Multiple scattering in porous media: Comparison with water saturated double porosity media a) H. Franklin and F. Luppe Laboratoire Ondes et Milieux Complexes, UMR CNRS 6294, Universit e du Havre, 75 rue Bellot, CS 80 540, 76058 Le Havre, France
J. M. Conoirb) UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
(Received 17 September 2013; revised 19 February 2014; accepted 31 March 2014) Multiple scattering in a poroelastic medium obeying Biot’s theory is studied; the scatterers are parallel identical cylindrical holes pierced at random in the medium. The paper focuses first on the influence, on the effective wavenumbers, of the mode conversions that occur at each scattering event. The effect of the holes on the dispersion curves is then examined for two different values of the ratio of their radius to the pores mean radius. Depending on the latter, the dispersion curves of the pierced material are compared, for the fast and shear waves, with those of either a more porous C 2014 Acoustical Society of America. medium or a double porosity medium. V [http://dx.doi.org/10.1121/1.4871182] PACS number(s): 43.20.Jr, 43.20.Gp [ANN]
I. INTRODUCTION
Many applications of wave propagation in poroelastic media containing random distributions of inhomogeneities may be found, from geophysical exploration,1–3 to ultrasonic evaluation of biological tissues such as bones.4 In such cases, the inhomogeneities are either solid inclusions, cracks, or a second network of larger pores. In this paper, the watersaturated poroelastic medium is supposed to obey Biot’s theory and the inhomogeneities are infinitely long water filled identical cylinders perpendicular to the direction of propagation that are larger than the pores. It will be referred to as the perforated poroelastic medium (PPM), and compared to either a more porous medium (MPM) obeying Biot’s theory as well, or to a double porosity medium (DPM), depending on the size of the holes. The PPM is a typical problem of multiple scattering as originally studied by Foldy.5 In such problems, it is well known that for small enough concentrations of scatterers the physical medium may be replaced by an effective homogeneous medium in which coherent waves propagate. These waves represent the acoustic field averaged over all possible configurations of scatterers. The propagation of each coherent wave is governed by a frequency dependent complex effective wavenumber. While in recent years there has been renewed interest and progress in the study of multiple scattering when the host medium is an ideal fluid,6–11 there have been much fewer works when more than one wave may propagate in the host medium, as is the case of a porous medium12,13 that obeys Biot’s theory.14 No theory was available to take into account the conversion modes that occur at
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected] b) Also at: CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France. J. Acoust. Soc. Am. 135 (5), May 2014
Pages: 2513–2522
each scattering event in a multiple scattering process. The problem has been overcome recently15 for an elastic host medium in which two acoustic waves propagate. The theory described in Ref. 15 can be readily applied to host media in which any number of waves propagate,16 and it is applied here to the calculation of the effective wavenumbers associated to the three coherent waves that propagate in the PPM. Increased absorption is a well-known effect of fractures in a porous medium, as described in continuum theories dealing with double porosity media.1–3 Such media have been intensively studied in the past decade for air saturated ones,17,18 and Olny and Boutin19 have emphasized that they can achieve larger sound absorption than single porosity materials. There are two scales in a DPM; one is associated to micropores and the other to (randomly distributed) larger pores. These two networks of pores are interconnected. In the PPM described above, the pores are interconnected, and the cylindrical holes (larger pores) are separated from each other. The macroscopic porosity they induce is somewhat connected through