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Published in Vadose Zone Journal 3:1200-1206 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

SPECIAL SECTION: HYDROGEOPHYSICS

Electrical Streaming Potential Measured at the Ground Surface

Forward Modeling and Inversion Issues for Monitoring Infiltration and Characterizing the Vadose Zone

Pascal Sailhac*, Mathieu Darnet and Guy Marquis

Eacute;quipe de Proche Surface, École et Observatoire des Sciences de la Terre–Institut de Physique du Globe de Strasbourg (CNRS UMR7516) 5, rue René Descartes– F-67084 Strasbourg (FRANCE)
* Corresponding author (pascal.sailhac{at}eost.u-strasbg.fr)

Received 30 January 2004.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 APPENDIX:
REFERENCES
 
Field estimation of the soil water flux has direct application for water resource management. Standard methods like tensiometry or time domain reflectometry are often difficult to use because of subsurface heterogeneity, whereas noninvasive tools such as electrical resistance tomography, nuclear magnetic resonance, or ground penetrating radar are limited to the estimation of the water content. We present an electrical method that provides water flux estimates: streaming potential (SP) monitoring. This cost-effective tool may help to estimate the nature of the flow process (infiltration or evaporation) in the vadose zone. We discuss interpretation strategies in terms of numerical modeling of both hydraulic and electric processes in the vadose zone and propose an inversion scheme that allows the soil hydraulic parameters to be estimated from in situ infiltration experiments.

Abbreviations: EK, electrokinetic • SP, streaming potential


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 APPENDIX:
REFERENCES
 
ELECTRICAL STREAMING potential monitoring has improved during the last decade and has been shown to be a useful tool for groundwater flow characterization. Streaming potential monitoring is a low-cost technique that requires only electrodes connected to a data logger to make noninvasive measurements related to groundwater flow. Its sensitivity to water flow (and not only to water content) gives the SP method a distinct advantage compared with other noninvasive techniques. For instance, Buselli and Lu (2001) showed that SP monitoring can efficiently detect seepage where other electrical and electromagnetic methods (except induced polarization) can only resolve geological structures.

Streaming potential measurements have been used for decades to study subsurface fluid flow for hydrological and geothermal applications (e.g., Ogilvy et al., 1969; Abaza and Clyde 1969; Bogoslovsky and Ogilvy 1970; Corwin and Hoover 1979; Sill, 1983). In these early papers, SP signals were explained by electrokinetic coupling between flow through porous media and electric polarization of the double layer located at the pore–liquid interface, on the basis of concepts first introduced by Helmholtz in the mid 19th century. Later, theoretical advances contributed to a better understanding of SP data in terms of thermodynamics of multiphase flow in porous media (Neev and Yeatts 1989; Li et al., 1995; Revil et al., 1999; del Rio and Whitaker 2001). Elaborate interpretation procedures based on integral transforms, numerical modeling, and inversion techniques have also been developed (Fournier 1989; Gibert and Pessel 2001; Sailhac and Marquis 2001; Titov et al., 2002). Recent studies have proposed methodologies to estimate key hydrologic parameters such as the depth of the water table, the hydraulic conductivity, and soil moisture front infiltration dynamics (Revil et al., 2003; Darnet et al., 2003; Darnet and Marquis 2004).

Darnet and Marquis (2004) presented one of the first quantitative applications of SP monitoring to vadose zone processes in their estimation of water retention curve parameters. They used a one-dimensional finite-difference code to model SP and tensiometric time series, thus enabling them to determine van Genuchten–Mualem-type soil hydraulic parameters. In this study we extend their SP monitoring interpretation methodology to the estimation of soil hydraulic properties during two-dimensional infiltration under steady-state conditions. We employ analytic formulations similar to those of Philip (1971) and Zhang et al. (2000) to compute two-dimensional infiltration from a line source and adopt an approach similar to that of Darnet and Marquis (2004) to compute synthetic SP data. Here we assume a homogeneous Gardner-type soil in which the unsaturated hydraulic conductivity has an exponential dependence on the pressure head. This allows linearization of Richards' equation through a Kirchhoff transformation (Gardner, 1958). One can therefore obtain analytic solutions, given reasonable approximations, for the hydraulic potentials in real soils (e.g., Revol et al., 1997; Zhang et al., 2000).

