Electrical conductivity of NaCl-H₂O fluid in the crust

Abstract

Ionic electrical conductivity of NaCl-H₂O fluid as a function of pressure (0.2–2.0 GPa), temperature (673–2000 K), and NaCl concentration (0.6–9.6 wt %) was investigated using molecular dynamics (MD) simulations. Conductivity versus NaCl concentration has a nonlinear relationship due to the presence of electrically neutral ion pairs in concentrated solutions. The calculated conductivity at 0.6 wt % NaCl is consistent with the available experimental data, and the calculated conductivity at higher temperatures shows a greater degree of pressure dependence. The major factors controlling the conductivity are the density of the NaCl-H₂O fluid and the permittivity of solvent H₂O. A purely empirical equation for deriving the conductivity was proposed. Highly conductive zones below a depth of 35 km in the middle portion of the continental crust can be interpreted by the presence of NaCl-H₂O fluid with the salinity ranging from 0.2 to 7.0 wt %. A highly conductive zone observed at a depth of 20 to 40 km above the subducting oceanic crust in Cascadia can be explained by the presence of low-salinity (0.5 wt %) NaCl-H₂O fluid possibly generated by the dehydration of basalt.

Key Points

• Electrical conductivity of NaCl-H₂O fluid is calculated by MD simulations
• Conductivity is controlled by the density and permittivity of fluids
• Presence of NaCl-H₂O fluid explains the highly conductive zones in the crust

1 Introduction

Aqueous fluids in the Earth's crust exert pore pressure and chemical corrosion on rocks and faults. Estimating the volume and distribution of these aqueous fluids is crucial for the prediction of earthquakes. At ambient conditions, aqueous fluids have a high conductivity relative to dry rocks and minerals. On this basis, the distribution of aqueous fluids has been investigated using geophysical observations of electrical conductivity in the Earth's crust. Highly conductive zones have been reported in the middle portion of the Earth's crust, arc, and subduction zones [Ichihara et al., 2014; Ichiki et al., 2009; McGary et al., 2014; Ogawa et al., 2001; Shankland and Ander, 1983; Soyer and Unsworth, 2006; Worzewski et al., 2011]. These high conductivity zones, however, cannot be simply interpreted to reflect the presence of aqueous fluids without proper consideration of the effects of elevated temperature and pressure on the fluid conductivity.

The stability and electrical conductivity of rocks [Fuji-ta et al., 2004; Kariya and Shankland, 1983; Olhoeft, 1981], minerals [Gasc et al., 2011; Guo et al., 2011; Guo and Yoshino, 2014; Reynard et al., 2011; Yang, 2011], graphite films on grain boundaries [Frost et al., 1989; Glover, 1996; Yoshino and Noritake, 2011], partial melts [ten Grotenhuis et al., 2005; Hermance, 1979; Lebedev and Khitarov, 1964; Roberts and Tyburczy, 1999], saline fluids [Glover and Vine, 1994; Guo et al., 2015; Hyndman and Hyndman, 1968; Hyndman and Shearer, 1989; Nesbitt, 1993; Shimojuku et al., 2012, 2014], and CO₂-H₂O fluid [Nesbitt, 1993] have been investigated as explanations for the highly conductive zones. However, the conductive zones likely represent a composite of these materials, in which aqueous fluids are considered to be a dominant cause of the high conductivity [e.g., Guo et al., 2015; Hyndman and Shearer, 1989; Shankland and Ander, 1983], and this would be more plausible at subduction zone, since the dehydration of hydrous minerals would occur in the subducting slab [Ichiki et al., 2009; McGary et al., 2014; Reynard et al., 2011].

If we are to explain elevated crustal conductivities by aqueous fluids, then we need to determine the conductivities of fluids that are likely to be present under appropriate conditions and then determine whether the salinity and volume fraction required makes geological sense. In the Earth's crust, the major dissolved ions in aqueous fluids are Na⁺ and Cl⁻ ions, as estimated from fluid inclusions [e.g., Roedder, 1984] and from deep crustal rock samples [e.g., Bucher and Stober, 2010]. To date, only low pressure (<400 MPa) conductivity values for the simplest and most common binary fluid (NaCl-H₂O) in the Earth's crust have been available [Bannard, 1975; Gruszkiewicz and Wood, 1997; Ho et al., 1994, 2000; Quist and Marshall, 1968]. The conductivity of KCl-H₂O fluid, which is considered to be an analogue of NaCl-H₂O fluid, is available for salt concentrations higher than that of NaCl-H₂O fluids [Hwang et al., 1970; Ucok, 1979], as reviewed by Nesbitt [1993]. However, the data are also limited to low pressures (<300 MPa), and there is no evidence that KCl-H₂O fluid can be used as a substitute for NaCl-H₂O fluid at elevated temperatures and pressures. Nesbitt [1993] concluded that the maximum conductivity observed in the midcrust cannot be explained by the presence of saline fluids. However, this conclusion needs to be evaluated by investigating the conductivity of NaCl-H₂O fluid as the major saline fluid in the Earth's crust.

Molecular dynamics (MD) simulations are a powerful technique for deriving the ionic conductivity of a solution. The method explicitly simulates ion and H₂O molecule dynamics at various salt concentrations, temperatures, and pressures; therefore, the physics behind the ionic conductivity can be interpreted from the static and dynamic properties of ions and H₂O molecules. MD simulations can be categorized by two different approaches. First-principles (FP) MD treats electron-electron interactions based on quantum mechanics [Marx and Hutter, 2009] and is suitable for reproducing electron-electron interactions, even though it makes a number of approximations. However, the method has a high computational cost, and the number of atoms and the simulation time are limited to several hundred atoms and subnanoseconds, respectively. The limited size of the simulation cell results in poor reproducibility of the dynamics of H₂O molecules in the liquid state [Kuehne et al., 2009; Kuo et al., 2004; Yeh and Hummer, 2004]; therefore, the method is unrealistic for simulating the conductivity of aqueous fluids. The classical MD approach approximates interatomic interactions using a simple analytical form. This approximation often reduces the transferability of the method between conditions other than those of the parameters fitted. However, if the transferability is confirmed at the targeted conditions, the low computational cost makes it possible to simulate the ionic conductivity of aqueous fluids. In this study, we employed classical MD simulations to derive the ionic conductivity of NaCl-H₂O fluid at elevated temperatures and pressures.