the microscopic porosity of the porous medium, but in a much weaker way than in a DPM. The other main difference is that, from the three-dimensional point of view, the DPM is isotropic at both scales (microscopic and macroscopic), while the PPM is strongly anisotropic, due to the fact that the largest (cylindrical) pores are all parallel. We shall investigate the effect of the difference between the PPM and the DPM on the increased attenuation of the fast and shear waves. Section II is devoted to the multiple scattering model for itself. It is briefly recalled in Sec. II A, along with the hypotheses that underlie it; its range of validity in the numerical case of the PPM under study is defined in Sec. II B through the study of the scattering by a single cylinder. The influence of the mode conversions on the effective wavenumbers is studied in Sec. II C. The velocity and attenuation curves of the coherent waves are discussed in Sec. III for
0001-4966/2014/135(5)/2513/10/$30.00
C 2014 Acoustical Society of America V
2513
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.138.73.68 On: Sat, 20 Dec 2014 14:35:52
two different ratios of the cylinders to the micropores radii, and Sec. IV is devoted to the comparison with either a MPM or a DPM. II. OVERVIEW OF THE MULTIPLE SCATTERING MODEL AND ITS CONDITIONS OF VALIDITY A. The model
The model used is that described in Ref. 15 in the case of cylindrical scatterers in an elastic matrix. In a porous medium that obeys Biot theory, the wavenumbers are complex, even in the absence of scatterers, while they were real in the elastic medium. However this does not raise any new difficulty, as all the series and integrals one has to deal with are still well defined and convergent, and so we need to focus here only on the main assumptions of the theory and on its main results. The generalized Fikioris and Waterman equations in the case of an elastic matrix15 consist in two sets of homogeneous linear equations. Equating their determinants to zero provides the dispersion equation of the coherent waves. The two sets are readily extended to three in the case of a porous Biot matrix that supports the propagation of a fast ðL1 Þ longitudinal wave, a slow ðL2 Þ one, and a transverse ðTÞ wave:
0
Nma ðnÞ ¼ nbJm0 ðnbÞHmð1Þ ðka bÞ ka bJm ðnbÞHmð1Þ ðka bÞ:
In Eq. (2), b is the radius of exclusion introduced in Fikioris and Waterman’s hole correction [see Eq. (7) in Ref. 15]; it is greater than twice the radius a of the cylinders in order to forbid any overlapping of scatterers. Jm and Nm are the Bessel and Neumann functions of order m, and Hmð1Þ is the Hankel function of the first kind and order m. The prime over J or H denotes the derivative with respect to the argument of the function. The next step in order to get an explicit approximate expression of the three effective wavenumbers na is to assume n2a close enough to ka2 , so that a formal asymptotic expansion of n2a in terms of e ¼ 4in0 2 ð2Þ n2a ¼ ka2 þ eyð1Þ a þ e ya þ …
(3)
might be truncated at some order at “low enough” concentration. Equations (52) and (54)–(58) of Ref. 15 provide then ð0Þ yð1Þ a ¼ Maa ;
þ1 X X 2pn0 a Tmba Abm Nmn ðna Þ ¼ 0; Aan 2 2 na ka b¼L1 ;L2 ;T m¼1
(4)
ð1Þ yð2Þ a ¼ Maa þ
ð0Þ
X
ð0Þ
Mab Mba
b ¼ L1 ; L2 ; T b 6¼ a
8a 2 fL1 ; L2 ; Tg:
(2)
ka2 kb2
;
(5)
(1) with
The Aan ’s are the unknowns of the equations and are related to the amplitudes of the coherent fields. ka is the wavenumber of the a-wave in the absence of scatterers and na , which we are looking for, is that of the coherent wave. n0 is the number of cylinders per unit surface, Tmba is the mth component of the diagonal scattering matrix of each single cylinder for a b-wave scattering into an a-wave (see Ref. 20 for its expression), and
ð0Þ
Mab ¼
þ1 X
Tnab ;
(6)
n¼1
ð1Þ ¼ Maa
þ1 X
þ1 X
X
b
Tnba Qnm ðka ÞTmab ;
(7)
m¼1 m¼1 b¼L1 ;L2 ;T
b Q nm ðka Þ
0 ð1Þ ð1Þ0 i p2 ka bJmn ðka bÞHmn ðkb bÞ kb bJmn ðka bÞHmn ðkb bÞ 1 ¼ if b 6¼ a ka2 kb2
(8)
a Q nm ðka Þ
" # ð1Þ p ððka bÞ2 ðm nÞ2 ÞJmn ðka bÞHmn ðka bÞ ¼ i 2 : 0 ð1Þ0 4ka þka2 b2 Jmn ðka bÞHmn ðka bÞ
(9)
and