We first recall the fundamentals of SP theory in terms of thermodynamics using Onsager's coupling relations (Onsager, 1931) and then develop our technique for modeling two-dimensional infiltration and computing the SP response. We follow with an analysis of our approach's sensitivity to variations of soil hydraulic parameters before illustrating our methodology with a numerical example.


   METHODS
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 APPENDIX:
REFERENCES
 
Streaming Potential
Streaming potentials are electric potentials generated by electrokinetic (EK) processes when an electrolyte flows in a porous medium. In what follows, we consider the macroscopic, steady-state electric field measured near the ground surface.

Consider a series of potentials defined as macroscopic quantities that relate to thermodynamic forces used in the soil science literature (e.g., Hillel, 1998): the matric or capillary potential {Phi} (m3 s–1 m–1), the pressure head {Psi} (m), the osmotic potential {Psi}o (m), the gravitational head {Psi}g (m), the piezometric head {Psi}p = {Psi} + {Psi}g (m), and the electric potential {Psi}e (V). Using near-equilibrium thermodynamics and assuming that the phenomenological coupling parameters form a symmetric matrix (Onsager, 1931; Overbeek, 1960), we can express the total electric current in terms of these macroscopic fluxes and the temperature T (K). Let us consider the total electric current:

[1]

By inspection, we find from Ohm's Law that Le = –{sigma} (S m–1), where {sigma} is the electric conductivity; while the Peltier effect implies that LTe = –T{pi}{sigma} (Wm s–1), where {pi} is the Peltier heat of thermo-electricity, and electrokinetic coupling implies Lec = –C'{sigma} (V S m–2), where C (V Pa–1) (or C' = C{rho}fg [V m–1]) is the EK coupling parameter, {rho}f is the density of fluid, and g is gravity; and Leo is electro-osmotic diffusivity.

Neglecting the effects of thermal and concentration gradients for the remainder of this demonstration, conservation of electric current in steady state yields

[2]

Therefore,

[3]

In a homogeneous medium, both the electrical conductivity {sigma} and the EK coupling C'{sigma} are constant, in which case Eq. [3] reduces to a simple Poisson equation {nabla}2{Psi}e = –C'{nabla}2{Psi}p. In this case, one classically considers the total electric potential {Psi}T = {Psi}e + C'{Psi}p that obeys Laplace's equation (e.g., Fitterman 1978). In a heterogeneous medium where the electrical conductivity or the EK coupling are not constant, Eq. [3] can be seen as a complex Poisson's equation containing gradients of both log({sigma}) and C':

[4]
where

[5]

Direct modeling of SP consists of solving Eq. [4] as an electric potential problem with some distribution of the conductivity {sigma} and the source Sec. The source Sec is responsible for primary electric potential variations, while {nabla}{Psi}e·{nabla}log({sigma}) is a term for secondary electric sources caused by electric conductivity gradients parallel to the electric field. For applications to the vadose zone, we also consider the critical threshold of saturation for the onset of flow. Above the critical no-flow saturation level, no flow occurs and the EK source Sec vanishes.

A relationship between the electrical conductivity and the water content is necessary for solving Eq. [4]. For simplicity we consider an unsaturated soil that is not dry and has relatively low clay content. Thus we are able to neglect surface conductivity effects and, assuming that the saturation is not very low, Archie's Law is valid. Archie's Law is given by

[6]
where {sigma}f is the electrical conductivity (S m–1) of the fluid, F is the electrical formation factor (dimensionless) related to porosity, n is the saturation exponent (dimensionless), and Se is the effective saturation (dimensionless), which is given by

[7]
where {theta} is the volumetric moisture content, and {theta}r and {theta}s are the residual and saturated values of moisture content, respectively.

