P.A. Duvillard , G. Ménard, and L. Ravanel
1EDYTEM, Université Grenoble Alpes, Université Savoie Mont‐Blanc, CNRS, UMR CNRS 5204, Le Bourget du Lac, France,2Université Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, Grenoble, France,
3Department of Mining and Metallurgical Engineering, Yazd University, Yazd, Iran,4Department of Geology, University of Otago, Dunedin, New Zealand
Abstract Spectral induced polarization spectra were carried out on three graphitic schists and two graphitic sandstones. The microstructural arrangement of graphite of two graphitic schists was studied with thin sections using transmitted and reflected light optical and electron microscopic methods. Chemical maps of selected areas confirm the presence of carbon. The complex conductivity spectra were measured in the frequency range 10 mHz to 45 kHz and in the temperature range +20 °C down to−15 °C. The measured spectra arefitted with a double Cole‐Cole complex conductivity model with one component associated with the polarization of graphite and the second component associated with the Maxwell‐Wagner polarization.
The Cole‐Cole exponent and the chargeability are observed to be almost independent of temperature including in freezing conditions. The conductivity and relaxation time are dependent on the temperature in a predictable way. As long as the temperature decreases, the electrical conductivity decreases and the relaxation time increases. Afinite element model is able to reproduce the observed results. In this model, we consider an intragrain polarization mechanism for the graphite and a change of the conductivity of the background material modeled with an exponential freezing curve. One of the core sample (a black schist), very rich in graphite, appears to be characterized by a very high conductivity (approximately 30 S/m). Two induced polarization profiles are discussed in the area of Thorens. The model is applied to the chargeability data to map the volumetric content of graphite.
1. Introduction
Induced polarization is a geophysical (galvanometric) method able to image the low‐frequency polarization properties of rocks (e.g., Dahlin & Leroux, 2012; Schumberger, 1920). By“low frequency,”we typically refer to frequencies below ~10 kHz. In this frequency range, the type of polarization mechanisms we are dealing with are nondielectric in nature. Induced polarization is indeed related to the accumulation or depletion of electrical charges at some characteristic length scales (typically grain/pore sizes) of the porous material and electro‐diffusion mechanisms under the application of a primary electricalfield (e.g., Marshall & Madden, 1959; Olhoeft, 1985). Induced polarization has a long history in mining geophysics as an efficient geophysi- cal method to localize ores (e.g., Mahan et al., 1986; Schlumberger, 1920). That said, graphite‐type rocks have not been thoroughly investigated in this context.
Schists are typically medium‐grade metamorphic rocks with distinctly planar tectonite fabrics, that is, foliations (Passchier & Trouw, 2005). Schist foliations are defined by the presence of through going sheets of phyllosilicate minerals where individual grains are large enough to be seen by the naked eye. This mineral arrangement typically also imparts a tendency to break easily parallel to foliation planes (fissility). This arrangement is also responsible for an anisotropy of mechanical strength that is just one physical property affected by the foliation. It is reasonable to expect other physical properties of schists to be similarly anisotropic. The planar phyllosilicates that most commonly define the fabrics of these rocks are micas, but graphite‐rich schists are also common with some black schists very rich in graphite formed by the medium‐grade metamorphism of sediments initially rich in organic matter (Landis, 1971).
Graphite is a mineral consisting of layers of graphene made of carbon atoms bonded together in a hexagonal two‐dimensional structure. Temperature has to exceed 400 °C to form crystalline graphite and the level of crystallinity depends on the degree of metamorphosis (Simandl et al., 2015). Graphite is considered one of the top 30 critical minerals in 2017 by the European Union (European Commission, 2017) for its use in
©2019. American Geophysical Union.
All Rights Reserved.
Key Points:
• Graphite‐bearing rocks exhibit anomalous electrical properties
• New experimental data are shown to characterize the electrical properties of these rocks
• A model is developed to explain the new observations
Correspondence to:
A. Revil,
andre.revil@univ‐smb.fr
Citation:
Abdulsamad, F., Revil, A., Ghorbani, A., Toy, V., Kirilova, M., Coperey, A., et al. (2019). Complex conductivity of graphitic schists and sandstones.
