Reconstruction of the Water Table from Self-Potential Data: A Bayesian Approach

Reconstruction of the Water Table from Self-PotentialData: A Bayesian Approachby A. Jardani1,2, A. Revil3,4, W. Barrash5, A. Crespy6, E. Rizzo7, S. Straface8, M. Cardiff9, B. Malama5,C. Miller5,10, and T. Johnson11

AbstractGround water flow associated with pumping and injection tests generates self-potential signals that can be

measured at the ground surface and used to estimate the pattern of ground water flow at depth. We propose aninversion of the self-potential signals that accounts for the heterogeneous nature of the aquifer and a relationshipbetween the electrical resistivity and the streaming current coupling coefficient. We recast the inversion of theself-potential data into a Bayesian framework. Synthetic tests are performed showing the advantage in using self-potential signals in addition to in situ measurements of the potentiometric levels to reconstruct the shape of thewater table. This methodology is applied to a new data set from a series of coordinated hydraulic tomography,self-potential, and electrical resistivity tomography experiments performed at the Boise HydrogeophysicalResearch Site, Idaho. In particular, we examine one of the dipole hydraulic tests and its reciprocal to show thesensitivity of the self-potential signals to variations of the potentiometric levels under steady-state conditions.However, because of the high pumping rate, the response was also influenced by the Reynolds number, especiallynear the pumping well for a given test. Ground water flow in the inertial laminar flow regime is responsible fornonlinearity that is not yet accounted for in self-potential tomography. Numerical modeling addresses the sensitiv-ity of the self-potential response to this problem.

Some figures in this paper are available in color in the online version of the paper.

IntroductionKnowledge of the geometry and the physical proper-

ties of aquifers is required to quantitatively estimatewater resources and to protect them from various possiblesources of contamination (de Marsily 1986). The flow ofground water is generally studied using in situ measure-ments from a set of observation wells. Unfortunately,because of their cost, usually only a few wells are avail-able for a given field test or monitoring program, result-ing in relatively scarce information to infer the pattern ofground water flow or the distribution of hydraulic param-eters. To remedy this problem, low-cost geophysicalmethods that are sensitive to ground water flow, and/orfeatures that significantly affect ground water flow, arerequired. Methods such as georadar and the DC-electricalresistivity tomography add information on the geometryof the water table and the water content distribution in theground (Binley et al. 2002; Kemna et al. 2002; Day-Lewis et al. 2005). These methods can be implementednot only at the ground surface but also in performingcross-hole tomography (Linde et al. 2006).

1Colorado School of Mines, Department of Geophysics,Golden, CO 80401.

2Bureau d‘Etudes ALISE, 76160 Saint-Jacques-sur-Darnetal,France.

3Corresponding author: Colorado School of Mines, Depart-ment of Geophysics, 1500 Illinois St., Golden, CO 80401; (303)250-8194; fax: (303) 273-3478; [email protected]

4INSU-CNRS LGIT UMR 5559, Universite de Savoie, EquipeVolcans, Le Bourget-du-Lac, 73376 France.

5Center for Geophysical Investigation of the Shallow Sub-surface, Boise State University, Boise, ID 83725.

6CNRS, Universite Aix Marseille III, Aix en Provence, 13545France.

7CNR-IMAA, Hydrogeophysics Laboratory, 85052 MarsicoNuovo (PZ), Italy.

8Dipartimento di Difesa del Suolo, Universita’ della Calabria,87036 Rende (CS), Italy.

9Stanford University, Stanford, CA 94305.10MSE Technology Applications Inc., Butte, MT 59701.11Idaho National Laboratory, Idaho Falls, ID 83415-2107.Received April 2008, accepted September 2008.Copyright ª 2008 The Author(s)Journal compilationª2008NationalGroundWater Association.doi: 10.1111/j.1745-6584.2008.00513.x

NGWA.org Vol. 47, No. 2—GROUND WATER—March–April 2009 (pages 213–227) 213

The self-potential method remains, however, the onlymethod that can simultaneously characterize the geometryand the dynamics of ground water flow in real time (Suskiet al. 2006; Jardani et al. 2007b, 2008). Indeed, the naturalelectrical potential resulting from the flow of ground water(called the streaming potential) is directly related to themovement of water within the aquifer. In addition, thecharacteristic time associated with the transport of the elec-tromagnetic information is small enough to consider thediffusion of the electromagnetic disturbances as a quasi-instantaneous phenomenon (Revil et al. 2003).

Quincke (1859) was the first to demonstrate that theflow of water through a capillary or a porous mediumgenerates an electric field of electrokinetic nature due tothe drag of the diffuse part of the electrical double-layercoating in the surface of the pores. Bogoslovsky andOgilvy (1973) and recently Revil et al. (2002a), Suskiet al. (2004), Rizzo et al. (2004), Titov et al. (2005), andMaineult et al. (2008) showed that an electric field ofelectrokinetic nature can be measured at the surface of theearth during infiltration or pumping tests. When water isflowing through a porous material, a streaming currentdensity is produced in response to the drag of the excessof electrical charges existing in the vicinity of the min-eral-water interface in the so-called diffuse layer (Ishidoand Mizutani 1981; Revil et al. 2003). The divergence ofthis streaming current density creates an electrical fieldthat can be measured at the ground surface of the earth.This electrical field is also influenced by the electricalresistivity distribution of the medium.

Unfortunately, the streaming potential is not the onlycomponent of the self-potential signals. Other componentsinclude electroredox phenomena (Naudet et al. 2004;Rizzo et al. 2004; Castermant et al. 2008) and diffusionand thermoelectric phenomena that are both related to thegradient of the chemical potential of ions in the pore waterof the porous material (Revil 1999; Revil and Linde 2006).

