Transport parameters and sound propagation in an air-saturated sand

Transport parameters and sound propagation in an air-saturated sand

Julian Tizianel, Jean F. Allard,

Bernard Castagne`de, Christophe Ayrault, Michel Henry, and Alexei Moussatov

Laboratoire d’acoustique, UMR CNRS 6613, Universite´ du Maine, Avenue Olivier Messiaen 72085 Le Mans Cedex 9, France

Antoine Gedeon

Laboratoire de chimie des surfaces, Universite´ Pierre et Marie Curie, 4 place Jussieu, Tour 55, 75230 Paris Cedex 05, France

共

Received 26 April 1999; accepted for publication 10 August 1999

Measurements at ultrasonic frequencies of the transmission coefficient and the sound speed in layers of quarry sand saturated by air and helium are performed at different pressures. The measurement of surface impedance at audible frequencies is also performed for air-saturated layers. Evaluation of transport parameters can be obtained from these measurements. Close sets of parameters can be obtained at high and low frequencies with the model by Pride et al.

Phys. Rev. B 47, 4964

1993

.

© 1999 American Institute of Physics.

S0021-8979

99

01322-5

I. INTRODUCTION

Acoustical properties of ground considered as a porous medium saturated by air are very important for environmen- tal studies concerning outdoor sound propagation. A pioneer work for the oil industry has been performed by Biot,

for porous media saturated by heavy fluids, who has shown that two compressional waves could propagate in a fluid satu- rated porous medium. The description of sound propagation has been improved by Johnson, Koplik, and Dashen,

who use a finite set of well-defined transport parameters. This work is a first step in the study of sound propagation in natural porous materials saturated by air with the transport parameters used by Johnson, Koplik, and Dashen,

and other parameters defined later. The medium is an air-saturated quarry sand

sable de Loire provided by Socie´te´ Baglione du Mans, carrie`re de Spay route d’Aulnays, 72700 Aulnays France

. The grain size distribution of this sand is indicated in Fig. 1. The dispersion is very large. Sound propagation in air-saturated packing of glass beads of diameter close to 1.5 mm has been investigated recently.

The viscous coupling between air and beads is sufficiently small and the mass of the beads is large enough for the beads to remain motionless when acoustic waves propagate in air saturating the pore space. This wave exists in air-saturated sand. It would not be modified if the sand were consolidated and is one of the two compressional Biot waves.

The second Biot wave could be created by a mechanical excitation of the consolidated sand.

The qualifications ‘‘slow wave’’ and ‘‘fast wave’’ are not adequate

and should be replaced by ‘‘air born’’ and ‘‘frame born wave.’’ A simple version of the full Biot theory can be used to describe the air born wave, the air inside the pore space can be replaced by an equivalent free fluid character- ized by a complex density which takes into account the in- ertial and viscous forces, and a complex dynamic compress- ibility which is the actual compressibility of air modified by

the thermal exchanges with the sand. In this work, the com- plex density is called the effective density, ␳

, related to the dynamic tortuosity ␣ of Johnson, Koplik, and Dashen

by

␳

⫽ ␣␳ ^{, where} ␳ is the density of air. The model by Johnson, Koplik, and Dashen

is used in Ref. 4 to predict the effective density, which is given by

the time dependence is exp( ⫺ i ␻ ^t)]

␳

⫽ ␳ ␣

冋 ^{1 ⫹ ␣}

^{i ␾␩} k

␳␻ 冉 ^{1 ⫺ i 4}

^␣

^{␾ k}

^␳␻ ␩ 冊

册 ^,

¹

where ␩ is the viscosity of air, ␾ is the porosity, k

the viscous permeability, ␣

is the tortuosity,

the viscous characteristic dimension, k

, ␣

, and

depending only on the geometry of the porous frame, and ␻ is the radian fre- quency. The viscous dimension

has been defined by Johnson, Koplik, and Schwartz.

It can be shown

that for a porous material with cylindrical pores having identical radii,

⌳

is equal to the radius of the pores, and

can be considered as a measure of the sizes of dynamically connected pores for the actual porous medium, as long as the viscous interaction between the frame and the air inside the porous medium is considered. A precise definition of tortuosity is also given by Johnson, Koplik, and Dashen,

and references therein. The model by Lafarge

is used for the bulk modulus K which is given by

K ⫽ ␥ ^P

␤

with

2

␤ ⫽ ␥ ⫺

␥ ⫺ 1

冋 ^{1 ⫹ k}

^{⬘ i Pr ␩␾ ␳␻} 冉 ^{1 ⫺ i 4k}

^{⬘ ⬘}

^{Pr ␾}

^{␳␻ ␩} 冊

册

^,

where Pr is the Prandl number, ␥ the ratio of the specific heats, P the atmospheric pressure,

⬘ the characteristic ther- mal dimension, and k

⬘ is the thermal permeability which has been defined in previous articles.

