Allègre et al. recently presented new experimental data regarding the dependence of the streaming potential coupling coefficient with the saturation of the water phase. Such experiments are important to model the self-potential response associated with the flow of water in the vadose zone and the electroseismic/seismoelectric conversions in unsaturated porous media. However, the approach used to interpret the data is questionable and the conclusions reached by Allègre et al. likely incorrec

Linde, N.

Revil, A.

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Geophys. J. Int. (2011) 186, 113–114 doi: 10.1111/j.1365-246X.2010.04850.xGJIGeomagnetism,rockmagnetismandpalaeomagnetismComment on ‘Streaming potential dependence on water-contentin Fontainebleau sand’ by V. Alle`gre, L. Jouniaux, F. Lehmannand P. SailhacA. Revil1,2 and N. Linde31Colorado School of Mines, Department of Geophysics, Golden, 80401, CO, USA. E-mail: arevil@mines.edu2CNRS-UMR 5559-LGIT, Universite´ de Savoie, Equipe volcan, 73376 Le-Bourget-du-Lac, France3University of Lausanne, Institute of Geophysics, 1015 Lausanne, Switzerland. E-mail: niklas.linde@unil.chAccepted 2010 October 13. Received 2010 October 13; in original form 2010 August 11SUMMARYAlle`gre et al. recently presented new experimental data regarding the dependence of thestreaming potential coupling coefficient with the saturation of the water phase. Such exper-iments are important to model the self-potential response associated with the flow of waterin the vadose zone and the electroseismic/seismoelectric conversions in unsaturated porousmedia. However, the approach used to interpret the data is questionable and the conclusionsreached by Alle`gre et al. likely incorrectKey words: Electrical properties; Magnetic and electrical properties; Hydrogeophysics;Permeability and porosity; Fracture and flow.The study of electrokinetic properties of rocks has recently at-tracted the attention of researchers because of their applicationsto monitoring and characterizing the vadose zone as well as oil andgas reservoirs. Therefore, the high-quality drainage experimentsof Fontainbleau sand presented by Alle`gre et al. (2010) are wel-come. Unfortunately, the approach employed to interpret the data isunsuitable and none of the conclusions in Alle`gre et al. (2010) de-scribing how the voltage coupling coefficient varies with saturationare credible. Linde et al. (2007) demonstrated for a similar exper-iment how the self-potential data can be interpreted by extendingthe approach of Sill (1983) to unsaturated conditions. We addresshow to analyse such data sets to avoid misinterpretation and confu-sion for future studies. The approach to analyse data employed byAlle`gre et al. (2010) is not new, but adds to an increasingly impor-tant number of articles (e.g. Thony et al. 1997; Darnet & Marquis2004; Sailhac et al. 2004) focusing on self-potential signals in theunsaturated zone but ignoring the well-established physics of whereself-potential source currents are generated and how these sourcecurrents create measurable electrical fields.To start, several of the basic equations of Alle`gre et al. (2010)are wrong. A minus sign is missing in eq. (1) before the matrixof material properties. Therefore Ohm’s law, Darcy’s law and boththe electrokinetic terms are incorrect. The second term of eq. (3)is not equal to zero because of steady-state conditions (all the datadiscussed in Alle`gre et al. (2010) were acquired outside steady-state conditions), but because we are in the quasi-static limit ofMaxwell’s equations for which we consider that the diffusion ofelectromagnetic disturbances is nearly instantaneous. It is not statedthat eq. (4) on which Alle`gre et al. (2010) base their analysis is onlyvalid in the condition J = 0 (J being the total current densityincluding both the conductive and convective terms), which is onlya reasonable assumption for 1-D applications.In Alle`gre et al. (2010), the (streaming) voltage coupling coef-ficient is determined by local measurements of the pressure andelectrical potential at different heights in the sand column. Thisapparent voltage coupling coefficient is claimed to be the true onearising