Inverse estimation of the permeability of porous materials using experimental data via reflected waves at low frequencies

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Inverse estimation of the permeability of porous

materials using experimental data via reflected waves at low frequencies

A Berbiche, Mustapha Sadouki, Z.E.A Fellah, Erick Ogam, Mohamed Fellah, Farid Mitri, Claude Depollier

To cite this version:

A Berbiche, Mustapha Sadouki, Z.E.A Fellah, Erick Ogam, Mohamed Fellah, et al.. Inverse estimation of the permeability of porous materials using experimental data via reflected waves at low frequencies.

Journal of Applied Physics, American Institute of Physics, 2016, 119 (014906), �10.1063/1.4939073�.

�hal-01318155�

rials using experimental data via reeted waves at low

frequenies.

A. Berbihe

Laboratoire de Physique Théorique,Faulté de Physique, USTHB, BP 32El Alia, Bab Ezzouar

16111, Algérie.

Laboratoire de Physique Théorique,Faulté de Physique, USTHB, BP 32El Alia,Bab Ezzouar

16111, Algérie.

M. Sadouki

Faulté des Sienes et Tehnique, Université deKhemis MilianaBP 44225 Ain Dea, Algérie.

Z.E.A. Fellah and E. Ogam

LMA, CNRS,UPR7051, Aix-MarseilleUniv,Centrale Marseille, F-13402MarseilleCedex 20,

Frane

M. Fellah

Laboratoire de Physique Théorique,Faulté de Physique, USTHB, BP 32El Alia, Bab Ezzouar

16111, Algérie.

F.G. Mitri

Chevron,Area 52 -ETC, 5 Bisbee Ct., Santa Fe, New Mexio 87508, United States.

C. Depollier

LUNAM Universite du Maine. UMR CNRS 6613 Laboratoire d'Aoustique de l'Universite du

Maine UFR STSAvenue O. Messiaen 72085 Le MansCEDEX 09 Frane.

1

1. Present:MPEI,Krasnokazarmennaya14Mosow111250Russia.

ofair-saturatedporousmaterials.Inthis method,asimpliedexpressionofthereetionoe-

ientisderivedintheDary'sregime(lowfrequenyrange),whihdoesnotdependonfrequeny

andporosity.Numerialsimulationsshowthatthereetionoeient ofaporousmaterial an

beapproximated by its simplied expression obtained from its Taylordevelopment to the rst

order.Thisapproximationisgoodespeiallyforresistivematerials(oflowpermeability)andfor

the lower frequenies. The permeability is reonstruted by solving the inverse problem using

wavesreetedbyplasti foamsamples,atdierentfrequenybandwidths intheDaryregime.

Theproposedmethodhasthe advantageofbeingsimpleompared totheonventionalmethods

thatuseexperimentalreeted data,andisomplementary tothetransmissivitymethodwhih

ismore adaptedto low resistivematerials (highpermeability).

The aousti haraterization of air saturated porous media

1−5

is very important for the

preditionofsoundpropagationinsuhmedia.Indeed,thesematerialsarewidelyusedtoredue

noise inbuildings,vehiles andairraft.

Severalnon-aoustiparameters

1,2,6

desribesoundpropagationinair-saturatedporousma-

terials. Thephysis of aousti wave propagation inporous materials is dierent depending on

the frequeny domain. The high and low frequeny ranges

1,2,6

are dened by omparing the

visous and thermal skin thiknesses δ = (2η/ωρ)^{1/2 and} δ^′ = (2η/ωρP_r)^{1/2 with the eetive}

radius of the pores r ⁽ρ ^{is the density of the saturating uid;} ω ^{the pulsation frequeny;} P_r

the Prandtl number and η ^{is the uid visosity). The high frequeny domain is dened when}

thevisous

6

and thermal

1

skinthiknesses aresmall omparedto theporeradius.Otherwise,it

is the lowfrequeny domain. IntheDary'sregime (owwithout inertial eet) orresponding

to low frequenies

7

,thevisous stati permeability k₀ ^{is themost important parameter desri-}

bingthevisouslossesausedbytheuid/strutureexhanges.Thisparameterisrelatedtothe

spei ow resistivity

3,5 σ ^{by the relation :} k₀ =η/σ^{. In the high frequeny range, the main}

parameters inthis domain arethe tortuosity

1

,visous

6

and thermal harateristi lengths

1,3

.

