castermant j. redox potential distribution inferred from self potential measurements associated with...

8/13/2019 Castermant J. Redox Potential Distribution Inferred From Self Potential Measurements Associated With the Corrosi

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Geophysical Prospecting,2008,56, 269282 doi:10.1111/j.1365-2478.2007.00675.x

Redox potential distribution inferred from self-potential measurementsassociated with the corrosion of a burden metallic body

J. Castermant,1 C.A. Mendonca,2 A. Revil,3,4 F. Trolard,1 G. Bourrie1

and N. Linde5

1INRA, UR 1119, Geochimie des Sol et des Eaux, F13545 Aix en Provence, France, 2 Instituto de Astronomia, Geof sica e Ciencias

Atmosf ericas, S ao Paulo, Brazil, 3 Colorado School of Mines, Department Of Geophysics, Golden, CO, USA, 4 CNRS- LGIT (UMR 5559),

University of Savoie, Equipe Volcan, Chamb ery, France, and5 Swiss Federal Institute of Technology, Institute of Geophysics, Zurich,

Switzerland

Received July 2007, revision accepted October 2007

A B S T R A C T

Negative self-potential anomalies can be generated at the ground surface by ore bod-

ies and ground water contaminated with organic compounds. These anomalies are

connected to the distribution of the redox potential of the ground water. To study

the relationship between redox and self-potential anomalies, a controlled sandbox

experiment was performed. We used a metallic iron bar inserted in the left-hand side

of a thin Plexiglas sandbox filled with a calibrated sand infiltrated by an electrolyte.

The self-potential signals were measured at the surface of the tank (at different time

lapses) using a pair of non-polarizing electrodes. The self-potential, the redox poten-

tial, and the pH were also measured inside the tank on a regular grid at the end of the

experiment. The self-potential distribution sampled after six weeks presents a strong

negative anomaly in the vicinity of the top part of the iron bar with a peak amplitude

of 82 mV. The resulting distributions of the pH, redox, and self-potentials were

interpreted in terms of a geobattery model combined with a description of the elec-

trochemical mechanisms and reactions occurring at the surface of the iron bar. The

corrosion of iron yields the formation of a resistive crust of fougerite at the surfaceof the bar. The corrosion modifies both the pH and the redox potential in the vicinity

of the iron bar. The distribution of the self-potential is solved with Poissons equation

with a source term given by the divergence of a source current density at the surface

of the bar. In turn, this current density is related to the distribution of the redox

potential and electrical resistivity in the vicinity of the iron bar. A least-squares inver-

sion method of the self-potential data, using a 2D finite difference simulation of the

forward problem, was developed to retrieve the distribution of the redox potential.

I N T R O D U C T I O N

With the self-potential method, the distribution of the electri-

cal potential at the surface of the Earth (or in boreholes) is

measured with respect to a reference electrode ideally placed

at infinity (e.g. Sato and Mooney 1960; Nourbehecht 1963).

This method evidences polarization processes occurring at

E-mail: [email protected]

depth. For example, the occurrence of strong negative self-

potential anomalies associated with the presence of ore de-posits has been known since the nineteenth century (e.g. Fox

1830; Blviken and Logn 1975; Thornber 1975a,b; Blviken

1978; Bigalke and Grabner 1997; Bigalke, Junge and Zulauf

2004). The amplitude of these anomalies usually reaches

a few hundred millivolts. Goldie (2002) reported a self-

potential anomaly amounting to 10.2 V associated with the

Yanacocha high sulfidation gold deposit in Peru. Negative

C2008 European Association of Geoscientists & Engineers 269


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270 J. Castermantet al.

self-potential signals of several hundred mV have also been

observed in association with contaminant plumes, rich in or-

ganic matter, associated with leakage from municipal landfills

(see Hammanet al. 1997; Nyquist and Corey 2002; Naudet,

Revil and Bottero 2003; Arora et al. 2007).

The interpretation of self-potential signals is complicated

by the fact that there are several contributions to these

signals. During the last two decades, the most investigated

contribution of self-potential signals has been the streaming

potential, which is the electric field associated with the flow

of the ground water. The underlying physics of this contribu-

tion is now fairly well-established (see Fitterman 1979; Ishido

and Pritchett 1999; Revil and Leroy 2001; Revil, Saracco and

Labazuy 2003a; Maineult, Bernabe and Ackerer 2005; Suski

et al. 2006). This contribution can be used to interpret self-

potential signals in terms of pattern of the ground water flow

(see Massenet and Pham 1985; Fournier 1989; Jardani et al.

