Laboratory Investigation of Heterogeneous Metamorphic Rocks and Their Spatial Distribution of Thermal Conductivity

Abstract

Assessing the variation in the thermal conductivity of heterogeneous rock materials can be critical when upscaling models to simulate geothermal system operation, especially for petrothermal systems, where conduction dominates over convection. This study’s objective was to evaluate heterogeneity effects when assessing the thermal conductivity of geological materials, in this case, metamorphic rocks from Kuujjuaq (Canada), where the installation of a ground-coupled heat pump system is expected. Four core samples of gneissic rocks were analyzed in detail and compared to results obtained from a thermal response test. Thermal conductivity measurements in dry conditions were performed on the cylindrical surface of the samples with an optical thermal conductivity scanner. The 2D thermal conductivity spatial distribution was obtained by ordinary kriging interpolation method and used for numerical modeling to simulate steady-state conductive heat transfer along the sample vertical direction. Then, the effective thermal conductivity was computed according to Fourier’s law, using the simulated temperature to investigate the effect of scale variation with the heterogeneity. Results indicate the importance of distinguishing between the sample section’s effective thermal conductivity and local average thermal conductivity. Significant scale effects were identified with a variation ratio comprised between −10% and +16% when varying the length of the sample section. The representative elementary volume for the effective thermal conductivity was determined equivalent to half of the sample length. This volume gave a thermal conductivity that is equal to the harmonic mean of the laboratory-assessed values with a relative error <5%. A comparison between the in situ and laboratory-assessed thermal conductivity indicates that the thermal conductivity inferred from the thermal response test is adequate for sizing a geothermal system, assuming a range of variability equivalent to 1.5 times its standard deviation.

1. Introduction

Thermal conductivity (TC) is a thermal property of rocks that influences heat transfer mechanisms and thus affects the calculation of the Earth natural heat flow or the simulation of geothermal energy systems. It is important to adequately estimate TC with an assessment that is representative of the rock heterogeneity. Several studies have shown that TC is a property that varies according to a number of factors, including mineral composition, grain size and orientation, the presence of fractures or cracks, rock structure or matrix porosity [1,2], the degree of water saturation [2,3,4], as well as the pressure and temperature conditions [1,5,6]. In addition, various methods are available to evaluate this property in situ at the well or field scale. One of these methods is thermal response tests (TRT; e.g., [7,8]), which provide a transient and bulk assessment that is thought to be characteristic of materials within the test radius of influence varying from the decimeter to the meter scale [9]. Rock samples can also be analyzed in the laboratory using the steady-state divided bar method (e.g., [10,11]) or various transient methods such as the modified transient plane source [12] or the optical thermal conductivity scanner (TCS) [13]. Laboratory TC analysis is made at a smaller scale than in situ field assessment and varies from the millimeter to the decimeter depending on the sample size and the method used. In this respect, the question of scale must be considered: is the TC estimation subject to the effect of rock heterogeneity when the length of the sample is varied? In other words, is the effect of heterogeneity scale-dependent?

Scale effect, referring to the influence of sample size on the measurement of a quantity assumed to be intrinsic, poses a constant challenge for material science (e.g., [14]). In the geothermal field, this issue is not well documented. Luo et al. [15] and Raymond et al. [16] reported that TC evaluated on a centimeter scale with samples tends to be higher than TC evaluated on a meter scale with TRTs. Jorand et al. [17] demonstrated how variations in rock layering and microfractures at the subcentimeter scale may influence TC values at the decimeter scale. The intrinsic heterogeneity of crystalline and sedimentary rocks caused by the above-mentioned properties may influence TC values when assessed at different scales. Popov et al. [2,18] introduced the thermal heterogeneity factor F to characterize the variability in texture, structure, and composition of the sample, defined as the maximum difference in TC along a scanning line divided by the average value. This factor was used by Li et al. [19] to experimentally investigate the influences of mechanical damage and water saturation on the distributed TC of granite. This factor is a useful metric to rapidly quantify TC heterogeneity, but a more detailed approach is needed to quantify the influence of heterogeneity on conductive heat transfer which is the dominant mechanism assumed for the design of closed-loop geothermal systems (e.g., [20,21,22]).

Kuujjuaq is an isolated northern community that currently relies on diesel as the main energy source and is planning to install its first geothermal heat pump system. The recent works of Cavalerie et al. [23] reported that 40 to 60 borehole heat exchangers at depths between 160 and 200 m would be required to implement such a system and fulfill the heating needs of a community building. The system sizing was based on the average TC inferred from a TRT that has been conducted by Géotherma Solutions Inc. [24]. Due to its remote location and subarctic climate, the installation of this geothermal system is projected to cost several million dollars. Investigating TC variability is essential for optimizing the design and performance of the geothermal system in this unique context. It is imperative to determine whether the TC value from the TRT is representative of the geological environment and adequate for sizing the geothermal system. Thus, this paper’s objective was to fully characterize the heterogenous nature of rock TC for a suite of metamorphic core samples obtained from a geothermal exploration borehole drilled in Kuujjuaq and evaluate if scale effects can be detected when considering the millimetric to decimetric scale. The outcome will provide the Kuujjuaq community with an appropriate range of TC to consider in the design of a ground-coupled heat pump system. The present study builds on the work of Jorand et al. [17], where the methodology has been originally adapted for cylindrical core samples and lies in the combination of TC assessments obtained along various parallel lines on a cylindrical surface. This detailed TC assessment is then analyzed under length variations, which is a new approach to characterize the rock TC heterogeneity in more detail. Results are compared to those obtained from the in situ TRT assessment and recommendations are given for the design of the geothermal system.