Journal of Geophysical Research: Solid Earth,124, 8223–8249. https://doi.org/
10.1029/2019JB017628
Received 3 MAR 2019 Accepted 31 JUL 2019
Accepted article online 7 AUG 2019 Published online 22 AUG 2019
lubricants, electrodes, solar panels, and batteries. Since graphite is a semi- metal, its electrical properties (like those of semiconductors and metals) have a strong influence on the conductivity and polarization properties of porous media in which they are located (Revil, Florsch, et al., 2015, Revil, Abdel Aal, et al., 2015; Revil, Coperey, Mao, et al., 2018). While it is broadly recognized that graphite is an excellent electrical conductor (e.g., Pierson, 1993), few works have addressed its polarization properties.
The only work, to our knowledge, about the complex conductivity of gra- phitic schists is the recent paper by Börner et al. (2018). They measured the complex conductivity of a black schist and an augen gneiss both col- lected in the Main Central Thrust shear zone in the Himalayas of central Nepal. Their results underline the extraordinary high chargeability and anisotropy of the black schist with important consequences for interpret- ing magnetotelluric data in these environments. That said, this work remains very qualitative with respect to the body of fundamental knowl- edge accumulated over the last years in understanding the mechanisms of polarization of rocks with semiconductors, semimetals, and metals (Mao & Revil, 2016; Misra et al., 2016a, 2016b; Revil, Coperey, Mao, et al., 2018).
Our primary goal is in this paper is to develop a model of the complex conductivity of these rocks analyz- ing the effect of graphite on the complex conductivity of the whole rock (conduction and polarization). In order to test this model, we performed complex conductivity measurements onfive core samples from the Alps (France and Switzerland). The spectra were obtained in the temperature +20 °C to−15 °C in order to better constrain the underlying physics of the conduction and polarization processes. We also per- formed an electrical conductivity experiment on pure graphite and polarization experiments at different volumetric contents of graphite in sand. In addition, we performed numerical experiments showing how the observed spectra can explain numerically the polarization process. In this case, we consider the electro‐migration and charge accumulation in the graphite particles, themselves dispersed in a back- ground material made of mineral (nonconducting) grains and pore water. Finally, we present induced polarizationfield data in which we image the conductivity and chargeability over two 600‐m‐long profiles carried out in Thorens (French Alps) in an area where permafrost and graphitic schists are known to be present.
2. Complex Conductivity of Rocks
2.1. A General Complex Conductivity Expression
In induced polarization, a rock is characterized in the frequency domain by its conductivity amplitude and its phase lag between the current and the electricalfield. In this section, we consider an isotropic material made of metallic and nonmetallic grains and liquid pore water (Figure 1). The amplitude and phase can be recast into a complex‐valued conductivity. The in‐phase (real) component characterizes the ability of rocks to conduct an electrical current while its quadrature (out‐of‐phase, imaginary) conductivity describes the ability of rocks to reversibly store electrical charges (Olhoeft, 1981). Typically, the complex conductivity of a rock entering into Ampère's law is written as
σ*¼σ∞ 1− M iωτ ð Þc
; (1)
whereσ* denotes the complex conductivity,ωis the angular frequency (rad/s),σ∞denotes the instantaneous (high‐frequency) conductivity of the rock (in S/m),σ0denotes the DC (steady state, direct current) conduc- tivity (S/m),M= (σ∞−σ0)/σ∞(dimensionless) denotes its chargeability,τ(in s) is the Cole‐Cole relaxation time, andc(dimensionless) denotes the Cole‐Cole exponent measuring the broadness of the distribution of the relaxation times (Cole & Cole, 1941).
A generalization of this equation fork‐induced polarization distributions plus an explicit relationship for the Maxwell Wagner polarization can be written as
Figure 1.Sketch of the mixture between a background (made of nonmetal- lic grains and pore liquid water plus eventually some ice) and graphite. The background material can be partially saturated at a water contentθ.
