Electrical and thermal behavior of unsaturated soils: experimental results Marie Nouveau, Gilles Grandjean, Philippe Leroy, Mickael Philippe, Estelle Hedri, Hassan Boukcim PII: DOI: Reference:

S0926-9851(16)30073-8 doi: 10.1016/j.jappgeo.2016.03.019 APPGEO 2942

To appear in:

Journal of Applied Geophysics

Received date: Revised date: Accepted date:

16 July 2015 17 February 2016 23 March 2016

Please cite this article as: Nouveau, Marie, Grandjean, Gilles, Leroy, Philippe, Philippe, Mickael, Hedri, Estelle, Boukcim, Hassan, Electrical and thermal behavior of unsaturated soils: experimental results, Journal of Applied Geophysics (2016), doi: 10.1016/j.jappgeo.2016.03.019

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

Electrical and thermal behavior of unsaturated soils:

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experimental results

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Marie Nouveau1, Gilles Grandjean1, Philippe Leroy1, Mickael Philippe1, Estelle Hedri2, Hassan Boukcim2 1

BRGM, Orléans, France

2

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Valorhiz, Montpellier, France

Abstract – When soil is affected by a heat source, some of its properties are modified, and in

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particular, the electrical resistivity due to changes in water content. As a result, these changes affect the thermal properties of soil, i.e., its thermal conductivity and diffusivity. We experimentally examine the changes in electrical resistivity and thermal conductivity for four soils with different grain size

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distributions and clay content over a wide range of temperatures, from 20 to 100 °C. This temperature

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range corresponds to the thermal conditions in the vicinity of a buried high voltage cable or a geothermal system. Experiments were conducted at the field scale, at a geothermal test facility, and in

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the laboratory using geophysical devices and probing systems. The results show that the electrical resistivity decreases and the thermal conductivity increases with temperature up to a critical temperature depending on soil types. At this critical temperature, the air volume in the pore space increases with temperature, and the resulting electrical resistivity also increases. For higher

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temperatures , the thermal conductivity increases sharply with temperature up to a second temperature limit. Beyond it, the thermal conductivity drops drastically. This limit corresponds to the temperature at which most of the water evaporates from the soil pore space. Once the evaporation is completed, the thermal conductivity stabilizes. To explain these experimental results, we modeled the electrical resistivity variations with temperature and water content in the temperature range 20 - 100°C, showing that two critical temperatures influence the main processes occurring during heating at temperatures below 100 °C. Key words: soils, thermal conductivity, electrical resistivity, moisture, experimental setup Nomenclature: : Campbell’s law delta parameter (% °C-1) : Soil porosity (%)

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

: Effective electrical resistivity (.m)  : Electrical resistivity of the pore water (.m)

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: Volumetric water content (%)

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a: Tortuosity factor (dimensionless)

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D: grain diameter (mm) F: Electrical formation factor (dimensionless)

n: Saturation exponent (dimensionless)

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m: Cementation exponent (dimensionless) R : RMS value (dimensionless)

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Sw: Moisture or water content (%) T, T1, T2: Temperature (°C)

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K: Thermal conductivity (W m-1 K-1)

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

Introduction Knowing the thermal properties of a soil is useful in many geo-environmental and industrial applications, including geothermal energy and the use of buried electrical cables to meet a growing

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demand for energy worldwide (Drefke et al., 2015, Jorand et al., 2015; Spitler and Bernier 2011).

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Buried electrical cables have to carry an electrical power level defined by the needs of customers but are constrained by the physical properties of the cable and its surroundings. This maximal power

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defines the cable specifications, which depend on the discharge capacity of the energy transformed into heat by the Joule effect and transmitted to the soil. This transmission capacity depends on the thermal interactions between the cable and the soil. A soil with a low thermal conductivity and

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diffusivity will have difficulty with evacuating the heat from the cable, thus leading to possible overheating of the cable. Therefore, to avoid this problem, the thermal behavior of a soil must be

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known accurately (Oclon et al., 2015; Salata et al., 2015).