A number of recent studies have shown the highinterest in the self-potential method for applications ingeohydrological problems. These applications include (1)the identification of the direction and rate of groundwater flow in the vadose zone and in shallow unconfinedaquifers (Doussan et al. 2002; Revil et al. 2003); (2) thedelineation of hydrothermal bodies (Finizola et al. 2002);(3) redox fronts associated with contaminant plumes(Naudet et al. 2004); (4) the localization of sinkholes inkarstic areas (Jardani et al. 2006, 2007a, 2008); (5) theinterpretation of pumping and infiltration tests (Rizzo et al.2004; Maineult et al. 2008); (6) the study of anisotropy infractured media (Wishart et al. 2006, 2008); and (7) thedetermination of the capillary pressure curve and relativepermeability of the capillary fringe using harmonic pump-ing tests (Maineult et al. 2008; Revil et al. 2008).

Two models have been developed in the literature inthe past two decades. The first school of thought is basedon a linear relationship observed, in some cases, betweenthe thickness of the vadose zone and the self-potentialdata measured at the ground surface. This approach iscalled the Aubert model (Linde et al. 2007). This

empirical relationship is valid only when the contrast ofelectrical resistivity between the vadose zone and theunconfined aquifer is important and when the vadose zoneis fairly homogeneous. The second school of thought hasproposed a model in which the divergence of the sourceof current is located at the water table (Fournier 1989;Birch 1993). These sources can be modeled by electricaldipoles normal to the water table. The intensity of thesedipoles is proportional to the potentiometric levels withrespect to a reference level (Revil et al. 2002a).

These two approaches have been used recently byLinde et al. (2007) in a Bayesian framework to recon-struct the shape of the water table at the scale of a smallcatchment. However, none of these approaches accountproperly for the heterogeneity of the coupling coefficientand electrical resistivity distribution. In the presentarticle, we propose a general approach free of all theprevious assumptions. This method is based on numeri-cally solving the Poisson equation for the streaming poten-tial and the hydraulic heads under steady-state conditions.However, we will show that this approach also validatessome of the assumptions made in our previous works.

Forward Modeling

Field Equations in the Viscous Laminar Flow RegimeSelf-potential signals associated with the flow of

ground water are due to the drag of excess electrical chargecontained in the pore water and resulting from the existenceof the electrical diffuse layer at the pore water-mineralinterface. In an isotropic heterogeneous media, the total cur-rent density j (in A/m2) is given by the sum of a conductivecurrent (Ohm’s law) and a source current of electrokineticnature (Ishido and Mizutani 1981; Revil et al. 2007):

j ¼ �rru� Lrh ð1Þ

where h is the hydraulic head (in m), u is the electrical(self-) potential (in V), r is the electrical conductivity ofthe porous material (in S/m), and L is the streaming cur-rent coupling coefficient (in A/m2). The term jS ¼ �Lrhis named the streaming current density and acts asa source of current density (Revil and Leroy 2004; Revilet al. 2005).

At Reynolds numbers between 1 and few hundred(the critical value corresponding to turbulence), the flowis said to be in the inertial laminar flow regime. Whenthe Reynolds number is much less than 1, the flowregime is called the viscous laminar flow regime. We willsee below that the interpretation of the self-potentialsignals associated with a dipole test requires that the dis-tinction be made between these two regimes. In the vis-cous-laminar flow regime, all the material properties(especially the streaming current coupling coefficient andthe permeability) are independent of the pressure gradient.

The electrical conductivity of a water-saturated porousmaterial results from two contributions: one in the bulkpore space and one along the surface of the pores (calledthe surface conductivity; Leroy and Revil 2004). These

214 A. Jardani et al. GROUND WATER 47, no. 2: 213–227 NGWA.org

two contributions do not act in parallel (Bernabe and Revil1995; Revil et al. 1998, 2002b). An important point is thatif the surface conductivity can be neglected with respect tothe electrical conductivity of the brine (i.e., for small Du-khin numbers, see Crespy et al. [2007]; Boleve et al.[2007a]), then both the streaming current coupling coeffi-cient L and the electrical conductivity r are inversely pro-portional to the electrical formation factor (Revil et al.1999). Therefore, tomography of the resistivity can be usedto assess the distribution of the streaming current couplingcoefficient. This point will be developed further below.

The streaming potential coupling coefficient isdefined by:

C ¼�@u@h

�j¼0

¼ �L

rð2Þ

This coefficient is usually the electrokinetic couplingcoefficient that is measured in the laboratory (see the fol-lowing). It is here expressed in millivolts per meter ofhydraulic head (it is usually expressed in millivolts perpascal by electrochemists).