σ*¼σ∞ 1−∑K
k¼1
Mk
1þðiωτkÞck
; (2)
Mk¼σ∞k−σ0k
σ∞k
; (3)
σ0¼σ∞ 1−∑K
k¼1Mk
; (4)
whereMk,ck, andτkdenote the chargeability, the Cole‐Cole exponent, and the relaxation time for processk for a total number ofKpolarization processes. For the two polarization processes, we have
σ*¼σ∞ 1− M1
1þðiωτ1Þc1− M2
1þðiωτ2Þc2
; (5)
σ∞¼σ∞1 þσ∞2; (6)
σ0¼σ∞ð1−M1−M2Þ; (7)
Figure 2.Sketch of the polarization of the nonmetallic grains and graphite. (a) The nonmetallic grain such as clays pos- sesses an electrical double layer. Under the influence of an external electricalfieldE0, this electrical double layer polarizes.
(b) Polarization of graphite. The charges carriers inside the grain (delocalized electrons and p‐holes) polarize the grain giving then rise to the formation offield‐induced diffuse layers (DL). Thefirst column defined the instantaneous con- ductivityσ∞(where all charges carriers participate in the conduction process) while the second column defines the DC conductivityσ0(part of charges are no more available for conduction that is whyσ∞>σ0).
where M1 and M2 are the chargeabilities (dimensionless), c1 and c2 are the two Cole‐Cole exponents (dimensionless), andτ1andτ2are the relaxation times (expressed in s). The index 1 and 2 refer to two dispersion processes, respectively. With these relationships, we have
σ0≠σ01þσ02; (8)
andσ∞1≠σ02. If the total chargeability is defined byM= (σ∞−σ0)/σ∞, we haveM1+M2=Mfrom equation (7) Figure 3.Pictures of cut faces of thefive core samples. (a) Valt1 core sample (graphitic schist). (b) Valt2 core sample (gra- phitic schist). (c) ValT3 core sample (carbonized black schist). (d) S1 (graphitic tight sandstone 1). (e) S2 (graphitic tight sandstone 2). All the core samples are from Val Thorens (France) except core sample S2, which is from Mont‐Fort (Swiss Alps, Switzerland). In (a)–(c) the traces of the schist foliations are highlighted by white dashed lines, and mica (m.) and graphite (gr.) pods are annotated.
Table 1
Petrophysical Parameters of the Core Samples
Sample Type Origin CEC (meq/100 g) Porosityϕ(‐) σ′(S/m) at 1 Hz (20 °C) φm(‐)
ValT1a Graphitic schist (zone 1) French Alps (Val Thorens) 2.1 0.042 0.00037 0.117
ValT1b Graphitic schist (zone 1) French Alps (Val Thorens) 2.1 0.042 0.015 0.117
ValT2a Graphitic schist (zone 1) French Alps (Val Thorens) 2.4 0.016 0.000687 0.089
ValT2b Graphitic schist (zone 1) French Alps (Val Thorens) 2.4 0.016 0.00171 0.089
ValT3 Schist carbonized French Alps (Val Thorens) 5.6 0.147 39.64 Unknown
S1 Graphitic tight sandstone French Alps (Val Thorens) 1.2 0.02 0.000151 0.054
S2 Graphitic tight sandstone Switzerland Alps (Mont‐Fort) 0.9 0.026 0.000505 0.031
Note. S1 and S2 correspond to two tight sandstones. Samples ValT1, ValT2, and ValT3 correspond to three graphitic schists. The volume fraction of graphite is determined from equation (18), which is validated by the data set shown in Figure 6c. The in‐phase conductivity is reported at a pore water conductivity of 0.019 S/m (25 °C).
and since 0≤M≤1, we have 0≤M1+M2≤1. In our model, one of the two mechanisms of polarization (for instance 1) would correspond to the polarization of graphite while the second mechanism (therefore mechanism 2) would correspond to the polarization of the electrical double layer of the insulating grains Figure 4.Sample ValT1. (a) Backscattered electron (BSE) image and (b–f) element maps acquired by electron dispersive spectroscopy (EDS) on a Zeiss Sigma SEM, operating at 8.5‐mm WD, 15‐keV accelerating voltage. Mapped element is labeled at bottom left of each image, and mineral phases consistent with the mapped mineralogy are also labeled.
