Effect on the thermal conductivity inhomogeneity of clay-bearing sandstone subjected to drying–wetting process

Thermal conductivity of rock is one of the important parameters to understand the heat conduction process in interior of the earth. The study of the effect on the thermal conductivity of clay-bearing sandstone subjected to drying–wetting process is of great significance to many geological and geotechnical engineering issues. In this study, drying–wetting cycle experiments on clay-bearing sandstone were carried out, including three times of drying and two times of water saturation treatment. The thermal conductivity of clay-bearing sandstone after each treatment was measured by transient hot wire method, and the thermal inhomogeneity was analyzed. The results indicate that the drying–wetting process leads to the decrease of the average thermal conductivity of clay-bearing sandstone, while the increase of thermal heterogeneity factor. Base on the results of 3D scanning and SEM, it is found that the development of pores and microcracks during the drying–wetting process is the main reason for the average thermal conductivity decreased and the thermal inhomogeneity increased. Further analysis shows that the interaction between clay minerals and water leads to the destruction of rock matrix structure, resulting in the increase of primary pores, the formation of new pores and secondary microcracks in clay-bearing sandstone. In addition, the numerical results show that the pore leads to the significant weakening of rock heat transfer effect, and the temperature field tends to be heterogeneous distribution. The research results can provide reference for the evaluation of thermal conductivity of rock mass in deep engineering.