The continuity equation for the electrical charge isr � j ¼ 0. The combination of this continuity equationand the constitutive equation, Equation (1), yields a Pois-son’s equation:

r � ðrruÞ ¼ J ð3Þ

where J ¼ �r�ðLrhÞ is the volumetric source term (inA/m3) for any self-potential station (a nonpolarizing elec-trode) located outside the source volume in which flowoccurs. In order to gain a better understanding of the vari-ous contributions of the source term of Equation (3), it isinsightful to expand J in terms of its contributions. Thesource term of Equation (3) can be written as J ¼�Lr2h�rL � rh. Using the Darcy constitutive equationu ¼ �Krh (K is the hydraulic conductivity in m/s), theLaplacian of the hydraulic head is given, in steady-stateconditions, by:

r2h ¼ �Q

Kdðr� rSÞ � rln K � rh ð4Þ

where Q is the pumping or infiltration rate (in m3/s), rS isthe position of the source where water is infiltrated orremoved from the ground, dðr� rSÞ is the Kroneckerdelta (d½r� rS� ¼ 1 when r ¼ rS and d½r� rS� ¼ 0 forr 6¼ rS). Therefore, the source term in the electrostaticPoisson equation can be written as:

J ¼ LQ

Kdðr� rSÞ � rL � rh 1 Lrln K � rh ð5Þ

There are three source terms in Equation (5). Thefirst one is associated with the pumping/injection rate.The second source term is associated with the heterogene-ity of the streaming current coupling coefficient of theground in the presence of a hydraulic gradient. As ex-plained above, this heterogeneity can be assessed inde-pendently with an electrical resistivity tomogram. Note

that the tomogram of the electrical resistivity (and henceL) is smoother than what we actually expect in the realworld due to regularization of the inverse problem inelectrical resistivity tomography. This regularizationwould decrease the value of rL. Conditional realizationsof L, using the best estimate and the posterior covariance,could be generated, and in general, these will generate‘‘rougher’’ L-fields than what the best estimate gives. Thispossibility is, however, not explored in the present work.The third source term is associated with heterogeneity ofthe permeability field.

If we consider N sources and sinks of water supply orremoval inside the ground, the distribution of the elec-trical potential is given by solving the following Poisson’sequation:

r2u ¼ �XNi¼1

CQi

Kdðr� rSi Þ �

1

rrL � rh

�rln r � ru� Crln K � rh ð6Þ

which shows that spatial variation of electrical conductiv-ity can also be considered as an additional self-potentialsource.

Field Equations in the Inertial Laminar Flow RegimeA number of recent works (Watanabe and Katagishi

2006; Kuwano et al. 2006; Revil 2007; Boleve et al.2007a) have studied the effect of the Reynolds numberon the magnitude of the streaming potential. For acapillary of radius R, U being the strength of the seepagevelocity, the Reynolds number is defined by (e.g.,Batchelor 1972):

Re ¼ qf UR=gf ð7Þ

where qf and gf are the mass density and dynamic shearviscosity of the pore fluid. In the inertial laminar flowregime (Re . 1), the apparent streaming current couplingcoefficient and the apparent hydraulic conductivity aregiven by (Boleve et al. 2007a):

L ¼ L0=ð1 1 Re Þ ð8Þ

K ¼ K0=ð1 1 Re Þ ð9Þ

where L0 and K0 are the value of the streaming current cou-pling coefficient and hydraulic conductivity in the viscouslaminar flow regime characterized by Re� 1. In particular,

K0 ¼ d20qf g=�24F3gf

�ð10Þ

from Revil and Cathles (1999), where d0 is the meangrain diameter, F is the electrical formation factor, qf isthe mass density of the pore fluid, g is the acceleration ofthe gravity, and gf is the dynamic shear viscosity of thepore fluid, and

L0 ¼ QVK0 ð11Þ

NGWA.org A. Jardani et al. GROUND WATER 47, no. 2: 213–227 215

(Boleve et al. 2007a, 2007b), where QV represents theexcess of electrical charge of the diffuse layer per unitpore volume (Revil and Leroy 2004; Revil et al. 2005;Revil and Linde 2006).

According to Boleve et al. (2007a), the Reynoldsnumber can be determined from the following expression:

Re ¼ 1

2

� ffiffiffiffiffiffiffiffiffiffiffiffi1 1 c

p� 1

�ð12Þ

c ¼bq2f g

g2f

d30

FðF � 1Þ3rh ð13Þ

where b ’ 2.25 3 10�3 is a numerical constant and cis a function of both the texture and head gradient. TheReynolds number depends on rh, so in this regime, thecurrent density and the Darcy velocity are nonlinear func-tions of the pressure heads, and the apparent hydraulicconductivity and the streaming current coupling coef-ficient depend also on the gradient of the hydraulic head.

Numerical ModelingThe electrokinetic equations can be combined with

the equations describing the flow of ground water ina deformable porous medium and solved with the finite-element method (e.g., Revil et al. 1999), the volume ele-ment method (Sheffer and Oldenburg 2007), or using thefinite-difference approach (Titov et al. 2005). Other meth-ods, like the boundary element method, have been used inelectroencephalography to solve for the electric and mag-netic fields in a piecewise homogeneous domain.

The normal component of the total current densityvanishes at the surface of the earth because air is insulat-ing. In self-potential monitoring, it is also important toremember that all the electrodes are connected to a refer-ence electrode for which the potential is equal to zero bydefinition. The three-dimensional (3D) forward modelwas solved with Comsol Multiphysics 3.4 (Comsol 2008)coupled with Matlab. The Darcy equation was solvedwith the Earth Science package, while the Poisson equa-tion for the self-potential was solved with the electromag-netic package. In the forward numerical modeling, it isvery important to remove the potential obtained at thelocation of the reference electrode in order to compare themeasured self-potential and the self-potential values re-sulting from numerical modeling.

Whatever the method used to discretize the forwardmodel, the model is linear between the head values andthe self-potential signals as long as the flow occurs in theviscous laminar flow regime. In this regime, Equation (2)can be written as:

u ¼ G h ð14Þ

where h is a column vector of hydraulic heads, u is a col-umn vector of the resulting self-potential at the groundsurface, and G is the kernel matrix that depends on the

distributions of the electrical conductivity and streamingcurrent coupling coefficient and the distance betweeneach elementary source and the self-potential stations.

