J Holzhauer
email: julia.holzhauer@univ-pau.fr

D Brito

C Bordes

Y Brun

B Guatarbes
email: bertrand.guatarbes@univ-pau.fr

Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand

Keywords: Seismics, Monitoring, Reservoir Geophysics, Acoustics, Electromagnetics

SUMMARY

Under certain circumstances, seismic propagation within porous media may be associated to a conversion of mechanical into electromagnetical energy known as a seismoelectromagnetic phenomemon. The propagation of fast compressional P -waves is more specifically associated to manifestations of a seismoelectric field linked to fluid flow within the pores. The analysis of seismoelectric phenomena, which requires combining the theory of electrokinetics to Biot's theory of poroelasticity, provides us with a transfer function noted E/ü that links the coseismic seismoelectric field E to the seismic acceleration ü. In order to measure the transfer function, we have developed an experimental set-up enabling seismoelectric laboratory observation in unconsolidated quartz sand within the kilohertz range. The investigation focused on the impact of fluid conductivity and water saturation over the coseismic seismoelectric field. During the experiment, special attention was given to the accuracy of electric field measurements. We concluded that, in order to obtain a reliable estimate of the electric field amplitude, the dipole from which the potential differences are measured should be of much smaller length than the wavelength of the propagating seismic field. Time-lapse monitoring of the seismic velocities and seismoelectric transfer functions were performed during imbibition and drainage experiments. In all cases, the quantitative analysis of the seismoelectric transfer function E/ü was in good agreement with theoretical predictions. While investigating saturation variations from the residual water saturation to full saturation, we showed that the E/ü ratio undergoes a switch in polarity at a particular saturation S * , also implying a sign change of the filtration, traducing a reversal of the relative fluid displacement with respect to the frame. This sign change at critical saturation S * stresses a particular behaviour of the poroelastic medium: the dropping of the coseismic electric field to zero traduces the absence of relative pore/fluid displacements representative of a Biot dynamically compatible medium. We concluded from our experimental study in loose sand that measurements of the coseismic seismoelectric coupling may provide information on fluid distribution within the pores, and that the reversal of the seismoelectric field may be used as an indicator of the dynamically compatible state of the medium.

INTRODUCTION

Seismoelectromagnetic effects may be observed when a seismic wave propagates within a porous medium formed from a solid frame filled with a fluid of low to moderate electrical conductivity.Under such conditions, an electrical double layer forms at the mineral-solution interface due to cations adsorption, thus enabling the setting of an electrokinetic coupling. The mechanically driven movements of the electrolyte relatively to the frame consequently result in a charge separation, producing a propagating electromagnetic (EM) field said to be "coseismic". According to the nature of the original seismic excitation, the propagating coseismic field may be overwhelmingly electric or magnetic; P-waves for instance are usually associated to coseismic seismoelectric (SE) fields. In any cases, seismoelectromagnetic manifestations will propagate similarly to the supporting seismic wave and will be affected by the properties -of either hydraulic, mechanical or electrical naturecharacterizing the medium in its close vicinity. When the incident acoustic wave meets a discontinuity affecting the pore space in any geometrical or chemical manner (should it be a change in matrix density or a brutal variation of fluid conductivity), seismoelectromagnetic effects of a second kind, referred to as "interfacial" and propagating at EM-speed, might be generated. Such effects have been reported at field scale [START_REF] Haines | Seismoelectric imaging of shallow targets[END_REF][START_REF] Dupuis | Seismoelectric imaging of the vadose zone of a sand aquifer[END_REF]) and mesoscopic scale [START_REF] Jougnot | Seismoelectric effects due to mesoscopic heterogeneities[END_REF][START_REF] Monachesi | An analytical study of seismoelectric signals produced by 1-D mesoscopic heterogeneities[END_REF][START_REF] Grobbe | Seismo-electromagnetic thin-bed responses: Natural signal enhancements?[END_REF]. Note that as any physical coupling, seismoelectrics has a counter-effect known as electroosmosis, in which the application of an EM field induces a seismic propagation within a porous medium [START_REF] Jouniaux | A review on electrokinetically induced seismo-electrics, electro-seismics, and seismo-magnetics for earth sciences[END_REF]. This phenomenon was first reproduced at field-scale by [START_REF] Thompson | Geophysical applications of electrokinetic conversion[END_REF] before being observed at lab-scale by [START_REF] Zhu | Experimental studies of electrokinetic conversions in fluid-saturated borehole models[END_REF].

The possibility of combining seismic and electric surveys (electric field monitoring during seismic excitation) has been addressed as early as the 1930s. Following observations by [START_REF] Blau | Conductivity dependence of seismoelectric wave phenomena in fluid-saturated sediments[END_REF], [START_REF] Thompson | The seismic electric effect[END_REF] suggested using their combination as an exploration tool (producing what he refers to as the seismic-electric effect). A few years later [START_REF] Ivanov | Effect of electrization of earth layers by elastic waves passing through them[END_REF][START_REF] Ivanov | The seismoelectric effect of the second kind[END_REF], having run similar experiments in the field, suspected the observed signals to result from an electrokinetic coupling effect that he called seismic-electric effect of the second kind in distinction to piezoelectricity. [START_REF] Frenkel | Orientation and rupture of linear macromolecules in dilute solutions under the influence of viscous flow[END_REF], attempted to develop a quantitative theory to explain Ivanov's observations. He formulated the first complete set of equations governing the acoustics of isotropic porous media. Yet it was Biot who, a decade later, achieved the first fully valid theory of poroelasticity. Biot (1956a,b) divided his treatment of the linear theory of porous media acoustics into two distinct frequencydomains, delimited with regard to the validity of Poiseuille flow assumption by introducing the Biot critical frequency fBiot. He stated the existence of one rotational (S -type) and two dilatational (P -type) waves, subsequently referred to as P -f ast and P -slow, the latter being highly dispersive and diffusive. In a later paper [START_REF] Biot | Generalized theory of acoustic propagation in porous dissipative media[END_REF] would emphasize the role this slow wave could play in electrokinetics, as it enhances fluid velocity within the pores. The next theoretical step to couple Biot's theory with electrokinetics while accounting for Onsager's reciprocity was taken by [START_REF] Neev | Electrokinetic effects in fluid-saturated poroelastic media[END_REF]. Yet, in accordance with Frenkel's approach, they did not consider the full set of Maxwell's equation, leading them to ignore seismomagnetic effect. The breakthrough of seismoelectromagnetics was finally enabled by Pride's formulation of the underlying theory.

The theoretical background for seismoelectromagnetic phe-nomena was proposed by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] as a set of equations based on Biot's original theory including the electrokinetic coupling as well as Maxwell's equations. In the light of this complete theory, the first dynamic transfer functions for the coseismic seismoelectric field were proposed a few years later by [START_REF] Pride | Electroseismic wave properties[END_REF] for both transverse and longitudinal waves. Using low-frequency assumptions, [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF] proposed a linear expression linking the coseismic electric field to seismic acceleration of P -waves, applicable in particular to field measurements at seismic frequencies. Lately, seismic interferometry methods using Green functions (Wapenaar et al. 2006;[START_REF] Wapenaar | Green's function representations for seismic interferometry[END_REF][START_REF] Slob | Electromagnetic Green's functions retrieval by cross-correlation and cross-convolution in media with losses[END_REF]) have been adapted to provide the impulsive seismoelectric response of the porous medium equivalent to usual transfer functions. This method was first implemented for interfacial [START_REF] Wapenaar | Seismic and electromagnetic controlled-source interferometry in dissipative media[END_REF][START_REF] De Ridder | Interferometric seismoelectric Green's function representations[END_REF]) and coseismic [START_REF] Schoemaker | Experimental validation of the electrokinetic theory and development of seismoelectric interferometry by crosscorrelation[END_REF]) seismoelectric before being extended by [START_REF] Gao | Seismoelectromagnetic waves radiated by a double couple source in a saturated porous medium[END_REF] to the seismomagnetic aspects.

Pride's theory being originally formulated for a fully-saturated porous medium, its adaptation to partially saturated conditions constitutes an important development offering new perspectives. While Biot's poroelasticity relations may be adjusted by defining effective fluid properties [START_REF] Wood | A textbook of sound[END_REF][START_REF] Teja | Generalized corresponding states method for the viscosities of liquid mixtures[END_REF][START_REF] Brie | Shear sonic interpretation in gas-bearing sands[END_REF], saturation-dependence of electrokinetics is still discussed. It has been the subject of numerous theoretical developments initially proposed for the continuous spontaneous potential, some based on volume-averaging methods [START_REF] Linde | Streaming current generation in two-phase flow conditions[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF]) others on capillary models [START_REF] Jackson | Characterization of multiphase electrokinetic coupling using a bundle of capillary tubes model[END_REF][START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF], the former highlighting the role of the electrolyte wettability while the latter focuses on the thickness of the electrical double layer. Yet, to account for experimental observations, further empirical laws are often required [START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF][START_REF] Allègre | A 1-D modelling of streaming potential dependence on water content during drainage experiment in sand[END_REF]. Eventually, the question of partial saturation has been directly broached under a seismoelectric angle either based on Pride's classical approach [START_REF] Warden | Seismoelectric wave propagation numerical modelling in partially saturated materials[END_REF], or on an alternative reformulation using electrokinetic couplings as a function of charge density [START_REF] Revil | Seismoelectric response of heavy oil reservoirs: theory and numerical modelling[END_REF][START_REF] Revil | Coupled hydromechanical and electromagnetic disturbances in unsaturated porous materials[END_REF]Revil et al. 2013).

The first seismoelectric laboratory experiments dealt with the impact of fluid properties on seismoelectric observations [START_REF] Parkhomenko | A study of the influence of moisture on the magnitude of the seismoelectric effect in sedimentary rocks by a laboratory method[END_REF][START_REF] Parkhomenko | Electrification phenomena in rocks[END_REF][START_REF] Parkhomenko | Borehole and laboratory studies of the seismoelectric effect of the second kind in rocks[END_REF]. These types of laboratory experiments were eventually resumed by [START_REF] Ageeva | E-effect in rocks (data from laboratory experiments)[END_REF] who introduced additional frequency variations. They observed at various frequencies that the ratio of the electric field to the liquid-phase pressure changed directly with water content and residual saturation on the one hand, and inversely with salinity, porosity and permeability on the other. In the meanwhile, an innovative laboratory apparatus of borehole geometry, involving Stoneley waves, enabled to observe coseismic and interfacial seismoelectric signals as well as electroosmosis [START_REF] Zhu | Experimental studies of electrokinetic conversions in fluid-saturated borehole models[END_REF]). On their borehole laboratory model, [START_REF] Zhu | Seismoelectric and seismomagnetic measurements in fractured borehole models[END_REF] finally reported the observation of seismomagnetic coseismic signals using a Hall-effect sensor. Contemporaneously, quantitative measurements led by [START_REF] Bordes | Etude expérimentale des phénomènes transitoires sismo-électromagnétiques: Mise en oeuvre au Laboratoire Souterrain de Rustrel[END_REF] in an underground low-noise laboratory confirmed experimentally the existence of the coseismic seismomagnetic field and its dependence on shear-waves, as predicted by Pride's theory [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluidfilled Fontainebleau sand[END_REF][START_REF] Bordes | Evidence of the theoretically predicted seismo-magnetic conversion[END_REF].