Introduction
Thermal conductivity of rock is one of the important thermophysical parameters in geoscience and geotechnical engineering, such as disposal of high-level radioactive waste (Kujundžić et al. 2012;Zhao et al. 2018Zhao et al. , 2019, exploitation of geothermal reservoir (Ramazanova et al. 2012;Hofmeister et al. 2014), recovery rate evaluation of heavy oil reservoir (Sun et al. 2017;Popov et al. 2013). In addition, the influence of drying-wetting process on the physical and mechanical properties of rock has always been an important research topic in the field of geoscience. Due to the in-situ rock is affected by precipitation, evaporation and groundwater level fluctuation for a long time, the drying-wetting process will inevitably lead to structural damage, deterioration of physical properties and mechanical properties of rock (Wen et al. 2015;Zhao et al. 2017;Xie et al. 2019;Guo et al. 2021;Wang et al. 2020). Therefore, the heat conduction characteristics of rocks in dry and wet conditions cannot be ignored, especially for rocks with high clay mineral content (Guo et al. 2017).
Effective thermal conductivity (ETC) of rock is considered as a function of rock matrix thermal conductivity (MTC), pore fluid thermal conductivity (FTC) and porosity (Albert et al. 2017a). Therefore, the thermal conductivity of rock is controlled by mineralogical composition (mineral type, particle contact, content and distribution), pore fluid (fluid type, content, distribution) and pore structure 1 3 328 Page 2 of 17 (porosity, pore diameter, pore distribution and connectivity) (Brigaud and Vasseur 1989;Clauser and Huenges. 1995;Popov et al. 2003;Abdulagatova et al. 2009;Sun et al. 2017;Jia et al. 2019;Abid et al. 2014;Shen et al. 2018a, b). At present, most of the published articles focus on the study of ETC in both dry and water-saturated conditions, or the quantitative relationship between ETC and moisture content or water saturation (Jorand et al. 2011;Nagaraju et al. 2014;Dochkov et al. 2014;Esteban et al. 2015;Luo et al. 2016;Yuan et al. 2021). However, some studies have shown that the drying-wetting process has a significant effect on the heat conduction characteristics of sedimentary rocks with high clay mineral content, and there is a big error in thermal conductivity before and after drying-wetting treatment. In this case, if the dry and wet environment of the rock is not fully considered, and the laboratory measurements represent the in-situ conditions, error of thermal conductivity characterization will be converted into error of heat flux calculation (Nagaraju et al. 2014). Fuchs et al. (2013) evaluated the accuracy of the two-component (rock matrix and pores) models that are used widely in rock thermal conductivity, including arithmetic mean, geometric mean, harmonic mean, Hashin and Shtrikman mean, and effective-medium theory mean. To improve the fitting of all models, the corresponding correction equations are calculated. The "modified" geometric mean provides the most satisfactory results and constitutes a universally applicable sedimentary rock model. Finally, a lithology-specific conversion equation is provided, which allows the calculation of water-saturated ETC from dry measured ETC and porosity. Albert et al. (2017b) found that water saturation affects not only ETC, but also the heat transfer behavior of mudstone matrix, thus revealing the effect of water saturation on the small-scale heterogeneity of thermal conductivity. Kämmlein and Stollhofen. (2019a,b) found that the difference in absolute MTC values of sandstones under different measurement conditions (air saturation or water saturation) may be related to the formation of authigenic kaolinite in pores, which results from the alteration of alkaline feldspar. Their research also shows that repeated drying and vacuum saturation can lead to the disintegration of clay minerals or the destruction of the lattice of clay minerals. Guo et al. (2020a) carried out an experimental study on three kinds of sandstones with different clay mineral contents. The results show that the drying-wetting process leads to microcracks in the matrix structure of sandstone, which leads to a significant decrease in ETC.
In the past few decades, some in-situ methods have been developed to determine the heat conduction performance of rock mass (Wilhelm 1990;Williams and Anderson 1990;Burkhardt et al. 1995;Pribnow and Sass 1995). However, its measurement accuracy cannot be compared with indoor experiments. Therefore, laboratory measurement is still the most reliable method to obtain thermal conductivity, including steady state method (protective hot plate method and divide bar method) and transient method (linear heat source method, plane heat source method and optical scanning method) (Popov et al. 2016;Guo et al. 2020b). Among them, the transient method is widely used because of its fastmeasuring speed and high accuracy. However, the detection depth and range of the transient device for the measuring surface of rock samples are limited. As a result, the measurement results only represent the thermal conductivity in range of centimeters or millimeters on the sample scale. For example, the Hot Disk TPS 2500 s experimental device used by Zhao et al. (2016) has a detection depth of about 10 mm. Pimienta et al. (2018) measured the 5 mm area of the specimen surface diameter by using the thermal conductivity scanner (TCS). Meanwhile, rock is a kind of anisotropic and heterogeneous material, and its thermal conductivity will change with the variation of spatial position. Based on the thermal conductivity of 89 rock samples collected in the literature, Deming (1994) proposed an empirical relationship between thermal conductivity anisotropy and thermal conductivity perpendicular to bedding. Davis et al. (2007) used a line source device to measure the thermal conductivity of 74 sedimentary rock, metamorphic sedimentary rock and granite samples, the anisotropy of thermal conductivity is calculated and analyzed. The thermal conductivity of two heterogeneous rock samples were measured by Jorand et al. (2013), and the effects of mineral inclusions, bedding and open microcracks on the distribution of thermal conductivity heterogeneity were studied. Haffen et al. (2017) obtained the porosity map by measuring the thermal conductivity map of granite and sandstone. The results allow thermal conductivity and porosity to be quantified at a millimeter resolution on the sample scale. Li et al. (2020aLi et al. ( , b, 2021 has studied the changes of thermal conductivity distribution on the granite sample scale before and after mechanical damage and heat treatment. The research results can provide a reference for the study of the evolution mechanism of thermal conductivity in engineering rock mass under deep high-temperature and high-pressure environment. Wu et al. (2021) measured the thermal conductivity of granite, rhyolite, sandstone and mudstone, focused on the relationship between rock type, microstructure, mineralogical composition and thermal conductivity anisotropy, and established the correction equation of rock thermal conductivity anisotropy.
At present, the mechanism of the effect on the thermal conductivity of clay-bearing sandstone subjected to drying-wetting process have not been fully understood. Therefore, this paper focuses on the effect on the thermal conductivity inhomogeneity of clay-bearing sandstone subjected to drying-wetting process. The clay-bearing sandstone was dried for three times and water-saturated for two times, and the thermal conductivity of clay-bearing sandstone after each treatment was measured by transient hot wire method. Meanwhile, the changes of structure on the sample during the drying-wetting process were obtained by 3D scanning and SEM. On this basis, the numerical model is used to further reveal the influence mechanism of pore on rock heat conduction.

Samples preparation
Clay-bearing sandstone is one of the most common rocks of coal mine top floor in China, and its thermophysical parameters are very important in the thermal load calculation and cooling system design of heat hazard mine. In this paper, the clay-bearing sandstone was taken from a coal mine in Shandong Province of China. The geomorphology unit of the mine field is the Quaternary piedmont alluvial plain, with flat terrain, and the ground elevation is generally about + 0.84 m to + 7.97 m. The strata from old to new are Mesozoic cretaceous, Cenozoic Paleogene and Quaternary. The water abundance of aquifer in the mine is weak to medium, and there is no obvious hydraulic connection between the aquifers. The water content is mainly static reserves. The stratum at the sampling position in this study belongs to the Quaternary system, and the original rock is less affected by groundwater. All disc samples have complete structure and no obvious bedding.
Twenty disk samples with a diameter of 50 mm and a height of 25 mm were drilled on natural rocks and grouped in pairs. Before the drying-wetting cycle treatment of the clay-bearing sandstone samples, the mineralogical composition and basic physical properties of the samples in the natural state were tested. All the tests were carried out at room temperature and standard atmospheric pressure. Among them, the mineralogical composition is analyzed by use D/ max-2500 X-Ray diffractometer, and the average P-wave velocity of the sample was obtained by using MC-6310 nonmetallic ultrasonic detector. The results show that the content of quartz and feldspar in the sample is 39.4% and 14.4% respectively, while the content of clay minerals (kaolinite, illite, and chlorite) reaches 46.2% (Table 1). Moreover, the average density and P-wave velocity of natural samples are 2.43 g/cm 3 and 1890 m/s, respectively.