Photomicrographs in (g and i) plane‐polarized light (PPL) and (h and j) reflected light highlight the two microstructures that dominate in these samples. (g and h) Mode 1: variable thickness and sometimes microfolded layers comprising large euhedral graphite (gr) grains surrounded by quartz (qtz) and chlorite (cht) pressure fringes. (i and j) Mode 2: disseminated graphite plus phyllosilicates (phy) with a weak foliation.
forming the skeleton of the background material (Figure 2). A third polar- ization mechanism would correspond to the Maxwell‐Wagner polariza- tion (see Appendix A). The goal of the next subsections is to describe a model of polarization for these two components. In our experiments, we will focus however on thisfirst contribution associated with graphite since a small amount of graphite (say 1% vol.) is enough to mask the back- ground polarization.
2.2. The Contribution Associated With the Background
The instantaneous conductivity is the conductivity of the material just after the application of an electricalfield (see Figure 2; high‐frequency asymptotic conductivity in frequency domain induced polarization).
The DC or stationary conductivity denotes a smaller conductivity including the effect of charge carrier blockage responsible for polariza- tion (Figure 2). Using a volume‐averaging approach, Revil (2013) obtained the following expressions for the high‐and low‐frequency con- ductivities of the background material (i.e., for the instantaneous con- ductivity and DC conductivity of the background material; see Figure 2),
σ∞2 ¼1
Fσwþ 1
Fϕ ρgBCEC; (9)
σ02¼1
Fσwþ 1
Fϕ ρgðB−λÞCEC: (10) In these equations, σw (in S/m) denotes the pore water conductivity (which depends on both salinity and temperature), F (dimensionless) the intrinsic formation factor related to the connected porosity ϕ (dimensionless) by thefirst Archie's lawF=ϕ‐mwherem(dimension- less) is called the first Archie exponent or porosity exponent (Archie, 1942), ρg is the grain density (in kg/m3, usually ρg = 2,650 kg/m3), and the cation exchange capacity (CEC) denotes the cation exchange capacity (in C/kg and often expressed in meq/100 g with 1 meq/100 g = 963.20 °C/kg). The CEC is mainly sensitive to the clay type (e.g., kaolinite, illite, smectite) and the weight fraction of these clay minerals in the rock. In equations (9) and (10), the quantity B (expressed in m2·s−1·V−1) denotes the apparent mobility of the counter- ions for surface conduction and the quantityλ (expressed in m2·s−1·V
−1) denotes the apparent mobility of the counterions for the polariza- tion associated with the quadrature conductivity (see Revil et al., 2017, and references therein). A dimensionless number R is also intro- duced with
R¼λ
B; (11)
(see Revil et al., 2017 for further explanations). This number is intro- duced because it corresponds to the maximum value of the chargeabil- ity in absence of metallic particles. From previous studies (e.g., Ghorbani et al., 2018), we have Β (Na+, 25 °C) = 3.1 ± 0.3 × 10–
9 m–2·s–1·V–1 and λ (Na+, 25 °C) = 3.0 ± 0.7 × 10–10 m–2·s–1·V–1. These two quantities have been determined using large data sets of rock samples (including hundreds of core samples) and are therefore Figure 5.Representative Raman spectrum of graphite. D1, G, and D2 bands
arefitted and illustrated. R2ratio and the estimated maximum metamorphic temperature (371 °C) are also provided.
Figure 6.Equipment for the complex conductivity measurements. (a) Thermostat bath KISS K6 used for the temperature measurements.