We have previously developed a flexible and induced point charge (FIPC) H₂O model for simulating supercritical H₂O fluids [Sakuma et al., 2013] and tested the physical properties of NaCl-H₂O fluids at elevated temperatures and pressures [Sakuma and Ichiki, 2015]. In this study, we further investigated the behavior of electrical conductivity of NaCl-H₂O fluid as a function of pressure, temperature, and NaCl concentration. Our MD simulations cover 0.2 to 2.0 GPa, 673 to 2000 K, and 0.1 to 1.8 mol/kg NaCl concentrations. The physics behind the ionic conductivity was interpreted from the macroscopic (e.g., density and permittivity) and atomistic properties (e.g., self-diffusion coefficient, hydration number, and number of ion pairs) of fluids. Low fluid conductivity observed for quartz-NaCl-H₂O [Shimojuku et al., 2014] and albite-NaCl-H₂O systems [Guo et al., 2015] was interpreted as the presence of ionic species dissolved from the rocks by comparing our MD data. The obtained conductivities were plotted based on three geotherm models for covering various temperature gradient conditions. The contribution of bulk NaCl-H₂O fluid to conductivity variations in the Earth's crust is discussed by comparing the electromagnetic images of crust, arc, and subduction zones. These conductivity data and the interpretation of the results provide a useful resource for deriving the realistic conductivity of complex mineral-fluid systems.

2 Computational Methods

2.1 MD Simulations

The three-dimensional Ewald summation method was used for calculating the Coulomb interaction [Ewald, 1921]. The potential models for H₂O, Na⁺, and Cl⁻ are as described in our previous simulations for NaCl-H₂O fluid [Sakuma and Ichiki, 2015; Sakuma et al., 2013]. The details regarding the potential models are explained in the supporting information.

The short-range repulsion and van der Waals terms were disregarded for distances of >50% of the cell dimension. Radial distribution functions (RDFs) were employed for the corrections of energy and pressure, required for a finite cutoff radius [Frenkel and Smit, 2002]. The differential equation for the motion of atoms was converted to a difference equation using the velocity Verlet algorithm. The time increment was 0.1 fs. The energy conservation (ΔE/E = 10⁻⁵/ps) was tested to validate the code by simulations of a microcanonical ensemble. Equilibrium volume and density under various pT conditions were obtained by constant number of atoms, pressure, and temperature (NpT) ensemble simulations. The temperature was controlled by a velocity-scaling method as an approximated Gaussian thermostat yielding the canonical ensemble. Canonical ensemble simulations were performed using the equilibrium volume for calculating the electrical conductivity. An in-house program based on the MXDORTO code [Sakuma and Kawamura, 2011] was used for all MD simulations.

Unit cells (Figure 1) were composed of >2000 H₂O molecules and NaCl-H₂O binary fluids (Table 1). The number of molecules was validated to obtain a reasonable density, as discussed in Sakuma and Ichiki [2015]. At least three different randomly generated initial configurations were employed for averaging the obtained properties of the NaCl solutions. Each run simulated the total MD process for 150–400 ps, which was composed of the first convergence stage (50 ps) and the subsequent equilibrium stage (100–350 ps). The convergence stage simulated equilibrium, and the equilibrium stage was used for calculating conductivity.

Figure 1

Snapshots of NaCl-H₂O fluid at T = 973 K and p = 0.8 GPa for three different salt concentrations. Solid lines indicate the shape of the unit cell. The red and white cylinders denote H₂O molecules. Small and large spheres denote Na⁺ and Cl⁻, respectively.

Table 1. H₂O (n_H2O) and NaCl (n_NaCl) Atoms in a Unit Cell

NaCl Concentration	nH2O	nNaCl	Atoms in Unit Cell
0.58 wt % (0.1 mol/kg)	2222	4	6674
3.38 wt % (0.6 mol/kg)	2035	22	6149
9.52 wt % (1.8 mol/kg)	2035	66	6237

2.2 Ionic Conductivity

Ionic conductivity (σ_e) was derived using the Green-Kubo relation [Frenkel and Smit, 2002]:

(1)

with

(2)

where V is the volume of a unit cell, k_B is Boltzmann constant, T is temperature, j^el(t) is the electrical current at time t, q_i is the partial charge of ion i, and v_i is the velocity of ion i. The integration of equation 1 was performed up to 1 ps by confirming the convergence of the autocorrelation function. In this study, we calculated the electrical conductivity at direct current (DC) using equation 1. The conductivity calculated by MD simulations corresponds to the same low-frequency range measured by the electromagnetic experiments, if there is no dielectric relaxation mechanism at lower frequency.

2.3 Self-Diffusion Coefficient

The self-diffusion coefficients (D_S) of ions and H₂O were calculated using the mean square displacement [Frenkel and Smit, 2002] as follows:

(3)

where N is the number of targeted ions in a unit cell, t is time, and r_i(t) is the position vector of the ith ion. A linear slope from 2 to 5 ps of the gradient of the mean square displacement was used to calculate D_S.

