Fatima Zahraa Kachkouch

Abdellah Alem

Alain Tinel

Hervé Franklin

Huaqing Wang

Effects of clogging on ultrasonic transmission through saturated granular single and double porosity media

Keywords: clogging, granular media, single porosity, double porosity, acoustic waves

The clogging caused by the deposition of suspended particles modifies in depth several mechanical parameters (permeability, porosity, bulk moduli, etc.) of porous media and influences the acoustic behaviour. Consequently, at a fixed position in the medium, changes in phase velocity and attenuation are observed in the amplitude of the temporal signal of the transmitted waves. In this work, ultrasonic techniques are presented both for detecting and measuring the clogging in a water saturated porous medium. The acoustic measurements from the clogged samples are compared with the deposition profiles obtained at the end of injection experiments.

Moreover, links are established between on the one hand phase velocity and the total porosity, and on the other hand the transmitted signal energy and the variation of porosity as consequence of particle deposition.

Introduction

During the flow of suspended fine particles through single or double porosity poroelastic media (referred to hereafter by SPP and DPP), several phenomena are likely to occur, causing adverse consequences on the proper functioning of the structures. It has been shown that the clogging of recovery wells is due to the presence of fine particles in the water [START_REF] Moghadasi | Model study on the kinetics of oil field formation damage due to salt precipitation from injection[END_REF]. The decrease in permeability [START_REF] Alem | Filtration of kaolinite particles in a saturated porous medium: hydrodynamic effects[END_REF], one of the consequences resulting from the clogging, can affect the proper functioning of the structure. Conversely, the detachment of particles and their transport by water flow can provoke internal erosion. This phenomenon is hardly detectable in its first manifestations.

The paper presents a nondestructive ultrasonic method for detecting the accumulation of deposited particles in SPP and DPP. The acoustic properties of the porous material must be related to the mechanical parameters. In the case of clean media (not crossed by a turbid flow), Kohout et al. [START_REF] Kohout | Ultrasonic Determination of Porosity in Homogeneous Ceramic Samples[END_REF] showed that the propagation velocities of acoustic waves depend on the porosity of the material. Based on the velocity of the slow longitudinal wave, tortuosity was determined at high frequency [START_REF] Allard | Evaluation of tortuosity in acoustic porous materials saturated by air[END_REF][START_REF] Melon | Correlation between tortuosity and transmission coefficient of porous media at high frequency[END_REF], and permeability at low frequency [START_REF] Fellah | Measuring permeability of porous materials at low frequency range via acoustic transmitted waves[END_REF].

The coupling between a destructive method discussed later in the paper and the ultrasonic method requires at first understanding how the acoustic waves propagate through such complex media. The model describing acoustic wave propagation in fluid saturated SPP media is due to Biot [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF]. The travelling of waves through poroelastic slabs allows extracting certain poroelastic parameters [START_REF] Jocker | Ultrasonic measurements on poroelastic slabs: Determination of reflection and transmission coefficients and processing for Biot input parameters[END_REF]. Lauriks et al. [START_REF] Lauriks | Characterization of porous acoustic materials[END_REF] have developed techniques for determining the physical parameters and elastic properties of the porous medium at high-frequency domain.

The majority of the aforementioned studies on acoustic wave propagation and/or fine particle filtration focused exclusively on SPP media. However, porous structures present in nature, contain fractures (macroporosity) and a finer porosity in the matrix (microporosity). This is why Biot's SPP model was extended to the case of a DPP medium [START_REF] Warren | The behavior of naturally fractured reservoirs[END_REF][START_REF] Wilson | A double porosity model for acoustic wave propagation in fractured porous rock[END_REF][START_REF] Berryman | Elastic wave propagation and attenuation in a doubleporosity dual-permeability medium[END_REF]. From this later model, recent studies on double porosity were led. Dai et al. [START_REF] Dai | Reflection and transmission of elastic waves from the interface of fluid-saturated porous solid and a double porosity solid[END_REF] studied reflection and transmission coefficients at a fluid-DPP interface. Kachkouch et al. [START_REF] Kachkouch | Reflection and transmission characteristics of a layer obeying the two pressure field poroelastic phenomenological model of Berryman and Wang[END_REF] studied the dispersion curves of a double porosity plate. Using the transmission coefficient, frame bulk and shear moduli of double porosity media were estimated [START_REF] Bai | Estimating frame bulk and shear moduli of two double porosity layers by ultrasound transmission[END_REF]. Note that other approaches ignored here [START_REF] Arbogast | Derivation of the double porosity model of single phase flow via homogenization theory[END_REF][START_REF] Auriault | Deformable porous media with double porosity[END_REF], derived alternative equations for DPP media from homogenization methods.

Our study considers both the theory of Biot (for the non-porous glass beads based medium) and of Berryman and Wang (for the porous grains based medium). The acoustic behavior in such materials strongly depends on the interaction between the two phases which have very different porosities, and on the friction between contacting grains [START_REF] Chotiros | Shear wave attenuation and micro-fluidics in water saturated sands and glass beads[END_REF]. For longitudinal waves, Kimura [START_REF] Kimura | Experimental validation and applications of a modified gap stiffness model for granular marine sediments[END_REF] developed a model showing that Biot's theory remains valid until 0 0.5 < k b ( 0 k is the wave number in water and b the diameter of the grains). That means working at low frequencies domain where it is known that the wavelengths involved are much greater than the characteristic pore sizes, and where it is also known that the nonlinear effects resulting from the friction between the grains can be neglected. In our case, considering a granular medium with a diameter of grains close to 3.5 mm and pores sizes detailed in section 2.1, means working at a frequency range around 40 kHz. Thus, the use of transducers with 44 kHz central frequency as we do in this paper means that we accept the measurements obtained as valid in the context of continuous media. Beyond approximately this frequency range, the effects of multiple scattering should also be considered.