The thermal dimension

⌳

⬘ is also equal to the radius of the pores for a porous

a兲Electronic mail: bal@laum.univ-lemans.fr

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0021-8979/99/86(10)/5829/6/$15.00 © 1999 American Institute of Physics

material with cylindrical pores having identical radii.

It can be considered as a measure of the sizes of the pores as long as the thermal exchanges between the frame and the air are considered. The two parameters

and

⬘ are not equal for usual porous materials and

⬘ is larger than

. For a smooth frame surface,

⬘ is given by

⌳

⬘ ⫽ 2V

S ,

3

where V is the air content and S is the air-frame contact surface where the thermal exchanges occur, in a given vol- ume of porous material. This parameter is related to the cor- relation length p

defined by Debye, Anderson, and Brumberger,

and Ma¨tzler,

which can be written

p

⫽ 2

1 ⫺ ␾

⬘ ^.

⁴

The characteristic impedance Z

and the wave number k

in the air saturating the sand are k

⫽ ␻ ⁽ ␳

/K)

, Z

⫽ ( ␳

K)

. In Ref. 4, measurements at audible and ultra- sonic frequencies are compared with predictions obtained from Eqs.

1

–

2

with the same set of parameters,

,

⬘ ^,

␣

^{, k}

, k

⬘ ^{, and} ␾ . It has been shown by Pride, Morgan, and Gangi

that at low frequencies the effective density given by Eq.

1

should be modified, mainly because the model by Johnson et al. does not provide the exact limit for Re( ␳

) when ␻ → 0. In the present work, the model by Pride, Mor- gan, and Gangi

is used, where ␳

is given by

␳

⫽ ␣

␳ 冋 ^{1 ⫹ C i ␻ ¯ f}

^␻

册 ^.

⁵

In this equation, C ⫽ 8 ␩␣

( ␴

␾ ^), ␻ ^¯ ⫽ ␻

␳ ^/(8 ␩ ^{), and} f ( ␻ )is given by

f

␻

1 ⫺ p ⫹ p

1 ⫺ iC

␻ ^{¯ /2p}

.

6

When p ⫽ 1, Eqs.

1

and

5

are identical. The low fre- quency limits of Im( ␳

) in Eqs.

1

and

5

are always the same. A reasonable choice for the range of variation of p is

the interval

1/2,1

. Previous studies have been performed on water saturated non consolidated sands. A synthesis of these works is given by Chotiros.

Other works have been per- formed by Johnson et al.

on synthetic consolidated granular materials, and Gist

on natural sand stones, in the context of the Biot theory. For water saturated nonconsoli- dated sands, the Biot theory does not give a precise descrip- tion of the experimental results, and there is not a clear status for the rigidity of the porous frame

which is an important parameter in that case. A study of sound propagation in the simple context of air-saturated natural sand can provide in- formation which are hidden by the complexity of the physi- cal mechanisms when the saturating fluid is water. In Secs. II and III nonacoustical and ultrasonic measurements on the air saturated and helium saturated sand of the transport param- eters used in the modeling are performed. Measurements in the audible frequency range for the air saturated sand are presented in Sec. IV.

II. NON ACOUSTICAL MEASUREMENTS

The viscous permeability k

is evaluated with a manom- eter and a calibrated air flow resistance, and the porosity ␾ from the measurement of the air content in a given volume of sand. The tortuosity ␣

is evaluated by successively measur- ing the resistivity of a conducting fluid r

and of the porous material saturated by the fluid r

, and using the relation

␣

⫽ ␾ ^r _r

f

.