After [START_REF] Zhu | Experimental studies of seismoelectric conversions in fluid-saturated porous media[END_REF] and [START_REF] Wood | A textbook of sound[END_REF] had investigated experimentally the dependence of coseismic electric amplitudes to fluid conductivity, [START_REF] Blau | Conductivity dependence of seismoelectric wave phenomena in fluid-saturated sediments[END_REF] confirmed the expected tendency of electric amplitudes to diminish with increasing conductivity on a sand column experiment. Finally [START_REF] Zhu | Experimental measurements of the streaming potential and seismoelectric conversion in Berea sandstone[END_REF] conducted extensive and quantitative dynamic measurements of seismoelectric coupling coefficients under varying conductivities (0.1 -0.3 S • m -1 ): the seismoelectric coupling would decrease with increasing conductivity and increasing frequency. According to recent extensive measurements by [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] regarding the dependence of the coseismic seismoelectric signal on water saturation within the range [0.3 -0.9], the seismoelectric ratio E/ü showed a rather stable behaviour, despite large saturation changes expected to induce correspondingly large variations of the medium bulk conductivity.

Hence, in spite of its electrokinetic nature, seismoelectric effects have generally been expected to show some aptitudes for hydraulic characterization of porous media [START_REF] Revil | The Seismoelectric Method: Theory and Application[END_REF]. Consequently, as part of the developing geophysical methods, seismoelectrics may eventually become a tool for reservoir characterization combining seismic and electric imaging abilities. Such perspective requires a comprehensive understanding of the underlying phenomena and their dependence to the involved medium parameters, as well as the preliminary development of a reliable quantitative measuring procedure. With this prospect, the goal of the present study is to perform quantitative measurements of the coseismic seismoelectric field for comparison to theoretical predictions, that is the study of the dynamic transfer function E/ü.

Following the fundamental studies by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] and [START_REF] Pride | Electroseismic wave properties[END_REF], developments by [START_REF] Warden | Seismoelectric wave propagation numerical modelling in partially saturated materials[END_REF] and [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] extended the original formulation of the seismoelectric dynamic transfer function, initially written for saturated media, to partially saturated states. This step was taken by introducing the model for electrokinetic coupling dependence on saturation developed by [START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF], along with a mechanical characterization of the biphasic fluid as a bulk effective fluid (the first topic has been discussed by [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF], while the second will be more largely exposed in this article). This led to the relation between the acceleration ü due to fast P -waves, and its associated coseismic electric field E:

E ü (ω, Sw, σ f , . . .) = i ρL ω˜ Hs 2 -ρ Cs 2 -ρ f (1)
where ω is the angular frequency, Sw is the water saturation, σ f is the fluid conductivity, ρ is the effective density, L is the dynamic seismoelectric coupling coefficient, ˜ is the effective electric permittivity, H and C are poroelastic moduli of the medium, s is the slowness of the fast compressional P -wave, ρ is the bulk density of the porous medium and ρ f is the effective fluid density. For the sake of clarity, the parameters dependence as a function of (ω, Sw, . . . ) is not explicitly written on the right hand side of eq. ( 1). Note that the original seismoelectric transfer function and the electrokinetic adjustment to partial saturation resulting in eq. ( 1) were both based on the assumption of an electrical double layer much thinner than the pore section [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF].

As detailed in [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF], we considered the frequency-dependent transfer function E/ü as expressed in eq. ( 1) to depend on several parameters of the medium: some of them are related to the fluid phase, others depend on the solid phase and finally further parameters characterize the frame. Our present purpose is to quantitatively evaluate E/ü in a medium where the parameters can be inferred, either from direct experimental measurements or from theoretical estimations. We have therefore chosen to develop a metric sandbox experiment after initial works by [START_REF] Barrière | Laboratory monitoring of P waves in partially saturated sand[END_REF] and [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]. With its moderate propagation velocity under sub-saturated conditions and a granular matrix easily described by models, unconsolidated quartz sand appears particularly adapted to our purpose [START_REF] Holzhauer | Seismic and seismoelectric monitoring of an unconsolidated porous medium[END_REF]). Yet, from a practical point of view, wavelengths being decimetric in subsaturated sand will grow metric in a fully-saturated medium. The characteristics of the experiment, as well as the physical properties of the sand are described in Part 2.

With all parameters of the medium involved in eq. ( 1) being reasonably known, our quantification attempt of the coseismic seismoelectric transfer function further requires high accuracy in acceleration ü and electric field E measurements at any given point of the sandbox. In this prospect, we have used calibrated accelerometers, and combined their results to those obtained on appropriately reconstructed dipoles which geometry is described in Part 3. Good knowledge of the sand physical parameters coupled to a reliable measurements routine will provides us with a precise estimation of the ratio E/ü for a given set of parameters during an experiment.

In order to validate our experimental approach in Part 4, we compare our measurements to theoretical predictions while varying the fluid conductivity σ f , a key parameter for the amplitude of E/ü, the other parameters remaining roughly constant throughout the experiments. Having concluded to the agreement between measurements and model predictions, we vary a more challenging parameter regarding experimental conditions, namely water saturation Sw. Related imbibition and drainage experiments are reported in Part 5. Part 6 eventually summarizes our results and concludes on our quantitative approach of the dynamic transfer function.

A MONITORED SEISMOELECTRIC EXPERIMENT

Description of the global set-up

We conceived a cubic sandbox from sealed plywood elements; a hole disposed on one side of the box allows the punch of a pneumatic seismic source to pass through. The inside is accessible by the open top. We arranged 3 cm-thick acoustic foam slabs at the bottom as well as on the opposed and lateral walls to the source in order to prevent boundary reflection effects. The dimension of the sandbox is 53 × 50 cm 2 . It is filled with a 50 cm high sand-layer; the emission point of the seismic source is located at the center of a 50 × 50 cm 2 panel. Four injection wells placed at each corner of the sandbox allow for bottom imbibition. These wells are connected to elevated water barrels in an attempt to improve saturation degree by an increased water column. The average quantity of water required for initial imbibition would range by 60 liters. A global view of the experiment is given in Fig. 1a).

[Figure 1 about here.] Since preliminary seismoelectric measurements demonstrated that the coseismic seismoelectric E field would be barely assessable for offsets much larger than 20 cm, all our captors were placed within a distance of 30 cm to the source. By this choice of offsets, we notably differed from previous seismic [START_REF] Barrière | Laboratory monitoring of P waves in partially saturated sand[END_REF] and seismoelectric [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]) laboratory analysis. While enabling high captor density (20 accelerometers within a maximal distance of 30 cm to the source), this receiver distribution required disposing the transducers on two parallel lines. A thin central space, aligned with the source, was left free to host an electrode array of 30 electrodes created in agreement with the specific offset-requirements. On this central line, electrodes were placed between offsets of 3 cm and 30 cm. Accelerometers were implanted on either side of this central line with a centimetric lateral offset as sketched in Fig. 2. Each accelerometer was systematically placed in regard to an electrode (see Fig. 1b) and Fig. 2). The sandbox was further equipped with moisture sensors surrounding the measurement line, with the purpose of recollecting saturation information.

[Figure 2 about here.]

The porous medium

The loose sand we used in our experiment is the same material as the one presented in the studies by [START_REF] Barrière | Laboratory monitoring of P waves in partially saturated sand[END_REF] and [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]: namely pure silica sand (SiO2 content superior to 98%) showing unidisperse granulometry of 250 µm and grain density around 2635 -2660 kg • m -3 . While hydraulic properties (as porosity, formation factor and permeability) were directly measured on small sand probes, other parameters, of electrical or mechanical nature, have been inferred from models or previous studies. The whole set of parameters and properties used in the present study is listed in Table 1 and follows the notations defined in appendices A and B from [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF].

We estimated the zeta potential ζ, a parameter characterizing the electrical double layer, by extrapolating an experimental observation while using a ζ model depending on concentration C0. On the one hand, [START_REF] Pride | Electrokinetic dissipation induced by seismic waves[END_REF] showed the dependence of ζ vs log(C0) to be linear, referring to various laboratory measurements in quartz sands. We then noticed that the slopes obtained from the different datasets were close to an averaged value of about 0.026 mV.l.mol -1 whereas the ζ(0) seemed to be a specific property of the medium. On the other hand, measurements by Nazarova-Cherrière (2014) showed the zeta potential in saturated Landes sand to be ζ = -35 mV (NaCl electrolyte, σ f 8.8 mS.m -1 ). Eventually, we used the relation ζ(C0) = 0.044+0.026 logC0 (in mV ) obtained by combining both model and measurements (see Table 1 for details).

[Table 1 about here.]

Pneumatic seismic source

The home-made pneumatic source was specifically designed to investigate the frequency range for which the medium should be most affected by Biot's losses, corresponding to the vicinity of the Biot frequency:

fBiot = φη f 2πγ0ρ f k0 (2)
estimated at approximately 2 kHz under sub-saturated conditions. In eq. ( 2) φ, γ0, k0 are respectively the porosity, the tortuosity and the permeability of the porous medium, while η f and ρ f stand correspondingly for the viscosity and density of the effective fluid. This source was made of a piston supplied with compressed air for one part, and a fixed frame that maintains the steel hitting plate by four screws equipped with shock-absorbers for another part (see Fig. 1a) and 1b) ). The hitting punch showed by 0.5 cm within the box through the hole arranged on one side of the sandbox to transmit the impulsion from the pneumatic source to the medium. As an example of typical accelerations measured at the hitting plate (usually ranging from 1 km • s -2 up to 5 km • s -2 ), Fig. 3 shows a seismic record, yet only remotely indicative of the energy transmitted to the sand. This figure also demonstrates the excellent reproducibility of the seismic source during a given experiment, for which the stack of 25 consecutive pulses remains extremely close to the signal generated by one single shot. In Fig. 3b) we may also appreciate the wide frequency spectrum content of the pneumatic source around the kilohertz. The resonance frequency, observed at the source in both time and frequency records at approximately 19 kHz, is related to a characteristic frequency of the source system; that high-frequency component is not observed on the accelerometers buried within the sand. The original impulse in Fig. 3, filtered by a low-frequency Butterworth filter of degree 8 and cutoff frequency 25 kHz that eliminates the high frequency component, offers a more realistic view of the signal efficiently propagating within the sand.