To carry out the experiments, an elongated rectangular column intended to contain the porous materials was manufactured. Since clogging does not manifest in the same way along the column -it is more important at the entrance and decreases with depth -the experimental set up developed by Bai et al. [START_REF] Bai | Ultrasonic characterization of particle retention by a double porosity medium[END_REF] has been improved: (i) by changing the disposition of the column from vertical to horizontal, implying an horizontal flow which better corresponds to the flow occurring in water retention structures, see Fig. 2, (ii) by allowing a translation of the transducers along the column (from the inlet to the outlet of the flow). Notice that gravity may be a more significant force for particle deposition for horizontal flow than for down flow as it has been shown in previous research (Chrysikopoulos et al. [START_REF] Chrysikopoulos | Effect of Gravity on Colloid Transport through Water-Saturated Columns Packed with Glass Beads: Modeling and Experiments[END_REF]).

In section 2, the three materials used in this paper and the equations governing the acoustic wave propagation in SPP and DPP media are recalled. The experimental setup combining the destructive method and the ultrasonic measurements is then described. Section 3 is devoted to the equations of particle filtration and to the results of the destructive injection method. A follow-up of clogging in space (deposition profiles) and in time (restitution curves) is achieved. Section 4 is concerned with the results of ultrasonic detection of deposition. The velocities of the fastest longitudinal wave are estimated at first for the clean medium and next during injection.

Comparison with the theoretical values is led for the clean medium only. The variation of the signal energy and attenuation are represented based on the decrease of the transmitted signal amplitude. Finally, section 5 confronts the two methods and allows establishing links between phase velocities and the total porosity at the end of the injection process on the one hand, and between energy and the porosity variation due to deposition on the other hand.

Background on porous media

Description of samples

The SPP medium (also referred to as GB) is formed by stacking spherical glass beads of diameter 3.5 mm. The random arrangement of the beads creates empty spaces called pores which will be filled with water. Also, two kinds of DPP media are investigated. The first -Tobermoriteis a stack of porous grains of semi-natural material Tobermorite 11 Å (porous cement grains irregularly shaped and sized from 3.15 mm to 4 mm obtained by sieving) (Figs. 1c,1d). The second -Robu -is a stack of porous grains of Borosilicate glass (VitraPOR® Sinterfilter & Laboratory Glassware) (Figs. 1a,1b). This latter is widely used in research field because of its resistance to heat, corrosion and thermal shock. Robu and Tobermorite present a dual porosity: macroporosity between the grains and microporosity inside the grains. The total porosities of the three samples measured using air pycnometer D5550-6 are 39%, 60% and 70% for GB, Robu and Tobermorite, respectively. Porosity and pore size distributions in the grains were determined with mercury intrusion porosimetry. Microporosity is 21% of the total volume for Robu, and 31% of the total volume for Tobermorite. For Robu pore sizes ranged between 6 μm and 147 μm with h10 = 39 μm, h50 = 54 μm and h90 = 100 μm and for Tobermorite between 0.08 μm and 100 μm with h10=0.06 μm, h50=1.35 μm and h90=14 μm (h10, h50 and h90 are pore diameters corresponding respectively to 10%, 50% and 90% of the cumulative pore size distribution). Measurements allow to estimate a macroporosity of 39% of the total volume both for Robu and Tobermorite.

Several physical parameters characterize each of the materials described above (the numerical values are taken, for Glass beads from Kimura [START_REF] Kimura | Frame bulk modulus of porous granular marine sediments[END_REF], and for the two DPP media from Bai et al. [START_REF] Bai | Ultrasonic characterization of particle retention by a double porosity medium[END_REF]). Most of these parameters are interdependent. For example, permeability is one of the most important parameters in a porous medium and is directly related to the size of the grains [START_REF] Kohout | Ultrasonic Determination of Porosity in Homogeneous Ceramic Samples[END_REF] and to the porosities.

Poroelastic waves

Biot's theory for a SPP medium

The porous medium is made up of two separately connected phases, the elastic matrix (skeleton) and the fluid that occupies the pore space. Let us introduce f ρ the mass density of fluid, η the fluid viscosity, s ρ the mass density of the elastic constituent, φ the porosity and k the permeability of the skeletal frame. The mass density of the porous medium is ( )

1 s f ρ φ ρ φρ = - + . Let u, U and ( ) φ = - w
U u be the solid displacement, the pore fluid displacement and the relative fluid to solid displacement vectors, respectively. The constitutive equations for a porous saturated isotropic medium are ( )

2 2 I H + C I τ µ ε ξ µε   = - +   , ( 1
)
f I P C M ε ξ = - + , ( 2
)
where τ is the stress tensor, f P the fluid pressure, ( )

2 T ε = ∇ + ∇ u u
the strain tensor ( ∇ is the gradient operator), I the unit tensor, . ξ = -∇ w the increment of fluid content and ( ) .

I tr ε ε u = = ∇
the dilatation of the volume element attached to the frame. The constants µ , H , M and C represent poroelastic moduli (Stoll et al. [START_REF] Stoll | Reflection of acoustic waves at a water-sediment interface[END_REF]).