7

Another parameter, the thermal characteristic dimension

⬘ ^, has been previously evaluated for synthetic materials with the Brunauer–Emmett–Teller

BET

method

which can be used to measure S and to evaluate

⬘ ^{from Eq.}

3

. The presence of dust, in the natural sand, and of a surface rough- ness with a length scale small compared to the viscous skin depth makes the effective equivalent smooth surface where the thermal exchanges occur much smaller than the actual surface S. The dimension

⬘ evaluated with the BET method is smaller than 1 ␮ m, and this result is not compatible with acoustical measurements. Another method of measuring

⬘ ^, developed by Leclaire, Swift, and Horoshenkov

after Root- are and Prezlow,

has been used. The method is based on the work dW required to raise a liquid in a capillary. Let ␸ be the contact angle for the wetting liquid

dW ⫽ ␴ ^cos ␸ ^dA,

8

where ␴ is the surface tension and dA the surface area cov- ered by the liquid along the pore wall during the process.

This work is equal to the product PdV of the pressure P required to move the liquid, and the following relation can be written

␴ ^cos ␸ ^dA ⫽ PdV.

9

Integration of Eq.

8

to the total pore volume V

yields the total pore surface area

S ⫽ 1

␴ ^cos ␸

PdV.

10

FIG. 1. Grain size distribution of the sand. The height of the rectangles indicates the reduced mass of the grains in the interval.

5830 J. Appl. Phys., Vol. 86, No. 10, 15 November 1999 Tizianelet al.

Measurements with water on calibrated beads indicate that an estimation of the pore surface area with an error between 20% and 10% is obtained using cos ␸ ⫽ 1 in Eq.

10

. An example of the variation of pressure as a function of the extracted volume of water is presented in Fig. 2. The actual extracted water volume is smaller than the total water vol- ume which completely saturates the sand, equal to 74 ml

the total volume for the water saturated sand is 200 ml

. The value for

⬘ obtained from a series of measurements is

⌳

⬘ ⫽ 80 ⫾ 10 ␮ m. The main part of the remaining volume may be related to dust and surface roughness saturation which do not contribute to the thermal exchanges: as indi- cated later, the values

⬘ ⫽ 90 ␮ m can be used successfully to predict the acoustical measurements. The nonacoustical mea- surements provide the following values for ␾ ^{, k}

, ␣

^{, and}

⌳

⬘ ^, ␾ ⫽ 0.37 ⫾ 0.01, k

⫽ (1.23 ⫾ 0.16) ⫻ 10

m

, ␣

⫽ 1.7

⫾ 0.1 and

⬘ ⫽ 80 ⫾ 10 ␮ m. The variability of the parameters for the different samples studied is larger than for usual syn- thetic porous media.

III. ULTRASONIC MEASUREMENTS

The sound speed c

in air and helium saturating the sand and the transmission loss TL of sand layers in air and helium have been measured with ultrasonic pulses. Following previ- ous works by Nagy and Johnson,

and Ayrault et al.,

mea- surements are performed at different static pressures larger than the atmospheric pressure. There is no noticeable de- crease of the wavelength when pressure increases, and large diffusion effects which occurs at high frequencies can be avoided.

At sufficiently high frequencies, the sound speed c

in air saturated sand is given with a good approximation by

c

⫽

␳ ^␥ ␣ ^P

冋 ^{1 ⫹ 冉 ␳␻ 2 ␩ 冊}

^⫻ 冉

^{1 ⫹}

^{␥ Pr ⫺}

^{1 ⬘} 冊册

^.

¹¹

The second term in the square brackets is much smaller than one. The air density ␳ is given by ␳ ⫽ M P/(RT), M being the molar mass, R the perfect gas constant and T the absolute

temperature. Let n be the refraction index, n ⫽ k

/k. The squared real part of the refraction index Re(n) ⫽ c/c

, c be- ing the sound speed in free air, can be written

关

Re

n

⫽ ␣

冋 ^{1 ⫹}

^{2 M ␩ ␻ RT P} 冉

^{1 ⫹ Pr ␥}

^⫺

^{1 ⬘} 冊 册 ^.

¹²

In this equation, the parameters are independent on P, and P and ␻ play the same role, i.e., n depends linearly on P

. Let W be the amplitude of the transmitted pressure field re- lated to an unit amplitude incident wave impinging on a layer at normal incidence. In the present work, the transmis- sion loss TL ⫽ ln(1/

W

). The transmission loss at high fre- quencies can be written

TL ⫽ ln 冉 ^{1 4 ⫹ ␾ ␾ / ␣ / ␣}

冊 ^{⫹ l 冉 ␻ 2 ␩␣ ␥ P}

^冊

^冉

^{1 ⫹ Pr ␥}

^⫺

^{1 ⬘ 冊 .}

¹³

In this equation TL also depends linearly on P

, l is the thickness of the sample. Measurements of n and TL as a function of P can only provide an estimation of 1/

( ␥

⫺ 1)/Pr

⬘ . For air, ( ␥ ⫺ 1)/Pr

⫽ 0.476 and

⬘ ^{is for} usual porous materials two or three times as large as

. Then variations of

around the physical values induce much large variations of n and TL than similar variations on

⬘ ^{. The} same equations are valid if helium is used instead of air, but ( ␥ ⫺ 1)/Pr

is then equal to 0.8088. Precise measurements with both gases can provide an evaluation of

and

⬘ ^for glass wools and reticulated foams.

