C Bordes

D Brito

J Holzhauer

L Jouniaux

S Garambois

M Dietrich

Laboratory measurements of coseismic fields: towards a validation of Pride's theory

Introduction

As discussed in previous chapters, seismoelectric phenomenon was first theoretically described in 1944 [START_REF] Frenkel | On the theory of seismic and seismoelectric phenomena in a moist soil[END_REF]. More recently, in his reference paper, [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] developed the whole set of governing equations in a saturated medium. These equations couple the Biot theory for seismic propagation in porous medium [Biot, 1956a,b] and Maxwell's equations for electromagnetism, via fluid and charge transport equations using a volume averaging approach. Pride's full seismoelectric analytical formulation has been largely used in recent years for numerical computations [START_REF] Garambois | Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid-saturated stratified porous media[END_REF][START_REF] Guan | Finite-difference modeling of the electroseismic logging in a fluid-saturated porous formation[END_REF]Zyserman et al., 2010;[START_REF] Santos | Numerical electroseismic modeling: a finite element approach[END_REF][START_REF] Santos | Numerical electroseismic modeling: a finite element approach[END_REF][START_REF] Warden | Seismoelectric wave propagation numerical modelling in partially saturated materials[END_REF][START_REF] Jouniaux | Reply to comment by a. revil on" seismo-electrics, electro-seismics, and seismo-magnetics for earth sciences" by l. jouniaux and f. zyserman[END_REF]] in order to discuss potential applications of seismoelectrics as a geophysical probing method. In the last decade, seismoelectric phenomena were also discussed by considering electrokinetic couplings as a function of the 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][START_REF] Jougnot | Seismoelectric effects due to mesoscopic heterogeneities[END_REF]Revil et al., 2013].

The two main effects generated by electrokinetic coupling are i) coseismic field accompanying the seismic propagation and ii) electromagnetic disturbances generated at depth when seismic waves are crossing an interface. The seismoelectric interface conversion is often perceived as a promising tool for reservoir characterization since it is expected to combine both electrical and mechanical sensitivities [START_REF] Garambois | Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid-saturated stratified porous media[END_REF][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]. The coseismic part is therefore considered as a strong disturbance to be removed for enhancing the interface response [START_REF] Warden | Curvelet-based seismoelectric data processing[END_REF]. However, coseismic phenomenon may also be perceived as a direct observation of fluid motions occurring with seismic propagation. Indeed, these relative fluid displacements are involved in attenuation and dispersion of seismic waves as discussed in the original Biot theory and in many more recent poroelastic studies [Pride and Berryman, 2003a,b;[START_REF] Müller | One-dimensional random patchy saturation model for velocity and attenuation in porous rocks[END_REF][START_REF] Müller | Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks?a review[END_REF]. The quantitative interpretation of this coseismic contribution would be therefore a powerful and original approach for testing many of these poroelastic issues. In this paper, we aim to show this quantitative interpretation to be within reach, the first step being the validation of Pride's theory.

When seismic waves are travelling in a porous medium, the coseismic seismoelectric field E is expected to be coupled to all propagation modes. As predicted by Biot [1956a,b] body waves are travelling as fast P f , slow P s and S waves. The E field can therefore be written as a function of seismic displacements by using the Helmholtz decomposition [START_REF] Hu | Simulation of an acoustically induced electromagnetic field in a borehole embedded in a porous formation[END_REF]:

E = β P f A P f ∇Φ P f + β P s A P s ∇Φ P s + β S A S ∇ × Γ S (1)
where A i and β i (i = P f , P s or S) are the amplitudes and seismoelectric couplings of each mode. Φ P i are the potential functions of P waves (that are pure divergence) and Γ S is the potential vector of S waves (that is pure curl). Hence, the contributions of P waves are vanishing in seismomagnetic field that is obtained by the fundamental Maxwell equation:

H = - i ω µ ∇ × E = - i ω µ β S A S ∇ × ∇ × Γ S ( 2
)
where ω is the pulsation of the seismic wave whose time dependence is supposed to be e -iωt , and µ is the magnetic permeability of the porous medium (µ

µ 0 = 4π10 -7 V.s. A -1 .m -1
in non-metallic rocks). Indeed, equation (2) shows the seismomagnetic field to be generated by S waves when all body waves are supposed to contribute to the seismoelectric field. Nevertheless, the effectiveness of S waves in seismoelectric field must be discussed relatively to the contribution of P waves, the simplest way to address this issue being the transfer function approach as proposed by [START_REF] Pride | Electroseismic wave properties[END_REF]. Considering both fast and slow P waves, the seismoelectric field can be written as a function of local accelerations Üi following [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]:

E(ω) = ψ Pf (ω) ÜPf (ω) + ψ Ps (ω) ÜPs (ω) + ψ S (ω) ÜS (ω) (3)
where ψ Pi and ψ S are respectively the complex dynamic transfer functions for P (fast or slow) and S waves defined by:

ψ Pi (ω) = i ω ρ(ω)L(ω) ε(ω) H s 2 Pi (ω) -ρ Cs 2 Pi (ω) -ρ f (4) and ψ S (ω) = i ω µ ρ(ω)L(ω) G ρ f s 2 S (ω) -ρ/G s 2 S (ω) -µ ε(ω)
.