Wave equations for longitudinal and transverse waves were derived by Biot [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF] from the constitutive equations, the equations of flow through a porous medium and the equations of motion. Let us assume fields harmonic in space and time ( (

) i x t e ω - l
dependence with propagation along the horizontal x axis) where ω is the angular frequency, t the time and r i i l l l = + a complex wave numberr ω l is the phase velocity and 2 8.686 i r π× ×l l the attenuation coefficient. The dispersion relations are given by:

2 2 2 2 2 2 2 2 0 f f H C C m M i F k ρω ρ ω ρ ω ω ωη l l l l - - = - - - (3)
for longitudinal waves, and

2 2 2 2 2 0 f f m i F k µ ρω ρ ω ρ ω ω ωη l - = - - (4)
for transverse waves [START_REF] Stoll | Reflection of acoustic waves at a water-sediment interface[END_REF]. Above, the other to slow wave with strong attenuations (the existence of which was proved in Plona experiments [START_REF] Plona | Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies[END_REF]). Eq. ( 4) has a unique root. Longitudinal and transverse waves are decoupled.

Berryman and Wang extension of Biot's theory to DPP medium

In the equations of motion for fluid-saturated DPP media [START_REF] Berryman | Elastic wave propagation and attenuation in a doubleporosity dual-permeability medium[END_REF], u, ( )

1 U and ( ) 2 U
denote the displacement vector of the solid frame, the micropore fluid (phase 1) and the macropore fluid (phase 2), respectively. The relative fluid-solid displacements are ( )

( ) ( ) ( ) ( ) j j j j ν φ = - w U u , in micropores ( 1 j = ) and in macropores ( 2 j = ). The coefficient ( ) 1 φ ( ( ) 2
φ ) is the volume fraction of micropores in matrix (of fracture porosity) with ( )

2 1 φ = ; ( ) 1 ν ( ( ) 2 ν
) is the volume fraction occupied by matrix (by macropores) with ( ) (  )

1 j j ν = ∑ , so that ( ) ( ) 1
(1) (1) ( 2 ) ( 2 )
2 2

I I τ µ ε ξ ξ µ ε   = - - - +   H C C , (5) ( ) ( ) ( )
( )

1 12 22 23 1 2 13 23 33 2 I C C C P C C C P ε ξ ξ     -     =         -         . ( 6
)
The strain tensor ε and its trace I ε have been defined above, in sec. 2.2.1. In Eq. ( 5)

( ) ( ) ( ) 1 1 2 12 13 4 3 u K C C µ ν φ ν = + + + H , (7) ( ) ( ) ( ) ( ) ( )
1 1 1 1 2 22 23 u K B C C ν φ ν = - + + C , (8) ( ) ( ) ( ) ( ) ( )
2 2 1 1 2 23 33 u K B C C ν φ ν = - + + C . ( 9
)
where µ and u K represent the shear modulus for the skeletal frame and the undrained bulk modulus of the DPP medium, respectively. Note that u K depends on Skempton's pore-pressure build-up coefficients ( )

j B ( 1, 2 j =
) [START_REF] Berryman | Elastic wave propagation and attenuation in a doubleporosity dual-permeability medium[END_REF]. The constants H , ( )

1 C and ( ) 2 C
define moduli for the DPP medium considered as a continuum. More details on the way to compute the coefficients [START_REF] Bai | Ultrasonic characterization of particle retention by a double porosity medium[END_REF]) can be found in the paper of Dai et al. [START_REF] Dai | Reflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium[END_REF] with the ij C s corresponding to their ij A s. Eqs. [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF][START_REF] Jocker | Ultrasonic measurements on poroelastic slabs: Determination of reflection and transmission coefficients and processing for Biot input parameters[END_REF][START_REF] Lauriks | Characterization of porous acoustic materials[END_REF] represent the constitutive equations of isotropic DPP media.

( ) j B and ij C ( ij ji C C = with a dependence on the moduli u K , 1 K , 2 K , K , s K , ( ) 1 s K , µ given in Table 1 of Ref.
Following the same method as in sec. 2.2.1, wave equations for longitudinal and transverse waves were derived by Berryman and Wang [START_REF] Berryman | Elastic wave propagation and attenuation in a doubleporosity dual-permeability medium[END_REF] from the constitutive equations, the equations of motion and the equations of flow through a DPP medium. Assuming once again fields with a

( ) i x t e ω - l
dependence, the dispersion relation can be written as 0 q q q q q q q q q = (11)

for transverse waves. The elements ij t and ij q of the determinants are given in the Appendix.

They differ from that given by Dai et al. [START_REF] Dai | Reflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium[END_REF] or Berrryman and Wang [START_REF] Berryman | Elastic wave propagation and attenuation in a doubleporosity dual-permeability medium[END_REF]. The latters observed the presence of an additional longitudinal wave due to the presence of a second network of connected pores (micropores inside the grains). The theoretical proofs are not reproduced here; however, in sec. 4, we compute the phase velocities using the values of Table 1 of Ref. [START_REF] Bai | Ultrasonic characterization of particle retention by a double porosity medium[END_REF] and compare them with the experimental measurements.

Experimental setup

The first equipment allows for acoustic measurements, the second allows the estimation of particle retention in order to evaluate the clogging. The porous material (either SPP or DPP medium) is set in an aluminum rectangular column of dimensions 100 x 100 x 450 mm 3 (wall thickness is 2 mm). The column is placed horizontally to avoid any settling causing heterogeneity of the material and is fully immersed in a large water-filled tank for at least 24 hours so that the porous media are totally saturated. To perform ultrasonic measurements, the two immersion transducers (Ultran group GC50-D50) -Ø5 cm and central frequency 44 kHz -positioned opposite each other (without contact with the metal plates) were used. Using a mechanical translation system that maintains fixed the distance from the column, these transducers (one acting as the emitter of acoustic waves, the other as the receiver) can move along the horizontal axis (Fig. 2) (eight different positions are defined for the results). and flow velocity of 0.013 cm.s -1 . At the end of each experiment, the material (porous medium with deposited particles) filling the column was carefully extracted and divided into eight sections according to the transducers positions in order to measure the amount of retained particles along the medium. The mass of the deposited particles in each section was then determined according to the procedure described in Alem et al. [START_REF] Alem | Filtration of kaolinite particles in a saturated porous medium: hydrodynamic effects[END_REF]. For each section, an average value of retention defined as the ratio between the volume of the retained particles in the section and the volume of the section, was calculated. Retention profile along the porous medium was then determined.