Measurements have been performed with air and helium saturated sand. The fre- quencies are 50 kHz for air and 146 kHz for helium so that the wavelength is roughly the same at equal static pressure.

Measurements of c/c

and TL in air and helium are repre- sented in Figs. 3–6, as function of ( P/ P

)

, P

being the atmospheric pressure. The thickness of the sample is 2.3 cm in Figs. 3 and 4 and 3.1 cm in Figs. 5 and 6. The predicted c/c

and TL are represented for parameters, ␾ ^, ␣

,

⬘ ^{, ad-} justed inside the error intervals defined in the previous sec- tion,

⬘ ⫽ 90 ␮ ^m, ␣

⫽ 1.68, ␾ ⫽ 0.37 , and

has been ad-

FIG. 2. Pressure as a function of the volume of water extracted from water saturated sand. The total volume of water which saturates the sample is equal to 74 ml.

FIG. 3. The squared real part of the refraction index (c/c1)²as a function of ( P0/ P)^1/2in helium. — Prediction关Eq.共12兲兴.䊊Measurement.␣⬁⫽1.68,

⌳⫽37␮m,⌳⬘⫽90␮m,␾⫽0.37.

justed for each figure. It may be noted that excellent agreements can be obtained between prediction and measure- ment for Figs. 3, 5, and 6 but not for Fig. 4 where the pre- dicted and measured slopes are different. There is no justifi- cation of this difference in the context of this work. The two values of

for c/c

in air and helium in Figs. 3–5 are

37

␮ ^m,

38 ␮ m, and for the transmission loss in Figs. 4–6

⌳⫽

27 ␮ ^m,

28 ␮ m. When a weak diffusion occurs and slightly modifies the transmission loss via Im(k

), the effect on the c/c

can remain completely negligible.

The dif- ference between the adjusted

related to c/c

and TL can be interpreted as a consequence of a weak Rayleigh scatter- ing not taken into account in Eqs.

12

and

13

which in- creases the damping but leaves c

unchanged, and

38

␮ m, related to the measurement of c/c

in air and helium, will be considered as the result of the ultrasonic measure- ments. A noticeable change in

⬘ slightly modifies the evalu- ation of

. The value

⬘ ⫽ 90 ␮ m is a reasonable choice, in

all the previous studies

⬘ is in the range 2

, 3

. Neverthe- less the measurements of n in air and helium are not precise enough for

⬘ to be evaluated with a good precision.

IV. MEASUREMENTS IN THE AUDIBLE FREQUENCY RANGE

Measurements of the acoustic surface impedance Z

in a Kundt tube have been performed for two samples of thick- ness l ⫽ 1.7 and 3 cm. Measurements of the reduced surface impedance Z

/Z, Z being the characteristic impedance of air, are presented in Figs. 7 and 8 and compared with predictions obtained from Eqs.

2

–

5

and the relation

Z

⫽ iZ

␾ ^cot

^k

^, l

,

14

FIG. 4. Transmission loss as a function of ( P0/ P)^1/2in helium. — Predic- tion 关Eq. 共13兲兴. 䊊 Measurement. ␣⬁⫽1.68, ⌳⫽27 ␮^m, ⌳⬘⫽90 ␮^m,

␾⫽0.37.

FIG. 5. The squared real part of the refraction index (c/c1)²as a function of ( P0/ P)^1/2 in air. — Prediction 关Eq. 共12兲兴. 䊊 Measurement. ␣_⬁⫽1.68,

⌳⫽38␮m,⌳⬘⫽90␮m,␾⫽0.37.

FIG. 6. Transmission loss as a function of ( P0/ P)^1/2in air. — Prediction 关Eq.共13兲兴.䊊Measurement.␣_⬁⫽1.68,⌳⫽28␮m,⌳⬘⫽90␮m,␾⫽0.37.