(5)

In equations ( 4) and ( 5), ρ = φ ρ f + (1 -φ) ρ s is the total density of the porous medium composed of fluid and solid phases whose respective densities are ρ f and ρ s .

G is the shear modulus of the frame, H = K U + 4 3 G is the P wave modulus, K U is the undrained bulk modulus (known as "Gassman's modulus" as well), and C = BK U derives from the Skempton's coefficient B that can be deduced from the porosity φ and from fluid and solid bulk moduli K f and K s [START_REF] Barriere | Laboratory monitoring of p waves in partially saturated sand[END_REF]]. The slownesses s Pi (P waves)

and s S (S waves), including both Biot's and electrokinetic losses, will be given in equations ( 14). [START_REF] Pride | Electroseismic wave properties[END_REF] used the following formulations of the effective electrical permittivity:

ε(ω) = ε(ω) + i ω σ(ω) -ρ(ω)L 2 (ω) (6)
and of the effective density of the fluid in relative motion

ρ = i ω η f k (ω) , (7)
where η f is the dynamic viscosity of the fluid. In this formulation, the effect of Biot's losses is carried by the dynamic permeability [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF]:

k (ω) = k 0       1 -i ω ω c 4 m p 1 2 -i ω ω c       -1 , (8)
where k 0 is the intrinsic permeability of the medium and m p is a pore space term. The Biot pulsation ω c or frequency f c defines the limit between low and high frequency domains for which energy dissipation is respectively due to viscous or inertial flows:

ω c = 2π f c = η f F ρ f k 0 . ( 9
)
where F is the formation factor and ρ f is the fluid density. The frequency dependent electrokinetic coupling L(ω) in equations ( 4), ( 5) and ( 6) is defined as:

L(ω) = L 0       1 -i ω ω c m p 4 1 -2 d Λ 2 1 -i 3/2 d δ 2      -1 2 L 0 1 -i ω ω c m p 4 -1 2 . ( 10
)
In this equation, Λ = √ m P k 0 F is a pore-shape parameter, δ = η f /ω ρ f is the skin depth and d = 10 -6 δ √ ω/2πC is the thickness of the double layer [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]. We notice that neglecting both terms in parentheses of equation ( 10) gives a more handy expression of L and represents a very tiny error (less than few percents). Eventually, L 0 is the low frequency coupling that can be defined as:

L 0 = - C ek σ f F 1 - 2 d Λ - C ek σ f F . ( 11
)
where F is the formation factor from Archie's law and σ f is the fluid's conductivity. The C ek coefficient is often called the electrokinetic coefficient. It characterizes the linear relation between the potential gradient ∆V and the pressure gradient ∆P involved in a steady state fluid circulation (C ek = ∆V/∆P). This coefficient may be expressed as a function of the volumetric charge density and the hydraulic conductivity [START_REF] Bolève | A new formulation to compute self-potential signals associated with ground water flow[END_REF] but this dependence to hydraulic properties may be debated [START_REF] Jouniaux | Reply to comment by a. revil on" seismo-electrics, electro-seismics, and seismo-magnetics for earth sciences" by l. jouniaux and f. zyserman[END_REF]. For laminar flows, it can also be expressed by the Helmholtz-Smoluchowski equation [START_REF] Overbeek | Electrochemistry of the double layer[END_REF] on the condition that the surface conductivity can be neglected:

C ek = f ζ η f σ f ( 12
)
where ζ is the zeta potential itself depending on pH, mineral and fluid composition [Jouniaux and Zyserman, 2016]. By compiling numerous streaming potential studies in sand and sandstones, [START_REF] Jouniaux | Electrokinetics in earth sciences: a tutorial[END_REF] proposed the simple empirical relation C ek = -1.2 x 10 -8 σ -1 f directly linking the electrokinetic coefficient to the fluid conductivity in the σ f = [10 -3 -10 1 ] S/m range.

Eventually, the slownesses of longitudinal P and transverse S waves are given by [START_REF] Pride | Electroseismic wave properties[END_REF] for plane waves:

s P (ω) =        1 2 γ(ω) - 1 2 γ(ω) 2 - 4 ρ(ω) ρ M H -C 2 ρ t ρ + ρ(ω)L(ω) 2 ε(ω)        1/2 , s S (ω) = 1 2 ρ t G + 1 2 µ ε(ω) 1 + ρ(ω)L(ω) 2 ε(ω) + 1 2 ρ t G -µ ε(ω) 1 + ρ(ω)L(ω) 2 ε(ω) 2 -4µ ρ 2 f L(ω) 2 G        1/2 (13) with γ(ω) = ρM + ρ(ω)H 1 + ρ(ω)L(ω) 2 / ε(ω) -2ρ f C H M -C 2 .
From this set of equations and the parameters of table (1) we get the magnitudes of the seismoelectric transfer functions of P i (i = f for fast P and i = s for slow P)