Among the methods

3−5,7−24

developed to measure the permeability or the ow resistivity,

we distinguish between the so-alled diret methods

5,8−12

whih do not use sound waves, and

theindiretmethods

13−24

thatusesoundwavestransmittedorreetedbytheporousmaterial.

The pratial implementation of the diret methods ould be both omplex and expensive.

In this paperwe proposeto measure thevisous permeability (and resistivity) bydeveloping a

simplemethodusingexperimentaldataoftransientwavesreetedfromsamplesofairsaturated

porousmaterials ina tube. The advantage of this method is theuse of a simplied expression

of the reetion oeient whih does not depend on the frequeny or the porosity of the

material. Letus reallthat the lassial methods

19

using thereeted waves requiresthe prior

depends on the frequeny and porosity. We show in this study that the proposed method is

espeially well adapted for resistive porous materials at low frequenies. In this approah, a

tube is used for the aousti propagation of transient waves. A single mirophone

7,19−21

is

used for the measurement of experimental signals, therefore, no alibration is required. Other

tehniques useanimpedanetube,inwhih standing wavesaregenerated, and where two

13−17

or three

18

mirophones are used for experimental measurements. In this ase, a alibration of

themirophones isneessaryforagoodqualityof theresults.Thisaousti reetivitymethod

is analternative to the transmissivitymethod developed reently

7

,whih isadapted to porous

materials oflowresistivity(high permeability).

II. EQUIVALENT FLUID MODEL

The mostgeneral andomplete theoryforthe desriptionofaousti propagationinporous

mediaistheBiot's

25

theory.Thismodeltreatsbothindividualandoupledbehavioroftheframe

andporeuid.Energylossisonsideredtobeausedbythevisosityoftheporeuidasitmoves

relativetotheframe.Thetheorypreditstwoompressionalwaves:afastwave,wheretheuid

and solid move inphase, and a slow wave where uid and solidmove out of phase. Generally,

when the sound wave propagates through a porous material saturated with air, the struture

remainsstationaryandnon-deformablewithrespettotheaoustiexitation

1,2

.Thevibrations

ofthe solidframe anoftenbenegletedintheabsene ofdiretontatwiththesoundsoure,

so that the waves an be onsidered to propagate only in uid. This ase is desribed by the

equivalent-uidmodel

1,2,6,7

,whihisapartiularaseoftheBiotmodel,inwhihuid-struture

interations are taken into aount in two frequeny response fators :the dynami tortuosity

of themedium α(ω) ^{given byJohnson} et al^{6,and the dynami} ompressibilityof the airinthe porousmaterialβ(ω)^{givenbyAllard}et al^{1.Inthefrequenydomain,thesefatorsmultiplythe}

theuidinfree spaeasthe frequeny inreases.Speially, inthelowfrequeny domain,the

visouseetsareimportant inallthe porevolume,and theompressiondilatation yleinthe

porousmaterialisslowenoughtofavorthethermalinterationsbetweenuidandstruture

2

.At

thesametimethetemperatureoftheframeispratiallyunhangedbythepassageofthesound

wavebeauseofthehighvalueofitsspeiheat:theframeatsasathermostat

2

.Inaddition,

thethermal ondutivity of the solid ishigh, and theexess heat is immediately evauated by

the solid, whih therefore remains at the same temperature during the ompression dilatation

yle

1,2

.IntheDary'sregime

7,26,27

(very low-frequeny approximation), theexpressionsofthe

responsefatorsα(ω) ^and β(ω) ^whenω →0^{aregiven bytherelations26 :} α(ω) =− ηφ

ρk₀jω, β(ω) =γ. ⁽¹⁾

where k₀ ^{isthe stati} permeability, φ^{theporosityand}γ ^{theadiabati onstant.}

Consider ahomogeneousporousmaterial thatoupiestheregion0≤x≤L^.Asoundpulse

impinges normally on the medium. It generates an aousti pressure eld p ^{and an aousti}

veloity eld v ^{within the material. These aousti elds satisfy the following} equivalent-uid marosopiequations (along the x−^{axis)1 :}

ρα(ω)jωv= ∂p

∂x, β(ω)

K_a jωp= ∂v

∂x, ⁽²⁾

where, j² =−1^,ρ ^{isthe saturating uiddensity and} K_a ^{is the} ompressibilitymodulus of the uid.