FIG. 7. Real and imaginary part of the reduced surface impedance thickness⫽3 cm. — Prediction 关Eq. 共14兲兴. 䊊 Measurement. ␣_⬁⫽1.68,

⌳⫽30␮m,⌳⬘⫽90␮m,␾⫽0.37, k0⫽1.2 10^⫺10 m², k₀⬘⫽5 10^⫺10 m².

5832 J. Appl. Phys., Vol. 86, No. 10, 15 November 1999 Tizianelet al.

共

the surface impedance is the ratio, in the frequency domain, of the acoustic pressure to the acoustic velocity at the surface of the sample when a plane acoustic wave impinges upon the sample at normal incidence

. The measured values of Sec. II are used for

⬘ ^{, k}

, ␣

^, ␾ . A reasonable agreement is ob- tained for both thickness with

31 ␮ ^{m, k}

⬘ ⫽ 5

⫻ 10

m

, k

⫽ 1.226 ⫻ 10

m

, ␣

⫽ 1.68, ␾ ⫽ 0.37, and p ⫽ 0.6. It may be noted that the dependence on k

⬘ ^is negligible at frequencies larger than 500 Hz. At very low frequencies, the viscous dissipation is small despite the high flow resistivity because the acoustic velocity is equal to zero at the contact surface with the bottom of the tube and very small in the whole volume of the porous sample. Neverthe- less the thermal losses are also very small, the imaginary part of the compressibility, responsible for the thermal losses which contribute to Re(Z

), being proportional to k

⬘ ^{, and k}

⬘ like for the case of glass beads and other porous media, has roughly the same order of magnitude as k

. Much larger values of k

⬘ ⁽ ⬎ 10

) are related to a predicted Re(Z

) too large. Much smaller values of k

⬘ do not noticeably modify Re(Z

), the thermal losses in this case being small compared to the viscous losses for k

⬘

5 ⫻ 10

m

. No precise evaluation of k

⬘ can be obtained for the considered medium.

It may be noticed that if the model by Johnson et al., p ⫽ 1 in Eq.

3

, is used instead of p ⫽ 0.6, a reasonable agreement for the surface impedance can be obtained for

26 ␮ ^m.

V. COMPARISONS BETWEEN HIGH AND LOW FREQUENCY MEASUREMENTS

For

, the low frequency and the high frequency estima- tions are

31 ␮ ^{m and}

38 ␮ m, respectively. Due to the fast variation of the evaluated high frequency value of

as a function of ␣

, a slight decrease around 0.1 of ␣

is suffi- cient to obtain very close evaluations of

at high and low frequencies. Nevertheless it must be noted that for

38

␮ m, the coefficient C defined in the introduction is equal to

3.08, close to C ⫽ 2 and 2.5 obtained by Johnson for sintered beads. For

31 ␮ ^{m, C} ⫽ 4.6. It is well known that C is generally close to one and this last estimation is noticeably larger than usual. Then the value

38 ␮ m is probably a reasonable choice for

. This value is slightly too large for a precise prediction of the surface impedance at audible fre- quencies to be obtained. In Ref. 4, C ⫽ 5, which is roughly twice as large as the values obtained in Ref. 13. This differ- ence, which is only related to a variation of

by a factor

冑

2, could be due to the use of Eq.

5

with p ⫽ 1 at low frequencies, and to a less precise evaluation of

at high frequencies, the variable parameter being frequency in a re- stricted range of variation where diffusion is negligible, and not the static pressure like in the present work.

VI. CONCLUSION

Sound propagation in a sand of Loire saturated by dif- ferent gases has been studied under different aspects. Surface impedance at audible frequencies for air saturated samples, time of flight and transmission loss at ultrasonic frequencies for air and helium saturated samples have been measured and predicted with a recent model for the effective density. The important fact is that close sets of transport parameters have been obtained at high and low frequencies. These transport parameters have been evaluated for a nonconsolidated sand.

An important difference with synthetic materials concerns the measurements of the thermal characteristic dimension, which is not possible with the BET method. Another differ- ence, inherent to the low viscous permeability of the mate- rial, is that the thermal permeability cannot be evaluated with precision.

ACKNOWLEDGMENTS

The authors are grateful to S. Unterseh for laser sieve measurements, J. C. Rabadeux for grain size distribution measurements, and D. Lafarge for helpful discussions.

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5834 J. Appl. Phys., Vol. 86, No. 10, 15 November 1999 Tizianelet al.