When the soil is affected by a heat source, some of its properties are modified. In particular, the electrical resistivity due to changes in water content (Cosenza et al, 2003; Revil, 2000, 1999). As a

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result, these changes affect the thermal properties, i.e., the thermal conductivity and diffusivity, which

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allow the heat to be evacuated far from the heat source (Guéguen & Palciauskas, 1992; DeVries, 1953). These processes can reach a high level of complexity, depending on the structural organization of soils that can be composed of grains that are extremely different (from a physical and chemical

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point of view) surrounded by an aqueous solution containing a large variety of ions (Mc Quarrie, et al., 1992; Revil, et al., 1998). Understanding how these components vary according to the temperature conditions is an important challenge at the frontier of physics and chemistry.

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The changes in water content and of the thermal properties of a soil submitted to a temperature gradient are usually measured in the laboratory, but in some cases, sampling is not possible, and these changes are monitored in the field using borehole measurements (Silliman and Neuzil, 1990). Nevertheless, these measurements are not necessarily accurate and can instead be destructive and expensive (Revil, 2000). Another way to monitor these changes consists of using non-invasive and non-destructive geophysical methods (Revil, 1999; Cosenza et al., 2003). Indeed, these methods can provide information related to the macroscopic behavior of soil, integrating what happens at the microscopic scale, which is too complex to be directly observed in the field. Several studies have shown the contributions of these methods and what they can provide to interpret processes related to hydrologically (Archie, 1942), mechanically (Mc Quarrie, et al., 1992; Schön, 1996) or thermally (Cosenza, et al., 2003; Guéguen & Palciauskas, 1992; DeVries, 1963) varying conditions. In particular, Electrical Resistivity Tomography (ERT) was widely used in the last few decades to monitor water content variations in soils (Archie, 1942; Montoroi, et al., 1997; Schneider, 2010). In some cases, authors also show the impact of temperature changes on the electrical resistivity 3

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

(Campbell et al., 1949)Generally, temperature effects are removed from the measured  values in order to make the experiment independent from daily or seasonal variations (Samouelian et al., 2005). However, electrical resistivity measurements can also be used to estimate the thermal properties of a soil because of the relationship between electrical and thermal conductivity (Revil et al., 2000;

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Cosenza et al., 2003; Jougnot and Revil, 2010).

Different studies have attempted to demonstrate an empirical link between the electrical resistivity and

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thermal conductivity of soils (Sreedeep, et al., 2005). Even if a mathematical link exists between electrical resistivity and the thermal conductivity of metals (Wiedemann & Franz, 1853), it is still difficult to establish such a relation for soils because their complex characteristics (e.g., pore network

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structure and connectivity, mineralogical heterogeneity, chemical composition of pore water) are difficult to take into account. The objective of this work is to i) study, from an experimental point of view, the changes in electrical resistivity (inm and thermal conductivity (K in W m-1 K-1) of four

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soils over a wide range of temperatures (T in °C) corresponding to the heat production of a buried high voltage cable or geothermal system from 10 to 100 ° C and ii) understand the impact of decreased water saturation on these properties. Available data demonstrate the link between these quantities, but

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they are either not dedicated to soil materials (Wiedemann & Franz, 1853), do not consider very high

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temperature effects (T > 90°C) (Ma, et al., 2011; Tarnawski, et al., 2000), or separately consider the electrical resistivity and thermal conductivity measurements of the soils (Cosenza, et al., 2003; Revil,

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2000; Tarnawski, et al., 2000; Ma, et al., 2011).

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Experiments and results We first decided to work in the framework of an experimental geothermal platform giving access to a complete control of the physical conditions and measuring systems (Philippe, 2010). At this site, an electrical Resistivity tomography (ERT) device was operated in time-slice mode (resistivity measurements every three hours during a total time of 8 days), while an injection of a hot fluid warmed a part of the soil’s platform. Because of the particularities of this platform, maximal temperatures of 35°C were reached in the underground, but some temperature effects were nevertheless observed. In the next section, we present this “low temperature experiment”, detailing the setup, the data and the results in terms of the impact of warming on the soil physical properties. However, because the impacts of high temperature were not evaluated in this case, we completed our study with several laboratory tests where we measured these temperature effects for values higher than 90°C and for four types of soils (different particle size distributions and clay contents). In these conditions of the “high temperature experiments”, the water saturation (Sw in %) becomes an

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

important factor influencing the soil physical and chemical properties. Finally, several experimental curves showing the relationships between , K and T were produced and are discussed. In addition, we attempted to explain for the simplest cases that these correlations can be described by physical laws

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already proposed in the literature.