in the constitutive equations and is therefore compared tothe local saturation values of the water phase. As we know since Sill(1983), the electrical potential related to the flow of the pore wateris governed by a Poisson equation with a source term given by thedivergence of the source current density [obtained easily from eqs(2) and (3) of Alle`gre et al. (2010)]. During a drainage experiment,one of the main sources of the self-potential signals is located atthe bottom surface of the tube where the pore water drains freelyout of the sand as a result of gravity. This source arises becausethere is a discontinuity in the streaming potential coupling coeffi-cient through this interface (e.g. Crespy et al. 2008 who examineda similar problem). Therefore, the full set of equations should beused and solved together to determine a relationship for how thevoltage coupling coefficient varies with water saturation. Alle`greet al. (2010) oversimplify the problem by solving locally only theconstitutive equations and ignoring the rest of the experimentaldomain. This results in a sort of ‘apparent voltage coupling coef-ficient’ that bears little resemblance to the actual voltage couplingcoefficient. The correct approach for the forward problem is to firstsolve Richard’s equation (or another suitable equation for the flowproblem) and then to use this solution to determine the source termin the Poisson equation for the electrostatic potential, and finallyC© 2011 The Authors 113Geophysical Journal International C© 2011 RASGeophysical Journal International114 A. Revil and N. Lindeto solve this Poisson equation over the whole experimental domain(e.g. the column, a lysimeter or the whole subsurface) with suitablemacroscopic boundary conditions.A simple example that illustrates the lack of a physical basis inthe analysis of Alle`gre et al. (2010) for anything but 1-D systems isoffered by considering a pumping experiment with water injectionQ at a given depth d and with two electrodes located r1 and r2 awayfrom the borehole at the surface of a flat homogeneous earth withan atmospheric pressure boundary condition at the ground surface.A self-potential signal will be recorded between r1 and r2, but nopressure difference will be recorded. Using their eq. (4) to predictthe voltage coupling coefficient using these local measurementswill result in values of the voltage coupling coefficient that are±∞, which is obviously wrong. The same problem arises whenconsidering local measurements only of any experimental domain,except for laboratory samples in which a pressure difference isimposed on the sides and in which lateral variability normal to thepressure gradient is ignored (e.g., Guichet et al. 2003; Revil &Cerepi 2004).One can also see that the results of Alle`gre et al. (2010) arewrong by studying their fig. 6. Alle`gre et al. (2010) use a ho-mogeneous sand, so the inferred variation of the relative voltagecoupling coefficient should, for a consistent theory, be the same forany current pair assuming that differential compaction is of limitedimportance. Their fig. 6 shows how their relative voltage couplingcoefficient values increase gradually from the upper electrode pairsto the lower ones with a factor of approximately 5. This is un-physical as it should be close to 1 for a correct theory ignoringexperimental errors. Their apparent relative coupling coefficientsreach 200 at intermediate water saturations, which is in stark con-trast to any previously presented experimental work, including thework of Guichet et al. (2003). The main reason for these results isthat the authors assume that all measured voltages are of electroki-netic origin, which is clearly not the case as the SP signal doesn’treturn to the background after drainage (see their fig. 3). We haveobserved this behaviour for numerous drainage experiments and itis very likely related to electrode responses.The concept of electrokinetic water saturation introduced byAlle`gre et al. (2010) is a result of the incorrect modelling of theirexperiment. There is only one water saturation (properly definedby hydrogeologists) and a residual water saturation, which has beenshown to be the same for the relative permeability and the streamingpotential coupling coefficient by Linde et al. (2007) and Revil et al.