Inverse ModelingIn this section, we perform a Bayesian inverse calcu-

lation to estimate hydraulic heads from self-potential dataincorporating into the kernel information about the distri-bution of electrical resistivity. Geostatistical constraintswill also be used to account for spatial structure or het-erogeneity in the model. In the Appendix, we show thatthe a posteriori estimate of the hydraulic heads and theiruncertainties are given by:

h ¼ hprior 1 CmGTh�G C

mGT 1 C

d

��1�dobs � G hprior

�ið15Þ

r�h�} exp

hðh� hÞT ^

C�1m ðh� hÞ

ið16Þ

where the a priori model of hydraulic heads hprior is givenby the model of Fournier (Birch 1993). The a posterioricovariance matrix of the hydraulic heads is given by:

Cp ¼�^GT C�1

d

^G 1 C�1

m

��1

ð17Þ

where^G is the Frechet derivatives of G evaluated at h. In

the present case,^G ¼ G because the problem is linear.

However, this would not be the case if the effect of theReynolds number were considered. This matrix is the res-olution measure that quantifies the uncertainties about thehydraulic heads.

Inversion of self-potential data is an underdeterminedproblem, and an infinite set of hydraulic head distribu-tions can reproduce the same self-potential data measuredat the ground surface. To reduce the nonuniqueness of thesolution, constraints have to be used. The method ofinversion described above can incorporate constraintssuch as the adjustment of the data balanced by the covari-ance of the uncertainty on the data, the adjustment ofa model balanced by the uncertainty on the model param-eters, the weighting of the model parameters according totheir depth, and constraints provided by in situ measure-ments of the hydraulic heads in a set of piezometers.

The adjustment of a model balanced by the covarianceof the parameters allows one to reconstruct an anisotropichydraulic head data set (e.g., in the case where discontinu-ities are present in the ground like for karstic areas). Thesecovariances can be given as an a priori probability distribu-tion from the geology of the site or from its topography.The covariance matrix can be determined from a vario-gram. If we make u the variable considered, the value ofthe semi-variogram between two distant localizations ofd is calculated by (e.g., Deutsch and Journel 1992):

cðdÞ ¼ 1

2NðdÞXNðdÞi¼1

½uðxiÞ � uðxi 1 dÞ�2 ð18Þ


where N(d) is the number of pairs of observations ofseparation distance d. The assumption of stationarity ofthe second order of the semi-variogram (Matheron1965) implies that the semi-variogram is invariant bytranslation; that is, the semi-variogram will not dependon the particular location of the observation points. Itdepends only on the distance that separates them. Oncecalculated, the semi-variogram reveals three character-istics: (1) the range; (2) the sill; and (3) the nuggeteffect. The range is the distance to which twoobservations are not correlated anymore; the value ofthe semi-variogram for this distance corresponds to thevariance of the variable. The sill C is the value of thesemi-variogram once the range has been reached; thisvalue corresponds to the variance r2 of the variable. Thenugget effect represents very small-scale variation atseparation distances less than those available in thedata set of measurements.

There is a simple relationship between the model ofC(d) covariance and the model of the semi-variogram:

cðdÞ ¼ r2 � C�d�

ð19Þ

where C(d) is the covariogram (or autocorrelationfunction).

The self-potential signals are generated by the diver-gence of the streaming current density, which is in turnrelated to the flow of the ground water. We assume thatthis source is located at the water table. If capillaryfringes can be neglected (this will be the case for the fielddata reported below), the electrical resistivity tomographycan also be used to determine the domain of strong prob-ability of the presence of the source term for theself-potential signals. This solution makes it possible tocalculate the kernels quickly, and the solution will be rel-atively well localized.

For pumping tests, the semi-variogram of the potenti-ometric data can be determined from the semi-variogramof the self-potential measurements. Indeed, we will showthat the self-potential data follow a quasi-linear trendwith respect to the hydraulic heads.

Synthetic Tests

Flow in the Viscous Laminar Flow RegimeThe first model corresponds to a synthetic case in

which the flow of ground water occurs entirely in the vis-cous laminar flow regime. The geometry of the system isshown in Figure 1. The medium is characterized by twolayers and a rectangular heterogeneity placed in the vicin-ity of the pumping well. Properties of the three materialsare given in Table 1. The hydraulic problem is solved in3D with Comsol Multiphysics 3.4 assuming that the watertable is a free boundary and by imposing constant headson the sides of this system.

The numerical finite-element simulation of steady-state pumping is shown in Figure 2. The resulting self-potential anomaly at the ground surface is shown inFigure 3. Note that the self-potential signals recorded atthe ground surface are linearly correlated to the hydraulicheads at depth (Figure 4) in agreement with our previousmodels (Rizzo et al. 2004; Straface et al. 2007). There-fore, we use the variogram of the self-potential map to de-termine the semi-variogram of the a priori hydraulic heads(Figure 5). We are aware that this approach may be criti-cized. Indeed, by using the self-potential data to generatea priori information, one may argue that the self-potentialdata are being ‘‘double weighted’’ (i.e., we are using it bothto get a prior model and to update that model). An alter-native approach (not used here) would be to use only thea priori observed head data to set up the a priori model.

Figure 1. Synthetic model. The properties of the three materials are reported in Table 1. The base of the aquifer occurs ata constant depth of 40 m from the flat ground surface. The heterogeneity is a rectangular block with dimensions dh = 20 m,dx = 40 m, dy = 100 m. The initial depth to the top of the aquifer is 10 m. Pumping is performed from a partially penetratingwell (diameter of 1 m, thickness of the pumping interval is 8 m [black zone in well]) located at x = 150, y = 100. We imposea constant hydraulic head of 30 m above the base datum at all the side boundaries of the aquifer.