2. In Situ Thermal Conductivity Assessment at the Field Site

An in situ assessment of TC and an evaluation of the shallow geothermal potential was previously made by Géotherma Solutions Inc. [24] for the studied borehole in Kuujjuaq, Canada (Figure 1).

The in situ assessment consisted of a TRT that was carried out with a heating cable installed in the Forum well, filled by groundwater found at a level of 2 m below the ground surface. The well intercepted metamorphic rocks of the Canadian Shield, at the geological contact between the False Suite (north) and the Kaslac Complex (south), both belonging to the Southeastern Churchill Province [27]. The recovered cores were described by Miranda and Raymond [26]. The details of the method and the experimental setup of the TRT is explained by Géotherma Solutions Inc. [24]. The methodology of Vélez et al. [28] for TRT with a single continuous heating cable was followed. Heat was injected along a 141 m cable, at a constant rate of 7.02 ± 0.04 Wm⁻¹ for a duration of 36 h. Thermal recovery was subsequently monitored for 47 h. A total of twenty temperature sensors were tied along the heating cable with a space every 5 m for the first 110 m, and then by 20 m until 130 m. Theinitial ground temperature and the thermal evolution was used to assess the ground TC. The evaluation of TC was based on the ground temperature assessed during the recovery period. The TC, λ (W m⁻¹ K⁻¹), was inferred using the slope k (K), or the rate of changes in average ground temperature ΔT (K), considering the natural logarithmic injection time t, which was normalized forthe recovery period with:

$$k={\displaystyle \frac{\Delta T}{\Delta \mathrm{l}\mathrm{n}\left(\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$t-t_{\mathrm{o}\mathrm{f}\mathrm{f}}$}\right.\right)}}={\displaystyle \frac{q}{4\pi \lambda }}$$

(1)

and

$$\lambda ={\displaystyle \frac{q}{4\pi k}}$$

(2)

where q is the heat injection rate (W m⁻¹), and t_off (h) is the time when heat injection was stopped.

The derived TC ranges from 2 W m⁻¹ K⁻¹ to 3 W m⁻¹ K⁻¹ (Figure 2), with uncertainty ranging from 3.44% to 3.64%. The average is 2.67 W m⁻¹ K⁻¹ with a standard deviation of 0.25 W m⁻¹ K⁻¹ [24]. The obtained in situ TC values are assumed to be horizontal assessments since the heat injection is perpendicular to the cable, which is vertical during the TRT. The radius of thermal influence was between 0.6 m and 0.8 m and was calculated at the four core sample’s depths, which are subject to detail analysis below.

3. Sample Description

Four drill core samples were retained from the Kuujjuaq exploration well (Figure 2) in which the TRT was carried out. The cores are 47.6 mm (1.85 inch) in diameter and the length vary from 96 mm to 210 mm. Two cores were collected at 60 m depth (top—T and bottom—B) and the others at 90 m and 105 m, respectively (Figure 2). Unrolled 2D photographs of the cores were made using high-resolution color images (0.125 mm/pixel). The images were taken with a line-scan optical camera with an 80 mm field of view and incorporated into an automated multi-function core scanning instrument named Itrax core scanner from Cox Analytical Systems, Mölndal, Sweden [29].

The core samples T60, B60, and B105 consist of paragneiss of the False Suite, which are metamorphic rocks of sedimentary origin (Figure 2a,b). The core samples T60 and B60 consist of paragneisses with pink granite centimetric ribbons. They are characterized by a grayish granoblastic matrix composed of quartz (55 mode%), plagioclase (37 mode%), biotite flakes (5 mode%), and hornblende (3 mode%). The penetrative metamorphic foliation is marked by ferromagnesian minerals, which define a lepidoblastic texture (biotite) and a nematoblastic texture (hornblende). The samples consist of 40% pink granite ribbons in sharp to defuse contact with the matrix. The medium-grained granitic material is recrystallized. The accessory phases represent 1% of the modal composition and include magnetite, biotite, and hornblende.