and S waves in quartz sand and sandstone (figure 1). In all ψ Pi and ψ S curves, the low and high frequency limits are pretty obvious: each curve tends to a different asymptote appart of the Biot frequency. We particularly notice the low frequency limit of ψ Pi tending to a non-dynamic (linear) transfer function, as expected by Garambois and Dietrich [2001]. The magnitude ψ S is very low compared to that of ψ Pi , and it seems very difficult to observe the seismoelectric field associated to the propagation of shear wave. This is a very interesting point since, as discussed above, the contribution of P waves is negligible in seismomagnetic field: it theoretically seems that measuring both seismoelectric and seismomagnetic fields would be an original way for separating P and S waves in a mixed seismic propagation. We also notice the ψ Ps transfer function to be very large compared to ψ Pf . The Biot slow wave is therefore expected to be enhanced in seismoelectric fields, especially in the diffusive regime [START_REF] Garambois | Observation of the diffuse biot slow wave via its electrokinetic coupling: a numerical perspective[END_REF]. Indeed, measuring the seismoelectric transfer functions would be an original and promising way to experience the Biot slow wave, that is still poorly observed and understood [START_REF] Garambois | Observation of the diffuse biot slow wave via its electrokinetic coupling: a numerical perspective[END_REF].

In the purpose of providing original observations for these fundamental poroelasticity issues, we aim to validate the Pride theory by performing laboratory experiments under controlled conditions. In section 2, we discuss some issues or questions that might be en-

σ f S.m -1 10 -3
Viscosity of the fluid

η f Pa.s 10 -3 Electrokinetic coefficient C ek V.Pa -1 -2 × 10 -5
Pore space coefficient m p 6 countered when designing seismoelectric experiments. We show afterwards some results

based on experiments we performed during the last decade. In section 3, we come back on the seismoelectric and seismomagnetic measurements we realized in a low noise laboratory [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF][START_REF] Bordes | Evidence of the theoretically predicted seismo-magnetic conversion[END_REF]. Eventually, in section 4, we evoke the results obtained by [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] and [START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF] in order to quantitatively validate the transfer function approach, for various fluid's conductivities and water saturations.

Designing seismoelectric experiments

The quantitative study of seismoelectric transfer functions needs carefully designed experiments including simultaneous seismic and seismoelectric measurements. A welldesigned experiment is intended to be a downscaled analog of field surveys, and/or to involve the same dissipation processes as in real conditions. When dealing with Pride's theory, the key parameter will be the vicinity of the Biot frequency. For this purpose, a compromise has to be sought by balancing frequency and petrophysical properties: the nominal frequency has to be as close to/far from the Biot frequency as in field survey. In section 2.1, we show that the choice of sandbox experiments in the kiloHertz range is a comfortable compromise, by comparing the f / f c ratio for various possible experiments.

Quantitative measurements require the acquisition system to be carefully designed (section 2.2), seismic sensors to be well calibrated, the coupling with the porous sample to be the best as possible, and the electrodes and dipole length to be carefully chosen. Thus, we aim to check in section 2.3 the linearity of seismoelectric amplitude versus source energy.

The choice of metallic electrodes, is discussed in section 2.4 and the reconstruction of dipoles using measurements referred to a common electrode (reference electrode) is discussed in section 2.5. Eventually we show the linearity of potential gradient to be unsatisfied for large dipoles and we make some suggestions on the best dipole length in section 2.6.

On the choice of sandbox experiments

When dealing with Pride's formulation, that is based on Biot's theory, the choice of the frequency range must be balanced against petrophysical properties involved in the Biot frequency f c (see equation 9). Indeed, depending on the frequency f , relative fluid/solid motions may be controlled by viscous ( f / f c < 1) or inertial ( f / f c > 1) forces. The seismoelectric transfer functions are expected to reach their maximum magnitude in the viscous domain (low frequencies) and to drop at highest frequencies (see figure 1). In terms of seismoelectric transfer functions, field surveys at seismic frequencies might be ideal conditions provided that electrokinetic and signal-to-noise ratios are high enough.

Working at ultrasonic frequencies does not seem be the simplest way to experiment on seismoelectric phenomena. On the one hand because they stand in the inertial domain for which transfer functions are dropping, on the second hand because seismic waves are strongly damped due to combined scattering and attenuation. Nevertheless it is possible to work on such frequencies (higher than 20k H z), for example in sandstone, by dealing with small samples [Zhu et al., 2000]. In this case, the f / f c ratio would stand around 10 1 -10 2 (see table 2) and the observations might be a nice analog of borehole measurements performed in the same frequency range.

