.
In the diffuseregime, where transmission-typemeasurementsare not feasiblebecauseof the very high normalized attenuation coefficient, we have to determine the
complexacousticimpedancefrom reflection-typemeasurements.In the propagatingregime,we can usetransmission
measurementsto determinethe velocity and attenuation
3228
J. Acoust.Soc. Am., Vol. 93, No. 6, June 1993
the apparentvelocityandattenuationcoefficient,
whichare
correctedaccordingto d. Figure5 showsthe apparentvelocityandattenuationcoefficient
of the slowcompressional
wavein air-filledcemented
glassbeadspecimen
grade55 as
functionsof frequencyfor differentsamplethicknesses.
Becauseof the largerimpedancemismatchand the additional
phaseshift at lower frequencies,the apparentvelocity
dropswhile the apparentattenuationincreasesin thin sampies. In the propagatingregime (above approx. 60 kHz),
the correctionsare negligible.
II. EXPERIMENTAL
TECHNIQUE
AND RESULTS
Figure 6 showsthe blockdiagramof the experimental
system used in this study. It is based on a recently developed method usingthe transmissionof airborneultrasonic
wavesthroughthin platesof air-filledporousspecimens
to
investigatethe propagation parametersof the slow com-
pressional
wave.
• Standard
ultrasonic
NDE equipment
was usedwithout any particulareffortto obtainhigh generation or detectionsensitivity.The rather poor coupling
betweenthe appliedcontacttransducersand air resultedin
a rather low, but fairly constant, sensitivityover a wide
frequency range of 50-500 kHz. In certain cases,we replaced the ultrasonic transmitter and receiver by a cornPeter B. Nagy: Slow-wave propagationin air-filledsolids
3228
I
infirfite
4mm
0.8
3mm
2mm
Imm
0.6
0.6
0.4
0.2
0.:2
theoretical
expenmental
0
50
(a)
100
150
1oo
200
200
(a)
Frequency
[kHz]
300
400
500
600
Frequency
[kl-[z]
2O
] lnrn
2 mm
3 nlfiq
• 0.6
infinite
o 04
ß-• 0.2
theoretical
expertmental
<
o
0
50
(b)
100
150
200
0
Frequency
[kHz]
100
(b)
200
300
400
500
600
Frequency
[kHz]
FIG. 5. (a) Apparentvelocity
and(b) attenuation
coefficient
of theslow
compressional
wavein air-filledcemented
glassbeadspecimen
grade55 as
FIG. 7. Comparison
between
thetheoretically
predicted
andexperimentallymeasured
slowwavevelocities
in cemented
glassbeadspecimens
for
functionsof frequencyfor differentsamplethicknesses.
grades(a) 55 and (b) 90.
mercial tweeterand electretmicrophone,respectively,so
that measurementscould be done between 10 and 50 kHz,
tweenthe transducers
weredigitallystored.Then the com-
too. In orderto assurean acceptable
signal-to-noise
ratio,
puterselected
thefirstfivecyclesof thesignal,fromwhich
it determinedtheinsertionlossL i andinsertiondelayti. In
spiteof the relativelynarrowbandwidthof the tone-burst
excitation,the frequencyspectrumof the transmittedsignal wasappreciably
differentfrom that of the directtrans-
extensive
signal
averaging
wasused,
upto 105samples.
The
transmitterwas driven by a tone-burstof five cycles.The
receivedsignalswith andwithoutthe specimen
placedbe-
missionbecauseof the strongfrequency-dependent
attenuationof the specimen.
The insertionlosswasdetermined
Ultrasonic Transducers
2 x Panametrics V101-0,5MHz
/ I"
by Fouriertransforming
the goredsignalsandcalculating
the ratio betweenthe two spectraat the maximumof the
attenuatedone. The insertiondelay was determinedby
findingthe maximumof the cross-correlation
functionof
the two signals.The normalizedvelocityand apparentattenuation coefficient were calculated from the insertion de-
lay ti and lossLi by the followingsimpleequations:
Specimen
%
s•c•on
1
¾f l+tivf/d
Computer
PC/AT
FIG. 6. Blockdiagramof the experimental
system.