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1-Low temperature experiments

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The objective of this first series of tests was to observe the evolution of the electrical behavior of soils when the temperature varies in real conditions. The geothermal platform of BRGM (Orléans City, France) was used to set up this experiment because it offers all of the necessary infrastructure: a geothermal system associated with thermodynamic machinery, soil temperature measurements,

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piezometers, and a meteorological station. All of these components are controlled from a cabin dedicated to monitoring systems and data management (Fig. 1; Philippe, 2010). The heating is generated by horizontal exchangers arranged in serpentine, wherein a heated fluid (mono-propylene

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glycol / water mixture) flows at 1 m depth from the geothermal machinery. This machinery system controls the temperature inside the heat exchanger pipes in a range of -5 to 45°C. Optical fibers are installed throughout these exchangers at 0.5 m, 1 m and 1.5 m depths. A Raman distributed

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temperature sensing (DTS) device measured the temperature during the experiments in the 0.5 – 1.5 m

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depth range. The network of horizontal heating pipes (horizontal geothermal exchangers) is spread over a sandy-gritty soils well known as Sologne sand. This sand contains approximately 5% clay and

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33% coarse sand (> 2 mm) and has a porosity of approximately 25%. During our experiment, the ERT device was installed over a horizontal thermal exchanger (AB profile; Fig. 1). One-half of the profile was set on heated soils, whereas the other part of the profile was set on

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unheated soils; the latter situation was used as a reference. The electrical measurement system was composed of 48 electrodes with a spacing of 25 cm, and according to a dipole-dipole configuration: two electrodes, separated by a constant l spacing (m), are used to inject current into the ground, and two additional electrodes, also separated by l spacing, are moved along the survey line at distances from the current electrodes at multiples of the l spacing. Measurements were realized in time-lapse mode every three hours for a total time of 8 days. The resistivity measurements were synchronized with the DTS temperature monitoring. Vertical profiles of moisture content were also measured according to the boreholes available on the site. In addition, a thermocouple temperature sensor was set close to the meteorological station and at 3 cm depth in order to estimate the surface temperature, which experiences daily variations. The ERT profiles were processed and inverted with the Res2DInv software (Loke & Dahlin, 2002) to compare real resistivity cross-sections. The measurements started at a reference time (t0: 2013/05/27 at 3 pm) and were repeated every 3 hours for 8 days (up to 2013/06/04 at 8 am).

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Fig. 2a shows the inverted ERT profile obtained for maximal heating at time t: 2013/06/03 at 2 pm; Fig. 2b corresponds to the difference between this profile and that obtained at time t0. The maximal temperature reached in the heated part is approximately 34°C, whereas it stays at approximately 15°C elsewhere. In the same way, the electrical resistivity varies from 130 Ω m to 160 Ω m when moving

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from the heated to the non-heated part (positions from 3 to 7 m) at 1 m depth. The relationship

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between temperature T and electrical resistivity  is then observed: increasing temperature leads to

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decreasing resistivity. During the heating, some intermediate resistivity measurements showed a curve with quasi-linear behavior for higher temperatures (20 °C  T  30 °C; Fig. 3). This quasi-linear behavior of the  versus T relationship for higher temperatures can be described by the following

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equation (Campbell, et al., 1949):

Eq.1

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where  is the electrical resistivity at the temperature T, 25 is the reference electrical resistivity taken at T = 25 °C and  is the slope compensation (%°C-1). Generally, in the case of sandy soils,  is evaluated to approximately 2% °C-1 for weak temperature corrections (Campbell et al., 1949). In our

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case,  is estimated around 3 - 3.5% °C-1 in the last part of the curve (where the slope reaches 6.3). Our

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higher calculated value of may be due to the use of inverted resistivity data, which may present some uncertainties compared to values recorded in laboratory experiments. To study this issue in a more

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complete way, in the next section, we propose a new experimental setup, realized in the laboratory in controlled conditions. Electrical resistivity and also thermal conductivity values were measured to understand how they are correlated and how they vary for high temperature ranges (20 to 100 °C) and

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according to soil type and water content.