3 Results

3.1 Ionic Conductivity

3.1.1 Low Salt Concentrations

Classical MD results should be validated by comparison to experimental results. Quist and Marshall [1968] measured the electrical conductivity of NaCl-H₂O fluid at 0.001–0.1 mol/kg (0.58 wt % NaCl), 0.1 MPa–0.4 GPa, and 373–1073 K. Bannard [1975] measured the conductivity at 0.01–5.0 mol/L, 0.025–0.2 GPa, and 298–800 K. We compared our MD results at 0.58 wt % NaCl with the experimental results of 0.58 wt % NaCl (Figure 2 and Table S2). The results show that the error bars in our MD results are large owing to the small number of dissolved ions in the solution (Figure 1a). Our calculated values are consistent with those of experiments at low temperatures [Bannard, 1975; Quist and Marshall, 1968]; although some differences occur between two experiments at 673 K: our MD results lie in between these two experimental results. In relation to the conductivity of the Earth's crust, only changes of orders of magnitude are important; therefore, these small differences are acceptable for the purposes of this study. The consistency of our results with those of experiments implies that our MD simulations do an adequate job of predicting the behavior of NaCl-H₂O fluid at high pressures, temperatures, and salt concentrations.

Figure 2

Comparison of electrical conductivity calculated by molecular dynamics (MD) simulations and those obtained experimentally at the same salt concentration (0.58 wt % = 0.1 mol/kg NaCl). Solid lines denote experimental results at 673, 674, and 972 K. Empirical fitting curves are shown as dashed lines. Error bars indicate the standard deviation of calculated data.

At elevated temperatures (>673 K) and pressures (>0.4 GPa), the predicted conductivity increases with increasing pressure (at constant temperature), although the conductivity at 673 K is almost independent of pressure (Figure 2). The pressure dependence becomes significant with increasing temperature and the pressure of maximum conductivity increased with increasing temperature.

3.1.2 High Salt Concentrations

Ion conductivities at high salt concentrations (3.38 wt % and 9.52 wt % ) were calculated using the MD simulations (Figure 3 and Table S2) and compared with experimental results [Bannard, 1975]. The conductivity values predicted by the MD simulations are consistent with those obtained experimentally. The pressure dependence at constant temperature is similar to the result of the 0.58 wt % NaCl concentration. Conductivity increases with increasing pressures but becomes pressure independent at high pressures. The temperature dependence is significant at low pressures, and the conductivity decreases with increasing temperature as we explain in the next section.

Figure 3

Calculated ion conductivity for (a) 3.38 wt % and (b) 9.52 wt % NaCl-H₂O fluids. The solid lines indicate the interpolated values of Bannard [1975]. Empirical fitting curves are shown as dashed lines. Error bars indicate the standard deviation of calculated data.

A simple quadratic equation was used to describe the obtained conductivity as a function of NaCl concentrations (Figure 4). The large deviation of fitting at high temperatures implies that higher-order functions are required to describe the relationship between conductivity and NaCl concentration. This nonlinear relationship indicates the presence of electrically neutral ion pairs in the concentrated solutions. The number of ion pairs is discussed in the section 3.2.4. A simple expression used for estimating the conductivity of highly concentrated NaCl-H₂O fluid by extrapolating the low-salinity experimental data [Reynard et al., 2011] overestimates the conductivity as shown in Figure 4. Therefore, the salt concentration required to explain the geophysical observations is revisited later by using the conductivity of our MD simulations.

Figure 4

Ion conductivity as a function of NaCl concentration at (a) 673 K, (b) 973 K, and (c) 1573 K. The calculated conductivities are fitted to a simple quadratic function and shown as dashed lines. The solid lines indicate the empirical expression used for estimating the conductivity by extrapolating low-salinity experimental data [Reynard et al., 2011]. Error bars indicate the standard deviation of calculated data.

3.2 Physics of Ionic Conductivity

3.2.1 Arrhenius Plots of Ionic Conductivity

The electrical conductivity of rocks σ is often expressed by an Arrhenius law:

(4)

Here σ₀ is the preexponential factor, ΔH, is the activation enthalpy of a conduction mechanism, k_B is the Boltzmann constant, and T is the absolute temperature. A negative conductivity versus 1/T slope can be explained by an Arrhenius law. In general, the ΔH is positive, and the Arrhenius law indicates that the conduction increases with increasing T. It is important to use a plot of logσ_e versus 1/T (Arrhenius plot), because different operative conduction mechanism can be determined from changes in slope on the plot. At constant pressure (0.4, 1.0, and 2.0 GPa; Figure 5), conductivity slightly increases with decreasing temperature, which is consistent with the conductivity measurements of NaCl-H₂O fluids [Quist and Marshall, 1968] and brine-bearing crustal rocks [Guo et al., 2015; Shimojuku et al., 2014]. This behavior is particularly significant at low pressure. The Arrhenius law, however, cannot explain the positive slope of the Arrhenius plot. In such a case, the conduction mechanism is not the kinetic energy of charged ions in the fluid. If Na⁺ and Cl⁻ make ion pairs and move together, no conduction occurs in the fluids. Therefore, we considered that the conduction mechanism of H₂O-NaCl fluid at elevated temperature is not controlled only by the mobility of ions, but the number of ion pairs in the fluids.

Figure 5

Arrhenius plots of the ionic conductivity of NaCl-H₂O fluid at (a) 0.4, (b) 1.0, and (c) 2.0 GPa. Experimental results [Quist and Marshall, 1968] of 0.58 wt % were shown for a comparison as solid symbols.

3.2.2 Mobility of Na⁺ and Cl⁻

Ion mobility is one of the most important parameters for conductivity at temperatures below the critical point [Quist and Marshall, 1968].

Ionic conductivity is often described by the Nernst-Einstein equation:

(5)

Here c_i is the concentration of ion i, and D_i is the self-diffusion coefficient of ion i. This equation can be derived from the Green-Kubo equation 1 for the absence of interactions [Bunde et al., 2005] and indicates that the ionic conductivity without making ion pairs has a linear relationship with the self-diffusion coefficients under the isothermal conditions.