3229
J. Acoust. See. Am., Vol. 93, No. 6, June 1993
(24)
and
Ota=( Li-- To)/d.
Peter B. Nagy: Slow-wavepropagationin air-filledsolids
(25)
3229
40
40
--
theoretical
--
[] experimental
theoretical
--- linearasymptote
[] expertmental
cn 30
• 3o
20
.•.
• l0
•
0
100
200
(a)
300
400
500
lO
600
100
200
(d)
Frequency[kHz]
300
400
500
60O
500
600
Frequency[kHz]
4O
40
--
theoretical
theoretical
--- linearasymptote
linearasymptote
experimental
30
experimental
2O
10
• •o
o
0
lOO
o
200
(b)
300
400
500
600
(e)
100
200
300
400
Frequency
[kHz]
Frequency
[kHz]
FIG. 8. Comparison
betweenthe theoretically
predictedand experimentally measured
slowwaveattenuations
in cementedglassbeadspecimens
for grades(a) 15, (b) 40, (c) 55, (d) 90, and (e) 175.
40
--
theorehcal
--- linearasymptote
[] experimental
• 30
materialsof well-definedporestructureto demonstratethe
accuracyof the measurement.
Later, we shallpresentsimilar results on different natural rocks of much more com-
• 20
o
0
(C)
I00
200
30o
400
500
600
Frequency
[kHz]
The thicknesses
of the specimens
variedbetween1 and
5 mm to accommodatedifferent permeabilitiesover the
widest possiblefrequency range. Becauseof the very high
attenuationin thesesamples,resonancepeaksin the transmissionare generallyvery weak but sometimes,especially
at lower frequencies,faintly visible.The sampleswere ap-
proximately4 in. in diameter and, especiallythe rocks,
showedsignificantinhomogeneity
of the attenuationdistribution within this area.The presenteddata include50-250
measurementstaken at randomly chosenpositionsas the
frequencywas scannedup and down within the usable
range.
First, let us show some typical results on synthetic
3230
J. Acoust. Soc. Am., Vol. 93, No. 6, June 1993
plicatedpore structureto demonstratethe feasibilityof
ultrasonicevaluationof such less-permeable
formations,
too. The first seriesof experimentswere conductedon cementedglassbeadspecimens
previouslylistedin Table I.
Generally,we foundexcellentconsistency
betweenthe theoretically predicted and experimentallymeasuredslow
wave velocities.Figure 7 showstwo examplesof the comparisonbetweentheoreticaland experimentalresultsfor
grades55 and90. For the attenuationcoefficient,
the agreement is lessperfect.Figure 8 showsthe comparisonbetweenthe theoreticaland experimentalresultsfor grades
15 through 175. For the smallestpore size (grade 15), the
agreementis still acceptableindicating that the total atten-
uationisdominated
byviscous
losses
throughout
thewhole
frequencyrange. As the pore size is gradually increased,
viscous lossesdecreasewhile scattering lossesbecome
stronger.As we have shownabove (seeTable I), there is
not a frequency-window
whereboth viscousand scattering
lossesare simultaneously
negligiblein air-saturatedporous
materials. At first only at higher frequencies(grades 40
and 55) then throughout the whole frequency range
(grades90 and 175), the experimentallyobservedattenuation coefficient significantly overshootsthe theoretical
Peter B. Nagy: Slow-wave propagation in air-filled solids
3230
TABLE III. Estimatedmaterial and geometricalparametersof natural
came somewhatlarger due to inherent macroscopicinho-
rocks.
mogeneities
foundin mostnaturalrocks.TableIII liststhe
materialsusedin this part of the study as well as their
relevantphysicaland geometricalproperties.The pore
shapefactorratio is between0.2 and 0.3, i.e., significantly
lower than for the glassbeadspecimens.