Experimental procedure
As shown in Fig. 1(a), the samples are dried three times and water-saturated twice. Dry condition: the sample is heated in a drying box ( Fig. 1(b)) at 70 °C for 24 h, until the mass of the sample reaches a constant, and then cooled to room temperature in a drying dish. Water saturation condition: soak the sample in a container containing deionized water ( Fig. 1(c)) for more than 48 h until the water content of the sample reaches a constant (Guo et al. 2020a). In addition, to limit water evaporation and side damage of the samples, each sample is wrapped in plastic film except the measuring surface (Haffen et al. 2017). The mass and thermal conductivity of the samples after each drying-wetting treatment were measured by using ADAM electronic balance (range from 0 to 750 g, error 0001 g) ( Fig. 1(d)) and transient hot wire experimental device (XIATECH Instrument TC3000) ( Fig. 2(a)). Meanwhile, scanning electron microscope (Apreo SEM) ( Fig. 1(e)) and 3D optical scanning system ( Fig. 1(f)) were carried out to observe the microstructure and macroscopic of all samples in three times dry condition.

Thermal conductivity measurement methodology
In this study, the thermal conductivity of the sample was measured through the hot wire method suggested by International Society for Rock Mechanics (ISRM 1979). Guo et al. (2017Guo et al. ( , 2020a describes in detail the main equipment and technical parameters involved in the experimental system. The accuracy of the device can reach ± 3%, but the penetration depth and range of the sample are limited, it can only measure the hot wire radial 10 mm range ( Fig. 2(b)), the measurement results represent the thermal conductivity near the measuring line of the sample. Therefore, to study the heterogeneity of thermal conductivity distribution on the sample scale, the surfaces of each pair of samples are marked as surfaces 1, 2, 3 and 4, and the four surfaces are combined according to 1-3, 1-4, 2-3 and 2-4 ( Fig. 2(c)). In addition, because the length of the probe hot wire is 25 mm, five measuring lines are marked on each measuring surface, and the angle between the measuring lines is 72° ( Fig. 2(d)). When measuring thermal conductivity, place the hot wire between two measuring lines with the same label on the two measuring surfaces. Five data are collected from each measuring line, and the average value is used as the ETC. Therefore, each composite surface will get 5 ETC values, each pair of samples will get 20 ETC values, and a group of samples will get 200 ETC values. In addition, plexiglass and borosilicate glass are used to calibrate the experimental device before each pair of samples are measured. The thermal conductivity of the two calibration samples is 0.2035 W/mK and 1.1450 W/mK respectively.

3D optical scanning system
Figure 1(f) is the core component of 3D optical scanning system. Its detailed technical parameters are shown in Table 2. Based on the principle of structured light binocular vision, the 3D scanner is highly integrated by two digital cameras, structured light projector and the central control circuit "opto-mechatronics". In the experiment, by scanning the measuring surface of the sample, the computer can obtain the 3D point cloud data of the measuring surface. The commercial software GEOMAGIC is used to