[Figure 3 In our attempt to compare coseismic electric data to their corresponding seismic arrivals, special attention was given to spacial precision for measurements of both types. This concern steered the conception of a suitable electrode array, designed to improve precision on electric measurement localisation. This array was assembled from 30 stainless steel rods of equal length, acting as electrodes. In conformity with previous field-scale seismoelectric studies using stainless steel electrodes [START_REF] Beamish | Characteristics of near-surface electrokinetic coupling[END_REF][START_REF] Strahser | Polarisation and slowness of seismoelectric signals: a case study[END_REF][START_REF] Strahser | Dependence of seismoelectric amplitudes on water content[END_REF], we verified that polarisation effects did not affect seismoelectric measurements. Control measurements conducted at similar fluid conductivity and saturation degree within a one-week interval, showed that we did not experience noticeable drift of the electric field, thus validating our choice of electrodes.

In order to keep the terminal connections out of the medium, chosen rods were of 50 cm length. Rigidifying units for the electrode array were kept away from the measurement plane to avoid the apparition of any guided waves. Each rod was then covered with a heat shrink tube on its whole length except for 0.5 cm at each of its ends, one end being the measuring tip while the other authorized the electric potential to be transmitted to the acquisition device.

We decided to record each potential difference relatively to the furthest electrode of the array taken as a common reference (see the difference in electric potential measurement Vi -V ref in Fig. 2); this procedure validated by [START_REF] Bordes | Etude expérimentale des phénomènes transitoires sismo-électromagnétiques: Mise en oeuvre au Laboratoire Souterrain de Rustrel[END_REF], would ultimately facilitate the reconstruction of dipoles of any given length. We took the smallest electrode inter-trace we could manage in order to allow for very small dipoles, the smallest dipole-length being here of 0.9 cm. Finally, the relative position of accelerometers and electrode was anticipated as to match the offset of each accelerometer either with the center or with the tip of an electric dipole.

Accelerometers length being of 1.56 cm, we spaced our accelerometers by 3.6 cm and implanted them on either side of the central line with a lateral offset of 1 cm (see Fig. 2). The electrode array of length 27 cm, was surrounded by 7 water sensors, 2 of them arranged vertically at the middle of the horizontal planes located 10 cm beneath and 10 cm above the measuring plane, while 5 further captors were disposed within the measuring plane at a distance of approximately 10 cm from the measuring line, as indicated in Fig. 2. These captors, acting as capacitance probes, are sensitive to the dielectric permittivity of the surrounding medium on a volume of approximately 12 cm 3 with a precision of ±3 to ±5 %. The sensors have been calibrated for unconsolidated Landes sand; calibration methodology can be found in [START_REF] Barrière | Atténuation et dispersion des ondes P en milieu partiellement saturé : approche expérimentale[END_REF]. Finally, in our experiments, water saturation values Sw were obtained by averaging the synchronous measurements of the 7 water moisture probes.

The acquisition chain

The acquisition chain was composed of a dynamic signal acquisition device providing 32 simultaneous 24-bits analog inputs, 13 channels being dedicated to seismic recording while the remaining 19 channels were reserved for electric measurements. Electric recording procedure was completed by a home-made preamplifier of input impedance 1 GΩ, meant to enable reliable measurements of electric potentials. This preamplifier also applied a high-pass filter cutting the recordings at 10 Hz. Saturation information were recorded thanks to an auxiliary device.

MEASURING ACCURATELY THE COSEISMIC ELECTRIC FIELD

General points and assumptions on the electric field

In coseismic seismoelectric field studies, we are dealing with a transient electric field, propagating concomitantly to its seismic support. In our experiments we consider the usual electric measurement derived from a potential difference between two points, upon the relation E = -∇V where E is the vectorial electric field and V is the electric potential. Thus, ∆Vij is measured between two electrodes ei and ej

as ∆Vij = (Vi -V ref ) -(Vj -V ref ) = Vi -Vj,
forming the so-called dipole, the potential at each point being measured with respect to common reference V ref (see Fig. 2). ∆Vij is then divided by the dipole-length l dip to retrieve the electric field according to the relation:

Eij = -∆Vij/l dip . (3)
Our experiment was precisely designed to investigate the stability of electric field measurements obtained from potential differences. During the experiments, potentials Vi were measured as a function of time at each electrode with respect to the common electrode. While electric acquisitions relatively to a common reference enabled us to reconstruct dipoles of any possible length, supernumerary electrodes offered the chance to translate a dipole of given length relatively to its corresponding accelerometer. In order to identify which dipole geometry was most appropriate as to accurately quantify the electric field at any given point, we checked the characteristics of the measured electric potential difference ∆V at a given point against some characteristics of the inducing seismic field. Due to the coseismic origin of the seismoelectric field (eq. ( 1)), and the linear relation of E to the potential (eq. ( 3)), it is expected that:

• the first signal in electric records should coincide in time with the first arrival in corresponding seismic records;

• the frequency contents characterizing seismic and seismoelectric fields should be relatively close, since dynamic effects are negligible within the kilohertz range [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF];

• the amplitude of the reconstructed electric field at any given point of the medium should ideally not depend on the selected dipole-length.

Effect of dipole geometry on electric potential waveform

The effect of dipole geometry on the measurement of seismoelectric fields has been a pending issue ever since this phenomenon regained attention in the 90's. Various authors [START_REF] Beamish | Characteristics of near-surface electrokinetic coupling[END_REF][START_REF] Strahser | Polarisation and slowness of seismoelectric signals: a case study[END_REF]) investigated, at a given point, what influence the spacial distribution of electrodes may have on the estimate of electric field amplitudes. Regarding our experiment, we investigated the potential differences measured for two distinct dipole geometries as represented in Fig. 4a) and 4b) for the electrode e5.8 cm located at offset 5.8 cm: the electrode array allows to investigate both the effect of dipole-length and dipole geometry relatively to the associated accelerometer located also at offset 5.8 cm. We have distinguished two cases of dipole reconstructions: in Fig. 4a) dipoles involving common mid-point geometry ∆VM (reconstructed symmetrically to the mid-point electrode e5.8 cm) and in Fig. 4b) dipoles involving common first electrode ∆VF (all dipoles having the electrode e5.8 cm as their first electrode). Dipoles associated to ∆VM and ∆VF are then varied in length according to the value of l dip .

Raw measurements of electric potentials ∆VM and ∆VF given by both dipole geometries of varying lengths are compared respectively in Fig. 4c) and 4d) where the electric potentials are represented as a function of time at offset 5.8 cm. For the common mid-point geometry, both the voltage and the time location of the maxima are highly dependent on dipole-length as seen in Fig. 4c). In this aspect, the electric signal behaves as expected since the larger the mid-point dipole we consider, the closer to the seismic source the first electrode will be. Conversely, for the firstelectrode geometry in Fig. 4d), we note that the envelope of the first electric signal evolves significantly in shape and in amplitude only as the dipole-length l dip increases from 0.9 cm to 2.7 cm. For l dip ≥ 2.7 cm the waveform of the first electric arrival remains very similar.

In Fig. 4e) and 4f), we considered the same records as in 4c) and 4d) and compared them to their seismic counterpart ü measured at the middle location of the dipole (common mid-point geometry) or at the first electrode (first-electrode geometry), their locations corresponding both to offset 5.8 cm. In this representation, seismic and electric signals are both normalized to 1 at the location of their respective maximum as to facilitate their comparison. It appears clearly that the best agreement in arrival-time for seismic and electric signals is found for the common mid-point geometry, shown in Fig. 4e). Regarding the frequency content, the shorter the dipole length l dip , the closer the electric waveform will be to that of the seismic first arrival for both common mid-point and first-electrode geometries (see Fig. 4e) and 4f) respectively.)

[Figure 4 about here.]

Role of dipole-length with relation to dominant seismic wavelength

In order to generalize to all offsets the observations performed at offset 5.8 cm in Fig. 4, we focused on a chosen experiment performed at Sw = 0.94 and σ f = 2.5 mS • m -1 from which we systematically picked the first maximum in electric records ∆VM,F for offsets within [3 cm -11.2 cm] and for reconstructed dipole lengths 0.9 cm ≤ l dip ≤ 11.2 cm. We then expressed l dip relatively to a local wavelength value λ = Vp/fnom estimated from the seismic first-arrival velocity VP , and from the central frequency fnom of the first seismic arrival. The averaged wavelengths λ for our sub-saturated experiments is approximately of 17 ± 3 cm.

Results for the mid-point dipole geometry are shown on the left column in Fig. 5a), 5c) and 5e). As previously observed at offset 5.8 cm in Fig. 4c), the maximum of the electric potential ∆VM increases nearly as a function of the dipole length l dip /λ for all offsets. The electric field |EM | in Fig. 5c) derived from ∆VM in Fig. 5a) seems to be relatively constant as a function of l dip /λ for all offsets, meaning eq. ( 3) is well-satisfied. Hence, the mid-point dipole geometry appears suitable to derive a stable electric field at any given point. Finally, in Fig. 5e), we first picked the time for the maximum amplitude of the electric and the seismic signals respectively, and then represented the ratio tmax(∆V )/tmax(ü). Since the electric field is supposed to be coseismic, tmax(∆V )/tmax(ü) should be very close to a value of 1. It appears in Fig. 5e) that if one wants to study a pure coseismic electric field, i.e. electric and seismic time records evolving in phase, then one must favour mid-point electric dipole of length l dip /λ ≤ 1/5.

Distinctly, the same approach on first-electrode geometry in Fig. 5b), 5d) and 5f) leads to different conclusions. ∆VF in Fig. 5b) first increases linearly as a function of l dip /λ before reaching a plateau for l dip ≥ λ/5. In both cases, the derived electric field |EF | in Fig. 5d) strongly varies as a function of l dip /λ, for all considered offsets. Finally, Fig. 5f) demonstrates that the first-electrode geometry is not best suited to obtain an electric field coinciding in time with the seismic field, since tmax(∆V )/tmax(ü) differs noticeably from 1 at any given offset and for any dipole-length.

[Figure 5 about here.]

Definition of the reference dipole geometry

To conclude on the most appropriate dipole for punctual electric field measurement in our experimental set-up, we compiled the electric field derived from the maxima in potential differences obtained over a variety of experiments in the first-electrode configuration with l dip = 0.9 cm (smallest in the experiment) and in the mid-point dipole case with l dip = 1.8 cm (being the smallest of its sort as well). In Fig. 6, the experimental data points remarkably align along the identity slope over two orders of magnitude of electric field variations. It supports the idea that, with regards to the sole amplitude of the electric field at any given point, it is equivalent to use the first-electrode dipole or the common mid-point dipole as long as we remain within the range of very limited dipole-lengths.

[Figure 6 about here.]

We conclude that the best match between seismic and seismoelectric waveforms is obtained for relatively small dipole-lengths of any geometry conforming with l dip ≤ λ/5. Yet characteristic arrival-times in seismic signals are in better agreement with those observed for the smallest mid-point dipoles rather than for the smallest first-electrode dipole. We accordingly choose to consider the common mid-point geometry. As a consequence, for the rest of our investigations the electric field will be determined on the smallest mid-point dipole, i.e. a dipole of length 1.8 cm and of mid-point geometry.