2-High temperature experiments After observing the evolution of the electrical resistivity of a soil up to 35 °C, we proposed new experiments for studying the behavior of the electrical resistivity and thermal conductivity of soils in a higher temperature range (15 – 100 °C). For this purpose, various monitoring devices were used in controlled conditions to measure several physical parameters: electrical resistivity (), thermal conductivity (K), temperature (T) and moisture (Sw). Due to the difficulties in correctly integrating measuring sensors inside the warming system, in the first stage, we developed a protocol to measure  and K parameters versus temperature for different types of soils. Then, we focused our attention on a specific soil to measure the effects of moisture on the electrical resistivity variations during the warming. a-Measurements of  and K vs. T 6

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

Four types of soils (Tab. 1), described according to the GTR classification (LCPC-SETRA, 1992), were sampled and prepared for the experiments. They were selected for different reasons: -

BRGM: this soil is the BRGM’s geothermal platform (Philippe, 2010)and refers to the in situ

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experiment described in the first section; This soil is classified as GTR B4 and is mainly

-

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composed of clayey sand (median grain diameter D50 = 1.331 mm);

Fontainebleau: this material is a white pure sand and is classified as GTR D1/B1 (D50 = 204

-

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µm);

Lauret: this soil is mainly composed of sandy and silty clays and classified as GTR C1A2 (D60 = 20 µm);

Cheverny: this soil is essentially composed of clays and is classified as GTR A4 (D80 = 80

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-

µm).

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Soil samples were prepared in the same way to avoid all possible artifacts during measurements. In particular, we removed coarse elements (gravels, stones) bigger than 2 cm; we stabilized the water content and compaction of all samples according to the Proctor test so that each sample could be considered in the same physical state. A TDR (Time Domain Reflectometry) probe was used to

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measure water content at the beginning and end of the experiment. This device was unable to resist

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high temperatures and was thus not placed inside the oven.

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The laboratory setup including the different sensors is shown in Fig. 4. A Wenner electrical quadrupole (A-B-M-N) was set in the middle of the bowl containing the soil samples (Fig. 4a) and was placed in an oven (Fig. 4b). The temperature (Pt100 probe) and thermal conductivity sensors (KD2 Pro with SH1 device) were set aside from the electrical device to avoid any interactions during the

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measurements. The acquisition system driving the different sensors and recording the measurements was located outside of the oven. We started measurements at ambient temperature (15 °C); then the oven was programmed to 130 °C, to reach gradually (+4°C per hour) more than 100 °C inside the soil samples. Measurements were taken regularly, every ~5mn, for a total time ranging from 20 to 24 hours. Each measurement was carried out over a time window of one minute to be sure the stationary conditions are reached. The results are presented in Figure 5. The variations in electrical resistivity  and thermal conductivity K with temperature T are presented in Figures 4a and 4b, respectively, for the different soil types considered. The visibility of the curves of the different soils changes depending on the nature of the soil: it is attenuated for sandy shale and shale (Lauret and Cheverny soils), where amplitudes of electrical and thermal curves are strongly reduced compared to other types of soils. However, these changes in electrical and thermal curves are well pronounced for Fontainebleau sand and BRGM’s platform clayey sand. In the case of Fontainebleau sand, the poor cohesion of the sand leads to contact issues between the sand and the measurement probes, causing the measurement of thermal 7

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

conductivity to be more difficult than for other soil types. As a consequence, the curve of the thermal conductivity of the Fontainebleau sand appears to be noisier and less regular than those of other soils. In these curves,  decreases and K increases with T, until a temperature value that depends on the soil type is reached (Fig. 5). From this particular temperature (T1),  stops decreasing and K increases

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more with temperature. Then, after a second temperature (T2) is reached,  increases and K decreases

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sharply with temperature. These two temperatures are called the critical temperatures (T1 and T2,

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respectively). b-Measurement of  and Sw vs T

To understand the relationship between soil electrical resistivity and its physical conditions, additional

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measurements were performed on the BRGM’s sandy soil in the laboratory. More specifically, this experiment aimed to model the evolution of the soil electrical resistivity with respect to both

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temperature and water content. The experimental setup is similar to the experimental setup used for the previous experiment. A TDR sensor recorded the soil water content at different temperatures. The results are shown in Fig. 6. The electrical variations with temperature are close to the values depicted

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in Fig. 5. Nevertheless, the higher water content of the BRGM’s soil during the water content and

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resistivity measurements is responsible for the lower soil resistivity values compared to the resistivity values presented in Fig. 5. The water content drops drastically when the temperature reaches a value of 84 °C. This temperature corresponds to the T1 limit observed in Fig. 5 and to the temperature where

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the resistivity shows a minimum value. The T2 limit is reached at the end of the observed data at approximately 96 °C, and corresponds to a considerable increase in the resistivity.

