Clarisse Bordes
email: clarisse.bordes@univ-pau.fr

Laurence Jouniaux
email: laurence.jouniaux@eost.u-strasbg.fr

Stéphane Garambois
email: stéphane.garambois@ujf-grenoble.fr

Michel Dietrich
email: michel.dietrich@ujf-grenoble.fr

Jean-Pierre Pozzi
email: pozzi@geologie.ens.fr

Stéphane Gaffet
email: gaffet@geoazur.unice.fr

Evidence of the theoretically predicted seismo-magnetic conversion

Seismo-electromagnetic phenomena in porous media arise from seismic wave-induced fluid motion in the pore space, which perturbs the equilibrium of the electric double layer. This paper describes with details the original experimental apparatus built within the ultra-shielded chamber of the Low Noise Underground Laboratory of Rustrel (France). We measured seismo-magnetic conversions in moist sand using two induction magnetometers, and a pneumatic seismic source to generate the seismic wave propagation. We ensured to avoid the magnetometer vibrations, which could induce strong disturbances from induction origin. Interpretation of the data is improved by an analytical description of phase velocities for fast (P f ) and slow (P s ) longitudinal modes, transverse mode (S) as well as the extensional mode due to the cylindrical geometry of the sample.

The purpose of this paper is to provide elements to measure correctly co-seismic seismomagnetic fields and to specify their amplitude. The seismic arrivals recorded in the sample showing a 1200-1300 m.s -1 velocity have been associated to P and extensional waves. The measured seismomagnetic arrivals show a velocity of about 800 m.s -1 close to the calculated phase velocity of S waves. Therefore we show that the seismomagnetic field is associated to the transverse part of the propagation, as theoretically predicted by Pride (1994), but never measured up to now.

Moreover, the combined experimental and analytical approaches lead us to the conclusion that the measured seismo-magnetic field is probably about 0.035 nT for a 1 m.s -2 seismic source acceleration (0.1 g).

Introduction

Seismic wave propagation in fluid-filled porous media generates electromagnetic phenomena due to electrokinetic couplings at the pore scale. These conversions are due to the relative motions of electrolyte ions with respect to the mineral surface (grains). Origin of these phenomena is generally attributed to an electrical double [START_REF] Overbeek | Electrochemistry of the double layer[END_REF] or triple-layer [START_REF] Davis | Surface ionization and complexation at the oxide/water interface[END_REF] at the rock/electrolyte interface. As a consequence, seismic wave propagation in the porous media can be accompanied by transient electric and magnetic fields created in the homogeneous porous medium. Existing investigations of the seismo-electromagnetic effects account for two main kinds of electrokinetic conversions:

1. In a homogeneous media, the pore fluid dragging ions of the electrolyte causes an electrical convection current. This current is naturally balanced by an equal and opposite conduction current that establishes an electrical potential gradient. Under an oscillating pressure source, the potential will linearly change in amplitude and sign with the pressure. This electrical potential will be seen as a wave travelling with the pressure, but since the total current is zero, no electric nor magnetic field will be observed outside the area of disturbances. This phenomena is the coseismic part of the seismo-electromagnetic phenomena.

2. On the other hand, when the seismic wave crosses an interface, an electrical current imbalance can be induced by the contrast of porous media properties (Garambois et al., 2002). This acts as the source for an electromagnetic disturbance, traveling with high speed and spreading throughout the formation.

Seismo-electromagnetic effects are especially appealing for environmental or oil exploration as they open the possibility to characterize the fluids contained in the subsurface or in reservoir rocks with the resolution of seismic methods. The second kind of seismo-electromagnetic conversion occurs at a contrast in electrical impedance, even if there is no contrast in seismic impedance. Interfaces between permeable or fractured zones may be detected by the measurement of either seismic to electric conversion [START_REF] Thompson | Geophysical applications of electrokinetic conversion[END_REF] or electric to seismic conversion [START_REF] Thompson | Field tests of electroseismic hydrocarbon detection[END_REF]. However, since the induced signals are low-amplitude signals, this method may not be efficient to characterize very deep formations, except in boreholes or using very high powerful seismic sources.

