#### 3.1. Redox Potential

[8] The relative electron activity of water is defined as *p*ɛ = −log{e^{−}} where {e^{−}} represents the potential electron activity in the pore water phase [*Holstetler*, 1984; *Thorstenson*, 1984]. In a reducing system, the tendency to donate electrons, or electron activity, is relatively large and *p*ɛ is low. The opposite holds true in oxidizing systems. In a reduction reaction, an oxidized species Ox reacts with *n* electrons to form a reduced species Red. The half-reaction Ox + *ne*^{−} Red is characterized by a reaction constant *K* = {Red}/{Ox}{e^{−}}^{n}. Because there is no electron in the pore water, the previous reduction reaction has to be coupled with an oxidation reaction (typically for reference purpose the oxidation of hydrogen). This leads to,

where *p*ɛ^{0} the standard electron activity of the actual reduction half-reaction when coupled to the oxidation of hydrogen under standard conditions [*Christensen et al.*, 2000]. The redox potential is defined through the Nernst equation by,

where *T* is the absolute temperature in K, *e* is the elementary charge of the electron, and *k*_{b} is the Boltzmann constant. Usually, the measurement of the redox potential is performed with platinum electrodes in equilibrium with the Fe(II,III) redox couple with a possible contamination by the O_{2}/H_{2}O redox couple at low concentration in iron (below 10^{−5} M) [*Nordstrom*, 2000]. Equilibrium implies that one can use the Nernst equation to determine the value of the redox potential. No other redox couples (with the exception of U(IV,VI) in natural aquatic systems) are known to produce an equilibrium potential at an electrode surface [*Nordstrom*, 2000]. The concept of the redox potential should be therefore defined in terms of redox couples and reactions.

#### 3.2. A Linear Idealized Geobattery Model

[9] Various geobattery models have been published in the geophysical literature since the seminal work by *Sato and Mooney* [1960]. *Sato and Mooney* [1960] developed a model to explain the self-potential signals associated with massive ore bodies in the context of exploration geophysics. *Bigalke and Grabner* [1997] later proposed a nonlinear model based on the Butler-Volmer equation, a classical nonlinear equation between the current and the voltage used to describe the source current density at the surface of metallic electrodes. This approach was recently applied by *Mendonça* [2008] for developing an inversion algorithm to retrieve the source of current associated with ore bodies. The reason for the nonlinearity in the geobattery model of *Bigalke and Grabner* [1997] is that the exchange of electrons to and from metallic bodies requires certain activation energies, which translate into potential losses [*Bockris et al.*, 1970]. The larger the current flowing from the surrounding ground, the larger these potential losses, and the larger the nonlinear behavior of the geobattery.

[10] We consider a massive ore body embedded in the conductive ground as shown in Figure 2a. Electrons within the ore body have a high mobility but do not exist in the surrounding rock mass. The presence of the ions controls the electrical conductivity in the surrounding rock mass. Because the fugacity of oxygen (the concentration of O_{2} dissolved in water) decreases with depth, the redox potential has, in the far field of the ore body, a strong dependence with the depth *z*. In the vicinity of the ore body, the distribution of the redox potential can be more complex because of contribution of interfacial processes and possibly biological activity [*Kelley et al.*, 2006]. *Castermant et al.* [2008] performed a laboratory experiment using a corroding iron bar in a sandbox. This experiment serves as an analog for a buried ore. They noticed a difference between the far field and the near-field distributions of the redox potential in the vicinity of a corroding iron bar. The near-field distribution of the redox potential was influenced by anodic and cathodic reaction.

[11] Let us consider for example the corrosion of an ore body like pyrite FeS_{2}. Reactions S(-II) and S(0) in pyrite coupled to release of SO_{4}^{2−} and Fe^{2+} at depth, coupled to (2) the reduction of oxygen near the oxic/anoxic interface (typically the water table). The soluble Fe released during the anodic reaction at depth can then eventually react, through advective, dispersive, and electromigration transport, with oxygen at the water table. It is therefore responsible for the distribution of the redox potential in the vicinity of the ore body. This mechanism can be summarized by the following reactions. At depth at the surface of the ore body, the following half-reaction occurs,

which is an abiotic half-reaction pulled along by sinks for electrons and Fe^{2+} at the oxic/anoxic interface. At the cathode, possibly in the vadose zone (Figure 2), we can have the following reactions,

Reaction (4) corresponds to the half-reaction associated with the electrons provided by the ore body. Reaction (5) is a redox reaction in the solution with the microorganisms being potentially able to accelerate this reaction depending on the pH of the solution (low pHs favor the reaction). Reaction (6) is the Fe(III) oxide mineral precipitation, which is an abiotic reaction. It is important to realize that the vertical redox gradient in the vicinity of the ore body, in addition to being influenced by reactions associated with the corrosion of the ore body (equations (3)–(6)), could be also influenced by redox reactions (possibly microbially catalyzed) that are unrelated to ore body corrosion, e.g., reactions associated with degradation of organic matter in the aquifer sediments surrounding the ore body in the way envisioned below in section 6. In this case, the ore body would serve as a conductor for transfer of electrons released during these reactions from depth to the oxic/anoxic interface [*Bigalke and Grabner*, 1997].