The averaged self-diffusion coefficients D_S of ions in NaCl-H₂O fluid at three different salt concentrations were investigated (Figure 6). The O and H atoms form H₂O molecules and therefore have almost identical values. The D_S values of O and H increase with increasing temperature, particularly at low pressures. Absolute values of D_S for O and H show no significant difference in these salinities. The D_S values of Cl⁻ and Na⁺ increase with increasing temperature and decrease with increasing pressure; however, the effect of pressure is smaller than that of O and H atoms. This behavior does not correlate with the pressure and temperature dependence of ionic conductivity (Figures 2 and 3), indicating that density and the presence of electrically neutral ion pairs are the key factors controlling the ion conductivity at these conditions. Assuming that the self-diffusion of atoms can be described by the Arrhenius law, the activation enthalpies are estimated to be 10.4–14.3 kJ/mol for Cl ions, 13.0–15.2 kJ/mol for Na ions, and 15.6–19.5 kJ/mol for H₂O molecules. The variation of the activation enthalpy is ascribed to the pressure dependence. The activation energies of Na and Cl ions increase with increasing pressure, while that of H₂O decreases with increasing pressure as shown in Figure S1 in the supporting information. The effect of salinity is insignificant for all atoms.

Figure 6

Self-diffusion coefficients of ions for different NaCl concentrations. Isotherms lines are provided as a guide. Error bars indicate the standard deviation of calculated data.

3.2.3 Density

The density of NaCl-H₂O fluid is an important property for conductivity [Quist and Marshall, 1968] since the increased number of ions per unit of volume seems to increase the conductivity. The density largely increases with increasing pressure (at p < 1.0 GPa), while the gradient gradually decreases at pressures of p > 2.0 GPa as shown in Figure 7 [Sakuma and Ichiki, 2015]. This behavior appears to be correlated with conductivity; however, the saturation of conductivity at high pressure cannot be explained solely by changes in density. For instance, the isotherm of conductivity at 673 K and 0.58 wt % shown in Figure 2 slightly increases from 0.3 to 1.0 GPa, and the conductivity slightly decreases from 1.0 to 2.0 GPa. This behavior is inconsistent with the increased density from 0.3 to 2.0 GPa as shown in Figure 7. Increased density affects the increase of conductivity for all pressure studied here; however, ionic conductivity can be explained by the mixed effects of density, number of ion pairs, and mobility of ions. The inconsistency at high pressure indicates that the effect of increased number of ion pairs is larger than the effect of density at high pressures.

Figure 7

Density of NaCl-H₂O fluids calculated by MD simulations (symbols) and empirical equation of state (EoS) (solid lines). The detailed discussion of the MD results and empirical EoS was described in previous literature [Sakuma and Ichiki, 2015].

3.2.4 Presence of Electrically Neutral Ion Pairs in the Solution

The nonlinear relationship between conductivity and NaCl concentration (Figure 4) possibly relates to the degree of electrically neutral ion pairs in the solution, since the Na⁺ and Cl⁻ ions can only transport charge if they do not form charge-neutralized ion pairs. The number of nearest counter ions reported by our previous research [Sakuma and Ichiki, 2015], at the same temperatures, pressures, and ion concentrations, is reprinted in Figure 8. The number of Cl ions surrounds a Na ion (bottom in Figure 8) increases with increasing salt concentration, which is consistent with the nonlinear relationship between conductivity and salt concentration (Figure 4). Note that the number is not the actual number of ion pairs. When the number is two, for instance, it refers the two different species (Na₂Cl₂ and NaCl₂)_. The number increases with increasing temperature, while the number of surrounding H₂O molecules decreases with increasing temperature (Figures 8a (top) to 8c (top) and 8a (middle) to 8c (middle)). This indicates that the making of ion pairs is energetically stable compared to the ions dissolved in H₂O as an ion. Such behavior can be explained by the change in the permittivity of the H₂O solvent [Sakuma et al., 2013]. Thus, the results show that the permittivity is the most important factor in controlling the solubility of ions in the solvent. The permittivity decreases with increasing temperature and decreasing pressure; therefore, the ion pairs are stabilized under low permittivity conditions. The permittivity of H₂O at 673 K and 0.34 GPa is high relative to those at 973 K and 1573 K, indicating the solubility of NaCl is large at 673 K and 0.34 GPa. The pressure dependence of the solubility of ions is less clear for 673 K than those at 973 K and 1573 K, because the permittivity of H₂O at 673 K is sufficiently high in this pressure range for dissolving the NaCl.

Figure 8

Running coordination numbers of Na-O (n_Na-O), Cl-H (n_Cl-H), and Na-Cl pairs (n_Na-Cl) determined by integrating the radial distribution functions up to the first minimum distance at (a) 0.58 wt %, (b) 3.38 wt %, and (c) 9.52 wt % NaCl. The lines are intended as a guide for the reader to indicate the isotherms. Error bars indicate the standard deviation of calculated data. Reprinted from Sakuma and Ichiki [2015], copyright 2015, with permission from John Wily and Sons Ltd.

4 Discussion

4.1 Effect of Proton Diffusion on Conductivity

The conductivity of pure H₂O at elevated temperatures and pressures was investigated using conductivity measurements [Holzapfel, 1969], and conductivity was shown to increase with increasing temperatures and pressures. The main conduction mechanism in pure H₂O should be proton diffusion. Assuming lithostatic conditions (density = 2.7 g/cm³) and surface heat flow of 50 mW/m², the pressure and temperature at 38 km of the continental crust can be estimated to be 1 GPa and 770 K, respectively. The conductivity by proton diffusion at these conditions can be estimated as 0.03 S/m [Holzapfel, 1969]. Therefore, the contribution of the proton diffusion to the conductivity of NaCl-H₂O fluid should be limited to dilute salt solutions. Relative to the ionic conductivity 5 S/m of 0.6 wt % NaCl-H₂O fluid, 0.03 S/m of the proton diffusion differs 2 orders of magnitude.