Over the past decades, field experiments were conducted to characterize these seismo-electromagnetic phenomena [START_REF] Thompson | The seismic-electric effect[END_REF][START_REF] Martner | The electroseismic effect[END_REF][START_REF] Broding | Field experiments on the electroseismic effect[END_REF][START_REF] Long | Field measurement of the electroseismic response[END_REF]. Successful field experiments performed in recent years [START_REF] Butler | Substraction of powerline harmonics from geophysical records[END_REF][START_REF] Thompson | Geophysical applications of electrokinetic conversion[END_REF][START_REF] Russell | Seismoelectric exploration[END_REF][START_REF] Mikhailenko | Mathematical modeling of seismomagnetic effects arising in the seismic wave motion in the earth's constant magnetic field[END_REF][START_REF] Beamish | Characteristics of near surface electrokinetic coupling[END_REF][START_REF] Hunt | Borehole elektrokinetic responses in fracture dominated hydraulically conductive zones[END_REF][START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF][START_REF] Kepic | Enhancing the seismoelectric method via a virtual shot gather[END_REF][START_REF] Thompson | Field tests of electroseismic hydrocarbon detection[END_REF][START_REF] Dupuis | Vertical seismoelectric profiling in a borehole penetrating glaciofluvial sediments[END_REF]Haines et al., 2007a,b;[START_REF] Dupuis | Seismoelectric imaging of the vadose zone of a sand aquifer[END_REF][START_REF] Strahser | Polarisation and slowness of seismoelectric signals: a case study[END_REF] have stimulated new interest in this particular mechanism. As described by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], analytical interpretation of these phenomena needs to connect the theory of [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid: I. low frequency range[END_REF] for the seismic wave propagation in a two phases medium with the Maxwell's equations, using dynamic electrokinetic couplings. These analytical developments opened the possibility to numerically simulate the electrokinetic coupling phenomena in homogeneous or layered saturated media [START_REF] Haartsen | Electroseismic waves from point sources in layered media[END_REF][START_REF] Haartsen | Dynamic streaming currents from seismic point sources in homogeneous poroelastic media[END_REF][START_REF] Singer | Modelling of electromagnetic fields in thin heterogeneous layers with application to field generation by volcanoestheory and example[END_REF][START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF], 2002) with applications to reservoir geophysics [START_REF] Hu | Simulation of an acoustically induced electromagnetic field in a borehole embedded in a porous formation, Borehole Acoustics and Logging Reservoir Delineation consortis[END_REF][START_REF] Singer | Electrokinetic logging has the potential to measure the permeability, Society of Petrophysicists and Well Log Analysts[END_REF][START_REF] Jackson | Development and application of new downhole technology to detect water encroachment toward intelligent wells[END_REF][START_REF] Saunders | A new numerical model of electrokinetic potential response during hydrocarbon recovery[END_REF]. Some laboratory studies exist on the zero-frequency limit, often referred to as the streaming potential [START_REF] Ishido | Experimental and theoretical basis of electrokinetic phenomena in rock water systems and its applications to geophysics[END_REF][START_REF] Pozzi | Electrical effects of fluid circulation in sediments and seismic prediction[END_REF]Jouniaux & Pozzi, 1995a[START_REF] Mikhailenko | Mathematical modeling of seismomagnetic effects arising in the seismic wave motion in the earth's constant magnetic field[END_REF][START_REF] Pengra | Determination of rock properties by low frequency ac electrokinetics[END_REF]Lorne et al., 1999a,b;[START_REF] Jouniaux | Streaming potential in volcanic rocks from mount peleé[END_REF][START_REF] Doussan | Variations of self-potential and unsaturated flow with time in sandy loam and clay loam soils[END_REF][START_REF] Hase | Zeta potential measurement of volcanic rocks from aso caldera[END_REF][START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF][START_REF] Guichet | Modification of streaming potential by precipitation of calcite in a sandwater system: laboratory measurements in the ph range from 4 to 12[END_REF]. Applying a sinusoidal fluid pressure, some authors studied the frequency-dependence of the streaming potential [START_REF] Packard | Streaming potentials across capillaries for sinusoidal pressure[END_REF][START_REF] Cooke | Study of electrokinetic effects using sinusoidal pressure and voltage[END_REF][START_REF] Reppert | Frequencydependent streaming potentials[END_REF]. These measurements on sand or core samples showed that the electrokinetic effect is particularly sensitive to the fluid conductivity, to the fluid-pH, and to the water-saturation state. However, few labora-tory experiments have been performed on dynamic seismo-electric conversions [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] Gaskarov | The seismoelectric effect in rocks and the preconditions for its application in geological prospecting work[END_REF][START_REF] Chandler | Transient streaming potential measurements on fluidsaturated porous structures: An experimental verification of biot's slow wave in the quasi-static limit[END_REF][START_REF] Mironov | Seismoelectric effect in rocks containing gas or fluid hydrocarbon (english translation)[END_REF][START_REF] Jiang | A method for measuring electrokinetic coefficients of porous media and its potential application in hydrocarbon exploration[END_REF][START_REF] Ageeva | E-effect in rocks[END_REF][START_REF] Zhu | Experimental studies of seismoelectric conversions in fluid-saturated porous media[END_REF] some of them focusing on laboratory borehole measurements [START_REF] Zhu | Experimental studies of electrokinetic conversions in fluid-saturated borehole models[END_REF][START_REF] Zhu | Crosshole seismoelectric measurements in borehole models with fractures[END_REF], 2005). Recently [START_REF] Chen | Experimental studies of seismoelectric effects in fluidsaturated porous media[END_REF] as well as [START_REF] Block | Conductivity dependence of seismoelectric wave phenomena in fluid-saturated sediments[END_REF] confirmed that a seismic wave crossing an interface induces an electromagnetic field, with electrokinetic origin, by measuring the associated electric field.

Most of these field and laboratory studies have concentrated on measurements of electrical fields as they require simple and cheap instrumentation.

The investigation of seismo-magnetic fields has received much less attention, mainly because of the complex apparatus for measuring low magnetic fluxes.

As suggested by [START_REF] Pride | Electroseismic wave properties[END_REF], the seismo-electric field E has to be associated to the whole seismic field whereas the seismo-magnetic field B is coupled to transverse modes (S waves). To our knowledge, no study succeeded to confirm the theoretically predicted seismo-magnetic field: neither its existence nor its magnitude. In a recent short paper [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF] announced that seismo-magnetic couplings were measurable at least in low magnetic noise environment and consistent with these theoretical predictions. In this paper, the seismic, seismo-electric and seismo-magnetic field were compared, but these preliminary results were not fully interpreted and seismo-magnetic amplitudes were not given.

To go further, this paper describes with details the original experimental apparatus and presents arguments to give seismo-magnetic amplitudes. This experiment was built in the ultra-shielded chamber of the Low Noise Underground Laboratory of Rustrel (France) to measure seismo-electromagnetics of the co-seismic type. It is shown that very weak residual vibrations can generate strong disturbing co-seismic signals that could be misinterpreted. We point out that devices for seismo-magnetic couplings measurements have to be very carefully designed for experimental measurements as well as for field investigations.

The major difficulty blocking access to seismomagnetic amplitudes is due to magnetometers, which are sensitive both to radial and azimuthal components of the magnetic field. As this dependence differs strongly according to the component (10 V /nT for radial and 10 mV /nT for azimuthal component), the access to real amplitudes can be a difficult task. However, analytical developments adapted to our experiment configuration showed that the radial component is negligible. It allowed to provide an estimation of the measured seismomagnetic field amplitude around 0.035 nT for a 1 m.s -2 seismic source.

Seismic to electromagnetic field conversion: theoretical background

The equations governing the coupled seismic and electromagnetic wave propagation in a fluid-saturated porous medium have been developed by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF].

In his work, this author derived a complete set of nine macroscopic equations, grouping into four Maxwell's and three poroelastic equations which are connected by two transport relations [ [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] equations ( 250) and ( 251)]:

J = σ(ω)E + L(ω) -∇p + ω 2 ρ f u s (1) -iωw = L(ω)E + k(ω) η -∇p + ω 2 ρ f u s (2)
In equation 1, the macroscopic electrical current density J is written as the sum of the average conduction and streaming current densities. Similarly, the fluid flux w of equation 2 is separated into electrically and mechanically induced contributions. The electrical fields and mechanical forces that generate the current density J and fluid flux w are, respectively, E and (-∇p+iω

2 ρ f u s ),
where p is the pore-fluid pressure, u s is the solid displacement, and E is the electric field. In the above relationships, ρ f is the pore-fluid density, η is the fluid's shear viscosity, and ω is the angular frequency. The most important parameter in equations 1 and 2 is the complex and frequency-dependent electrokinetic coupling coefficient L(ω), which describes the coupling between the seismic and electromagnetic fields [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF][START_REF] Reppert | Frequencydependent streaming potentials[END_REF]:

L(ω) = L 0 1 -i ω ω c m 4 1 -2 d Λ 2 1 -i 3/2 d ωρ f η 2 -1 2 (3)
where the static electrokinetic coupling is given by:

L 0 = - φ α ∞ ε rf ζ η 1 -2 d Λ (4)
where m and Λ are geometrical parameters of the pores given in table C, φ is the porosity, α ∞ the tortuosity, d the Debye length and ε rf the dielectric constant of the fluid. The critical angular frequency ω c defined in the Biot's theory separates the viscous and inertial flow domains. Its value depends on the permeability k 0 :

ω c = φ α ∞ k 0 η ρ f (5)
The remaining two coefficients σ(ω) and k(ω) of equations 1 and 2, represent the electric conductivity and dynamic permeability of the porous material, respectively. These coefficients are detailed in appendix A.1. The most influencing parameter on the electrokinetic coupling coefficient (equations 3 and 4) is the electrical potential at the rock/fluid interface (called zeta potential), itself depending on fluid composition, pH, and formation factor [START_REF] Davis | Surface ionization and complexation at the oxide/water interface[END_REF][START_REF] Ishido | Experimental and theoretical basis of electrokinetic phenomena in rock water systems and its applications to geophysics[END_REF]Jouniaux & Pozzi, 1995a;Lorne et al., 1999a;[START_REF] Jouniaux | Streaming potential in volcanic rocks from mount peleé[END_REF][START_REF] Guichet | Modification of streaming potential by precipitation of calcite in a sandwater system: laboratory measurements in the ph range from 4 to 12[END_REF].

In light of Pride's theory, [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF] expressed the electric and magnetic fields E and H as a function of the seismic displacement u. In case of plane waves propagating in the homogeneous medium at seismic frequencies (ω < ω c ), they found:

E ≃ 1 σ f ε 0 ρ f κ f ζ η 1 - ρ ρ f C H ü (6)
for displacements associated to the longitudinal fast (type I) P-waves [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid: I. low frequency range[END_REF][START_REF] Biot | Mechanics of deformation and acoustic propagation in porous media[END_REF], and

| H | ≃ φ α ∞ ε 0 ρ f κ f |ζ| η G ρ | u| (7)
for displacements associated to transverse SH-and SV -waves.

Note that the seismo-magnetic field is coupled to the transverse propagation mode. The in-line component E of the electric field accompanying the compressional waves is therefore approximately proportional to the grain acceleration. In expressions 6 and 7, ε 0 is the vacuum's dielectric constant and σ f represents the fluid's electric conductivity, G is the shear modulus of the framework, φ is the porosity, and ρ is the bulk density. The tortuosity α ∞ represents a geometrical factor depending on the pore structure and orientation, and is usually taken equal to the product of the porosity by the formation factor. The definitions of the C and H moduli are those of [START_REF] Biot | Mechanics of deformation and acoustic propagation in porous media[END_REF] (appendix A.1).

However, boundary conditions can produce significant effects on the seismic propagation and have to be taken into account for the seismo-electromagnetic interpretations. In the experiment described afterwards, the sand is placed in a cylindrical column. Therefore extensional modes have to be taken into account. As an illustration, analytical solutions of the seismo-electromagnetic fields are developped in appendix B. Moreover, we will show in section 5 that the vertical seismic excitation should induce an azimuthal magnetic field in such a configuration.

3 Experimental setup et al., 1999;[START_REF] Gaffet | Simultaneous seismic and magnetic measurements in the low-noise underground laboratory (lsbb) of rustrel, france, during the 2001 january 26 indian earthquake[END_REF]. To preserve these outstanding conditions, sen-sitive parts of the experiment including sample, seismic source and sensors, are located within the ultra-shielded chamber. On the contrary, the instruments generating electromagnetic disturbances (electronic devices) are placed outside the chamber, within the corridor (figure 1).

[Figure 1 about here.]

Characteristics of the porous medium

Physico-chemical properties of the mineral grains are of the highest importance on seismo-electromagnetic couplings. Especially, the grains have to be poorly soluble to avoid increasing of the fluid conductivity and decreasing of the electrokinetic coupling (see equation 6). Consequently, we chose a highly referenced medium for mechanical, electrical and chemical properties composed with 99 % of silica. A Plexyglas column 104 cm long and 8 cm diameter (figure 2, part A) is filled with Fontainebleau sand that can be imbibed from below with demineralized water to obtain high electrokinetic couplings [START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF]Lorne et al., 1999a). The sand was highly compacted by about 1500 hammer shots during the filling in order to limit the high frequency attenuation. The measured permeability of the sand is 5.8 10 -12 m 2 , its electrical resistivity is 22 kΩ.m, and the water conductivity is 3.1 mS.m -1 with a pH of 6.55 at 20.5 • C.

The bulk density of the sand (1.77 10 3 kg.m -3 ) has been measured by weighing during the filling. By comparing to the bulk density of silica (2.65 10 3 kg.m -3 ), we obtain the total porosity of the sand (33%) which is constant along the sample. On the other hand, the volume of water imbibed in the sand does not exceed 1.9 L. The sand has been fluid filled by imbibition from a water tank connected to the lower extremity. This procedure does not allow to acquire a fully saturated sample, but rather in the range of 80 to 95%.

Experimental setup within the ultra-shielded chamber

Seismic wave propagation is generated by a pneumatic seismic source [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF] able to reproduce a large number of identical impacts within a few minutes. This system was carefully designed to be free of electromagnetic noise and to allow stacking in order to improve the signal to noise ratio. The impact is obtained by launching a ruby ball on a granite plate. The time source function is recorded by an accelerometer fixed on the plate and exhibits a broad band spectra from 150 Hz to 10 kHz [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF]. Seismic excitation is vertically polarized and the seismic propagation contains longitudinal (compressive), transverse (shear) and extensional modes.

[Figure 2 [Figure 3 about here.]

Remote control of the experimental setup

To avoid strong electromagnetic disturbances from electronic devices, the whole experiment was remote controlled from the corridor of the laboratory (figure 3). It includes the seismic source driving, the charge amplifier of the accelerometer, and the acquisition system. All measurements were performed using a 24 bit seismic recorder (Geometrics StrataVisor NZ) using a 21 µs time sampling rate. The source driving system consists in an electrovalve supplied by a pulse generator (24 V /256 ms) whose input is protected from a possible discharge of the coil by a freewheel diode. This driving system is enclosed in a ground connected shielding box to avoid the transmission of electromagnetic disturbances from the electrovalve to the acquisition wires.