[12] The distribution of the redox potential should thus be viewed as a general source mechanism that drives electrical current flow inside the ore body because, from a thermodynamic standpoint, a gradient of the chemical potential of charge carriers (here the electrons) corresponds to a driving force for an electrical current density A sketch of the equivalent circuit is shown in Figure 2b. Two models of battery are possible. In one, the ore body serves directly as a source of electrons, vis-à-vis equation (3), that flow from depth to the shallow subsurface through the ore body.Equation (3) describes a source of electrons originating from the oxidation of 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 transfer of electrons that originate from redox reactions (potentially microbially catalyzed) that take place away from the ore body and have nothing to do with the corrosion of the ore body. Both models can coexist. For example, the 14 electrons released in half-reaction (3) can pass directly through the ore body to oxygen at the aerobic/anaerobic interface, while the electrons released during the oxidation of Fe^{2+} in solution at or near the ore body/water interface (as illustrated in Figure 2) can also be transmitted to the shallow subsurface through the ore body.

[13] In the low-frequency limit of the Maxwell equations, the electric field **E** and total current density **j** obey:

Equation (7) implies that the electrical field can be derived directly from an electrostatic potential *ψ* with **E** = −∇*ψ*. Equation (8) means that in steady state conduction, the total current density is conservative.

[14] Outside the ore body, the classical Ohm's law holds, and therefore the current density is given by **j** = *σ***E**, where *σ* is the electrical conductivity of the surrounding material. Inside the ore body, an extra source current density exists. In a linear geobattery model, this source current density is,

where **E**_{e} is the thermodynamic force driving the transfer of charges inside the ore body. Outside the ore body, **E**_{e} = 0. This suggests that the electrical circuit under consideration is formed by a generator (the ore body), and an external part (the conductive ground outside the ore body). From the standpoint of potential field theory [*Blakely*, 1995], the ore body is a source of current and the goal of a geophysical survey is to establish a relationship between the localization (and geometry) of this source of current and the measurements of the resulting self-potential signals in boreholes or at the ground surface of the Earth. In turn, such a forward relationship can be used within inversion frameworks to estimate the characteristics of the source (position, geometry, distribution of the redox potential at its boundaries, operating half-reactions).

[15] The electromotive force is defined as the voltage of the driving force of the source current density between the terminal points of the generator (the cathode in the upper side and the anode in the lower side, see Figure 2a),

where *d***l** is a length unit vector along a current line, (+) describes the anode, and (−) the cathode (Figure 2). In our case, the thermodynamic force driving the transfer of charges inside the ore body is of chemical origin. This force is the gradient of the chemical potential of the electrons, hence the redox potential. Therefore, we have **E**_{e} = −∇*E*_{H} where *E*_{H} is the redox potential (a similar relationship exists for ionic charge carriers [see *Revil and Linde*, 2006; *Revil et al.*, 2009]). Inserting **E**_{e} = −∇*E*_{H} into equation (10) yields *E*_{emf} = *E*_{H}^{(−)} − *E*_{H}^{(+)}. Therefore the electromotive force is, at first approximation, the difference of the value of the redox potential at the terminal points of the battery.

[16] In identifying the electromotive force, there are two approximations that are made that are worth commenting on. The first approximation concerns the geometry of the system, where we consider here only the terminal points of the system while obviously the true distribution of the redox potential along the surface of the ore should be carefully considered (see *Castermant et al.* [2008] for a detailed numerical forward and inverse modeling). The second point is that we treat here the overall “potential constraint” from an ideal perspective (i.e., with no losses). For example, in the case described above, the electromotive force between electron donors and acceptors is calculated with the Nernst equation as a function of the amount of oxidized and reduced compound present in the system in the vicinity of the ore body. However, losses are known to exist between the electron donors and the ore body (and similarly between the ore body and the electron acceptors). These losses can be described by activation losses, electrolyte-resistance losses, and diffusion losses [see *Bockris et al.*, 1970; *Logan et al.*, 2006; *Biesheuvel et al.*, 2009].