The EK coupling coefficient C' is related to the electrical conductivity {sigma} (S m–1) of unsaturated porous media by the Helmholtz–Smoluchowski equation (Smoluchowski, 1905):

[8]
where {rho}f, {epsilon}f, and {eta}f are, respectively, the density (kg m–3), dielectric permittivity (F m–1), and dynamic viscosity (Pa s) of the fluid, g is gravity (m s–2), {zeta} is the zeta potential (V), and F is the electrical formation factor (dimensionless). For typical groundwater conductivities, C' ranges from –1 to –15 mV m–1 (Revil et al., 2003). Darnet and Marquis (2004) estimated C' {approx} –40 mV m–1 by interpreting SP data acquired by Doussan et al. (2002) for a sandy loam during rainfall.

Note that Archie's Law and the Helmholtz–Smoluchowski equation follow a first-order approximation for low Dukhin's number (relatively low surface electric conductivity). This is a realistic approximation only in unsaturated soil having a relatively low clay content in which partial saturation and the electric conductivity of the pore water are not very low. In a more general model of unsaturated soil, one may consider surface electric conductivity effects (Lorne et al., 1999; Revil et al., 1999; Lyklema 2001, 2003; Guichet et al., 2003).

Consider the pressure head {Psi} = {Psi}p + z, where z is positive downward. Equations [4] and [5] give the following equation for the streaming potential {phi} = {Psi}e:

[9]

Hereafter we assume that the soil at some fixed moisture content is homogeneous (in terms of electrical properties). Thus, both the electrical conductivity {sigma}S and the electrokinetic coupling coefficient C'S at saturation are constant. Equation [9] can then be simplified using Eq. [6] through [8] to

[10]

Equation [10] is a diffusion equation in which the source term is C'S{nabla}2{Psi} and where diffusion is controlled by the relative electric conductivity, which equals effective saturation to the power n: {sigma}/{sigma}S = Sen.

Steady-State Modeling: Two-Dimensional Infiltration from Line Source
Following the same approach as Zhang et al. (2000) for TDR modeling, we use the analytic formula of hydraulic potentials for two-dimensional steady infiltration from a surface line source obtained by Philip (1971) to develop analytic expressions for modeling the SP. We develop analytic expressions for function parameters in Eq. [10]: the effective water content Se that gives electrical conductivity variations and the Laplacian of the pressure head {nabla}2{Psi} that gives the EK sources. For simplicity, we only consider a surface line source and a surface SP profile perpendicular to the source; extensions to SP anomalies caused by steady infiltration from sources at arbitrary depth and sloping boundaries can be similarly derived from expressions of the matrix flux potential (Raats 1972; Philip and Knight 1997).

The matric flux potential, {Phi} (m3 s–1 m–1) for a surface line source perpendicular to the (x, z) plane and located at (x = 0, z = 0) is given by (Philip 1971)

[11]
where q (m3 s–1 m–1) is the constant source strength due to source, z is positive downward, {kappa}n is the modified Bessel function of the third kind of order n, and X = {alpha}x/2 and Z = {alpha}z/2 are dimensionless variables. The soil sorptive number, {alpha} (m–1), is related to the coarseness of the soil (reciprocal of the capillary length) and is defined by an exponential relation between the unsaturated hydraulic conductivity and the pressure head {Psi} (Gardner,1958):

[12]
where KS (m s–1) is the hydraulic conductivity at saturation.

Similar to Zhang et al. (2000), we use the soil water content to pressure head relationship of Russo (1988):

[13]
where Se is the effective water content (as defined in Eq. [7]), and m is Mualem's constitutive parameter of the soil (Mualem, 1976). Figure 1 illustrates Russo's equation for m = 0.5. While Zhang et al. (2000) assumed that at infinite distance from the origin the effective saturation is zero (i.e., pressure head equal to –{infty}), here we shift the matric flux potential {Phi} of Eq. [11] by {Phi}0 to obtain realistic saturations when considering infiltration in wet unsaturated soils. The pressure head {Psi} is related to the matric flux potential {Phi} through

[14]
where {alpha}{Phi}0 = KSexp({alpha}{psi}0) and {Psi}0 is the pressure head at infinity (Zhang et al. [2000] assumed {Phi}0 = 0 and {Psi}0 = –{infty}).