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Interpretation and discussion 1-Low temperature experiments During the BRGM’s geothermal platform experiment, the electrical resistivity decreased with increasing temperature, whereas no water content variations were observed (Fig. 2 and Fig. 3). This suggests that the highest temperature reached during the warming was not high enough to reach the liquid-vapor transition of water. The variations in electrical resistivity with temperature are consistent with Campbell’s law (Campbell, et al., 1949; Eq.1), which states that the decrease in resistivity is typically a few percent per degree Celsius. Campbell et al. (1949) estimated the change to be approximately 2% °C-1, but in our case, the compensation slope has a value of 3.5% °C-1, which does not match the values found in the literature for the BRGM’s soil (e.g., 1.9% °C-1    2.5% °C-1) (Ma, et al., 2011). This difference in the value of the compensation slope may be explained by our use of inverted resistivity data from the field to estimate the compensation slope, whereas compensation slopes reported in the literature are based on resistivity values of soils recorded in the laboratory. 8

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

Inverted resistivity data from field measurements present more uncertainties than the resistivity values recorded in the laboratory because of the variation in the solution in the inversion process (Samouelian et al., 2005). In addition, the increase in the molecular agitation due to the decrease in the water viscosity with temperature is responsible for the decrease in the water resistivity and resulting soil

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resistivity with temperature (Revil, 2000; Samouelian et al., 2005; Zimmerman et al., 2012). This difference might be explained by the fact that our measurements were performed in laboratory

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conditions, which are able to impose high temperatures, instead of field conditions, where temperatures are limited to a maximum of approximately 35°C.

2-High temperature experiments

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In the resistivity curves,  decreases with temperature up to a first critical temperature that depends on the soil type (Fig. 5a). This behavior of the electrical resistivity can be easily observed for the BRGM

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soil, where a critical temperature T1 equal to approximately 85 °C is observed. A similar critical temperature is also observed for clayey, Lauret and Cheverny soils, but these soils, which contain more conductive clays, are less resistive and exhibit smaller resistivity variations compared to the

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BRGM soil. In addition, the Fontainebleau sand exhibits a lower critical temperature (T1 = 70 °C,

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approximately). This difference of T1 between Fontainebleau and the three other soils may be explained by the presence of microporous clays in the three other soils. Clays are characterized by high specific surface areas (between 10 and 1000 m2 g-1; Revil and Leroy, 2004) and the presence of

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charged nano/micropores (of nanometric to micrometric thickness) that can retain a large quantity of water compared to larger macropores (of micrometric to millimetric thickness; Jougnot and Revil, 2010). Therefore, the water content of Fontainebleau sand may decrease more rapidly with

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temperature (due to water evaporation) compared to the water content of the other soils containing clays. This causes the resistivity of Fontainebleau sand to increase at lower temperatures compared to the resistivity of the other soils (due to the increasing air volume). In addition, the compensation slopes ( calculated for the different soils (Eq. 1) are roughly equal to 2% in the temperature range of 20 °C - 50 °C, confirming that the higher values obtained in the field ( = 3.5% °C-1) may be due to the uncertainties associated with the inversion process of the apparent resistivity data. When temperature increases beyond the critical temperature T1, the electrical resistivity stabilizes (Fig. 5a) and then increases sharply with temperature up to second critical temperature T2, equal to approximately 100°C for the BRGM soil and 80°C for the Fontainebleau sand. The water content may decrease significantly in these temperature conditions (T1  T  T2), leading to increasing resistivity, but the electrical resistivity of the pore water also decreases with temperature because of the resulting decrease in the viscosity of the water and the increase in the molecular agitation (Samouelian et al., 2005). Therefore, when the soils are submitted to temperatures between the two critical temperatures, there may be a competition between two opposite effects of temperature on soil resistivity, leading to a 9

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

stabilization of the resulting resistivity. At the second critical temperature, the soils may lose a large quantity of water because of high water evaporation (Mermoud, 2006). The second critical temperature of Fontainebleau sand is smaller than those of the other soils, probably because Fontainebleau sand possesses only weakly charged macropores that more easily lose their water

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content compared to soils containing clays with charged micropores. In addition, the second critical

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liquid water in atmospheric conditions (Nukiyama, 1966).