The inident pⁱ(t) ^{and reeted} p^r(t) ^{elds are related in the time domain by the reetion}

sattering operator

19,20 R ^:

p^r(x, t) = Z _t

0

R(τ)pⁱ

t−τ+ x c₀

dτ. ⁽³⁾

The temporal operator kernel R ^{is alulated by taking the inverse Fourier transform of the}

R(ω) =˜ 1−Y²(ω)

sinh (jκ(ω)L)

2Y(ω) cosh (jκ(ω)L) + (1 +Y²(ω)) sinh (jκ(ω)L), ⁽⁴⁾

where :

Y(ω) =φ s

β(ω)

α(ω), ^and κ(ω) =ω s

ρα(ω)β(ω) K_a .

Usingtheexpressions(1)ofthedynamitortuosityandompressibility,weobtainthefollowing

expressionfor the reetion oeient :

R(ω) =˜ 1−C₁²ω

sinh LC₂√ jω 2C₁√

jωcosh LC₂√ jω

+ 1 +C₁²jω

sinh LC₂√

jω, ⁽⁵⁾

where

C₁=

sγρk₀φ

η , C₂ =

r γηφ

K_ak₀ ⁽⁶⁾

By doing the Taylor series expansion of thereetion oeient (Appendix. B), when the fre-

quenytends tozero (ω →0^{),we obtain:} R(ω) =˜ 1

1 +_LC^2C1

2

! 1−jω

ω_c +...

, ⁽⁷⁾

where :

ω_c=

3

1 + _LC^2C1

2

2LC₁C₂

1 + _LC^3C1

2 + 3 ^C12

L²C2²

. ⁽⁸⁾

As a rst approximation, at very low frequenies, the reetion oeient (7) is given by the

rstterm :

R˜ = 1

1 +_LC^2C1₂ = 1 1 + ^2k_Lη⁰√

ρKa

(9)

Thissimpliedexpressionofthereetionoeientisindependentofthefrequenyandporosity

of the material,and dependsonly onits statipermeability.

Figure1showsthevariationofthemodulusofthereetionoeient(Eq.5)withtheporosity,

reetion oeients given by Eq.(9) and Eq.(5), respetively. It an be seen that for the low

frequenies, the two reetionoeientsoinide.

Themodulusof the reetionoeient (7)is given by

|R(ω)˜ |= 1 1 +_LC^2C1

2

!v u u t 1 +

ω ω_c

2!

(10)

Letusstudythevaluesofω/ω_c^and|R(ω)˜ |^{forvariousvaluesoffrequenyandofow}resistivity. For theapproximation (9) of the reetion oeient to be valid, it isneessary that ω/ω_c ^be

very small ompared to 1. Table I shows values of ω/ω_c ^and | R(ω)˜ | ^{for various values of}

frequeny andow resistivityfor a porousmediumof thiknessL= 5^{m and porosity} φ= 0.9^.

It an be seen that for the same value of the ow resistivity, thevalues of |R(ω)˜ | ^{are almost}

onstant and those of ω/ω_c ^{are small ompared to 1, espeially for low frequenies and high}

resistivities.

Note thatthe"smallness" ondition obtained when ω/ω_c ≪1 ^{is dierent from theone usedin}

therelations(1)ofthedynamitortuosityandompressibility.Aording towhatispostulated

inRef. 27 and later mathematially justiedin Ref.28, thedynami tortuosity α(ω) ^{an have}

pole-expansionwithone ofthe polesverylose to 0.Thisdistaneisthe'smallness' implied by

Eq. (1).

Inthe time domain,theexpressionof thereetionoperator dened by Eq.(3)is given by:

R(t) = 1

1 +^2k_Lη⁰√ ρK_a

!

δ(t), ⁽¹¹⁾

where δ(t) ^{isthe Dirafuntion.}

In the next paragraph, we ompare the expression (5) of the reetion oeient, with its

simplied expression (9), using numerial simulations of signals (Eq. 3) reeted by a slab of

air-saturated porous material.