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  		Fig. 1. Plots of Russo's model of effective saturation vs. pressure head for Mualem's parameter m = 0.5 in Eq. [13] (left), and the shift in the matric flux potential caused by a non-zero background effective saturation (right). The cross is for a soil in which effective saturation before infiltration is Se0 = 0.75; The corresponding shift relative to the analytic expression of {Phi} in Eq. [11] is {Phi}0 = 0.11KS/{alpha} (for m = 0.5).

 
For two-dimensional infiltration from a surface line source, Eq. [10] is a diffusion equation with relative conductivity {sigma}/{sigma}S = Sen and a source term C'S{nabla}2{Psi} that depends analytically on {Phi}(x, z) (as detailed in the Appendix):

[15]

[16]
where {Phi}2(x,z) (m) is a function similar to {Phi}(x,z) except that {kappa}0 in Eq. [11] has been replaced by {kappa}2, while JZ(x,z) (m s–1) is the vertical water flux density given by

[17]

To calculate the SP anomalies near the surface, we use an implicit finite-difference algorithm (Mufti 1976) to numerically solve Eq. [10] in which the unsaturated electric conductivity and electrokinetic sources are derived from the analytic formula for steady-state two-dimensional infiltration from a line source.

Parameter Sensitivity
The analytic expressions developed above involve five independent parameters: {alpha} (m–1), q/KS (m), {Phi}0/KS (m), C'S (V m–1), and 2n/(2 + m) (dimensionless). The first three parameters are related to the soil hydraulic properties and the last two parameters are hydroelectric coupling parameters. {Phi}0/KS is related to the initial effective water saturation Se0 (using Eq. [15]). The effect of Se0 on the hydraulic potential has not, to our knowledge, been considered before in the literature.

To test the usefulness of the inversion of SP data to determine soil hydraulic and electrokinetic parameters, we compute a parameter sensitivity function similar to that used by Simunek and van Genuchten (1996) but consistent with the interpretation of Kabala (2001). We use our forward modeling scheme to calculate the effect on SP measurements of a 1% change in each parameter: {alpha}, q/KS, or Se0. The sensitivity to other parameters {Phi}0/KS, C'S and 2n/(2 + m), could be calculated similarly, but they are not necessary since {Phi}0/KS simply corresponds to Se0 (see Eq. [15]), C'S is merely a multiplying factor whose sensitivity is obvious, and 2n/(2 + m) relies on the Archie's Law exponent n, which may be estimated using electrical conductivity measurements on soil samples in the laboratory. For simplicity, we first consider only the sensitivity of the maximum value of the SP horizontal gradient measured at a depth of 10 cm (here labeled {chi}). Its sensitivity to some parameter ß is defined as:

[18]
where {Delta}ß = 0.01ß.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
SUMMARY AND CONCLUSIONS
 APPENDIX:
REFERENCES
 
Steady-State Modeling: Two-Dimensional Infiltration from a Line Source
Hydraulic potentials and their resulting effective soil water saturation and electric SP values were calculated for a range of soil parameters similar to those studied by Zhang et al. (2000):

The electrokinetic coupling parameter at saturation C'S is only a multiplier and is arbitrarily set to unity (C'S = –1mV/m). Modeled electric potentials and electric fields must be multiplied by the actual C'S values to be comparable with field data. In addition, the initial effective saturation Se0 ranges between 0.5 and 0.9.