EFFECT OF FLUID CONDUCTIVITY ON TRANSFER FUNCTIONS

A sensitivity analysis performed over the various mechanical, electrical and hydraulic properties of the medium showed that one of the most important effect on the amplitude of the dynamic seismoelectric transfer function presented in eq. ( 1), |E/ü(ω, Sw, σ f , . . .)|, was produced by a change in fluid conductivity [START_REF] Holzhauer | Seismic and seismoelectric monitoring of an unconsolidated porous medium[END_REF]. Transfer functions |E/ü(Sw, ω)| obtained for a saturation of Sw = 0.95 and three different conductivities σ f are shown in Fig. 7 as a function of frequency, all other parameters of Table 1 remaining constant. As expected [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF][START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]) a low-frequency plateau is followed by a gradual decrease of the transfer function in the vicinity of the Biot frequency. Fig. 7 confirms that a slight change in fluid conductivity σ f impacts significantly the amplitude of the transfer function.

[Figure 7 about here.]

The fluid conductivity also happens to be the most adjustable parameter within a sandbox experiment: as a fluid property, its change demands no great operation but to patiently equilibrate the medium towards the wanted conductivity value by continuous water circulation. In the following, we focus on measuring that seismoelectric transfer function during the experiments, in an attempt to characterize the variation of this function as fluid conductivity σ f is changing.

As soon as the 70's, [START_REF] Parkhomenko | Borehole and laboratory studies of the seismoelectric effect of the second kind in rocks[END_REF] noted in their conclusions that "as the degree of mineralization of the solution saturating the rock increases, the magnitude of the E-effect is reduced approximately exponentially" for experiments conducted on partially saturated sand having NaCl conductivities ranging from 47 mS • m -1 to 19 S • m -1 . This effect was particularly brought to light in the low-frequency approximation of the coseismic transfer function given by [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF], which proved the dependence of the coseismic transfer function to be inversely proportional to fluid conductivity. Within the last decade, further similar studies have been conducted either on sand and glass beads [START_REF] Blau | Conductivity dependence of seismoelectric wave phenomena in fluid-saturated sediments[END_REF] or on Berea sandstone [START_REF] Zhu | Experimental measurements of the streaming potential and seismoelectric conversion in Berea sandstone[END_REF] for frequencies reaching some tens of kilohertz.

In our experiment, investigation of the transfer function dependence on fluid conductivity was conducted following initial imbibition. The medium was first equilibrated with demineralized water for a couple of hours, eventually giving the measurement at a lowest value of σ f = 1.7 mS • m -1 . Fluid conductivity was then controlled by progressive addition of NaCl salts to eventually cover fluid conductivities ranging from 2.5 mS • m -1 to 10 mS • m -1 . Throughout this process, the conductivity of the fluid was repeatedly measured with a conductimeter within the four injection wells at the corners of the sandbox; the homogeneity of the fluid conductivity within the box was verified by another measurement at the at the top of the sand layer. In Fig. 8, we present three acquisitions realized within some days after initial imbibition of the medium. Beside stacking, the sole treatment applied to these data consists in electric reconstruction of the 1.8 cm mid-point dipoles. Each electric record in Fig. 8 is compared to its synchronous seismic, both signals being scaled relatively to one another in order to be compared.

From the less to the most saline experiment, water saturation stayed within the 0.95 ± 0.02 range, the P -wave velocity ranging by 165 ± 20 m • s -1 . Some discrepancies remain however between the seismoelectric and seismic velocities, the latter being always slightly higher than the former when determined on first-arrival ba-sis. Yet, the most interesting observation in Fig. 8 is that, while seismic remains mostly invariant in amplitudes throughout the experiments, electric amplitudes decrease drastically, almost by one order of magnitude, as fluid conductivity increases from 1.7 mS •m -1 to 7.7 mS • m -1 . We also note that our experimental values of |E/ü| evolves as expected with respect to the fluid conductivity σ f : as awaited the coseismic ratio decreases as conductivity increases.

[Figure 8 about here.] Fig. 9 presents a comparison between the measured ratio |E/ü(Sw, ω)| and the corresponding theoretical conductivitydependent dynamic transfer function, computed after eq. ( 1). For a given conductivity, experimental |E/ü| values are shown at various offsets together with their corresponding averaged values and standard deviations. Like in [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF], values of |E/ü| vary noticeably as a function of offsets, although these variations are not predicted from the theoretical expression of the coseismic electric field as presented in eq. ( 1). This dispersion of amplitude ratios might therefore be seen as an experimental bias, being eventually used for the estimation of uncertainties by standard deviation.

The frequency signature of the first seismic and electric signals being grossly encompassed within the [0.5 -2 kHz] range, in Fig. 9 we represented the dynamic transfer function for these two bounding frequencies at a saturation Sw = 0.95. Although the error bars on experimental data are quite significant, the agreement between averaged experimental and theoretical points is convincing; in particular, the decreasing trend of |E/ü| vs σ f is wellretrieved from experimental data.

We finally added to Fig. 9 an averaged value of the experimental transfer function measured in [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] at further offsets than in the present study, i.e. at offsets of 20, 30 and 40 cm. These measurements were performed with first-electrode geometry and a value of l dip /λ 0.4 (λ being the typical wavelength defined from the central frequency and apparent velocity of the seismic first arrival in time records). Based on previous study regarding the dipole lengths (Part 3), more specifically focusing on the firstelectrode measurements shown in Fig 5 d), we can infer that the electric amplitude related to the point shown in Fig. 9 taken from [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] could be underestimated by a factor between 2 and 4. Based upon that correction, the point from [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] would match the trend of |E/ü| vs σ f established in Fig. 9 of the present study .

[Figure 9 about here.] As a conclusion, the mid-point geometry, when associated to very short dipoles, provides some |E/ü| measurements very close to theoretical predictions based on Pride's theory generalised to effective medium under partial saturation conditions. Eventually, the chosen dipole geometry seems to enable accurate and quantitative measurements of the transfer functions whatever the fluid conductivity.

THE ROLE OF WATER CONTENT ON COSEISMIC SEISMOELECTRIC TRANSFER FUNCTIONS: A FULL SATURATION RANGE ANALYSIS

Experimental observations

Part 4 having validated the use of eq. ( 1) to estimate the seismoelectric transfer function |E/ü| by varying one influent parameter σ f , we now address its validity with respect to a further parameter of important impact: the water saturation degree Sw. Our study relies on three rounds of experiments during which saturation variations were closely monitored. These three rounds are represented in terms of measured P -wave velocities in Fig. 10. First, we acquired data related to an initial imbibition starting from dry sand: it provided us with seismic properties of the medium under dry and sub-saturated conditions. The medium was then put to a rest for one month, with occasional fluid re-equilibrations by water circulation. The achievement of full saturation launched the second round of experiments, consisting in a monitored drainage process from Sw = 1 to Sw 0.3 over 11 hours. The medium was subsequently submitted to a rapid and poorly documented cycle of imbibition-drainage, not reported here, in a failed attempt to rereach immediate full saturation before the medium had rested long. Eventually, this failed attempt was followed by a third experimental round monitoring re-imbibition with progressive addition of water from the residual water saturation Sw0 0.25 to Sw 0.9.

[Figure 10 about here.] A series of velocity values describing this set of three experiments is to be found in Fig. 10. These velocity values were estimated by linear regression on basis of first-arrival time-picking between offsets 10 to 23 cm. As a striking feature of Fig. 10, we note the hysteretic behaviour of the measured seismic velocities during the three experiments. Indeed, while low-saturation velocities for the drainage and secondary imbibition tend to superpose, they do not converge towards the initial dry sand velocity value preceding first imbibition. Similarly, for higher saturation degrees such as Sw > 0.5, drainage velocity values tend to be greater than those for imbibition at comparable saturation degrees. This type of behaviour has repeatedly been reported in literature, testifying of higher velocities while draining than while imbibing. [START_REF] Walton | The effective elastic moduli of a random packing of spheres[END_REF] and [START_REF] Barrière | Laboratory monitoring of P waves in partially saturated sand[END_REF] associated this phenomenon to a weakening of the frame when injecting fluid during imbibition. Alternatively, [START_REF] Knight | A laboratory study of the dependence of elastic wave velocities on pore scale fluid distribution[END_REF] and [START_REF] Cadoret | Influence of frequency and fluid distribution on elastic wave velocities in partially saturated limestones[END_REF] attributed this discrepancy to a homogeneity loss of the effective fluid (air+water) while drying, in comparison to the homogeneity level experienced during imbibition (often under depressurization). According to them, while fluid and gas can coexist within a pore during the imbibition phase and favour homogeneity of the medium, drainage would rather see that a pore is either filled with or emptied from its water, according to its aspect ratio. Characteristics of the effective fluid were indeed involved under the form of an adaptive effective fluid modulus K f in the calculation of the corresponding velocity models presented as solid and dashed lines in Fig. 10 (respectively standing for the best fits and misfits of 10%). These models are resulting from least-square joint inversion of the VP (Sw) data in question, and their related saturationdependent E/ü ratios (see Fig. 12 to come) in the context of the the partially saturated seismoelectric model presented in eq. ( 1); the models for effective-fluid-modulus description will be further explained in this last section.

The evolution of seismic and electric records with water saturation during the time-lapse monitored drainage, second of the three cycles of experiments, is now presented in Fig. 11. A sign inversion of the coseismic electric field with respect to the seismic acceleration was observed during that drainage. In our experimental data, while the seismic waveform evolves much with saturation, the first arrival remains always negative as recorded by accelerometers and as expected for an initial compression. On the contrary, the electric field appears to reverse its sign during the drainage course, leading to a sign inversion of the experimentally measured E/ü.

Despite large errors due to a poor signal-to-noise ratio and a possible DC shift of the electric field with respect to the zero baseline, we could determine that the sign change happens for Sw 0.6 on observations made at offsets 3.9 cm and 5.8 cm. At further offsets (> 5.8 cm) the limited number of available stacks (from 5 to 25), imposed by the time-lapse nature of the experiment, was unfortunately not sufficient to provide reliable information on the coseismic seismoelectric signals.

[Figure 11 about here.]

In order to gain some insights into the origin of the sign inversion of the dynamic transfer function, we focused on the sign of E/ü at offset 3.9 cm, that offset being the bestdocumented throughout the whole time-lapse monitored experiment. In Fig. 12, we consider all data associated to a saturation information (drainage or imbibition) by representing the E/ü ratio vs water saturation Sw. This compilation encompasses data acquired at various conductivities during initial imbibition, as well as data acquired at constant conductivity σ f = 7.2 mS • m -1 during the drainage phase and a following re-imbibition. In order to be studied relatively to their saturation dependence, E/ü ratios determined for initial imbibition, while varying conductivity σ f , were rescaled into their expected values at 7.2 mS•m -1 by using eq. ( 1).

[Figure 12 about here.]