Nevertheless, the f / f c ratio for field studies at seismic frequencies stands around 10 -3 (viscous domain) and upscaling the results from acoustic measurements to field sur-

Solving impedance issues for electric measurements

When dealing with quantitative measurements of electrical potentials, the acquisition system must be carefully chosen and follow basic precautions. Indeed, its input impedance R in must be very high compared to the impedance between the electrodes R di p . The later can be easily measured for various length of dipoles: in the case the assumption R di p R in would be verified, the quantitative measurement of ∆V may be envisaged safely. In the opposite case, the equivalent resistance would be R

-1 = R -1 di p + R -1
in and the acquisition system would act as a tension divider. It would therefore be necessary to correct the measurement ∆V m in order to obtain the real potential difference ∆V by:

∆V = ∆V m R di p + R in R in (14)
Nevertheless, when dealing with various fluid conductivities and/or saturations, changes in R di p may be very strong and the correction may become unrealistic. Thus, the best solution would be to increase the input impedance until 0.1 to 1 GΩ by custom-made preamplifiers that may also include gain or digital filtering [START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] .

2.3 Checking the linearity of the coseismic effect for variable source magnitude

As shown in section 1, the conversion from seismic to electric energy follows a dynamic transfer function dropping with frequency (figure 1). A seismic source which waveform would be reproducible is therefore expected to be converted into a seismoelectric field which waveform should be reproducible as well. Eventually, changes in source's magnitude is expected to involve the same changes in seismoelectric amplitudes with a linear relation. This assumption is very important when stacking or normalizing processing are used, especially when dealing with mechanical sources that are less reproducible than piezoelectric sources. When designing the experiment [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF][START_REF] Bordes | Evidence of the theoretically predicted seismo-magnetic conversion[END_REF],

checked the maximum amplitudes recorded on a dipole for various source magnitudes, all other parameters being constant. As expected, the potential difference increases quasilinearly with the source magnitude (figure 2). Thus, it makes sense normalizing seismic and seismoelectric records by the source magnitude. This normalization, that is possible only by systematically recording the source signal, allows a quantitative interpretation of amplitudes, even when changes in source magnitudes cannot be avoided.

Unpolarizable electrodes versus metallic electrodes

The question, whether or not non-polarising electrodes are needed in seismoelectric measurements, is often discussed. Metallic electrodes are generally not admitted for spontaneous potential surveys since they may be affected by biases due to polarisation effects. Indeed, these effects can be particularly strong at very low frequency when metallic electrodes are kept in a surrounding electrolyte for a very long time. Nevertheless, seismoelectrics consist in measuring transient potential gradients on very short times (typically few milliseconds) and polarisation issues cannot be addressed with the same point of view.

In field and laboratory seismoelectric studies, both electrode systems are used, assuming the polarisation effect to be poor. This issue has been addressed at field scale by

Max. amplitude of the seismic source (g) Max. amplitude of ΔV (mV) Beamish [1999] who used electrodes of three types: 1) stainless-steel (standard) electrodes, 2) lead rods, 3) non-polarizing Cu/CuSO4 electrodes. He concluded that the received voltages appear largely independent of electrode type as long as the ground contact resistance was no issue. Hence he would advocate for porous-pot electrodes only in arid environments, or advice to water the metal rods. Many other field studies used simple metallic electrodes on the surface or in boreholes [START_REF] Butler | Measurement of the seismoelectric response from a shallow boundary[END_REF][START_REF] Mikhailov | Using borehole electroseismic measurements to detect and characterize fractured (permeable) zones[END_REF][START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF][START_REF] Haines | Seismoelectric imaging of shallow targets[END_REF][START_REF] Strahser | Polarisation and slowness of seismoelectric signals: a case study[END_REF][START_REF] Dupuis | Vertical seismoelectric profiling in a borehole penetrating glaciofluvial sediments[END_REF][START_REF] Dupuis | Seismoelectric imaging of the vadose zone of a sand aquifer[END_REF]. Many laboratory studies were performed with metallic electrodes as well: [START_REF] Chen | Experimental studies of seismoelectric effects in fluidsaturated porous media[END_REF][START_REF] Zhu | Experimental studies of electrokinetic conversions in fluid-saturated borehole models[END_REF]Zhu et al., , 2000;;[START_REF] Zhu | Seismoelectric and seismomagnetic measurements in fractured borehole models[END_REF][START_REF] Dukhin | Seismoelectric effect: A nonisochoric streaming current. 1. experiment[END_REF][START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF]. Some authors used Ag/AgCl non-polarising electrodes in the laboratory [START_REF] Block | Conductivity dependence of seismoelectric wave phenomena in fluid-saturated sediments[END_REF][START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF][START_REF] Bordes | Evidence of the theoretically predicted seismo-magnetic conversion[END_REF][START_REF] Schakel | Laboratory measurements and theoretical modeling of seismoelectric interface response and coseismic wave fields[END_REF][START_REF] Smeulders | Seismoelectric conversion for the detection of porous medium interfaces between wetting and nonwetting fluids[END_REF] but did not discuss their absolute necessity, except [START_REF] Zhu | Experimental measurements of the streaming potential and seismoelectric conversion in berea sandstone[END_REF] who measured streaming potential and seismoelectric conversion on the same experiment.

Beamish's conclusions were confirmed by [START_REF] Bordes | Etude expérimentale des phénomènes transitoires sismoélectromagnétiques: Mise en oeuvre au laboratoire souterrain de rustrel, pays d'apt[END_REF] by comparing seismoelectric waveforms obtained by Ag/AgCl and silver electrodes. The datasets shown in figure 3 were obtained in two distinct experiments. The experimental apparatus was a vertical column of quartz sand, filled of water from the lower extremity [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF].