The shortest sample B90 is a diorite that is potentially part of the Kaslac Complex or an undetermined geological unit of intermediate mafic rocks within the Kuujjuaq region (Figure 2c). Sample B90 consists of a fine-grained granoblastic diorite interlayered with a metatexite and is cut by an injection of coarse granite. The diorite is mainly composed of biotite and hornblende (65 mode%), plagioclase and granoblastic quartz (34 mode%), and pyrite (1 mode%). The metamorphic foliation is defined by the alignment of lepidoblastic biotite and nematoblastic hornblende. The metatexite shows alternations of melanosome and fine-grained granoblastic quartzofeldspathic ribbons incorporating 10% of lepidoblastic biotite and coarser discontinuous hololeucocratic ribbons. The coarse granite is composed of feldspar (70 mode%), quartz (29 mode%), and hornblende (1 mode%). It is partially recrystallized.

Unlike the other cores, the longest sample B105 is a whitish migmatized paragneiss. Its matrix consists of quartz (60 mode%) and feldspar (30 mode%), finely recrystallized, and 10% of ferromagnesian minerals. The penetrative and spaced metamorphic foliation is folded and marked by a preferential orientation of biotite lamellae, defining a lepidoblastic texture. Fold plans are outlined by a discrete brittle planar fabric. The metamorphic foliation is also highlighted by 35% of discontinuous hololeucocratic ribbons of tonalitic composition (quartz, 25 mode%; plagioclase, 75 mode%), slightly coarser than the matrix but still recrystallized. Microcracks intersecting the foliation were also observed, but it is difficult to determine their origin as it may have occurred naturally or been induced by drilling (Figure 2b,d).

4. Methods

The following steps were followed to analyze the variation in TC due to heterogeneity: 1—a detailed TC assessment was performed on core samples; 2—X-ray microfluorescence (XRF) and magnetic susceptibility were analyzed using an Itrax core scanner; 3—the correlations between the TC and the geochemical elements analyzed from XRF, and with the magnetic susceptibility were verified; 4—a 2D model of the spatial distribution of the assessed TC was generated; 5—the 2D TC model was imported into COMSOL (Version 6.2) and conductive heat transfer was simulated to calculate the effective TC, evaluated at different scales; and 6—laboratory results were compared with the in situ TC inferred from the TRT.

4.1. Detail Thermal Conductivity Assessment

The TC was evaluated along 16 parallel lines, which were positioned on the cylindrical surface spaced approximately 9.3 mm (Figure 3). The optical TCS from LGM Company, Lippmann and Rauen GbR, Achim, Germany, was used, which relies on the method developed by Popov et al. [13,18]. The number of scan lines was determined as a function of the core’s diameter to guarantee the optimum measurement resolution, and to ensure that the distance between lines considered the influence of the thermal radius beam and avoided overlapping punctual optical scanning measurements. When evaluating TC, the heat source applied to a point is diffused on a circular plane with a radius of thermal influence of ~10 mm [17,30]. The core was marked using a 3D-printed template on one end, and the sample is blackened prior to TC evaluation to ensure homogeneous absorption of the light irradiation (Figure 3). The “TC mode only” was used, allowing an accuracy of ±3% with a resolution of 2 mm and at a recommended scanning speed of 5mm.s⁻¹ [31].

4.2. X-Ray Fluorescence (XRF) and Magnetic Susceptibility

An Itrax core scanner was used to complement the petrographic description and to better characterize the heterogeneity (Figure 4). It allows us to acquire high-resolution geochemical elements composition profiles and a semi-quantitative magnetic susceptibility analysis [32]. When a chemical element is bombarded with X-rays at a given energy level, the element emits secondary radiation that is characteristic of the excited atom and proportional to the number of atoms of a given specie element in the sample. For each element, after normalization, to negate closed-sum effects, its chemical profile is expressed in peak areas and the amplitude of the peaks is, in theory, proportional to the concentration of the element in the sample [33]. However, these results are semi-quantitative [32]. The Itrax was set up with a 2 mm spatial resolution, the same as the TCS, with 10 s exposition time. The X-ray source uses a molybdenum target operated under a voltage of 35 kV and a current of 55 mA, which is suitable for most elements [32]. The chemical composition and magnetic susceptibility of each sample was compared to the TC to evaluate if there is a correlation between these parameters that would be related to the effect of scale-dependent heterogeneity. In this study, the Itrax analysis was performed along a chosen scanline of each sample (Figure 3a). The scan line #12 was selected as it was considered to be located in the middle of the sample images, based on the configuration of the scan line positioning and order in Figure 3a.

5. Numerical Modeling

5.1. Interpolation of Thermal Conductivity

The ordinary kriging method was used to interpolate values between the TC assessments and define a 2D TC spatial distribution map at the core surface. This method was chosen based on the statistical distribution of the assessed TC, confirmed with the semivariogram for each sample [34], and taking into account the prediction depends only on the distance and direction between the data points. This minimizes the prediction errors, which were themselves estimated [35,36]. In ArcGIS Pro (Version 3.5), we used the interpolation tool of the Spatial Analyst Tools set to complete the kringing. Different interpolations were tried with increased numbers of scan lines along which TC is inferred. Interpolation with the highest number of scan lines (16 for a cylindrical sample, Figure 3a) was retained at the end to make sure the interpolation results are the most representative of the samples.