With the exception of the permeabilityof the Bereasandstone
specimen,
Ko
Type
Cavallo
q6
buff
massilion
Sunset blush
massilion
Berea
r,•
s•
(mm)
0.15
0.7
2.6
0.3
2.4
0.15
0.6
2.8
0.3
2.1
0.6
2.3
0.3
1.7
0.2
3.2
0.2
1.5
Copper-variegated
0.13
sandstone
0.12
sandstone
d
(10 12m2)
predictionandapproaches
a linearasymptote[dashedlines
in Fig. 8(b)-(e)].
The primary purposeof our experimentson synthetic
porousmaterialswas to demonstratethe accuracyof the
measuringsystemand to verifythe feasibilityof Attenborough'ssimple model for random but statisticallywelldefinedpermeableformations.Our next stepwas the adaptationof this techniqueto natural rocksof relatively
high permeabilitybetween100 and 1000 mDarcy. Basically, the resultswere fairly similar to thoseobtainedfor
syntheticmaterials,althoughthe scatterof the data be-
--
ß
• 0.8.
[
which was measured at the Lawrence
Livermore
National
Laboratory,all parameterswereadjustedto obtainthe best
agreementbetweenthe analyticalresultsand the experimental data. Later we plan to determinethe three basic
material parameters,namelythe porosity,the permeability, and the tortuosity,by separatemeasurements
and adjust onlytheporeshapefactorratio,whichis theonlytruly
independentacousticparameter.Figures9-12 show the
normalizedslow wave velocityand attenuationcoefficient
in differentnatural rocks.Again, the experimentallymeasuredvelocityis consistent
with the analyticalresultswhile
the attenuation coefficientexhibits higher-than-predicted
valuesand more or lesslinear frequencydependence
[the
strangebehaviorof the calculated"apparent"attenuation
coefficientat low frequencies
is causedby the increasing
contributionof the impedancemismatchin the total insertion loss,as it was shownin Fig. 5(b)].
--
lheorefical
[I
expemnenlal
lheoretical
experimental
>
0.6
g 0.6
0.4
0.4
-• 0.2
0.2
c
0
0
50
100
150
0
50
Frequency
[kHz]
100
150
Frequency
[kH2]
25
20
I
theoretical
"•
• 20
experimental
theoretical
[ ! experimental
._o 15
u
10
<
0
0
50
100
150
Frequency
[Id-lz]
FIG. 9. Normalized slow wave velocityand attenuationcoefficientin a
2.4-ram-thickCavallobuff massilionsandstone
specimen.
3231
d. Acoust. Sec. Am., Vol. 93, No. 6, June 1993
0
50
100
150
Frequency[kHzl
FIG. 10. Normalized slow wave velocityand attenuationcoefficientin a
2.l-mm-thick Sunsetblushmassilionsandstone
specimen.
Peter B. Nagy: Slow-wave propagationin air-filledsolids
3231
theoretical
--
expenmemal
experimental
0.6
•> 0.6
0.4
--• 0.4
0.2
-•
0
50
theoretical
100
0.2
150
0
50
Frequency
[kHz]
100
150
Frequency
[kHz]
25
25
theoretical
Iheorefical
experimental
experimental
'lit i
• 20
O
10
O
50
1O0
150
Frequency
[Idqz]
FIG. 11. Normalized slow wave velocity and attenuation coefficientin a
1.7-mm-thickcoppervariegatedsandstonespecimen.
0
50
100
150
Frequency[kHz]
FIG. 12. Normalizedslowwave velocityand attenuationcoefficientin a
1.5-mm-thickBereasandstonespecimen.
III. DISCUSSION
Transmissionof airborne ultrasonic waves through
thin air-filled porousplateswas used to study slow wave
propagationin permeablesolids.In the diffuseregime,i.e.,
at low frequencies,the velocityand the attenuationcoefficient contain the sameinformationon the permeableformation. The attenuation is linearly while the velocity is
inversely
proportional
tothesquare
rootof•6s2•/%.
