Effect of Temperature on the Thermal Conductivity of Rocks and Its Implication for In Situ Correction

College of Geosciences, China University of Petroleum, Beijing, Beijing 102249, China State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing, Beijing 102249, China Tarim Oilfield Company, PetroChina, Korla, Xinjiang 841000, China Exploration and Development Research Institute, PetroChina Southwest Oil & Gasfield Company, Chengdu, Sichuan 610041, China PetroChina Hangzhou Institute of Geology, Hangzhou 310023, China


Introduction
Thermal conductivity of rocks is an essential parameter in many fields such as constraining the dynamic evolution of planets, exploring the thermal state of basins, and utilizing geothermal resource. A large number of studies have been conducted on the calculation of heat flow and temperature in deep strata with demand for the exact data of thermal conductivity in situ, which cannot, however, be directly measured because of the limitation of current technology. Moreover, theoretic calculation is also impossible on account of extreme complicated structure of rocks. To date, experimental measurements in high temperature or pressure are the sole approach to attain situ thermal conductivity of rocks. There exist only a few studies focusing on the factors influencing thermal conductivity for rocks in recent years. [1][2][3][4][5]. Initially, Birch and Clark [6] measured the λ of several rocks by adopting the steady method with rising temperature. The experimental results showed that the λ of all samples were between 1.67 and 5.86 W/m·K, and the λ of feldspar aggregates and glasses with low-conductivity increased as temperature rise, whereas those samples having a high conductivity showed a negative temperature dependence. A simple equation, λ = 1/ðA + B × TÞ, to describe the relationship between the λ and T was proposed by Seipold (1998), in which A and B, controlled by the lithology, are the fitting coefficients. This study seemed to reveal a generalized conclusion that rocks with a relatively high thermal conductivity exhibit a stronger decrease in thermal conductivity with increasing temperature than those with lower λ.
Vosteen and Schellschmidt [7] obtained the λ of magmatic, metamorphic, and sedimentary rocks using transient methods. Their results indicated that the value of λ at 0°C governs the temperature dependence of λ in rocks, consistent with Seipold (1998). Abdulagatov et al. [8] employed a steady method to determinate λ of five rocks (sandstone, limestone, amphibolite, granulite, and pyroxene-granulite) at temperature between 0°C and 150°C and high pressure simultaneously. Their results demonstrated a nearly linear temperature dependence of the λ. Although the above-mentioned studies provide useful insights into the temperature dependence of λ for rocks, the knowledge of temperature effect on λ need to be improved.
The aims of the present study are (1) to develop a novel model to predict λ as a function of temperature and (2) to provide a basis for the correction of λ at high temperature.

Samples and Methods
2.1. Sample Information. Twenty-eight rock specimens of outcrops are collected, including twelve carbonates, five clastic rocks, six intrusive rocks, and five volcanic rocks. To ensure the universality and impersonality of experimental results, these opted samples are not restricted in one local region, but locates in a wide range of sedimentary basins or structural belts, involving North China Plain, Sichuan Basin, Tarim Basin, Songliao Basin, etc. To rule out rock structure effect on thermal conductivity, all samples chosen for the study were homogeneous and isotropic at hand scale, with no visible bedding or cracks. All specimens were deposited in a dry environment at room temperature and atmospheric pressure. Each sample was prepared as two cuboids of 50 mm length, 50 mm width, and 15 mm thickness for thermal conductivity measurements (Figure 1). The surfaces of the processed cuboids must be as flat as possible. Even though there is a little microirregularity on the sample's surface, a strong contact thermal resistance can be produced leading to fluctuating results. To ensure this, each sample surface in contact with the probe (introduced in the following text) must be polished. Detailed descriptions and physical parameters of the analyzed specimens are listed in Table 1.