[17] Under the simplifying assumptions discussed above, we now explore the difference of electrical potential between two arbitrary points (P_{1} and P_{2}) of the conductive ground outside the ore body:

where *ρ* = 1/*σ* is the electrical resistivity of the ground outside the ore body (in ohm m). If P_{1} and P_{2} correspond to the terminal points of the ore body and we consider only the external part of the circuit, we obtain:

where *I* is the electrical current crossing the surface of the ore body and *R*_{e} is the external resistance of the ground. Note that *I* is a dependent variable *I* = *I*(*E*_{H}*;R*_{e}*;R*_{i}).

[18] We now consider the internal part of the circuit with the electromotive force. The generalized Ohm's law **j** = *σ* (**E** + **E**_{e}) yields **E** = *ρ***j** − **E**_{e}. The difference of electrical potential between two generic points P_{1} and P_{2} is:

In considering the whole circuit with P_{1} equal to the minus pole and P_{2} equal to the plus pole, we obtain *E*_{emf} = *I*(*R*_{i} + *R*_{e}). Inserting this relationship in equation (12), we obtain,

Usually the external resistance of the ground is much greater than the internal resistance of the ore body (*R*_{e} ≫ *R*_{i}) and therefore, the difference of electrical potential between the terminal points of the ore body is nearly equal to the electromotive force:

Using *E*_{emf} = *E*_{H}^{(−)} − *E*_{H}^{(+)}, this yields,

Therefore measurements should show electrical potential readings that are more negative in domains characterized by high redox potentials by comparison with domains characterized by low value of the redox potential (see *Timm and Möller* [2001] for a laboratory example). In the case where a crust of resistive oxidation products forms at the surface of the anode (see *Castermant et al.* [2008] for a laboratory example), according to equation (16) the difference of electrical potential between the terminal points is expected to be always smaller than the difference of redox potential between the terminal points.

[19] Assuming *R*_{e} ≫ *R*_{i} holds, equation (18) demonstrates the important point that the difference of electrical potential between the terminal points is equal to minus the difference of the redox potential between these points.

#### 3.3. Dipolar Nature of the Self-Potential Field

[20] Determining the electromotive force is only the first step of our analysis. The second step is to derive expressions for the electrical potential at any observation point located, for example, at the ground surface or in a borehole. In this section, we adopt the standpoint of potential field theory [*Blakely*, 1995], which is a classical approach used in geophysics to study quasi-static fields that can be derived from a scalar potential. The total current density is written as **j** = *σ***E** + **j**_{S} where **E** = −∇*ψ* is the quasi-static electrical field, *ψ* is the self-potential, and **j**_{S} = −*σ*∇*E*_{H} is the source current density. The continuity equation is given by equation (8). These equations together yield a Poisson equation for the electrical potential:

This very simple partial differential equation has two source terms on its right-hand side. The first term corresponds to the divergence of the source current density. This term is called the primary source term. The second source term corresponds to the change in the electrical potential distribution associated with heterogeneities in the distribution of the electrical resistivity. This second term corresponds to the secondary sources of electrical potential disturbances. It implies that any rigorous interpretation of the self-potential field should include a description of the electrical resistivity distribution of the medium. This information can be obtained in turn from geophysical measurements (DC-resistivity or EM-inductive methods) or by interpreting the (hydro)geological architecture of the system in terms of an electrical resistivity distribution.

[21] We note that equation (19) is also similar to the classical Poisson equation of electrostatics in a vacuum if we define an apparent charge density *ρ*_{e} as,

We annotate *S* as the ground surface. We note Ω as the volume of the ore body and *∂*Ω as the interface between the ore body and the surrounding porous material. The boundary condition at the ground surface (air is a perfect insulator) is _{S} · ∇*ψ* = 0, on *S* where _{S} the outward unit vector is normal to the ground surface.

[22] We demonstrate in Appendix A that the electrical field created in the ground by an ore body is dominated mainly by the dipolar component of the field in the far field. A laboratory validation of this concept has been illustrated recently by *Castermant et al.* [2008]. They show that the corrosion of a metallic body in a sandbox produces both a negative self-potential pole close to the top surface of the tank and a positive self-potential pole at depth (see also *Mendonça* [2008] for a discussion related to the gold deposit of the Yanacocha district in Peru). The dipolar nature of the redox potential was also recognized in the sandbox experiment performed by *Naudet and Revil* [2005] using the power law decrease of the electrical potential with distance from the source.