Since
thestaticpermeability
% andtheporeshape
factorratiosp
alwaysoccurin the samecombinationthroughthe normalized pore radius•, they cannotbe separatedby acoustical
geometryof the pore spaceand the surfaceroughness
of
the porechannels.Our experimental
resultson glassbead
specimens
indicatethat the slopeof the attenuationcoefficient versusfrequencycurve approachesthe sameact/af
•0.033 dB/mm kHz value at high frequencyfor all
grades.Thesesamplesare cementedfrom sphericalglass
beadsof differentdiametersand all haveapproximatelythe
same porosity,tortuosity,and pore geometry(grade designationsusedto identifyEP brand porousstructuresdenote the maximum interstitialpore diameterin microns).
Since the high-frequencyasymptoteof the slow wave ve-
measurements.
Instead,
K0/• wasused
to define
a new locity
isv(high),•250
m/s(vf•333m/s,ro0•1.79),
the
materialparameter,the so-calledacousticpermeabilityKa,
whichcanbe determinedfrom the low-frequency
propagation parametersof the slow compressional
wave. Comparison of the acousticpermeabilityKa and the static permeability % couldyield valuableinformationon the geometry
of the pore space.
In the propagatingregime, i.e., at high frequencies,
only the velocityseemsto be consistentwith the theoretical
predictions.Anyway, the velocityby itself is sufficientto
determinethe high-frequency
tortuosityr• of the material
while the attenuationcoefficientwould provideonly redun-
normalizedattenuationis constantat a,•8.3 dB.
Since,at leastin the upperpart of the frequencyrange,
the grain size is comparableto the wavelength,the observedexcessattenuation could be causedby elastic scattering. For example,the diameter of the cementedglass
beadsin grade175is d,•600 ttm, i.e., kdg•>labove60
kHz. Unfortunately, at this point, there seemsto be no
theoretical prediction available in the literature for the
scatteringinducedattenuationof the slow compressional
wave in permeablesolids.On the other hand, evenwithout
further calculations,we can postulatefrom the linear fredantinformation
ons2•/K0•6,
which
is better
determinedquencydependenceof the attenuationcoefficientthat it has
to be independentof the characteristicdimensionof the
from the low-frequencybehavior.The high-frequencyattenuation contains valuable additional information on the
scatterer.This is becausethe dimensionalityof the problem
3232
J. Acoust.Soc. Am., Vol. 93, No. 6, June 1993
Peter B. Nagy:Slow-wavepropagationin air-filledsolids
3232
requiresthat the normalizedattenuationbe independent
of
the sizeof thescattererwhenit is independent
of the wavelength.This conclusionis at leastnot contradictedby our
experimentalresultsin the self-similarglassbead specimens,where the grain sizechangedapproximatelyone order of magnitude without any significantchangein the
high-frequencyslopeof the attenuationversusfrequency
curve. Of course,this apparentconfirmationmight be a
pure coincidenceand cannot prove by itself that the observedexcessattenuationis causedby elastic scattering.
Actually, it would be rather unusualif the scatteringinducedattenuationturnedout to be independentof the scatterer's size.
ducedexcessattenuationis much higherin water-saturated
samplesthan in air-saturatedones.Although there is very
little knownaboutthe actualfrequencydependence
of the
slowwaveattenuationin water-saturated
poroussolids,the
predictedlinear relationshipseemsto be in goodagreement
with previouslypublisheddata by Plona and Winkler (see
lines B and C in Fig. 4 of Ref. 24). In air-saturatedsamples, the surfaceroughnessinducedexcessattenuationis
more difficultto separatefrom scatteringlosseswhich also
start to givesignificantcontributionsto the observedtotal
attenuationimmediatelyabove the diffuse regime. Obviously, further analyticaleffortsare neededto developapprophate modelsfor theseattenuationmechanismsso that
The excessattenuation could be also causedby increasedviscousdrag due to the rather unevenpore geometry or surfaceroughness.At very high frequencies,where
the viscousskin depth•5is so much smallerthan the pore
the measured
radiusapthatit becomes
comparable
to thepore-wall
sur-
The propagationof slow compressional
wavesin airsaturatedpermeablesolids was studied by experimental
face roughness
h, the viscousdrag sharplyincreases
and
the attenuationof the slow compressional
wave increases
with frequencymuch fasterthan the expectedsquare-root
relation. This is becausethe peaksof the surfaceprofile
protrudeinto a regionwhere the fluid velocityis already
significanttherebygreatlyincreasingthe friction between
the solid frame and the movingfluid column.In a recent
data
can be evaluated
in terms
of micro-
scopicgeometricalpropertiesof the permeableformation.
