Review: Some low-frequency electrical methods for subsurface characterization and monitoring in...

Review: Some low-frequency electrical methods for subsurface

characterization and monitoring in hydrogeology

A. Revil & M. Karaoulis & T. Johnson & A. Kemna

Abstract Low-frequency geoelectrical methods includemainly self-potential, resistivity, and induced polarizationtechniques, which have potential in many environmental andhydrogeological applications. They provide complementaryinformation to each other and to in-situ measurements. Theself-potential method is a passive measurement of theelectrical response associated with the in-situ generation ofelectrical current due to the flow of pore water in porousmedia, a salinity gradient, and/or the concentration of redox-active species. Under some conditions, this method can beused to visualize groundwater flow, to determine permeabil-ity, and to detect preferential flow paths. Electrical resistivityis dependent on the water content, the temperature, thesalinity of the pore water, and the clay content andmineralogy. Time-lapse resistivity can be used toassess the permeability and dispersivity distributionsand to monitor contaminant plumes. Induced polariza-tion characterizes the ability of rocks to reversiblystore electrical energy. It can be used to image

permeability and to monitor chemistry of the porewater–minerals interface. These geophysical methods,reviewed in this paper, should always be used inconcert with additional in-situ measurements (e.g. in-situpumping tests, chemical measurements of the pore water),for instance through joint inversion schemes, which is anarea of fertile on-going research.

Keywords Geophysical methods . Groundwaterhydraulics . Groundwater monitoring . Hydraulicproperties . Unsaturated zone . Review

Introduction

Hydrogeophysics is presently a field of fertile researchin which non-intrusive or minimally intrusive (geo-physical) methods are used to assess parameters ofhydrogeological importance including the distributionof transport properties (permeability, transmissivity,storativity, dispersivity), water content, water quality(the detection of various types of contaminants), andbiological activity (see Rubin and Hubbard 2005;Hubbard and Linde 2011). While there are a numberof geophysical electromagnetic methods operating in abroad range of frequencies (near direct current (DC) toGHz), the objective here is to provide a basicdescription of selected, low-frequency (below 10 kHz)geoelectrical methods and their use in hydrogeology.These methods include one passive method, the self-potential method, and two active methods, the DCresistivity and induced polarization methods. Thesemethods have been selected because they sharecharacteristics both in their physics and in the waythey are implemented in the field.

The self-potential method is a passive method similarto electroencephalography in medical imaging. It isdirectly sensitive to the flow of the groundwater and tothe chemistry of both the pore water and the pore-water/mineral interface. The DC resistivity method is an activegeophysical method aimed at imaging the resistivity (theresistance of a material to the flow of an electrical current)of the subsurface or its inverse, the electrical conductivity.The induced polarization method is a natural extension ofthe DC resistivity method to time-varying fields. In

Received: 3 January 2011 /Accepted: 8 December 2011Published online: 10 February 2012

* Springer-Verlag 2012

A. Revil :M. KaraoulisColorado School of Mines, Department of Geophysics,Green Center,1500 Illinois Street, 80401( Golden, CO, USA

M. Karaoulise-mail: [email protected]

A. Revil ())LGIT, UMR 5559, CNRS, Equipe Volcan,Université de Savoie, 73376( Le Bourget-du-lac Cedex, Francee-mail: [email protected]

T. JohnsonPacific Northwest National Laboratory, P.O. Box 999, Richland,WA 99352( USAe-mail: [email protected]

A. KemnaDepartment of Geodynamics and Geophysics,University of Bonn,Nussallee 8, 53115 Bonn, Germanye-mail: [email protected]

Hydrogeology Journal (2012) 20: 617–658 DOI 10.1007/s10040-011-0819-x

addition to resistivity, this method can be used to imagethe ability of porous materials to store reversibly electricalenergy (like a capacitance). The frequency-domain in-duced polarization (or SIP, for spectral induced polariza-tion) method provides a conductivity and a phase lagbetween the current and the electrical field, which can bewritten as a complex electrical conductivity. Anotherapproach is to measure the polarization of porousmaterials in the time domain, looking at the change in“secondary” voltage after the shutdown of an injectedcurrent. The key parameter provided by time-domaininduced polarization is the chargeability. Both thephase and the chargeability are essentially a measureof polarization relative to conduction. The complexconductivity and the chargeability are sensitive toparameters that can be used to describe the permeabil-ity (Börner 1992, 2006; Kemna 2000; Revil andFlorsch 2010) and the chemistry at the pore-water/mineralinterface (Vaudelet et al. 2011a, b). They are also verysensitive to the presence of metallic particles likepyrite (e.g., Pelton et al. 1978).

Although all of these three methods are classicalmethods in applied geophysics (see, e.g., Rubin andHubbard 2005), a number of recent advances regardinghydrogeological applications have not been covered bypreviously published review papers or books. Thereforethe goal here is to provide an up-to-date overview.Methods based on electromagnetic induction (e.g., Nobes1996; Tezkan 1999; Pellerin 2002) or the more recentlypursued seismoelectrical method (e.g., Dupuis et al. 2009;Jardani et al. 2010) will not be discussed. The main goalof the present paper is to offer hydrogeologists a simpledescription of the low-frequency geoelectrical methodsand their usefulness in solving practical hydrogeologicalproblems. It is firmly pointed out that the ability ofgeophysical methods to assess subsurface propertiesshould not be oversold. For instance the way electricalcurrent flows in a rock or a soil is vastly different from theway groundwater flows in the same network of pores andcracks. Therefore the imaging of electrical conductivity(DC resistivity imaging) cannot be used directly to assessa property like permeability. That said, complex conduc-tivity can be used to assess textural parameters, which inturn can be used to estimate permeability. Typicallygeophysical methods have to be used in concert withother data, for instance via joint inversion schemes, asthey provide spatialized information that is complementaryto local (in-situ) measurements in wells (fluid pressure andpore water chemistry). Note also that these in-situmeasurements have their own uncertainty as drilling awell generally creates a disturbance of the subsurfaceproperties in the vicinity of the well, which in turnaffects direct measurements. On the other hand, thereis a gain to be had in monitoring a dynamic process(groundwater flow, tracer test, etc.) non-intrusively ornon-destructively. The present paper describes how thethree low-frequency geoelectrical methods introducedin the previous can be used to assess hydrogeologicalproperties as well as the limitations of the methods.

The self-potential method

PrincipleThe self-potential method is a passive geophysical methodinvolving the measurement of the electrical potentialdistribution at a set of measurement points (stations).The measurements are performed using non-polarizingelectrodes as shown in Fig. 1a and b. The difference of theelectrical potential between two electrodes is measured byusing a voltmeter with a high sensitivity (at least 0.1 mV)and a high input impedance (typically ∼10–100 MOhmfor soils to 1,000 GOhm on permafrost). Figure 1c isshowing the record of the electrical potential differencebetween one electrode located close to an injection welland a reference electrode located several tens of metersway (further than the radius of influence of the well). Thedata are the raw data (no filtering applied). One can seethe self-potential response associated with the pumpingtests and the recovery following the shutdown of thepump. To perform accurate self-potential measurements,the impedance of the voltmeter has to be at least tentimes higher than the impedance of the groundbetween the two electrodes in order to avoid leakageof current in the voltmeter. The voltmeter has to becalibrated regularly against known resistances to checkthe accuracy of the voltage measurements for a broadrange of resistance values.

Self-potential mapping consists of establishing a mapof the self-potential at the ground surface using a referenceelectrode as a fixed base. For instance two non-polarizing(Petiau) Pb/PbCl2 electrodes (Petiau 2000) can be used(see Fig. 1b), one fixed and one to scan the electricalpotential at the ground surface. The fixed (reference)electrode is kept in a small hole filled with bentonite mud.Because the presence of the mud modifies the electricalpotential at the contact between the electrode and theground, the potential of this station is arbitrary and shouldnot be used for creating any self-potential map thatinvolves all the other data. Adding salty water to improvethe contact between the electrode and the ground shouldbe avoided, especially for monitoring purposes, becauseevaporation of the water changes the salinity of the porewater, generating spurious potential changes over time inthe vicinity of the surface of the electrodes. The movingelectrode is used to measure the electrical potential at a setof stations referenced in space with a global positioningsystem (2 m accuracy in x and y is usually good enoughfor most applications). Prior to and after the measure-ments, the difference of voltage between the referenceelectrode and the scanning electrode has to be checked byputting the electrodes one against the other through theirporous membranes (for instance in Fig. 1b, the porousmembrane is made of wood) and by measuring theirdifference of electrical potential. The drift of the voltagebetween the electrodes should be kept as small as possibleover time (e.g., < 2 mV day–1 is an acceptable drift forself-potential mapping). The potential map is, therefore, amap relative to the (unknown) potential at the base station.Actually, like all scalar potentials in physics, the self-

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potential is defined to an additive constant. Only theelectrical field (which is the gradient of the electricalpotential in the low-frequency limit of the Maxwellequations) is well-defined. At the interface between theground and the atmosphere, the electrical field is tangen-tial to the ground surface because air can be considered asan insulator.

It is possible to use the self-potential method as amonitoring method. In this case, a multi-electrode array isused and connected to a multichannel or multiplexedvoltmeter as shown in Fig. 1a. This approach iscompletely analogous to what is done in electroencepha-lography (EEG) for medical applications (Fig. 1). In EEG,a network of electrodes is used to monitor the change inthe distribution of the electrical potential on the scalp inorder to localize the active part of the brain where

electrical currents are manifested along the synapsesbetween neurons (Grech et al. 2008). Even if the electricalpotentials are measured at several hundred Hertz in EEG,the same fundamental (quasi-static) equation applies. Thisequation will be described in the following In the case ofgeoelectrical self-potential signals, the main sources ofcurrents in the ground are closely connected to theexistence of the electrical double layer at the pore-water/mineral interface (described in Fig. 2).

Before going further in the description of the self-potential signals, it is worth describing the concept of theelectrical double layer at the pore-water/mineral interfacebecause of its importance in all that follows. Figure 2sketches a silica grain coated by an electrical double layer.When a mineral like silica is in contact with water, itssurface gets charged because of chemical reactions

Fig. 1 Typical recording system to monitor the self-potential response associated with a pumping test. a Picture showing the network ofelectrodes and the recording multielectrode voltmeter. b Petiau non-polarizing electrodes. c Typical raw data for the self-potential (modifiedfrom Jardani et al. 2009). Phase I corresponds to the data obtained prior to the start of the pumping test; II is the transient phase duringpumping; III is the steady-state phase; IV is the rapidly changing portion of the recovery phase; V corresponds to the slowly changing tosteady-state portion(s) of the recovery phase. Before being analyzed, these data need to be detrended, filtered, and shifted in such a way thatthe potential prior to the start of pumping (t=0) is equal to zero for each electrode

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between the surface sites and the pore water. For instancethe silanol groups, >SiOH (where > refers to the mineralframework), loose a proton in contact with water togenerate negative surface sites (>SiO–). The resultingmineral surface charge is pH-dependent, being typicallynegative at pH 7. At this pH, the surface charge attractsthe ions of positive sign (counterions) and repels the ionsof the same sign (co-ions). It results in the formation of aso-called diffuse layer characterized by an excess ofcounterions and a depletion of co-ions (Fig. 2). Inaddition, some ions can be sorbed directly on the mineralsurface forming the so-called Stern layer. The Stern layeris therefore comprised between the o-plane (mineralsurface) and the d-plane, which is the inner plane of theelectrical diffuse layer on Fig. 2. The term “electricaldouble layer” is a generic name describing this electro-chemical system. The term electrical-triple-layer model(TLM) is often used when different types of sorptionphenomena are considered at the level of the Stern layer.

Note that in some clay minerals, like smectite for example,the solid particles have a negative surface charge that isdirectly related to isomorphic substitutions in theircrystalline network, in addition to a surface-charge densityassociated with acid-base reactions.

There are two fundamental consequences associatedwith the existence of the electrical double layer: (1) thepore water is never neutral, and (2) there is an excess ofelectrical conductivity in the vicinity of the pore-water/mineral interface. The first consequence is fundamentalto understand the occurrence of electrical currentsassociated with the flow of the pore water and,therefore, self-potential signals. The second consequence iscrucial to understand electrical conductivity and chargeabil-ity of porous materials for DC resistivity and inducedpolarization studies.

Discussed next is the origin of noise in the measure-ments of the self-potential signals. In self-potentialmapping, noise can originate from different transient

Fig. 2 Sketch of the electrical double layer at the pore-water/mineral interface coating a spherical grain (modified from Revil and Florsch2010). The local conductivity σ(χ) depends on the local distance χ from the charged surface of the mineral. The pore water is characterizedby a volumetric charge density QV corresponding to the charge of the diffuse layer per unit pore volume—in Coulombs (C) m–3. The Sternlayer is responsible for the excess surface conductivity ∑S (in Siemens, S) with respect to the conductivity of the pore water σf, while thediffuse layer is responsible for the excess surface conductivity ∑d. These surface conductivities are sometimes called specific surfaceconductance because of their dimension but they are true surface conductivities. The Stern layer is comprised between the o-plane (mineralsurface) and the d-plane, which is the inner plane of the electrical diffuse layer (OHP stands for outer Helmholtz plane). The diffuse layerextends from the d-plane into the pores. M+ stands for the metal cations and A– for the anions. In the present case (negatively charged mineralsurface), M+ are the counterions, while A– are the co-ions

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sources. These include induction of telluric currentsoccurring inside the conductive Earth due to transientcurrent flow in the ionosphere, lightning, the presence oflarge cumulus clouds, and induction in the groundassociated with powerlines as well as direct currentinjection associated with cathodic mise-à-la-masse(grounding) used to prevent corrosion of metallic pipesfor instance (see Corwin and Hoover 1979). Someexamples of spurious self-potential signals of culturalorigin are shown in Fig. 3, for instance related to themetallic casing of a piezometer (Fig. 3a) or cathodicprotection (Fig. 3b). Transient signals can be filtered out ifa fixed dipole is used to record the electrical signals duringmapping (or, in the case of telluric noise, a magnetometerto record correlated magnetic signals).