2006a,b,c, 2007; Lindeet al.2007b). The theory of streamingpotential was also recently extended to two-phase flow condi-

tions by Lindeet al. (2007a) and Revilet al.(2007) and in the

inertial laminar flow regime (corresponding to high Reynolds

numbers) by Revil (2007) and Bol `eveet al. (2007). It has ap-

plications in a number of areas ranging from the detection

of hydromechanical disturbances in volcanoes or active faults

(Revilet al. 2003a; Darnet, Marquis and Sailhac 2006; Ishido

and Pritchett 1999; Finizolaet al.2002; Finizolaet al. 2003),

the study of potential leakage in dams and embankments (e.g.,

Bol`eve et al. 2007; Sheffer and Oldenburg 2007), and the anal-

ysis of pumping tests (Rizzo et al. 2004; Titov et al. 2005;

Strafaceet al.2007).

The electro-redox component of the self-potential signals

is associated with redox reactions (e.g. Bigalke and Grabner

1997). Thermoelectric and electro-diffusion effects, associated

with gradients in the electrochemical potentials of the charge

carriers (ions and electrons), are two other contributions

(e.g. Corwin and Hoover 1979; Maineult et al. 2005, 2006;

Revil and Linde 2006). Note that faults can be identified with

the self-potential method because they are main pathways (or

seals) for the flow of the ground water (Revil and Pezard 1998)

or because of vein mineralizations (e.g. graphite) along the

fault plane (Bigalke and Grabner 1997). The nature of the re-lationship between the distribution of the self-potential signals

and the distribution of the redox potential at depth has been

recently debated by several authors (e.g. Nyquist and Corey

2002; Arora et al. 2007). To explain self-potential anoma-

lies observed with the occurrence of organic-rich contaminant

plumes (e.g. Naudet et al. 2003), Arora et al. (2007) and Linde

and Revil (2007) introduced a linear relationship between the

source current density and the gradient of the redox potential.

The assumption made by Naudet and Revil (2005) and Arora

et al. (2007) that biofilms of bacteria can transmit electrons

has recently been validated in the laboratory by Ntarlagiannis

et al. (2007). This model was successfully applied to invert

the distribution of the redox potential over the contaminant

plume of Entressen in the South of France (Linde and Revil

2007). However, this model has never been tested with respect

to the corrosion of a metallic body.

As self-potential anomalies include an electrical signature of

ongoing redox reaction processes occurring at depth, it should

be possible to invert self-potential signals to obtain informa-

tion related to these redox processes. The possibility to invert

self-potential data in terms of the distribution of the redox po-

tentials is important in ore prospection and in environmental

applications where the self-potential method can be used as

a non-intrusive sensor of the distribution of the redox poten-

tial over contaminant plumes after removal of the streamingpotential component (e.g. Naudet et al. 2003, 2004; Naudet

and Revil 2005; Maineult, Bernabe and Ackerer 2006; Arora

et al.2007). It can be also used to locate metallic pipes in the

ground and abandoned boreholes because of the corrosion of

their metallic casing. In contaminated shallow aquifers, the

redox potential is usually measured in a set of boreholes. This

is both time-consuming and expensive, and does not allow a

dense sampling of the subsurface. Furthermore, the physical

meaning of redox potential estimates from in situ measure-

ments is considered to be uncertain because of the introduc-

tion of oxygen in the system and perturbations of the redox

reactions in the vicinity of the boreholes (Christensen et al.

2000). Furthermore, it is not clear that geostatistical analysis

of a few redox potential data collected locally is indicative of

electrochemical conditions at larger scales (Stoll, Bigalke and

Grabner 1995).

The development of algorithms to localize the causative

source of self-potential signals is not new. Early works were

based on analytical solutions for simple geometries (e.g.

Nourbehecht 1963; Paul 1965; Fitterman1976; Rao and Babu

1984). Algorithms have recently been developed to invert self-

potential signals in terms of electrochemical source parameters

at depth. Minsley, Sogade and Morgan (2007) developed analgorithm to invert self-potential signals in terms of the distri-

bution of the volumetric current (the divergence of the source

current density) at depth by solving Poissons equation for the

self-potential. Revil, Ehouarne and Thyreault (2001) proposed

a cross-correlation algorithm to localize the intersection of ore

bodies with the water table. Using the physical model devel-

oped by Arora et al. (2007), Linde and Revil (2007) solved

C2008 European Association of Geoscientists & Engineers,Geophysical Prospecting,56, 269282


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Redox potential distribution inferred from self-potential measurements 271

the Poisson equation to determine the distribution of the re-

dox potential at depth over the contaminant plume associated

with the presence of a municipal landfill. Mendonca (in press)

developed an algorithm to invert self-potential signals to de-

lineate the position of ore bodies at depth using the geobattery

model developed by Stoll et al. (1995). These models gener-

ally include the distribution of the electrical resistivity as a

prioriinformation in the inversion process. This is definitively

important in the case of ore deposits because of the strong

contrast in the electrical resistivity between the ore body and

the host material.