The main drawback of this apparatus is the fluid accumulation at the bottom of the column with a possible saturation gradient along the sample. This effect may be particularly strong in the upper part of the column. In the following, we therefore focus the comparison between both types of electrodes on offsets larger than 40cm.

In this experiment, potential gradients were measured by referring to a common electrode located 95cm from the source, and the seismoelectric field was calculated as the ratio of potential gradient and dipole length (E = -∆V/∆x). All the signals shown in this figure were normalized by the trace maximum (for a clearer view of waveforms) and by the source amplitude (for correcting the source non-reproducibility). The maximum of the seismoelectric amplitudes is shown in the right part for a quantitative comparison.

The saturation gradient along the column might explain the increase of seismoelectric amplitudes. Eventually the comparison of these datasets shows both signals to be very close in amplitude and waveform, the strong differences observed in later parts being probably due to a lack of reproducibility of the mechanical source. We conclude from this test both non-polarising and metallic electrodes to be well suited for seismoelectric measurements, as initially suggested by [START_REF] Beamish | Characteristics of near-surface electrokinetic coupling[END_REF].

Local measurement versus dipole reconstruction

In field measurements, the seismoelectric acquisition consists of dipoles arrays, each dipole being composed of two electrodes located on either side of a seismic sensor. The dipole length l is chosen in order to provide the best signal to noise ratio, as large dipoles are tending to favour both seismoelectric signals and electrical ambiant noise. In this case, the local potential gradient measured by electrodes located at the offset x is: The main drawback of the local measurement is that datasets cannot be rearranged for testing other dipole geometries, since dipoles are completely independent from each others. A possible alternative is the measurement by referring to a common electrode.

∆V loc (x) = V (x - l 2 ) -V (x + l 2 ) ( 15
Hence, locating the electrodes every distance l, the potential gradient at offset x measured by the reconstructed dipole of length l would be:

∆V rec (x) = V (x - l 2 ) -V (ref) -V (x + l 2 ) -V (ref) ≡ ∆V loc (x). (16)
Indeed, the reconstructed and local dipoles are expected to provide the same signal, which can be easily checked by operating both acquisitions, as shown in figure 4. This simple test shows both acquisition systems to be consistent, even if some discrepancies are observed in the later part of the signal. This test confirms the reference electrode system to be an interesting alternative to the local measurement.

From the initial reference measurement, it is afterwards possible to provide many datasets corresponding to other electrode configurations, for example with various dipole length (l, 2l, 3l...). Another disadvantage of the local dipoles is that many more electrodes have to be set up. It is therefore possible to process many more traces by using the reconstructed dipoles, for the same experimental setup, that is a very interesting point for understanding the various wavefronts travelling in the sample.

On the best electrode configuration

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, the influence of the spacial distribution of electrodes on the estimate of electric field amplitudes. Indeed, estimating the local electric field by using its definition E = -∇V assumes the potential gradients to be measured between two points sufficiently close. In this case, isopotential curves are expected to be almost parallel between electrodes, i.e. the variations of potential are gradual and almost linear (null divergence). If this assumption is completed, the local amplitude of the electric field

E x = -∆V/∆x (17)
should not depend on the selected dipole-length, and the ∆V = f (l di p ) curve should be linear (∆x = l di p ).

When dealing with seismoelectric fields, the shape of isopotential curves is completely driven by the original seismic wave and its characteristic wavelength λ. At a given time, signal reaching electrodes of dipoles larger than a quarter wavelength might be out of phase: the isopotential curves within the dipoles might be ungradual (bumps and holes) and the E x = -∆V/∆x relation might be not adapted. This statement can be checked by using reconstructed dipoles of various lengths as proposed by [START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF]. In this experiment, seismoelectric potential gradients at various offsets were recorded by a dense electrodes array, allowing many different dipole-length l di p . In figure 5, the maximum amplitudes of the first arrival ∆V are displayed as a function of l di p /λ ratio and clearly show the linear relation to be obtained only for dipole smaller than a quarter wavelength (even a fifth for some traces). Eventually, [START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF] concluded that the smaller the dipole length is, the better the measurement of the electric field will be, as long as the signal to noise ratio remains sensible.

Measuring the seismomagnetic field

Whether seismoelectric measurements were performed in the laboratory or in the field, the measurement of the seismomagnetic field was rarely addressed [START_REF] Zhu | Seismoelectric and seismomagnetic measurements in fractured borehole models[END_REF][START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF][START_REF] Bordes | Evidence of the theoretically predicted seismo-magnetic conversion[END_REF]. As shown in section 1, P waves are pure divergence and the corresponding seismoelectric field has no magnetic counterpart. The seismomagnetic field B is then expected to be coupled to the sole S waves, whose seismoelectric couplings are very low (figure 1). Indeed, B is expected to be of very weak magnitude and the measurement seems possible only under very low electromagnetic magnetic noise.