Spatial noise can be associated with strong heteroge-neity of the resistivity distribution in the shallow subsur-face (the first centimeters to meters). The latter can be dueto dry conditions and/or the presence of more resistivebodies (like stones) embedded in a more conductivematrix like a fine-grained soil with higher moisture

content. This spatial noise can be filtered out if measure-ments are made with a high spatial density (e.g., fivemeasurements over a square meter at a given station).Generally, Fourier or wavelet-based filters can be usedto remove noise in a self-potential map (see, e.g.,Moreau et al. 1996 for a filtering approach based onorthogonal wavelets).

Other artifacts can be associated with the non-polariz-ing electrodes themselves, which have an inner potentialthat is temperature dependent. Therefore, having twosimilar electrodes at two different temperatures yields adifference in their electrical potentials. This effect can becorrected if the temperature is measured at the place wherethe self-potential measurements are made, with theelectrodes being in thermal equilibrium with their envi-ronment, and if the temperature coefficient of the electrodetype is known (e.g., 0.2 mV/°C for Petiau electrodes, seePetiau 2000). Usually, Petiau electrodes (Fig. 1b) arepreferred to Cu/CuSO4 electrodes, which have a substan-tially higher temperature dependence (0.7 to 0.9 mV/°C,see Antelman 1989).

Fig. 3 Type of spurious self-potential anomalies associated with cultural (anthropogenic) activity. a Self-potential anomaly associatedwith the metallic casing of a piezometer. The anomaly (−24 mV) due to redox effect associated with the corrosion of the metallic body. bSelf-potential anomaly associated with the cathodic protection of a gas pipe at the Entressen landfill in the south of France. c Self-potentialanomaly associated with grounded monitoring instruments (to avoid their damage by lightning) at the Bemidji USGS site in Minnesota(USA) in 2010

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Short historyThe history of self-potential signals starts with the firstmeasurements made by Robert W. Fox (Fox 1830) inCornwall (UK) over sulfide vein mineralization. Veryquickly the self-potential method was used qualitativelyfor the prospection of ore bodies. Field measurementswere drastically improved with the development of non-polarizing electrodes for geophysical applications, first byMC Matteucci at the Greenwich Observatory (Matteucci1865). Thanks to non-polarizing pot electrodes, Matteucciobserved in 1847 that earth currents in telegraph wireswere correlated with the aurora borealis (Spies 1996).Non-polarizing electrodes were later used by Carl Barusfor the exploration of ore bodies, especially the Comstockgold lode in Nevada, USA (Barus 1882; Rust 1938). Thefirst discovery of an ore body of commercial interest withthe self-potential method was done in 1906 in Norway byMuenster (see Rust 1938). Sato and Mooney (1960), intheir seminal paper, made a synthesis of all availableinformation and proposed their famous “geobatterymodel”. In this model, the corrosion of an ore body andthe distribution of the redox potential with depth (associ-ated with the change of oxygen concentration with depth)explains the occurrence of self-potential anomalies at theground surface (Sato and Mooney 1960; Revil et al. 2001;Castermant et al. 2008). This model was refined later byStoll et al. (1995) and Bigalke and Grabner (1997), usinga non-linear model known to electrochemists as theButler-Volmer model (Bockris and Reddy 1970;Mendonça 2008), and is discussed further in the following.This model describes electrical potential losses betweenelectron donors and the metallic body, and the metallic bodyand electron acceptors.

Recently, a similar battery model was proposed byArora et al. (2007) and Linde and Revil (2007) fororganic-rich contaminant plumes in which a sequence ofredox reactions occur and biotic electron nanowires,existing between bacteria in a biofilm, may be involvedin the process and, therefore, the redox reactions decoupleover space. The difference between the abiotic and bioticmodels of geobatteries is that in the biotic model, there areno potential losses because the bacterial biofilms play therole of catalysts in the transfer of electrons betweenelectron donors and electron acceptors (Revil et al.2010a). These concepts of geobatteries were also testedand confirmed in the laboratory (Timm and Möller 2001;Naudet and Revil 2005; Castermant et al. 2008).

A second contribution to self-potential signals isrelated to concentration gradients of ionic species in thepore water, the so-called diffusion current or potential. It wasfirst commercially used in geophysics by Schlumberger(1920) especially as a downhole measurement tool fordiagraphy (Schlumberger et al. 1932, 1933).

A third contribution to the occurrence of self-potentialsignals is related to groundwater flow. As shown in Fig. 2,most natural porous media have an excess of charge intheir pore water (in the diffuse layer) to balance the fixedcharge occurring on their mineral surface. The streamingcurrent associated with the flow of the pore water

corresponds to the electrical current generated by the dragof the excess charges of the diffuse layer. The charge ofthe diffuse layer balances exactly the charge attached tothe mineral surface in such a way that the porous material,as a whole, is neutral. The advection of a net charge in afixed framework attached to the grains is by definition acurrent density (flow of electrical charges per unit surfacearea per unit time). The physics of the streaming potentialtakes its root in the experimental work done by Quincke(1859), and a theoretical expression was obtained 20 yearslater by Helmholz (1879) for glass capillaries. In the mid-time, Bachmetjew in 1894 was probably the first toobserve a self-potential field associated with the motion ofthe groundwater through sands in Germany. Nourbehecht(1963) developed linear constitutive equations for thecoupled generalized Darcy and Ohm laws in porousmedia. Later, Pride (1994) gave a comprehensive theory ofthese “electrokinetic” effects by upscaling the Nernst-Planckand Navier-Stokes equations using a volume-averagingmethod. Bogoslovsky and Ogilvy (1972) gave a semi-quantitative treatment of the self-potential signature ofpreferential drainage near dams (see also Gex 1993). Rizzoet al. (2004) applied this approach to pumping testsfollowing the ideas developed by Semenov in the 1980s inthe former USSR. The self-potential method has also beenapplied successfully in geothermal exploration (see a reviewof earlier works by Corwin and Hoover 1979, and the morerecent work by Ishido 1989, 2004; Ishido and Pritchett 1999,and Richards et al. 2010).

Underlying physicsAs mentioned in the previous section, the occurrence ofremote self-potential signals is associated with theoccurrence of electrical currents in the ground. Theexistence of an electrical current in the ground is relatedin turn to thermodynamic non-equilibrium situationsaffecting the transport of charge carriers in the ground.Two main contributions usually dominate the sourcecurrent density: the streaming current associated with theflow of the pore water and the diffusion currentsassociated with gradients in the chemical potentials ofcharge carriers (electrons and ions).

As mentioned in the preceding, the streaming current isassociated with the advective transport of the excess ofelectrical charges existing in the pore water of a porousmaterial in the so-called diffuse layer (Fig. 2). Because ofthe existence of the electrical diffuse layer, the pore waterof a porous material is usually non-neutral and carries anexcess of electrical charges at pH above the so-calledpoint of zero charge—PZC, with pH(PZC)=3 for silica—and an excess of negative charges at lower pH values. Theflow of water through the porous material drags thisexcess of charge and is, therefore, responsible for anelectrical current called the streaming current.

A second type of contribution to the source currentdensity is related to the occurrence of gradients inconcentrations of charge carriers (ions and electrons).The diffusion current density is related to molecular

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diffusion of ionic charge carriers. For example, a solutionof NaCl provides the same amount of cations (Na+) andanions (Cl−) in water. In a concentration (salinity)gradient, the anion Cl− has a higher mobility (intrinsicdiffusion coefficient) than the cation Na+. In a salinitygradient, this difference in mobility is responsible for asource of current, which is driven by the gradient of thechemical potential of the salt. This chemical potentialgradient is actually proportional to the salinity gradient.The source current density is controlled by the differenceof mobility between the anions and the cations (e.g.,Nourbehecht 1963; Revil 1999, and references therein). Inwater, this source current density is counterbalanced by aconduction current density and the resulting electrical fieldis called the diffusional electrical field. In a porousmaterial, the same type of coupling between the diffusionflux and the current density exists but the concentrationsof cations and anions are also controlled by the presenceof the electrical diffuse layer (Revil 1999; Woodruff et al.2010). The resulting source current density can besubstantially stronger than in a pure electrolyte and theresulting electrical field is often called the membranepotential or membrane electrical field (Ellis and Singer2007; Woodruff et al. 2010).

The relative electron activity of water is defined aspε = − log{e−}, where {e−} represents the potentialelectron activity in the pore-water phase (Hostetler 1984;Thorstenson 1984). In a reducing system, the tendency todonate electrons, or electron activity, is relatively largeand pε is low. The opposite holds true in oxidizingsystems. In a reduction reaction, an oxidized species Oxreacts with n electrons to form a reduced species Red. Thehalf-reaction Ox + ne− → Red is characterized by areaction constant K = {Red} / {Ox} {e−}n. Because thereis no electron in the pore water, the previous reductionreaction has to be coupled with an oxidation reaction(typically for reference purpose the oxidation of hydrogen).This leads to

p" ¼ p"0 þ 1

nlog

Oxf gRedf g

� �; ð1Þ

where pε0 is the standard electron activity of the actualreduction half-reaction when coupled to the oxidation ofhydrogen under standard conditions (Christensen et al.2000). The redox potential is defined through the Nernstequation by

EH ¼ 2:3kbT

ep"; ð2Þ

where T is the absolute temperature in K, e is theelementary charge of the electron, and kb is the Boltzmannconstant. Consider a massive ore body embedded in theconductive ground. Electrons within the ore body have ahigh mobility but do not exist in the surrounding rockmass. The presence of the ions in the pore water controlsthe electrical conductivity in the surrounding rock mass.

Because the fugacity of oxygen (the concentration of O2

dissolved in water) decreases with depth, the redoxpotential has, in the far field of the ore body, a strongdependence with the depth z. In the vicinity of the orebody, the distribution of the redox potential can be morecomplex because of contribution of interfacial processesand possibly biological activity. The near-field distributionof the redox potential is influenced by anodic and cathodicreactions.

Let us consider for example the corrosion of an orebody like pyrite (FeS2). Reactions involve (1) theoxidation of the S(-II) and S(0) in pyrite coupled torelease of SO4

2− and Fe2+ at depth, coupled to (2) thereduction of oxygen near the oxic-anoxic interface(typically the water table). The soluble Fe released duringthe anodic reaction at depth can then eventually react,through advective, dispersive, and electromigration trans-port, with oxygen at the water table. It is thereforeresponsible for the distribution of the redox potential in thevicinity of the ore body. This mechanism can be summarizedby the following reactions. At depth at the surface of the orebody, the following half-reaction occurs,

FeS2 þ 8H2O ⇄ Fe2þ þ 2SO2�4 þ 14e� þ 16Hþ; ð3Þ

which is an abiotic half-reaction pulled along by sinks forelectrons and Fe2+ at the oxic-anoxic interface. At thecathode, possibly in the vadose zone, one can have thefollowing reactions

14 e� þ 3:5 O2 þ 14 Hþ ⇄ 7 H2O; ð4Þ

4 Fe2þ þ O2 þ 4 Hþ ⇄ 4 Fe3þ þ 2 H2O; ð5Þ

4 Fe3þ þ 12 H2O ⇄ 4 Fe OHð Þ3 þ 12 Hþ: ð6ÞEquation (4) corresponds to the half-reaction associated

with the electrons provided by the ore body. Equation (5)is a redox reaction in the solution with the microorganismsbeing potentially able to accelerate this reaction dependingon the pH of the solution (low pHs favor the reaction).Equation (6) is the Fe(III) oxide mineral precipitation,which is an abiotic reaction. It is important to realize thatthe vertical redox gradient in the vicinity of the ore body,in addition to being influenced by reactions associatedwith the corrosion of the ore body (Eqs. 3–6), could bealso influenced by redox reactions (possibly microbiallycatalyzed) that are unrelated to ore body corrosion, e.g.,reactions associated with degradation of organic matter inthe aquifer sediments surrounding the ore body in the wayenvisioned in the following. In this case, the ore bodywould serve as a conductor for transfer of electronsreleased during these reactions from depth to the oxic-anoxic interface.

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The distribution of the redox potential should thus beviewed as a general source mechanism that drives electricalcurrent flow inside the ore body because, from a thermody-namic standpoint, a gradient of the chemical potential ofcharge carriers (here the electrons) corresponds to a drivingforce for an electrical current density. Two models of batteryare possible. In the first, the ore body serves directly as asource of electrons, vis-à-vis Eq. (3), that flow from depth tothe shallow subsurface through the ore body. Equation (3)describes a source of electrons originating from the oxidationof the ore body. This corresponds to an “active electrode”model. In the second model, called the “passive electrode”model, the ore body serves simply as a conductor for transferof electrons that originate from redox reactions (potentiallymicrobially catalyzed) that take place away from the orebody and have nothing to do with the corrosion of the orebody. Both models can co-exist. For example, the 14electrons released in half-reaction (Eq. 3) can pass directlythrough the ore body to oxygen at the aerobic-anaerobicinterface, whilst the electrons released during the oxidationof Fe2+ in solution at or near the ore body-water interface canalso be transmitted to the shallow subsurface through the orebody. Because of the existence of potential losses betweenthe electron donors and the metallic body, for instance, thegoverning constitutive current/force equation is the non-linear Butler-Volmer equation (Bockris and Reddy 1970;Mendonça 2008). There are cases for which the relationshipbetween the source current density and the redox potentialgradient is linear, in the presence of catalysts that lower theenergy barriers between the electron donors and the metallicbody or between the metallic body and electron acceptors.For instance, bacteria can be very efficient catalysts for redoxreactions and in the presence of a biofilm. The relationshipbetween the current density and the gradient of the redoxpotential is found to be linear (Revil et al. 2010a). The effectsshould not be confused with electrodic voltages associatedwith changes in the surface chemistry of the electrodeswhich are not self-potential signals (Slater et al. 2008;Williams et al. 2010).