Figure 2 shows cross sections for {alpha} = 12 m–1, Se0 = 0.6, and q/KS {0.001, 0.005, 0.025, 0.1 m}. The infiltration is essentially vertical for large q/KS and lateral for small q/KS. They illustrate typical experiments that would be run for SP-based soil parameter estimation. Only the source strength q varies from one test to the other; the soil parameters {alpha}, Se0, and KS remain the same. We can see that for relatively small source strengths, q/KS, spatial variations in the effective saturation Se appear only at very small scales (<10 cm when {alpha} = 12 m–1 and Se0 = 0.6); however, SP shows variations at larger scales (>10 cm). Figure 2 also shows the horizontal electric field Ex and its derivative E'x at the 10-cm depth that could be measured by simply using tens of unpolarizable electrodes.



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  		Fig. 2. Cross-sections of four source strength infiltration simulations ({alpha} = 12 m–1 and Se0 = 0.6, and q/KS ranges from 1 mm to 10 cm), showing effective water saturation Se and electrokinetic potential V (mV) vs. depth (from 0 to 60 cm), and its first- and second-order horizontal derivatives at a depth of 10 cm (horizontal electric field Ex [mV cm–1] and its derivative E'x [mV cm–2]). Please note that the distributions are symmetric with respect to the x = 0 vertical plane, and that V and E'x are even and Ex is odd.

 
To illustrate how SP depends on typical unsaturated hydraulic parameters, Fig. 3 shows the maximum horizontal electric field {chi} at a depth of 10 cm vs. q/KS for selected values of {alpha} and Se0. Their relation depends mainly on the electrical conductivity (or effective saturation; see Eq. [6]). Values of {chi} increase with increasing q and reach a few millivolts per centimeter; they are in the range of vertical SP gradients observed during one-dimensional infiltration experiments by Doussan et al. (2002).



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  		Fig. 3. Plots of the electric field Ex maxima {chi} vs. q/KS. Results are for a depth of 10 cm, and for constant Gardner soil parameters {alpha} equal to 4, 8, 12, or 16 m–1 (top), and constant background effective water saturation Se0 equal to 0.6, 0.7, or 0.8 (bottom). Note that the maxima increase with increasing q/KS and {alpha}, and decrease with increasing Se0.)

 
Parameter Sensitivity Results
The sensitivity of the maximum SP gradient {chi} was calculated for {q/KS = 0.025 m, Se0 = 0.6}, {{alpha} = 12 m–1, Se0 = 0.6}, and {{alpha} = 12 m–1, q/KS = 0.025 m}. Figure 4 shows the sensitivity coefficients µ({alpha}), µ(q/KS), and µ(Se0) (mV m–1). All three sensitivity curves show relatively low values. The sensitivity to Se0 was the largest (1–5% variation on {chi}), the sensitivity to {alpha} was the smallest ({approx}0.1%), and the sensitivity to q/KS was intermediate ({approx}0.2–1%). In relation to parameter estimation by the inversion of SP data, we expected that Se0 and KS (or q/KS) could be determined, but that {alpha} might be more difficult to estimate with reasonable accuracy. However, given the accuracy of unpolarizable electrodes (about 0.2–0.5 mV), the sensitivity of {chi} to {alpha} (around 1–3 mV m–1) was above realistic detection levels.



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  		Fig. 4. Sensitivity of the maximum SP gradient {chi} to {alpha} (m–1), q/KS (m), and Se0.

 

   SUMMARY AND CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
APPENDIX:
REFERENCES
 
Previous methodologies developed for the determination of soil hydraulic parameters from SP data are mostly limited by the assumption of having a hydraulically and electrically homogeneous medium. The one-dimensional infiltration simulations of Darnet and Marquis (2004) constituted one of the first applications of SP to flow in the vadose zone that successfully estimated the water retention parameters from both rainfall and SP time series. In the two-dimensional infiltration modeling scheme presented here, SP data need to be acquired only once, provided the two-dimensional infiltration source is well known. These two studies show that one can run either unsteady or steady-state infiltration experiments to estimate unsaturated soil hydraulic parameters from SP measurements.