Laboratory experiments have been conducted by Bigalke

and Grabner (1997) to investigate ore deposits, Timm and

Moller (2001) and Maineult et al. (2006) for liquid-liquid

redox reactions and Naudet and Revil (2005) to study

bacteria-mediated redox processes associated with contami-

nant plumes. However, a comprehensive validation of the in-

verted redox potential has not been performed to date. A fieldvalidation is of course the ultimate goal of such types of inves-

tigation but prior to that, laboratory validations are necessary.

Indeed, the baseline electrochemistry and geometry are known

in a controlled sandbox experiment. All forcing functions and

errors can be controlled. None of the laboratory experiments

discussed above can be used to test these redox potential al-

gorithms. We present therefore below a sandbox experiment

to evaluate the relationship between self-potential signals and

redox potentials and to investigate the effectiveness of the in-

version of self-potential data to retrieve the distribution of the

redox potential.

T H E G E O B A T T E R Y M O D E L

To explain qualitatively the self-potential anomalies associ-

ated with ore bodies, a geobattery model was introduced

in the seminal paper of Sato and Mooney (1960) (Fig. 1).

This model, later revisited by Sivenas and Beales (1982), Stoll

et al. (1995) and Bigalke and Grabner (1997), consider both

the distribution of the redox potential and the kinetics of the

chemical reactions at the surface of the metallic particles. The

ore body participates directly in the two half-cell reactions

of the electrochemical cell, which consists of anodic (oxidiz-ing) and cathodic (reducing) reactions. These reactions are

located at the bottom and the top of the ore body, respectively

(Fig. 1). According to Sivenas and Beales (1982), the deep

anodic reaction corresponds to the galvanic corrosion of the

metallic body. The cathodic reaction is the reduction of oxy-

gen radicals at the top of the body. The driving force of the

electrochemical cell is atmospheric oxygen dissolved in the

Cathode

Anode

e-

e-

e-e-

Ore body

Earth's surface

Ox

z

+

EH = 0

EH < 0

EH > 0

Redox

potential

Fe2+ Fe3+ + e-

O2+4H++4e-2H2O

zero potential line

e-

Figure 1 Sketch of the geobattery model of Sato and Mooney (1960).

The electric field is established when the gradient of the redox poten-

tial is connected by an electronically conductive body. Reduction of

oxygen near the surface and oxidation of the metallic body at depth

are responsible for the generationof a netcurrent insidethe body. This

source current is responsible for the generation of an electric field in

the surrounding conductive medium.

ground water. The ore body serves as an electronic conductor

to transfer electrons through the system.To connect the electrical current density created by the cor-

rosion of the iron bar in the field to the distribution of the

redox potential, it is necessary to use Maxwell equations in

their quasi-static limit (Stollet al.1995; Bigalke and Grabner

1997). The total current density j is the sum of Ohms law

plus a net source current density jS. Therefore, the electrical

potential is a solution of a Poisson equation with a source term

corresponding to the divergence of the source current density

at the surface of the metallic particles (e.g. Stoll et al. 1995;

Bigalke and Grabner 1997),

j = 0, (1)

j = + jS, (2)

where jS (in A m2) is the source or driving current density

occurring in the conductive medium, is the self-potential

(in V), and (in S m1) is the electrical conductivity of the

medium.



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Stoll et al. (1995) and Bigalke and Grabner (1997) de-

rived an inert electrode model, which is based on the classical

electrochemical model for metallic electrodes of Bockris and

Reddy (1970). This model is based on the non-linear Butler-

Volmer equation between the electrical potential and the cur-

rent density generated at the surface of the metallic body. Once

linearized, this equation yields,

jS = j0nF

RT(EH + Em ) , (3)

wherej0 is the exchange current density at the surface of the

metallic body,nis the number of molar equivalent transferred

for the given exchange reaction between the electronic con-

ductor and the surrounding medium, Fis the Faraday constant

(9.65 104 C mol1), R isthe gas constant (8.31 J K1mol1),

Tis the temperature (in K), EHis the redox potential (in V),

andEmis the redox potential of the metallic conductor (in V).

The redox potential of the metallic conductor corresponds

to the chemical potential of the electrons inside this body. Be-

cause the resistivity of the metallic conductor is very small,

Em has approximately a constant value. The current density

jS equals zero everywhere except at the surface of the iron

bar. Mendonca (in press) presented an inverse procedure to

determine the causative distribution of source currents in the

ground responsible for the observed distribution of the self-

potential at the ground surface and assuming a known electri-

cal conductivity model.

