In this article, a modeling extension for the description of wave propagation in porous media at low-mid frequencies is introduced. To better characterize the viscous and inertial interactions between the fluid and the structure in this regime, two additional terms described by two parameters α1 and α2 are taken into account in the representation of the dynamic tortuosity in a Laurent-series on frequency. The model limitations are discussed. A sensitivity analysis is performed, showing that the influence of α1 and α2 on the acoustic response of porous media is significant. A general Bayesian inference is then conducted to infer, simultaneously, the posterior probability densities of the model parameters. The proposed method is based on the measurement of waves transmitted by a slab of rigid porous material, using a temporal model for the direct and inverse transmission problem. Bayesian inference results obtained on three different porous materials are presented, which suggests that the two additional parameters are accessible and help reduce systematic errors in the identification of other parameters: porosity, static viscous permeability, static viscous tortuosity, static thermal permeability, and static thermal tortuosity.

1.
J.
Allard
and
N.
Atalla
,
Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials
, 2nd ed. (
John Wiley & Sons
,
New York
,
2009
).
2.
J.
Bear
,
Dynamics of Fluids in Porous Media
(
Courier Corporation
,
New York
,
2013
).
3.
K.
Vafai
,
Porous Media: Applications in Biological Systems and Biotechnology
(
CRC Press
,
Boca Raton, FL
,
2010
).
4.
D. L.
Johnson
,
J.
Koplik
, and
R.
Dashen
, “
Theory of dynamic permeability and tortuosity in fluid-saturated porous media,”
J. Fluid Mech.
176
,
379
402
(
1987
).
5.
A.
Norris
, “
On the viscodynamic operator in Biot's equations of poroelasticity
,”
J. Wave Mat. Interact.
1
,
365
380
(
1986
).
6.
D.
Lafarge
, “
Propagation du son dans les matériaux poreux à structure rigide saturés par un fluide viscothermique: Définition de paramètres géométriques, analogie electromagnétique, temps de relaxation” (“Sound propagation in rigid porous materials saturated with a viscothermal fluid–geometrical parameters, electromagnetic analogy, relaxation times and ‘universality theory’ ”)
, Ph.D. thesis, Université du Maine,
1993
, available at http://cyberdoc.univ-lemans.fr/theses/1993/1993LEMA1009.pdf (Last viewed November 1, 2018).
7.
D.
Lafarge
,
P.
Lemarinier
,
J. F.
Allard
, and
V.
Tarnow
, “
Dynamic compressibility of air in porous structures at audible frequencies
,”
J. Acoust. Soc. Am.
102
(
4
),
1995
2006
(
1997
).
8.
L. L.
Beranek
, “
Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets
,”
J. Acoust. Soc. Am.
19
(
4
),
556
568
(
1947
).
9.
R. L.
Brown
and
R. H.
Bolt
, “
The measurement of flow resistance of porous acoustic materials
,”
"J. Acoust. Soc. Am.
13
(
4
),
337
344
(
1942
).
10.
Z. E. A.
Fellah
,
S.
Berger
,
W.
Lauriks
,
C.
Depollier
,
C.
Aristegui
, and
J.-Y.
Chapelon
, “
Measuring the porosity and the tortuosity of porous materials via reflected waves at oblique incidence
,”
J. Acoust. Soc. Am.
113
(
5
),
2424
2433
(
2003
).
11.
P.
Leclaire
,
L.
Kelders
,
W.
Lauriks
,
M.
Melon
,
N.
Brown
, and
B.
Castagnede
, “
Determination of the viscous and thermal characteristic lengths of plastic foams by ultrasonic measurements in helium and air
,”
J. Appl. Phys.
80
(
4
),
2009
2012
(
1996
).
12.
K. V.
Horoshenkov
, “
A review of acoustical methods for porous material characterisation
,”
Int. J. Acoust. Vib.
22
,
92
103
(
2017
).
13.
K.
Attenborough
,
I.
Bashir
, and
S.
Taherzadeh
, “
Outdoor ground impedance models
,”
J. Acoust. Soc. Am.
129
(
5
),
2806
2819
(
2011
).
14.
Z. E. A.
Fellah
,
M.
Fellah
,
N.
Sebaa
,
W.
Lauriks
, and
C.
Depollier
, “
Measuring flow resistivity of porous materials at low frequencies range via acoustic transmitted waves
,”
J. Acoust. Soc. Am.
119
(
4
),
1926
1928
(
2006
).
15.
A.
Berbiche
,
M.
Sadouki
,
Z.
Fellah
,
E.
Ogam
,
M.
Fellah
,
F.
Mitri
, and
C.
Depollier
, “
Experimental determination of the viscous flow permeability of porous materials by measuring reflected low frequency acoustic waves
,”
J. Appl. Phys.
119
(
1
),
014906
(
2016
).
16.
Z. E. A.
Fellah
,
M.
Fellah
,
F.
Mitri
,
N.
Sebaa
,
W.
Lauriks
, and
C.
Dépollier
, “
Transient acoustic wave propagation in air-saturated porous media at low frequencies
,”
J. Appl. Phys.
102
(
8
),
084906
(
2007
).
17.
M.
Sadouki
,
M.
Fellah
,
Z. E. A.
Fellah
,
E.
Ogam
,
N.
Sebaa
,
F.
Mitri
, and
C.
Depollier
, “
Measuring static thermal permeability and inertial factor of rigid porous materials (L)
,”
J. Acoust. Soc. Am.
130
(
5
),
2627
2630
(
2011
).
18.
T. G.
Zieliński
, “
Normalized inverse characterization of sound absorbing rigid porous media
,”
J. Acoust. Soc. Am.
137
(
6
),
3232
3243
(
2015
).
19.
Y.
Champoux
and
J.-F.