Dimensions 210 × 400 × 546 mm: bath volume 4.5 L. (b) Impedance meter ZEL‐SIP04‐V02 used for complex conductivity meter (see Zimmermann et al., 2008, for further details). (c) Chargeability versus graphite content for seven mixes of graphite and silica sand. The chargeabilityMwas calculated from equation (3) by replacing the measured in‐phase conductivity at the high and low frequency (10 kHz and 10 mHz). In this case, the sand used for the background had a very small chargeability (~10−3).
reliable. It follows that the dimensionless numberR is typically around 0.09 ± 0.01 (independent of the temperature and saturation; Revil, Coperey, Deng, et al., 2018).
From equations (3), (9), and (10), we obtain the following expression of the chargeability of the background,
M2¼ ρgλCEC
ϕσwþρgBCEC: (12)
This equation shows explicitly the dependence between the background chargeabilityM2, the pore water conductivity, and the CEC. We also have the propertyM2≤R~0.09 ± 0.01. Therefore, the background char- geability is generally quite small (less than 10% or 100 mV/V).
The Cole‐Cole time constantτ2is associated with a characteristic pore sizeΛ(in m; see Revil & Florsch, 2010; Revil et al., 2012) according to
Figure 7.Complex conductivity spectra of samples S1 and S2. (a) In‐phase conductivity of sample S1. (b) Quadrature con- ductivity of sample S1. (c) In‐phase conductivity of sample S2 (the phase reaches ~100 mrad). (d) Quadrature conductivity of sample S2 (the phase reaches ~80 mrad at the peak frequency). The plain lines represent thefit of the data with a double Cole‐Cole as described in Appendix A. The Cole‐Cole parameters are reported in Tables 2–7. We did not try tofit the high frequencies because of electromagnetic effects and we did not try to interpret the Maxwell‐Wagner
polarization for the same reason. Only the data with an uncertainty smaller than 1% were considered in the Cole‐Colefit of the data.
τ2¼ Λ2
2DSð Þþ ; (13)
whereDSð Þþ denotes the diffusion coefficient of the counterions in the Stern layer (in m2/s). The value of this diffusion coefficient DSð Þþ should relate to the mobility of the counterions in the Stern layer,βSð Þþ, by the Nernst‐Einstein relationship,
DSð Þþ ¼kbTβSð Þþ
qð Þþ
; (14) where T denotes the absolute temperature (in K), kb denotes the Boltzmann constant (1.3806 × 10−23 m2 kg s−2 K−1), and |q(+)| is the charge of the counterions in the Stern layer coating the surface of the grains (|q(+)| =ewhereedenotes the elementary charge for Na+). Since each relaxa- tion time depends on a pore size, the Cole‐Cole exponent c2 measures the broadness of the pore size distribution.
Figure 8.Complex conductivity spectra of sample ValT2. (a) In‐phase conductivity (transverse component). (b) Quadrature conductivity (transverse component). The phase reaches ~200 mrad at its maximum. (c) In‐phase conduc- tivity (in‐plane component). (d) Quadrature conductivity (in‐plane component). The phase reaches ~300 mrad at the peak frequency. The plain lines represent thefit of the data with a double Cole‐Cole as described in Appendix A. The Cole‐Cole parameters are reported in Tables 2–7. The linear scale was used to better compare the data with the Cole‐Colefit. Only the data with an uncertainty smaller than 1% were considered in the Cole‐Colefit of the data.
2.3. The Contribution Associated With Graphite
For a mixture of nonmetallic grains, pore water, and metallic grains, Revil, Florsch, et al., (2015), Revil, Abdel Aal, et al., (2015) have demonstrated that the chargeability of the mixture is given by
M¼9
2φmþM2; (15)
whereφmdenote the volume fraction of metallic particles in the medium. In addition, the instantaneous and the steady state conductivities of the mixture are related to the instantaneous and steady state conductivities of the background material according to (Revil, Florsch, et al., 2015)
σ∞¼σ∞2ð1þ3φmþ…Þ; (16) σ0¼σ02 1−3
2φmþ…
: (17)
Using equations (5), (6), and (8), this implies that
Figure 9.Complex conductivity spectra of sample ValT1. (a) In‐phase conductivity (transverse component). (b) Quadrature conductivity (transverse component). The phase reaches ~400 mrad at its peak frequency. (c) In‐phase con- ductivity (in‐plane component). (d) Quadrature conductivity (in‐plane component). The phase reaches ~600 mrad at the peak frequency. The plain lines represent thefit of the data with a double Cole‐Cole as described in Appendix A. The Cole‐ Cole parameters are reported in Tables 2–7. Only the data with an uncertainty smaller than 1% were considered in the Cole‐Colefit of the data.