An alternative model considers a linearrelationship between

the source current density and the redox potential (Arora

et al.2007; Linde and Revil 2007),

jS = EH, (4)

2.0 m

0.5 m

7 cm

Iron barControled sandbox

Reference

VMetrix MX20

Roving electrode

Figure 2 Sketch of the Plexiglas controlled sandbox experiment.

whereis the electrical conductivity of the volume character-

ized by a rapid change in the redox potential. This conductiv-

ity can be different from the conductivity of the host medium,

as discussed below. We name this model the active electrode

model.

In the quasi-static limit of the Maxwell equations, equa-

tions (1) and (4) yield,

() = js , (5)

where for the two models described above, we have,

1

Re(EH + Em ) = js , (6)

(EH) = js , (7)

respectively, and where Re = RT/(gj0nF) is the electrode re-

sistance (expressed in ) in the inert electrode model and g

is the specific surface area of the surface of the metallic body

(in m1).

M A TE R I A L S A N D M E T H O D S

The experiment was conducted in the 2 m 50 cm 7 cm

Plexiglas sandbox shown in Fig. 2. This tank was filled with a

well-calibrated commercial sieved sand. This sand has a log-

normal grain size distribution, between 100 and 160 m. Its

mean grain size is 132m. The porosity of the sand is 0.34

0.01. X-Ray analysis shows that it is composed of 95% silica,

4% orthoclase feldspar and less than 1% albite. Note that the

same type of sand was used in the experiments reported by

Naudet and Revil (2005), Suski, Rizzo and Revil (2004) and

recently by Lindeet al. (2007).



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The sand was mixed with an aqueous solution (0.01 M KCl,

2103 M NaOH) and was used to fill the tank completely

by taking care to avoid air entrapment. The use of NaOH

was intended to accelerate the process of corrosion. However,

despite the fact that the initial pH of the solution was 10.5,

the equilibrium pH of the pore water obtained by mixing the

solution with the sand was found to be equal to 7.0.

At equilibrium, the conductivity of the pore water was equal

to 0.164 S m1. The electrical formation factor of the sand is

4.3 0.1 (see Suskiet al.2004). Therefore, the conductivity

of the saturated sand is 0.038 S m1. This is in agreement

with measurements made on some samples saturated with

the same pore water. Using an impedance meter, we obtain

a conductivity of the saturated sand equal to 0.039 S m1at

4 kHz (26 m).

The sand in the tank was left to compact and to equilibrate

with the pore water for 24 hours. A small amount of formalde-

hyde (135L per 1 L of solution) was also added to the porewater to impede the development of micro-organisms during

the experiment. Previous experiments showed that the pres-

ence of formaldehyde does not influence the measurement of

the redox and self-potentials.

After 24 hours, a cleaned iron bar was introduced vertically

at the left-hand side of the tank (Fig. 2). The bar is a rect-

angular piece of iron with a thickness of 2 cm and a height

of 50 cm. It was left in contact with the bottom of the tank.

The upper boundary was exposed to the air, fixing the value

of the redox potential at 680 mV. The bar was not in contact

Iron bar

20 16060 14012010080 18040

Measurement ports

b.

Distance (in cm)

Ref

0

-20

-40

-60

-80a.Self-potential (in mV)

Distribution at t= 0+

Distribution at t= 6 weeks

20 16060 14012010080 18040

Self-potential(inmV)

0

10

20

30

40

50

Depth(incm)

200

Figure 3 Sketch of the sandbox experiment. (a) Self-potential profiles (in mV) at the top surface of the tank att = 0+ andt = 6 weeks. (b) Side

view of the saturated sandbox used for the experiment. Ref indicates the position of the reference electrode.

with the front and back sides of the tank (Fig. 2). Thereafter

the bar was left to corrode for six weeks at room temperature

(242 C). Therefore, we can assume that there was no flow

of water in or out of the tank, as the water level was kept

constant during the experiment.

All self-potential signals were measured with reference to a

non-polarizing electrode located at the right-hand side at the

surface of the tank (Figs 2 and 3). This electrode is called the

reference electrode (Figs 2 and 3). Measurements were taken

at the surface of the tank (z = 0) every 5 cm at times t0 = 0+

(defined as the time corresponding to the introduction of the

iron bar), four weeks and six weeks after the introduction of

the bar.

After six weeks, we also measured the distributions of the

self-potential signals, the redox potential and the pH at dif-

ferent depths and distances from the iron bar (Fig. 4). The

self-potentials were measured with Ag/AgCl non-polarizing

electrodes (REF321/XR300 from Radiometer Analytical, witha diameter of 5 mm) and a calibrated voltmeter (MX-20 from

Metrix with a sensitivity of 0.1 mV and an internal impedance

of 100 M). The reference electrode for the self-potential

measurements should be ideally placed at infinity, where by

definition the electrical potential falls to zero. We placed the

reference electrode at the right-hand side of the tank to be far

from the perturbed zone (Figs 2 to 4).