4 Seismo-magnetic measurements 4.1 Necessity of exceptional care for seismo-magnetic measurements Seismo-magnetic measurements need carefully designed experimental apparatus to avoid electromagnetic disturbances. Especially, it is essential to avoid those due to the seismic triggering. All disturbances occuring using the seismic source trigger would be stacked and could distort or mask real signals. Indeed as the permanent magnetic field is 10 times smaller than in usual conditions, the moving of conductive or magnetic material can generate some strong disturbances from induction origin. Thus, the stainless steel ball and the metallic spring we choose at the beginning of the experiment was replaced by a ruby ball and a rubber band in the final free noise seismic source. Moreover, all metallic screws of the experimental apparatus were replaced by nylon screws. Only the copper tube of the seismic source, poorly moving and non-magnetic, was preserved to be ground-connected and avoid an electrostatic charge accumulation.

Once this simple care taken, most serious disturbances occur when the magnetometer standing have mechanical contacts with the sample. Indeed, if vibrations due to the seismic propagation are transmitted to the magnetometers, a transient magnetic field is measured but is not related to any electrokinetic origin. These signals, due to the transient moving of the magnetometer in the permanent magnetic field, can not be distinguished from electrokinetically generated seismo-magnetic field.

The experimental apparatus was anticipated to avoid these disturbances:

the magnetometers and the sample were fixed on two differents stands (figure 2, part A). Thereby, there is no contact between the sample and the magnetometers (figure 2, part B). However, residual seismic waves can be transmitted to the floor and then to the magnetometer stand. To illustrate this phenomena, we recorded the horizontal and vertical residual vibrations of the upper magnetometer occuring after one shot (figure 4). This simple test clearly shows that the magnetometer's vibrations are very weak and measurable by accelerometers (< 2.10 -3 m.s -2 on the horizontal but negligible on the vertical accelerometer, part A). In this case, corresponding magnetic signals show a strong 80 Hz/8 mV signal probably corresponding to the resonant frequency of the magnetometers stands. This phenomena can be strongly attenuated by introducing a soundproofing protection on the floor (figure 4 part B) but cannot be totally cancelled (in the range of 2 mV). In the next section, it will be shown that these disturbances are delayed with respect to the seismic propagation in the sample and are out of the time window of interest (0-2 ms).

[Figure 4 about here.]

Results

The existence of the theoretically predicted seismo-magnetic conversions can be studied by comparing seismic and magnetic signals in dry and moist sand.

Such a comparison allows to separate as an evidence coherent magnetic signals identified as disturbances from those generated by electrokinetic effects. Indeed, the electrokinetic effects occur only when water is present [START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF], as shown in the figure 5. Seismo-magnetic sections presented were obtained by stacking 100 records of the upper magnetometer at the seven possible locations.

Corresponding seismic records obtained on an identical apparatus are given to show the comparison of seismic and seismo-magnetic propagations. The grey zone is jointly defined for seismic and magnetic sections only to highlight the location of first arrivals.

All sections are trace by trace normalized to emphasize the propagation phenomena. This figure clearly shows that coherent magnetic arrivals appear when the sand is fluid-filled. This comparison of measurements in dry and moist sand gives arguments to show the electrokinetic origin of the transient magnetic field. These seismo-magnetic signals are the first kind of the seismo-magnetic conversion, and are travelling with the seismic wave within the moist sand.

[Figure 5 about here.]

Comments on amplitudes

In order to find a representative event in the seismic and magnetic sections, we picked the negative polarity of first arrival as it is the cleanest in the seismomagnetic records. The maximum negative amplitudes were measured within the grey area and plotted on the right of each section. Furthermore, the occurrence of seismomagnetic phenomena in fluid-filling sand is highlighted by the comparison of the first arrival amplitude (black line) with the average noise level recorded before the impact (grey line). In order to respect a meticulous interpretation of the data, the maximum amplitude values correspond to the output of the magnetometer (mV ). Indeed, the discussion on real magnetic amplitudes needs a complete analytical discussion given in section 5

As evoked above, [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF] showed that seismo-magnetic signals are proportional to the S-waves grain velocity in the low frequency approximation. Using the same procedure, we show in appendix B that the azimuthal seismo-magnetic field should be proportional to the radial grain velocity associated to transverse waves. In the moist sand the amplitude of the seismo-magnetic signal decreases from the top to the bottom of the column, mainly because the seismic energy decreases from the impact seismic source.

Full waveform of seismic and seismo-magnetic signals are similar, however, a complete comparison of magnetic and seismic fields would be not appropriate in this case. Indeed, the seismic source generates P , S and extensional waves, but the different seismic modes can not be separated because of the too small sam-ple length and because accelerometers measure the axial component of the grain acceleration. Thus, it is impossible to choose an event in the seismic signals that could be directly associated to S waves and compared to the seismo-magnetic field.

The comparison of the magnetic amplitudes in dry and moist sand shows as an evidence that the seismo-magnetic conversions are very weak. A comparison of maximum amplitudes of the grey zones in magnetic signals show that signals in moist sand (0.3 mV at 10 cm) are only 6 times larger than in dry sand (0.05 mV at 10 cm). It is interesting to note that the signal to noise ratio is better in moist sand, partly due to a lower noise level in moist sand. Indeed, real noise amplitudes measured before the triggering are significantly lower in moist (0.02 mV ) than in dry sand (0.05 mV ).

The conversion to real magnetic amplitude values needs the assumption of a purely azimuthal or radial field. According to the major polarization of the magnetic field, this amplitude could vary from 0.035 pT to 0.035 nT for a 1 m.s -2 seismic source. This topic will be discussed in the light of the analytical approach of seismo-electromagnetic fields of the section 5.

Comments on propagation velocities

One of the most significant results is obtained by comparing seismic and seismomagnetic propagation velocities by picking first breaks on enlarged views of the seismic and magnetic sections (figure 6). A careful analysis in the [0 -1.5 ms]

window of the magnetic records on dry and moist sand shows a residual disturbance arising at t = 0 s. This phenomenon exhibits amplitude of the same order as the signal. Then, it is invisible on the one shot record but appears after the stacking procedure. It is due to the seismic triggering and can be due to crosstalk (from time source function recording) or to electromagnetic disturbances by the impact. This disturbance has to be eliminated, or at least attenuated, to enable the computing of the apparent velocity of the seismo-magnetic wave within the compacted sand.In order to minimize this phenomenon, the lower magnetometer was used as a reference. The subtraction of the reference from each point of measurement improves the data by minimizing the distur-bance and unmasking the first breaks of interest (figure 6).

This subtraction enables us to deduce the apparent velocity of the seismomagnetic field (791 ± 80 m.s -1 ) with a better signal to noise ratio. This velocity can be compared to the significantly faster seismic velocities. The measured seismic velocities are almost the same in dry (1201 ± 85 m.s -1 ) and moist sand (1328 ± 94 m.s -1 ). These values are consistent with P -waves velocities measured by [START_REF] Elliott | Compressional velocities of partially saturated, unconsolidated sands[END_REF] in partially saturated compacted sample (Ottawa sand, porosity 30 %).