Fig. 2. Experimental setup for acoustic and mechanical analyses of the clogging phenomenon.

The column containing the medium is fixed and the transducers move along the horizontal x axis (eight positions are shown corresponding to different layers having roughly the same properties in the clogged medium). The distance between the two transducers is 36 cm. Figure 3b shows that particle deposition is non-uniformly distributed in the pore space.

Evaluation of particle deposition using the destructive method

Restitution curves and deposition profiles

Indeed, the deposition is more important in the first centimeters near the entrance of the medium and decreases with depth. The comparison of the retention profiles between the three porous media shows that retention at a fixed depth is globally highest for Robu, sligtly lower for Tobermorite, and is the weakest for GB. This observation is more marked for the first ten centimeters of the medium where the deposition is more important. Despite the high total porosity (70%) and microporosity (31%) of Tobermorite, the small sizes of a large part of micropores in Tobermorite grains suggests less access of suspended particles to these pores leading to less probability of their capture. However, for Robu, the sizes of the micropores are large enough that the particles in suspension can access a large part of these pores which explains the higher values of deposition.

Let σ be the local retention defined as the volume of particles deposited per unit volume of the porous medium, U the Darcy velocity, C the concentration of suspended particles, z the axial distance along the column, t the time, and

/ c t t z U = -φ
the time corrected for the initial filtration at different positions. Neglecting diffusion and assuming a small rate of change in porosity, the process of capture of injected SusP into a porous medium can be described, for one dimensional flow, by a mass conservation and rate equations for the particles in its simplest form as

0 c C U t z ∂ ∂ + = ∂ ∂ σ , 0 c U C t ∂ = ∂ σ λ ψ ( 12
)
0 λ is the first order filtration coefficient of the medium. ψ is a dimensionless retention function that accounts for time and depth dependent deposition, it is expressed as:

( )

exp t - = β ψ ζ α (13)
Where ( ) 12) into the first equation of Eq. ( 12), a linear first order solution is derived as follows:

0 0 C C z ∂ + = ∂ λ ψ (14)
Using the boundary condition ( ) 0 0, c C t C = , the following solution of Eq. ( 14) is found :

( ) 0 50 0 exp( ) 0 , c d D t c C z t C e -Λ = α (15)
where ( )

0 0 / 1 Λ = - λ β and 1 0 1 D -   = -   β ζ
. Inserting the expression for C (Eq. ( 15)) in the second equation of Eq. ( 12), a simple linear first order differential equation is obtained.

Considering the initial condition ( )

,0 0 z = σ
, the solution of this differential equation is obtained.

( ) ( )

0 50 0 0 50 0 exp( ) 50 0 1 , c d D d D t c U z t e e d D - -Λ -Λ -   = -   β α β ζ σ α (16)
In Eqs. 15 and 16, the initial stage (first two pore volumes), where the rate of recovered particles increases from 0 to the plateau, has been neglected. Eqs. 15 and 16 are used to evaluate the evolution with time of the restitution at the outlet of the medium, and the deposition profile in the porous medium at the end of each injection test.

The comparison between experimental and calculated results of the restitution (Fig. 3a) and the deposition profiles (Fig. 3b) shows globally good agreement, suggesting that the model, despite its simplicity, provides a satisfactory prediction of particle removal by the SPP or the DPP media.

For each medium, an average initial filtration coefficient 0 λ is calculated ( )

1 0 0 50 0 0 50 1 1 1 L L z d dz L d L - -   -   = + =     -     ∫ β β ζ λ λ λ β ( 17
) 0 λ (in cm -1
) is the highest for Robu (0.042), slightly lower for Tobermorite (0.040), and is the weakest for GB medium (0.022). This confirms that the filtration capacity, thanks to additional retention sites available in microporosity, is larger for the DPP media (Robu and Tobermorite respectively) than the SPP medium (GB). Values of the coefficient α (in s -1 ) indicate a rate of the filtration capacity drop with time in the decreasing order GB (-1.5x10 -5 ), Tobermorite (-5.1x10 -6 ) and Robu (-5.1x10 -6 ). Knowing at any time the deposition profile in the medium is very interesting because the evolution of the permeability (not investigated in this paper) depends not only on the mass of deposited particles but especially on the deposition profile in the medium (Alem et al. [START_REF] Alem | Filtration of kaolinite particles in a saturated porous medium: hydrodynamic effects[END_REF]). This result confirms the availability of more capture sites in the DPP medium and the accessibility of SusP to the microporosity of this medium.

Ultrasonic detection of particle deposition

Temporal signals and phase velocities

The study of the acoustic wave transmission through porous media allows tracing the evolution of temporal signals in a bandwidth determined according to the central frequency of the transducer (Fig. 4). The translation of the transducers along the horizontal axis makes it possible 4c and4d).

From these curves, the attenuation of the signal transmitted through the turbid medium (after injection), describing the energy loss caused by the particle deposition, was determined. In order to compare experimental results to the theoretical ones, phase velocities corresponding to the propagation of the first longitudinal wave in the three materials were calculated; their evolutions with the frequency are presented in Fig. 5d. The comparison was made between the theoretical values taken in the domain not far from the central frequency (Fig. 5d) and the experimental ones (Figs. 5a, 5b and 5c) taken at t = 0 (before injection). One observes that the velocity increases with frequency abruptly up to about 5 kHz and remains quasi-constant for GB as well as for Robu. At t = 0, the values are identical to the experimental values with a difference of a few m/s. For Tobermorite, the phase velocity varies exponentially and continues to increase with frequency; its value for the central frequency ( 1635 / V m s ≈

) confirms that obtained experimentally before injection (Fig. 5b).