IV. CONCLUSIONS
means between 10 and 500 kHz. Due to the excellent sen-
sitivity of the suggestedexperimental technique, lowpermeabilitymaterialsincluding natural rocks can be inspected,too. Currently, the threshold sensitivity of our
systemis approximately100mDarcy. In the low-frequency
(diffuse) regime, the experimentalresults are consistent
paper,
25GistusedNorris'analytical
technique
26for the with theoreticalpredictions.The phasevelocityand attendynamic permeability of porous formations to show that
uation coefficientare determinedby the "acoustic"permethere is a transitionfrequencyrangewherethe normalized ability of the specimenandboth increaseproportionallyto
attenuationcoefficient
of the slowwaveis roughlyindepen- the squareroot of frequency.In the high-frequency(propdentof frequency.
Bothbelow(ap•<S)
andabove(S•<h) agating) regime, the experimentalresults are consistent
this transitionrange,the attenuationcoefficient
is propor- with the theoreticalpredictionsfor the phasevelocitybut
tional to the squareroot of frequency.In the transition not for the attenuationcoefficient.The phasevelocityasfrequency
range(hK•5•<ap),
theflowpattern
of thefluidis ymptoticallyapproachesa maximumvalue determinedby
determinedby the pore geometry,but the viscousfriction
the tortuosityof the specimenwhile the attenuationcoefis greatly increasedby surfaceroughness.Although such ficientbecomeslinearly proportionalto frequencyinstead
'conditions
aremoretypical
forwatersaturation,
thesame of the expectedsquare-rootrelationship.It was suggested
effectmight contributeto the observedexcessattenuation that the observeddiscrepancyis partly due to irregular
in air-filled materials,too. For example,Waetzmann and
pore geometryand partly to pore-wallroughness,which
Wenkefoundmore then 50 yearsagothat the attenuation substantiallyreduce the high-frequencydynamic permecoefficientof airborneguidedwavesin ductswith rough ability of the specimens.In addition, scatteringlossesalso
wallsdeviatesfrom the theoreticallypredictedsquare-root give significantcontributionsto the observedtotal attenufrequencydependenceand the excessattenuationis lination, since,for air saturation,there is no frequency"winearlyproportional
to frequency?
dow" where both viscousand scatteringlossesare weak
In the low-frequencydiffuseregime, the normalized simultaneously.Further theoretical efforts are needed to
attenuation is constant at 2rr. At the transition between the
incorporatethese excessattenuation mechanismsinto exdiffuseand propagatingregimes,the normalizedattenua- isting analyticalmodels.
tion startsto decrease
and approaches
an inversesquareroot frequencydependence.At very high frequencies, ACKNOWLEDGMENTS
where the viscousskin depth becomesnegligibleto the
This work was sponsored
by the U.S. Departmentof
surfaceroughness,
thenormalizedattenuationwouldagain
Energy, Basic Energy SciencesGrant No. DE-FG-02assumean inversesquare-rootfrequencydependence
but
87ER13749.A000.The authorwishesto acknowledge
the
for the veryhighscattering
losses.
In liquid-saturated
specvaluablecontributionsof Gabor Blaho,SuquinMeng, and
imens,the attenuationcoefficientcan drop to a very low
Laszlo Adler to this effort.
value before scatteringlossesbecomeimportant. In airsaturatedspecimens,there is not such a wide window for
more or less attenuation-free,dispersion-freeslow wave
propagationand the attenuationcoefficient
cannotsignificantlydrop beforescatteringlossesstart to dominate.Consequently,the relative role of the surfaceroughnessin3233
J. Acoust.Soc. Am., Vol. 93, No. 6, June 1993
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3234