Table 1Material Properties Used for the First

Synthetic Model

Material K (m/s) q (ohm/m) L (A/m2)

1 10�6 10 53 10�4

2 10�4 20 53 10�4

3 10�5 100 53 10�4


This semi-variogram is fitted with a Gaussian model.Then, we used the methodology developed previously toinvert the a posteriori probability density of the hydraulicheads; the result is shown in Figure 6. Figure 7 is a plotof the true hydraulic heads of the synthetic model vs. theinverted hydraulic heads, which shows the high degree ofcorrelation; in this regard, the scatter is the result ofnumerical noise.

Flow in the Inertial Laminar Flow RegimeA second synthetic case is developed to study the

self-potential response around a pumping well in a casewhere Re . 1 in the vicinity of the pumping well. In thiscase, the flow of ground water in the vicinity of thepumping well occurs in the inertial laminar flow regime.The geometry is the same as that in the previous case.The material properties we used are collected in Table 2.We used material properties that favor a high Reynoldsnumber close to the pumping well. In Figure 8, we plotthe self-potential at the ground surface when the effectof the Reynolds number is not taken into account. In

Figure 9, we plot (1) the self-potential at the ground sur-face when the effect of the Reynolds number is taken intoaccount and also (2) the Reynolds number. The result isstriking. When the effect of the Reynolds number is takeninto account, the self-potential anomaly is no longer cen-tered on the pumping well. We will show later that thisobservation is in agreement with the field results.

Field ApplicationIn June 2007, we performed a series of coordinated

hydraulic tomography, self-potential, and electrical resis-tivity tomography field experiments at the Boise

3029.9529.929.8529.829.7529.729.6529.60.0

0.5

1.0

1.5

2.0

Hydraulic head (in m)

Sel

f pot

entia

l (in

mV

)

ϕ−ϕ0 = Ca (h − h0)

Ca = -4.7 mV/m,h0 = 30 m

Figure 4. Correlation between the self-potential data at theground surface and the hydraulic heads computed in thewells shown in Figure 2.

Lag distance (in m)

Sem

i-va

riog

ram

(in

mV

2 )

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0

0 10 20 30 40 50 60 70 80 90 100 110

Figure 5. Semi-variogram of the a priori hydraulic headmodel determined from the linear relationship u � u0 = Ca

(h – h0). The semi-variogram is fitted with a Gaussian model.

Figure 3. Distribution of self-potential at the ground surfaceunder steady-state conditions. Note the nonsymmetric distri-bution of the electrical equipotentials associated with thepresence of the heterogeneity.

Figure 2. Distribution of the hydraulic heads (as metersabove base datum) under steady-state conditions (the pump-ing rate is 10�5 m3/s). The filled circles correspond to thepositions of a set of arbitrarily located observation wells.


Hydrogeophysical Research Site (BHRS). This site islocated on a gravel bar adjacent to the Boise River, 15km from downtown Boise, Idaho. The BHRS has beenestablished to develop combined hydrogeological andgeophysical methods to determine the distributions ofhydraulic properties (e.g., transmissivity, storativity,

specific yield) of naturally heterogeneous aquifers in theshallow subsurface. Eighteen fully penetrating wells havebeen cored and completed at the BHRS to provide accessfor detailed characterization and testing (Clement et al.1999; Barrash et al. 2006).

The aquifer at the BHRS consists of Pleistocene toHolocene coarse fluvial deposits that are unconsolidatedand unaltered and that are underlain by a red clay forma-tion (Barrash and Reboulet 2004). The thickness of thefluvial deposits is in the range of 18 to 20 m, and satu-rated thickness of the aquifer is generally about 16 to 17m. The aquifer at the BHRS is known to be hetero-geneous with a multiscale and hierarchical sedimentaryorganization (Barrash and Clemo 2002) whereby theaquifer as a whole includes layers, and lenses within lay-ers, that can be recognized with a variety of geologic,hydrologic, and geophysical methods (e.g., Reboulet andBarrash 2003; Tronicke et al. 2004; Clement and Barrash2006; Clement et al. 2006; Ernst et al. 2007; Irving et al.2007).

Self-Potential and Resistivity MeasurementsTen dipole tests were performed in which water was

pumped from one well and injected into another. For thepresent investigation, we use data from two dipole testswith wells C1 and C4 as pumping and injection wells(alternatively). The self-potential signals were measuredat the ground surface in a set of 88 electrodes withrespect to a reference electrode placed 90 m from theregion of investigation (Figure 10; seven additional elec-trodes were placed in boreholes). The dashed line inFigure 10 separates two regions of high and low trans-missivity determined from the previous studies.

29.9 30.0 30.1 30.2 30.3 30.4 30.5 30.6Inverted values of the hydraulic heads (m)

29.9

30.0

30.1

30.2

30.3

30.4

30.5

30.6

Tru

e va

lues

of

the

hydr

aulic

hea

ds (

m)

Figure 7. True value of the hydraulic head vs. the invertedvalues of the hydraulic heads under steady-state conditionsof pumping (R2 = 0.97).

Table 2Material Properties Used for the Second Set of

Synthetic Models

Material K (m/s) q (ohm/m) L (A/m2)

1 5 3 10�3 10 4 3 10�4

2 5 3 10�3 20 2 3 10�4

3 10�5 100 5 3 10�4

Figure 8. Influence of the Reynolds number. Self-potential dis-tribution obtained by neglecting the effect of the Reynoldsnumber. Note that the self-potential anomaly is centered on thepumping well (white circle). The pumping rate is 10�5 m3/s.