Methods.
Over the past decades, methods in thermal analyses have been substantially improved in accuracy and expanded to dozens of secondary approaches. According to the status of temperature field inside the measured samples, these experimental techniques can be classified into two groups. (1) Steady methods: these ones feature a steady temperature field across the sample throughout the measurement process and were extensively employed in the past. High accuracy is their most prominent strength. However, its price is rather time-consuming and expensive. It will take five to seven hours to wait for the establishment of steady temperature field. For these disadvantages, steady methods have been hardly employed in the studies of rocks or minerals in recent years. (2) Transient methods: contrary to steady methods, transient methods have a variable temperature field in the interior of samples throughout transient measurements. By contrast, an outstanding upside of transient methods is time-saving and cheaper. The measurement process duration typically ranges from a couple of seconds to a few minutes. In the past, they are less accurate than steady methods. By decades of improvement and development, transient methods have been accurate enough for the determination of rocks now. Most researchers preferred to them, so the reliability of these approaches have been thus validated. The major transient methods are as follows: (1) optical scanning (uncertainty 2%) [9], (2) need-probe (uncertainty 5%) [10], (3) hot wire (uncertainty 4%) [11,12], and (4) transient plane source (TPS) (uncertainty 3%) [13]. In this study, TPS method was employed.
The TPS method is based on the principle of plane transient heat conduction in an infinite medium, which ensures that heat flow transporting through the specimen cannot reach its boundaries. After the measured sample is fixed in the sample-holder, waiting time of three minutes at least was required to achieve the thermal equilibrium between the measured specimen and the probe, before each starting of measurement. Upon reaching this state, the specimen was heated by the probe, producing a thermal disturbance in the temperature field within the specimen. Since the thermal conductivities of materials are diverse, the corresponding temperature rise varies. Specimens with good thermal conducting properties are characterized by rapid heat transport inside the specimens, and the probe temperature rise is relatively slow and low. In contrast, for specimens with good thermal insulation properties, the probe temperature rises rapidly, and heat is transported slowly.

Geofluids
The temperature rise (Δ T) and the dimensionless time function (D s ðτÞ) can be fitted as a linear function (Equation (1)). The slope of the straight line obtained is a function of the value of λ in the measured specimen. The corresponding temperature rise can be expressed as: where Δ TðτÞ is the mean temperature rise in the probe, P 0 is the total output power of the probe, r is the radius of the probe, λ is the thermal conductivity of the specimen, τ is a dimensionless function, depending on time, α is the thermal diffusivity of the specimen, t is the measurement time from the start of the transient recording, and θ is the characteristic time. According to Equation (1), λ can be calculated as the following: where k, which is known, represents the slope of the line.

Equipment and Measurements.
In this work, a Hot Disk TPS2500S (Figure 2(a)) was adopted to measure the λ value of the analyzed specimens. The apparatus measures thermal conductivity of materials ranging from 0.005 to 500 W/m·K in accuracy and reproducibility better than 3% and 1%, respectively. The core component of the apparatus is a thin double spiral probe, etched out of a thin nickel foil (Figure 2(b)), which is sandwiched between two sheets of polyimide electrical insulator. A constant direct current is applied to the probe, causing its surface temperature to rise and the electric resistance of the nickel wire inside the probe increases simultaneously. The temperature of the surface of the specimen is associated with the resistance of nickel wire based on the known equation constraining temperature and 3 Geofluids resistance. The probe acts as both the heat source and a temperature recorder.
Prior to thermal conductivity measurements, the specimens were soaked in an acetone solution for 3 h in order to remove oil at the surface, cleaned using an ultrasonic cleaner, and placed in an oven at 50°C for 48 h to dry. Subsequently, the specimens were baked at 100°C for 5 h to remove adsorbed water. Thermal conductivity measurements were conducted at 25°C, 50°C, 100°C, 150°C, 200°C, 250°C, and 300°C. The temperature was controlled by the PF Oven ( Figure 2(c)), which is capable of controlling temperature within a range of just ±0.5°C for 30 min. Herein, we set the interval time between two adjacent measurements as ten minutes, which is more than three minutes of official recommendations. Two specimens were tightly clamped together to ensure the inexistence of an air gap between the specimens and the probe (Figure 2(d)). Three repeated measurements are conducted at each temperature to check the reproducibility, and their averages are regarded as the final result. Generally, one temperature sequence measurement required at least twelve hours.