Heat conduction model of porous rock
In this study, it is assumed that the rock is a structure composed of continuous and dispersed phases, the dispersed phase is randomly generated by Monte Carlo method and embedded into the continuous phase, and a heat conduction model of porous rock is established. Monte Carlo simulation is a stochastic simulation method, which needs to test the number of Monte Carlo simulations that need to obtain stable output statistics by observing the influence of the average value of the simulation results and the coefficient of variation (COV) on the number of simulations (Liu et al. 2018). The specific modeling process is shown in Fig. 3(a). The size of the continuous phase is set to 100 × 100 mm, and the dispersed phase is defined as a circle with a diameter of 1-5 mm (Li et al. 2020a, b). Based on the steadystate heat transfer analysis, the finite element model considers the one-dimensional heat conduction process shown in Fig. 3(b). Constant temperatures of T 1 and T 2 ( T 1 > T 2 ) are applied to the upper and lower boundaries of the square model, respectively. The left and right boundaries are set to thermal insulation. Due to the temperature difference, the heat flux will flow through the model specimen from top to bottom. The specimen is heterogeneity, and the heat flux density will change with the spatial position after reaching a steady state, so it is necessary to use the homogenization method to calculate its ETC (Chen et al. 2015): where, A is the area of the 2D specimen, a is the area of the i microelement, n is the total number of microelements. For the finite element method, the microelement can be regarded as an element. Thus, Eq. (1) can be expressed as Eq. (2): where, q i is the heat flux of unit i. Figure 4 is the experimental result of thermal conductivity of all samples in both dry and water-saturated conditions. The thermal conductivity of Dry I, Dry II and Dry III are from 1.671W/mK to 2.640W/mK, and the standard deviations are 0.1894W/mK, 0.1972W/mK and 0.2368W/mK, respectively. The thermal conductivity of Saturation I and Saturation II ranged from 1.986W/mK to 2.793 W/mK, and the standard deviations were 0.1287W/mK and 0.1741W/mK, respectively. Compared with the drying condition, the thermal conductivity of many samples increased slightly in watersaturated condition, the reason is the thermal conductivity of water (0.6W/mK) was significantly higher than that of air (0.026W/mK) (Nagaraju et al. 2014;Popov et al. 2003). However, due to the high clay mineral content, the bound water cannot be completely evaporated at the temperature of 70 °C. Due to the residual water content, the increase of thermal conductivity in partially water-saturated condition is not obvious, even lower than that in dry condition (Jorand et al. 2011). In addition, it should be noted that the average thermal conductivity showed an obvious downward trend in Dry I, Dry II and Dry III conditions. Compared with Dry I, the average thermal conductivity decreased by 0.143W/mK in Dry II condition, with an average decline rate is 6.23%. Meanwhile, the average thermal conductivity decreased by 0.298W/mK in Dry III condition, with an average decline rate is 12.97%. Figure 5 is the thermal conductivity distribution map. It reflects the characteristic of thermal conductivity inhomogeneity of the all sample. The X axis represents the number  of samples and the Y axis represents 20 thermal conductivities measured for each pair of samples. The thermal conductivity distribution on the sample scale is heterogeneous in each state, which is caused by the heterogeneity of sandstone structure. For example, crystal anisotropy in minerals, structural anisotropy caused by rock particle arrangement (Jorand et al. 2013). In addition, there are obvious differences in thermal conductivity distribution under the three times dry condition. As shown in Fig. 6, the proportion of the thermal conductivity between 1.6W/mK and 2.0W/mK in three times dry condition is 0%, 21% and 57.5%, respectively. The proportion of the thermal conductivity between 2.0W/mK and 2.3W/mK in three times dry condition is 52%, 55.5% and 28.5%, respectively. The proportion of the thermal conductivity between 2.3W/mK and 2.7W/mK in three times dry condition is 48%, 23.5% and 13.5%, respectively. It can be seen that while the drying-wetting process leads to the decrease of the overall thermal conductivity, the distribution range also increases. Popov et al. (2003) defined the rock thermal inhomogeneity factor (β), which is used to describe the inhomogeneity of rock thermal conductivity on the millimeter scale, which is calculated by Eq. (3):