The image corresponds to the unrolled 2D surface of the core, such that each TC point is located using the coordinates in the Z direction of the core (i.e., along the length), and in the φ direction along the circumference of the core where the angular position is converted into arc length.

5.2. Heat Transfer Simulations

The simulations were based on conductive heat transfer only without internal heat generation or storage. Fourier’s law (Equation (3)) was numerically solved under steady-state conditions, and resulted in a 2D temperature field influenced by the sample’s TC distribution. The heat conduction simulated in the samples is thus expressed by:

$$Q=-\lambda \nabla T$$

(3)

where $Q$ (W m⁻²) is the heat flux, $\lambda$(W m⁻¹ K⁻¹) is the interpolated TC, and $\nabla$T (K m⁻¹) is the temperature gradient.

The 2D model had the same dimensions as the 2D interpolated TC spatial distribution map. The COMSOL predefined mesh size “extremely fine” was used to ensure the temperature solution is independent of the element sizes (Figure 5). This created meshes with up to 10⁴ elements along the middle cutline.

In COMSOL, the 2D interpolated TC distribution is used as input property and assigned to the model. Boundary conditions were set for the calculation of the effective TC or ${\lambda }_{\mathrm{e}}$ in the Z direction along the vertical axes of the core samples (Figure 6), corresponding to the direction of the scan lines. It does not make physical sense to simulate heat transfer in the φ (radial) direction. A cylindrical geometry would have been required for the simulation of circumferential heat conduction, based on the TC distribution measured along the circumference. However, we chose 2D simulations in a plane to better visualize the results. Heat transfer is simulated under steady-state conditions, with the heat flow $Q$ (Wm⁻²) set at one boundary and a constant temperature T₀ (°C) at the opposite boundary (Figure 6). A heat flow value of 150 W m⁻² was chosen and was sufficiently high to produce a temperature difference of at least 10 °C between the imposed temperature T₀ and the calculated temperature at the base of the sample. This heat flow value impacts the computed temperature distribution but is independent of the effective TC inferred with the numerical simulations as the input TC distribution is independent of temperature in our model. The heat flow, which is constant across the sample, is proportional to the temperature gradient multiplied by the effective TC. Different heat flow values were tried for the boundary condition with preliminary simulations and resulted in the same effective TC values.

6. Effective Thermal Conductivity Calculation Under Scale Variation

To analyze the effect of the heterogeneity on the TC, the effective TC or ${\lambda }_{\mathrm{e}}$ was recalculated for each section of the core sample using Equation (4) deduced from Fourier’s law:

$${\lambda }_{\mathrm{e}}={\displaystyle \frac{LQ}{T_{1-}{T}_0}}$$

(4)

where L (mm) is the length of the sample section along the Z axis, aligned in the direction of simulated heat flow $Q$ in Figure 6. T₀ and T₁ (°C) are the imposed and computed temperature distribution that bounded the sample section.

The following analysis was carried out by choosing a cut line located at the middle of the 2D-simulated temperature distribution of the models. We averaged the temperature profile every 2 mm.

The section length L was varied along the Z direction, according to the following approaches:

• Cumulatively, i.e., the variable L is incremented/decremented in one direction (Figure 7a).
• Randomly, where sections with a variable length are defined and each section is randomly moved along the temperature profile (Figure 7b). To ensure that the TCS measurements can be compared to the numerical simulation results, the smallest section length must not be shorter than the minimum length required for TC analysis with the optical scanner, which is 40 mm [18]. The section length was chosen to be a fraction of the sample length, with the smallest being one quarter. Thus, the minimum sample length is at least 160 mm, well above the recommendation for TCS analysis.

The variation ratio (%) was calculated, since it shows as a function of the section length the extent of effective TC variability in relation to the effective mean TC of the total sample length. The higher this value is, the greater the dispersion of the TC along the core sample.

The relative error ${\epsilon }_{\mathrm{L}}$ (%) was calculated using Equation (5) to define the existence of a representative elementary volume (REV) [17]. The REV is based on the section length L, allowing the effective TC to be the closest to the harmonic average TC ${\lambda }_{\mathrm{h}\mathrm{a}\mathrm{r}}$ (W m⁻¹K⁻¹) of the entire sample.

$${\epsilon }_{\mathrm{L}}={\displaystyle \frac{{|\lambda }_{\mathrm{e}}-{\lambda }_{\mathrm{h}\mathrm{a}\mathrm{r}}|}{{\lambda }_{\mathrm{h}\mathrm{a}\mathrm{r}}}}$$

(5)

7. Comparison with In Situ Thermal Conductivity from TRT

The TC is inferred vertically with numerical simulation and horizontally with the TRT and TCS. Given no major evidence of TC anisotropy, the samples are assumed to be isotropic, which allowed for a pertinent comparison between these methods. This comparison aimed to determine whether in situ and sample effective TC differ substantially and if they can be considered equally representative of local heterogeneities.