Results
Experimental results are listed in Table 2 and shown in Figure 3. At room temperature (25°C), the average thermal conductivity is 5:03 ± 1:20 W/m · K and 3:43 ± 1:48 W/m · K for carbonate and clastic samples, which is obviously greater than that of the intrusive and volcanic rocks, 2:09 ± 0:29W/ m · K and 1:57 ± 0:41 W/m · K, respectively. Standard deviation shows a more concentrated distribution of thermal conductivity for intrusive and volcanic rocks at room temperature, as confirmed by Figure 3. Thermal conductivity of most samples decreases as temperature rises except for a few intrusive and volcanic samples. Although majority of samples exhibit a decrease in thermal conductivity with temperature increase, there are considerable discrepancy in the slope dλ/dT and the extent of decrease from 25°C to 300°C among these samples. As shown in Figure 3, the intrusive and volcanic samples exhibit a gentler decrease compared with the others.

Major Influencing Factors of Rock Thermal Conductivity.
Thermal conductivity of rocks at a fixed temperature is dominated by internal factors, including composition, porosity, and structure. From Figure 3, the dolomite rocks have approximately twice the thermal conductivity of the marble and limestone. Such a difference can be explained by their composition. Dolomite, a major mineral in dolomite rocks, has a lager thermal conductivity of 13.16 W/m·K as compared to 8.58 W/m·K of the calcite, a main mineral in limestones [14]. However, thermal conductivity of dolomite and 4 Geofluids calcite is much more than their corresponding rocks. This is because there exists thermal resistance on the grain boundary as well as imperfection and defect inside the crystal within a rock. For clastic rocks, the large value of 6.25 W/m·K may be attributed to the incorporation of quartz. Quartz is a common mineral in clastic rocks and has a considerable thermal conductivity of up to 18.37 W/m·K [15]. Compared with the carbonate and clastic rocks, intrusive and volcanic rocks alike exhibit a lower thermal conductivity ranging from 1.10 W/m·K to 2.56 W/m·K and more concentrated distribution. The low thermal conductivity may be due to the low thermal conductivity of the minerals composed of igneous rocks. For example, feldspar, the most common and abundant mineral in Earth's crust, has a thermal conductivity just from 1.50 to 2.50 W/m·K [15]. The concentration of thermal conductivity can be because different types of igneous rocks have same types of forming-rock minerals. For instance, many igneous rocks have the same minerals of feldspar, biotite, and quartz. Though the range of thermal conductivity is different for the four types of rocks, their range largely over-lap each other. Therefore, lithology cannot be exactly distinguished only by relying on thermal conductivity.