Forward and inverse modelingOne of the most impressive advances made in self-potential interpretation over the past few years has beenthe inversion of the self-potential signals to locate theirsource in the ground. This has applications in mappinggroundwater flow and redox potential. The inversion ofself-potential signals is in itself not new. However, most ofthe classical methods were based on polarized spheres(Yungul 1950), dipole current lines (Paul 1965), sourceand sink pairs (Hase et al. 2005), or other simplegeometries (Nourbehecht 1963; Fitterman and Corwin1982; Rozycki et al. 2006). Such types of methods havealso been used (and abused) to interpret other potentialfield problems in magnetism and in gravity. Despite thefact that these methods are useful, little information isgained about the source itself, and the physics of theprimary flow problem (using the terminology of Sill 1983)is sometimes missed.

Recently, more sophisticated methods have emergedusing deterministic inversion with Tikhonov regularizationto reduce the non-uniqueness of the inverse problem(Jardani et al. 2007a, b, 2008; Minsley et al. 2007a, b;Linde and Revil 2007; Mendonça 2008). Under someconditions, the inverse problem can be parameterized insuch a way that stochastic inversion within a Bayesianframework can be used to determine the posteriorprobability density of material properties like the perme-ability of a set of geological units to reconstruct the flowof the groundwater (Jardani et al. 2009; Jardani and Revil2009; Revil and Jardani 2010).

Next, a framework to model self-potential data isprovided. In a porous material saturated by a brine, thetotal electrical current density j (expressed in A m−2)represents the flux of electrical charge (C m−2 s−1). Thiscurrent density is given by (Sill 1983)

j ¼ �0 Eþ jS; ð7Þwhere E is the electrical field (in V m−1) (in the quasi-static limit of the Maxwell equations written as E=-∇= ,where = is the electrical potential expressed in V), σ0 isthe DC electrical conductivity of the porous material(in S m−1), and jS is a source current density (in A m−2)associated with any potential disturbance that can affectthe movement of charge carriers. Equation 7 stands for ageneralized Ohm’s law, which is valid whatever theconductivity of the brine is; the first term on the right-hand side of Eq. 7 corresponds to the classical conductionterm (classical Ohm’s law defining the condution currentdensity). In addition to the constitutive equation, Eq. (7),one needs a continuity equation for the current density inorder to determine a field equation for the electrostaticpotential = . In the magneto-quasi-static limit of theMaxwell equations, for which the displacement currentis neglected, the continuity equation for the currentdensity is

r � j ¼ 0 ð8ÞThis means that the total current density is conservative

(all the current entering a control volume has to leave it,and there is no storage of charges inside the controlvolume).

Revil and Linde (2006) built a model to capture boththe effect of porosity and the multi-component ioniccharacteristics of the electrolyte upon the source currentdensity jS. For a brine-saturated porous material, the totalsource current density is given by (Revil and Linde 2006,Eq. 182)

jS ¼ QVu� kbTXNi¼1

Ti�0

qir ln if g; ð9Þ

where kb is the Boltzmann constant (1.381×10–23 J K−1),{i} is the activity of species i (taken equal to the

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concentration of species i for an ideal solution), QV is the(effective) volumetric (moveable) charge of the diffuselayer per unit pore volume at saturation of the water phase(expressed in C m−3, see also Eqs. 18 and 19 inSection The DC resistivity method in the following), urepresents the Darcy velocity (in m s−1), qi is the charge ofspecies i (in C), and Ti (dimensionless) is the macroscopicHittorf number of the ionic species i in the porous material(that is, the fraction of electrical current carried by thisspecies). In the case of clay minerals for instance, a largefraction of the cations adhere to the negatively chargedmineral surfaces but the interior of the pore space stillcontains more cations than anions (in such a way that acontrol volume of clay materials with its pore water isneutral). In this case, the moveable effective chargedensity QV is positive and the streaming current positivein the flow direction.

Equation (9) can be easily generalized for unsaturatedmaterials, yielding

jS ¼QV

swu� kbT

XNi¼1

Ti swð Þ�0 swð Þqi

r ln if g; ð10Þ

where 0 ≤ sw ≤ 1 represents the (relative) water saturation(see Linde et al. 2007, and Revil et al. 2007, for theelectrokinetic component, and Revil 1999, and Woodruffet al. 2010, for the electrochemical component).

Revil (1999) showed that the gradient of the logarithmof the activity of the salt is equivalent to the gradient ofthe logarithm of the conductivity of the salt. Using thischange of variable, one can rewrite the total source currentfor a binary electrolyte like NaCl as

jS ¼QV

swu� kbT

e�0 2T þð Þ � 1� �r ln�w; ð11Þ

where e is the elementary charge of the electron (1.6×10–19 C)and T(+) is the so-called macroscopic Hittorf number of thecation. This macroscopic Hittorf number is simply theconductivity contribution due to the cations divided by thetotal conductivity of the material. In the laboratory, the excessof charge per unit pore volume can be obtained through themeasurement of the streaming potential coupling coefficientC(often expressed in mV m−1). This coeffficient is defined byC ¼ @y=@hð Þsw;j¼0 and corresponds therefore, for a coresample, to the change of electrical potential Δ=measured at the end faces of the core sample for achange of the difference of hydraulic head Δh whenthe total current j is zero and there is no change in theconductivity of the pore water. For saturated conditions, thecoupling coefficient is given by C ¼ QVK=s0. The reader isdirected to Bolève et al. (2007) and references therein for theexperimental determination of this coupling coefficient. Themacroscopic Hittorf number T(+) can be also measured in thelaboratory through diffusion (membrane) potential measure-ments (Thomas 1976). It is obtained by looking at thechange of electrical potentials associated with a change ofsalinity with no flow allowed.

Combining Eq. (7) with the continuity equation for thecharge, Eq. (8), the self-potential field = is the solution tothe Poisson equation

r � s0ryð Þ ¼ r � jS ; ð12Þwith the source current density = = ∇·js (in A m−3).Equation (12) is the fundamental field equation in theinterpretation of self-potential signals. It is stating that anelectrical potential distribution is created by a source termcorresponding to the divergence of a source currentdensity. This (in-situ) source current density depends onthe flow and chemistry of the pore water through the useof Eq. (10) for instance. The distribution of the electricalpotential is also controlled by the distribution of theelectrical conductivity σ0. Note that the existence of asource current density jS requires that the systemworks in thermodynamic disequilibrium. That said, itis especially important to discriminate between“steady-state” and “equilibrium” situations. Since self-potential signals are generally observed to be persistentand stable over time in a number of geologicalenvironments, this means that these environments areusually working in a steady state of thermodynamicdisequilibrium.

There are two ways to use self-potential signals toimage the ground. The first one has not attracted too muchattention so far. It consists to measure the potential = , todetermine = = ∇·js (by modeling the groundwater flowand computing the streaming current density, whichrequires knowledge of the hydraulic conductivity fieldand the hydraulic boundary conditions), and to thendetermine the distribution of the electrical conductivityσ0 by use of Eq. (12). This inverse problem is non-linearand equivalent to the one in electrical resistivity tomog-raphy (see section The DC resistivity method in thefollowing), except that internal sources of current are usedrather than imposed currents. Slob et al. (2010) recentlyproposed to use a cross-correlation approach of self-potential fluctuations to determine the resistivity between aset of electrodes. Jardani and Revil (2009) used a stochasticapproach to determine the probability density distributionsof the resistivity values of a set of geological units. In bothcases, self-potential signals and their fluctuations, therefore,can be used to image the resistivity contrasts of the groundwithout injecting any current.

A more classical approach is, however, to measure = ,to assume that the spatial or spatio-temporal distributionof σ0 is known (or independently measured throughelectrical resistivity tomography), and to then compute =or jS from Eq. (12). The corresponding inverse problem islinear and a variety of methods can be used to solve it.However, like for any potential field problem (e.g.,gravity, magnetism), the inverse problem is ill-posed andgenerally underdetermined. The source of current can bespatially localized or, conversely, supported by a largevolume. In the second case, a variety of methods can beapplied including wavelet or cross-correlation techniques,or methods based on the minimization of a cost function.

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This self-potential inverse problem is fundamentallysimilar to the one in EEG because, in both cases, thesame underlying linear Poisson equation is solved (Grechet al. 2008).

ApplicationsSill (1983) used a finite-difference scheme to performthe first numerical modeling of the self-potential signalsassociated with the flow of groundwater. Fournier (1989)used potential field theory to develop a relationshipbetween the self-potential signals measured at theground surface and the position of the water table foran unconfined aquifer. A number of recent developmentsinclude the extension of the streaming potential theory tounsaturated flow (Revil et al. 2007; Linde et al. 2007), tocases with high Reynolds numbers (Kuwano et al. 2006,2007; Bolève et al. 2007), and to the development offorward and inverse modeling approaches for a widerange of applications such as to reconstruct the shape ofthe water table or to model the flow of water in thevadose zone. The inclusion of the effect of highReynolds numbers (>1) is required to model groundwa-ter flow in the inertial, turbulent flow regime duringpumping tests in high-permeability materials or to studyleakages associated with internal erosion of earth dams,for instance.

With the development of a new generation of veryreliable non-polarizing electrodes (Petiau 2000), the self-potential method has been used to perform a number ofremarkable breakthroughs in near-surface geophysics.There are several very promising quantitative investiga-tions reported in the recent literature showing clear self-potential signals associated with groundwater flow in fieldconditions. For example, Perrier et al. (1998) recordedelectrical potential variations associated with the varia-tions of the level of two lakes in the French Alps and theresulting groundwater flow and deformation over a periodof several years. Once the redox component of the self-potential signals was removed (this component wasassociated with the presence of graphite in the sedimen-tary formations), the residual self-potential signals showedan excellent correlation with the time variation of thedifference in elevation of the two lakes. Rizzo et al. (2004)and Titov et al. (2005) studied electrical signals associatedwith the recovery phase of pumping tests performed inan unconfined aquifer (see also Bogoslovsky andOgilvy 1973). An example of self-potential signalsassociated with a pumping test is shown in Fig. 4.These self-potential signals were used to determine thedistribution of the hydraulic transmissivity of theaquifer by Straface et al. (2007).

Wishart et al. (2006) used self-potential signals todetermine the anisotropy of transmissive fractures in afractured aquifer. Jardani et al. (2006a, b) used the self-potential method to locate sinkholes and crypto-sinkholeson a karstic plateau. Revil et al. (2005) used the self-potential method to image a paleochannel of a river in itsestuary. Suski et al. (2006) validated the physics of

streaming potential in the field by monitoring thegroundwater flow resulting from the infiltration of waterin a ditch. In this experiment, all the material propertiesinvolved in the coupled hydro-electrical problem wereindependently measured in the laboratory using coresamples and used to model successfully the electricalresponse associated with the infiltration of the groundwa-ter in the field. This study suggests that the physics ofstreaming potential/current is the same at different scales.

While self-potential mapping has been applied for along time to detect preferential fluid flow pathways inembankments and earth dams (Bogoslovsky and Ogilvy1973; Gex 1980; Merkler et al. 1989; Wilt and Corwin1989; AlSaigh et al. 1994; Sheffer and Howie 2001,2003), it has recently emerged as a powerful quantitativemethod in determining flow properties in such environ-ments (Rozycki et al. 2006; Sheffer and Oldenburg 2007;Rozycki 2009; Bolève et al. 2009). The self-potentialmethod was used by Rozycki et al. (2006) to diagnosequantitatively leakages in dams and embankments.Kulessa et al. (2003a, b) showed that high self-potentialsignals are generated during Earth tide deformation ofglaciers and groundwater flow in permeable channelsorganized between the glaciers and the underlying substra-tum. Bolève et al. (2009) developed a scheme to locateleakages in embankments and to determine the flow rate inthe leaking area (Fig. 5). Figure 5 shows a self-potential mapon the dry side of an embankment. The occurrence of anegative self-potential anomaly is used to locate a leakingarea and to assess the flow rate through inverse modeling.

While the self-potential signals are usually measured atvery low frequency (typically one measurement perminute to follow pumping tests, Jardani et al. 2008), thereare some applications requiring the use of higher samplingfrequencies (still under the quasi-static assumption for thesignals). An example is provided by Haas and Revil(2009) who recorded self-potential electrical bursts associ-ated with the occurrence of Haines jumps during drainageand imbibition of a porous sand.

The self-potential method has also been applied togeothermal systems to image the patterns of groundwaterflow and to determine the permeability of the geologicalformations. Figure 6 shows an example from the geother-mal field of Cerro Prieto in Baja California, Mexico(Jardani and Revil 2009). The Cerro Prieto geothermalfield is one of the main geothermal fields in the world. Itsgeology is composed of alluvial fans and shale intercala-tions (Fig. 6a). The self-potential field shows a dipolaranomaly (Fig. 6b). Electrical resistivity tomography wasused to define the main geometry of the system (Fig. 6c).The geometry of the geological system was assumed to beknown (the position of the geological unit boundaries andfaults) from prior drilling and geophysical information(seismic, electromagnetic (EM) and large-scale DC resis-tivity surveys). A Markov chain Monte Carlo (McMC)approach was used to sample the posterior probabilitydensity of the material properties for each geological unitand fault using surface self-potential measurements andin-situ temperature measurements in boreholes. This

626


modeling approach can be also used to test the depen-dence of the reconstructed material properties on the priorgeological knowledge.