Now, if the frequency is low enough for the asymptotic expansions for small arguments of the Bessel and Hankel functions in Eqs. (8) and (9) to be used, Eq. (7) turns to
ð1Þ ¼ Maa
þ1 þ1 X 1 1 X jm nj Tnaa Tmaa þ 2 ðka Þ2 n¼1 m¼1
2514
X b ¼ L1 ; L2 ; T b 6¼ a
þ1 þ1 jmnj X X 1 ka Tnba Tmab ka2 kb2 n¼1 m¼1 kb
J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
X k2 b ¼ L1 ; L2 ; T a b 6¼ a
þ1 þ1 X X 1 T ba T ab ; 2 kb n¼1 m¼1 n m
(10)
Franklin et al.: Multiple scattering in porous media
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.138.73.68 On: Sat, 20 Dec 2014 14:35:52
and Eqs. (3)–(5) finally provide the low concentration and low frequency approximations of the effective wavenumbers that will be used in Sec. III, ! 2 na n0 a n20 að0Þ n30 aðcÞ ¼ 1 þ 2 d1 þ 4 d2 þ d2 þO 6 ; ka ka ka ka (11) with da1 ¼ 4i
þ1 X
Tnaa ;
(12)
n¼1 að0Þ
d2
¼ 8
þ1 X þ1 X
TABLE I. Values of the physical parameters of QF20 and water (from Ref. 21). Physical parameters
Symbols (units)
Values
Bulk modulus of grains Dried frame bulk modulus Dried frame shear modulus Solid density Bulk modulus of water Density of water Viscosity of saturating water Porosity Permeability Pore mean radius Tortuosity
Kr (Pa) Kb (Pa) l (Pa) qS (kg m3) K0 (Pa) q0 (kg m3) g (kg m1 s1) b k (m2) ap (m) s
36.6 109 9.47 109 7.63 109 2760 2.22 109 1000 1.14 103 0.402 1.68 1011 3.26 105 1.89
jm nj Tmaa Tnaa ;
m¼1 n¼1 aðcÞ
d2
X
¼ 16
dab 2 ;
(13)
b ¼ L1 ; L2 ; T b 6¼ a
dab 2 ¼
jmnj ka2 ka Tmba Tnab : 2 2 k k k b a b m¼1 n¼1 þ1 X þ1 X
(14) ð Þ
As noticed already in Ref. 15, the da1 and da2 0 terms have the same formal expressions as in the case of an ideal fluid as the host medium, and involve only scattering coefficients Tnaa that describe the scattering of an a-wave into an a-wave. The possibility of mode conversions occurring at each scattering event appears explicitly in the coupling term aðcÞ only, through the scattering coefficients Tnab , with d2 b 6¼ a. However, one should keep in mind that the value of Tnaa depends nonetheless on those of the Tnab coefficients with b 6¼ a, through the continuity conditions20 on the interface of the scatterer. All the scattered b-waves of amplitude Tnab propagate from one scatterer to another; those with large =mðkb Þ=
J. M. Conoirb) UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
(Received 17 September 2013; revised 19 February 2014; accepted 31 March 2014) Multiple scattering in a poroelastic medium obeying Biot’s theory is studied; the scatterers are parallel identical cylindrical holes pierced at random in the medium. The paper focuses first on the influence, on the effective wavenumbers, of the mode conversions that occur at each scattering event. The effect of the holes on the dispersion curves is then examined for two different values of the ratio of their radius to the pores mean radius. Depending on the latter, the dispersion curves of the pierced material are compared, for the fast and shear waves, with those of either a more porous C 2014 Acoustical Society of America. medium or a double porosity medium. V [http://dx.doi.org/10.1121/1.4871182] PACS number(s): 43.20.Jr, 43.20.Gp [ANN]
I. INTRODUCTION
Many applications of wave propagation in poroelastic media containing random distributions of inhomogeneities may be found, from geophysical exploration,1–3 to ultrasonic evaluation of biological tissues such as bones.4 In such cases, the inhomogeneities are either solid inclusions, cracks, or a second network of larger pores. In this paper, the watersaturated poroelastic medium is supposed to obey Biot’s theory and the inhomogeneities are infinitely long water filled identical cylinders perpendicular to the direction of propagation that are larger than the pores. It will be referred to as the perforated poroelastic medium (PPM), and compared to either a more porous medium (MPM) obeying Biot’s theory as well, or to a double porosity medium (DPM), depending on the size of the holes. The PPM is a typical problem of multiple scattering as originally studied by Foldy.5 In such problems, it is well known that for small enough concentrations of scatterers the physical medium may be replaced by an effective homogeneous