4.2.
Effect of Temperature on Thermal Conductivity. The λ of almost all rocks, as presented in Figure 3, decreases with increasing temperature, consistent with expectations. Different from the previous studies [5,[16][17][18], a unique phenomenon, that the λ of some intrusive and volcanic rocks firstly increases and then decreases with rising temperature, can be observed from Figures 3(c) and 3(d). This will be discussed in detail below. The λ of carbonate and clastic rocks have a more evident decrease than intrusive and volcanic rocks. For carbonate and clastic rocks, the mean values of the temperature derivative of λ are from -1.32 to -0.26 W/m·K/100°C and from -1.11 to -0.07 W/m·K/100°C, respectively. In contrast, the λ of intrusive and volcanic rocks is universally insensitive to temperature, frequently presenting a slight increase or an increase followed by decrease.
The mean values of the temperature derivatives for intrusive and volcanic rocks are only -0.15 to +0.02 W/m·K/100°C and -0.16 to +0.02 W/m·K/100°C, respectively. The positive mean value demonstrates that the λ at 300°C is greater than the value at room temperature, meaning a slight increase of λ. The rise in λ should stem from the incorporation of amorphous materials, due to rapid crystallization of magma. Amorphous materials, the opposite of crystals, are solids that lack the long-range order, which is a typical characteristic of a crystal. It has been considered that the thermal conductivity of amorphous materials increases as temperature rise [19,20]. Consequently, we speculate that rocks for which thermal conductivity initially increases and then decreases with temperature may be the mixtures of crystalline and amorphous constituents. Generally, amorphous constituents are just incorporated into volcanic rocks [21] due to rapid crystallization process, which is caused by the huge difference in temperature between the magma and the context at ground level. But a similar tendency to the volcanic rocks has been observed in intrusive rocks, which is normally produced in subsurface having a greater temperature than the ground level. This may occur because the intrusive rocks studied crystallize at swallow depths and, thus, at lower temperature, as may lead to a little formation of amorphous constituents in despite of being in underground. Additionally, an asymptotical convergence of λ can be observed for carbonate and clastic rocks with increasing temperature, in agreement with previous studies [1,3,8]. The standard deviation of λ at room temperature in carbonate and clastic rocks was 1.25 W/m·K and 1.65 W/m·K, respectively. At a temperature of 300°C, the corresponding value decreased to 0.44 W/m·K and 0.54 W/m·K, respectively. Based on the shape of the curves of λ − T in Figure 3, we obtain an empirical rule showing that the curves of λ − T, regardless of lithology, are governed by the value of λ at 25°C. This rule has been plotted in Figure 4. These curves shown in Figure 4 can be classified into three groups. (1) The λ of the specimens with the λ values higher 4.5 W/m·K at room temperature strongly decrease, and the absolute value of the temperature derivative of thermal conductivity (ⅆλ/dT) gradually decreases. The resulting curves have a  (3) The λ of the samples having values below 2.5 W/m·K either initially increase and then decrease or slightly increase. Irrespective of the trend observed in the last case, the influence of temperature is both minor. Experimental results analogous to the first group have been also observed in the studies headed by Seipold (1998) and Vosteen [7]. Therefore, the concave pattern shown in the first group may be a universal characteristic for those rocks with a high thermal conductivity (>4.5 W/m·K). By contrast, the second and third cases have not been explicitly found in past publications yet. Unfortunately, the statistics have failed to be logically explained from a physical perspective at present. Despite all this, the rule can still provide a distinctive thinking that the λ of rocks at high temperature can be statistically reflected though its λ at room temperature.
According to the foregoing thought, the temperature effect can be quantified by defining a new term of α t , which means the decrease rate of λ at temperature of t: in which α t is the rate of decrease at temperature of t, λ ð25Þ is the thermal conductivity at 25°C, and λ t is the thermal conductivity at temperature of t. Based on the present data, we calculated the α t at 50°C, 100°C, 150°C, 200°C, 250°C, and 300°C for each sample. The obtained results are plotted in Figure 5. Equation y = ax + bð1,/,xÞ + c was employed to fit the results, obtaining fitting parameters a, b, and c ( Table 3). The R-squared at different temperature are all nearly 0.90 except the temperature of 50°C.

Geofluids
The generality of the method is validated by applying them to some independent data set from ( [6,22]. The Figure 6 presents the comparation of experimentally measured values with estimated values by current method. As a whole, the estimated values are consistent with measured values. Although some slight difference exists, this is possible due to the uncertainty of experiments. These discrepancies range from -4.0% to +3.5% with an average of 0.09%, which is low enough for the estimation of thermal conductivity in the deep of a basin.