M1¼9
2φm; (18)
σ∞1 ¼3φmσ∞2: (19) Finally, from equations (18) and (19), we have
σ01¼σ∞1 1−9 2φm
¼3φm 1−9 2φm
σ∞2: (20)
The last parameter to obtain is the relaxation timeτ1(in s). It is given by (Revil, Coperey, Mao, et al., 2018)
τ1≈a2e2Cm
kbTσ∞2 ; (21) whereadenotes the diameter of the metallic particle (in m),Cmdenotes the concentration of the charge carriers in the solid metallic particle (in m−3), andethe elementary charge (1.6 × 10−19°C).
2.4. Influence of Temperature and Permafrost
We discuss now the temperature dependence of the complex conductivity above the freezing temperature (typically but not necessarily around 0 °C). Following Vinegar and Waxman (1984), the pore water conductivity and mobilities B and λ have all the following linear temperature dependence:
Figure 10.Spectral induced polarization of sample ValT3 (black schist). (a) In‐phase conductivity spectra in the direction and normal to the foliation plane. (b) Phase spectra in the direction and normal to the foliation plane. Note the high conductivity and small polarization along the foliation plane. This is consistent with graphite above a percolation threshold in this direction. Note the low conductivity and high polarization normal to the foliation plane consistent with disconnected graphite in this direction.
Figure 11.Electrical conductivity at 1 Hz of sample ValT3 (black schist) along the foliation plane. This sample exhibits an exceptionally high elec- trical conductivity (comprised between 30 and 40 S/m), quite temperature independent, and likely associated with graphite being above a percolation threshold through the core sample. The error bars are here determined from the uncertainty associated with the three cycles used at each frequency.
Θð Þ ¼T Θð ÞT0 ½1þαTðT−T0Þ; (22) whereT0 and Tare the reference temperature (T0 = 25 °C) and the temperature (in °C), respectively;
Θ(T) corresponds toσw(T),B(T), orλ(T); andΘ(T0) corresponds to the same property atT0, and the sen- sitivityαTis in the range 0.019–0.022/°C (e.g., Revil et al., 2017). According to equation (22), the conduc- tivity goes to zero at a temperature of−25 °C, remarkably close to the so‐called eutectic temperatureTE
close to −21 °C for NaCl. Actually the eutectic temperature is exactly predicted for αT = 1/(TE +T0) = 0.0217/°C. Reaching the eutectic temperature leads to the simultaneous crystallization of ice and salt. In equation (22), the temperature dependence of the pore water conductivity is controlled by the temperature dependence of the ionic mobilities of the cations and anions. Taking equations (9) and (10) (together with equation (22)) into equations (1)–(4), the temperature dependence of the complex con- ductivity (i.e., in‐phase and quadrature conductivities) is therefore imposed by the thermal dependence of charge carrier mobilities.
Equations (2)–(4) need to show explicitly the dependence of the different properties with the water content.