The pH was measured with a calibrated pH meter (pH

330/SET-1 WTW from Fisher Scientific) and an electrode

PH/T SENTIX 41. The redox potential was measured with



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Figure 4 Distribution of the self-potential (a), distribution of the redox potential (b), and distribution of the pH (c) at the end of the experiment

(distributions are shown 6 weeks after the introduction of the iron bar). Note that the self-potential signals exhibits a clear dipolar distribution

with a negative pole located near the surface of the tank and a positive pole located at depth.

redox combination electrodes (InLab501 from Mettler

Toledo). The measured values (EAg/AgCl) were converted to the

normal hydrogen electrode (ENHEor EH), according to the re-

lationship ENHE = EAg/AgCl + 208.56 mV (Macaskill and Bates

1978).

We did not observe any self potential anomalies directly

after the introduction of the iron bar (Fig. 3a). A negative self-

potential anomaly started to develop at the surface of the tank

in the days following the introduction of the bar. The peak

of this anomaly is clearly associated with the presence of the

iron bar. After 6 weeks, the amplitude of this anomaly reached

82 mV in the vicinity of the bar. The polarity of the anomaly

agrees with field observations (e.g. Sato and Mooney 1960)

and the amplitude is similar to those reported in the laboratory

by Bigalke and Grabner (1997).

The self-potential distribution shown on Fig. 4 has a dipolar

character, with a small positive anomaly located in the bottom

part of the iron bar. A stronger negative anomaly is located in

the vicinity of the top part of the iron bar (Fig. 4a). This ob-

servation is also consistent with the data reported by Bigalke



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Figure 5 Picture showing the presence of lepidocrocite on the upper

part of the iron bar from the tank at the end of the experiment after

6 weeks (the surface of the initial bar was totally clean).

and Grabner (1997) and the classical theory of the geobattery

(Sato and Mooney 1960). After six weeks, the iron bar was re-

moved from the tank and evidence of corrosion was observed

on its surface (Fig. 5).

The distribution of the redox potential (Fig. 4b) appears to

consist in two contributions. The far field contribution (at dis-

tancesx > 20 cm from the iron bar) varies mainly with depth

and ranges from +260 mV at the top surface of the tank to

40 mV at the bottom of the tank. The strong gradient of

the redox potential within the upper 20 cm implies an abrupt

change from oxidizing to reducing conditions (Fig. 4b). This

feature can be observed after 6 weeks while initially, the dis-

solved oxygen concentration was constant inside the tank. It

is likely that the chemical reactions at the surface of the iron

bar and the degradation of a small amount of organic matter

observed inside the sand consumed the dissolved oxygen of

the pore solution. The slowness of the diffusion of oxygen in

the tank is therefore responsible for a vertical gradient in the

redox potential in the upper 20 cm of the tank.

In the vicinity of the iron bar (at x < 20 cm), there is a

strong perturbation of the redox potential with respect to the

far-field distribution. The redox potential is strongly negativein the immediate vicinity of the iron bar (in the range of200

to 250 mV). We observe that the colour of the sand in the

vicinity of the iron bar turns into a blue-greenish colour over

time. If the sand is exposed to air, this colour turns to ochre.

This behaviour is typical of the formation of fougerite (the

green rust is transformed into lepidocrocite when exposed to

air, see Trolard etal. 1997; Trolard 2006; Trolard etal. 2007).

Far from the iron bar, the pH is equal to the initial pH of

the pore water solution (pH = 7.0) (Fig. 2c). In the vicinity

of the iron bar, the pH is alkaline and equal to 9.0. The pH

gradient is mainly horizontal. An explanation for this trend

will be attempted in Section 4.

G E O C H E M I S T R Y

We now describe the electrochemistry leading to the corrosion

of the iron bar. The corrosion of iron results in the formation

of a green rust called fougerite (e.g. Srinivasan et al. 1996;

Trolard 2006). In our experiment, the values of the redox po-

tential and the pH measured in the vicinity of the iron bar are

compatible with the stability domain of fougerite reported by

Trolardet al.(1997). The first step in corroding iron in a neu-

tral aqueous solution is the formation of Fe(OH)2 according

to the following sequence of reactions:

Fe Fe2++2e, (8)

Fe2++2OH Fe(OH)2. (9)

Then Fe(OH)2, which is unstable, is oxidized to ferric oxyhy-

droxide depending on the anion in the solution (mainly lepi-

docrocite, goethite, or magnetite). Fougerite is an intermediate

component in these reactions that can be written as (Trolard

et al.1997; Trolard, 2006; Trolardet al.2007):