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temperature of the BRGM soil (T2 = 100 °C, approximately) corresponds to the boiling temperature of

The behavior of the thermal conductivity of soils can also be characterized by the two critical temperatures previously presented. Indeed, the thermal conductivity of soils increases with

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temperature when the first critical temperature is reached and increases more sharply with temperature when the second critical temperature is reached. Soils having a high sand content present a higher thermal conductivity than soils having a lower sand content because of the high thermal conductivity

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of quartz (K = 8  1 W m-1 °C-1) compared to other minerals, like clays (K = 2  1 W m-1 °C-1; Revil, 2000). When T1  T  T2, the soils may retain most of the initial liquid water in their pore space, and the thermal conductivity of liquid water increases with temperature because of the decrease in the

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water viscosity with temperature (Ramires et al., 1995). In addition, water evaporation may increase

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the thermal conductivity of the soils because of the vapor flow (Hiraiwa & Kasubuchi, 2000). Nevertheless, when the soils are subjected to temperatures higher than the second critical temperature,

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their thermal conductivity drops sharply with temperature, probably because of the high water evaporation under these temperatures conditions and the resulting break of the water bridges at the surface of the grains (Mermoud, 2006; Hamamoto et al., 2010). The thermal conductivity measurements clearly show the second critical temperature, which is lower for the Fontainebleau sand,

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and Cheverny and Lauret soils (T2 = 89, 89, and 93 °C, respectively) compared to the BRGM soil (T2 = 96 °C). The observed differences in T2 between the different soils may be explained by their different petrophysical properties (e.g., porosity, clay and sand content) and pore water chemical compositions (Cosenza et al., 2003; Revil, 2000). When the temperature increases slightly (a few degrees Celsius) beyond the second critical temperature, the thermal conductivity begins to stabilize because most of the pore water is evaporated (Mermoud, 2006; Fig. 5b). At these very high temperatures (T > 90 °C), the thermal conductivity of Cheverny shale soil can be more stable and higher than the thermal conductivity of Fontainebleau sand because nano/microporous clay minerals can retain a large quantity of water, even at high temperatures. The soil of the BRGM’s geothermal platform has the highest thermal conductivity at the highest temperatures (T > 90 °C). Therefore, the experimental results show that the electrical resistivity method can be used to monitor the thermal properties of soils because of the relationship between electrical and thermal conductivities, and two critical temperatures can influence the main processes occurring in the pore space of soil during heating. The thermal conductivity of microporous shaley sands and shales is more 10

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stable with temperature than the thermal conductivity of macroporous clayey sands and sandstones. Nevertheless, clayey sands and sandstones have higher thermal conductivities than shales when the temperature is lower than the second critical temperature. Therefore, in these temperature conditions, soils containing a large sand content can better evacuate the heat than soils containing a large clay

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content. It is therefore crucial to have information about these critical temperatures to choose the

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adapted material to avoid overheating of the cable, which depends on the temperature of the soil. The

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drastic drop of the thermal conductivity of the soil can be detected using electrical resistivity methods because it is correlated with the high increase in the measured electrical resistivity with temperature. To physically express the effects of temperature on physical soil properties, we attempt in the

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following section to model the resistivity variations in the BRGM soil as a function of its petrophysical properties and water content, showing the importance of the two critical temperatures previously presented for describing the behavior of the resistivity of the soils with temperature. Such a

3-Sample resistivity modeling

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relation can help to better link the electrical resistivity to the thermal conductivity of the soils.

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As explained in the first section, in most studies the compensation slope  is estimated as ~ 2% °C-1

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(Campbell, et al., 1949; Ma, et al., 2011) when the temperature interval ranges approximately from 10 to 40 °C. In that case, the relation between soil electrical resistivity and temperature can be considered as linear. According to our last resistivity experiments, we can suppose that for higher temperatures,

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i.e., 40 °C < T < 100 °C, this relationship is nonlinear, and thus,  varies according to T. To understand this feature in a more physical approach, we propose to use standard equations to model our experimental resistivity curves.