The self-potential method also represents a very usefultool to diagnose various types of contaminant plumes.Naudet et al. (2003, 2004) and Naudet and Revil (2005)provided a way to separate the redox and streamingcontributions to the self-potential signals over a contaminantplume (both contributions were simply additive) and were

able to quantitatively connect the measured self-potentialsignals to the distribution of the redox potential of acontaminated unconfined aquifer associated with a munici-pal landfill (Fig. 7). These observations were later analyzedby Arora et al. (2007) who proposed a biogeobattery modelthat was modeled recently by Revil et al. (2010a). Understress (i.e., lack rate of electron acceptors), bacteria can beconnected by conductive pili (Fig. 8c), which can transferelectrons from the oxidation of the organic matter (in

Fig. 4 Self-potential anomaly associated with a pumping test in a confined aquifer (modified from Rizzo et al. 2004). a Description of thefield test with the position of the pumping well and the piezometers. b Recovery phase following the shutdown of the pump. c Electricalresistivity tomogram showing the resistivity distribution. d Self-potential time series during the recovery phase. e Total change of self-potential versus total change in the hydraulic head between the steady-state pumping conditions and the complete relaxation of thehydraulic head

627


reduced environments) as shown in Fig. 8a (biogeobatterycombining matallic particles and bacteria like in a microbialbiofuel cell) or in Fig. 8b (biogeobbatery with bacteria only).Usually oxygen would serve as terminal electron acceptor

(Fig. 8). However, some contaminant plumes rich in organicmatters do not always show large self-potential anomalies. Inthis case, the bacteria does not play a direct role in the sourcecurrent density. A small current density is still associated

Fig. 5 Self-potential associated with earth dams and embankments (modified from Bolève et al. 2009). a Example of an embankment withknown leakage. The presence of dying roots and tunnels made by animals like nutria are responsible for leakages as well as internal erosionassociated with suffusion or piping, for instance. b Self-potential map realized on the dry flank of the embankment. The presence ofcharacteristic self-potential anomalies reveals the presence of a flow path. c–d Modeling of the self-potential signals associated with theleak occurring below the vertical seal. The approach can be used to estimate the flow rate of water non-intrusively

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with the gradient of the chemical potential of the ionicspecies that are present in the pore water.

The DC resistivity method

PrincipleThe DC electrical conductivity σ0 (expressed in S m−1) ofa porous rock is the reciprocal of the DC electricalresistivity ρ0 (in ohm m), i.e., σ0=1/ρ0. The electricalconductivity represents the ease with which the electricalcurrent is flowing through the Earth. An electrical currentrepresents the flux of charge carriers through a givencross-sectional area. These charge carriers can be ions(cations and anions) and electrons. In an electrical field,the positive charge carriers move in the direction of the

electrical field while the negative charge carriers move inthe opposite direction. At the macroscopic scale, thefundamental constitutive equation governing electrical con-duction is Ohm’s law, which relates the conduction currentdensity j (in A m−2) to the electrical field E (in V m−1):

j ¼ �0 E; ð13Þ

E ¼ �ry ; ð14Þwhere = is the electrical potential (expressed in V). Thesecond term of Eq. 7 is not considered in Eq. (13) becausethe self-potential field is automatically canceled during theresistivity measurements. Equation (14) is used to satisfy∇ × E = 0 in the low-frequency limit of the Maxwell

Fig. 6 Cerro Prieto, Mexico, is one of the main geothermal fields in the world. a Geology; b The self-potential field (measured by Corwinet al. 1979); c Electrical resistivity tomography used to define the main geometry of the system; d Temperature; e Result of the stochasticinverse modeling to determine the groundwater flow model honoring all the constraints and observations of this hydrogeological problem.Ref indicates the position of the reference electrode (zero voltage). Modified from Jardani and Revil (2009)

629


equations (as the curl of the gradient of a scalar is alwaysnull). The continuity equation can be written as

r � j ¼ =; ð15Þwhere = corresponds to a volumetric charge source term(= > 0) or sink term (= < 0) and is expressed in A m−3.

This source of charges represents the amount of electricalcharges added or retrieved per unit volume of the porousmaterial and per unit time. It can be seen as a current perunit surface area of the control volume. Outside areaswhere the current is injected or retrieved, the continuityequation is simply ∇·j = 0. Therefore outside source/sinkvolumes associated with the current electrodes, there is no

Fig. 7 Analysis of the self-potential field at the Entressen landfill in the south of France. a The measured self-potential field can bedecomposed into two contributions. b The first contribution is the streaming potential field associated with the flow of groundwater in ashallow unconfined aquifer. c The second contribution is the redox component assumed to be associated with the presence of organiccontaminants. d The first contribution is obtained by correlating the self-potential field with hydraulic heads outside the contaminant fieldand using a kriged water table map to determine the streaming potential contribution over the plume. Once this contribution is removedfrom the measured self-potential field, the residual self-potential map is correlated with the in-situ measurements of the redox potential.Modified from Arora et al. (2007)

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storage of electrical charges inside the porous material andthe current entering a control volume also has to leave thiscontrol volume. This leads to the Poisson equation:

r � s0ryð Þ ¼ 0; ð16Þwhich simplifies to the Laplace equation in piecewisehomogeneous domains.

Equation (16) can be solved with the same boundaryconditions prevailing for the self-potential problem. Thenormal component of the conduction current density andthe normal component of the electrical field are both zeroat the ground-air interface. After providing a brief historyof the method, this equation is solved in the followingsection for a homogenous half-space. The dependence ofthe electrical conductivity upon the textural properties ofthe porous material and the properties of the pore fluid arediscussed next.

Short historyThe DC resistivity method is based on Ohm’s law, namedafter Georg Simon Ohm, who first discovered the directproportionality between the potential difference appliedacross a conductor and the resultant electrical current

(Ohm 1827). An early history of the DC resistivitymethod can be found in Rust (1938). The idea to useelectrical resistivity to locate mineral deposits goes backto Fred Brown in 1883 and to Ambronn in 1913 fordownhole measurements (see Rust 1938). Frank Wennerof the US Bureau of Standards wrote two papers (Wenner1912, 1915) in which he introduced the concept ofapparent resistivity with the idea of exploring larger depthsas the distance between the two current electrodes isincreased. These papers were published a few years beforethe papers published by Schlumberger, who also introducedthe same concepts in France with the four electrodetechnique (Schlumberger 1920). The idea of electricalresistivity tomography was developed by Tikhonov in theUSSR in 1943 (Tikhonov 1943). Further discussion aboutDC resistivity tomography is provided in the following.

Underlying physicsA water-saturated rock is, at least, a two-phase compositeconsisting of a solid mineral phase and a network of pores,which are saturated by an electrolyte. Usually most of thegrains are not conductive (exceptions concern somemetallic grains, veins like graphite, and some non-oxidized sulfides). In this case, the conductivity is

Fig. 8 Sketch of two possible electron transfer mechanisms in a contaminant plume. a Model I, in which the presence of a conductivemineral facilitates electronic conduction. b Model II, in which only bacteria populations are connected by conductive pili. At the “bacterialanode”, electrons are gained through the oxidation of the organic matter, iron oxides, or Fe-bearing phyllosilicates. The electrons areconveyed to the “bacterial cathode” through a network of conductive pili. At the “bacterial cathode”, the reduction of oxygen and the nitrateprevails as electron acceptors. In this system, bacteria act as catalysts. The transport of electrons through the anode to the cathode of themicrobattery may involve different bacterial communities and different electron transfer mechanisms including external electron shuttles. cShewanella oneidensis with pili, showing the complexity of the 3D organization of pili. The picture shows how pili can be interconnected.The picture also shows that several pili can start from a given bacterium (images were provided by Y. Gorby and were acquired inassociation with the study by Ntarlagiannis et al. (2007) using a field emission electron microscope). Modified from Revil et al. (2010a)

631


restricted to the pore water plus an additional componentoccurring along the pore-water/mineral interface in the so-called electrical double layer (shown in Fig. 2). Thissecond component is called surface conductivity. In thefollowing, for its simplicity and robustness, a variant ofthe classical Waxman and Smits (1968) model is pre-sented. However, other models based on the differentialeffective medium approach could represent alternativeoptions (see Revil et al. 1998, and Gelius and Wang 2008,and references therein). For a partially saturated porousrock with a relative water saturation sw (sw=1 representsthe fully water saturated case), a simple linear equationcan be written for the DC-electrical conductivity,

s0 ¼ swn

Fsw þ b þð Þ

QV

sw

� �; ð17Þ

QV ¼ ð1� f ÞQV : ð18Þ

QV ¼ �S1� ff

� �CEC; ð19Þ

where σw is the brine conductivity (S m−1), β(+) is themobility of the cations in the pore water (m2 s−1V−1), QV

and QV is the charge per unit pore volume of, respective-ly, the electrical double layer (Stern and diffuse layer) andthe diffuse layer (outer part of the electrical double layer)(both in C m−3), f is the fraction of counterions in theStern layer, which corresponds to the inner part of theelectrical double layer (Fig. 2), ρS is the mass density ofthe solid phase, and ϕ is the porosity. The fraction ofcounterions f is typically equal to 0.98 for kaolinite, 0.85for illite, and 0.90 for smectite (see Leroy et al. 2008;Leroy and Revil 2009). CEC is the cation exchangecapacity of the clay fraction, F is the electrical formationfactor (dimensionless) given by Archie’s law, F = ϕ−m

(Archie 1942), m is called the cementation exponent(Archie’s first exponent), n is the saturation exponent(Archie’s second exponent). The cation exchange capacityrepresents the quantity of exchangeable sites at the surfaceof a mineral per unit mass of the solid phase. It isclassically measured by potentiometric titration usingcations with a strong affinity for the mineral surface(Kitsopoulos 1999; Revil et al. 2002).

For a multicomponent electrolyte, the pore waterconductivity is related to the mobility of the ions by,

sw ¼XNi¼1

qibiCi; ð20Þ

where qi is the charge of species i (in Coulomb, C), Ci isits concentration (in m−3), and βi is the mobility of speciesi. In the case of a simple 1:1 electrolyte (in m2s−1 V−1), the

conductivity is given by σw = eCw (β(+) + β(−)) where Cw

is the salinity of the pore water (usually reported in Mol l−1,but is expressed in m−3 in SI units). The temperaturedependence of ionic mobility is given by (Waxman andSmits 1968; Waxman and Thomas 1974; Sen and Goode1992; Revil et al. 1998),

b �ð ÞðTÞ ¼ b �ð Þ T0ð Þ 1þ aw T � T0ð Þ½ �: ð21Þwhere !w=0.023°C

−1. This relationship governs both thetemperature dependence of the connate water and surfaceconductivities. This temperature dependence is sometimesuseful to correct field data for the influence of thetemperature changes (e.g., Hayley et al. 2007).

Equation (17) is, however, only valid at very lowfrequencies (below the frequency of relaxation of theelectrical double layer around the grains for instance, asexplained in section The induced polarization (IP)method). Electrical conductivity is indeed a frequency-dependent property. This is a fundamental point whenconsidering that electrical conductivity is usually mea-sured at a few kHz in the laboratory and around 1-10 Hzin the field. The frequency dependence of electricalconductivity is investigated by the induced polarizationmethod. At high frequencies (above the largest frequencyof relaxation, corresponding to the smallest grains, butbelow the so-called Maxwell-Wagner polarization band),the electrical conductivity can be written as

s1 ¼ swn

Fsw þ b þð Þ

QV

sw

� �: ð22Þ

This means that at high frequencies, both the diffuse layerand the Stern layer contribute to the surface conductivity,while only the diffuse layer contributes to surface conduc-tivity at low frequencies (Revil and Florsch 2010).

An alternative way to write the electrical conductivity isto use the specific surface conductance of the electricaldiffuse layer, ∑d, and the specific surface conductance of theStern layer, ∑S (both in S). These two surfaceconductivities express the excess conductivity of the diffuseand Stern layer integrated over the thickness of the diffuseand Stern layer, respectively (Fig. 2). They can be computedfrom an electrical double layer model associated with acomplexation model of the pore water interface (seeVaudelet et al. 2011a, b, for some examples). One can write(see Pride 1994 for the saturated case)

s0 ¼ swn

Fsw þ 2

LswSd

� �; ð23Þ

s1 ¼ swn

Fsw þ 2

LswSd þ SS� ��

; ð24Þ

where Λ (in m) is the characteristic pore size (in m; seePride 1994, and references therein).

632


In the special cases for which metallic minerals arepresent in a formation (e.g., sulfides), in addition,electronic conduction phenomena must be taken intoaccount (Fig. 9). It also seems that biofilms may generatean excess conductivity because of conductive nanowiresconnecting the bacteria as discussed by Reguera et al.(2005) for Geobacter and Gorby et al. (2006) forShewanella oneidensis. This electronic conduction mech-anism is presently the subject of ongoing researchinvestigations, but is likely to be associated with atunneling effect associated with the presence of packedhemes in these pili (e.g., Reguera et al. 2006, 2007; Revilet al. 2010a). A heme is composed of an iron atom locatedin the center of a large heterocyclic organic ring. Theheme iron serves as a source or sink of electrons duringelectron transfer or redox reactions in biological systems.

In addition to this unique characteristic, bacteria have anelectrical double layer similar to the one reported in Fig. 2for mineral grains. This electrical double layer is thereforeresponsible for surface conductivity. Enhancement of theelectrical conductivity, possibly due to biological activity,was shown by Atekwana et al. (2004) at a hydrocarbon-contaminated site (see Fig. 9). Atekwana and colleaguesargued that degradation byproducts drive the increase influid conductivity (and perhaps metal oxides) that aresubsequently detected by macroscopic electrical conduc-tivity measurements.

Forward and inverse modelingA typical acquisition system for DC resistivity measure-ments is shown in Fig. 10. It comprises a resistivity meter,

a.a.

b.b.