4.2 Implications for Fluid Conductivity Change With Depth

Estimating the electrical conductivity of pure NaCl-H₂O fluid in the Earth's crust is important when considering the contribution of NaCl-H₂O fluid to subsurface conductivity profiles, as measured by the magnetotellulic (MT) technique and is useful for further considerations of more realistic and more complex rock-fluid systems. In order to calculate the electrical conductivity of NaCl-H₂O fluid along a geotherm model, we used an analytical form fitted to the MD results at high pressures. The purely empirical equation is as follows:

(6)

(7)

(8)

where σ_e is conductivity (S/m), p is pressure (MPa), T is temperature (K), and c is NaCl concentration (wt %). The parameters α_i (I = 1, 2, 3), β_ij (i, j = 1, 2, 3), and γ_ijk (i, j, k = 1, 2, 3) were obtained by least square fitting to the MD results. The best fit parameters are given in Table 2. This fitted equation is purely empirical, and therefore, the extrapolation to low (<0.2 GPa) and high pressures (>2 GPa), low (<673 K) and high temperatures (>2000 K), and salt concentrations (>10 wt %) cannot be guaranteed. Our MD results cannot cover all of the pT conditions considered here; for instance, conductivity at elevated temperatures (>1573 K) and 3.38 wt % NaCl was not calculated; therefore, the fitting equation and parameters may be unreliable at these conditions.

Table 2. Parameters of Conductivity (Equation 8)

Parametera	Value
γ111 (wt %^−2 K^−2 MPa^−2 S m^−1)	−2.76823 × 10^−12
γ112 (wt %^−1 K^−2 MPa^−2 S m^−1)	2.86668 × 10^−11
γ113 (K^−2 MPa^−2 S m^−1)	−1.01120 × 10^−11
γ121 (wt %^−2 K^−1 MPa^−2 S m^−1)	6.32515 × 10^−9
γ122 (wt %^−1 K^−1 MPa^−2 S m^−1)	−6.35950 × 10^−8
γ123 (K^−1 MPa^−2 S m^−1)	2.14326 × 10^−8
γ131 (wt %^−2 MPa^−2 S m^−1)	−2.92588 × 10^−6
γ132 (wt %^−1 MPa^−2 S m^−1)	2.69121 × 10^−5
γ133 (MPa^−2 S m^−1)	−9.20740 × 10^−6
γ211 (wt %^−2 K^−2 MPa^−1 S m^−1)	6.52051 × 10^−9
γ212 (wt %^−1 K^−2 MPa^−1 S m^−1)	−7.43514 × 10^−8
γ213 (K^−2 MPa^−1 S m^−1)	2.23618 × 10^−8
γ221 (wt %^−2 K^−1 MPa^−1 S m^−1)	−1.47966 × 10^−5
γ222 (wt %^−1 K^−1 MPa^−1 S m^−1)	1.67038 × 10^−4
γ223 (K^−1 MPa^−1 S m^−1)	−4.54299 × 10^−5
γ231 (wt %^−2 MPa^−1 S m^−1)	6.88977 × 10^−3
γ232 (wt %^−1 MPa^−1 S m^−1)	−7.25629 × 10^−2
γ233 (MPa^−1 S m^−1)	1.89836 × 10^−2
γ311 (wt %^−2 K^−2 S m^−1)	−2.60077 × 10^−6
γ312 (wt %^−1 K^−2 S m^−1)	3.64027 × 10^−5
γ313 (K^−2 S m^−1)	−7.50611 × 10^−6
γ321 (wt %^−2 K^−1 S m^−1)	6.12874 × 10^−3
γ322 (wt %^−1 K^−1 S m^−1)	−9.01143 × 10^−2
γ323 (K^−1 S m^−1)	1.51621 × 10^−2
γ331 (wt %^−2 S m^−1)	−3.17282
γ332 (wt %^−1 S m^−1)	50.2186
γ333 (S m^−1)	−6.22277

• ^a Parameters obtained by the equations resulting from molecular dynamics (MD) simulations.

Experimental results [Bannard, 1975] for NaCl-H₂O fluid at lower temperatures (<600 K) and pressures (<200 MPa) were fitted to the following equation:

(9)

(10)

where δ_i (I = 1, 2, 3, 4, 5) and ϕ_ij (i, j = 1, 2, 3, 4, 5) are the fitting parameters (Table 3). In equations 9 and 10, the pressure dependence of the conductivity was neglected because no significant pressure dependence was observed at temperatures of <600 K [Bannard, 1975; Quist and Marshall, 1968]. Fitting was performed to the experimental conductivity data at p = 200 MPa, T < 600 K, and NaCl of 0.1–3.0 mol/L, as reported by Bannard [1975]. Therefore, this purely empirical equation can be guaranteed over these conditions.

Table 3. Parameters of Conductivity (Equation 10)

Parametera	Value
ϕ11 (wt %^−3 K^−4 S m^−1)	−1.61994 × 10^−12
ϕ12 (wt %^−2 K^−4 S m^−1)	4.32808 × 10^−11
ϕ13 (wt %^−1 K^−4 S m^−1)	1.15235 × 10^−10
ϕ14 (K^−4 S m^−1)	2.52257 × 10^−10
ϕ21 (wt %^−3 K^−3 S m^−1)	1.88235 × 10^−9
ϕ22 (wt %^−2 K^−3 S m^−1)	−5.82409 × 10^−8
ϕ23 (wt %^−1 K^−3 S m^−1)	−3.37538 × 10^−7
ϕ24 (K^−3 S m^−1)	−4.53779 × 10^−7
ϕ31 (wt %^−3 K^−2 S m^−1)	−5.65158 × 10^−7
ϕ32 (wt %^−2 K^−2 S m^−1)	2.70538 × 10^−5
ϕ33 (wt %^−1 K^−2 S m^−1)	2.40270 × 10^−4
ϕ34 (K^−2 S m^−1)	2.97574 × 10^−4
ϕ41 (wt %^−3 K^−1 S m^−1)	4.64690 × 10^−5
ϕ42 (wt %^−2 K^−1 S m^−1)	−6.70560 × 10^−3
ϕ43 (wt %^−1 K^−1 S m^−1)	−2.69091 × 10^−2
ϕ44 (K^−1 S m^−1)	−8.37212 × 10^−2
ϕ51 (wt %^−3 S m^−1)	2.58834 × 10^−3
ϕ52 (wt %^−2 S m^−1)	6.92510×10^−1
ϕ53 (wt %^−1 S m^−1)	−3.22923
ϕ54 (S m^−1)	8.48091

• ^a Parameters obtained by fitting equations to experimental conductivity data [Bannard, 1975].