Fe(OH)2+xA [FeII1xFe

IIIx(OH)2]

x+[xA]x+xe, (10)

FeII

1xFeIIIx (OH)2

x

+

[xA]x+OH FeOOH+xA

+(1x)e+H2O, (11)

where A is an interlayer anion that compensates forthe excess

of positive charge of the layer due to the partial oxidation of

FeII toFeIII. In this experiment, either Cl orOH are possible

candidates for A (Trolardet al. 1997; Simon 1998; Trolard

2006; Trolardet al. 2007). However, OH can be dismissed

because its concentration is too small (see the value of the pH

in the vicinity of the bar on Fig. 4).

Oxygen gas, O2, plays the role of electron acceptor accord-

ing to,

1/4O2+1/2H2O + e

OH, (12)

so that the global reaction is,

Fe + 3/4O2+1/2H2O FeOOH. (13)

Since atmospheric oxygen is permanently diffusing into the

tank, chemical equilibrium is never established. The observed



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horizontal gradient of the value of the pH in the tank (see

Fig. 4c) is likely related to the generation of OH in the vicinity

of the iron bar (see reaction 12) and by diffusion of OH from

this area towards the right-hand side of the tank. To check if

this is possible, we consider the value of the diffusivity of OH

in water:Df(OH) = 5.3109 m2 s1 (Samson, Marchand

and Snyder 2003). The tortuosity of the pore space in the sand

is given by the product between the formation factor and the

porosity and is equal to 1.7. For a time constant of 6 weeks,

the characteristic diffusion length of OH in the sandbox is

L 2(Df)1/2 20 cm. This order of magnitude is compatible

with the distance between the pH front and the bar (40 cm,

see Fig. 4c).

To summarize this section, theiron barwas originally placed

in an environment in which there was no pre-existing gradient

in the redox potential distribution. At the beginning of the

experiment, the pH andthe redox potential were constant over

the entire volume of the tank. Chemical reactionsat thesurfaceof the iron bar are responsible for its corrosion. This corrosion

creates a perturbationof thedistribution of the redox potential

in the vicinity of the iron bar, a depletion in the concentration

of oxygen inside the tank, a basic pH front diffusing inside

the tank and the formation of a crust on the surface of the

bar. This crust is probably responsible for an increase of the

resistivity at the surface of the iron bar.

I N V E R S E M O D E L L I N G

Inversion or source localization of self-potential signals is a

relatively new field (see Revil et al. 2004; Jardani, Dupontand Revil 2006b; Minsley et al. 2007; Mendonca in press).

In essence, it is however very similar to the inverse problem

arising in gravity andmagnetism. In this section, we develop an

inverse modelling approach that uses the twomodelsdiscussed

in Section 2 and connecting the self-potential signals to the

redox potential distribution.

Figure 6 Resistivity model used for the inversion of the redox potential.

Estimation of the volumetric current density

The inversion of the self-potential data in terms of the dis-

tribution of a redox potential is a two-step process, which

comprises the determination of the distribution of the source

current density and then the determination of the redox po-

tential. We note qS = js as the volumetric source cur-rent density (in A m3). Expressions for js are given in

Section 2. Using a finite-difference formulation, the Poisson

equation can be written in matrix form as,

Au = q, (14)

where A is a NNmatrix, u an N-dimensional vector with

the unknown potential values, andqanN-dimensional vector

with thevolumetric source of current. Thesources areassumed

to be located at the surface of the iron bar, and the conductivity

model is assumed to be known (see below). Because the tank

is thin (7 cm), we use the 2D finite-difference formulation

described in Mufti et al. (1976) to solve Poissons equation.

This means the electric field only has components along the

xy-plain of the tank.

We considerM station points at nodesk(i) (i = (1, . . . ,M).

The theoretical values of the potentials at these stations corre-

spond to the vector u0 = Qu, where Q i s an (MN) matrix. All

the entries of linek of this matrix are null except for the kth

entry which is equal to 1. The source terms in the mesh nodes

can be assembled in aM-dimensional vectorq such that,

u0 = Rq, (15)

where the (M N) resistance matrixRis given byR = QA1.This matrix samples the measured self-potential stations where

self-potential data have been obtained along the top surface of

the tank with a reference point at infinity. In the present case,

we assume that the reference electrode is far enough from the

iron bar to be considered at infinity. This is a good approx-

imation because the electrical potential distribution is flat in



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the vicinity of the reference electrode (see Figs 3 and 4). The

matrixR corresponds to the Green functions connecting the

self-potential response to a unit current source at a given po-

sition and accounting for the electrical resistivity distribution

of the medium.