The [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF][START_REF] Bordes | Evidence of the theoretically predicted seismo-magnetic conversion[END_REF] experiments were therefore performed in a low noise underground laboratory (LSBB, Rustrel France) whose noise level was measured by [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 be lower than 2 f T/ √ H z above 10 H z. The experimental apparatus was made of a porous sample, a seismic source, electric dipoles, two homemade fluxgate magnetometers and accelerometers located within the ultrashielded chamber. All electronic devices were located outside the chamber for avoiding electromagnetic disturbances from the instruments. Seismic and seismoelectric measurements were simultaneously performed in a common experiment whereas seismomagnetic field was recorded in a dedicated experiment. The experimental apparatus was carefully designed for avoiding any vibrations of sensors or metallic elements. Reproducibility was also ensured by using a controlled setup procedure, including sand packing and water filling.

These experiments eventually showed both seismoelectric and seismomagnetic field to be measurable in this very favorable environment (figure 6). Their electrokinetic origin was confirmed by checking that no coherent arrival was recorded in dry sand. In "fluid filled" sand, the first arrival of seismic and seismoelectric fields have the apparent velocity of a P wave [START_REF] Bordes | First laboratory measurements of seismo-magnetic conversions in fluid-filled fontainebleau sand[END_REF] in a partially saturated compacted sand ( 1300 m/s).

As for the seismomagnetic field, its apparent velocity is consistent with S waves propagating in the same medium ( 900 m/s). This experiment was an interesting step in the validation of Pride's theory since it confirms the existence of a measurable seismomagnetic field, most likely with an S wave apparent velocity. Nevertheless, measuring the 3 components of all fields would be a valuable improvement, since it would enable the evaluation of wave polarisation and would therefore confirm the identification of P and S waves. Gaskarov [1971] 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 approxi- mately exponentially". This effect was afterwards 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].

Within the last decade, further similar studies have been conducted either on sand and glass beads [START_REF] Block | 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. Investigation of the transfer function dependence on fluid conductivity was also conducted by [START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF] for the quantitative validation of Pride's transfer functions (figure 7). In this experiment, the fluid conductivity was controlled by progressive addition of NaCl salts to eventually cover fluid conductivities ranging from 2.5 mS • m -1

to 10 mS • m -1 . As σ f increases, the seismoelectric amplitude drastically decreases (almost by one order of magnitude) when seismic amplitude remains mostly invariant, and the |E/ ü| is therefore damping very fast. Eventually, experimental observations are consistent with theoretical predictions, both in order of magnitude and conductivity dependence of seismoelectric transfer functions.

Effect of water saturation in sands and dynamic compatibility phenomenon

Although the dependence of seismoelectromagnetic signals on fluid parameters (fluid's conductivity, pH, viscosity) were numerically and/or experimentally addressed, the impact of partial saturation has been rarely studied. However, full saturation is often not achieved in reservoirs, since they generally contain at least a few percents of gas or oil, strongly tion of the dominant frequency (blue: offset=20cm, red: offset=30cm and green: offset=40cm 8). These studies showed that the transfer functions behave as a plateau in the S w = [0.3 -0.9] range as long as the fluid distribution remains homogeneous (all experiments from A and first imbibition from B). The seismoelectric transfer function at shortest offset was clearly lower than those measured at further ones in experiment A. This observation was not fully understood, but near-field effects were suspected. Eventually, the Pride theory extended to partial saturation succeeded in predicting the order of magnitude of the transfer functions.

By investigating the full saturation case, [START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF] obtained an original observation: the sign of the transfer function was changing at a specific saturation S * related to the fluid distribution. This phenomenon can be recovered in theoretical transfer functions by accounting for fluid heterogeneities via Brie's formulation for the effective bulk modulus of the fluid [START_REF] Brie | Shear sonic interpretation in gas-bearing sands[END_REF]. Hence, S * is expected to be lower for an heterogeneous fluid (i.e. laid out in patches) than for an homogeneous onde (fine mixture of gas and water). Actually, the water was sucked out very rapidly during the drainage experiment and the formation of patches is very likely.

The S * saturation corresponds to the peculiar state of "dynamic compatibility" predicted by Biot [1956a]. In his original paper, Biot evoked: "A remarkable property is the possible existence of a wave such that no relative motion occurs between fluid and solid".

This condition is obtained when an equilibrium between elastic and dynamic constants is achieved. As suggested by [START_REF] Hu | The electric field induced by the fast p-wave and its nonexistence in a dynamically compatible porous medium[END_REF] this condition can be observed at a critical porosity, but it can be reached at a critical saturation as well. Eventually, under the dynamic compatibility condition, neither attenuation nor seismoelectric field may be observed since both solid and fluid are moving in-phase. The polarity change in seismoelectric transfer functions obtained by [START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF] is therefore an original observation of this poorly experienced phenomenon.