An important question when deriving conductivity at depth is whether or not conductivity profiles are affected by the hydrostatic or lithostatic pressure models employed. Hydrostatic pressure models assume that fluid is connected from the surface to depth. However, such connection of fluid from surface to lower crust seems to be unrealistic due to the presence of impermeable layer. Therefore, a lithostatic pressure model should be investigated for considering the wide range of fluid fraction in the crust, arc, and subduction zones. The network of fluid at elevated temperature can be determined by the dihedral angle and fluid fraction [von Bargen and Waff, 1986]. Fluid can establish channels at extremely small fluid fraction <0.01 if a dihedral angle is <60°. The critical dihedral angle for achieving the interconnection of fluid increases with increasing fluid fraction; for example, the dihedral angle is ~80° for a 5% fluid fraction. The dihedral angle of the quartz-water system in the crust ranges from 60° to 80° [Holness, 1995]; therefore, the fluid fraction is critical for establishing interconnection. In this study, the pressures of the lithostatic model were calculated using the averaged density of the crust (2.7 g/cm³).

Depending on the targeted area, several geotherm models have been proposed for describing the temperature gradient with depth [Iwamori, 2007; Omori et al., 2009; Peacock et al., 2002; Shankland and Ander, 1983; Turcotte and Schubert, 2002]. Here we calculated the conductivity along three representative geotherm models (3.3 K/km, 6.7 K/km, and 30 K/km).

The results show that conductivity at lithostatic conditions (Figure 9) increases with depth, with the maximum observed using the high-geotherm models (6.7 K/km and 30 K/km). Because pore is considered to be unable to keep its autonomous shape below lower crust, we show only lithostatic pressure gauge here. A maximum appeared using all geotherm models, and the depth of the maximum became shallower with increasing temperature gradient. The change in conductivity, amounting to an order of magnitude, can contribute to the change (of 2 orders of magnitude) observed by the MT method in old continental crust. Constant salt concentrations were assumed for calculating the conductivity profile. This may not be realistic because the salt concentration of near-surface aqueous fluids would be lower due to mixing with surface H₂O (e.g., precipitation). The connectivity and fluid fraction should drastically affect the electrical conductivity observed using the MT method, and therefore, in future work a fluid-rock composite model should be developed for interpreting MT observations from shallow to deep crust.

Figure 9

Electrical conductivity profiles along three geotherm models at different lithostatic conditions. The dashed lines indicate the extrapolation of experimental fits by neglecting the pressure dependence.

4.3 Comparison With the Bulk Conductivity of Brine-Bearing Rocks

Recent electrical conductivity measurements of rock-H₂O-NaCl system at high p and T [Guo et al., 2015; Shimojuku et al., 2014] reported that the fluid conductivity derived from the extrapolation to 100% fluid was significantly lower than the conductivity of bulk fluid experiments of low salinity [Quist and Marshall, 1968]. This low conductivity may be ascribed to the difference between fluid-rock composite models and real samples, but the low conductivity cannot be explained only by the difference [Guo et al., 2015; Shimojuku et al., 2014]. It is worthwhile to compare the conductivity of high salinity fluid with our MD simulations for understanding the mechanisms of low fluid conductivity. The fluid conductivity of MD simulations and experiments are plotted in Figure 10. Our MD results are in agreement with available experimental data [Quist and Marshall, 1968] by extrapolating the experimental data to 1.0 GPa. The experimental values at p = 400 MPa and T = 775 K of 0.05 mol/kg and 777 K of 0.1 mol/kg were plotted as extrapolated values at 1 GPa by assuming that the pressure dependence can be neglected over 400 MPa at these temperatures. There are large discrepancy between the MD simulations and the experiments conducted for brine-bearing quartzite [Shimojuku et al., 2014]. The experiments of albite-H₂O-NaCl system [Guo et al., 2015] has large discrepancies at low NaCl concentration (<6 wt %), but the value at 10.5 wt % is similar to our MD results. The solubility of albite decreases with increasing salinity of H2O-NaCl fluids [Shmulovich et al., 2001]. This means that the fluid conductivity can be mainly characterized by the Na⁺ and Cl⁻ at high salinity for alibite-H₂O-NaCl system.

Figure 10

Fluid conductivities of MD simulations (solid line) and extrapolation of the experiments [Quist and Marshall, 1968] (open circles). Fluid conductivities derived from bulk conductivities of brine-bearing quartzite [Shimojuku et al., 2014] and albite [Guo et al., 2015] were shown as crosses and solid squares, respectively.

Here we discuss the mechanism of low fluid conductivity of quartz-H₂O-NaCl fluid measurements [Shimojuku et al., 2014]. One possibility for the discrepancy may be the adsorption of Na⁺ and/or Cl⁻ onto the surfaces of quartz grains. The adsorption density of Na⁺ on quartz was determined at pH from 1 to 12 by streaming potential measurements [Li and De Bruyn, 1966]. Assuming that the mean particle radius of quartz is 10 µm, fluid volume fraction is 0.2, salinity is 1.2 wt %, and maximum adsorption density of Na⁺ is 3.5×10⁻¹⁰ mol/cm², the adsorbed Na⁺ is only 2.3% of total Na⁺ in the fluid. Therefore, the effect of adsorption on quartz seems to have little effect on the conductivity of the experiments. Although the adsorption density of Na⁺ may be enhanced at elevated temperature and pressure, the effect should decrease with increasing salinity, since most adsorption site should be occupied at high salinity. However, such behavior cannot be observed from the experiments [Shimojuku et al., 2014].