A B C

0 20 40 60 80 100 0 10 20 30 40 50 60

composition Electrical conductivity (mS m )

2225

2275

Ele

vati

on (

m a

sl)

c. Electrical conductivity profileUnsaturated

Residual LNAPL

Free LNAPL

Saturated / Dissolved LNAPL

Sand

Gravel

Clay

-1

a. b. SEM pictures

Fig. 9 Possible enhancement of electrical conductivity at the water table due to biofilms. a–b SEM pictures showing a dense network ofbacteria with “pili-like” structures in the capillary fringe at the hydrocarbon-contaminated site of Carson City, Nevada, USA. It is notknownwhether these pili are nanowires or not. Images were acquired using a JEOL JXM6400 scanning electronmicroscope systemwith an EvexEDS X-ray microanalysis system. c Soil lithology, soil conditions, and conductivity profiles of the petroleum-contaminated site. Percentages ofsilt and clay, sand, and gravel (A). Residual, free, and dissolved LNAPL phases (B). Electrical conductivity expressed in milliSiemens per meter(C) (modified from Atekwana et al. 2004). The inverted triangle denotes the water table. Modified from Revil et al. (2010a)

633


a battery, a switching box, several cables, and stainlesssteel electrodes. A good resistivity meter has to becalibrated regularly to check the accuracy of the measure-ments for a broad range of resistances. For eachmeasurement, two current electrodes are used: one toinject the current into the subsurface or a sample and theother to retrieve the same amount of current from it (notethat in effect a single electrode cannot carry current into amedium). By convention, these electrodes are named A

and B, respectively. The electrical field is measured with(at least) two other electrodes (M and N) called thepotential electrodes. The current electrodes cannot be usedat the same time to measure the associated drop of theelectrical potential in the Earth, due to the contactimpedance across the electrodes, especially at lowfrequencies. Therefore two potential electrodes (M andN) are used for that purpose (Fig. 11). The resistance R (inohm) measured across a cylindrical core sample of

Fig. 10 Typical instrumentation for DC resistivity and time-domain induced polarization. a The system comprises the resistivity meter andthe switching box with the battery. b–c The cables (1.26 km with 64 electrodes and a take-out interval of 20 m in the present case, eightreels). A typical acquisition takes 2 h; however, it strongly depends on the number of available measurement channels in the instrument. dThe contact resistance between the stainless steel electrodes and the ground is decreased by adding salty water and/or bentonite. e Reels forthe cable

634


Fig. 11 Principle of resistivity measurement in the field. a Electrode arrangement: each measurement involves two current electrodes (Aand B) and two potential electrodes (M and N). b Classical electrode arrays used in electrical resistivity surveying (a is the minimumdistance between two electrodes, n is an integer for measurements with mutiple distance of a). The arrays include surface arrays (includingthe square array) and borehole arrays. c Construction of a so-called pseudosection (apparent resistivity section) using a dipole-dipole arrayalong a profile. The pseudo-section is a collection of apparent resistivity data plotted as a function of pseudo-depths. d 2D versus 3Dacquisition arrays of electrodes

635


resistivity ρ (in ohm m), length L, and cross section A isgiven by R=ρ L/A, where g=A/L (in m) is called thegeometric factor. For field acquisition, the geometricfactor depends on the position of the electrodes asdiscussed in the following. The resistance is obtained byapplying Ohm’s law, in the form U=R I, where U is thevoltage (difference of potential) measured between M andN and I is the strength of the injected current (Fig. 11).

A typical acquisition setup for field investigations(Fig. 11) is discussed next. One of the two currentelectrodes is positive (A) and sends the current I (source)into the ground while the other is negative (B) andretrieves the current,I (sink). Assuming a homogeneous,semi-infinite space (so-called half-space) of resistivity ρand assuming that the current electrodes represent idealpoint electrodes (which may be more or less violated inpractice, for instance in favor of good coupling conditions,by sticking electrodes deep into the ground), the potentialmeasured at a point P at distances rA and rB from A andB, respectively, can be readily found by solving theLaplace equation corresponding to Eq. (16) using sphericalcoordinates. For a half-space, this yields

yðPÞ ¼ Ir2p

1

rA� 1

rB

� �: ð25Þ

Consider AM is the distance between the electrodes Aand M, AN the distance between the electrodes A and N,BM the distance between B and M, and BN the distancebetween B and N. The potential at M and N, respectively,due to A and B can be found using Eq. (25) and thesuperposition principle:

yðMÞ ¼ Ir2p

1

AM� 1

BM

� �; ð26Þ

yðNÞ ¼ Ir2p

1

AN� 1

BN

� �: ð27Þ

Therefore the drop of potential between electrodesM and N for a homogenous earth with resistivity ρ isgiven by

U ¼ yðMÞ � yðNÞ ¼ Irg: ð28Þ

with

1

g¼ 2p

1

AM� 1

BM� 1

ANþ 1

BN

� �: ð29Þ

For the case of borehole current electrodes, each termon the right-hand side of Eq. (29) has to expanded like

1/AM → (1/2) (1/AM + 1/A′M), where A′ denotes themirrored position of the respective current electrode abovethe surface, i.e., for an electrode at depth z the no-flowboundary condition at the surface is fulfilled by consideringa (fictive) mirror source of same strength and polarity atposition –z (i.e., above the surface). In Eq. (29), g (in m) isthe geometrical factor for the array of electrodes, relating themeasured resistance, R=U/I, to the resistivity ρ. In thegeneral case of a heterogeneous ground, for any possiblefour-electrode arrangement, the geometrical factor, whenmultiplied with the measured resistance R, yields theso-called apparent resistivity ρa,

ra ¼ Rg; ð30Þwhich represents a weighted average of the true resistivityof the subsurface. By definition, the apparent resistivity isequal to the true resistivity in the case of a homogeneousground. Generally, however, the true resistivity is onlyobtained after inversion of the apparent resistivity data(see the following section).

The way in which the current and potential electro-des are arranged on the Earth’s surface is called aresistivity array. A large number of electrode arrayshave been suggested in the literature (see Fig. 11), butonly a few are extensively used. The main character-istic of an array is its geometrical factor, which isuniquely related to the respective distances between theprobes. Some arrays like the Wenner array are moresuitable for tabular media and some others, like thedipole-dipole array, are more suitable to image verticalfaults.

Electrodes can also be placed inside boreholes toincrease the resolution at depth, typically using dipole-dipole and pole-dipole arrays (e.g., Zhou and Greenhalgh2000). The choice of a particular resistivity array for asurvey is based upon considerations regarding theoreticaladvantages and drawbacks of the array and its signal-to-noise ratio (e.g., Ward 1990). More recently, approacheshave been suggested to compute optimized measurementprotocols, i.e., which provide best resolution, based forinstance on sensitivity criteria (e.g., Stummer et al. 2004).

There has been a great improvement in resistivityinstruments over the years. Initial instruments were simplevoltmeters connected manually to the different potentialelectrodes. The whole procedure was therefore verytime-consuming. Modern equipment uses automaticswitching boxes (see Fig. 10) allowing a large numberof measurements to be taken in a short period of time.Multi-channel instruments can also improve the speedof the acquisition by measuring several potentialdifferences between electrode pairs simultaneously.

Inversion, or inverse modeling, is the procedure to gofrom a pseudosection of apparent resistivity data (a simplegeometrical arrangement of the measured data accountingfor the systematically varying signal support volumes ofthe individual four-electrode configurations in a resistivitysurvey; see, e.g., Binley and Kemna 2005) to an invertedre s i s t i v i ty image , a l so ca l l ed a tomogram.

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Correspondingly, the approach to construct an electricaltomogram is also referred to as electrical resistivitytomography (ERT). Inverse modeling is usually performedwith deterministic approaches. These approaches look forretrieving the true resistivity distribution of the groundfrom the apparent resistivity data. In other words, inversemodeling seeks to find a resistivity model that explains (or“predicts”) the given field measurements. It is obvious thatthe initial step of inverse modeling is to perform forwardmodeling (what are the apparent resistivity data for agiven resistivity distribution?). This is done by solvingnumerically the Poisson equation, Eq. (10) with a currentsource term for point electrodes on the right-hand side(see, e.g., Binley and Kemna 2005), for the electricalpotential. There are many references on how to solve thePoisson equation in both 2D and 3D, using numericalmethods like the finite-element method, the finite-differencemethod, or the boundary element method (e.g., Coggon1971; Dey and Morrison 1979a, b; Rijo 1977, 1984;Pridmore et al. 1981; Tsourlos 1995; Xu et al. 1998).

A number of deterministic approaches have beenproposed to solve geophysical inverse problems includingthe least-squares method (e.g., Lines and Treitel 1984) andmethods using a smoothness constraint for regularization(Constable et al. 1987; deGroot-Hedlin and Constable1990). These approaches have been adopted to solve theresistivity inverse problem in both 2D (e.g., Sasaki 1992;LaBrecque et al. 1996) and 3D (e.g., Park and Van 1991;Sasaki 1994; Zhang et al. 1995) A good review of thesetype of approaches for resistivity imaging, includingoptions for image appraisal, can be found in Binley andKemna (2005). Recent inversion methods involve also theuse of stochastic techniques (e.g., Malinverno and Torres-Verdin 2000; Andersen et al. 2001; Yeh et al. 2002;Ramirez et al. 2005). An example of inversion of anapparent resistivity pseudosection is shown in Fig. 12. Theinversion was performed using a deterministic inversionwith a smoothness constraint as regularization. Asexpected, the positions of the heterogeneities are retrievedbut they look smoother than the true heterogeneities. Thelatter problem can be reduced by appropriate modificationof the regularization term (see, e.g., Blaschek et al. 2008,and references therein).

ApplicationsCross-hole and surface ERT have been used for abroad variety of applications in hydrogeology and forthe assessment of geohazards. Concerning geohazards,electrical resistivity has been used to image sinkholes(Jardani et al. 2007b), cavities (Smith 1986), landslides(Lapenna et al. 2005), and areas of mechanicalweaknesses and preferential fluid flow pathways thatare important to image for geotechnical applications(Ha et al. 2010).

ERT has also been applied to contamination problems.For instance Naudet et al. (2004) and Abu-Zeid et al(2004) have applied this method to the mapping of landfillleachates. Two old landfills were investigated by

Bernstone and Dahlin (1997) (see also Bernstone et al.2000; Yoon et al. 2003; Meads et al. 2003). ERT has alsobeen used to target the effect of specific contaminants likedense non-aqueous phase liquids (DNAPLs; Goes andMeekes 2004) and perchloroethylene (Daily et al. 1998).As will be shown later, induced polarization usuallyshould provide a better sensitivity to what is sorbed ontothe mineral surface, while ERT is more sensitive to thetotal dissolved solids (TDS) and the presence ofimmiscible and insulating phases in the pore space ofthe porous materials.

Time-lapse resistivity is a powerful technique for avariety of applications. This includes the monitoring ofwater flow in the vadose zone and, therefore, thedetermination of the hydrodynamic properties of thevadose zone (e.g., Daily et al. 1992; Suzuki and Higashi2001; Binley et al. 2002; Koestel et al. 2009). Legaz et al.(2009) showed that the displacement of water by steam ina hydrothermal system can be imaged with time-lapseERT. Slater et al. (2007a) demonstrated that time-lapseERT can also be used to monitor the production ofbiogenic gases in peat soils. Miller et al. (2007)demonstrated how time-lapse ERT can be used forwatershed characterization. Time-lapse electrical resistiv-ity tomography can be used to image the evolution of asalt plume as a function of time, as documented bynumerous applications (see review by Kemna et al.2006 and references therein). If combined withequivalent transport models, time-lapse ERT imagesof spatio-temporal solute tracer evolution can providequantitative information on flow and transport param-eters in heterogeneous aquifers (Kemna et al. 2002;Vanderborght et al. 2005).

Figure 13 shows a recent application to the 3D imagingof an active volcano (La Fossa di Vulcano, AelolianIslands, in Italy). The acquisition system used to obtainthese data is shown in Fig. 10. Six weeks in total wereneeded to collect 9,525 DC apparent resistivity measure-ments at a total of 958 electrode locations. This uniquedata set has been inverted in 3D using parallel processingon an unstructured tetrahedral mesh containing 75,549finite-element nodes and 398,208 elements to accuratelymodel the topography of the volcanic edifice (Johnson etal. 2010; Revil et al. 2010b).

Kemna et al. (2006) and Müller et al. (2010) analyzedtime-lapse DC resistivity data collected at the Krauthausentest site in Germany. Figure 14 shows time-lapse ERTimages resulting from the injection of a positive tracer(ionic strength and conductivity higher than the back-ground, left panel) and a negative tracer (ionic strengthand conductivity smaller than the background, rightpanel), into a heterogeneous aquifer. It shows a cross-section, perpendicular to the mean flow direction, withERT images at selected days after injection. Thelegend describes the relative changes of both electricalconductivity and resistivity. This time-lapse resistivityapproach can be used to track the evolution of the salttracer over time. Once this is done, a second inverseproblem can be set up to invert the underlying flow

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and transport properties (permeability, dispersivity)from the time-lapse electrical images based on an

established relationship between electrical conductivityand salt concentration.

Fig. 12 Example of electrical resistivity tomography (ERT) (often also simply referred to as resistivity imaging). a True resistivitydistribution. b Pseudosection of apparent resistivity. The pseudodepths are here shown as n a, where n is a multiple of the electrodeseparation a (see Fig. 11b). c Inverted resistivity section (resistivity tomogram or image). The triangles indicate positions of the fictitiouselectrodes

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The induced polarization (IP) method

PrinciplesInduced polarization is a geophysical method looking atlow-frequency polarization mechanisms in earth materialsassociated with the reversible storage of electrical charges.The term “induced” is misleading as there are noelectromagnetic (EM) inductive effects involved in theIP mechanisms. This is why some authors prefer theterminology “complex resistivity”. IP can be performedeither in the time domain or in the frequency domain(Fig. 15). For instance, in the time domain, an electricalcurrent is injected over a certain time in the groundbetween two current electrodes A and B. Then this current

is suddenly shut down, and the transient voltage decaybetween two potential electrodes M and N is recorded fora period of time. In principle all the arrays used forelectrical resistivity (see Fig. 11) can also be used forinduced polarization. However, they are not equivalentparticularly with respect to signal-to-noise ratio, theoccurrence of electromagnetic coupling effects (Millett1967; Dey and Morrison 1973; Coggon 1973, andHohmann 1973), or the occurrence of electrode polariza-tion effects (Dahlin 2000). Recent works have includedthe electromagnetic inductive effects in the modeling ofinduced polarization (see, e.g., Ingeman-Nielsen andBaumgartner 2006; Ghorbani et al. 2009). Frequency-domain induced polarization is practically limited to

Fig. 13 3D resistivity tomogram of La Fossa di Vulcano (Italy) showing the log of the electrical conductivity. a Photograph taken fromthe north. Craters: PN Punte Nere; FV Forgia Vecchia; PC Pietre Cotte; GC Gran Cratere. F1, F2, and F3: areas of intense fumarolicactivity. b–d. Resistivity tomogram and its interpretation (vertical exaggeration 2:1). Note the asymmetry of the conductivity distributionbetween the EW and SN profiles. e Percentage error distribution. Note the large errors associated with the inability to inject high currents inthe ground. Modified from Revil et al. 2010b and Johnson et al. 2010

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100 Hz as an upper frequency in the field. Above thisfrequency, typically EM induction effects start to domi-nate and to mask the IP signature. For the generalprinciples of both time-domain and frequency-domain IP,refer to the textbooks by Bertin and Loeb (1976) andSumner (1976).