The fact that any significative increase in seismic velocity with fluid filling is not observed is consistent with laboratory measurements of [START_REF] Knight | A laboratory study of the dependence of elastic wave velocities on pore scale fluid distribution[END_REF]. They showed that the behavior of the P -wave velocity with increasing water-saturation within sandstones strongly depends on the means used to saturate the porous medium. In the imbibition case, the P -wave velocity is increased only after 80% of water-saturation (and is constant below), whereas in drainage conditions the P -wave velocity is increased with water-content as soon as 45% of water saturation (see also [START_REF] Gist | Interpreting laboratory velocity measurements in partially gas-saturated rocks[END_REF]. Since the seismic velocities in our dry and moist sand are the same (given the associated errors), and because we are in imbibition conditions, we can deduce that the sand is partly saturated, probably with a maximum of 80%-85% of water saturation.

[Figure 6 about here.]

In the present section we proved that seismo-magnetic phenomena in partly saturated sand have an electrokinetic origin. The velocity of the seismo-magnetic field is measured to be lower than the seismic velocity. In order to improve interpretations of velocities and to better understand seismomagnetic amplitudes, an analytical approach of the seismo-electromagnetic couplings including the geometrical characteristics of our experimental setup is proposed. Indeed, under a vertical seismic excitation in a cylindrical porous medium, specific modes can be generated due to the presence of the free cylindrical surface. In the next section we calculate the extensional phase velocity and deduce that the fast seismic arrivals are associated to the P and extensional waves, and that the slower seismo-magnetic arrivals are associated to S-wave.

5 Analytical approach for seismo-electromagnetic fields interpretations

The description of the complete seismic propagation needs to take into account the fast (P f ) and slow (P s ) longitudinal modes, the transverse modes (S) as well as the specific modes due to the cylindrical free surface. Considering the axial seismic excitation, we assume that these specific modes are only extension and we neglect flexion and torsion [START_REF] Dunn | Acoustic attenuation in fluid-saturated porous cylinders at low frequencies[END_REF]. The goal of this analytical approach is to evaluate expected phase velocities of the different waves, in order to identify which waves are converted into seismo-electromagnetic fields.

We show that V ext and V P f are very close unless the sand is not totally fluidfilled. Moreover, we show that in a cylindrical porous medium under a vertical excitation, the induced seismo-magnetic field should be azimuthal.

Seismic velocities of body waves

In order to compute the phase velocity of the extensional mode, the bulk (K f r )

and shear (G) frame moduli of the sand have to be estimated. Then, K f r can be deduced from inversion of the measured P f wave velocity. Indeed, suitable moduli can be calculated from observations using expressions of phase velocities:

V P f = 1 Re(s P f ) V P s = 1 Re(s P s ) V S = 1 Re(s S ) (8)
where s P f , s P s and s S are slownesses of P f , P s and S waves from [START_REF] Pride | Electroseismic wave properties[END_REF] (equations 30, 31 and 33 of appendix A.2).

As it was evoked above, our fluid-filling method (imbibition) does not enable us to obtain a fully saturated sample and the seismic velocities in dry and moist sand are very close. Indeed, [START_REF] Knight | A laboratory study of the dependence of elastic wave velocities on pore scale fluid distribution[END_REF] showed that during the imbibition procedure, the P f wave velocity is quite constant until the saturation coefficient exceeds 85%. A computation of phase velocities needs to take into account the partial saturation. Since the velocity change is negligible in the 0 to 80% saturation range, a first order approximation of the frame modulus can be obtained by considering that the fluid is totally replaced by air. In this case, the computation of the P f -wave velocity from expressions 8 versus frame modulus shows that the measured 1250 m.s -1 velocity corresponds to a unique value of the frame modulus K f r = 1.2 10 9 P a (figure 7). This value is larger than the usually assumed values for sand, which are around [0.1 -0.5 GP a] [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF][START_REF] Karner | Subcritical compaction and yielding of granular quartz sand[END_REF][START_REF] Block | Coupled electrokinetic-biot theory and measurement techniques in sediment acoustics[END_REF], 2006). However this value is within the range [0 -4.2 GP a]

proposed by [START_REF] Hickey | Choosing biot parameters for modeling water-saturated sand[END_REF] for modeling water-saturated sand.

[Table 1 about here.]

On the other hand, the shear modulus G ≃ 1.32 10 9 P a can be inferred considering the ratio K f r /G = 0.9 from [START_REF] Murphy | Modulus decomposition of compressionnal and shear velocities in sand bodies[END_REF] for quartzic granular media. This computation shows that the S-wave velocity can be estimated to 860 m.s -1 from these deduced moduli in the dry sand (figure 7). Assuming that this first order inversion in dry sand can not introduce strong errors, the calculated frame bulk and shear moduli values will be used in the next section to compute the phase velocity of the extensional mode.

[Figure 7 about here.]

Extensional modes

Considering the vertical seismic source, some specific modes can be generated on the free cylindrical surface of the sample from interferences of P and S waves.

Extensional modes occuring in water saturated porous media have been early

studied by [START_REF] Gardner | Extensional waves in fluid-saturated porous cylinders[END_REF] for quantitative interpretation of seismic velocities in resonant bars. Using this approach, numerical computation of [START_REF] Dunn | Acoustic attenuation in fluid-saturated porous cylinders at low frequencies[END_REF] show specific dispersion and attenuation effects in cylindrical samples of sandstones. In this case, the phase velocity of extensional modes is smaller than P f but larger than S wave phase velocity. A first order approximation of the extensional phase velocity can be obtained from the bar velocity given by:

V ext = E ρ ( 9
)
where E is the Young's modulus usually defined in elastic media, and can be measured for rock samples as the slope of an axial stress versus axial strain curve from a deformation test. We can use this expression for the porous media by substituting the elastic moduli K by the equivalent Gassman moduli K G :

E = 9 K G G 3K G + G ( 10
)
where K G , the Gassman's modulus defined in appendix C and A.1, represents the equivalent K modulus for the porous media. A prediction of the phase velocities in the partially saturated sand can be obtained by using an average on the density ρ f and the bulk modulus K f versus the saturation coefficient

S w : ρ f (S w ) = S w ρ water + (1 -S w ) ρ air (11)
and

K f (S w ) = S w K water + 1 -S w K air -1 (12)
In the following expressions, we will take into account the partial saturation by substituting K f and ρ f by their average values. Finally, an estimation of the extension's phase velocity in the partially saturated sample can be computed from:

V ext (S w ) = 9 K G (S w ) G ρ(S w ) [ 3K G (S w ) + G ] (13)
A computation of equations 11, 12, 13, and 8 respectivly predicts 1250 m.s -1 and 1260 m.s -1 phase velocities for extensional and P waves in dry sand (figure 8). These velocities respectively increase to 1315 m.s -1 and 2050 m.s -1 for the water-saturated sand. This result is consistent with the work of [START_REF] Dunn | Acoustic attenuation in fluid-saturated porous cylinders at low frequencies[END_REF] showing a V ext /V S = 1.6 ratio. The P -wave velocity is strongly increased by water only when approaching the water-saturation state. Since the extensional waves are induced by interferences of S and P waves, the extensional wave is also fastened. Finally, this computation clearly shows that the P f and extension modes are very difficult to distinguish in partial saturation conditions, at least by considering the first break of seismic arrival. Thus, the seismic arrivals recorded in the sample showing a 1200/1300 m.s -1 velocity will be associated to P and extensional waves. Slow events with 800/900 m.s -1 velocities will be associated to S-waves.

[Figure 8 about here.]

Seismo-magnetic field polarization

As a complement of the seismic velocities analysis, a complete formulation of seismo-electromagnetic fields is required to propose a correct interpretation of the measurements, especially for the seismo-magnetic field polarization and magnitude. Seismo-electromagnetic fields can be expressed from seismic displacements via appropriate coefficients for each propagation mode [START_REF] Pride | Electroseismic wave properties[END_REF]. As reminded in appendix B, displacements associated to P f and P s waves are described by the scalar potentials φ f and φ s whereas S waves displacements can be represented by the vector potential Γ S .

The seismo-magnetic field is derived from the Maxwell equation:

H = - i ω µ (∇ × E) (14)
where µ is the magnetic permeability of the free space, and H and E are respectively the seismo-magnetic and seismo-electric fields. Indeed, the purely longitudinal part of the seismic propagation (P waves) does not contribute to the seismo-magnetic field and only S and extensional waves can produce coseismic magnetic effects. In appendix B, we showed that the seismo-magnetic field generated from S waves (H S ) and from extensional waves (H ext ) are expressed by:

H s,θ = - φ α ∞ ǫ 0 κ f ζ η ρ f G ρ du s,r dt (15)
and

H e,θ = - φ α ∞ ǫ 0 κ f ζ η ρ f E ρ du s,r dt (16)
Whatever the origin of the seismo-magnetic field is, the only non-zero component is the azimuthal one and the analytical prediction shows that the measured magnetic field should be azimuthal. Even if it is not possible to rule very weak radial perturbations, it is reasonable to assume that they are negligible.

These results are consistent with the study of [START_REF] Hu | Simulation of an acoustically induced electromagnetic field in a borehole embedded in a porous formation, Borehole Acoustics and Logging Reservoir Delineation consortis[END_REF] for a fluidfilled borehole and give fundamental arguments for the interpretation of the seismo-magnetic measurements. Indeed, it is possible to assume that the quantification of amplitudes needs to take into account the azimuthal sensitivity of the magnetometers.

Discussion and conclusion

The exceptional environment and facilities of the Low Noise Underground Laboratory of Rustrel gave us the possibility to build a carefully designed experimental setup for seismo-magnetic measurements in moist sand. The seismic arrivals recorded in the sample shows a 1328 ± 94 m.s -1 velocity and have been associated to P and extensional waves in the partially saturated sand. We calculated a S-wave velocity of 800 to 900 m.s -1 . The measured seismo-magnetic arrivals shows a velocity of about 791 ± 80 m.s -1 . Therefore, we propose that the seimo-magnetic field is electrokinetically induced and is associated to Swave propagation. This was theoretically predicted by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], but never measured up to now.

Magnetic measurements were performed with two induction magnetometers whose radial and azimuthal resolution are respectively around 10 V /nT and 10 mV /nT for a maximal amplitude of 0, 35 mV in output of the magnetometers. Applying the Pride's theory for seismo-electromagnetic couplings, we show that the seismo-magnetic field is purely azimuthal. Even if it is very difficult to rule out the possibility of very weak non-axial vibrations, generating radial magnetic field, it is reasonable to conclude that the measured amplitudes are in the order of 0.035 nT for a 1 m.s -2 seismic source. Further studies, implying the measurement of the three components of the magnetic field, are necessary to definitely conclude on the seismo-magnetic amplitudes. In order to better understand the seismo-magnetic conversions, further numerical, field and laboratory investigations are needed. To this end, the LSBB facilities give exceptional possibility to measure seismo-electromagnetic couplings in porous media.

Usual field seismic sources are explosives, vibroseis or sledgehammer blows which induce accelerations values in the range of 0.05 to 100 m.s -2 (table 2).

In the assumption of amplitudes about 0.03 nT for a 1 m.s -2 seismic source, field seismo-magnetic measurements could be performed out of the low noise environment. Nevertheless, extra care has to be taken to ensure the electrokinetic origin of the measurements. Especially, magnetometers have to be isolated from ground motions as well as from coherent electromagnetic noise. Thus, borehole measurements would be very difficult, because of unavoidable contacts between the tool and the mud transmitting ground vibrations. Nevertheless, subsurface measurements could be performed with ingenious experimental apparatus including a strong stacking procedure to improve the signal to noise ratio.

[Table 2 about here.] A Pride's theory for seismo-electromagnetic couplings

A.1 Mechanical and transport macroscopic equations

The Biot's theory is the correct general model governing poroelastic response [START_REF] Pride | Connecting theory to experiment in poroelasticity[END_REF][START_REF] Pain | A mixed finite-element method for solving the poroelastic biot equations with electrokinetic coupling[END_REF][START_REF] Masson | Finite difference modeling of biot's poroelastic equations at seismic frequencies[END_REF]. Poroelastic

∇ • τ B = -ω 2 (ρu s + ρ f w) (17) τ B = (K G ∇ • u s + C∇ • w) I + G ∇u s + ∇u s T - 2 3 ∇ • u s I (18) -p = C∇ • u s + M ∇ • w (19)
where τ B is the stress tensor, I is the identity matrix, u s is the displacement of the solid, and w is the average volume filtration. And K G , C, M and ∆ are poroelastic constants defined by:

K G = K f r + φK f + (1 + φ)K s ∆ 1 + ∆ (20) C = K f + K s ∆ 1 + ∆ (21) M = K f φ(1 + ∆) (22) ∆ = K f φK 2 s [(1 -φ)K s -K f r ] (23)
The frame bulk modulus K f r is the reverse of the compressibility, itself being a measure of the volumetric strain resulting from an applied change in pressure [START_REF] Hart | Laboratory measurements of a complete set of poroelastic moduli of berea sandstone and indiana limestone[END_REF].