The other possibility to explain the low conductivity is the increase of ion pairing of NaCl due to the presence of ionic species dissolved from the solid. Assuming that the solubility of SiO₂ is 1.2 mol/kg at 1 GPa and 1100 K estimated from an extrapolation of low-pressure experiments [Manning, 1994], the solubility is 2 times larger than the NaCl concentration of seawater (0.6 mol/kg). As discussed in Shimojuku et al. [2012], the conduction of salt-free H₂O-quartzite system at 1 GPa and 1100 K should be due to the presence of charged silicate species like Si₂O₂(OH)⁵⁻; however, the conductivity was much lower than H₂O-NaCl (3 wt %)-quartzite system [Shimojuku et al., 2014]. This indicates that the dominant species of dissolved Si should be a neutral species such as Si(OH)₄. In our preliminary MD simulations (not shown in this manuscript), the number of ion pairs of Na and Cl in H₂O did not change under the presence of Si(OH)₄ at ambient conditions. Therefore, the presence of dissolved species from quartz seems not to enhance the ion-pairing in the H₂O-NaCl fluid. Such neutral Si(OH)₄ species, however, would be stable by clustering in H₂O fluid, and these clusters may disturb the movement of ions in the fluids. MD simulations should investigate this hypothesis in future work.

4.4 Implications for the Salt Concentration and Fluid Fraction in the Crust, Arc, and Subduction Zones

Here the bulk conductivity of a fluid-rock composite is calculated to interpret the electromagnetic observations in the crust, arc, and subduction zones. To estimate the bulk conductivity of fluid-rock composite, Hashin-Shtrikman upper bound model (HS⁺) [Hashin and Shtrikman, 1962] is employed;

(11)

Here σ_bulk is the bulk conductivity, σ_e is the conductivity of NaCl-H₂O fluid, σ_s is the conductivity of solid, and Φ is the fluid fraction. This model assumes an homogenous distribution of fluid in rock and thin completely interconnected films at the grain boundaries [Schmeling, 1986; Waff, 1974]. The model is applicable for a dihedral angle <60° for the interconnection of fluid [von Bargen and Waff, 1986]. This condition would be achieved at the highly conductive zones of crust for SiO₂-NaCl-H₂O system [Watson and Brenan, 1987] and albite-SiO₂-H₂O system [Holness, 1995]. Here we estimate the plausible salinity of NaCl-H₂O fluids by comparing electromagnetic observations and our MD simulations.

4.4.1 Continental Lower Crust

We calculated the conductivity of NaCl-H₂O fluid at elevated temperature, pressure, and salinity to 10 wt %. A conductivity profile can be estimated by assuming the conductivity of host rock, fluid fraction, salinity, and geotherm. The conductivity of rocks less than 0.001 S/m and 1% porosity would be a plausible assumption for the continental lower crust [Hyndman and Shearer, 1989]. A geotherm model [Turcotte and Schubert, 1982, equations (4)–(31)] of surface heat flow 50 mW/m² was employed. Under these conditions, the geotherm and the bulk conductivity as a function of salinity was plotted in Figure 11. This results indicate that the presence of 0.2 to 7.0 wt % NaCl can explain the highly conductive zones of continental lower crust. These salinities are similar to that of seawater (3.4 wt %), so the range estimated seems to be acceptable. These concentrations, however, would be minimum values, since the Hashin-Shtrikman upper bound is based on the complete interconnection of fluids at grain boundaries. The connectivity of the continental lower crust should be estimated from the fluid fraction and dihedral angle. The dihedral angle of anorthite with H₂O fluids are less than 60° at p > 1 GPa [Yoshino et al., 2002]. These pressures correspond to the depth below 35 km and highly conductive zone was observed in the depth. Highly conductive zone at shallow depth (~20 km) may not be explained only by the presence of H₂O fluid, but it does not seem that our argument is unrealistic to the continental lower crust below 35 km. Increased salinity decreases the dihedral angle of quartz-H₂O-NaCl system [Laporte and Watson, 1991; Watson and Brenan, 1987] at elevated temperature and pressure. Such salinity effect should be investigated for anorthite-H₂O system. The increased conductivities with low salinity (<0.5 wt %) may be ascribed to the insufficient description of the empirical equation 6. These conductivities were calculated by using equation 6, and the conductivities with salinity <0.5 wt % was the extrapolation of our MD results. The calculated conductivities of 0.1, 0.2, and 0.3 wt % at 26 km were consistent with the extrapolation of experimental values; however, the values below 40 km depth should be validated by future research.

Figure 11

(a) Geotherm model used for calculating the bulk conductivity and (b) bulk conductivity of NaCl-H₂O fluid-rock composite along the geotherm of continental crust (solid lines). Right vertical axis of the geotherm is the depth. The conductivity was calculated by assuming the Hashin-Shtrikman upper bound, rock conductivity of 0.001 S/m, surface heat flow of 50 mW/m², averaged crust density of 2.7 g/cm³ for lithostatic pressure, and fluid fraction of 0.01. The salinities of NaCl ranged were shown in the top of the figure. The calculated salinities were 0.1, 0.2, 0.3, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, and 10.0 wt %. The discrepancy of lines at around 26 km is due to the difference of our MD results and the extrapolation of experimental results [Bannard, 1975]. Shaded area indicates the high conductivity range of continental crust [Hyndman and Shearer, 1989; Shankland and Ander, 1983].