Energy and wave attenuation

The for Robu and GB (Tobermorite is not presented here for simplification). The values of E ∆ at the entrance are very important compared to other positions. At the outlet, the difference in energy becomes almost zero, this is directly related to the curves of the deposition profiles (Fig. 3b).

Comparing the two media, the evolution of E ∆ with time is more pronounced in Robu because of the presence of micropores which are accessible to the particles. We can, therefore, confirm that particles penetrate in both macropores and micropores.

Another way to highlight the particle deposition influence on the acoustic properties is to consider the attenuation of the transmitted wave based on the well-known Beer's law ( )

1 0 ln / x r D A A - =
where r is the attenuation, D the distance between the two transducers, 0 A the reference signal amplitude taken here before injection at t = 0 and x A the signal amplitude during injection. These amplitudes are determined from the spectrum of the signal obtained by calculating the Fast Fourier Transform (FFT) (Figs. 4c and4d).

The attenuation of ultrasonic waves in porous media is caused by viscous friction due to the relative motion of the fluid to the frame (in the frequency range considered here, the other possible source of attenuation, the scattering due to the contact between the grains, can be neglected) between the fluid and the solid. One presents the attenuation of the first longitudinal wave with time (Figs. 6c and6d) for Robu and GB. There is an increase of the attenuation with time and a drop in space (from pos1 to pos8). The presence of particles in the pores generates a larger differential movement between the solid and the fluid phases, causing an increasing dissipation of energy and thus an augmentation in attenuation with time. In position 1, and as previously explained, a large part of particles that propagate in the medium are trapped in the first centimeters of the medium. This is why a very pronounced increase of attenuation is observed there. From the third position, this variation becomes less important, especially for the first hours of injection which the attenuation is almost constant.

Comparison between ultrasonic and destructive evaluations

In order to find a link between particle deposition and energy variation of the transmitted signal, we represent in Fig. 7, the spatial variation with depth of the deposition and of E ∆ the change occurring for the porosity of the medium as consequence of particle deposition explained by lower ultrasonic velocity and elastic constant of the deposition material compared to the porous medium. Since the retention is very high in the first positions, a decrease in porosity occurs. Consequently, the medium tends to an elastic one. This is why one observes higher values of the velocity after injection (especially in the entrance). The slope of the linear relation between velocity and porosity is a very good indicator of the rate of medium degradability due to deposition. This coefficient is very low for the two DPP media, but it is very high for the SPP medium. As previously underlined, the presence of microporosity in the DPP media provides additional sites for the capture of suspended particles. The decrease of the velocity with porosity was also observed by Kohout [START_REF] Kohout | Ultrasonic Determination of Porosity in Homogeneous Ceramic Samples[END_REF]. In his work, he studied several porous media with different porosities limited to 33%, contrary to our study where we reached porosities up to 70%.

The variation of the acoustic energy of the transmitted signal through the medium is simulated very satisfactorily by the following empirical relation q E p e ∆ ∆ = φ where ∆φ is the variation of the total porosity at time t , and p , q the calibration coefficients that depend on the characteristics of the medium. The coefficient q is an indicator of the degradation rate of the acoustic characteristics by the deposition. It is also a good indicator of the clogging velocity of the porous medium. The values of q are 2.036, 3.023 and 7.930 for Robu, Tobermorite and GB respectively, which means that the clogging in the SPP GB is faster than that in Tobermorite and in Robu. These measurements can also provide information about the evolution of the characteristics of the medium, and thus in this study of clogging. The ultrasonic method for determining the porosity of solid samples appears promising in the zone of the tested porosities.

After proper calibration, it should be possible to use it to determine the porosity of any porous medium. The measurement is rapid and the method is non-destructive.

Conclusions

The clogging phenomenon in porous media is quite complicated by the different processes that describe it. Deposition mainly affects porosity and permeability, which are the most important parameters in porous media, particularly in water retention structures. The increase in Concerning the clogging, despite the uncertainties in the measurements a relation is observed between the destructive evaluation and the non destructive ultrasonic detection based on wave transmission through the medium. The variation of acoustic properties (phase velocity, transmitted signal energy and wave attenuation) with the deposition has been presented. Indeed, the curves of the attenuation and the energy variation are directly related to the deposition profile.

The fast longitudinal velocity estimated by the flight time varies in time and in space as well with the clogging. A linear variation of the velocity with the total porosity obtained after injection is given, which is expected because the velocity depends strongly on the medium density, and therefore on its porosity. We have also observed an exponential dependence of the signal energy variation with the porosity variation due to deposition. From these results, one deduces the sensitivity of acoustic methods to the phenomenon of clogging.

A phenomenon not taken into account in our experiments is that of multiple scattering between beads which would result from the use of high frequencies (we took care to avoid this by choosing fairly low frequencies). During the flow of the turbid solution, some particles can be deposited on the grain surfaces; therefore, the multiple scattering could be influenced by the change in the cross section of the scatters due to the accumulation of the particles on their surfaces. In addition to the attenuation caused by this multiple scattering, another source of attenuation is due to the conversion of fast waves into slow waves in the matrix (seat of the micropores) during the flow (which is no longer of the Poiseuille type) induced by the passage of these waves [START_REF] Müller | A first-order statistical smoothing approximation for the coherent wave field in random porous media[END_REF] for example. All these phenomena gain importance at high frequencies and deserve to be examined closely during more detailed experimental studies. ) can be found in Ref. [START_REF] Berryman | Elastic wave propagation and attenuation in a doubleporosity dual-permeability medium[END_REF]. The mass densities ij ρ depend on the porosities and mass density of the fluid f ρ and solid s ρ but also on the three types of tortuosity: (1 ) τ (micropores), who showed that in wave propagation studies, the permeabilities accounting for the coupling between micropores and macropores can be neglected, resulting in simplified formulas.