For frequency-domain IP, the following notations areused: ω=2π f denotes the angular frequency, f denotes thefrequency (in Hz), and i = (−1)1/2 denotes the imaginaryunit. In spectral IP (SIP), both an impedance and a phaseare measured. The impedance is converted to a conduc-tivity through a geometrical factor (see section The DCresistivity method) while the phase does not depend on thegeometrical factor. The magnitude of the conductivity (|σ|)and the phase lag (φ) between the excitation current and

the resulting voltage in frequency-domain IP are related tothe real (in-phase) (σ′) and imaginary (out-of-phase orquadrature) (σ′′) components of the complex conductivity(σ*) according to

s� ¼ sj j exp i’ð Þ ¼ s 0 þ is 00; ð31Þwith |σ| = (σ′2 + σ″2)1/2 and tan φ = σ″/σ′. Accordingly,the phase depends both on electromigration (conduction)properties (through σ′) and charge storage (through σ"). Inother words, the phase is essentially a measure ofpolarization relative to conduction. It follows that forinterpretation purposes, it may be advantageous to plot thereal (in-phase) and imaginary (quadrature) conductivitiesrather than the magnitude of the conductivity and the

Fig. 14 Time-lapse ERT results of two salt tracer tests conducted in a heterogeneous aquifer at Krauthausen, Germany. The tomograms onthe left side correspond to a positive salt tracer test injection (injected brine with a higher conductivity than the background). The panels onthe right side correspond to a negative salt tracer test (injected fresh water with a lower conductivity than the background). The twosequences of tomograms (left and right sides) show the relative changes of electrical conductivity (left) and electrical resistivity (right) in across-section perpendicular to the mean flow direction at selected days after the salt tracer injections, which took place upstream. The whitedots indicate the position of the electrodes. From Kemna et al. (2006) (see also Müller et al. 2010). This information, if gathered in 3D, canbe used to invert the permeability and the dispersivity distributions

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phase. Note that in the definition in the preceding, thephase (of complex conductivity) is positive (since thevoltage lags behind the current). If the phase refers tocomplex resistivity (the inverse of complex conductivity),it is negative; however, it has the same absolute value asthe conductivity phase. The complex conductivity itself isjust a way to express a magnitude and a phase lag.Induced polarization is usually displayed as the resistivity(or conductivity) magnitude |ρ| = 1/|σ| (in ohm m) and thephase lag φ (in rad), or alternatively as the in-phaseconductivity σ′ and the quadrature conductivity σ″ (bothin S m−1), respectively. Because the phase lag is typicallyvery small (absolute value often<10 mrad), the quadratureconductivity is much smaller than the in-phase conductivityand the magnitude of the conductivity is close to the in-phasecomponent. The absolute phase can be substantially largerthan 10 mrad in resistive media (e.g., for unsaturated porousmaterials) or in presence of disseminated clays or sulfides(such as pyrite).

Induced polarization can be also described by acomplex resistivity ρ* or a complex permittivity ε*,which are related to the complex conductivity by

�� ¼ 1

��¼ iw"�: ð32Þ

This explains why this method is also called AC(alternating current) impedance spectroscopy or low-

frequency dielectrical spectroscopy in electrochemistryand colloidal sciences.

In spectral IP (SIP) measurements, the frequency ofthe current samples a range of frequencies typicallybetween 1 mHz and 100 Hz in the field, and down to1 μHz and up to 50 kHz in the laboratory. Both theamplitude of the voltage and the phase lag between thecurrent and the voltage are measured. The impedance(complex voltage to current ratio) is multiplied by thesame geometrical factor as used in DC resistivity toobtain the complex resistivity (inverse of complexconductivity) at each frequency. The phase lag is indepen-dent of the geometric factor.

To outline the theoretical basics of time-domain andfrequency-domain induced polarization, one can use thesimplest model of polarization called the Debye model.Under this assumption, the complex resistivity at theangular frequency ω is written as

r� ¼ r0 1� m 1� 1

1þ iwt

� �� ; ð33Þ

where ρ0 is the DC resistivity (in ohm m), m is here thechargeability (dimensionless), and C is the relaxation time(in s). In the time domain, and assuming that the current isinjected over a long period of time so that polarization hasreached a steady state, the voltage between M and N

Fig. 15 Principle of time-domain and frequency-domain induced polarization (IP). a Time-domain IP, in which the secondary decay ismeasured after the current shut-off. The result is usually described in terms of partial chargeabilities (usually expressed in ms). bFrequency-domain IP, in which the injected current is harmonic and the voltage between the electrodes M and N is measured at a samplingfrequency (which has nothing to do with the frequency of the harmonic current) that is high enough to measure accurately the small phaselag between the voltage and the current. The amplitude of the conductivity and the phase can be recasted into a complex conductivity. Thiscomplex conductivity is characterized by an in phase component (real part) and an out-of-phase or quadrature conductivity (imaginary part).The complex conductivity is usually measured for a range of frequencies

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Hydrogeology Journal (201 ) 20: 617–658 DOI 10.1007/s10040-011-0819-x2

decays exponentially with time after shut-off of theelectrical current (e.g., Florsch et al. 2010):

yðtÞ ¼ y1 exp � t

t

� ¼ y0

y1

y0exp � t

t

� ; ð34Þ

yðtÞ ¼ y0m exp � t

t

� ; ð35Þ

where =0 is the voltage between M and N measured justbefore the shut-down of the current and =1 is the voltagemeasured between M and N just after the shut-down of thecurrent. The voltage = 1 is also called the secondaryvoltage. Typically the normalized voltage is used:

~yðtÞ ¼ yðtÞy0

¼ m exp � t

t

� : ð36Þ

The parameter m in Eqs. (35) and (36) is actually thesame chargeability parameter as in Eq. (28) (Wait 1959). Itis defined by,

m ¼ r0 � r1r0

¼ y1

y0; ð37Þ

where ρ∞ is the high-frequency resistivity (for ωC ≫ 1)while ρ0 is the DC resistivity (for ωC ≪ 1). If we introducethe DC conductivity and high-frequency conductivity asσ0 = 1/ρ0 and σ∞ = 1/ρ∞, respectively, the chargeability isalso given by m = (σ∞−σ0)/σ∞. The in-phase andquadrature resistivities are given by

r0 ¼ Re r� ¼ r0 þ r0 � r1ð Þ w2t2

1þ w2t2; ð38Þ

r00 ¼ Im r� ¼ � r0 � r1ð Þ wt1þ w2t2

; ð39Þ

where Re( . ) and Im( . ) represent the real and imaginarycomponents of the argument. For the complex conductiv-ity, the real and imaginary components are given by

�0 ¼ �0

�02 þ �002;ð40Þ

�00 ¼ ��00

�02 þ �002:ð41Þ

Usually, the polarization of a rock is not described by asingle relaxation time but by a distribution of relaxationtimes. A very popular model accounting for this is theCole-Cole model (Fuoss and Kirkwood 1941; Cole andCole 1941), also used also for mechanical problems (Revil

et al. 2006). This model is based on assuming thefollowing (normalized) distribution of relaxation times(Cole and Cole 1941),

gðtÞ ¼ 1

2ptsin pða � 1Þ½ �

cosh a lnðt=t0Þ½ � � cos pða � 1Þ½ � ; ð42Þ

ð1

0

gðtÞdt ¼ 1; ð43Þ

where a is called the Cole-Cole exponent.This distribution has a bell-shaped symmetry around

the time constant C0, like the log normal distribution, buthas much longer tails. The Debye case is recovered in thespecial case of !=1. In the time domain and using theCole-Cole distribution of relaxation times, the normalizedvoltage decay is given by (Pelton et al. 1978)

~yðtÞ ¼ yðtÞy0

¼ mX1j¼0

ð�1Þ tt0

� ja

Gð1þ jaÞ ; ð44Þ

where Γ is Euler’s Gamma function defined by

GðxÞ ¼ð1

0

ux�1e�udu: ð45Þ

As mentioned by Florsch et al. (2010), the previousseries converges in all cases. However, if t>C , it is betterto employ the series provided by Hilfer (2002).

In the frequency domain, the complex resistivity andthe complex conductivity can be obtained by convolvingthe Debye function with the distribution of relaxationtimes. Using a Cole-Cole distribution of relaxation times,this yields

r ¼ r0 1� m 1� 1

1þ ðiwt0Þa� ��

; ð46Þ

s ¼ s1 1� m

1þ iwt0ð Þa� �

; ð47Þ

In the general linear case, the distribution of relaxationtimes g(C ) can be anything (possibly a multimodal normal-ized distribution). The complex conductivity can then bewritten as

� ¼ �1 þ �0 � �1ð Þð1

0

gðtÞ1þ iwt

dt; ð48Þ

where g(C ) is assumed to be normalized. The relationshipbetween the Cole-Cole relaxation time C 0 and the

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frequency of the phase peak is wpeak ¼ r0=r1ð Þ1 2a= ð1=t0Þ(Pelton et al. 1983). The real and imaginary parts of theconductivity are given by the following expressions,

�0 ¼ �1 þ �0 � �1ð Þð1

0

gðtÞ1þ w2t2

dt; ð49Þ

�00 ¼ � �0 � �1ð Þð1

0

gðtÞwt1þ w2t2

dt: ð50Þ

In the following, the chargeability is described, which isthe most common parameter reported in time-domain IP.Tong et al. (2006a, b) provide a time-domain analysis of thevoltage decay curve for an arbitrary distribution of relaxationtimes. The voltage is given by a convolution of the Debyebehavior with the normalized distribution of relaxation times:

~yðtÞ ¼ yðtÞy0

¼ mgðtÞ � exp � t

t

� ; ð51Þ

where the convolution product is given by,

f ðtÞ � hðtÞ ¼ð1

0

f ðtÞhðt � tÞdt; ð52Þ

The partial chargeability (usually expressed in ms) isdefined as

Mi;iþ1 ¼ 1

y0

ðtiþ1

ti

yðtÞdt ¼ðtiþ1

ti

~yðtÞdt; ð53Þ

For the Debye and Cole-Cole models, the partialchargeabilities are given by (Florsch et al. 2010)

Mi;iþ1 ¼ mtðe�ti=t � e�tiþ1=tÞ; ð54Þ

Mi;iþ1 ¼ mðtiþ1 � tiÞ � mtGð2þ aÞ

� tiþ1

t

� aþ1

� tit

� aþ1� �

þ :::; ð55Þ

with ti+1> ti. In the limit ti< ti+1<< C (early times of thevoltage decay curve), both expressions reduce to the sameasymptotic limit given by

Mi;iþ1 mðtiþ1 � tiÞ: ð56ÞThis shows explicitly the relationship between the

partial chargeability M and the intrinsic chargeability m.

Like for the phase, the chargeability is essentially ameasure of polarization relative to conduction.

Short historyA review of the early IP literature (mostly related to theexploration of ore deposits) can be found in Rust (1938),Bertin and Loeb (1976), Sumner (1976), and Seigel et al.(2007). In this section, a historical perspective is given onthe modelling of induced polarization of a porous materialmade of grains with an electrical double layer at the pore-water/mineral interface. The case of IP effects associatedwith metallic bodies is discussed in the last part of thissection. Silicate minerals commonly carry an intrinsicnegative charge due to isomorphic substitutions within thecrystal lattice and acid/base reactions at the mineralsurfaces. The surface charge is balanced by counterionssurrounding the surface. The most strongly boundcounterions are located in the Stern layer and forminner-sphere or outer-sphere complexes with the surfaceas shown in Fig. 2. A silica grain immersed in water istherefore surrounded by an electrical double layer, whichcomprises the Stern layer of weakly sorbed counterionsand co-ions and a Gouy-Chapman diffuse layer ofcounterions (Fig. 2). Modeling the induced polarizationresponse of granular media is a two-step process involvingfirst the modelling of the dipole moment of a single grainand then upscaling the solution for a water-saturated packof grains or a colloidal suspension. The two operations arecompletely decoupled. Usually the volume-averagingapproach or the differential effective medium approachare suitable tools for the upscaling part of the modelling(e.g., Cosenza et al. 2009), the first part being related tothe electrochemistry of the electrical double layer underthe influence of an alternating electrical field. Thesubsequent discussion focuses on the modelling of thedipole moment of a single grain or particle coated with anelectrical double layer.

The mobility of the counterions and co-ions in theelectrical double layer (Stern and diffuse layers) isresponsible for the so-called surface conductivity, whichcan be understood as an excess electrical conductivitycoating the surface of the grains as discussed earlier. Thefrequency range of the alternating electrical field isconsidered to be low enough (typically below 100 Hz)such that the Maxwell-Wagner polarization (e.g., Chenand Or 2006; Bertrand and Endres 2009) can be neglected.