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Campbell’s law (Eq.1) and Archie’s law (Archie, 1942) are used to model the resistivity behavior of the BRGM soil as a function of water content and temperature. Archie’s law is written as follows:

Eq.2 where andw are the electrical (effective) resistivity of the soil and the electrical resistivity of the pore water, respectively, F is the electrical formation factor, Sw is the water saturation (fraction of the voids filled with liquid water), and n is the saturation exponent of the liquid water phase. The electrical formation factor F is given by

Eq.3

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where a is the tortuosity factor,  is the soil porosity (volume of the voids divided by the total volume of the sample), and m is the cementation exponent (depending on the grain shape). Archie’s law assumes that the solid phase is non-conductive compared to the liquid phase. Therefore,

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this law is adapted for describing the resistivity of soils presenting a low electrical surface

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conductivity, such as sandstones or clayey sands, but it is not adapted for describing the resistivity of shales (Revil et al., 1998). The parameters (a, m, n) depend on the type of soil. In practice, (a, m, n)

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are mostly taken in the ranges of ([0.2, 1.5], [1.3, 2.2], [1.8, 2.2]) for common soils (Schön, 1996; Revil, 1999). Using these values of the parameters (a, m, n), Archie’s law (Eq. 1 and Eq. 2) can be written as a function of the volumetric water content:  Sw. The following equation is obtained for

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m  n:



Eq.4

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Campbell’s law (Eq.1) is used to calculate the resistivity of the pore water as a function of the water resistivity at T = 25 °C (reference) and the temperature T (Revil et al., 1998):

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Eq. 5

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It is now possible to model the resistivity of the BRGM soil as a function of the water content and the temperature by combining Archie’s and Campbell’s laws (Eq. 4 and Eq. 5). The following equation is

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obtained:



Eq. 6

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The resistivity of the pore water at a temperature of 25 °C (w25) and Archie’s law parameters at low and high temperatures were fitted using Eq. 6 and the resistivity measurements (Fig. 6). Table 2 gives the values of parameters used in Eq.3 to Eq.6. Figure 6 shows the theoretical electrical resistivity curve obtained using Eq. 6, superimposed with the resistivity observations and the moisture content values measured during the experiment. Note that the two electrical resistivity curves are consistent. The difference between the two curves is very low, on the order of 4% of the measured data according to the L1 norm,

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addition, the T1 limit (T1 = 84 °C) corresponds well with the decreasing water content observed in the interpolated  curve. However, we should remain critical of our resistivity model because i) it is based on Campbell and Archie’s law, and not validated for some soils containing clays, and ii) some petrophysical parameters such as a, m, and n were estimated from the literature and thus contain some uncertainties. A perspective of this study should be to test and validate this modeling approach to other soil types.

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Conclusion The experiments, realized on the BRGM’s geothermal platform and in the laboratory, allowed for studying the relationships between the thermal conductivity, electrical resistivity, temperature and

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water content of soils with different particle size distributions and clay contents. To obtain such

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measurements, different probing systems were used, at the platform scale (Raman TDS for temperature, electrical resistivity tomography for resistivity) and the sample scale (Pt100 for

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temperature, TDR for water content, SH1 KD2Pro for thermal conductivity, electrical quadrupole for resistivity). As a result, we finally identified two main properties of soils, depending on their temperature:

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1) At lower temperatures (between 20 and 70 - 85 °C), when the soil water content remains constant, the soil electrical resistivity decreases with increasing temperature because of the

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resulting decrease in the pore water viscosity and resistivity. Under these temperature conditions, the thermal conductivity increases with temperature. 2) At higher temperatures (between 85 and 100 °C), when the water present in the pore space

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significantly evaporates, the soil electrical resistivity rises abruptly due to the increasing air

temperature.

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content. In these temperature conditions, the thermal conductivity decreases drastically with

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Soils with a high sand content and low clay content (sandstones and clayey sands) present higher electrical resistivity and thermal conductivity variations than soils with a higher clay content (shaley sands and shales) because the sandstones and clayey sands have macropores and the thermal

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conductivity of quartz is significantly higher than that of clays. To better understand these behaviors, we developed simple models to simulate the variations in the soil electrical resistivity as a function of the temperature and water content. For this purpose, we used the equations of Archie and Campbell to fit observations with modeled data, and we thus confirmed our experimental observations. However, these results are only validated in the framework of this study, and it would be interesting to test the proposed resistivity model of other soil types featured with different properties in terms of compaction, porosity, composition, etc. A deeper thermodynamic study could also clarify some points for explaining such results in detail. From a practical point of view, these results clarify the phenomena occurring around the buried high voltage cables: below a first critical temperature, the electrical resistivity and thermal conductivity are a function of the temperature: initially decreasing and then increasing with increasing temperature. Then, between the first and a second critical temperature, two phenomena occur in the pore water, evaporation and condensation, which produce notable effects on the electrical and thermal behaviors. Then, above the second critical temperature, 13