8. Results

8.1. Thermal Conductivity Assessment Results and 2D Spatial Distribution

A 64 TC profiles, i.e., 16 profiles per samples, were recorded. Detailed data are available in the Supplementary Material provided with the article. The point-by-point average of the 16 TC profiles, for each sample, is presented in Figure 8. The resulting TC of the samples varies between 2.6 W m⁻¹ K⁻¹ and 3.1 W m⁻¹ K⁻¹, with the largest range found in sample B90. The assessed TC profile along the scan line #12 has a wider range, i.e., from 2.3 W m⁻¹ K⁻¹ to 3.8 W m⁻¹ K⁻¹, but nearly coincides with the TC obtained by using the arithmetic (${\lambda }_{\mathrm{a}\mathrm{r}\mathrm{i}}$), harmonic (${\lambda }_{\mathrm{h}\mathrm{a}\mathrm{r}}$_r), and geometric (${\lambda }_{\mathrm{g}\mathrm{e}\mathrm{o}}$) means of the 16 profiles (Figure 8b). The different average TC (${\lambda }_{\mathrm{a}\mathrm{r}\mathrm{i}}$), (${\lambda }_{\mathrm{h}\mathrm{a}\mathrm{r}}$), (${\lambda }_{\mathrm{g}\mathrm{e}\mathrm{o}}$) varied less than ±0.01W m⁻¹ K⁻¹ (Figure 8b). The average thermal heterogeneity factor F was comprised between 0.098 and 0.394, and the maximum was found in sample B90 (

Table 1. Average thermal conductivity of the samples assessed with the TCS with their associated coefficient of variation (G) [18] and thermal heterogeneity factor.

Sample ID	Sample Length (mm)		Average (W m−1 K−1)			G (%)	Thermal Heterogeneity Factor F
	Total Length	Length Analyzed	Arithmetic	Harmonic	Geometric
T60	160	150	2.86	2.86	2.86	2.10	0.098
B60	149	136	2.81	2.80	2.81	3.36	0.152
B90	96	84	2.86	2.83	2.84	10.53	0.394
B105	210	186	2.74	2.73	2.74	3.59	0.158

). The samples have therefore a low to high level of heterogeneity.

The 2D maps of TC (Figure 9) show various zones of heterogeneity. The TC ranges from 2.13 W m⁻¹ K⁻¹ to 4.29 W m⁻¹ K⁻¹, with an increment of 0.2 W m⁻¹ K⁻¹ between the classes. The highest TC of 4.29 W m⁻¹ K⁻¹ as well as the lowest TC of 2.13 W m⁻¹ K⁻¹ were found in the sample B90, which has the highest arithmetic mean value of 2.85 W m⁻¹ K⁻¹ (Table 1).

8.2. XRF and Magnetic Susceptibility Results

The XRF analysis allowed us to qualitatively confirm the presence of the geochemical elements contained in the minerals of the samples. In fact, in the paragneiss samples (T60, B60, and B105), the range of the peak values for the major elements Si, Al, and K is relatively higher than that of the Fe, S, Ca, Ti, and Cr. In contrast, these last major element peaks are higher in the sample B90, which has a mafic composition. The sample B90 also showed the highest spectrum contrast profile along the chosen scan line, due to high heterogeneity between its coarse granite injection and the metatexite layer, compared to the three other samples. The coarse granite injection seems to contain more Si, Al, and Ca than the metatexite layer, which oppositely contains more S, K, Fe, Ti, Cr, and Mn. The samples T60, B60, and B105 showed some high peaks for S and Fe observed at certain locations in the sample, which correspond to the sulfur minerals. Visually, there is no direct correlation of the TC profile with each elemental profile, although the Si profile appears to most closely resemble the TC profile. The peaks or the low points along the elemental profiles do not align well with those of the TC profiles. The same disagreement was observed with the magnetic susceptibility profile (Figure 10). In addition, the coefficient of correlation between the TC and magnetic susceptibility and TC against geochemical ‘counts’ from XRF analysis is 0.01 < R² < 0.2, signifying zero to very low correlation. XRF peaks do not inform on volumetric concentration of elements, which can make correlation with TC difficult.

8.3. Effective Thermal Conductivity Under Variation of Scale

The resulting effective TC calculated from the numerical heat flow simulations (Figure 11 and Figure 12), regardless of how the scale was varied, showed the following:

• TC profiles assessed with the TCS are comparable to the one calculated with numerical simulation results, although the latter is smoother, particularly where the fluctuations are of low amplitude, and the amplitudes are reduced at the highly heterogeneous sections, for example, from length Z = 5 to 45 mm in sample B90 or Z = 28 to 180 mm in sample B105. This is because the TC assessed at a point with the millimeter scale can be different from that calculated for a section at the centimeter scale involving heat transfer over a greater distance.
• The shorter the section, the more the effective TC is over- or underestimated, depending on the heterogeneity of the section. The variation ratio varies with the section length. For the four studied samples, it varied from −10% to +16%, in relation to the TC value calculated for the entire length of the sample.
  When the section length is randomly moved along Z direction, the result shows the following:

  2.1

  The effective TC profile calculated with numerical simulation results has the same trend as that assessed with the TCS but is not fully comparable; the section length tends to smooth the local heterogeneity.