Determination of the effective fluid moduli and relation to homogeneity degree

Having determined VP vs Sw (Fig. 10 at offset 3.9 cm) and E/ü vs Sw (Fig. 12) during the time-lapse monitored experiments, we intended to understand these measurements in the context of dynamic transfer functions under partial saturation conditions. First, we noticed that the velocity variations shown for the three cycles in Fig. 10, call for a necessary change in the properties of the effective fluid during the experiments, combined to a modification in the solid frame consolidation of the porous medium. These changes involve the incompressibility of the drained solid frame KD and the effective fluid modulus model K f (Sw) .

Concerning the bulk modulus KD, the initial value of 25.5 M P a proposed by [START_REF] Barrière | Laboratory monitoring of P waves in partially saturated sand[END_REF], deduced from [START_REF] Walton | The effective elastic moduli of a random packing of spheres[END_REF] developments on grain-contact theory, is well-adapted to account for initial imbibition, for which low-saturation velocity plateau was estimated at 170 ± 5 m•s -1 , based on measured velocities at extreme saturation degrees Sw = 0 and Sw = [0.9 -0.95]. For drainage and re-imbibition data, this well-monitored velocity has increased to 230 ± 10 m • s -1 . To reproduce such plateau value, bulk modulus KD had to be increased to 50 M P a. A possible explanation to this increase between the initial imbibition and the following cycles could involve a consolidation process of the porous frame as residual water produces surface tension [START_REF] Gallipoli | An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour[END_REF].

Regarding the effective fluid modulus variations with saturation K f (Sw), we considered two models. The first model is the so-called Reuss average [START_REF] Wood | A textbook of sound[END_REF], classically associated to isostress conditions [START_REF] Mavko | The rock physics handbook: Tools for seismic analysis of porous media[END_REF] and hence adapted to a homogeneous effective fluid:

1 K f (Sw) = 1 KR(Sw) = 1 -Sw Kg + Sw Kw (4)
where Kw and Kg are respectively the liquid water-phase and gaseous air-phase elastic moduli (see Table 1). The Reuss model is particularly suited to well-homogenized porous media, for which heterogeneities are small in comparison to the wavelength. The second model, identified as the Brie model, is calculated as a saturation power-law of chosen exponent e [START_REF] Brie | Shear sonic interpretation in gas-bearing sands[END_REF]:

K f (Sw) = KB e(Sw ) = (Kw -Kg)Sw e + Kg (5)
This adaptable empirical law has been used with moderate e exponent (e O(10)) to account for inhomogeneous fluid conditions: in this respect, a parallel to patchy saturation was drawn in [START_REF] Carcione | Physics and seismic modeling for monitoring CO 2 storage[END_REF] and [START_REF] Dvorkin | Identifying patchy saturation from well logs[END_REF], Brie's model offering the advantage of being very straightforward as its implementation requires no preliminary knowledge on patches size and distribution. Interestingly, we note that a Brie model with exponent e = 1 corresponds to the Voigt arithmetic average [START_REF] Voigt | Textbook of crystal physics[END_REF], particularly suited to characterize isostrain conditions [START_REF] Mavko | The rock physics handbook: Tools for seismic analysis of porous media[END_REF]. It defines a upper bound for modulus of the multiphasic fluids, despite being poorly adapted to their description. High values of the e exponent on the other hand (e O( 100)) are bestsuited to model homogeneous medium. Note that as e increases towards higher values, KB e(Sw ) comes closer to the Reuss model KR(Sw) before eventually surpassing it.

We have tested Pride's model for velocity and seismoelectric transfer function using the Reuss definition for effective fluid modulus KR(Sw) coupled with the properties listed in Table 1. The results, mapped as black plain lines in Fig. 10 and Fig. 12, reveals that the cycle which is better described as an homogeneous effective fluid of Reuss type is the initial imbibition represented in both figures as blue data points. Using KD = 2.5 × 10 7 P a (see Table 1), the Reuss model offers a fair estimate of our experimental velocities [START_REF] Barrière | Laboratory monitoring of P waves in partially saturated sand[END_REF]). In Fig. 12 however, the E/ü data related to the first imbibition cycle are not that well-explained by the Reuss model.

In an attempt to gain more information on the homogeneity degree of the medium during these time-lapse monitored experiments realized in three periods, we inverted jointly seismic velocities VP and local estimates of the coseismic seismoelectric transfer function E/ü respectively presented in Fig. 10 and Fig. 12. This least-square inversion was led in the theoretical context of the partial saturation model evoked in eq. ( 1). The inversion was run taking the Brie exponent e as the only free parameter, all further physical properties being given in Table 1. Results of the least-square inversion are presented in Fig. 10 and Fig. 12 for each monitored cycle. In those figures, each best-inverted solution (plain line) is bracketed by a couple of functions (dashed lines) giving solutions with a 10% error in the misfit function compared to the best solution. The inverted models, although failing in following precisely the data, seem to qualitatively account for the three types of behaviour encountered in VP and E/ü during the three cycles. As a result, the inverted e exponent appeared much more sensitive to the E/ü data rather than to acceleration measurements; that high sensitivity to electric measurement is explained by the high variability of the E/ü function vs Sw with respect to the variable exponent e, whereas the change of the VP function vs Sw with e are less important and consequently less discriminant regarding the inversion process.

The inversion of first imbibition data confirms a high homogeneity level of the medium during initial imbibition since the inverted exponent is e = 61 ± 11. It is important however to stress that this inversion is rather poorly constrained given that the E/ü values were only measured at very high saturation Sw. A Brie exponent of e = 9 ± 2 was obtained for the inversion regarding the drainage, that case being the most-constrained with the maximum of data in Fig. 10 and Fig. 12, resulting in a very small variation of the inverted exponent. The quite low value of the inverted e exponent reveals a relatively heterogeneous medium during drainage. The Brie model for the effective fluid modulus K f = KB9 fits the E/ü experimental data particularly well, inclusive of the sign inversion occurring around Sw 0.6. The inversion of the second imbibition data lead to an exponent e = 32 ± 5, intermediate between the initial imbibition and the drainage, traducing an homogeneity increase of the effective fluid [START_REF] Knight | A laboratory study of the dependence of elastic wave velocities on pore scale fluid distribution[END_REF][START_REF] Cadoret | Influence of frequency and fluid distribution on elastic wave velocities in partially saturated limestones[END_REF], yet not as good as when starting from dry material. This might be due to the trapping of air bubbles within small pores during the refilling process. For this case corresponding to e = 32, the change in sign would be expected for a saturation degree approaching 0.9. Though we might identify an onset of this sign change when looking at experimental points over 0.8 saturation, we were never able to reach the saturation break point, despite how long we waited and how often we put the fluid to circulate.

Investigation on the origin of the sign change in E/ü

Finally, the sign change of the transfer function visible in Fig. 12 was fully experienced only once during the drainage phase, while initial imbibition resulting in Sw > 0.95 offered strictly negative ratios and secondary imbibition jamming at Sw = 0.88 gave overwhelmingly positive ratios. We have investigated the origin of the sign change by considering eq. ( 1) as a function of Sw. This equation can also be rewritten as:

E ü (ω, Sw, σ f , . . .) = ρL iω˜ β = - ρL iω˜ H(Sw)s 2 (Sw, ω) -ρ(Sw) C(Sw)s 2 (Sw, ω) -ρ f (Sw) , (6)
where the new β-term is a mechanical coupling related to the fluid/matrix displacement ratio [START_REF] Pride | Electroseismic wave properties[END_REF]. That βterm of sheer mechanical origin is the only term to change sign as a function of saturation in eq. ( 6). This change of sign occurs at a particular saturation S * where, from eq. ( 6),

H(S * )s 2 (S * , ω) -ρ(S * ) = 0. (7)
It means that the phase velocity of the fast P -wave should be:

VP (S * ) = 1/s(S * ) = H(S * ) ρ(S * ) , (8)
an expression valid at all frequencies, no frequency-dependence being observed. Since β = 0 at S * , there should be no fluid/matrix relative motion, and the energy dissipation of the "Biot" type induced by macroscopic fluid flow should vanish. In Fig. 13a), we computed the inverse of the seismic quality factor Q -1 (Sw) with regard to the Reuss and Brie models for effective fluid modulus K f (Sw). As expected, for all curves Q -1 is put to zero at critical saturation S * , thus confirming that dissipation and attenuation effects vanish at that particular point. The eventuality of such manifestations had been theoretically addressed by [START_REF] Hu | Attenuation and seismoelectric characteristics of dynamically compatible porous media[END_REF] in the context of seismoelectrics. In their parametrical study, they considered a fully saturated porous medium of varying porosity and identified a porosity degree for which no mechanical losses were expected. That particular fluid/solid association was acknowledged as a dynamically compatible medium, as defined by Biot (1956a).

[Figure 13 about here.]

To complete our investigation on the origin of the sign change, we studied the relation between fluid and frame displacements (respectively u f and u) within the porous medium. As defined by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], the filtration w = φ (u f -u) characterizes the relative motion between phases and is related to the frame displacement by w = -βu. Hence, the fluid and frame component of the displacements in the direction of the seismic propagation are linked by the relation:

u f u = 1 - β φ . (9)
Fig. 13b) displays the u f /u ratio vs Sw for the same models as presented in Fig. 13a). Not surprisingly, u f /u reaches unity at critical saturation S * . At this point, u f is strictly equal to u and the filtration disappears, the frame being displaced in phase with the fluid as the seismic wave propagates: neither attenuation, dispersion nor seismoelectromagnetic coupling can occur. For Sw < S * , the fluid displacement is greater than the frame displacement induced by seismic wave propagation (u f /u > 1 and w > 0); for Sw > S * instead, the absolute fluid displacement is shorter than that of the frame (u f /u < 1 and w < 0). As the sign of the filtration term w determines the sign of the potential difference ∆V , it ultimately governs the sign of the dynamic transfer function E/ü.

Although the singular value Sw = S * is observed distinctly through an electrokinetic measurement, the origin of the filtration reversal is to be found in mechanical properties. When we consider partially saturated sand, mechanical properties change indeed dramatically with Sw. In such medium, KD KS and the undrained modulus can be reasonably approximated by [START_REF] Pride | Relationships between seismic and hydrological properties[END_REF]:

KU (Sw) KD + K f (Sw) φ (10)
Hence, the P -wave modulus can be expressed as:

H(Sw) = KU (Sw) + 4 3 G (KD + 4 3 G) + K f (Sw) φ . (11)
From this equation, we define two P -wave moduli H f r and H f , respectively associated to the frame and fluid contributions:

H f r = KD + 4 3 G and H f (Sw) = K f (Sw) φ . (12)
H, H f r and H f (Sw) are computed vs Sw in Fig. 13c) for only two of the models presented in Fig. 13a) and 13b), with the purpose of clarity. For the lowest saturation degrees, H is dominated by H f r since the saturating fluid is mostly a very compressible gas. When saturation progressively increases, the modulus H f increases as well, following the rapid evolution of K f (Sw) expressed by the Brie effective models in Fig. 13c). The critical saturations S * , denoted as stars in Fig. 13, are always reached in close vicinity to a saturation degree where H f r and H f equally contribute to H. For larger saturation degrees, the P -wave modulus H is dominated by the fluid contribution H f .