The first mechanistic model describing the effect of analternating electrical field upon the polarization of a singlegrain was proposed by Schwarz (1962) (see also Swan2000). Gerhard Schwarz replaced the electrical doublelayer with a single layer of counterions. These counterionswere considered to be able to move tangentially along thesurface under the influence of a surface electrochemicalpotential gradient. In his model, the counterions weretherefore unable to get exchanged between the bulksolution and the double layer. As the result of hismodeling, the surface conductivity was described by aDebye relaxation with no DC component. His approach

643


was later extended by Schurr (1964) who introduced a DCcomponent to surface conductivity. As mentioned in thetwo first sentences of his abstract, Schwarz (1962) wasinterested in modeling the polarization of the electricaldouble layer and not the Stern layer: “It is proposed thatthe low-frequency dielectrical dispersion observed withcolloidal particles suspended in electrolytes is due to thepolarization of the counterion atmosphere around theparticles by an external electrical field (…). The displace-ment of counterions in the double layer is shown to beequivalent to the existence of a surface capacitancedisplaying a diffusion-controlled relaxation.” It wasshown later that his treatment of the polarization of theelectrical double layer is incorrect.

The second chapter in the development of the modelingof the polarization of a single particle immersed in anelectrolyte was developed in the Soviet Union by Dukhinand Shilov (1974) and several years later in USA byFixman (1980). Dukhin and Shilov (Shilov and Dukhin1970; Dukhin and Shilov 1974, 2002) developed alinearized polarization model for a particle surroundedby a Gouy-Chapman diffuse layer. They considered thatthe Stern layer does not polarize and does not contributeto surface conductivity. The model of Dukhin and Shilovfor the polarization of the diffuse layer of a sphericalparticle in a 1:1 electrolyte is often called the “standardmodel” in electrochemistry (see Grosse 2009, and refer-ences therein). Dukhin and Shilov demonstrated that thetreatment of Schwarz and Schurr of the polarization of theelectrical double layer of a single particle was inappropri-ate. The standard model was developed and applied tocolloidal suspensions by a large number of researchers(Chelidze et al. 1977; Grosse 2009) and generalized tonon-symmetric cases by Grosse (2009). It involves twocomponents in the dipole moment, one being related to thedisplacement of the ionic cloud of the diffuse layer arounda fixed solid particle in the direction of the electrical field,and a diffusion (or membrane) polarization mechanismassociated with the diffusion of the salt around the particle(Fig. 16).

It was quickly realized, however, that this standardmodel was unable to explain all experimental data (Rosenand Saville 1991). The first section of the paper byRazilov and Dukhin (1995) was for example entitled“Urgency of further development of the theory of low-frequency polarization of the electrical double layer”. Itwas indeed found, in the beginning of the 1980s, that thecases for which the standard model applies are more theexception than the rule.

With the work of Zukoski and Saville (1986a, b) on theso-called dynamic Stern layer, researchers like Lyklema(2002) started to realize that the inner portion of theelectrical double layer could be the key to understandingthe discrepancy between experimental data and the modelsand not only for induced polarization (Barchini andSaville 1995). There were indeed a number of experimen-tal data that were not explained by classical electrokinetictheories, which were all assuming an immobile Sternlayer. However, working on a complete theory of induced

polarization of the electrical double layer proved to beextremely challenging from a mathematical standpoint(Razilov and Dukhin 1995). It was also perceived thatwhile the work of Schwarz and Schurr was inappropriateto model the double layer, it could be used to model thepolarization of the Stern layer (de Lima and Sharma1992), assuming that the kinetics of sorption/desorptionis slow. In geophysics, however, the standard modelhas been used in most publications (de Lima andSharma 1992; Lesmes and Morgan 2001) until recently(Leroy et al. 2008).

In geophysics, a number of researchers have used amembrane polarization model to explain induced polari-zation data without always pointing at the connectionbetween membrane polarization and the standard model ofDukhin and Shilov. The seminal work by Marshall andMadden (1959) was a key influence in geophysics.Marshall and Madden introduced a membrane polarizationmodel to understand the influence of pore throats incharged porous media upon induced polarization (theterminology “membrane polarization” comes from the factthat this theory is an extension to the frequency or timedomain of the classical membrane polarization theory inself-potential). These “bottlenecks” could potentiallyresult in polarization of the diffuse layer. While thismodel is perfectly correct from a physical standpoint, itwas then promoted by a number of geophysicists thatmembrane polarization is necessarily associated withthroats in charged porous media while double layerpolarization models are associated with grains. This is,however, untrue and membrane polarization exists forboth grains and throats. Vinegar and Waxman (1984)recognized that two mechanisms working in series may beimportant in explaining the induced polarization of shalysands: a complex surface conductivity at the surface of theminerals (possibly in the Stern layer) and a bulk contributionin the pore space corresponding to membrane polarization.The membrane polarization model was developedfurther by Konstantin Titov and co-workers (Titov etal. 2002; Tarasov and Titov 2007).

Recently, Revil and colleagues have argued inseveral papers (Leroy et al. 2008; Revil and Florsch2010; Schmutz et al. 2010; Vaudelet et al. 2011a, b;Revil and Skold 2011) that the main polarizationmechanism in silica sands seems to be associated withthe Stern layer (Fig. 16) and that the models ofSchwarz can be applied to understand this contribution.The discussion comes back more explicitly on thiscontribution in the next section.

Olhoeft (1985) argued for the existence of another(non-linear) polarization mechanism related to ion ex-change at the surface of minerals. To the authors’knowledge, this mechanism has never been modeled orproven to exist for non-metallic particles. Its possibilitycould be associated with the kinetics of sorption/desorp-tion of the counterions and co-ions in the Stern layer(Fig. 1). However, because the kinetics of sorption isgenerally very slow (on the order of several tens ofminutes to several hours), it is likely that this mechanism

644


could only influence induced polarization below 1 mHz.This mechanism is, therefore, not discussed further in thefollowing.

Underlying physicsFollowing Leroy et al. (2008); Revil and Florsch (2010),and Jougnot et al. (2010), a model is proposed here thataccounts for the distribution of the polarization lengthscales. The distribution of polarization length scales isalways broader than that predicted simply by the grain-size distribution, as discussed in detail by Leroy et al.(2008), who demonstrated the importance of the rough-ness of the grains in IP. Typically, one observes a plateau

in the phase spectrum for a broad frequency band betweenthe relaxation of the bigger grains at low frequencies andthe Maxwell-Wagner polarization at few kHz (see, e.g.,Vinegar and Waxman 1984). Mineral grains like sandgrains have generally a rough surface (Hyslip and Vallejo1997; Drolon et al. 2003). If these surface irregularities areassociated with polarization length scales, this means thata broad distribution of relaxation times is expected, thelargest relaxation times being limited by those associ-ated with the size of the grains themselves. In materialscience, constant-phase elements in equivalent circuitsare used extensively for fitting impedance data, andthis behavior is generally attributed to surface rough-ness (McDonald 1987). The idea that the surface

Fig. 16 Polarization mechanisms in the absence of electronic conductors in a granular sand. a Mechanisms: Two distinct mechanismsseem to prevail—The low frequency polarization seems to be related to the electrical double layer while the higher frequency polarization iscalled the Maxwell-Wagner polarization and is out of the scope of the present paper. The electrical double layer polarization includes threecontributions, the polarization of the Stern layer, the polarization of the diffuse layer, and the membrane polarization. b Typical spectrumfor the phase. The frequency peak is related to the characteristic time τ0, which is in turn related to the grain diameter d0 and the diffusioncoefficient D(+) of the counterions of the Stern layer

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roughness of the particles affects the complex conductivityof porous materials has been discussed by several authors(LeMéhauté and Crépy 1983;Wong 1987; Bordi et al. 1993;Cametti et al. 1995).

The complex effective conductivity and the effectivephase lag of the porous material are given by (de Limaand Sharma 1992),

s � ðwÞ ¼ seff ðwÞ þ iw"eff ðwÞ; ð57Þ

’ ¼ atanw"effseff

� �; ð58Þ

respectively, and where the effective conductivity and theeffective permittivity are defined by

seff ðwÞ ¼ Re s � ðwÞ½ �; ð59Þ

"eff ðwÞ ¼ Im s � ðwÞ=w½ �; ð60ÞThe effective complex conductivity is given by (Revil

and Florsch 2010)

s � ðwÞ ¼ 1

Fsf � þðF � 1ÞsS � ðwÞ½ �: ð61Þ

The complex conductivity of the solid grain σS* isgiven by

sS� ¼ sS0 þ iðsS

00 þ w"SÞ; ð62Þand for frequencies much below the relaxation fre-quency of water, the complex conductivity of water,s�f , is given by

sf� ¼ sf0 þ iw"f ; ð63Þ

where εf denotes the permittivity of pure water andεf=(80±1)ε0; εS denotes the permittivity of silica andεS=(4.6±0.8)ε0; and ε0 denotes vacuum permittivity(8.854×10−12 F m−1). The distribution of the polariza-tion length scales can be represented by the Cole-Coledistribution,

f ðdÞ ¼ 1

pdsin pð1� aÞ½ �

cosh 2a ln d=d50ð Þ½ � � cos pð1� aÞ½ � ; ð64Þ

ðþ1

0

f ðdÞdd ¼ 1:; ð65Þ

where d50 represents the median of the polarization lengthscale distribution and ! is the Cole-Cole exponent (0≤!≤

1, !=1 corresponds to the Debye model). In this case,surface conductivity is described by the Cole-Cole model,

sS ¼ s1S þ s0

S � s1S

1þ iwtð Þa ; ð66Þ

s0S ¼ 4

d0Sd; ð67Þ

s1S ¼ 4

d0Sd þ SS� �

: ð68Þ

For shaly sands, the term (4/d0)Σd can be replaced by

ð2=3Þ½f =ð1� f Þ�bðþÞQV and the term (4/d0)ΣS can be

replaced by ð2=3Þ½½f =ð1� ½f Þ�bðþÞS fQV where bðþÞ

S denotesthe mobility of the counterions in the Stern layer (Revil2012). This model is combined with the effectiveconductivity model in Eq. (61), and it implies that theconductivity itself is given by a Cole-Cole model. Thismay provide a physical foundation to the Cole-Cole modeldescribed in the previous section. Such a model accountsfor the broad distribution of relaxation times between therelaxation time of the grain and the relaxation time of theMaxwell-Wagner polarization.

There are some other potential explanations to thetypically broad distribution of relaxation times. For instance,if there are non-linear interactions between the dipolemoments of each grain, this can broaden the apparentdistribution of polarization length scales (D. Lesmes,personal communication, 2010). A pore-scale model wouldbe needed to check this prediction for a collection of grainsimmersed in an electrolyte. Another possibility to explain thebroadness of the relaxation time distribution is to account formembrane polarization. It is known that for a given grain-size distribution, membrane polarization produces a distri-bution that is broader than the Stern layer polarization (deLima and Sharma 1992).

Induced polarization is very sensitive also to thepresence of disseminated metallic particles in the ground.Pelton et al. (1978) used for instance the experimental dataobtained by Christoph Grissemann at Innsbruck University(Austria) using artificial mixtures of sulfides, quartz, andcement to investigate the role of pyrite on the frequency-domain induced polarization response in a frequency rangeof 4 to 10 kHz. The square of the grain size of the pyritegrains and their concentration have both an influence on theapparent induced polarization spectra. Florsch et al. (2010)used for example IP to quantify the volume and mass of slagheaps at an archeological site (see alsoWeller et al. 2000, andUllrich et al. 2007). Nordsiek and Weller (2008) and Florschet al. (2010) found a linear relationship between thechargeability and the slag concentration that they used toquantify the amount of slag. In the monitoring of contam-inant plumes, the precipitation of metallic particles

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associated with microbial-induced mineralization processescan have a substantial influence on the IP response. As seenin the following, these observations can be used forenvironmental purposes.

Ntarlagiannis et al. (2010) showed that in the absenceof microbial activity, an SIP response can be associatedwith abiotic FeS precipitation. They demonstrated that theSIP signals are diagnostic of FeS mineralization and canbe differentiated from SIP signals from biomineralizationprocesses. They found that the imaginary conductivity andthe normalized chargeability show a nearly linear depen-dence on the amount of FeS precipitating out of solution,above a threshold value of 0.006 g for their experimentalconditions. Therefore, for time-lapse experiments, thesuggestion here is to couple SIP modeling with reactivetransport modeling to model the different contributions tolow-frequency polarization. The IP contribution associatedwith the precipitation of metallic particles is expected tobe non-linear in nature because at the microscopic scale,the transfer of charge is controlled by a non-linear (Butler-Volmer) equation between the transfer of charge and thevoltage drop between the solution and the metallic particles(Bockris and Reddy 1970). During biodegradation, the IPsignature associated with the formation of FeS mineraliza-tion is expected to be substantially stronger than the IPsignature associated with the presence of biofilms. Thismakes induced polarization a suitable monitoring anddecision-making tool for sustainable remediation of metalsin contaminated groundwater (Williams et al. 2009;Ntarlagiannis et al. 2010). Clearly there is a gap toconnect reactive transport models to IP changes that could beused in turn to image andmonitor the causative mechanisms.Two examples of induced polarization applied to thedetection of a contaminant and the monitoring of anamendment are discussed in the following.

Forward and inverse modelingThe forward modeling of IP data can be directly related tothe modeling of DC resistivity data, i.e., the solution ofthe Poisson equation Eq. (16) with a current source termfor point electrodes on the right-hand side (see, e.g.,Binley and Kemna 2005, and corresponding references inthe previous section The DC resistivity method). For time-domain IP data, it is common to adopt the model by Seigel(1959) and account for the effect of subsurface chargeabilitym in the forward modeling in terms of a reduced effectiveconductivity σm=(1–m)σ, expressing a reduced currentdensity in the presence of polarization dipoles, where σ isthe DC conductivity (without polarization). Apparent char-geability data ma are then computed by solving two DCforward problems, one for σ and one for σm, according to,

ma ¼ f ðsmÞ � f ðsÞf ðsmÞ ; ð69Þ

where f denotes the DC resistivity forward operator. Notethat Eq. (69) reflects the definition of intrinsic charge-ability given in Eq. (37).