One may question the limitations of using SP data under the circumstances of a noisy electric field environment. Indeed, SP time series contain additional electric field fluctuations of telluric and human origins (e.g., hourly variations caused by magnetic storms and temperature changes, or intermittent power supplies and radio emissions). In practice, applications to unsteady cases require ad hoc data reduction to provide time series of the streaming potentials. A much simpler treatment of the data is needed for the steady-state case. In fact, SP time series can be averaged over a period of one or more days to provide the steady-state streaming potential profile necessary for estimation of soil parameters.

Our approach to the hydraulic problem, using analytical solutions, shows a dependency of the SP response on several unsaturated soil parameters, listed here in decreasing order of sensitivity: the electrokinetic coupling parameter at saturation C'S, the effective soil water saturation prior to the infiltration experiment (Se0), the ratio of the constant source strength to the hydraulic conductivity at saturation (q/KS), the soil sorptive number {alpha}, Mualem's parameter m, and finally Archie's Law exponent n. In principle, all of these parameters could be constrained by inverting EK data obtained during a series of infiltration experiments with varying source strength q. On the basis of a preliminary sensitivity analysis, we believe that KS, {alpha}, and Se0 can be constrained by SP data inversion. Further numerical experiments, as well as field and laboratory experiments, are necessary to establish the validity and relevance of the technique. We expect that the present analytical developments will be realistic for applications where the area of the infiltration test is reasonably homogeneous with respect to parameters {alpha}, KS, Se0, m, n, and C'S. For soils with a high degree of heterogeneity, numerical modeling will be necessary.


   APPENDIX:
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 APPENDIX:
REFERENCES
 
EXPRESSION FOR THE ELECTROKINETIC SOURCE TERM IN THE MATRIX FLUX POTENTIAL OF PHILIP (1971)
Deriving Eq. [15] to [17] requires several steps and assumptions detailed in this appendix. Our approach can be used by others to obtain the partial derivatives for other {Psi}–Se and {sigma}–Se relationships. First, Eq. [15] results by substituting {Psi} given by Eq. [14] into the {Psi}–Se relationship of Russo given by Eq. [13].

Equations [16] and [17] may be derived by considering the right-hand side (the source term) of Eq. [10]. Using Eq. [14], this term becomes

[A.1]

Moreover, since {Phi} satisfies the Richards' equation, its Laplacian [({partial}2/{partial}x2) + ({partial}2/{partial}z2)]{Phi} is related to ({partial}/{partial}z){Phi} through

[A.2]

Thus Eq. [A.1] can be expressed in terms of first-order derivatives of {Phi} only:

[A.3]

Let us now calculate gradients {Phi} in Eq. [11] using dimensionless variables X = {alpha}x/2 and Z = {alpha}z/2.

The horizontal derivative of {Phi} is ({partial}/{partial}x){Phi} = [({alpha}/2)({partial}/{partial}X)]{Phi}:

[A.4]
where {Phi} is given by Eq. [11] and

The expansion in Eq. [A.4] uses the relationship

which is based on the property of modified Bessel functions of the second type that for all x: d{kappa}0(x)/dx = –{kappa}1(x) and {kappa}2(x) = {kappa}0(x) + 2{kappa}1(x)/x.

The vertical derivative of {Phi} is similarly ({partial}/{partial}z){Phi} = [({alpha}/2) ({partial}/{partial}Z)]{Phi}:

[A.5]
where {Phi} is given by Eq. [11], and where we again used the property that for all x, d{kappa}0(x)/dx = –{kappa}1(x).

Using the vertical flux Jz = –({partial}/{partial}z){Phi}, in which K = {alpha}({Phi} + {Phi}0) is the unsaturated hydraulic conductivity, and expressions for the gradients in Eq. [A.4] and [A.5], Eq. [A.3] can finally be written in the form of Eq. [16], from which the expression for Jz as given by Eq. [17] can be derived.


   ACKNOWLEDGMENTS
 
Financial support for this work was provided by CNRS/INSU ACI Eau Sol et Environnement. We also wish to thank the reviewer André Revil, an anonymous reviewer, and Associate Editor Andrew Binley, for their fruitful comments. This is EOST-IPGS publication number 2004.16-UMR7516.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 APPENDIX:
REFERENCES
 




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