In the last decade, the potential of seismoelectrics for the geophysical characterization of porous media has been proven by the spectacular development of numerical, field and laboratory studies in litterature. The coseismic part of the seismoelectric field is often considered to be devoid of interest since it comes with the seismic field. Nevertheless, this phenomenon carries the signature of relative fluid motion within the pores that is involved in attenuation and dispersion of the seismic waves. A quantitative study of these seismoelectric fields may actually provide a powerful tool for experiencing various phenomena described by poroelastic theories. We would advise the transfer function approach as the best way to study the coseismic coupling since it neutralizes the effect of any changes in seismic amplitudes when monitoring any parameter variations.

This quantitative interpretation requires well designed experiments including calibrated seismic measurements and electric dipole which length and location must be carefully chosen. The choice of the frequency f must be balanced against the Biot frequency f c in order to get a f / f c ratio close to that of field studies. We showed the sandbox experiments in the kiloHertz range to be a reasonable analog of field studies at seismic frequencies. We would recommend to measure the potential difference by referring to a common reference electrode located as far as possible from the source: this method provides large datasets that can be rearranged for various dipole length. Metallic electrodes may be used, and the best dipole length seems to be smaller than a fifth wavelength for ensuring the fundamental relation E = -∆V/∆x to be relevant.

In light of these recommendations, we performed a series of laboratory experiments in order to test some of Pride's expectations. For instance, by comparing seismomagnetic and seismoelectric fields in a sand column, we confirmed their respective coupling to S and fast P waves. We also showed the magnitude of theoretical transfer functions obtained from Pride's theory to be consistent with laboratory measurements, even under variable fluid conductivity and water saturation. We were also able to observe the "dynamic compatibility" predicted by Biot, this peculiar state in which both solid and fluid phases are moving in-phase.

All these experiments confirmed the coseismic part of seismoelectric fields to provide an interesting and powerful tool for investigating the effect of pore fluid on the seismic propagation in porous media. Many other studies might be considered. For exam-ple, the study of patchy saturation might show a particular effect on seismoelectric phenomenon combining both coseismic and interface conversions at patches boundaries [START_REF] Jougnot | Seismoelectric effects due to mesoscopic heterogeneities[END_REF]. Experiments focussing on very short offsets might also provide original observations of the diffusive part of the Biot slow wave [START_REF] Garambois | Observation of the diffuse biot slow wave via its electrokinetic coupling: a numerical perspective[END_REF].

In future experiments, special efforts will have to be deployed for obtaining dynamical transfer functions, ideally on a large range of frequencies. Eventually, the recent development of various codes for the computation of the seismoelectric effect should provide great opportunities for improving the interpretation of both laboratory and field data.

Figure 1 .

 1 Figure 1. Magnitudes of dynamic transfer functions of slow P (ψ Ps ), fast P (ψ Pf ) and S waves (ψ S ) computed for water filled quartz sand (black curves) and sandstone (red curves), and respective Biot's frequencies f c .

Figure 2 .

 2 Figure2. Checking the linearity of the coseismic effect in Fontainebleau sand for variable source magnitude (pneumatic device) [adapted from[START_REF] Bordes | Etude expérimentale des phénomènes transitoires sismoélectromagnétiques: Mise en oeuvre au laboratoire souterrain de rustrel, pays d'apt[END_REF]

Figure 3 .

 3 Figure 3. Comparison of seismoelectric signals obtained by unpolarizable and silver electrodes [adapted from[START_REF] Bordes | Etude expérimentale des phénomènes transitoires sismoélectromagnétiques: Mise en oeuvre au laboratoire souterrain de rustrel, pays d'apt[END_REF]

  x)=ΔV ref (x-l/2)-ΔV ref (x+l/2)

Figure 4 .

 4 Figure 4. Comparison of local SE measurements with their reconstructed waform using a reference electrode [adapted from[START_REF] Bordes | Etude expérimentale des phénomènes transitoires sismoélectromagnétiques: Mise en oeuvre au laboratoire souterrain de rustrel, pays d'apt[END_REF]

Figure 5 .

 5 Figure 5. Checking the validity of seismoelectric field reconstruction from potential difference measurements [adapted from[START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF]

4

  Quantitative validation of Pride's transfer functions 4.1 Checking the effect of fluid's conductivity Fluid conductivity is the most adjustable parameter in laboratory experiments: its change demands no great operation but to equilibrate the medium towards the wanted conductivity value by continuous water circulation. As early as the 70's, Parkhomenko and

Figure 6 .

 6 Figure 6. Seismic, seismoelectric and seismomagnetic signals recorded along a cylindrical sample in dry and fluid filled sand [adapted from[START_REF] Bordes | Etude expérimentale des phénomènes transitoires sismoélectromagnétiques: Mise en oeuvre au laboratoire souterrain de rustrel, pays d'apt[END_REF]

Figure 7 .

 7 Figure 7. Seismoelectric transfer function as a function of fluid's conductivity measured in a sandbox experiment [adapted from Holzhauer et al., 2016]. Theoretical curves at 0.5 k H z (dashed line) and 2 k H z (solid line) were obtained using the dynamic formulation of ψ Pf including the Jackson [2010] model for saturation dependence of the electrokinetic coefficient.

Table 1 .