Assuming that thefirst and second Archie exponents are equal to each other, that is,n=m(see Revil, 2013), and assuming the segregation of the salt in the liquid water phase, these equations can be written as
σ∞2 ¼θm−1ϕσwþρgBCEC
; (23)
σ02¼θm−1hϕσwþρgðB−λÞCECi
: (24)
Below freezing conditions, the water content of the liquid water is changing with the temperature according to a freezing curve. Duvillard et al. (2018) and Coperey et al. (2019) used an exponential freezing curve for the volumetric water contentθ(T) written as
Table 2
Sample ValT1a, Cole‐Cole Parameters Resulted From Fitting the Laboratory Measurements With Double Cole‐Cole Model
Temperature (°C) σ∞(S/m) M1(‐) MMW(‐) c1(‐) cMW(‐) τ1(s) τMW(s) RMS (%)
20 2.18E−03 0.569 0.278 0.67 0.21 9.81E−07 4.41E−07 2.72E−01
15 2.01E−03 0.581 0.275 0.70 0.21 1.13E−06 4.46E−07 2.86E−01
10 1.95E−03 0.588 0.285 0.74 0.22 1.12E−06 3.08E−07 3.03E−01
5 1.90E−03 0.663 0.228 0.67 0.20 7.54E−07 2.96E−07 2.73E−01
2 1.75E−03 0.617 0.277 0.70 0.20 1.05E−06 8.77E−08 3.07E−01
0 1.74E−03 0.651 0.250 0.74 0.22 9.75E−07 2.92E−07 2.52E−01
−2 7.24E−04 0.751 0.226 0.47 0.02 3.12E−06 1.08E−07 1.07E+00
−5 4.52E−04 0.528 0.360 0.52 0.31 1.16E−05 2.55E−07 1.56E+00
−8 2.99E−04 0.495 0.399 0.52 0.43 2.28E−05 2.09E−06 1.74E+00
−10 1.90E−04 0.477 0.410 0.53 0.67 8.99E−05 5.55E−06 1.92E+00
−15 1.43E−04 0.304 0.612 0.58 0.69 3.62E−04 6.19E−06 1.73E+00
Table 3
Sample ValT1b, Cole‐Cole Parameters Resulted From Fitting the Laboratory Measurements With Double Cole‐Cole Model
Temperature (°C) σ∞(S/m) M1(‐) MMW(‐) c1(‐) cMW(‐) τ1(s) τMW(s) RMS (%)
20 1.00E−01 0.551 0.380 0.60 0.62 1.31E−02 8.34E−06 3.62E−01
15 9.99E−02 0.515 0.428 0.60 0.53 1.33E−02 6.17E−06 3.73E−01
10 9.98E−02 0.463 0.488 0.59 0.52 1.38E−02 4.35E−06 3.71E−01
5 9.93E−02 0.511 0.443 0.60 0.56 1.17E−02 9.45E−06 3.86E−01
2 9.51E−02 0.496 0.463 0.59 0.56 1.18E−02 6.53E−06 3.42E−01
0 9.99E−02 0.501 0.460 0.60 0.63 1.18E−02 9.27E−06 5.62E−01
−2 9.99E−02 0.515 0.449 0.59 0.64 1.12E−02 1.15E−05 3.38E−01
−5 9.17E−02 0.501 0.465 0.59 0.63 1.19E−02 9.97E−06 2.95E−01
−8 8.98E−02 0.459 0.514 0.59 0.76 1.20E−02 1.24E−05 3.22E−01
−10 7.61E−02 0.452 0.519 0.59 0.62 1.59E−02 1.34E−05 2.82E−01
−15 4.77E−02 0.455 0.515 0.61 0.49 2.55E−02 3.37E−05 2.73E−01
θð Þ ¼T ðϕ−θrÞexp −T−TF
TC
þθr;T≤TF
ϕ; T>TF
8<
: ; (25) which has the advantage to only require three parameters,TF(the liquidus or freezing point),TC(a charac- teristic temperature), andθr(the residual water content).
3. Laboratory Experiments
3.1. Core Sample Characterization
We consider in this study four samples from the French Alps (Val Thorens, ~2,850 m above sea level) and one sample from the Swiss Alps (Mont Fort, 3,328 m above sea level). Three of these samples are graphitic (black) schists (ValT1, ValT2, and ValT3) and two are graphitic tight sandstones (S1 and S2). The complex conductivity of the two graphitic schists (ValT1 and ValT2) was measured both normal to foliation plane (experiments labeled ValT1.a and ValT2.a) and along it (experiments labeled ValT1b and ValT2b). The sam- ples are shown in Figure 3.