4.4.2 Subduction Zone and Arc

The effects of partial melt are beyond the scope of this study, and therefore, we discuss high conductivity zones at low temperature (<900 K). Conductivity distribution of the Cascadia subduction zone is one of the most investigated area [Evans et al., 2014; McGary et al., 2014; Soyer and Unsworth, 2006]. A highly conductive zone (~0.03 S/m) was observed at a depth of 20~40 km above the subducting oceanic crust [McGary et al., 2014; Soyer and Unsworth, 2006]. The temperature of the zone is estimated to be 673 to 773 K [Peacock et al., 2002], and the pressure is estimated to be 1.0 GPa (assuming an average density of 3.0 g/cm³). Since this zone has been interpreted by the hydrated basalt-eclogite transition [McGary et al., 2014], the solid conductivity was assumed to be 10⁻⁴ S/m of basalt [Kariya and Shankland, 1983]. According to the Hashin-Shtrikman upper bound model, a conductivity 0.03 S/m of NaCl-H₂O fluid-rock composite can be achieved by the presence of NaCl-H₂O fluid of the salinity higher than 0.5 wt % under the fluid fraction of 0.01 as shown in Figure 12. The salinity is smaller than seawater (3.4 wt %), and this may be explained only by the fluid release of the basalt-eclogite transition.

Figure 12

Bulk conductivity as a function of fluid fraction and salinity for discussing the fluid in the Cascadia subduction zone. The salinity was calculated from 0.1 to 2.0 wt % with an increment of 0.1 wt % to 1.0 wt % as shown from blue to red lines.

The presence of fluids in intraplate earthquake zones of northern east Japan back arc is implied by MT observations [Ogawa et al., 2001]. The depth of the high conductivity zone (~1 S/m) is located from 10 to 15 km depth, and concentrated seismicity near the rims of the conductive zones may be related with the presence of fluids. A temperature of 573 K at a depth of 10 km can be estimated from heat flow and thermal gradient measurements [Tanaka and Ishikawa, 2002] by assuming a thermal gradient of 30 K/km. The solid basalt conductivity would be less than 10⁻⁴ S/m [Kariya and Shankland, 1983], and the rock conductivity of 10⁻⁴ S/m was used in these calculations. To explain the 1 S/m conductivity, 4 wt % NaCl concentration is required at a fluid fraction of 0.05 as shown in Figure 13. The fluid fraction of 0.05 was estimated [Uyeshima, 2005] by using the seismic velocity data [Matsubara et al., 2004] and a theoretical model of solid-liquid composite [Takei, 2002]. The concentration is similar to the seawater and such seawater-like fluid may be present in this area.

Figure 13

Bulk conductivity as a function of fluid fraction and salinity for discussing the fluid of the northern east Japan back arc. The salinity was calculated from 1.0 to 10.0 wt % with an increment of 0.1 wt %.

5 Conclusions

In this study, we investigated the ionic electrical conductivity of NaCl-H₂O fluid as a function of pressure (0.2–2.0 GPa), temperature (673–2000 K), and NaCl concentration (0.6–9.6 wt %) by using classical MD simulations. The calculated conductivity at 0.6 wt % NaCl, low temperature, and low pressure was consistent with the available experimental data. The calculated conductivity at elevated temperatures (>673 K) and pressures (>0.4 GPa) increases with increasing pressures, and a broad peak was observed at 673 K. The pressure dependence was significant with increasing temperature. The maximum conductivity appears to be independent of temperature. The relationship between conductivity and NaCl concentration was nonlinear, which is explained by the presence of electrically neutral ion pairs in the concentrated solutions. The major factors controlling conductivity are the density of NaCl-H₂O fluid and the permittivity of solvent H₂O. However, ion mobility was not a key factor in conductivity.

The obtained conductivity was plotted based on three geotherm models, and the contribution of bulk NaCl-H₂O fluid to the conductivity variations in the Earth's crust was discussed. Highly conductive zones below a depth of 35 km in the continental lower crust can be interpreted by the presence of NaCl-H₂O fluid with the concentration ranging from 0.2 to 7.0 wt %. A highly conductive zone (~0.03 S/m) observed at a depth of 20~40 km above the subducting oceanic crust in Cascadia can be interpreted by the presence of low concentrated (0.5 wt %) NaCl-H₂O fluid possibly generated by the dehydration of basalt. Fluids in the back arc of northern east Japan can have NaCl concentration similar to the seawater. These conductivity data and the interpretation of the results provide a useful resource for deriving the realistic conductivity of complex mineral-fluid systems.

Acknowledgments

We acknowledge N. Nishiyama for the discussion. This research was partially supported by JSPS KAKENHI (grant 23740390 and 26610172). This research was performed under the project Grant-in-Aid for Scientific Research on Innovative Areas “Geofluids.” Interatomic potential functions, parameters, data supporting Figures 2 and 3, and activation enthalpy as a function of pressure are available as equations S1 and S2, Tables S1 and S2, and Figure S1 in the supporting information. This manuscript was greatly improved by the comments of three anonymous reviewers and associate editor.

Appendix A: Comparison of NaCl-H₂O Fluid With KCl-H₂O Fluid

KCl-H₂O fluid has been used as an analogue for NaCl-H₂O fluid due to the similar conductivity at ambient conditions. However, it remains important to compare the conductivity at elevated temperatures and pressures. The conductivity profiles of KCl-H₂O [Hwang et al., 1970] and NaCl-H₂O fluids [Bannard, 1975], assuming lithostatic conditions, are shown in Figure A1. Since the pressure range was limited to <0.3 and <0.2 GPa for KCl-H₂O and NaCl-H₂O fluids, respectively, the depth of the data was limited to <12.0 km. The comparison shows that the fluids behave in a similar manner and that KCl-H₂O fluid can be used as an analogue for NaCl-H₂O fluid at conditions shallower than 12.0 km. However, the extrapolation to deeper conditions remains untested and represents an important avenue for future work.

Figure A1

Comparison of NaCl-H₂O and KCl-H₂O fluids along three geotherm models. Experimental conductivity data were sourced from Bannard [1975] and Hwang et al. [1970].