Consider two samples F1 and F2 of air-saturated plasti foam, having the same thikness

andporosity,respetively;L= 5 ^m,and φ= 0.9^.Thepermeabilityof thesetwosamples takes

two dierent values. F1 is less resistive (more permeable) than F2. The international system

of units for permeability is m

2

. A pratial unit for permeability is the Dary (D^{), (1 Dary}

=0.97×10⁻¹²m^2).The permeabilityvalueof F1is:k₀ = 6185.6D^;(ow resistivity:σ = 3000

N m

−4

s), and of F2; k₀ = 61.86D ^(ow resistivity : σ = 3.10^{5 N m−4s) . In this paragraph,}

a omparison between simulated reeted signals omputed with dierent expressions of the

reetion oeients(Eqs. 5 and 9) isgiven for the samples F1 and F2. We showunder whih

onditions we an use the simplied expression of the reetion oeient (Eq. 9), instead of

using the more general expression (frequeny dependent, Eq. 5). Fig. 1 shows a omparison

between two simulated reeted signals omputed with dierent expressions of the reetion

oeients for sample F1. The frequeny bandwidth of the inident signal used for this rst

numerial simulation is (300-600)Hz. The rst signal (solid line) orresponds to the simulated

reetedsignalusingexpression(5)ofthereetionoeient,andtheseondone(dashedline)

using the relation (9). We note that for this frequeny range (300-600)Hz, the reeted waves

preditedbythe two terms ofthereetion oeient aresigniantly dierent. An important

shift is observed between the two signals; 40% ^{for the amplitude, and 0.6}% ^{for the phase. By}

making the same omparison with sample F2, whih is less permeable than F1, the results

giveningure2showaslightdierene between thetwosimulated signals(shiftof10%^forthe

amplitude,andof0.05%^{forphase).Weanonludethatthe}approximation(9)ofthereetion

oeient ismuh moreaurate when theporous mediumis lesspermeable (more resistive).

Another test is performed by taking an inident signal with lower frequenies (40-70)Hz.

The reeted signals alulated from equations (5) and (9), are ompared in gures 3 and 4,

for samples F1 and F2, respetively, in the frequeny domain (40-70)Hz. These omparisons

urvesorrespondingto F1and F2.Indeed,thesimpliedexpressionofthereetionoeient

givenbyequation(9)isdevelopedinverylowfrequenies.Thisstudyshowedthatthesimplied

expression(9)givesthesameresultsasexpression(5)forthelowerfrequenies,espeiallyforthe

lesspermeable (more resistive) materials. It would be more advantageous to use thesimplied

expression (Eq. 9) of the reetion oeient, sine it does not depend on the frequeny or

porosity,and issimpler.

IV. INVERSION OF EXPERIMENTAL REFLECTED DATA

The study presentedin theprevious setion shows that thereetion oeient at low fre-

quenyrangeanbesimpliedtoanexpressionwhihisindependentoffrequenyandporosity.

Thisapproximation ofthe reetion oeient iseven more valuable when theporousmedium

is resistive, i.e., withlow permeability. Therefore,thesimplied expressionof thereetion o-

eient depends only on the permeability of the material. In this part of this work, we show

howtoharaterize aporousmaterialsaturatedwithairinthelowfrequenyregime, bysolving

theinverseproblemusingexperimental dataof reetedwaves.Thesimpliedexpressionofthe

reetion oeient willbeused toobtain thepermeabilityof theporousmedium.

Thebasiinverseproblemassoiatedwithaslabofporousmaterialmaybestatedasfollows:

from the measurement of the signals reeted outside theslab, theobjetive is to haraterize

themedium.Ouraimistodeterminethestativisouspermeabilityk0 ^{ofthematerialbysolving}

the inverse problem for waves reeted by theslab of porous material. The inverse problem is

to ndthepermeabilityof thematerialwhihminimizes thefuntion

U(k₀) = Z _t

0

p^r(x, t)−p^r_exp(x, t)2

, ⁽¹²⁾

where p^r(x, t) ^{is the reeted wave predited by Eq. 3 and} p^r_exp(x, t) ^{is the} experimentally

determined reeted signal. The analytial method of solving the inverse problem using the

an be found whih minimizes

U(k₀) =

i=N

X

i=1

(p^r(x, t_i)−p^r_exp(x, t_i))², ⁽¹³⁾

whereinp^r(x, t_i)_i=1,2,...N^{isthedisretesetofvaluesofthesimulatedreetedsignaland}p^r_exp(x, t_i)_i=1,2,...N

isthe disrete set ofvalues ofthe experimentalreeted signal.