Inverting self-potential data is a typical potential field prob-

lem and the solution of such types of problems is known to

be ill-posed and non-unique (Tikhonov and Arsenin 1977).

It is therefore important to add additional constraints to re-

duce the space of the solution. The criteria of data misfit

and model objective function place different and competing

requirements on the models. These objective functions can

be balanced using Tikhonov regularization (Tikhonov and

Arsenin 1977), through the definition of a global objective

function,G,

G =

Wdu0 R

Tq

2+ a

Waq

2+ r

Wr q

2, (16)

where T is transpose and whereAf2 = fTATAfdenotes the

Euclidian norm, a and r are regularization parameters un-

der the constraint that (0 < a < and 0 < r < ), u0

is vector of Nelements corresponding to the self-potential

measurements at the surface of the tank (and in boreholes if

needed), andWd= diag {1/1, . . . , 1/N} is a square diagonal

weightingN Nmatrix (elements along the diagonal of this

matrix are the reciprocals of the standard deviations i of the

data). We consider that the probability distribution of the self-

potential measurements is Gaussian (see Lindeet al.2007 for

a field example). The matrices Wa and Wr are regularizing

weighting matrices that impose absolute and relative prox-imity constraints. The linear damping operator Wa imposes

deviations from a zero source term and Wr imposes smooth-

ness between adjacent source terms. The Laplacian operator

used to impose smoothness,Wm, is given by Zhdanov (2002).

The distribution of the source current in the tank can be es-

timated from the least squares solution of the inverse problem

(see Menke 1989; Mendonca in press),

q = Su0, (17)

S RTR + a Wa + r Wr

1RT. (18)

The parametersaand rare regularization parameters. A

popular approach for choosing these regularization param-

eters is the L-curve criterion (Hansen 1998). The L-curve

is a plot of the norm of the regularized smoothing solu-

tionsWa q

and Wr q versus the norm of the residuals

of data misfit functionWd(u0 RTq)

. These dependenceshave an L-shaped form, which reflects the heuristics that for

0

10

20

30

40

50

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Current at the surface of the bar (in mA)

Depthinthetank(incm)

0 20 40 60 80 100 120 140 160 180 200

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Distance (cm)

Self-potential(mV)

residual

Measured self-potential data

Best least square fit

Result from the inversion

Interpolation

Figure 7 Results from the inversion with the active electrode model.

(a) Curve fit of the self-potential data resulting from the inverted cur-

rent source model. The noise level of the self-potential data was as-

sumed to be 0.6 mV (i.e., the mean value of readings for the stations

located between 70 and 130 cm). (b) Current source model at the

surface of the iron bar resulting from the inversion with the activeelectrode model.

large regularization parameters the residual increases without

reducing the model norm of the solution much, while for small

regularization parameters the norm of the solutions increases

rapidly without much decrease in the data residual. Thus, the

best regularization parameter should lie on the corner of the



10/14


Figure 8 Distribution of the redox potential

obtained from self-potential current sources

by using different boundary conditions (a,

b, c) and the distribution of the mea-

sured redox potential (d). We assume =

0.0025 S m1 (400 m) for the conductiv-

ity of the crust coating the iron bar.

L-curves. We choose parameters in the interval [max/1000,

max], with max being the maximum singular value of the re-

sistance matrix R. This allows a and r to be large with-

out severely deteriorating the data fit compared with the case

a = r = 0.Under the assumptions of uncorrelated, additive, zero-mean

noise, the standard deviation for thei-th estimate of the volu-

metric current source density is:

E {qi } =

Di (SS

T), (19)

where is the mean value of the standard deviation of the

data andDi(SST) is the i-th diagonal term of the matrixSST.

We use = 0.6 mV as determined from the mean standard

deviation value for the readings between 70 and 130 cm from

the iron bar at the top surface of the tank. Note that the error

associated with the model itself is not accounted for.

The algorithm was applied to the self-potential datarecorded at the top surface of the tank, six weeks after the

introduction of the iron bar. For the resistivity model (see

Fig. 6), we assume that the resistivity of the iron bar is 0.001

m. The iron bar is coated with a thin layer (1 cm) of resistive

crust (400 m). At the end of the sandbox experiment, we

removed the iron bar and attempted to measure the resistivity

of the crust (due to fougerite, see Fig. 5). The resistivity we



11/14


obtained was in the order of 400 200 m) but there is a

lot of uncertainty associated with this measurement. The sat-

urated sand has a resistivity of 25m (see Section 3). To solve

the problem, we used Dirichlet boundary conditions.

With the self-potential data and the assumptions made

above, the distribution of the inverted current along the iron

bar is shown in Fig. 7. Figure 7(a) shows the best fit to the

self-potential data measured at the top surface of the tank.