medium in which coherent waves propagate. These waves represent the acoustic field averaged over all possible configurations of scatterers. The propagation of each coherent wave is governed by a frequency dependent complex effective wavenumber. While in recent years there has been renewed interest and progress in the study of multiple scattering when the host medium is an ideal fluid,6–11 there have been much fewer works when more than one wave may propagate in the host medium, as is the case of a porous medium12,13 that obeys Biot’s theory.14 No theory was available to take into account the conversion modes that occur at
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected] b) Also at: CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France. J. Acoust. Soc. Am. 135 (5), May 2014
Pages: 2513–2522
each scattering event in a multiple scattering process. The problem has been overcome recently15 for an elastic host medium in which two acoustic waves propagate. The theory described in Ref. 15 can be readily applied to host media in which any number of waves propagate,16 and it is applied here to the calculation of the effective wavenumbers associated to the three coherent waves that propagate in the PPM. Increased absorption is a well-known effect of fractures in a porous medium, as described in continuum theories dealing with double porosity media.1–3 Such media have been intensively studied in the past decade for air saturated ones,17,18 and Olny and Boutin19 have emphasized that they can achieve larger sound absorption than single porosity materials. There are two scales in a DPM; one is associated to micropores and the other to (randomly distributed) larger pores. These two networks of pores are interconnected. In the PPM described above, the pores are interconnected, and the cylindrical holes (larger pores) are separated from each other. The macroscopic porosity they induce is somewhat connected through the microscopic porosity of the porous medium, but in a much weaker way than in a DPM. The other main difference is that, from the three-dimensional point of view, the DPM is isotropic at both scales (microscopic and macroscopic), while the PPM is strongly anisotropic, due to the fact that the largest (cylindrical) pores are all parallel. We shall investigate the effect of the difference between the PPM and the DPM on the increased attenuation of the fast and shear waves. Section II is devoted to the multiple scattering model for itself. It is briefly recalled in Sec. II A, along with the hypotheses that underlie it; its range of validity in the numerical case of the PPM under study is defined in Sec. II B through the study of the scattering by a single cylinder. The influence of the mode conversions on the effective wavenumbers is studied in Sec. II C. The velocity and attenuation curves of the coherent waves are discussed in Sec. III for
0001-4966/2014/135(5)/2513/10/$30.00
C 2014 Acoustical Society of America V
2513
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.138.73.68 On: Sat, 20 Dec 2014 14:35:52
two different ratios of the cylinders to the micropores radii, and Sec. IV is devoted to the comparison with either a MPM or a DPM. II. OVERVIEW OF THE MULTIPLE SCATTERING MODEL AND ITS CONDITIONS OF VALIDITY A. The model
The model used is that described in Ref. 15 in the case of cylindrical scatterers in an elastic matrix. In a porous medium that obeys Biot theory, the wavenumbers are complex, even in the absence of scatterers, while they were real in the elastic medium. However this does not raise any new difficulty, as all the series and integrals one has to deal with are still well defined and convergent, and so we need to focus here only on the main assumptions of the theory and on its main results. The generalized Fikioris and Waterman equations in the case of an elastic matrix15 consist in two sets of homogeneous linear equations. Equating their determinants to zero provides the dispersion equation of the coherent waves. The two sets are readily extended to three in the case of a porous Biot matrix that supports the propagation of a fast ðL1 Þ longitudinal wave, a slow ðL2 Þ one, and a transverse ðTÞ wave:
0
Nma ðnÞ ¼ nbJm0 ðnbÞHmð1Þ ðka bÞ ka bJm ðnbÞHmð1Þ ðka bÞ:
In Eq. (2), b is the radius of exclusion introduced in Fikioris and Waterman’s hole correction [see Eq. (7) in Ref. 15]; it is greater than twice the radius a of the cylinders in order to forbid any overlapping of scatterers. Jm and Nm are the Bessel and Neumann functions of order m, and Hmð1Þ is the Hankel function of the first kind and order m. The prime over J or H denotes the derivative with respect to the argument of the function. The next step in order to get an explicit approximate expression of the three effective wavenumbers na is to assume n2a close enough to ka2 , so that a formal asymptotic expansion of n2a in terms of e ¼ 4in0 2 ð2Þ n2a ¼ ka2 þ eyð1Þ a þ e ya þ …
(3)
might be truncated at some order at “low enough” concentration. Equations (52) and (54)–(58) of Ref. 15 provide then ð0Þ yð1Þ a ¼ Maa ;
þ1 X X 2pn0 a Tmba Abm Nmn ðna Þ ¼ 0; Aan 2 2 na ka b¼L1 ;L2 ;T m¼1
(4)
ð1Þ yð2Þ a ¼ Maa þ
ð0Þ
X
ð0Þ
Mab Mba
b ¼ L1 ; L2 ; T b 6¼ a
8a 2 fL1 ; L2 ; Tg:
(2)
ka2 kb2
;
(5)
(1) with
The Aan ’s are the unknowns of the equations and are related to the amplitudes of the coherent fields. ka is the wavenumber of the a-wave in the absence of scatterers and na , which we are looking for, is that of the coherent wave. n0 is the number of cylinders per unit surface, Tmba is the mth component of the diagonal scattering matrix of each single cylinder for a b-wave scattering into an a-wave (see Ref. 20 for its expression), and
ð0Þ
Mab ¼
þ1 X
Tnab ;
(6)
n¼1
ð1Þ ¼ Maa
þ1 X
þ1 X
X
b
Tnba Qnm ðka ÞTmab ;
(7)
m¼1 m¼1 b¼L1 ;L2 ;T
b Q nm ðka Þ
0 ð1Þ ð1Þ0 i p2 ka bJmn ðka bÞHmn ðkb bÞ kb bJmn ðka bÞHmn ðkb bÞ 1 ¼ if b 6¼ a ka2 kb2
(8)
a Q nm ðka Þ
" # ð1Þ p ððka bÞ2 ðm nÞ2 ÞJmn ðka bÞHmn ðka bÞ ¼ i 2 : 0 ð1Þ0 4ka þka2 b2 Jmn ðka bÞHmn ðka bÞ
(9)
and
Now, if the frequency is low enough for the asymptotic expansions for small arguments of the Bessel and Hankel functions in Eqs. (8) and (9) to be used, Eq. (7) turns to
ð1Þ ¼ Maa
þ1 þ1 X 1 1 X jm nj Tnaa Tmaa þ 2 ðka Þ2 n¼1 m¼1
2514
X b ¼ L1 ; L2 ; T b 6¼ a
þ1 þ1 jmnj X X 1 ka Tnba Tmab ka2 kb2 n¼1 m¼1 kb
J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
X k2 b ¼ L1 ; L2 ; T a b 6¼ a
þ1 þ1 X X 1 T ba T ab ; 2 kb n¼1 m¼1 n m
(10)
Franklin et al.: Multiple scattering in porous media
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.138.73.68 On: Sat, 20 Dec 2014 14:35:52
and Eqs. (3)–(5) finally provide the low concentration and low frequency approximations of the effective wavenumbers that will be used in Sec. III, ! 2 na n0 a n20 að0Þ n30 aðcÞ ¼ 1 þ 2 d1 þ 4 d2 þ d2 þO 6 ; ka ka ka ka (11) with da1 ¼ 4i
þ1 X
Tnaa ;
(12)
n¼1 að0Þ
d2
¼ 8
þ1 X þ1 X
TABLE I. Values of the physical parameters of QF20 and water (from Ref. 21). Physical parameters
Symbols (units)
Values
Bulk modulus of grains Dried frame bulk modulus Dried frame shear modulus Solid density Bulk modulus of water Density of water Viscosity of saturating water Porosity Permeability Pore mean radius Tortuosity
Kr (Pa) Kb (Pa) l (Pa) qS (kg m3) K0 (Pa) q0 (kg m3) g (kg m1 s1) b k (m2) ap (m) s
36.6 109 9.47 109 7.63 109 2760 2.22 109 1000 1.14 103 0.402 1.68 1011 3.26 105 1.89
jm nj Tmaa Tnaa ;
m¼1 n¼1 aðcÞ
d2
X
¼ 16
dab 2 ;
(13)
b ¼ L1 ; L2 ; T b 6¼ a
dab 2 ¼
jmnj ka2 ka Tmba Tnab : 2 2 k k k b a b m¼1 n¼1 þ1 X þ1 X
(14) ð Þ
As noticed already in Ref. 15, the da1 and da2 0 terms have the same formal expressions as in the case of an ideal fluid as the host medium, and involve only scattering coefficients Tnaa that describe the scattering of an a-wave into an a-wave. The possibility of mode conversions occurring at each scattering event appears explicitly in the coupling term aðcÞ only, through the scattering coefficients Tnab , with d2 b 6¼ a. However, one should keep in mind that the value of Tnaa depends nonetheless on those of the Tnab coefficients with b 6¼ a, through the continuity conditions20 on the interface of the scatterer. All the scattered b-waves of amplitude Tnab propagate from one scatterer to another; those with large =mðkb Þ=