Results

Our analysis involving poromechanical moduli gives conclusive evidences about the origin of the polarity shift observed in seismoelectric transfer functions: the shift occurs at a critical saturation degree S * marking a transition between two different mechanical regimes. The first regime, corresponding to relatively low water saturations, is characterized by a dominant incompressibility of the frame, implying a bigger displacement of the fluid with respect to the frame (w > 0) as the seismic wave propagates. Conversely, at relatively larger saturation degrees, the incompressibility of the fluid overtakes that of the frame, causing the frame displacement to become larger than that of the fluid (w < 0) as the seismic wavefront passes.

It is of importance to note that this change in polarity E/ü is very specific to unconsolidated porous media filled with gaswater mixture, combining a highly variable effective fluid modulus K f (Sw) with a very low frame modulus such as KD KS. It would be hardly observable in a consolidated medium such as sandstone for which fluid and frame moduli monotonously verify K f (Sw)

KD. Nor would it occur in unconsolidated sand filled by an effective fluid such as an oil-water mixture, for which the fluid modulus would remain relatively stable while Sw varies. Despite these restrictions, our study offers an experimental evidence for what has been defined by Biot (1956a) as the "dynamic compatibility condition", which has long been considered as a theoretical object of study [START_REF] Burridge | The fundamental solution in dynamic poroelasticity[END_REF][START_REF] Simon | An analytical solution for the transient response of saturated porous elastic solids[END_REF][START_REF] Mesgouez | Transient mechanical wave propagation in semi-infinite porous media using a finite element approach[END_REF]), yet had, to our knowledge, never been directly observed. Further questions remain as how to find a theoretical explanation unifying our observations with the projections from [START_REF] Hu | Attenuation and seismoelectric characteristics of dynamically compatible porous media[END_REF], the first requiring partial saturation while the other assumed a fully saturated media. The answer stands possibly in relation to the thermodynamics of capillary pressure in porous media as developed by Wei & Muraleetharan (2002a,b).

The purpose of our study was to achieve high spatial and temporal resolution while measuring the seismoelectric coupling on a medium submitted to important changes in fluid conductivity and saturation, up to saturation completion. Special attention was given to acceleration and electric potential measurements in order to derive accurate estimations of the dynamic transfer function E/ü. Experimental measurements, performed at frequencies in the kilohertz range, were compared to the theoretical framework for coseismic seismoelectric established by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] and [START_REF] Pride | Electroseismic wave properties[END_REF] and extended to partial saturation conditions by [START_REF] Warden | Seismoelectric wave propagation numerical modelling in partially saturated materials[END_REF] and [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF].

The high-resolution electrode array placed in our sandbox led to convincing conclusions regarding the relevant dipole length l dip for electric field measurements. We clearly showed that l dip should necessarily be of maximum length λ/5, λ being the wavelength of the propagating seismic wave. Should this condition not be satisfied, the value of the local electric field derived as -∆V /l dip may be underestimated, thus impacting the transfer function determination. We also demonstrated that the best-suited dipole geometry for coseismic transfer function estimation should be centered on the point where the corresponding seismic acceleration is measured. A direct comparison of seismic and electric field waveforms showed that, for first-electrode dipole geometry, we may observe a tradeoff between seismic and electric arrival-times, although not altering the amplitudes of the electric voltage with respect to the seismic field. While these experimental results on dipole geometry are rather conclusive, a numerical study would be particularly relevant to test other possible geometries of electrodes (e.g. multipoles) at laboratory and field scales.

From the theory, we expected fluid conductivity σ f to have a strong impact on the amplitude of the seismoelectric transfer function. Therefore we performed a set of experiments under varying conductivities while all other parameters, in particular saturation degree Sw, remained fixed. We thus checked a well-established result that is the decrease of the seismoelectric transfer function E/ü as fluid conductivity increases. Quantitatively, the experimental points measured at saturation degrees close to Sw 0.95, for various conductivities, were quite remarkably predicted by the coseismic seismoelectric model at partial saturation involving the saturation-dependent electrokinetic coupling model of [START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF]. That positive agreement between experimental data and theory legitimated our further use of the transfer function adapted to partially saturated conditions under an effective fluid approach.

Saturation degree Sw being another key parameter, we monitored the same experimental set-up under varying water content and performed a full saturation range monitoring over a couple of imbibition and drainage sequences. The time-picking of seismic first arrivals led to hysteretic observations classical for unconsolidated media: seismic velocity values VP from imbibition and drainage do not superpose, an effect we attributed to changing mechanical properties of the partially saturated sand. During those imbibitiondrainage cycles, we simultaneously compiled the measurements of E/ü vs Sw taken at the closest offset to the source. We then proceeded to the joint inversion of the saturation-dependent VP and E/ü values in the least-square sense on basis of eq. ( 1). The adjustable variable during this inversion was Brie's effective fluid modulus KB e through its exponent e, as it traduces the degree of homogeneity of the multiphasic fluid within the porous medium. The VP and E/ü measurements were satisfactorily explained by the inversion, yet the inverted coefficient e seemed to be much bet-ter constrained by the electric data than by the velocity measurements. The first imbibition, starting from dry sand, appeared to be achieved under highly homogeneous fluid distribution as attested by the fitting effective modulus (high e exponent). On the contrary, the drainage revealed to be quite heterogeneous (low e exponent), possibly indicating preferred paths and patches during the draining process. Eventually, the following imbibition testified from a more homogeneous medium as compared to the drainage.

We reported a peculiar observation during the drainage phase: for a saturation value S * close to Sw 0.5, the function E/ü experienced a sign change, also predicted by our calculations. Searching for the origin of that event, we concluded that it arises from a purely mechanical cause rather than from an electrokinetic phenomenon. At this critical saturation degree S * , we showed that the P -wave modulus H is equally supported by the frame and the fluid phase, implying that frame displacement equals fluid displacement as the seismic wave goes through, causing the seismic attenuation of the "Biot" type and the coseismic seimoelectric field to vanish in the absence of filtration. This effect long known as "Biot's dynamic compatibility" has often been considered as an hypothetical object of study: the present saturation-dependent analysis in partially saturated sand constitutes a very original observation of this phenomenon. Further computations predicted that for a given material, critical saturation S * would change according to imbibition and drainage phases, hence participating to the observed hysteresis in link with its connection to fluid homogeneity issues.

Considering that the shift in the polarity of E/ü coincides with a non-attenuated seismic wave, we expect the monitoring of both the seismoelectric field and the seismic attenuation to be capable of detecting critical saturation S * . Interestingly, the coseismic seismoelectric signal may provide better access to fluid distribution than seismic attenuation does, and that for two reasons. First, the seismoelectric analysis requires simple time-picking to monitor the polarity of first arrivals, while attenuation calculation needs further assumptions on propagation geometry and geometrical spreading. Second, the change in coseismic seismoelectric signals is more marked (sign change) than that observed in seismic attenuation data (gradual decrease and increase with no sign change), hence facilitating its observability.

Finally, this study shows that propagating coseismic seismoelectric fields may be accurately measured by potential differences and might be strongly influenced by fluid heterogeneity. On this last topic, experimental apparatus comparable to our sandbox experiment could be more systematically used to gain further datasets with broader scope, as to investigate the effect of different heterogeneities types. The achievement of a joint spectral analysis on both seismic and seismoelectric fields would constitute a strong improvement towards the dynamic interpretation of seismoelectric measurements. However, further experimental studies demand additional theoretical and numerical developments, including a better understanding of the role patchy saturation may play in seismoelectric couplings [START_REF] Müller | Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks -A review[END_REF][START_REF] Rubino | Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks[END_REF][START_REF] Dupuy | Influence of frequency and saturation on avo attributes for patchy saturated rocks[END_REF][START_REF] Jougnot | Seismoelectric effects due to mesoscopic heterogeneities[END_REF].

ACKNOWLEDGMENTS

JH thanks the French Ministry of Higher Education and Research for providing a PhD funding (2011)(2012)(2013)(2014). The authors thank Total for providing an additional grant during the PhD thesis. This study has also benefited from the ANR programme of the French Government, especially via the TRANSEK project. The authors thank S. Garambois (ISTerre, France), J. Cresson (LMAP, Pau), V. Poydenot and P. Sénéchal (LFC-R, Pau) for their constructive discussions.

LIST OF FIGURES

1

Pictures of the experimental set-up. a) Global view of the sandbox with pneumatic source (hold by the red frame). b) Detail of the pneumatic source. The piston on the right hand side is moved forward towards the hitting plate, where a transducer, in the upper left, records the acceleration. c) Partial emptying of the sandbox after a measuring campaign: half of the measuring plane is covered in a 25 cm thick sand-layer while the other half is visible. In particular, one sees the vertical electrodes measuring the electric field in the same horizontal plane where the accelerometers, also visible though buried in the sand, measure the seismic field while the other half is covered by a 25 cm thick sand-layer 2 Schematic view of the sandbox from above. The hitting-plate of the pneumatic source is pictured in dark gray at the right side of the sandbox. The electrode array, in line with the hitting plate, is shown as a series of 30 aligned blue circles, saving the yellow colour for the common reference electrode. The electric potential at position i is measured with respect to the common reference as Vi -V ref . The 20 accelerometers disposed on two parallel lines are visible as small rectangles. üi is the acceleration measurement at position i. A last accelerometer is placed on the punch for direct recording of the source acceleration. The experimental set-up also includes 4 injections wells pictured as large circles in the corners of the sandbox as well as 7 water sensors, from which 5 lay within the measuring plane sketched as blue rectangles and further 2 belong to the vertical plane (not seen here).

3 a) Acceleration measurements ü vs time recorded at the hitting plate during a series of shots by the home-made pneumatic source. The 25 superimposed shots were obtained for Sw = 0.95 and σ f = 2.5 mS • m -1 . In red, the averaged signal in time filtered by a low pass Butterworth filter of degree 8 and cutoff frequency 25 kHz. b) In black, the corresponding 25 spectra associated to the signals shown in a). In red, the averaged spectrum subjected to the same filtering process as in a).

4

a) and b): Schematic representations of electric and seismic acquisition arrays in the experimental sandbox, for two geometries of dipole reconstruction at offset 5.8 cm. Each electrode rod is sketched as a blue circle. In a), electric potential differences ∆V are reconstructed from dipoles of varying dipole-length l dip , all centered on e5.8 cm. These dipoles representing the mid-point geometry are labeled ∆VM . In b), dipoles share e5.8 cm as a common first electrode and are labeled ∆VF ; the last electrode in yellow is the common reference electrode. c) and d): Experimental dataset of potential differences ∆VM and ∆VF at offset 5.8 cm measured vs time, for varying dipole length l dip . e) and f): Same experimental electric dataset as in c) and d) normalized in order to compare electric potentials ∆VM and ∆VF with respect to seismic acceleration ü recorded at the same offset 5.8 cm. Corresponding colours in thick lines traduce identical dipole lengths in electric data through a) to f).