  2.2

  The longer the section length (L = 1/2), the closer the calculated effective TC is to the average TC for the whole sample assessed with TCS, generally the harmonic mean. Moreover, the lowest relative error on effective TC (${\epsilon }_{\mathrm{L}}$ <5%) is obtained with L = 1/2, which can correspond to the REV of each sample (

  Table 2. Representative elementary volume estimates.

  Sample ID	Section Length (mm)		Thermal Heterogeneity Factor F	G (%)	εL (%)
  T60	L = 1/2	80	0.041	1.43	1.45
  	L = 1/3	50	0.058	2.00	8.80
  	L = 1/4	37.5	0.077	2.46	2.53
  B60	L = 1/2	68	0.046	1.72	2.51
  	L = 1/3	44	0.090	2.91	3.48
  	L = 1/4	34	0.109	3.35	3.17
  B90	L = 1/2	42	0.122	3.38	4.10
  	L = 1/3	28	0.165	5.20	4.23
  	L = 1/4	21	0.214	6.40	6.54
  B105	L = 1/2	93	0.647	1.79	0.54
  	L = 1/3	62	0.567	1.65	4.48
  	L = 1/4	46.5	0.763	1.69	4.38

  G = standard deviation divided by mean [18], ε_L = relative error according to Equation (5). The representative elementary volume and corresponding G and F values for each sample is shown in bold.

  ). This indicates that a REV [17] might exist for these samples, but is of greater scale than that of the samples. The shorter the section (L = 1/3 and L = 1/4) length, the more the effective TC is over- or underestimated. However, the effective TC of the section follows the heterogeneity of the sample.
• On Figure 12b, at the position 70–80 mm, the TC values overlap, which indicates that the scale has no effect on effective TC calculation, i.e., the calculated effective TC from numerical simulation results is the same, whatever the length of the section. It is because that section is located at a symmetrically opposed TC variation. Thus, by expanding the section length, the addition of more TC values on one side is always compensated by the new values included on the opposite side, so that the average value does not vary.

9. Comparison of In Situ and Laboratory-Assessed TC

The discrepancy between the average TC assessed with the TCS and the calculated effective TC from numerical simulations varied from one sample to another when compared to the in situ value inferred with the TRT at the corresponding depth (Figure 13). From the surface to 70 m depth, the TRT profile shows significant TC variations, which suggests the presence of metric scale heterogeneities. At 60 m depth, both TCS and calculated TC values are significantly greater than the TRT results. From 75 m to 95 m depth, the TC values from TRT remain constant. The TCS assessments on sample B90 slightly underestimate TC relative to TRT but the result remains within the TRT error bars. From 100 m depth, the TC obtained with the TRT decreases. Both the TCS and calculated TC values for sample B105 are lower than the TC from the TRT. The spatial resolution of TCS and calculated TC at the millimeter to centimeter scale detects the presence of heterogeneities that influence the global trend of the TC value obtained with the TRT, which is characteristic of the meter scale. For all samples, the calculated effective TC tends to be lower than the laboratory-assessed TC.

However, the average TC obtained with each method of investigation is comparable when considering the investigated total length (Table 2). The average thermal heterogeneity factor F of 0.242 calculated over the depth of 60 to 105 m is comparable to that assessed with the TCS (F = 0.201) over the total length of the four samples. The heterogeneity factor from the simulation is the lowest. The G value is also the lowest from the simulation results and decreases with the investigated length.

10. Discussion and Conclusions

This study analyzes the variation in TC under the effect of scale-dependent heterogeneity. The in situ TC was inferred from a TRT with a heating cable of 141 m in length with temperature measurements from 20 to 130 m depth. Four samples with 55.6 cm total length were subsequently selected along the 60 m to 105 m interval for detailed TC laboratory analysis with the optical scanning method (TCS). Finally, the effective TC of these same samples was calculated from 2D numerical heat transfer simulations. The thermal radius of influence of each method, TRT, TCS, and numerical model, are metric, millimetric, and centimetric, respectively.

The four samples were macroscopically described, photographed, and XRF analysis was made to give a first-order evaluation of the sample’s intrinsic heterogeneity. However, we recognized that the number of samples limited the representativeness of the heterogeneity assessment within the rock at Kuujjuaq. The XRF analysis was limited to detecting primarily heavy elements as Itrax is not reliable for elements lighter than Al due to their low fluorescence yield and strong absorption of their characteristic X-rays by air or matrix. This poses a challenge for accurately characterizing ferromagnesian rocks such as those in Kuujjuaq.