Implication for In Situ Thermal Conductivity.
In the studies on temperature and heat flow of a basin, the situ thermal conductivity has long been roughly evaluated. Obviously, it is unclear. Studying thermal property of rocks under high temperature or pressure can provide better advices for these researches demanding precise thermal conductivity in situ. By rating the present findings to the previous temperature data published by others, the temperature effect of thermal conductivity of the deep carbonates in Tarim Basin has been estimated. From the paper [2], we collected the raw data of thermal conductivity for carbonate rocks in Tarim Basin from 4000 m to 7000 m depth, involving ten limestones and four dolomites (Table 4). In real strata condition, temperature, pressure, and fluid exert an impact on thermal conductivity. Temperature is negative, and pressure and fluid are positive [19,21,23,24]. Their combination makes the determination of in situ thermal conductivity more complicated and problematic. For a given rock, the characteristics of rocks such as porosity and mineral are closely linked with the influ-ence of external factors [25,26]. According to the previous studies [27,28], the effect of fluid on thermal conductivity depends chiefly on the porosity of rocks. The rocks with larger porosity will have a more remarkable increase in thermal conductivity. Further, the distribution and movement of fluid and thermal conductivity in dry conditions are also associated with the dependence of thermal conductivity. Especially for the rocks with low thermal conductivity in dry conditions, thermal conductivity can even be double at 100% water saturation [24]. Among the referred rocks, the five samples containing fluid have a small porosity ranging from 2.5% to 5.0%. Hence, these samples are treated as dry ones. The effect of pressure depends on porosity and elastic modulus [29,30]. For the rocks having a large porosity, thermal conductivity will significantly increase. Generally, the pressure dependence of thermal conductivity can be divided two stages [4,17,31]. In the first, pressure can close the pores and cracks inside rocks, in which thermal conductivity will considerably expand. In the second, the pores and cracks are closed, and pressure increases thermal conductivity by enhancing the efficiency of heating transfer in crystal lattice [21], in which the increase of thermal conductivity is relatively small and slow. Because the porosity does not pronouncedly change from underground to ground, the pressure forced by the above strata is assumed just to act on the second stage. The pressure of rock matrix can be calculated by Equation (4).
in which P is the pressure of rock matrix, ρ rock is the mean density of rocks of 2800 kg/m 3 , ρ water is water density of 1000 kg/m 3 , g is net acceleration of 9.8 N/kg, and H is the depth. By a simple calculation, P is 141 MPa at 8000 m.
According to the related studies [4,8,21,32], pressure can increase thermal conductivity from 1% to 5% for the rocks with low porosity at the pressure of 150 MPa in the second stage. If the depth of rocks is shallow, the effect will be more insignificant. The order of magnitude for increase in pressure is less than that of temperature. Therefore, the effect of pressure can be neglected in this correction. Thermal conductivity of these samples is calculated at 50°C, 100°C, 150°C, 200°C, 250°C, and 300°C, using the parameters shown in Table 3. The results are illustrated in Figure 7. Since the in situ temperature of the samples is unknown, their temperature can only be obtained by calculation. Temperature at a certain depth can be formulated as: where T is temperature at a certain depth, T 0 is land surface temperature, D is depth, and gradT is geothermal gradient.
In the light of the previous related studies, the gradT and T 0 in Tarim Basin is taken to 20.7°C/km and 18°C [33], respectively. So, T is calculated (as shown in Table 4). Each calculated temperature value is substituted to the corresponding curve to attain a point of intersection in Figure 8. Its vertical coordinate is the corrected value of thermal conductivity (as exhibited in Table 4). The comparison between   Figure 4: Schematic diagram of the three groups of thermal conductivity deduced as a function of temperature. Blue denotes samples with the λ higher 4.5 W/m·K at room temperature; red denotes samples with the λ between 2.5 and 3.5 W/m·K at room temperature; green denotes the samples with λ below 2.5 W/m·K at room temperature. 7 Geofluids the corrected and measured values are plotted in Figure 6. A conclusion can be easily reached. The value of thermal conductivity at room temperature is a more critical factor to govern the extent of temperature effect rather than the depth.
The limestones with low original thermal conductivity are slightly influenced by temperature even though they are at considerable depth. By contrast, the dolomites with high thermal conductivity have a remarkable decrease in thermal   The corrected results are examined by continuous temperature logging data and thermal conductivity. The temperature profile and stratigraphic column of the well Zhong 4 in Tarim Basin are collected (Figure 9). Temperature gradient decreases considerably at 5400 m depth from 22.9°C/km to 14.7°C/km, which is derived from the great discrepancy between the upper and lower strata in thermal conductivity. The upper strata is composed of sandstone, mudstone, and limestone. According to the previous data of thermal conductivity in the Tazhong area of Tarim Basin [34] and the thickness of corresponding rocks, the average thermal conductivity of the upper strata is attained by calculating the weighted harmonic mean, and it is 2.34 W/m·K. The lower strata of dolomite has a mean thermal conductivity of 4.03 W/m·K. Then, the heat flow can be acquired, being 53.59 mW/m 2 and 59.24 mW/m 2 through the upper and lower strata, respectively. The difference stems from the uncorrected thermal conductivity of the lower strata. Because of the hot temperature and greater thermal conductivity in lower strata, it is necessary to take into temperature effect on thermal conductivity account. Contrary to this, the upper strata have a low thermal conductivity, and the circumstance temperature is also low. Both two factors commonly lead to the weak temperature effect. According to the foregoing method, thermal conductivity of the lower strata is corrected as 3.25 W/m·K. With the corrected value, the recalculated heat flow is 47.78 mW/m 2 , slightly less than that of the upper strata. This is due to the radioactive heat generation in rocks. Radiogenic heat production rate of rocks in Tarim Basin approximates 1.17 μW/m 3 [35]. Based on this, the contribution of radiogenic heat of the upper strata to heat flow can be obtained by the radiogenic heat production rate multiplied by the thickness of strata, and it is 6.43 mW/m 2 . Adding 6.43 mW/m 2 to 47.78 mW/m 2 , the total heat flow is obtained and 54.21 mW/m 2 , which is essentially equal to the value of 53.59 mW/m 2 . The agreement indicates that it is necessary to correct thermal conductivity of rocks under high temperature, especially for those rocks with great thermal conductivity.