Figure 6. Reconstruction of the a posteriori distribution ofthe hydraulic heads.


Each electrode (stainless and nonpolarizing) wasplaced inside a hole 10 cm deep and filled with a moist-ened bentonite and gypsum mixture to ensure good con-tact between the electrode and the ground, and stoneswere placed above the electrodes. Measurements of theself-potential signals were carried out with a Keithley2701 multichannel voltmeter, and we used nonpolarizingPb/PbCl2 (Petiau) electrodes (Perrier et al. 1998). Thevoltmeter was connected to a laptop computer where thedata were recorded. All the electrodes were scanned dur-ing a period of 30 s.

Here, we note that in-field temperature changescaused drift in the measurements. To account for thisdrift, an additional reference electrode was located 50 mfrom the middle of the investigated region, and tempera-ture of the packing medium around the reference electrodewas measured periodically. According to SDEC (the com-pany manufacturing the Petiau electrodes), the tempera-ture dependence of these electrodes is 0.210 mV/�C.During the day, we measured variations of temperatureranging from a few degrees Celsius to 15 �C. A differ-ence in temperature of 10 �C is responsible for a drift of

the self-potential measurement of 2 mV, and thereforethis effect cannot be neglected.

Figure 11a shows the time variation of the measuredself-potentials (raw data) on one electrode during thedipole test. This potential measurement is divided intofive time intervals denoted I to V. The data can be consid-ered to be the sum of the time variation due to the pump-ing/injection test and its recovery plus a drift. The driftitself is believed to be generated by the variation of thetemperature over time and can be adjusted by a poly-nomial of the third order with the coefficients of thispolynomial fitted in time intervals I, III, and V. This trendis then removed from the raw data, which gives the datashown in Figure 11b. These data are used to build self-potential maps like the one shown in Figure 12. Rizzoet al. (2004) used a filtering operation to improve thesignal-to-noise ratio of these data. However, we foundthat the self-potential (SP) data collected during thedipole tests at the BHRS in 2007 are of high quality.The standard deviation can be determined from the

Figure 10. Sketch showing the position of the Petiau electro-des used for the self-potential monitoring (represented bystars), the stainless steel electrodes for the resistivity tomo-graphy (open circles), and the position of wells (the filledcircles) in the central area of the BHRS. The self-potentialmap corresponds to the self-potential distribution prior todipole pumping/injection tests. Relatively lower self-potentialcorresponds with portions of the aquifer that include sandchannel deposits (i.e., relatively higher transmissivity orthickness-averaged hydraulic conductivity). The dashed lineseparates these two areas of relatively higher and lowertransmissivities.

Figure 9. Influence of the Reynolds number. (a) Distributionof self-potential determined with accounting for the distribu-tion of the Reynolds number. Note that the self-potentialanomaly is not centered on the pumping well (white dot),and its magnitude is smaller than that in Figure 8. (b) Distri-bution of the Reynolds number determined from thehydraulic head gradient. The pumping rate is 10�5 m3/s.


self-potential data that were recorded for time prior to thedipole test. The stack of all these data can be used todetermine the mean and the variance prior the dipole test.The standard deviation is typically of 0.1 mV. Conse-quently, no filtering was applied to these data.

Under steady-state conditions, Rizzo et al. (2004)and Suski et al. (2006) demonstrated that the self-potential data exhibit a linear relationship with the depthof the water table. A plot of the self-potential data asa function of the hydraulic heads is shown in Figure 13and will be interpreted in the next sections.

In addition to the self-potential data, electrical resis-tivity measurements were also collected during the steady-state conditions of the dipole tests. These measurementswere taken with stainless steel electrodes (their positionsare shown in Figure 10) and an IRIS-PRO System. Anexample of resistivity tomogram is shown in Figure 14.

Laboratory Measurements of the Material PropertiesHydraulic permeability, electrical resistivity, and

streaming potential coupling coefficient were measuredon 16 core samples from wells at the BHRS. Only the

-6

-5

-4

-3

-2

-1

0

1

Sel

f P

ote

nti

al (

in m

V)

-60 -40 -20 20 40 60 80 100 120 140

Time (min)

78

79

80

81

82

83

84

85

86

87

Sel

f P

ote

nti

al (

in m

V)

a. b.

0 -60 -40 -20 20 40 60 80 100 120 140

Time (min)

0

I II III IV V

Figure 11. Example of self-potential data during a dipole pumping/injection test. (a) Raw data. (b) Detrended data. We canrecognize five phases in the pumping test: I corresponds to the data obtained prior to the start of the pumping test; II is thetransient phase during pumping; III is the steady-state phase; IV is the rapidly changing portion of the recovery phase; and Vcorresponds to the slowly changing to steady-state portion(s) of the recovery phase.

Figure 12. Self-potential maps obtained under steady-state conditions (phase III of Figure 11 for the pumping/injection testsC4 pumping-C1 injection (a) and for the reciprocal test C1 pumping-C4 injection (b) Note that for the pumping well in bothtests, the self-potential anomaly is not centered on the pumping well itself like in the synthetic case (Figure 9a). However, theself-potential anomaly associated with the injection well is centered on the injection well for both tests.


fine fraction of the material (,5 mm) was used.The methodology used to measure these properties is thesame as that reported by Suski et al. (2006) and Boleveet al. (2007a) and will not be repeated here. The resultsare reported in Table 3. The average value of the stream-ing potential coupling coefficient is �13.2 mV/m fora brine conductivity of 0.0221 6 0.0005 S/m (at 25 �C),which is the conductivity of the ground water measured inone of the wells at the BHRS during the field experimentsin June 2007. Because

C ¼ �L

rð20Þ

from Equations (1) and (2), we can use the electricalresistivity tomography and the value C ¼ �13.2 mV/m to

map the distribution of L in 3D from the electrical resis-tivity and Equation (27).

