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 Electro-osmotic coefficients are determined experimentally and compared to numerical data. The numerical model assumes random packings of grains of various shapes; the prediction corresponds to a modified analytical solution. A special set up was designed to measure all relevant properties independently as functions of the porosity and electrolyte concentration. Two model clays and a compact natural clay were used. Good agreement was found between experimental and numerical results, indicating that the electro-osmotic coefficient can be predicted from conductivity and permeability.
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 Electro-osmotic phenomena occur when an electrolyte flows along electrified solid surfaces. These phenomena are important in a large number of industrial fields such as cosmetics, paintings, mining, civil engineering [Keijzer al., 1999; Laursen, 1997]. In geophysics it has been suggested that anomalous electric and magnetic fields observed before earthquakes and volcanic activity could be generated by the electrokinetic effect [Ishido and Mizutani, 1981; Zlotnicki and Le Mouel, 1990] induced by water flow resulting from internal stresses, thermal buoyancy effects and meteoritic waters. In order to model these effects (see the pioneering contributions of Nourbehecht  and Fitterman ; see also Adler et al. ), it is important to estimate the electro-osmotic coefficient L12, which is equal to L21 because of the Onsager reciprocal relation.
 The major objective of the present experiments was to check our numerical prediction which indicated that conductivity and permeability can be used to estimate L21 [Coelho et al., 1996]. Moreover, the numerical prediction corresponds to a modified analytical solution originally valid for circular Poiseuille flows which relates these three quantities. In the next section, this analytical solution is briefly reviewed. The materials, the experimental procedure and the experimental cell are described in Section 3. Section 4 is devoted to the analysis of the experimental data and to their comparison with the analytical solution.
 Consider a dilute electrolyte which fills the pores of a porous medium whose surface is charged. In electro-osmosis, the medium is subjected to an external electric field E. As a result, a body force appears within the Debye layer close to the wall, which sets the ions of this region into motion. Because of the fluid viscosity μ, an overall seepage velocity U is generated in the absence of any macroscopic pressure gradient ∇P. Hence, application of an external electric field E results in the flow of an electric current I and in an electro-osmotic velocity U proportional to E, if E is sufficiently weak.
 On the other hand, when a macroscopic pressure gradient is applied to the porous medium, the fluid percolates with a Darcy velocity U. Moreover, the ions within the Debye layer are set into motion with a disturbed distribution. This results in a macroscopic electric current density I in the absence of any external electric field.
 When the two “forcing factors” E and ∇P are applied simultaneously to the porous medium, one obtains
where σ is the electric conductivity tensor, K the permeability tensor and L12 = L21t are the electro-osmotic tensor coefficients. For isotropic media, these tensors reduce to scalars denoted by the usual letters; for instance, K stands for K.
 In order to determine the tensors K, σ and L21, three coupled partial differential equations are solved. The electrical potential, the ionic concentrations and the velocities are solutions of the Poisson-Boltzmann, the convection diffusion and the Stokes equations, respectively.
 These equations were numerically solved [Coelho et al., 1996] for a large variety of geometries, such as cubic and orthorhombic arrays of spherical and ellipsoidal particles, random packings of such particles and reconstructed media.
where σ∞ is the fluid conductivity and σ the macroscopic conductivity of the medium filled by the electrolyte. Therefore, Coelho et al.  represented their numerical results as
where εel, ζ, κ−1 are the fluid permittivity, the zeta potential and the Debye-Hückel length, respectively.
 The representation equation (3) was used to display the numerical data in Figure 6. The numerical data are close to the predictions derived from the analytical solution for circular Poiseuille flows of the Poisson-Bolztmann, the convection diffusion and the Stokes equations which govern the local fields
where the radius of the tube is replaced by Λ′.
3. Experimental Methods
 Because our objective is to verify equation (4) experimentally, we have to measure independently all the quantities which appear in this formula.
3.1. Electro-osmotic Cell
 Cells similar to the one used here are detailed by Mammar et al.  and Rosanne et al. . The cell consists of a circular poly(methyl methacrylate) (PMMA) cylinder of internal diameter equal to 15 mm, and is inserted into a stainless steel frame. The clay sample material is located between two sintered bronze plates with a pore size between 40 and 90 μm; the plate thickness is 4.5 mm.