Allard
, “
Dynamic tortuosity and bulk modulus in air-saturated porous media
,”
J. Appl. Acoust.
70
(
4
),
1975
1979
(
1991
).
20.
J.-D.
Chazot
,
E.
Zhang
, and
J.
Antoni
, “
Acoustical and mechanical characterization of poroelastic materials using a Bayesian approach
,”
J. Acoust. Soc. Am.
131
(
6
),
4584
4595
(
2012
).
21.
M.
Niskanen
,
J.-P.
Groby
,
A.
Duclos
,
O.
Dazel
,
J.
Le Roux
,
N.
Poulain
,
T.
Huttunen
, and
T.
Lähivaara
, “
Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements
,”
J. Acoust. Soc. Am.
142
(
4
),
2407
2418
(
2017
).
22.
R.
Roncen
,
Z. E. A.
Fellah
,
F.
Simon
,
E.
Piot
,
M.
Fellah
,
E.
Ogam
, and
C.
Depollier
, “
Bayesian inference for the ultrasonic characterization of rigid porous materials using reflected waves by the first interface
,”
J. Acoust. Soc. Am.
144
(
1
),
210
221
(
2018
).
23.
J.
Kergomard
,
D.
Lafarge
, and
J.
Gilbert
, “
Transients in porous media: Exact and modelled time-domain Green's functions
,”
Acta Acust. united Acust.
99
(
4
),
557
571
(
2013
).
24.
M. A.
Biot
, “
Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range
,”
J. Acoust. Soc. Am.
28
(
2
),
168
178
(
1956
).
25.
C.
Zwikker
and
C. W.
Kosten
,
Sound Absorbing Materials
(
Elsevier
,
New York
,
1949
).
26.
Z. E. A.
Fellah
and
C.
Depollier
, “
Transient acoustic wave propagation in rigid porous media: A time-domain approach
,”
J. Acoust. Soc. Am.
107
(
2
),
683
688
(
2000
).
27.
C.-Y.
Lee
,
M. J.
Leamy
, and
J. H.
Nadler
, “
Frequency band structure and absorption predictions for multi-periodic acoustic composites
,”
J. Sound Vib.
329
(
10
),
1809
1822
(
2010
).
28.
C.
Boutin
and
C.
Geindreau
, “
Estimates and bounds of dynamic permeability of granular media
,”
J. Acoust. Soc. Am.
124
(
6
),
3576
3593
(
2008
).
29.
R. J.
Brown
, “
Connection between formation factor for electrical resistivity and fluid-solid coupling factor in Biot's equations for acoustic waves in fluid-filled porous media
,”
Geophysics
45
(
8
),
1269
1275
(
1980
).
30.
D.
Smeulders
,
R.
Eggels
, and
M.
Van Dongen
, “
Dynamic permeability: Reformulation of theory and new experimental and numerical data
,”
J. Fluid Mech.
245
,
211
227
(
1992
).
31.
R. C.
Smith
,
Uncertainty Quantification: Theory, Implementation, and Applications
(
SIAM
,
Philadelphia, PA
,
2013
), Vol. 12.
32.
A.
Tarantola
,
Inverse Problem Theory and Methods for Model Parameter Estimation
(
SIAM
,
Philadelphia, PA
,
2005
).
33.
J.
Kaipio
and
E.
Somersalo
,
Statistical and Computational Inverse Problems
(
Springer Science and Business Media
,
New York
,
2006
), Vol. 160.
34.
S.
Torquato
, “
Relationship between permeability and diffusion-controlled trapping constant of porous media
,”
Phys. Rev. Lett.
64
(
22
),
2644
2646
(
1990
).
35.
N.
Metropolis
,
A. W.
Rosenbluth
,
M. N.
Rosenbluth
,
A. H.
Teller
, and
E.
Teller
, “
Equation of state calculations by fast computing machines
,”
J. Chem. Phys.
21
(
6
),
1087
1092
(
1953
).
36.
W. K.
Hastings
, “
Monte Carlo sampling methods using Markov chains and their applications
,”
Biometrika
57
(
1
),
97
109
(
1970
).
37.
W. R.
Gilks
,
S.
Richardson
, and
D.
Spiegelhalter
,
Markov Chain Monte Carlo in Practice
(
CRC Press
,
Boca Raton, FL
,
1995
).
38.
E.
Laloy
and
J. A.
Vrugt
, “
High-dimensional posterior exploration of hydrologic models using multiple-try dream(ZS) and high-performance computing
,”
Water Resour. Res.
48
(
1
),
W01526
, doi: (
2012
).
39.
A.
Gelman
and
D. B.
Rubin
, “
Inference from iterative simulation using multiple sequences
,”
Statist. Sci.
7
(
4
),
457
472
(
1992
).
40.
Y.
Champoux
,
M. R.
Stinson
, and
G. A.
Daigle
, “
Air-based system for the measurement of porosity
,”
J. Acoust. Soc. Am.
89
(
2
),
910
916
(
1991
).
41.
D.
Bies
and
C. H.
Hansen
, “
Flow resistance information for acoustical design
,”
Appl. Acoust.
13
(
5
),
357
391
(
1980
).
42.
M.
Avellaneda
and
S.
Torquato
, “
Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media
,”
Phys. Fluids A
3
(
11
),
2529
2540
(
1991
).
43.
B.
Gurevich
and
M.
Schoenberg
, “
Interface conditions for Biot's equations of poroelasticity
,”
J. Acoust. Soc. Am.
105
(
5
),
2585
2589
(
1999
).
44.
Z. E. A.
Fellah
,
M.
Fellah
,
W.
Lauriks
, and
C.
Depollier
, “
Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material
,”
J. Acoust. Soc. Am.
113
(
1
),
61
72
(
2003
).
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