Interpretation of the DataFirst, we can plot the self-potential data, under

steady-state conditions, as a function of the hydraulicheads for both the dipole hydraulic test C4-C1 (pumpingfrom C4 and injecting into C1) and its reciprocal C1-C4(pumping from C1 and injecting into C4). According toRizzo et al. (2004) and Suski et al. (2006), we shouldobserve a linear relationship between these two variablesthat can be written as u� u0 ¼ Caðh� h0Þ, where Ca isan apparent coupling coefficient (in V/m), u0 is the refer-ence electrical potential (0 mV), and h0 is the referencehydraulic head. This is indeed the case with Ca ¼ �5.8

-6

-5

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8

Hydraulic heads (in m)Hydraulic heads (in m)

Sel

f-p

ote

nti

al (

in m

V)

-5

-4

-3

-2

-1

0

1

10 0.2 0.4 0.6 0.8

Sel

f-p

ote

nti

al (

in m

V)

a. b.

a b

Figure 13. Self-potential data from electrodes collocated with transducers in wells vs. the value of the hydraulic heads in thosewells during the steady-state portions of dipole tests. Data shown at three different times (50, 53, and 56 min after the begin-ning of the pumping/injection). (a) Pumping/injection test C4-C1. (b) Pumping/injection test C1-C4.

Figure 14. Result of the inversion of the resistivity data acquired before the injection-pumping experiment. (a) Inverted resis-tivity model. The positions of the electrodes are shown in Figure 10. (b) Distribution of electrical resistivity cut at the 3000ohmmeter isosurface.


mV/m for the dipole test C4-C1 and Ca ¼ �5.3 mV/m forthe dipole test C1-C4 (Figure 13). According to Suskiet al. (2006), Ca and C are approximately related by Ca ¼C/2. Because the average value of C is �13.4 6 3.4 mV/m, this yields Ca ¼ �6.7 6 1.7 mV/m, which can be con-sidered as a reasonable estimate of the value of the appar-ent coupling coefficient given above. Then, we use thisvalue of Ca to reconstruct the semi-variogram of the a priorihydraulic head model determined from the linear relation-ship u� u0 ¼ Caðh� h0Þ (Figure 15). The semi-variogramshown in Figure 15 is fitted with a Gaussian model.

The optimization of the self-potential data usingthe method described previously yields the a posterioriprobability density of the hydraulic heads shown inFigures 16a and 16b for dipole test C4-C1. However, thisestimate is wrong in the vicinity of the pumping wellbecause we did not account for the effect of the Reynoldsnumber. To obtain a better estimate, we took the a posteri-ori (optimized) probability density of the hydraulic headsreconstructed from the self-potential measurements onlyin the area close to the injection well while only thepotentiometric data are used in the vicinity of the injec-tion well. The result is shown in Figures 16c and 16d.This information can be used to determine the trans-missivity distribution in the aquifer in the vicinity of thepumping and injection wells.

ConclusionsWe have proposed a method for inverting self-poten-

tial data to determine the shape of the water table in

steady-state conditions of pumping or injection tests. Weinvert the Poisson equation in a probabilistic (Bayesian)framework and using also the available potentiometricdata in the inverse problem. This method has been testedsuccessfully on a synthetic case and was used to invertfor the shape of the water table during a dipole test andits reciprocal at the BHRS. The next step will be to

1.2

0.8

0.6

0.4

0.2

1.0

0

Lag distance (in m)

Sem

i-va

riog

ram

(in

mV

2 )

0 1 2 3 4 5 6 7 8 9

Figure 15. Semi-variogram of the a priori hydraulic headmodel determined from the linear relationship u � u0 = Ca

(h � h0). The semi-variogram is fitted with a Gaussian model(angle 55�).

Table 3Experimental Data Performed on Core Samples from the BHRS

Sample1Permeability

(m2)

CouplingCoefficient(mV/m)

ElectrolyteConductivity

(S/m)

SampleConductivity

(S/m)FormationFactor

CouplingCoefficient2

(mV/Pa)

A1: 11.55-12/M4 1.7413 10�11 �15.7 0.0183 4.443 10�03 6.9 �13.0A1: 26.9-27.5/M3 1.973 10�11 �20.1 0.0170 3.433 10�03 6.5 �15.5B2:26.35-27.0/M3 7.583 10�12 �20.0 0.0209 3.993 10�03 8.1 �18.9B2:38.4-39.0/M2 9.663 10�12 �15.0 0.0203 4.263 10�03 7.7 �13.8B2:39.65-40.2/F2 8.773 10�12 �17.2 0.0234 5.393 10�03 6.0 �18.2B2:40.2-40.65/F2 1.313 10�11 �13.0 0.0193 3.543 10�03 11.5 �11.3B2:41.85-42.4/M2 9.083 10�12 �14.2 0.0203 3.183 10�03 9.2 �13.0B2:47.0-47.7/M2 2.823 10�11 �13.5 0.0183 3.923 10�03 8.7 �11.2B5:10.8-11.15/B4 2.303 10�11 �12.9 0.0163 2.913 10�03 5.8 �9.5B5:55.15-56.0/M1 1.063 10�11 �15.2 0.0174 3.433 10�03 7.5 �12.0C1:16.45-17.0/B4 2.673 10�12 �9.1 0.0173 3.753 10�03 4.1 �7.1C1:17.75-18.1/M4 1.973 10�11 �14.9 0.0172 3.543 10�03 6.2 �11.6C1:34.1-35.0/M3 4.043 10�12 �9.1 0.0216 3.983 10�03 6.7 �8.9C4:10.75-11.05/S5 4.163 10�11 �19.4 0.0154 3.193 10�03 4.3 �13.5C4:11.05-12.0/S5 3.463 10�12 �9.6 0.0155 3.723 10�03 4.3 �6.7C5:55.12-56.0/M1 2.913 10�11 �17.0 0.0175 3.563 10�03 7.3 �13.5