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Nouveau et al., Electrical and thermal behavior of unsaturated soils: experimental results

the pore water is evaporated, producing drastic electrical and thermal variations: a strong increase and decrease in the electrical resistivity and thermal conductivity, respectively. The heat produced by the cable is thus a crucial parameter, and for avoiding overheating of the cable-soil system, the temperature has to be maintained below the second critical temperature so that water is present in the

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soil and heat evacuation can be preserved. Of course, the critical temperature is dependent on the soil

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type. Therefore, we suggest extending this work to other GTR soil classes to characterize these soils

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from electrical and thermal points of view.

Acknowledgements

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This project was funded by the BRGM and a project of “Investissement d’Avenir Carnot-PME”. The authors would like to thank Alain Tabbagh, Damien Jougnot and Quentin Vitale for fruitful

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discussions. They also thank the reviewers that gave pertinent comment for improving the quality of

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the manuscript.

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Zimmerman, G.H., Arcis, H., & Tremaine, P.R., 2012. Limiting Conductivities and Ion Association Constants of Aqueous NaCl under Hydrothermal Conditions: Experimental Data and Correlations.

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Table Fontainebleau

BRGM platform

Lauret

Cheverny

40

Natural conditions: 12

20

Dmax (mm)

42.5

15

> 50 mm

D50 (m)

204

1331

D < 2 mm (%)

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< 20

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Water content (%)

62

D < 80 µm (%)

< 35%

51.5

13

56

D < 2 µm (%) 39

GTR class

D1 Sand

Measured experimentally: 67

T2 (°C)

87

56

C1A2

A4

Clayey sand

Sandy shale

Shale2

84

85-95

80-95

97

~95

~90

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T1 (°C)

81

B4

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Soil type

40

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Porosity (%)

< 20

Table 1: Petrophysical properties of the four different soils used for the electrical resistivity and

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thermal conductivity experiments in the laboratory.

Low temperatures

High temperatures

T < 84 °C

T > 84 °C

1.3

1.3

2

2

Tortuosity factor a

0.35

0.35

Compensation slopes 

0.02

0.02

40

40

Formation factor F

1.15

1.15

Water content  (%)

12



Water saturation Sw (%)

30

/

Water resistivity at 25 °C (.m)

30

w25.Swn/F

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Cementation exponent m Saturation exponent n

Porosity 

Table 2: Soil parameters used in Eq.6 for modelling resistivity values fitting experimental measurements shown in Fig.6

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Figure captions

Figure 1: Schematic presentation of the BRGM’s geothermal platform showing the geothermal installations, including the heat pipe network, the optical fibers distributed at three depths, four piezometers and a meteorological station. The grey part of the calorific pipe network was not active.

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Figure 2: a) Inverted resistivity profile at time t: 2013/06/03-2pm, b) resistivity changes between the reference profile (at time t0 and t), and c) temperature computed from the DTS at time t. The 

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measurement accuracy is about 1%.

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Figure 3: Electrical resistivity () versus temperature (T) during the heating experiment realized on the

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BRGM’s geothermal platform. The measurements were recorded at 1 m depth. The measurement

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accuracy is indicated by error bars

Figure 4: a) Photo of the laboratory setup. Pt100: temperature sensor; TDR: Time Domain Reflectometry sensor; SH1: KD2 Pro thermal sensor; A-B-M-N: electrical quadrupole.

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Figure 5: Electrical resistivity (; a) and thermal conductivity (K; b) measured for four soil types in the temperature range of 20 – 100°C. T1 and T2 indicate the critical temperatures observed for the

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Fontainebleau case. The measurement accuracy is about 1% and 5% for  and K respectively.

Figure 6: Comparison between the theoretical and measured resistivity curves as a function of soil temperature, with evidence of the two critical temperatures and water content measurements. T1 is the beginning temperature of high evaporation of water in the pore space. T2 is the temperature when most of the water of the pore space has been evaporated.

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Highlights We experiment changes in electrical and thermal conductivity for four common soils

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Electrical resistivity decreases and thermal conductivity increases with temperature

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This phenomena is observed up to a critical temperature for each soil

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This limit corresponds to water evaporation from the soil pore space

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