(2007) (see also Revil & Cerepi 2004; Linde 2009; Jackson 2010).Alle`gre et al.’s (2010) so-called ‘electrokinetic residual saturation’is a result of the misinterpretation of the data as contributions thatare not of electrokinetic origin are used to infer a spatially and tem-porally variable voltage coupling coefficient as a function of watercontent.In conclusion, the experiment of Alle`gre et al. (2010) is very sim-ilar to the experiment of Linde et al. (2007). However, while Lindeet al. (2007) used the full set of field equations (obtained for boththe hydraulic and the electrical problems by combining the consti-tutive and continuity equations) to test a functional relationship ofhow the voltage coupling coefficient varies with water saturation, asubset of equations is inappropriately used in Alle`gre et al. (2010)without attempting to correct the data for signals that are not of elec-trokinetic origin. It is regrettable that the authors didn’t try to defineor test previously proposed functional models of how the voltagecoupling coefficient varies with water saturation. This would havemade their results less sensitive to data noise and it would be moreevident that electrode effects dominate the response for the latterpart of their experiment. The authors state in their conclusion that‘An unexpected behaviour of the electrokinetic coefficient has beenpresented in this work’. This ‘unexpected behaviour’ and the as-sociated statements and new concepts result from an incorrect useof electrokinetic theory lacking generality and an uncritical dataprocessing. A complete reinterpretation of this data set is necessaryto understand the streaming potential dependence on water-contentin Fontainebleau sand.REFERENCESAlle`gre, V., Jouniaux, L., Lehmann, F. & Sailhac, P., 2010. Streaming po-tential dependence on water-content in Fontainebleau sand, Geophys.J. Int., 182, 1248–1266, doi:10.1111/j.1365-246X.2010.04716.xCrespy, A., Revil, A., Linde, N., Byrdina, S., Jardani, A., Bole`ve, A. &Henry, P., 2008. Detection and localization of hydromechanical distur-bances in a sandbox using the self-potential method, J. geophys. Res.,113, B01205, doi: 10.1029/2007JB005042.Darnet, M. & Marquis, G., 2004. Modelling streaming potential(sp) signalsinduced by water movement in the vadose zone, J. Hydrol., 285, 114–124.Guichet, X., Jouniaux, L. & Pozzi, J.-P., 2003. Streaming potential of a sandcolumn in partial saturation conditions, J. geophys. Res., 108, 2141.Jackson, M.D., 2010. Multiphase electrokinetic coupling: insights intothe impact of fluid and charge distribution at the pore scale froma bundle of capillary tubes model, J. geophys. Res., 115, B07206,doi:10.1029/2009JB007092.Linde, N., 2009. Comment on ‘Characterization ofmultiphase electrokineticcoupling using a bundle of capillary tubes model’ by Mathew D. Jackson,J. geophys. Res., 114, B06209, doi:10.1029/2008JB005845.Linde, N., Jougnot, D., Revil, A., Matthai, S.K., Renard, D. & Doussan,C., 2007. Streaming current generation in two-phase flow conditions,Geophys. Res. Lett., 34, L03306, doi:10.1029/2006GL028878.Revil, A., & Cerepi, A., 2004. Streaming potential in two-phase flow con-dition, Geophys. Res. Lett., 31(11), L11605, doi:1029/2004GL020140.Revil, A., Linde, N., Cerepi, A., Jougnot, D., Matthai, S. & Finsterle, S.,2007. Electrokinetic coupling in unsaturated porous media, J. ColloidInterface Sci., 313, 315–327.Sailhac, P., Darnet, M. & Marquis, G., 2004. Electrical streaming potentialmeasured at the ground surface: forward modeling and inversion issuesfor monitoring infiltration and characterizing the vadose zone, VadoseZone J., 3, 1200–1206.Sill, W.R., 1983. Self-potential modeling from primary flows, Geophysics,48(1), 76–86.Thony, J.L., Morat, P., Vachaud, G. & Le Moue¨l, J.L., 1997. Field char-acterization of the relationship between electrical potential gradientsand soil water flux, C. R. Acad. Sci. Se´r. Ila Sci. Terre Plane`tes, 325,317–325.C© 2011 The Authors, GJI, 186, 113–114Geophysical Journal International C© 2011 RAS

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