The samples werefirst dried for 24 hr at about 50 °C and then saturated under vacuum with NaCl solutions of known conductivity (σw= 0.019 S/m at 25 °C; see Woodruff et al., 2014, for additional details regarding this procedure). The core samples were left in this solution for two weeks to complete saturation. The com- plex conductivity spectra were obtained over the temperature range of +20 °C to−15 °C. The samples were put in a bag and immersed in a thermally controlled bath (Kiss K6 from Huber). For each experiment, sam- ple and bath temperatures were recorded every minute with thermocouples in contact with the core sample and one sensor placed directly in the bath. We use K‐thermocouples and a data logger CR1000 from Campbell scientific for the thermal monitoring.
Table 5
Sample ValT2b, Cole‐Cole Parameters Resulted From Fitting the Laboratory Measurements With Double Cole‐Cole Model
Temperature (°C) σ∞(S/m) M1(‐) MMW(‐) c1(‐) cMW(‐) τ1(s) τMW(s) RMS (%)
20 3.92E−03 0.485 0.164 0.84 0.54 3.87E−02 6.55E−04 7.05E−01
15 2.70E−03 0.438 0.155 0.87 0.53 4.32E−02 5.03E−04 8.51E−01
10 2.01E−03 0.425 0.184 0.84 0.48 5.52E−02 1.93E−04 7.52E−01
5 1.48E−03 0.400 0.220 0.81 0.44 7.61E−02 8.95E−05 6.40E−01
2 1.24E−03 0.407 0.202 0.80 0.47 9.49E−02 8.19E−05 5.24E−01
0 1.14E−03 0.388 0.239 0.78 0.45 1.10E−01 3.28E−05 4.52E−01
−2 9.83E−04 0.420 0.212 0.77 0.59 1.31E−01 7.06E−05 3.05E−01
−5 8.13E−04 0.362 0.329 0.77 0.44 1.87E−01 5.24E−05 1.81E−01
−8 5.00E−04 0.445 0.202 0.75 0.61 2.41E−01 4.05E−04 1.60E−01
−10 3.97E−04 0.495 0.142 0.73 0.80 2.71E−01 1.04E−03 3.93E−01
−15 2.24E−04 0.492 0.155 0.72 0.82 4.66E−01 1.88E−03 5.67E−01
Table 4
Sample ValT2a, Cole‐Cole Parameters Resulted From Fitting the Laboratory Measurements With Double Cole‐Cole Model
Temperature (°C) σ∞(S/m) M1(‐) MMW(‐) c1(‐) cMW(‐) τ1(s) τMW(s) RMS (%)
20 2.47E−03 0.367 0.417 0.30 0.69 9.53E−04 2.92E−05 9.87E−02
15 2.31E−03 0.354 0.460 0.32 0.69 1.21E−03 2.56E−05 9.11E−02
10 2.00E−03 0.371 0.451 0.31 0.70 1.10E−03 3.12E−05 9.82E−02
5 1.86E−03 0.323 0.520 0.31 0.67 1.61E−03 2.43E−05 9.66E−02
2 1.67E−03 0.375 0.474 0.30 0.72 8.96E−04 2.80E−05 8.82E−02
0 1.55E−03 0.374 0.478 0.30 0.72 9.91E−04 2.93E−05 9.54E−02
−2 1.56E−03 0.321 0.550 0.31 0.69 1.39E−03 2.25E−05 9.64E−02
−5 1.14E−03 0.390 0.504 0.29 0.78 4.98E−04 2.84E−05 9.28E−02
−8 9.31E−04 0.412 0.497 0.28 0.78 3.06E−04 2.86E−05 9.96E−02
−10 8.18E−04 0.395 0.529 0.27 0.78 2.74E−04 2.74E−05 1.05E−01
−15 6.70E−04 0.332 0.613 0.27 0.79 2.76E−04 2.29E−05 1.10E−01