According to Albert et al. [3], a high thermal heterogeneity is defined by F > 0.35 and low thermal heterogeneity by F < 0.1. The thermal heterogeneity factor of the four samples analyzed lies between these values, although they have similar average TC (Table 1). The sample T60 is the most homogenous sample, whereas B60 and B105 are comparable. The F factor of the sample B90 is slightly higher because of the contrast between the intrusive and the melanosome sections. Moreover, B90 has the highest G value (10%), which is a sign of a heterogeneous and complex mineralogical arrangement. This is reflected by the TC profiles along a selected scan line (Figure 8 and Figure 10). Also, due to the heterogeneity, the average TC assessed along the middle of the sample is not comparable to that assessed for the entire sample, as shown in Figure 8.

There is no obvious correlation between the elemental peaks and the TC value, despite the measurement resolution being similar. This finding was based on the single chosen scan line at the middle of each sample, which could potentially limit the number of points for comparison. Nevertheless, the geochemical elements profile along the chosen scan line can help to identify the locally present dominant minerals. For example, the peaks in the TC and Si profiles and the flat line in the K profile of the sample B90 correspond to the granite injection that is rich in quartz (29 mode%) and poor in potassium feldspar (70 mode%). On the other hand, the K-Fe-Ti-Cr-and Mn peaks in the metatexite indicate that this portion of the sample is biotite and amphibolite-rich. The TC of these minerals reported in the literature is within the range of 2.02–3.15 W m⁻¹ K⁻¹ and 2.81–2.91 W m⁻¹ K⁻¹, respectively [1]. It is important to remember that TCS analysis does not allow the evaluation of mineral TC but the TC of a group of minerals. Further thin section analysis could help explain the distribution of elements at the mineral level (e.g., Ti in sphene or in rutile; K in biotite or in feldspar; punctual peaks of sulfide-bound S in paragneiss, etc.). Thin section analysis could also be used to evaluate the connectivity between phases and its influence on TC. In future work, analyzing more than one Itrax scan line per sample is recommended to better explain this miscorrelation.

The magnetic susceptibility profile can indicate the presence of ferromagnetic minerals such as magnetite and hematite, which are scattered in the samples and can act as a good thermal conductor, which can in turn influence the TC distribution if their volumetric concentration is sufficient. Therefore, a high peak in the magnetic susceptibility corresponds to an Fe peak but this does not directly translate in a TC peak even though the TC of magnetite is 4.61 W m⁻¹ K⁻¹ and that of hematite is 12.42 W m⁻¹ K⁻¹ [1]. TC peaks were not observed in our samples at the location of magnetic susceptibility peaks (Figure 10 and Figure 11). This is because Itrax analysis does not inform on volume concentration and the comparison is based on the single chosen scan line at the middle of each sample, which could potentially limit the analysis. Discrepancies are likely due to the global effect of the different mineral phases rather than the TC of a single mineral at the measurement point. The TCS and Itrax measurements were both conducted at a spatial resolution of 2 mm. However, the depth of investigation is slightly different. The depth of investigation is less than a mm for the XRF analysis [37] and depends on several factors including element and matrix effects, while it is on the order of less than 10 mm for the TCS [17,30]. This absence of correlation may be due to different depths of investigation and the volume fraction of magnetite and hematite that remains small, therefore not significantly influencing TC, but enough to be detected by the Itrax.

The 2D spatial distribution of the TC (Figure 9) clearly shows zones of heterogeneity in the rock. According to mineral TC values found in the literature [1], it should be possible to distinguish the repartition of the quartz-rich zones that should have high TC (>3 Wm⁻¹K⁻¹). This is evident in the sample B90 (Figure 9c). The sample’s pinkish portions (Figure 8) dominantly contain plagioclase and feldspar (aluminosilicates), minerals with TC falling within 2.04–2.68 W m⁻¹ K⁻¹ [1]. The sample’s dark zones with ferromagnesian minerals (Figure 8) are dominated by biotite, a mineral with low to medium TC of 2.02–3.14 W m⁻¹ K⁻¹, and amphibolite, with a moderate TC of 2.81–2.91 Wm⁻¹K⁻¹ [1] (Figure 8). Our analysis shows that the association of TC with the surface color cannot be made, due to the large variation in the TC within the group of minerals present in the sample. We can conclude that there is not necessarily a strong correspondence between the 2D photograph of the core sample and the interpolated 2D TC distribution for the analyzed metamorphic rocks. Similar results were obtained by Jorand et al. [17] for laminated clay stone and a sedimentary conglomerate on a flat surface of split cores. Finally, despite the millimetric scale of the TC assessment made with the TCS, the accuracy of the predicted model generated by the interpolation method depends on the density of the input values. By using the ordinary kriging, the number and the distance of the data can influence the resulting 2D TC distribution. In addition, the distribution of TC in the 2D images clearly shows an inline interpolation effect parallel to the acquisition direction, as observed in Figure 8a. In this study, the root mean square error decreased by about 0.25% from 8 to 16 scan lines. Hence, the better the resolution of the input data, the better the model [35,36].