In section 2 the equations ( 1) and ( 2) refere to complex dynamic parameters such as electrokinetic coupling L(ω), permeability k(ω) and effective conductivity σ(ω):

L(ω) = L 0 1 -i ω ω c m 4 1 -2 d Λ 2 1 -i 3/2 d ωρ f η 2 -1 2 (24)
where the static electrokinetic coupling is given by:

L 0 = - φ α ∞ ε rf ζ η 1 -2 d Λ (25)
where m and Λ are geometrical parameters of the pores given in table C. The critical angular frequency ω c separates the viscous and inertial flow domains and is defined by:

ω c = φ α ∞ k 0 η ρ f (26)
Assuming that the d/Λ term is negligible, L 0 parameter can be expressed versus electrokinetic coupling C ek from electrofiltration measurements [START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF]. Then:

L 0 = - φ α ∞ C ek σ f = -C ek σ r (27)
Expression of the effective conductivity of the material introduces surface conduction effects due to electrofiltration (C em ) and electro-osmotic (C os ) effects:

σ(ω) = φ σ f α ∞ 1 + 2 C em + C os (ω) σ f Λ (28)
At last, the dynamic permeabilty of the porous media [START_REF] Johnson | Theory of dynamic permeability in fluid saturated porous media[END_REF] depends on the observed frequency, especially for very high frequencies:

k(ω) = k 0 1 -i ω ω c 4 m -i ω ω c -1 (29)

A.2 Phase velocities of body waves

The phase velocities analysis used in this paper is based on the formulation of [START_REF] Pride | Electroseismic wave properties[END_REF] in the low frequency domain. This approach has been developed for seismoelectric studies, but in the case L(ω) = 0 the phase velocities are equivalent to the Biot's results (1956). Slowness are expressed in the case of plane longitudinal and transverse waves:

For the fast P-waves:

2s 2 P f = γ -γ 2 - 4ρρ M H -C 2 ρ t ρ + ρL(ω) 2 ε ( 30
)
For the slow P-waves:

2s 2 P s = γ + γ 2 - 4ρρ M H -C 2 ρ t ρ + ρL(ω) 2 ε ( 31
)
where

γ = ρM + ρH(1 + ρ L(ω) 2 ε ) -2ρ f C M H -C 2 (32)
For the S waves:

2s 2 S = ρ t G + µ 0 ε 1 + ρL(ω) 2 ε + ρ t G -µ 0 ε 1 + ρL(ω) 2 ε 2 -4µ 0 ρ 2 f L(ω) 2 G ( 33
)
where H = K G + 4G is an elastic modulus, ρ t the complex density and ρ the effective density given by:

ρ t = ρ - ρ 2 f ρ (34) ρ = i ω η k(ω) (35)
Attenuation effects of the Biot's theory are described by the imaginary part of the effective density ρ. In this expression ε corresponds to the effective permittivity introducing conduction and electro-osmotic dissipation:

ε = ε 0 ε rT + i ω σ(ω) -ρ(ω)L 2 (ω) (36)
B Seismo-electromagnetic fields in the porous cylindrical sample

Similarly to [START_REF] Hu | Simulation of an acoustically induced electromagnetic field in a borehole embedded in a porous formation, Borehole Acoustics and Logging Reservoir Delineation consortis[END_REF] study for a fluid-filled borehole, we consider an axisymetric porous system under vertical seismic excitation generating P-SV waves. According to [START_REF] Pride | Electroseismic wave properties[END_REF], four different propagation modes should be taken into account: fast and slow P waves (P f and P s ) as longitudinal modes and S and electromagnetic (em) waves as transverse modes. Nevertheless, we neglect electromagnetic modes generating electroosmotic displacements in the absence of significant source.

B.1 Seismomagnetic coupling from extensional modes

The total displacement u can be expressed in the (i r , i θ , i z ) coordinate system by:

u e = A P f ∇Φ P f + A Ps ∇Φ Ps + A S ∇ × (Γ S i θ ) (37)
where Φ P f and Φ Ps are scalar potentials and Γ S a vector potential satisfiying the Helmholtz decomposition with wavenumbers l j = ω s j . In the case of the vertical seismic source, the potentials can be expressed by:

   Φ P f = I 0 (η P f r) e ikez and Φ Ps = I 0 (η Ps r) e ikez Γ S = I 1 (η S r) e ikez (38)
where I i are the modified Bessel's function of order i and η j (j = P f , P s or S

) are related to the axial wavenumber k e of extensional waves (in the vertical direction z) by the relation:

η j = k 2 e -l 2 j ( 39
)
where l j = ωs j are the radial wavenumbers of each mode. Then, components of the displacement u can be calculated:

u e =      A P f η P f I 1 (η P f r) + A Ps η Ps I 1 (η Ps r) -i k e A s I 1 (η S r) 0 i k e A P f I 0 (η P f r) + i k e A Ps I 0 (η Ps r) + A s η s I 0 (η S r)      ( r, θ, z ) e ikez (40)
According to [START_REF] Pride | Electroseismic wave properties[END_REF], the seismoelectric field can be deduced via appropriate coefficients β j for each propagation mode:

E e =      β P f A P f η P f I 1 (η P f r) + β Ps A Ps η Ps I 1 (η Ps r) -i k e β S A s I 1 (η S r) 0 i k e β P f A P f I 0 (η P f r) + i k e β Ps A Ps I 0 (η Ps r) + β s A s η s I 0 (η S r)      ( r, θ, z ) e ikez (41)
where coefficients β j are given by:

β j =              i ωLρ ε Hs 2 j -ρ Cs 2 j -ρ f f or j = P f or P s -iωµρL G ρ f s 2 j -ρ G s 2 j -µε for j = S (42)
where ρ, C and M formation parameters are given by equations 21 and 22.

Then, the seismomagnetic field H can be deduced from classical Maxwell's equation:

H = - i ω µ (∇ × E) ( 43
)
where µ is the magnetic permeability of the free space. Combined with equation 6, this equation leads to:

H e =      0 1 iµ ω s 2 S A S β S I 1 (η S r) 0      ( r, θ, z ) e ikez (44)
By comparison with equation 40 we get:

H e,θ = 1 iµ β S s 2 S k e du s,r dt (45)
This equation shows that the seismomagnetic field associated to extensional waves is proportional to the grain velocity of S waves via the dynamic variables β S , s 2 S and k e .

B.2 Seismomagnetic coupling from S waves

On the other hand, seismomagnetic coupling can be generated by pure S waves.