6

Electric field amplitudes |EM | calculated on mid-point dipoles of length l dip = 1.8 cm vs electric field amplitudes values |EF | computed on first-electrode dipole of l dip = 0.9 cm length, at corresponding offsets. These data come from a variety of experiments performed at various saturation rates Sw and various fluid conductivity σ f : each symbol in the legend corresponds to a couple of parameters (Sw, σ f ) whereas the colour of the symbol gives the offset where the electric field has been measured. The dashed line represents the identity function.

7

Modulus of the transfer function |E/ü| as a function of frequency f computed from eq. ( 1), according to the medium parameters given in Table 1. The transfer function is computed for three various fluid conductivities at full saturation Sw = 1. The Biot frequency is shown as a red dotted line.

8

Acceleration ü (a to c) and electric field EM , (d to f) data, averaged on about 50 shots, obtained at various offsets for experiments performed under different fluid conductivities σ f , and represented vs time. Corresponding colours in thick lines traduce corresponding offsets in seismic and seismoelectric figures. Relative amplitudes are preserved as to give 50 m • s -2 per vertical division for the seismic field and 0.5 V • m -1 per division for the electric field. For the sake of data readability, electric curves were flipped in polarity.

9

Experimental and theoretical values |E/ü| represented vs fluid conductivity σ f . Experimental values of |E/ü| are shown with coloured dots for various offsets at a given fluid conductivity σ f . The averaged values and their standard deviation are represented as black crosses. Error bars on conductivity were estimated to be of the order 0.5 mS •m -1 . The dashed and continuous lines correspond to the dynamic seismoelectric transfer function from eq. ( 1), respectively computed for (Sw = 0.95, f = 0.5 kHz) and (Sw = 0.95, f = 2 kHz), all further physical properties matching those. An additional data point displayed at a conductivity value of 11.7 mS • m -1 was obtained from averaging experimental values taken from [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]. 10 P -wave velocities VP deduced from time-picking of the first seismic arrival during time-lapse monitored experiments with varying saturation Sw. The measurements were performed during initial imbibition, subsequent drainage and following reimbibition. Velocity time-picking was performed between offsets 10 to 23 cm. Error bars in velocity amplitudes were calculated from a linear regression of the time-picking vs offsets on a 80% confidence interval. The velocity models are represented by solid and dashed lines matching the colour of the corresponding experimental points. They were jointly obtained by least-square inversion on these VP data and on the E/ü data from Fig. 12, while adapting the effective fluid modulus K f . All the velocity models are computed at a frequency f = 1.5 kHz.

11 Evolution of seismic (blue) and electric (red) signals vs time as a function of changes in water saturation Sw, for a set of experimental data acquired during drainage. Note that while water saturation is determined by the origin ordinate of each seismic curve, the vertical divisions on this same axis enable to get back to the signals amplitudes. The scale is of 10 m • s -2 per vertical division for the seismic field, and 0.025 V • m -1 per vertical division for the electric field. The origin on the time axis coincides with the initial punch of the pneumatic source. Seismic signals were taken for the first and second receivers at respective offsets a) 4.8 cm and b) 5.8 cm; electric signals were was taken for the mid-point dipole of length 1.8 cm placed at offset a) 3.9 cm and b) 5.8 cm. Seismic signals in blue result from a stack 25. Electric signals in bold, dashed and light lines result respectively from stacks 50, 25 and 5. 12 Experimental values of local E/ü ratio estimated during time-lapse monitored experiments with varying saturation Sw. The measurements -performed during initial imbibition, subsequent drainage and following re-imbibition -are represented for a fluid conductivity of 7.2 mS • m -1 . We considered electric data acquired at offset 3.9 cm, combined to seismic amplitudes extrapolated to a corresponding offset. Systematic error bars on the amplitude of E/ü are of the order of 1 V •s 2 •m -2 based on the uncertainties in electric and seismic pickings. The horizontal error bars on saturation are computed from the difference between the minimum and maximum values of the recorded saturation by the capacitance probes. The estimates of the transfer function are represented by solid and dashed lines matching the colour of the corresponding experimental points. They were jointly obtained by least-square inversion on these E/ü ratios and on the associated VP data from Fig. 10, while adapting the effective fluid modulus K f . Models based on an effective fluid modulus of the Brie type with exponent e are noted KB e, while KR points at an effective fluid modulus based on the Reuss average. Since the dynamic transfer functions eq. ( 1) is complex, the theoretical functions in the present figure are drawn as sign(real(E/ü))× | E/ü |. All model predictions are computed at a frequency f = 1.5 kHz. 13 a) Inverse of the seismic quality factor Q -1 vs Sw, computed from eq. ( 1) using properties given in Table 1. Curves are obtained for four distinct estimations of the effective fluid modulus K f (Sw), as KR and KB e with e = [9,32,61]. For each case a star, matching the curve in colour, gives the exact location of the critical saturation S * for which attenuation vanishes. b) Ratio of the fluid over frame displacements u f / u vs saturation Sw. Filtration velocity w is positive when u f /u > 1 and negative when u f /u < 1. The displacements are computed for the same models as in a); note that corresponding critical saturations S * coincide with u f /u = 1 for which there is no filtration. c) H(Sw), H f r and H f (Sw) vs Sw computed for two distinct models of effective fluid modulus KB 9 and KB 61. Corresponding stars give the exact location of critical saturation S * .

V i -V ref Figure 2
. Schematic view of the sandbox from above. The hitting-plate of the pneumatic source is pictured in dark gray at the right side of the sandbox. The electrode array, in line with the hitting plate, is shown as a series of 30 aligned blue circles, saving the yellow colour for the common reference electrode. The electric potential at position i is measured with respect to the common reference as V i -V ref . The 20 accelerometers disposed on two parallel lines are visible as small red rectangles. üi is the acceleration measurement at position i. A last accelerometer is placed on the punch for direct recording of the source acceleration. The experimental set-up also includes 4 injections wells pictured as large circles in the corners of the sandbox as well as 7 water sensors, from which 5 lay within the measuring plane sketched as blue rectangles and further 2 belong to the vertical plane (not seen here). In a), electric potential differences ∆V are reconstructed from dipoles of varying dipole-length l dip , all centered on e 5.8 cm . These dipoles representing the mid-point geometry are labeled ∆V M . In b), dipoles share e 5.8 cm as a common first electrode and are labeled ∆V F ; the last electrode in yellow is the common reference electrode. c) and d): Experimental dataset of potential differences ∆V M and ∆V F at offset 5.8 cm measured vs time, for varying dipole length l dip . e) and f): Same experimental electric dataset as in c) and d) normalized in order to compare electric potentials ∆V M and ∆V F with respect to seismic acceleration ü recorded at the same offset 5.8 cm. Corresponding colours in thick lines traduce identical dipole lengths in electric data through a) to f). 1), according to the medium parameters given in Table 1. The transfer function is computed for three various fluid conductivities at full saturation Sw = 1. The Biot frequency is shown as a red dotted line.

f (H z) |E/ü| (V • s 2 • m -2 ) σ f = 2 mS/m σ f = 5 mS/m σ f = 8 mS/m f B iot
σ f (mS • m -1 ) | E/ü | (V 2 .s 2 .m -2
) 4.8 cm 5.8 cm 7.6 cm 8.6 cm 10.4 cm 11.4 cm The measurements were performed during initial imbibition, subsequent drainage and following re-imbibition. Velocity time-picking was performed between offsets 10 to 23 cm. Error bars in velocity amplitudes were calculated from a linear regression of the time-picking vs offsets on a 80% confidence interval. The velocity models are represented by solid and dashed lines matching the colour of the corresponding experimental points. They were jointly obtained by least-square inversion on these V P data and on the E/ü data from Fig. 12, while adapting the effective fluid modulus K f . All the velocity models are computed at a frequency f = 1.5 kHz.

f = K R K f = K B 61 K f = K B 9 K f = K B 32
S w E/ü (V • s 2 • m -2 ) First imbibition Drainage Second imbibition K f = K R K f = K B 61 K f = K B 9 K f = K B 32 Figure 12
. Experimental values of local E/ü ratio estimated during time-lapse monitored experiments with varying saturation Sw. The measurementsperformed during initial imbibition, subsequent drainage and following re-imbibition -are represented for a fluid conductivity of 7.2 mS •m -1 . We considered electric data acquired at offset 3.9 cm, combined to seismic amplitudes extrapolated to a corresponding offset. Systematic error bars on the amplitude of E/ü are of the order of 1 V • s 2 • m -2 based on the uncertainties in electric and seismic pickings. The horizontal error bars on saturation are computed from the difference between the minimum and maximum values of the recorded saturation by the capacitance probes. The estimates of the transfer function are represented by solid and dashed lines matching the colour of the corresponding experimental points. They were jointly obtained by least-square inversion on these E/ü ratios and on the associated V P data from Fig. 10, while adapting the effective fluid modulus K f . Models based on an effective fluid modulus of the Brie type with exponent e are noted K B e , while K R points at an effective fluid modulus based on the Reuss average. Since the dynamic transfer functions eq. ( 1 1) using properties given in Table 1. Curves are obtained for four distinct estimations of the effective fluid modulus K f (Sw), as K R and K B e with e = [9,32,61]. For each case a star, matching the curve in colour, gives the exact location of the critical saturation S * for which attenuation vanishes. b) Ratio of the fluid over frame displacements u f / u vs saturation Sw. Filtration velocity w is positive when u f /u > 1 and negative when u f /u < 1. The displacements are computed for the same models as in a); note that corresponding critical saturations S * coincide with u f /u = 1 for which there is no filtration. c) H(Sw), H f r and H f (Sw) vs Sw computed for two distinct models of effective fluid modulus K B 9 and K B 61 . Corresponding stars give the exact location of critical saturation S * .