Conclusion
From the measured thermal conductivity of twenty-eight samples, the following conclusion can be derived: (1) Thermal conductivity of samples at 25°C reveals that the average of λ for carbonate and clastic rocks is greater than that of intrusive rocks and volcanic rocks. In spite of it, various lithology rocks may still be overlap each other to a great extent due to numerous factors to exert a critical effect on λ of rocks  (2) The experiment under high temperature demonstrates that the λ − T curves can be divided into three categories relying on the value of λ at 25°C regardless of lithology. With a great λ above 4.5 W/m·K at 25°C, the rocks have strongly decreased λ with increasing temperature, and the λ − T curves gradually slow down. If lying from 2.5 W/m·K to 3.5 W/m·K, the λ shows a linear decrease. When below 2.5 W/m·K, the λ slightly increases from room temperature to 150°C and then decreases at remaining temperature range. The phenomenon may be explained by the fact that the silicate material with similar the λ at room temperature may exhibit analogous respondence of the λ to temperature. While deviating significantly in mineralogy, chemical structure, elements, etc., different types of rock can be taken as the silicatebearing material and therefore is constrained by the heat transfer law of silicate (3) The correction of thermal conductivity in Tarim Basin suggests that the effect of temperature on thermal conductivity depends strongly on its value at room temperature, instead of depth at which the sample lied. Although pressure and fluid can affect thermal conductivity as well, their effect can be ignored compared with that of temperature in present condition. The correction of thermal conductivity is necessary for those rocks with high thermal conductivity under high temperature   [22], and the remaining samples are from [6]. 10 Geofluids

Data Availability
The data used to support the findings of this study are available from the first author upon request. First author: Chi-Chen; email: chenchigeosci@foxmail.com.