Using the same procedure, we can obtain the expressions of seismoelectric and magnetic fields

E s =      -i k s I 1 (η S r) 0 η s I 0 (η S r)      ( r, θ, z ) A S β S e iksz (46)
and

H s =      0 1 iµ ω s 2 S A S β S I 1 (η S r) 0      ( r, θ, z ) e iksz (47)
leading to

H s,θ = 1 iµ β S s 2 S k s du s,r dt (48)
This expression shows that the seismomagnetic field is proportional to the grain velocity of s waves similarly to extensional mode (note the difference of coefficient with k s instead of k e ).

B.3 Low frequency approximation

As suggested by [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF], the low frequency approximation leads to a significantly simpler form of equation 48. Using the same procedure (not detailed here) we use the zero frequency value of the S wave slowness:

s 2 S = ρ t G (49)
Using the zero frequency electrokinetic coupling L 0 , the β S coefficient can be expressed by:

β S = iωµL 0 G ρ f ρ t (50)
Leading to the relation:

H s,θ = ωL 0 ρ f k s du s,r dt = H θ = L 0 ρ f G ρ du s,r dt (51)
Using the definition of the electrokinetic coupling:

L 0 = - φ α ∞ ǫ 0 κ f ζ η (52)
we get:

H s,θ = - φ α ∞ ǫ 0 κ f ζ η ρ f G ρ du s,r dt (53)
Using the same procedure, we get the expression of the seismomagnetic field associated to extensional modes:

H e,θ = - φ α ∞ ǫ 0 κ f ζ η ρ f E ρ du s,r dt (54)
where E is the Young's modulus of the sample defined by equation 10. The low frequency approximation shows that the seismomagnetic field is proportional to the radial grain velocity associated to S waves. The transfert function depends on the origin of the seismic propagation (S or extensional waves), via the phase velocities E/ρ and G/ρ. A simple comparison of the transfert functions from equations 48 and 53 leads to the conclusion that the error from low frequency approximation is lower than 2% in the range of 100 to 3000 Hz.

C Notations

[Table 3 about here.] Figure 6: Zoom on first break of the seismic and seismo-magnetic measurements (trace/trace normalised). The seismo-magnetic gradient representation improves the velocity resolution by partly substracting the disturbance occuring around t = 0 s. Apparent velocities in moist sand are measured to be 1328 ± 94 m.s -1 for seismic wave and 791 ± 80 m.s -1 for the seismo-magnetic wave.

Figure 7: Computation of the seismic velocities from versus frame modulus K f r in the dry and water saturated cases [START_REF] Pride | Electroseismic wave properties[END_REF] . Considering the measured P f velocity (1250 m.s -1 ), the corresponding frame modulus of the sand can be estimated to 1.2 GP a. The calculated S wave velocity, deduced from this K f r value, is near 860 m.s -1 . (S.m -1 ) (V.P a -1 ) 0.3 5.8 10 -12 3.1 10

  about here.] Seismo-magnetic couplings associated to the seismic propagation can be measured with two induction magnetometers. The first one, in the upper position, can be placed in 7 different locations whereas the second one is fixed as a reference position (figure 2, part A). Each magnetometer is divided into two elements (figure 2, part B) which can be splitted to be placed around the column without any contact with the sample. Each element consists in a magnetic core focussing magnetic fluxes. These fluxes generate induction currents in the coils that are finally amplified and recorded. These magnetometers are sensitive both to radial and azimuthal components of the magnetic field and the output consists in one channel. As a consequence, getting real amplitudes needs to expect or at least to assume a field polarization. Sensitivities are given in figure 2, part C, where grey and black lines represent respectively lower and upper magnetometer. The radial sensitivity was calibrated by measuring magnetic fluxes generated by a calibration coil placed in the symetry axis of the magnetometers. It is almost constant, nearly 10 V /nT , in the 100 to 1000 Hz range. The azimuthal sensitivity was calibrated by measuring fluxes generated by a vertical wire in the magnetometer's symetry axis and is rounded to 10 mV /nT in order to give an order of magnitude of azimuthal fields.

  response allows for the coupled interactions between the elastic deformation of a porous solid and the fluid flow and fluid accumulation in the same material. The Pride's theory for the seismo-electromagnetic couplings is based on the connexion of the Biot's and Maxwell's equations via electrokinetic couplings. This theory is therefore suitable for the fluid flow induced by seismic wave. Pride (1994) obtained macroscopic equations governing mechanical motion, charge transport and electromagnetism laws characterizing the full wave propagation field. All notations used in this section are explicited in appendix C. The mechanical equations are obtained via a generalized Biot's law taking into account the boundary conditions at the pore scale:

Figure 1 :Figure 4 :Figure 5 :

 145 Figure 1: Sensitive part of the experiment (sample and magnetometers) is located within the ultrashielded chamber of the laboratory whereas devices generating electromagnetic disturbances are located outside. The ambient magnetic noise is lower than 2 f T / √ Hz above 10 Hz within the chamber (adapted from http://lsbb.unice.fr)

Figure 8 :

 8 Figure 8: Comparison of the phase velocities of P f , extension and S waves versus saturation coefficient. The P and extensional waves can not be distinguished by velocity measurements

Table 1 :

 1 -3 -6545 10 -9 Values of the parameters used for computations. Electrokinetic coupling measured by[START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF] on Fontainebleau sand was converted to the measured fluid conductivity.

				Assumed parameters
		α	ρ f	ρs (silica)	η	Ks	K f
			(kg.m -3 ) (kg.m -3 )	(P a.s)	(P a)	(P a)
	Water 3	1000	2650	0.001	36 10 9 2.22 10 9
	Air	3	1.2		18 10 -6		0.15 10 6

Acknowledgements

This work was supported by the French National Scientific Research Center (CNRS), and "ACI Eau et Environnement" of the French Ministry of Research.

We are grateful to G. Waysand for helpul discussions and authorization to perform this experiment in the LSBB laboratory. We thank M. Auguste, D. Boyer, A. Cavaillou, G. Clerc, R. Guiguet, and Y. Orengo for their technical help.

List of Figures

Zoom on first break of the seismic and seismo-magnetic measurements (trace/trace normalised). The seismo-magnetic gradient representation improves the velocity resolution by partly substracting the disturbance occuring around t = 0 s. Apparent velocities in moist sand are measured to be 1328 ± 94 m.s -1 for seismic wave and 791 ± 80 m.s -1 for the seismo-magnetic wave. 43 7

Computation of the seismic velocities from versus frame modulus K f r in the dry and water saturated cases [START_REF] Pride | Electroseismic wave properties[END_REF] . Considering the measured P f velocity (1250 m.s -1 ), the corresponding frame modulus of the sand can be estimated to 1.2 GP a.