Figure 7(b) shows the distribution of the volumetric source

current density along the iron bar.

Determination of the redox potential

The distribution of the redox potential can be estimated

by computing a forward problem based on equation (7),

where the source termsqscorresponds to those determined in

Section 5.1.Figure 8 shows different redox distributions estimated with

the current density, for different levels of conditioning (i.e.

use of surface information only, surface information combined

with chosen points at depth and use of the whole set of data).

Note that the filled circles are the measurements used as con-

straints. In the case of Fig. 8(a), only surface redox values

were used. In the cases of Figs 8(b) and 8(c), redox values at

depth are included. Note that the model implies that EH =

0 is obtained at a depth of 20 cm, while it is located at a

depth of 10 cm. Despite of the lack of resolution near the

bottom of the tank, the algorithm is capable of capturing the

pattern of the redox potential distribution in the tank. Indeed,

the redox field seems to exhibit two components: one tank-

wide effect due to oxygen diffusion from atmosphere plus

an anomalous distribution due to the corrosion of the iron

bar. The anomalous component is recovered from the self-

potential current terms but the regional component is only

due to the boundary conditions. We believe that such a con-

dition does not prevent field applications in mineral and envi-

ronmental investigations because in both cases the anomalous

redox field plays a major role. In conclusion, we think that

our procedure is not very sensitive in retrieving the regional

field but seems effective in retrieving the anomalous redoxcomponent.

Figure 9(a) indicates that the measured data are compat-

ible with the inert electrode model. For which the location

of the source currents coincides with the surface of the iron

bar. The redox potential distribution used for this model was

taken from the distribution of the redox potential measured

in the far-field (between position 70 cm and 130 cm on the

0 5 10 15 20 25 30 35 40 45 501. 5

1

0. 5

0

0.5

Depth (in cm)

Electrode resistance ( )

51015

in

Self-

potentialcurrentsource(inmA)

0 20 40 60 80 100 120 140 160 180 20090

80

70

60

50

40

30

20

10

0

10

Distance along the surface of the tank (in cm)

SPanomaly(mV)

151015

Electrode resistance ( )in

a.

b.

Figure 9 Result from the inversion using the inert electrode model. (a)

Measured self-potential anomaly (full circles) and theoretical anoma-

lies provided by the inert electrode model with electrode resistance

of 1, 5, 10, and 15 . (b) Current along the surface of the iron bar

determined from the inversion of the self-potential data (black line)

and current distributions prescribed by inert electrode models with

electrode resistance of 1, 5, 10, and 15 (coloured lines).

profile). We expect that such a central portion of the profile

is not affected by boundary conditions and more conditioned

by the diffusion of the oxygen through the top of the tank.

Figure 9(b) shows inverted current model and current



12/14


distributions predicted by the inert electrode model based on

measured redox potentials. This indicates that equation (4) is

able to capture the main behaviour of the more complex inert

electrode model.

C O N C L U D I N G S T A T E M E N T S

We performed a controlled sandbox experiment to test the

geobattery model associated with the corrosion of a metallic

body. Themain results obtained in this work are(1) Thereduc-

tion of oxygen near the surface of the tank and the oxidation

of the metallic body at depth are responsible for a net current

inside the metallic bar. (2) The oxidation of the metallic body

produces fougerite, which modifies the redox potential and

the pH in the vicinity of the iron bar. Fougerite is also respon-

sible for the formation of a resistive crust at the surface of the

bar. (3) A dipolar self-potential anomaly is produced with a

positive pole at depth. This is in agreement with the predic-tion of the geobattery model of Sato and Mooney (1960). (4)

The strength of the self-potential positive pole is much smaller

than the negative pole located in the vicinity of the surface of

the tank. An algorithm was developed to invert the distribu-

tion of the redox potential at depth. The results are improved

if the inversion is conditioned to the value of the redox po-

tential in a few boreholes. Future works will combine spectral

induced polarization and self-potential data to provide better

constraints on the electrochemistry of redox processes occur-

ring at depth, especially when bacteria are assumed to play a

key role.

A C K N O W L E D G E M E N T S

We thank the Institut National de Recherche Agronomique

(INRA), the French National Research Council (CNRS) and

the Region Provence-Alpes-Cotes-dAzur (PACA) for their

support. We thank ADEME (Philippe Begassat) for its sup-

port and ANR-CNRS-INSU-ECCO for funding the project

POLARIS. The grant of Julien Castermant is supported by

Region PACA and INRA. We thank the two referees, Volker

Rath and Alexis Maineult and the Associate Editor, Oliver Rit-

ter, for their very helpful comments. We thank T. Young forhis support at CSM.

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