Fig. 1 .

 1 Fig. 1. (a) A grain of Robu (size is 3.5 mm), (b) view at MEB. (c) A grain of Tobermorite 11 Å (grain size can vary from 3.15 mm to 4 mm), (d) view at MEB.

1 ≥a 1 F

 11 ) represents the structure factor accounting for the apparent increase in fluid inertia caused by the tortuosity of the pores. is the pore size parameter) accounts for the fact that at higher frequencies, the fluid flow in the pores no longer respects the Poiseuille type flow. At low frequency, ( ) χ = and therefore disappears from the equations. Eq. (3) has two roots: one corresponds to the fast wave having moderate attenuations;

1 νφ.

 1 represents the volume fraction of micropores in the medium. The increments of fluid content are defined by Let τ be the stress tensor and ( ) fluid pressures. The constitutive relations for isotropic DPP media are

A

  generator (Sofranel model 5058 PR) delivers pulses of amplitude 300V and duration 200ns; the oscilloscope synchronization signal is also delivered by the generator. The received signals are digitized, averaged over 200 acquisitions by a Lecroy 9310 AL digital oscilloscope (400 MHz bandwidth) and then stored on a computer linked to the oscilloscope. The column was fed by reservoir containing water (pH of 6.7 ± 0.2) and suspended particles (SusP) (Portland kaolinite particles with average diameter of 11 µm) using a Cole-Parmer Master flex peristaltic pump with flow rate control. The suspension is injected through the porous medium over a longer time at constant flow. Six piezometers are placed on the column as indicated in Fig. 2 to measure the pore pressure variation along the porous medium. At the outlet of the column, the effluent was passed through a continuous flow turbidimeter (Kobold Instruments) to record the turbidity values in terms of Nephelometric Turbidity Units (NTUs). Suspended particles concentrations in the effluent, noted C, were determined with the help of correlations made between measured suspended particles concentrations in the water and values in NTU given by the turbidimeter. The SusP injection experiments were conducted with a constant injection concentration 0 1 / C g l =

Figure 3

 3 Figure 3 shows the experimental breakthrough curves (BTCs) of the injected SusP and the retention profiles in the three porous media. It also shows comparison between experimental and computed data according to the analytical solutions (Eqs. 15 and 16). The BTCs (Fig. 3a) are represented by the relative concentration

Fig. 3 .

 3 Fig. 3. Experimental and simulated (a) breakthrough curves (BTCs), and (b) deposition profiles.

  the median grain size of the porous medium. The empirical parameter β controls the shape of the spatial distribution of retained particles and accounts for the depth dependence of filtration coefficient. The term -β ζ describes depth-dependent retention (a decreasing rate of retention with depth). This term equals 1 for 0 = β and an exponential distribution of retained particles with depth is predicted similar to conventional filtration theory. On the other hand, for 0 > β , the retention profile exhibits an hyperexponential shape with a higher deposition rate close to the porous medium inlet. In this study, 0.432 = β was retained based on Bradford et al. [26]. The term explains filling phenomenon of retention sites and implies a decrease of retention capacity with time. Substituting the second equation of Eq. (

  to record the acoustic signals in eight different positions along the column. The materials are not quite arranged in the same way along the column (manual filling), thus the initial signals received on each position are slightly different, which turns out to be normal considering that the positioning of grains is done randomly. Figs. (4a and 4b) shows the change of the temporal signals amplitude (in Volts) for Robu (the changes for the two other materials are similar) before and after 2 hours and 7 hours of injection (for better readability only signals of positions 1 and 8 are shown). One notes that the signal amplitude decreases after injection, for all positions. The deposition of particles in the medium causes absorption implying energy losses and therefore a decrease in amplitude. However this diminution is less pronounced for position 8 (Fig. 4b), a consequence of the fact that particles are deposited less and less when approaching the outlet of the column. The spectral representation of the impulse responses are given in Figs. (

Fig. 4 .

 4 Fig. 4. Temporal signals of Robu for position 1 (a) and for position 8 (b). Spectral representation of the impulse response for position 1 (c) and for position 8 (d), before and during injection.

Fig. 5 .

 5 Fig. 5. Measured phase velocities of Robu (a), Tobermorite (b) and GB (c) before (t =0 hour) and during injection for four positions. Theoretical phase velocities (d) of Robu and Tobermorite (left axis) and GB (right axis).

2 E.

 2 temporal signals transmitted through the studied materials have been interpreted in order to model the clogging phenomenon. The signals in this case indicate the energy by which the waves have been transmitted through the porous media. A change of these signals after particle deposition is visualized (Fig. 4) and it is explained by a loss of energy transmitted through the medium during injection. From that, one introduces the signal energy which is frequently used in signal processing ( ) x t dt = ∫ where x (in volts) is the realized measurement modulus squared (in the present case, the amplitude of transmitted signal). As previously specified, the signals at each position along the horizontal axis of the model differ slightly because of the manual filling of the material in the column. For this reason, the calculation of the energy has been made with reference to the signal obtained before injection (clean medium). Therefore the evaluation of the deposition corresponds to the difference of the energies before ( b E ) and after ( a E ) Figs. (6a and 6b) show the variation of E ∆ with the x-position

Fig. 6 .