1Sample names identify well (e.g., A1), depth interval in feet from land surface (e.g., 11.55 to 12.0), lithotype (e.g., M for mixed cobble sizes dominate in the sample),and stratigraphic unit (e.g., 4 for unit 4). See Reboulet and Barrash (2003) and Barrash and Reboulet (2004).2Normalized to a pore water conductivity equal to 0.0221 S/m at 25 �C.


extend this approach to transient conditions to be able, inaddition to the transmissivity, to determine the storativity.

AcknowledgmentsWe thank Frederic Perrier for the use of his electro-

des and fruitful discussions. Terry Young is thanked forhis support at Mines. EPA grants X-96004601-0 andX-96004601-1 provided support for experimentation atthe BHRS and for Boise State University researchers. Wethank one anonymous referee and Fred Day-Lewis fortheir very constructive and useful reviews.

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AppendixIn this Appendix, we present the basis for the Bayes-

ian approach used in the main text. If we consider thattwo states A and B exist, the probability that state B willfollow, or overlie, state A, PðAjBÞ (where PðAjBÞ is calleda conditional probability), is the probability that bothstates occur, P(A, B) (joint probability), divided by theprobability that state A occurs: PðAjBÞ ¼ PðA;BÞ=PðBÞ.Similarly, we can write PðBjAÞ ¼ PðB;AÞ=PðAÞ. It fol-lows that the two conditional probabilities PðAjBÞ andPðBjAÞ are related to each other by Bayes’ theorem:

PðBjAÞ ¼P�AjBÞP

�B�

PðAÞ ðA1Þ

We note now h a vector column of M unknownhydraulic heads. The vector d represents a vector columnof N self-potential observations. Bayes’ theorem, writtenin terms of probability distribution between the modelparameters and the data, is written as:

PðhjdÞ ¼P�djhÞP

�h�

PðdÞðA2Þ

This can be written as PðhjdÞ}PðdjhÞPðhÞ. The a pos-teriori probability density on the space of models, rðhÞ,is therefore proportional to the product of two terms(Tarantola and Valette 1982):

rðhÞ}LðhÞqðhÞ ðA3Þ

where the likelihood function LðhÞ represents the prob-ability density associated with the fit of the data and thesecond term, qðhÞ, represents the a priori probability den-sity. This probability density incorporates informationabout the subsurface that is independent of the observeddata from which the inferences are being made. The goalof the Bayesian approach was to construct the probabilitydensity rðhÞ, where h is an a posteriori estimate of thehydraulic head distribution.


Equation (A3) is rather general. Let us considerthat all uncertainties in the problem can be describedby stationary Gaussian distributions. This yields thefollowing:

L�h�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pÞ�N

det Cd

sexp

��1

2

�Gh�dobs

�TC�1d

�Gh�dobs

�

ðA4Þ

q�h�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pÞ�M

det Cm

sexp

��1

2

�h� hprior

�TC�1m

�h� hprior

�

ðA5Þ

where G is the kernel or forward modeling operator thatconnects the self-potential data to a variation of thehydraulic head (see the Forward Modeling section ofthe main text), Cd is the data covariance matrix, dobsis the observed self-potential data, which are measured atthe ground surface, hprior is the a priori value of thehydraulic heads, Cm is the model covariance matrix, andN and M are the number of observations and the numberof model parameters, respectively.

In the case where all the uncertainties are Gaussian,the a posteriori probability density is the normalizedproduct of Equations (A4) and (A5). Because the forwardoperator K is linear (in the inertial laminar flow regime),this distribution is itself a Gaussian and is given fromEquation (A3) by:

rðhÞ} exp

��1

2

�G h� dobs

�T

Cd

�G h� dobs

�

�1

2

�h� hprior

�TC�1m

�h� hprior

�ðA6Þ

The matrices Cd and Cm incorporate the uncertaintiesin the data (modeling and observational errors) and un-certainties related to the hydraulic heads, respectively.Linde et al. (2007) showed from field data that a self-poten-tial measurement can be considered as a random processwith a probability density that is Gaussian, and they ex-plained how Cd can be obtained from mapping the self-potential data at the scale of a catchment. For self-potentialmonitoring, Cd can be obtained by looking at the noiselevel for all the data taken before the start of the dipole test.

We maximize the a posteriori probability density onthe model parameters (a procedure called MAP in the liter-ature, e.g., Gouveia and Scales 1997). Maximizing likeli-hood corresponds with minimizing the associated negativelog-likelihood. Consequently, we have to minimize the fol-lowing cost function,C ¼ �ln ðrðhÞÞ, given by:

2C ¼�G h� dobs

�TCd

�G h� dobs

�1

�h� hprior

�TC�1m

�h� hprior

�ðA7Þ

The maximization of the previous cost function isobtained by setting @C=@h ¼ 0 (Tarantola and Valette1982), which yields Equations (15) to (17) of the main text.


Reconstruction of the Water Table from Self-Potential Data: A Bayesian Approach

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