 The clay sample is separated from the bronze plates by polyvinylidene difluoride (PVDF) membranes (from Millipore); the order of magnitude of the pore diameter in the membranes is equal to 0.65 μm. The desired compaction pressure Pc is exerted on the bronze plates by a piston, which moves along the axis of the cell by turning a screw. Pc is smaller than 40 bars.
 The clay sample, the PVDF membranes, the two bronze plates, and the lower part of the PMMA cell are placed in a large dish which contains the electrolyte solution. The entire system is then degassed.
 The electrical potential is imposed on the bronze electrodes. Measurements must be made during the first few minutes of the experiments in order to avoid polarization of the electrodes.
 Three types of materials were used for the experiments, namely two model powders (muscovite mica and sodium montmorillonite) and a natural compact argilite extracted in East France from a Callovo-Oxfordian formation. Quantities relative to these materials will be indicated by the subscripts mi, mo and ar, respectively.
 The muscovite mica (Comptoir des Minéraux et Matières premières) is essentially composed of SiO2 (48%) and Al2O3 (34%). An analysis by SEM shows a lamellar structure with grains of diameters about 4 μm; these grains tend to aggregate into larger clusters.
 The sodium montmorillonite is a bentonite (OENO, France) whose ionic exchange capacity is equal to 80 meq/100 g. When immersed into water, this clay may double its volume and form a gel. This powder was also analyzed by SEM; the dimensions range between grains of 2 μm diameter and clusters of 10 μm diameter.
 A SEM analysis of the argilite powder shows an heterogeneous structure with many aggregates. Because of the careful crushing process to obtain powder from the original block, the average grain radius ranges from 1 to 10 μm.
 The densities ρ of the three clays were obtained by standard weight and volume measurements: ρmi = 3150 ± 150 kg/m3, ρmo = 2450 ± 180 kg/m3 and ρar = 2660 ± 150 kg/m3. The specific surface Ssp was measured on the two model powders by nitrogen adsorption with Pc = 0: Ssp,mi = 2.5 107 m−1 and Ssp,mo = 1.5 109 m−1.
 The solute was sodium chloride supplied by SIGMA (purity 99.5%). The solvent was pure water filtered by an HPCL Maxima unit. The concentration C ranged between 5 10−5 mol/l and 10−2 mol/l.
 The zeta potential ζ was estimated by measuring the electrophoretic mobility of clay particles in electrolyte solutions. Because of the range of particle dimensions and of C (i.e., of small κ−1), the Smoluchowski formula [Hunter, 1988] can be used for all particle shapes with an estimated precision of 10%
where ue is the electrophoretic mobility.
 Results for ζ in various NaCl solutions for the three studied materials are displayed in Figure 1. At a constant pH ∼ 5, ζmi and ζmo are seen to be independent of C, while ∣ζ∣ar increases sligthly with C. Note that montmorillonite is more charged than muscovite mica, which is in turn more charged than argilite. Some data due to Delgado et al.  are in excellent agreement with our measurements.
 The permeability K was measured by generating a steady flow, by means of a constant pressure difference ΔP ≈ 2.4 105 Pa (2.4 bar). Darcy's law is then used to determine K
where Q (m3/s) is the volumetric flow rate (obtained by measuring the liquid mass which flows through the sample during a given time), S is the sample crosssectional area and L the sample length.
 The sample conductivity σ was measured by a classical method. A constant dc voltage ΔV ∼ 1 V (Convergie SDST 50/1.5c power supply) was imposed between both bronze plates at a vanishingly small pressure difference (ΔP = 0). The generated electric current i was measured with a multimeter (Keitley model 175A). Then, if ΔV is not too large, Ohm's law is valid and σ is estimated by
where i0 is the current intensity at t = 0.
 The electric current was observed to decrease during the measurement time when the electric potential difference is set, probably due to the formation of polarization layers on the electrodes. Simultaneously, a water flow rate Qe is induced by ΔV, so that L21 is obtained by
The flow rate Qe was determined from the slope of the straight line fit of the collected liquid mass as a function of time. We consider the slope during the first three minutes of the experiment before Qe starts to decrease due to polarization.
 The pH was determined after each experiment in the liquid inside the cell and in the outlet liquid after permeation.
4. Experimental Results and Analysis
 Results are displayed in Figure 2. Kmo is significantly smaller than Kar and Kmi, which are relatively close one to another. It is interesting to note that for the same value of Pc, we have εar < εmi < εmo. This might be due to the less polydisperse character of the argilite powder. It was not possible to obtain porosities smaller than 0.4 for mica and 0.8 for montmorillonite with our experimental cell, because the cell was not designed to withstand large compaction pressures. It is important to note that the permeability obtained for the argilite original sample is in perfect agreement with Kar, as displayed in Figure 2.