 6 Fig. 6. Variation of energy for transmitted signals: Robu (a) and GB (b). Attenuation profiles for transmitted signals: Robu (c) and GB (d), after particle injection, for four positions.

  for the three materials. The accumulation of particles in the entrance of the column is confirmed by a great change in E ∆ after injection. The higher the change in E ∆ , the smaller the distance from the entrance. Near the outlet of the column, E ∆ continues to vary in the same way as the deposition profile for the three materials, however, from the fourth position approximately (at 20 cm), the variation becomes very slight compared to the entrance. One observes that E ∆ is very low in Tobermorite despite a high deposition (the deposition affects very few its acoustic characteristics).

Fig. 7 .

 7 Fig. 7. Comparison between deposition profiles (left axis) and energy variation profiles (right axis) along the horizontal x axis for the three materials.

Fig. 8 .

 8 Fig. 8. Phase velocity versus total porosity (a) and variation of energy for transmitted signals versus porosity variation (b), for Robu, Tobermorite and GB after particle injections (t = 6 hours).

  deposition with time causes a drop in permeability which can have adverse consequences on the proper functioning of a medium. Several parameters can be involved in the fine particles deposition in a saturated porous medium. The comparison of three different materials (two of them with double porosity and the third one with single porosity) made it possible to deduce the great importance of the medium structure in the particle deposition. The retention capacity of the medium depends closely on the tortuosity and the size (length and diameter) of the micropores (for Tobermorite, despite a very V = -1.0668 φ + 1708.3 R 2 = 0.9397 V = -13.516 φ + 2646.2 R 2 = 0.9169 ΔE = 1x10 -07 exp (7.93 φ ∆ ) R 2 = 0.9294 ΔE = 4 x10 -07 exp (3.0228 φ ∆ ) R 2 = 0.9501 high porosity, this capacity is less important compared to Robu which contains larger and deeper micropores).

;

;

τ

  (macropores) and τ (global). The drag coefficients ij b depend on the fluid viscosity, the porosities and the permeabilities. For the computation of the ij b s, we follow Berryman and Wang

Appendix

The elements ij t ( , 1, 2, 3

) in the dispersion equation, Eq. ( 10), are


Lemée

M Gharbi

C Davoisne

F Le Marrec

L Dupont

N Lemée

Disentangling elastic relaxation and ferroelectric domain contributions to in plane X-ray scattering profile: A necessity in strained ferroelectric superlattices

Keywords: Ferroelectrics, superlattice, thin films, domain structure, X-ray scattering

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Ferroelectric nanostructures are expected to contribute to increase storage densities for ferroelectric memories, and to device miniaturization. For fundamental physics, they constitute fascinating objects to understand ferroelectricity at the nanoscale, since low dimensionality gives rise to new phenomena. In particular as a ferroelectric film becomes thinner, the density of the domains and the domain walls increases and this can play a key role in the functional properties as, for example, in the negative capacitance effect which has recently attracted much attention [1]. Considering a uniformly out of plane polarized material, the polarization discontinuity at the film boundaries creates a depolarizing field that counteracts the polarization. When the screening by free charges is not effective, the stability of the polar phase is preserved by forming a 180° stripe domain of alternate up and down polarization that produces characteristic features in the x-ray scattering pattern [START_REF] Streiffer | Thin Films[END_REF]. Indeed, the phase contrast between the areas of up and down polarization produces a modulation of the structure factor perpendicular to the domain walls, and is at the origin of satellite peaks. This 180° domain structure has thus been evidenced in ferroelectric thin films [START_REF] Streiffer | Thin Films[END_REF]3] as well as in ferroelectric/paraelectric superlattices (SLs) [4 -5]. However, such in-plane modulation can also result from periodic structural defects [6 -7] or ferroelastic domains [8 -9] created to minimize the elastic energy. Here we illustrate this point through the study of a Pb(Zr0.2Ti0.8)O3 / SrTiO3 SL. Transmission electron microscopy (TEM) and x-ray diffraction (XRD) investigations are combined to disentangle the contributions of the elastic relaxation from the ferroelectric domains structure.

Experimental details

The Pb(Zr0.2Ti0.8)O3 / SrTiO3 SL was grown starting with a PZT layer, on a single crystal TiO2 terminated (001) SrTiO3 (STO) substrate, by pulsed laser deposition using a KrF excimer laser (λ = 248 nm) and ceramic targets of SrTiO3 (STO) and Pb1.1(Zr0.2Ti0.8)O3 (PZT 20/80). This SL, with wavelength 4.5 nm, is composed of 10 bilayers, each consisting of 5 unit cells of PZT and 6 of STO, which we designate as [PZT5/STO6]10. The PZT 20/80 and STO layers were deposited at 690 °C in 0.1 mbar of O2, with a laser fluence of 2 J.cm -2 and a pulse rate of 2 Hz. Structural analysis was performed on a high resolution D8 Discover Bruker diffractometer (Cu Kα1 radiation, 1.5406 Å). Temperature measurements were carried out in air from room temperature up to 600°C, using a high temperature XRD-chamber. All measurements were reproducible after thermal cycling. The lattice parameter was determined by tracking satellites peaks around the (002) reflection. Microstructural characterization was performed using a TECNAI F20-S-Twin operating at 200kV equipped with a Scanning Transmission Electron Microscope (STEM) function, and a High Angle Annular Dark Field (HAADF) camera giving an image contrast based on atomic number. The cross section samples were prepared by a focused ion beam (FIB) technique after depositing a carbon/platinum bilayer to protect the surface from sputtering.