Theexperimentalsetup

19

isshowninFig.5.Thetubelengthisadaptabletoavoidreetion,

and to permit the propagation of transient signals, aording to the desired frequeny range.

For measurements in the frequeny range (20-100)Hz, a length of 50m is suient but it is

useful to usean anehoi devie plaed at the end of the pipe. It is not important to keep the

pipe straight; it an be rolled in order to save spae without perturbations on experimental

signals. Thetube diameter is 5 m (theut-o of the tube f_c ∼4^{kHz).A sound soure Driver}

unit"Brand"onstituted byloudspeakerRealisti40-9000isused. Tone-burstsareprovided by

Standard Researh Systems Model DS345-30MHz synthesized funtion generator. The signals

areampliedand lteredusing modelSR650-Dualhannellter,StandfordResearh Systems.

The signals (inident and reeted) are measuredusing thesame mirophone (Bruel & ^Kjaer,

4190). Theinidentsignalismeasured byputtinga totalreetor

19

inthesame position asthe

poroussample.

The inverse problemis solved for four ylindrial samples ofplasti foam M1,M2, M3 and

M4(M1-M4) of adiameter of 5m. Theirporositiesand thiknesses aregiven inTable II. The

permeabilityvaluesobtainedbylassialmethods

5,10,7,19,20

aremarkedby∗^{inTables III,IV,V}

andVI.Thepermeabilityisinvertedusingexperimentalsignals reetedbytheporousmaterial

samples(M1-M4).Thevariationsintheostfuntionpresentonelear minimumorresponding

to the solution oftheinverse problem. Figs6-9 showthevariationof theost funtionU ^when

varyingthepermeabilityindierentfrequenybandwidths,forthesamples(M1-M4).Theresults

of the inverse problemaresummarized inTables III, IV,V andVI, inwhih inverted values of

between inverted permeability values obtained using the simplied expression (Eq. 9) of the

reetionoeientanditsgeneral expression(Eq.4)aregiveninTablesIII,IV,V andVI.The

averagevalues ofthese inverted dataaremarked by♯ ^{inTables III-VI,for eahporous sample.}

From these tables, it an be seen that the inverted values of permeability obtained using the

simplied expression of the reetion oeient (Eq. 9) are lose to those obtained using its

general expression(Eq.4). In addition,the averaged valuesof theoptimized permeabilities (k₀⁾

arealso lose to thoseobtained using lassialmethods (marked by ∗^).

We notied that the experimental data for the lower frequenies ontain muh more noise

relative to other measures. Thisis whywe deided to expand themeasures on small frequeny

bands(40-120)Hz, wheretheapproximationofthereetionoeientisstillvalid.Thesesmall

frequeny bandshave beenusedintheinversionproess to traedierent minimizationurves,

in the aim to obtain an good average value for the inverted permeability. The muh higher

frequenies for whih theapproximation of theoeient reetion is not valid were not taken

into aount inthe inversion.

The standard deviations of the experimental inverted values of the permeability are eva-

luated in Table VII for samples M1-M4. From this table, we an see that sample M3 has the

smallest standard deviation ompared to the other samples.Indeed, sample M3 hasthelowest

permeability value, and is therefore the most resistive material. We also see from Table VII

thataslongasthe samples areresistive (withlow permeability), thestandarddeviation islow.

Thisexperimental studyshows thattheinversionontheexperimentaldataispartiularly good

andaurate asthemedium ismoreresistive,andthuslesspermeable.Thisresultonrmsthe

validityof thetheoryofthe reetionoeient thatwe developed inthepreviousparagraph.

Aomparisonbetweenanexperimentalreetedsignalandsimulatedreetedsignalisgiven

in Figs. 10 (a-d) for the optimized values of the inverted permeabilities of the porous samples

(M1-M4),respetively.The frequenybandwidth ofthe inident signalsis (40-60)Hz. Itan be