Modeling of frequency-domain IP data, i.e., complex-valued impedance data, comprising magnitude and phase,can be accomplished by solving the same Poissonequation as in the DC case, but with a complex-valuedelectrical conductivity distribution and thus also a complex-valued electrical potential distribution for the given currentsource configurations. The computation is analogous to theDC case using for instance the finite-difference method(Weller et al. 1996) or the finite-element method (Kemna2000), but applying complex calculus throughout.

The same strategies used to invert DC resistivity dataapply to IP data (e.g., deGroot-Hedlin and Constable1990; see, e.g., Binley and Kemna 2005). For inversion oftime-domain IP data, first the DC resistivity inverseproblem is solved for the conductivity distribution σ. Ina subsequent step, the apparent chargeability data ma areinverted into a distribution of intrinsic chargeability musing Eq. (69) for the underlying forward modeling (e.g.,LaBrecque 1991; Loke et al. 2006). For the inversion offrequency-domain IP data, data misfit and model rough-ness term in the objective function can be directlyformulated for complex-valued data (impedances) andcomplex-valued model parameters and the optimizationproblem solved for the distribution of complex conduc-tivity (or complex resistivity) using complex calculus(Kemna and Binley 1996; Kemna et al. 2000). One pointthat needs special care during the inversion is the fact thatthe imaginary part of the complex impedance/resistivity istypically some orders smaller than the real part, makingthe complex-valued inversion less sensitive to the IPcomponent (imaginary part or phase). One approach toovercome this problem is to perform additional iterationsteps in the inversion solely for the IP component(imaginary part or phase) once the complex-valuedinversion scheme has converged (e.g., Kemna 2000).Recently this approach has been generalized to performtime-lapse SIP tomography (Karaoulis et al. 2011b)generalizing the approach developed by Karaoulis et al(2011a) for DC resistivity.

ApplicationsInduced polarization has been used to characterize thepermeability of porous rocks (Fig. 17, e.g., Börner et al.1996; Kemna 2000; Slater and Lesmes 2002) and thepresence of oil in the pore space of sediments (Fig. 18, seeVinegar and Waxman 1982, 1984). Figure 17 shows thatthe low-frequency relaxation time and the formation factorcan be used to predict the permeability of porous medialike sandstones within one order of magnitude (see Reviland Florsch 2010, and references therein). A review of theliterature regarding the connection of the induced polar-ization parameters with permeability is provided by Slater(2007). Revil and Florsch (2010) have recently provided amodel connecting permeability to IP parameters fornarrow and very broad distributions of polarization lengthscales. Figure 19 shows the distribution of the permeabil-ity using cross-borehole tomography at the Strasbourg-Entzheim test site in France (see Kemna 2000; Kemna et

647


al. 2004). This application is based on the permeability-IPparameter relationships by Börner et al. (1996) and Slaterand Lesmes (2002).

The classical application of induced polarization hasbeen the exploration of ore bodies (e.g., Seigel et al. 2007)and there is a vast amount of literature on this subjectwhich is outside the scope of the present article. Forenvironmental applications, induced polarization has beenapplied to the detection of contaminants (e.g., Douglas

and Oldenburg 1996; Vanhala 1997) and more recently tothe detection of biological activity (Ntarlagiannis et al.2005a, b; Davis et al. 2006).

Induced polarization has been shown to be sensitive tovarious types of contaminants from heavy metals toorganic contaminants. Aristodemou and Thomas-Betts(2000) applied induced polarization to a waste disposalsite, while Douglas and Oldenburg (1996) applied it tomine drainage problems. Previous works have shown thatthe induced polarization response of unsaturated porousmaterials changes with the saturation of the water phase(Vinegar and Waxman 1982, 1984; Titov et al. 2004;Ulrich and Slater 2004; Binley et al. 2005; Ghorbani et al.2008). Figure 18 shows that induced polarization is alsosensitive to the presence of oil, and therefore inducedpolarization can be used to monitor the presence ofhydrocarbons in both the vadose zone and shallowaquifers, as resulting from oil spills for instance. Olhoeft(1986) reported an increase of the phase of materialscontaining clay particles and contaminated with oil.Vanhala et al. (1992) described the spectral inducedpolarization signature associated with the presence oftoluene, heptane, and ethylene glycol in glacial tills. Theyobserved an increase of the magnitude of the phase in thepresence of these organic contaminants. Börner et al.(1993) observed an increase of the magnitude of the phasein clays in the presence of organic contaminants and adecrease of the magnitude of the phase for oil-contaminatedsandstones except for the case of benzene. Vanhala (1997)showed that the introduction of motor oil decreases themagnitude of the phase of glacial sediments (sands and tills).Cassiani et al. (2009) performed an investigation showingthe effect of the saturation of hydrocarbons upon inducedpolarization (using a non-wetting oil) but they investigatedonly a few extreme values in the oil relative saturation(volumetric fraction of the pore space occupied by the oilphase). Figure 20 shows a cross-borehole imaging resultfrom the Strasbourg-Entzheim site showing the in-phase(real) component, the quadrature (imaginary) component,

Fig. 17 Prediction of permeability k from induced polarizationdata. Comparison between the simplified model developed by Reviland Florsch (2010) and the data (relaxation time C0 from time-domain induced polarization and measured gas permeability) fromTong et al. (2006b, their Table 1). The predictions are accurateinside one order of magnitude over four decades without the needfor free parameters. The two solid lines parallel to the 1:1 linecorrespond to a variation of plus or minus one order of magnitudewith respect to the predicted trend

Fig. 18 Influence of the oil saturation upon the SIP response of oil-bearing sands. The experimental data are from Vinegar and Waxman(1984) and Schmutz et al. (2010). These sands are water wet. This means that the grains are in contact with water rather than with oil

648


and the phase of the complex conductivity (from Kemna etal. 2004). Note the presence of an anomaly in the phaseassociated with the presence of kerosene on top of the claylayer. It is also expected that the investigation of the phaseand imaginary conductivity in a broad range of frequencies(see Fig. 21) could be used to determine the distributionof the Cole-Cole parameters in the field and therefore

to gain more insight about dynamic processes occurring inthe ground.

Induced polarization is sensitive to various direct andindirect biological activity in soils. For instance, Ntarlagianniset al. (2005b) demonstrated that IP is directly sensitive to thepresence of microbial cells and Davis et al. (2006) showedthat time-lapse IP can be used to observe the formation of

Fig. 19 Distribution of estimated permeability k (in D) in the cross-borehole image plane at the Strasbourg-Entzheim site, France, asobtained from complex conductivity tomography: a using the complex conductivity-hydraulic conductivity relationship developed byBörner et al. (1996); b using the complex conductivity-hydraulic conductivity relationship found by Slater and Lesmes (2002). FromKemna et al. (2004)

Fig. 20 Cross-borehole imaging results from the Strasbourg-Entzheim site showing the in-phase (real) component, the quadrature(imaginary) component, and the phase of the complex conductivity. The geological log as determined from cored drillings and occurrenceof contamination as proved by chemical sample analyses shown for comparison. Solid circles indicate the positions of electrodes. FromKemna et al. (2004)

649


biofilms. Induced polarization is also sensitive to byproductsresulting from biological activity like sulfide mineral precip-itation associated with bacterial activity (Ntarlagiannis et al.2005a; Williams et al. 2005; Slater et al. 2007b; Personna etal. 2008). As explained recently by Ntarlagiannis et al.(2010), IP effects associated with bacteria and biofilms lead tosmall IP effects (<1.5 mrad in the phase). In contrast, theprecipitation of metallic minerals generates much largerinduced polarization responses (by one order of magnitude),the evolution of which could be imaged in a recent field study(Williams et al. 2009). A review of the recent literature inusing geoelectrical methods to assess biological activity insoils can be found in Atekwana and Slater (2009).

Time-lapse induced polarization (e.g., Karaoulis et al.2011b) is probably one of the most exciting newapproaches in geophysics. Even if the SIP data collectionis relatively slow, the duration of the data acquisition canbe reduced with multichannel equipment (using hundredsof channels) and contaminant plumes can be followed fordays, weeks, or even years. Figure 22 shows 3D time-lapse tomography and time-domain IP inversion resultsduring a bioremediation application. More details can befound in Johnson et al. (2010). The primary contaminantat the site was trichloroethylene (TCE). The site wasmonitored automatically and 3D ERT/IP images wereproduced at approximately 2-day intervals. At day zero, alactate based amendment was injected over the length ofthe well shown in the Day 5 tomogram. Time-lapsechanges in bulk conductivity show the plume sinking andspreading over a lower confining unit with time, and thendiluting at the end of the sequence. As explained in thepreceding, the chargeability characterizes the capacity of a

material to store electrical energy in terms of ionaccumulation. Time-lapse changes in chargeability arelikely caused by sorption in the electrical double layer andbio-geochemical processes induced by the amendment, butthis is not yet proven. The observed changes in thechargeability can be associated both to direct effect of thepore-water conductivity upon the resistivity—see Eq. (37),combined with Eq. (17)—and to the direct effect ofthe composition of the pore water upon electricaldouble layer properties.

Another recent case study showing the usefulness oftime-lapse spectral induced polarization is shown inFig. 23 from the work published recently by FloresOrozco et al. (2011). An acetate amendment experimentwas performed at the Department of Energy’s RifleIntegrated Field Research Challenge (IFRC) site nearRifle, Colorado (USA). This type of experiment hasproven to be efficient to remove uranium from ground-water by stimulating the growth and activity of indigenousGeobacter bacterial species. Spectral induced polarizationwas used as a non-intrusive method to monitor theirin-situ activity and the end products of their metabolism.The monitoring was performed during an extended period,over hundreds of days following the acetate injection.Changes in the phase lag of induced polarization results(see Fig. 23) reveal spatiotemporal changes correlatedwith increases in Fe(II) and precipitation of metal sulfides(e.g., FeS) following concurrent iron and sulfate reduc-tion. During the experiment, only modest changes wereobserved for the magnitude of the resistivity over themonitoring period. This indicates that the changes in thephase are driven by changes in the imaginary component

Fig. 21 Frequency dependence of resistivity magnitude and phase in the tomographic image plane, as determined by multifrequencycomplex resistivity inversion from the Strasbourg-Entzheim site: a geological cross-section; b resistivity magnitude; c Phase. The slicesshow the horizontal variation between the boreholes as well as the interpolated spectral variation from 0.125 Hz to 64 Hz at different depthlevels. Note that the frequency axis is descending, i.e., frequency decreases into the paper plane (labels are in log2 of the frequency in Hz).From Kemna (2000)

650


Fig. 22 3D time-lapse resistivity tomography and time-domain IP inversion results during a bioremediation application at a superfund sitein Brandywine, Maryland, USA. The primary contaminant at the site is trichloroethylene (TCE), and bioremediation was chosen as theremedial solution. From Johnson et al. (2010)

651


of the conductivity (not by the in-phase conductivity, φ ≈σ″/σ′ ). The largest phase anomalies (>40 mrad) wereobserved hundreds of days after halting acetate injection.This observation reflects the preservation of the reducedconditions in the aquifer, an important condition forensuring the long-term stability of the immobilized,redox-sensitive, uranium.

Many geochemical and hydrogeological processesoccurring in the shallow subsurface could be monitoredusing time-lapse spectral induced polarization. For instance,Ghorbani et al. (2008) monitored successfully a waterinfiltration experiment in the vadose zone using inducedpolarization measurements in the frequency-domain. Thedrawback in using SIP in monitoring dynamic processes isassociated with the long time required to collect field data, aswell as with more elaborated data analysis and processingcompared to conventional DC resistivity surveys.

Concluding statements

A review is provided of some recent applications of theself-potential, DC resistivity, and induced polarizationmethods through a set of applications of these methods toa variety of hydrogeological problems. Through theexamples, it is clear that the self-potential, resistivity,

and IP signals can be, in principle, measured with thesame equipment and that the derived parameters arecomplementary to each other to characterize redoxprocesses, permeability, and the distribution of inorganiccontaminants. The present grand challenges are: (1) to getbetter petrophysical models that incorporate the complexnature of the pore space characteristics and electrochemistryof the interface between the different phases of the porousmaterial; the inclusion of bacterial activity is important tounderstand changes in the biosphere;(2) to perform jointinversion and time-lapse tomography of geophysical datasets, for instance, the self-potential, induced polarization,and nuclear magnetic resonance (NMR) methods (e.g.,Grunewald and Knight 2011) are all well-suited for thetomography of permeability; and (3) to establish theconnection of geophysical data to reactive transport models.The complementarity of self-potential and complex resistiv-ity opens the door to a number of unexplored applications inhydrogeophysics, especially if these geophysical methodscan be connected with reactive transport models. They willneed to be coupled also with other geophysical methods(NMR, seismic, and electromagnetic methods) and in-situmeasurements (hydraulic head, chemistry, temperature, pH,redox active species, biological activity). Such data mergingcan be performed within a Bayesian framework as shown forinstance by Chen et al. (2009).

Fig. 23 Selected resistivity phase images for complex impedance data collected at different times and frequencies along a surface profileat the Department of Energy’s Rifle Integrated Field Research Challenge (IFRC) site near Rifle, Colorado (USA) in 2008 and 2009. Elapsedtime in days is referenced to the start of acetate injection in 2007 in the shallow aquifer of the site. The change in the phase (more than 20mrad) can be related to changes in the redox properties of the aquifer. From Flores Orozco et al. (2011)

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Acknowledgements We thank NSF for funding the SmartGeoEducational Program (Project IGERT: Intelligent Geosystems;DGE-0801692) and EPA (D. Werkema) for funding M. Karaoulis.We thank A. Flores Orozco and K. H. Williams for sharing Fig.23with us and M. Nabighian for bringing to our attention keyreferences about early works. K. Singha, L. Slater, and twoanonymous referees are thanked for their careful and very usefulreviews of this manuscript.

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