S w Q -1 K f = K R K f = K B 61 K f = K B 9 K f = K B 32 S *
S w u f /u = 1 -β/φ K f = K R K f = K B 61 K f = K B 9 K f = K B 32 S *

Parameter

(m 2 • s -1 • V -1 ) 1 -i ω ωc(Sw) m P 4 -1/2
Table 1. Properties of the experimental porous medium constituted of Landes sand saturated by a water-gas fluid mixture. The listed parameters are divided into four main classes of properties concerning the fluid, the grain, the frame and the fluid-solid interface. They all intervene within the model computation of the coseismic seismoelectric phenomena presented in eq. ( 1), a model originally described in [START_REF] Pride | Electroseismic wave properties[END_REF], before being extended to partial saturation in [START_REF] Warden | Seismoelectric wave propagation numerical modelling in partially saturated materials[END_REF] and [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]. Note that the electrolyte used in this Table is a N aCl solution: for that mixture we used the following relation between conductivity σ f and concentration C 0 [START_REF] Haartsen | Electroseismic waves from point sources in layered media[END_REF]): σ f (C 0 ) = e 2 (b N a + b Cl )C 0 N A × 1000 C 0 × 9.3 where e = 1.6 × 10 -19 C is the elementary electric charge, b N a = b Cl = 3 × 10 11 m • s -1 • N -1 are the ionic mobilities of cations and anions and N A = 6.022 × 10 23 mol -1 is the Avogadro number. The expression of L(Sw, ω) corresponds to a simplified expression given in [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] where ωc(Sw) = 2πf Biot (Sw), where f Biot (Sw) is the critical pulsation defined upon the properties of the effective fluid.

  b): Maximum of electric potential difference ∆V as a function of l dip /λ. a) presents mid-point potential ∆VM at various offsets while b) shows first-point potential ∆VF . c) and d): Electric field amplitudes deduced from potential differences shown in a) and b) using eq. (3), and giving respectively |EM | and |EF |. e) and f): Representation of time tmax(∆V ), picked at maximum amplitude of the electric signal, divided by time tmax(ü), picked at maximum amplitude of the seismic signal, as a function of l dip /λ. Mid-point potentials ∆VM are used in e) whereas first-point potential ∆VF are used in f).

Figure 1 .

 1 Figure 1. Pictures of the experimental set-up. a) Global view of the sandbox with pneumatic source (hold by the red frame). b) Detail of the pneumatic source.The piston on the right hand side is moved forward towards the hitting plate, where a transducer, in the upper left, records the acceleration. c) Partial emptying of the sandbox after a measuring campaign: half of the measuring plane is covered in a 25 cm thick sand-layer while the other half is visible. In particular, one sees the vertical electrodes measuring the electric field in the same horizontal plane where the accelerometers, also visible though buried in the sand, measure the seismic field while the other half is covered by a 25 cm thick sand-layer .

Figure 3

 3 Figure 3. a) Acceleration measurements ü vs time recorded at the hitting plate during a series of shots by the home-made pneumatic source. The 25 superimposed shots were obtained for Sw = 0.95 and σ f = 2.5 mS • m -1 . In red, the averaged signal in time filtered by a low pass Butterworth filter of degree 8 and cutoff frequency 25 kHz. b) In black, the corresponding 25 spectra associated to the signals shown in a). In red, the averaged spectrum subjected to the same filtering process as in a).

Figure 4 .

 4 Figure 4. a) and b): Schematic representations of electric and seismic acquisition arrays in the experimental sandbox, for two geometries of dipole reconstruction at offset 5.8 cm. Each electrode rod is sketched as a blue circle.In a), electric potential differences ∆V are reconstructed from dipoles of varying dipole-length l dip , all centered on e 5.8 cm . These dipoles representing the mid-point geometry are labeled ∆V M . In b), dipoles share e 5.8 cm as a common first electrode and are labeled ∆V F ; the last electrode in yellow is the common reference electrode. c) and d): Experimental dataset of potential differences ∆V M and ∆V F at offset 5.8 cm measured vs time, for varying dipole length l dip . e) and f): Same experimental electric dataset as in c) and d) normalized in order to compare electric potentials ∆V M and ∆V F with respect to seismic acceleration ü recorded at the same offset 5.8 cm. Corresponding colours in thick lines traduce identical dipole lengths in electric data through a) to f).

Figure 5 .

 5 Figure 5. a) and b): Maximum of electric potential difference ∆V as a function of l dip /λ. a) presents mid-point potential ∆V M at various offsets while b) shows first-point potential ∆V F . c) and d): Electric field amplitudes deduced from potential differences shown in a) and b) using eq. (3), and giving respectively |E M | and |E F |. e) and f): Representation of time tmax(∆V ), picked at maximum amplitude of the electric signal, divided by time tmax(ü), picked at maximum amplitude of the seismic signal, as a function of l dip /λ. Mid-point potentials ∆V M are used in e) whereas first-point potential ∆V F are used in f).

Figure 6 .

 6 Figure 6. Electric field amplitudes |E M | calculated on mid-point dipoles of length l dip = 1.8 cm vs electric field amplitudes values |E F | computed on first-electrode dipole of l dip = 0.9 cm length, at corresponding offsets. These data come from a variety of experiments performed at various saturation rates Sw and various fluid conductivity σ f : each symbol in the legend corresponds to a couple of parameters (Sw, σ f ) whereas the colour of the symbol gives the offset where the electric field has been measured. The dashed line represents the identity function.

Figure 7 .

 7 Figure 7. Modulus of the transfer function |E/ü| as a function of frequency f computed from eq. (1), according to the medium parameters given in Table1. The transfer function is computed for three various fluid conductivities at full saturation Sw = 1. The Biot frequency is shown as a red dotted line.

/Figure 8 .

 8 Figure 8. Acceleration ü (a to c) and electric field E M , (d to f) data, averaged on about 50 shots, obtained at various offsets for experiments performed under different fluid conductivities σ f , and represented vs time. Corresponding colours in thick lines traduce corresponding offsets in seismic and seismoelectric figures. Relative amplitudes are preserved as to give 50 m • s -2 per vertical division for the seismic field and 0.5 V • m -1 per division for the electric field.For the sake of data readability, electric curves were flipped in polarity.

Figure 9 .

 9 Figure 9. Experimental and theoretical values |E/ü| represented vs fluid conductivity σ f . Experimental values of |E/ü| are shown with coloured dots for various offsets at a given fluid conductivity σ f . The averaged values and their standard deviation are represented as black crosses. Error bars on conductivity were estimated to be of the order 0.5 mS • m -1 . The dashed and continuous lines correspond to the dynamic seismoelectric transfer function from eq. (1), respectively computed for (Sw = 0.95, f = 0.5 kHz) and (Sw = 0.95, f = 2 kHz), all further physical properties matching those. An additional data point displayed at a conductivity value of 11.7 mS • m -1 was obtained from averaging experimental values taken from[START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF].

Figure 10 .

 10 Figure10. P -wave velocities V P deduced from time-picking of the first seismic arrival during time-lapse monitored experiments with varying saturation Sw. The measurements were performed during initial imbibition, subsequent drainage and following re-imbibition. Velocity time-picking was performed between offsets 10 to 23 cm. Error bars in velocity amplitudes were calculated from a linear regression of the time-picking vs offsets on a 80% confidence interval. The velocity models are represented by solid and dashed lines matching the colour of the corresponding experimental points. They were jointly obtained by least-square inversion on these V P data and on the E/ü data from Fig.12, while adapting the effective fluid modulus K f . All the velocity models are computed at a frequency f = 1.5 kHz.

Figure 11 .

 11 Figure11. Evolution of seismic (blue) and electric (red) signals vs time as a function of changes in water saturation Sw, for a set of experimental data acquired during drainage. Note that while water saturation is determined by the origin ordinate of each seismic curve, the vertical divisions on this same axis enable to get back to the signals amplitudes. The scale is of 10 m • s -2 per vertical division for the seismic field, and 0.025 V • m -1 per vertical division for the electric field. The origin on the time axis coincides with the initial punch of the pneumatic source. Seismic signals were taken for the first and second receivers at respective offsets a) 4.8 cm and b) 5.8 cm; electric signals were was taken for the mid-point dipole of length 1.8 cm placed at offset a) 3.9 cm and b) 5.8 cm. Seismic signals in blue result from a stack 25. Electric signals in bold, dashed and light lines result respectively from stacks 50, 25 and 5.

  Figure12. Experimental values of local E/ü ratio estimated during time-lapse monitored experiments with varying saturation Sw. The measurementsperformed during initial imbibition, subsequent drainage and following re-imbibition -are represented for a fluid conductivity of 7.2 mS •m -1 . We considered electric data acquired at offset 3.9 cm, combined to seismic amplitudes extrapolated to a corresponding offset. Systematic error bars on the amplitude of E/ü are of the order of 1 V • s 2 • m -2 based on the uncertainties in electric and seismic pickings. The horizontal error bars on saturation are computed from the difference between the minimum and maximum values of the recorded saturation by the capacitance probes. The estimates of the transfer function are represented by solid and dashed lines matching the colour of the corresponding experimental points. They were jointly obtained by least-square inversion on these E/ü ratios and on the associated V P data from Fig.10, while adapting the effective fluid modulus K f . Models based on an effective fluid modulus of the Brie type with exponent e are noted K B e , while K R points at an effective fluid modulus based on the Reuss average. Since the dynamic transfer functions eq. (1) is complex, the theoretical functions in the present figure are drawn as sign(real(E/ü))× | E/ü |. All model predictions are computed at a frequency f = 1.5 kHz.

Figure 13

 13 Figure13. a) Inverse of the seismic quality factor Q -1 vs Sw, computed from eq. (1) using properties given in Table1. Curves are obtained for four distinct estimations of the effective fluid modulus K f (Sw), as K R and K B e with e =[9, 32, 61]. For each case a star, matching the curve in colour, gives the exact location of the critical saturation S * for which attenuation vanishes. b) Ratio of the fluid over frame displacements u f / u vs saturation Sw. Filtration velocity w is positive when u f /u > 1 and negative when u f /u < 1. The displacements are computed for the same models as in a); note that corresponding critical saturations S * coincide with u f /u = 1 for which there is no filtration. c) H(Sw), H f r and H f (Sw) vs Sw computed for two distinct models of effective fluid modulus K B 9 and K B 61 . Corresponding stars give the exact location of critical saturation S * .

C

  ek (V • P a -1 ) 7.14 × 10 -7 ζ(σ f ) σ f Bordes et al. (2015) Electrokinetic coef. C ek (Sw) = C ek (Sw = 1) × f (Sw) , ω) = L 0 (Sw)×from[START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]

LIST OF TABLES

Properties of the experimental porous medium constituted of Landes sand saturated by a water-gas fluid mixture. The listed parameters are divided into four main classes of properties concerning the fluid, the grain, the frame and the fluid-solid interface. They all intervene within the model computation of the coseismic seismoelectric phenomena presented in eq. ( 1), a model originally described in [START_REF] Pride | Electroseismic wave properties[END_REF], before being extended to partial saturation in [START_REF] Warden | Seismoelectric wave propagation numerical modelling in partially saturated materials[END_REF] and [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]. Note that the electrolyte used in this Table is a N aCl solution: for that mixture we used the following relation between conductivity σ f and concentration C0 [START_REF] Haartsen | Electroseismic waves from point sources in layered media[END_REF]): σ f (C0) = e 2 (bNa + b Cl )C0NA × 1000 C0 × 9.3 where e = 1.6 × 10 -19 C is the elementary electric charge, bNa = b Cl = 3 × 10 11 m • s -1 • N -1 are the ionic mobilities of cations and anions and NA = 6.022 × 10 23 mol -1 is the Avogadro number. The expression of L(Sw, ω) corresponds to a simplified expression given in [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] where ωc(Sw) = 2πfBiot(Sw), where fBiot(Sw) is the critical pulsation defined upon the properties of the effective fluid.