 The effects of Pc and κ were studied by measuring σ for several ε at fixed C, and for several C at fixed ε. It is more convenient to represent the experimental results in terms of the electric formation factor F, which is defined as the ratio between the fluid conductivity after permeation σfp and the porous medium conductivity σ
 The results obtained for the three powders are displayed in Figure 3. It appears that F depends only on ε and is independent of C. This means that for the studied materials, the influence of C on σ due to double layer effects is very small. For a given ε, F is of the same order for the three materials. As expected, F is a decreasing function of ε.
 Finally, the measurements obtained for 0.34 ≤ ε ≤ 0.85 can be compared to the numerical ones obtained for 0.1 ≤ ε ≤ 0.75 on various packings by Coelho et al.  (Figure 3). Such calculations were performed for monodisperse particles when surface effects are not considered. Numerical and experimental results which overlap for 0.34 ≤ ε ≤ 0.37 can be described by an Archie's law, F = ε−2. These results indicate that σ is essentially dominated by geometrical effects.
4.3. Electro-osmotic Coefficient
L21 was measured by the same procedure as σ. Results concerning L21 as a function of C are shown in Figure 4 for the three materials. εmi and εar have been chosen equal to 0.6, and εmo equal to 0.85. L21 is independent of C in the studied range for mica and argilite; the mean values were found to be equal to (3.1 ± 0.4) × 10−9 m2V−1s−1 and (3.3 ± 1.2) × 10−9 m2V−1s−1, respectively. Values of L21 are comparable for mica and argilite. However, L21,mo increases with C up to C < 10−3 mol/l and then reaches an almost constant value equal to (2.4 ± 0.3) × 10−9 m2V−1s−1. In spite of the larger value of εmo, L21,mo is smaller than L21,mi and L21,ar.
 Values of L21 for the three materials as a function of ε are displayed in Figure 5. The data were all measured at C = 10−4 mol/l. L21 increases with ε. Again, L21,mi is comparable to L21,ar. To obtain the same value of L21, εmo must be twice larger than εmi and εar.
 The experimental values L21 were first represented as functions of κΛ′ where Λ′ is the characteristic length scale defined by equation (2). In such a representation, L21 increases with κΛ′. For the same κΛ′ (<20), L21,ar is lower than L21,mi which in turn is lower than L21,mo. For large κΛ′ (≥20), the data are seen to tend to the same constant of about 3 × 10−9 m2V−1s−1 for the three materials. Results relative to the dependence of L21 on the type of clay are correlated to the zeta potential.
 In Figure 6, all the results are gathered in the dimensionless form L′21 as functions of κΛ′, and compared to the numerical data recalled in Section 2. By standard techniques, the experimental errors on κΛ′ and L′21 are estimated to be equal to 25% and 60%. These experimental errors are clearly overestimated when Figure 6 is considered.
 It is remarkable to note that the experimental data cluster around a single curve with very little dispersion. Figure 6 has to be compared with Figures 4 and 5 where the dimensional value L21 undergoes large variations. As expected, L′21 tends to zero for large κΛ′, i.e., for thin double layers, and data are close to Overbeek's formula [Overbeek, 1952]. Moreover, the comparison with numerical results is very successful; therefore, the experimental data are also well approximated by equation (4).
5. Concluding Remarks
 The relation (4) provides an extremely useful tool to predict the electro-osmotic coefficient L21 in a simple way, which avoids its tedious measurement. The potential user has to know K and F to estimate Λ′. These quantities can be estimated from other measurements or from existing correlations e.g., in particle packings, there are many correlations in the literature. One can then easily derive L′21 from equation (4); the dimensional value is known when κ and ζ are known. This scheme was actually followed by Adler  to estimate the electro-osmotic coefficient on a macroscopic scale. It should be emphasized that equation (4) is reliable because it has a theoretical justification and because it has been checked experimentally and numerically. Moreover, because of the square root in equation (2), the errors in the estimations of K and F are divided by two for Λ′.
 Finally, this approach is likely to apply to other porous media such as sandstones and to other coupling coefficients, as was already shown numerically by Marino et al. .
 This work was partly supported by grants from ANDRA.