Results and discussion

Figure 1 displays the XRD characterizations of a [PZT5/STO6]10 SL. The satellite peaks detected in the θ-2θ scan (Fig. 1a) and in the reciprocal space map (RSM) along the [00L] direction (Fig. 1b), are indicative of chemical modulation along the growth direction. Due to the in-plane lattice mismatch (-1.2%) between the PZT layers and STO substrate, elastic relaxation by surface roughening, plastic relaxation by dislocations or by the formation of ferroelastic domains can occur. Indeed when both elastic and depolarization effects compete with each other, it can lead to a polydomain structure consisting of polarized out of plane c-domains and in plane a-domains [9]. Here, however, the reflections are related only to ferroelectric c-domains. In-plane RSM (Fig. 1c) shows almost coherent growth of the SL to the substrate.

Due to the ferroelectric/paraelectric nature of the [PZT5/STO6]10 SL, a 180° stripe ferroelectric domain structure is expected to minimize the depolarizing field. These nanodomains can give rise to an in-plane satellite peaks detectable in x-ray scattering perpendicular to the domain walls [10]. The diffraction profile along the [H00] direction (Fig. 1d) indicates a complex profile including two types of in-plane modulation. At first sight, the two peaks close to the main Bragg peak could be interpreted as the signal of an in-plane modulation due to the 180° stripe nanodomain structure, corresponding to an average period of 34 nm. Nanodomains are expected to originate from the ferroelectric layers which consist of 5 unit cells. A domain period of 5 nm and 9 nm was respectively reported in a PTO5/STO5 SL [11] and in a tricolor SL [PT9/STO2/PZT10]20 [5],

showing the dependence of the domain period on the thickness of the ferroelectric layer.

The value of 34 nm is therefore well above what could be expected in a 5 unit cell thick PZT layer: the origin of this 34 nm period must arise from another source. In Fig. 3d (also visible in fig. 3c) a second modulation of very low intensity, around h = + /-0.09, characterizes a period of around 4.5 nm, coherent with a modulation due to the polar stripe nanodomain structure. The peaks are weak but the domain randomness contributes to their smearing out. [10]. To further explore the 34 nm periodicity, TEM investigations have been carried out on this SL to determine if it could be produced by an ordered arrangement of structural defects. The images displayed in Fig. 2 confirm the absence of a-domains, as no characteristic domain walls between a-and c-phases are evidenced [9]. In Fig. 2b, the arrangement of alternating white (higher atomic number Z, PZT layers) and grey layers (lower Z, STO layers), confirms the SL periodicity along the growth direction. An almost periodic wave-like modulation with an average period of around 34 nm (+/-8 nm) is also observed and is coherent with the value estimated from the XRD investigation. This modulation is undoubtedly at the origin of the two satellite peaks close to the Bragg peak in Fig. 1d. Such wave-like modulations have already been reported in SLs characterized by a significant mismatch between the materials involved in the heterostructure [12,13]. Indeed, in these structures, strain is relieved through elastic relaxation by surface roughening. All buried layers grown under stress contribute to the surface roughness. Indirect additional evidence for the existence of a polar nanodomain structure in the sample is our determination of the Curie temperature. In Fig 3 we plot the average c-axis lattice parameter as a function of temperature. A decrease of this parameter, due to the reduction of the out-of-plane component of polarization on approaching the cubic/tetragonal phase transition, is observed. The Curie temperature is around 375 °C and thus significantly lower than the reported value of 460°C for bulk PZT. This result is not common in epitaxially grown thin films since strain is reported to push the transition to higher temperature than in bulk. However bulk-like or even lower Tc values have been reported in c-axis oriented thin films [START_REF] Streiffer | Thin Films[END_REF] , [14] and more recently in ferroelectric/paraelectric PT-based SLs [15] , [16] despite epitaxial strain effects. Their common feature is a destabilization of the ferroelectricity by the depolarizing field and the resulting occurrence of 180° domains, at the origin of a lowering Tc which has also been justified theoretically , [17]. This feature is coherent with the detection of nanodomains by XRD.

Conclusion

The ferroelectric/paraelectric SLs constitute flexible model systems to investigate the effects of strain and electrostatics on ferroelectric nanostructures like 180° domain structure. XRD is commonly used to evidence 180° stripe domains through the detection of satellite peaks characteristics of the ferroelectric stripe period. Our XRD and TEM investigations performed on a PZT/STO SL, clearly show that in such samples characterized by a noticeable misfit strain, the detection of diffuse scattering peaks can originate either from ferroelectric 180° nanodomains or from modulation related to elastic relaxation. It is shown that TEM investigations are required to disentangle elastic relaxation and ferroelectric domain contributions to in plane scattering profiles.

Fig. 1 .

 1 Fig. 1. XRD investigations of a [PZT5/STO6]10 SL : (a) θ -2θ XRD : the asterisks indicate the satellite peaks, RSM around the (002) (b) and (103) reflection (c), (d) the diffraction profile along the [00H] direction corresponding to the dotted line on the RSM. The plain and dotted arrows indicate, respectively, ferroelectric domains and elastic relaxation. The RSM is plotted in reciprocal-lattice units with respect to the substrate lattice parameter (3.905 Å).

Fig. 2 .

 2 Fig. 2. (a) TEM micrograph of the PZT/STO SL in which the darker (greyer) layers correspond to PZT (STO) and b) associated STEM-HAADF image which is of opposite contrast to the TEM image.

Fig. 3 .

 3 Fig. 3. Temperature dependence of the average out-of-plane c axis lattice parameter for the ([PZT5/STO6]10) SL. The dotted line marks the Curie temperature of bulk PZT.

Acknowledgements

This work was supported by the region of Hauts de France and the European Fund for Economic and Regional Development FEDER. We thank Prof. M. G. Karkut for useful discussions.