Results obtained for heat transfer simulations made in this study are coherent with those obtained by Jorand et al. [17] when comparing the TC profile assessed with TCS to the one calculated for numerical simulation results. The analysis we originally introduced as a function of section length clearly shows the effect of a local versus global evaluation that was highlighted in the value of the effective TC. When the length of the section is increased along the selected scan line, the effective TC calculated is similar to the average TC of the whole sample assessed with the TCS (Figure 11). Furthermore, it seems that the cumulative length method highlights the smoothing effects of the local TC and heterogeneity variations during the numerical heat flow simulations. This effect is under the direction of the heat conduction propagation. The highest variation ratio was found in the sample B90 with the highest G value and highest thermal heterogeneity factor F, which means the length section and sample degree of heterogeneity have a significant effect on the value of TC.

When a sampling section, regardless of its length, is randomly moved along the sample Z axis, the effect of local heterogeneities is highlighted. Consequently, the effective TC profile is closer to the TCS profile (Figure 12). Hence, only a few sections along the sample have their effective TC values close to the average TC assessed with the TCS (Figure 12). Additionally, at a given location, when the effective TC is the same whatever the length of the section, this indicates that the variation in TC is symmetrically opposed and varying linearly. However, the higher the thermal heterogeneity factor, the greater the effect of scale, as shown by the sample B90, where none of the points overlap in the profile (Figure 12c). The relative error ${\epsilon }_{\mathrm{L}}$ in the effective TC for the REV estimates decreases with increasing section length and corresponding G and F values (Table 2).

The TC range at the millimeter scale obtained with the TCS can be reflected by that assessed at the meter scale obtained with the TRT when considering the thermal radius of influence and when the thermal heterogeneity coefficient is comparable (

Table 3. Comparison of the results between the different methods.

		TC Average (W m−1 K−1)
Method	Investigated Length (m)	Arithmetic	Harmonic	Geometric	Thermal Heterogeneity Factor F	G (%)
TRT	45	2.74	2.73	2.73	0.242	7.04
TCS	0.556	2.82	2.80	2.81	0.201	4.90
Simulation	0.556	2.83	2.82	2.82	0.123	2.73

G = standard deviation divided by mean [18].

). Otherwise, the in situ TC tends to be lower than that measured in the laboratory under dry conditions [15,16]. Water saturation can increase the thermal conductivity of porous rocks [38]. An average discrepancy of approximately 12% has been reported between in situ TC obtained from TRTs and laboratory TC evaluated with the TCS method on dry sedimentary rock samples with a porosity range above 0.1 [39]. This discrepancy was thought to highlight the impact of water saturation on the TC analysis of rocks with significant porosity, although the TC obtained with the two test methods exhibited a similar variation trend along the depth of the analyzed borehole. In our case, the hand samples did not show the visible porosity that is expected below 2–3%. The absence of visible porosity in our samples justifies TC analysis under dry conditions to simulate conductive heat transfer only. However, connected fractures networks can contribute to the total porosity and affect tests made in situ. Overall, the effect of water saturation on laboratory TC analysis is expected to be small because of the low porosity of our samples, while it can explain some differences with the in situ TC determined with the TRT.

The comparison between in situ, laboratory, and effective TC from numerical simulations did not consider anisotropy. This limitation is linked to our 2D rectangular model simulating heat transfer along the core axis only. This can potentially introduce bias when comparing results, although no strong evidence of anisotropic TC was present. Moreover, due to the smoothing effect, the effective TC deduced from the numerical heat transfer simulations at the decimetric scale tend to underestimate the local heterogeneity, which is better evaluated at the metric scale (Table 3). We conclude that the effective TC varies with heterogeneity and is scale dependent. The sample size used for this work was limited to the centimeter to decimeter scale, which might influence the results. By increasing the sample size to the meter scale, heterogeneity issues would be better considered in future studies.

In light of these findings, the average TC obtained from the TRT (2.67 W m⁻¹ K⁻¹) is advisable as it allows the representation of the mean TC condition associated with the geological environment along the borehole. The wider the radius of influence or the longer the depth of investigating, the better the heterogeneity is represented. The standard deviation of 0.25 W m⁻¹ K⁻¹ calculated from all TRT measurements is equivalent to approximately half to once the degree of variability associated with the scale effect deduced on samples (−10% to +16%). Considering this variability, sizing calculations for the Kuujjuaq ground-coupled heat pump system could be made with the average TC for the base case scenario, while worst- and best-case scenarios could consider the average TC ± 1 to 1.5 the standard deviation. This range would most likely represent the heterogenous ground conditions in which the geothermal system can operate.