 1 Parameters used for the computation of seismoelectric transfer functions of figure 1 in quartz sand and sandstone.

	Parameter	Notation	Unit	Value
				Waterfilled sand / sandstone
	Porosity	φ		0.4 / 0.15
	Formation factor	F		4 / 15
	Intrinsic permeability	k 0	m.s -2	10 -11 / 10 -13
	Bulk modulus of the solid	K s	GPa	0.02 / 1
	Bulk modulus of the fluid	K f	GPa	0.02 / 1
	Bulk modulus of the frame	K f r	GPa	0.02 / 1
	Shear modulus of the frame	G	GPa	0.012 / 0.6
	Density of the solid (quartz)	ρ s	kg.m -3	2650
	Density of the fluid (water)	ρ f	kg.m -3	1000
	Electrical conductivity of the fluid

Table 2 .

 2 Comparison of key parameters in field and laboratory seismoelectric studies. Computation of the Biot frequency was performed for waterfilled sandstone and sand whose properties are given in table 1.

		Field surveys	Ultrasonic	Sandbox	Sandbox
		waterfilled sandstone waterfilled sandstone S w =0.5	S w =1
	Frequency
	f (H z)	50	10 6	1000	1000
	Biot's frequency
	f c (H z)	1.6 × 10 4	1.6 × 10 4	800	4000
	f/f c	3.14 × 10 -3	63	1.26	2.51 × 10 -1

veys may be difficult. It is more recommended to use lower frequencies(1 -10 k H z), i.e. closer to the Biot frequency. For this purpose, sandbox experiments have the advantage to use lower f / f c values. Mechanical seismic sources may be driven by compressed air or weight drop, and are particularly well suited for emitting frequencies in the kiloHertz range. They generate higher energy than piezoelectric transducers with limited electromagnetic disturbances. Moreover, many shapes of sandbox experiments can be envisaged (depending on which propagation mode is required), and instrumentation is easy to install (seismic receivers, electrodes, saturation probes....) with satisfying mechanical coupling.

  ). In experiment B, theoretical curves were obtained by a joint least-square inversion of velocities and seismoelectric transfer functions in order to estimate the best exponent in Brie's model. The best fit (bold lines) is surrounded by the curves obtained for a 10% misfit.The effect of water saturation on seismoelectric transfer functions was studied in sandbox experiments by[START_REF] Bordes | Impact of water saturation on seismoelectric transfer functions: a laboratory study of coseismic phenomenon[END_REF] and[START_REF] Holzhauer | Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand[END_REF]. The goal was to compare laboratory data with theoretical predictions from[START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] theory, extended to partial saturation by an effective fluid model as proposed in[START_REF] Warden | Seismoelectric wave propagation numerical modelling in partially saturated materials[END_REF]. The

	Bordes et al. [2015] experiment (A) focused on the S w = [0.3 -0.9] range for various
	offsets, when Holzhauer et al. [2016] (experiment B) succeeded in reaching the full satura-
	tion (figure

Acknowledgments

This work was supported by the French Ministry of Higher Education and Research, the French National Research Agency (ANR Transek project), the Center for National Scientific Research (CNRS), the ISIFoR Institute, and the TOTAL company. We acknowledge technical staff involved in these studies: M. Auguste, D. Boyer, Y. Brun, A. Cavaillou, B.

Guatarbes, J-M Martin, E. Normandin,V. Poydenot and P. Sénéchal. Jouniaux, L., and F. Zyserman (2016), A review on electrokinetically induced seismoelectrics, electro-seismics, and seismo-magnetics for earth sciences, Solid Earth, 7(1),

249.

Masson, Y. J., and S. R. Pride (2007), Poroelastic finite difference modeling of seismic attenuation and dispersion due to mesoscopic-scale heterogeneity, Journal of Geophysical -24-Confidential manuscript submitted to AGU Books

Confidential manuscript submitted to AGU Books influencing mechanical [Bachrach and Nur, 1998;Bachrach et al., 1998;[START_REF] Rubino | Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks[END_REF] as well as electrical [START_REF] Archie | The electrical resistivity log as an aid in determining some reservoir characteristics[END_REF] and electrokinetic [START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[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][START_REF] Allegre | Streaming potential dependence on water-content in fontainebleau sand[END_REF][START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF][START_REF] Allègre | Modelling the streaming potential dependence on water content during drainage: 1. A 1D modelling of SP using finite element method[END_REF][START_REF] Allègre | Influence of water pressure dynamics and fluid flow on the streaming-potential response for unsaturated conditions[END_REF] properties of the medium. Field investigations were performed by [START_REF] Strahser | Dependence of seismoelectric amplitudes on water content[END_REF] who measured both the seismoelectric coupling and the electrical impedance between electrodes and suggested that the water content should modify seismoelectromagnetic couplings. In the laboratory, [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] measured the seismoelectric potential during water imbibition of dried rocks. Their results showed a dependence of the seismoelectric effect on water content, but they did not measure seismic displacements nor acceleration assuming that it should not vary with a reproducible seismic source. However it is admitted that seismic amplitudes should depend on water saturation due to specific dissipation phenomena Carcione