Inhomogeneity of thermal conductivity distribution
In this work, Eq. (3) is used to calculate the β value of 10 pairs of clay-bearing sandstone, which is used to characterize the degree of heterogeneity distribution of the thermal conductivity of each pair of samples in both dry and water-saturated conditions. Therefore, max , min and mean are the maximum, minimum and average thermal conductivity for each pair of samples in both dry and water-saturated conditions, respectively. Figure 7 shows β value calculation results. The range of β is from 0.2149 to 0.4736 in dry (3) = max − min mean condition and from 0.1378 to 0.3147 in water-saturated condition. The average value of β in water-saturated condition is significantly lower than that in dry condition, because the difference of thermal conductivity between water and sandstone matrix is much smaller than that between air and sandstone matrix, which will significantly reduce the distribution difference of thermal conductivity. It should be noted that the β values of all samples showed an increasing trend in both dry and water-saturated conditions. Among them, compared with Dry I condition, the average β value of increased by 0.0684 in Dry II condition, with an average increase rate is 26.86%. Meanwhile, the average β value of increased by 0.1727 in Dry II condition, with an average increase rate is 67.70%. Figure 8 shows the change of the standard deviation of the height of the measured surface under three times dry condition. Overall, the standard deviation is from 0.0252 mm to 0.0446 mm in Dry I condition. The standard deviation is from 0.0338 mm to 0.3416 mm in Dry II condition. The standard deviation is from 0.0644 mm to 0.5778 mm in Dry III condition. Compared with Dry I condition, all surface standard deviations are increased in both Dry II and Dry III conditions. Among them, the average standard deviation increased by 0.1221 mm in Dry II condition, with a rate of 353.22%. The average standard deviation increased by 0.2523 mm in Dry III condition, with a rate of 728.64%. However, for each pair of samples, compared with Dry I, the increase degree of height standard deviation of each measuring surface is different during drying-wetting process in both Dry II and Dry III conditions. For example, Fig. 9 is a macroscopic image of the four measured surfaces of the 1st pair of samples. The four surfaces were relatively smooth and flat in Dry I condition, and the standard deviations were 0.0294 mm, 0.0446 mm, 0.0325 mm and 0.0417 mm, respectively. Under Dry II condition, the standard deviations of surface 1 and surface 4 increased to 0.1141 mm and 0.0784 mm, respectively, but only slight damage and microcracks appeared in very small areas. The roughness of surface 2 and surface 3 changed little, the standard deviations were 0.0784 mm and 0.0653 mm, respectively. Under Dry III condition, the damage degree of surface 1 and surface 4 further increased, the damaged area further expanded, and more obvious depressions and damaged areas were formed. At the same time, some cracks will further expand or connect, and even form macroscopic cracks that run through the whole surface. The standard deviation of surface 1 and surface 4 increased to 0.4580 mm and 0.5674 mm, respectively. The roughness of surface 2 and surface 3 also increased, but the standard deviation was only 0.0893 mm and 0.0956 mm, which was caused by the shedding of loose mineral particles on the surface of the sample. Therefore, the damage degree of the sample is heterogeneous, the damage effect is concentrated in the limited area. Accordingly, the decrease degree of thermal conductivity in these areas will be significantly greater than that in other areas.

Effect of microstructure on thermal conductivity during drying-wetting process
The macroscopic damage on the sample is the result of the continuous accumulation of microscopic damage. However, in some places where there is no obvious macro-damage, there may also be microstructure damage. Figure 10 is the microstructure image of the measured surface obtained by electron microscope scanning. Under Dry I condition, the surface of sandstone is relatively flat, the cementation between mineral particles is firm, and the microstructure is relatively dense. The number of pores on the measured surface is small and the pore diameter is small. Under Dry II condition, the measuring surface begins to become rough, the cementation degree between mineral particles decreases, the relatively flat measuring surface appears curling phenomenon, and the pore diameter increases and new pores are produced at the same time. In addition, due to the infiltration of water into the primary pores of sandstone and scouring the pore wall, the cementation degree of mineral particles decreases further, becomes looser, and falls off and leaks along the pores in the drying-wetting process, resulting in slight damage to the measuring surface. Under Dry III condition, the size, number and connectivity of pores gradually increase, and microcracks occur, while the increased pore channels also provide more contact space for waterminerals contact. As a result, the structure of the measuring surface is further deteriorated, flakier, honeycomb structures and microcracks appear, and the damaged area is further increased. Meanwhile, we also used SEM images to observe the internal microstructure of the samples in Dry I, Dry II and Dry III condition. As shown in Fig. 11, the change of the internal microstructure is consistent with that of the measured surface, the number and diameter of pores increase with the increase of dry and wet times, accompanied by the initiation and propagation of microcracks.
Water-rock interaction is the main reason for the damage of rock structure in the process of drying and wetting. Among them, the physical interaction between water and rock (for example, softening and lubrication) leads to a loose connection between the matrix and cement, weakening the combination of rock grains. The chemical processes of water-rock interaction, including ion exchange, dissolution, hydrolysis, hydration, redox etc., contribute to the change of mineralogical composition. In this study, clay-bearing sandstone mainly contains quartz, feldspar and clay minerals. Mechanical behavior of sandstone deteriorated by hydrolysis of quartz (Hadizadeh and Law 1991;Zhou et al. 2018). For silicate rocks in wet conditions, water can be used as a corrosive solvent to hydrolyze quartz minerals. During the hydrolysis of Si-O-Si bond in quartz promotes the growth of subcritical cracks, because the stronger Si-O-Si bond is substituted by the much weaker hydrogen bonds (Michalske and Freiman 1982;Hadizadeh and Law 1991;Liang et al. 2023). Feldspar is a kind of negatively charged silicate mineral with spatial structure, which is easy to dissolve and hydrolyze when it comes into contact with water. Under the action of hydrochemistry, its plasticity and compressibility are enhanced,   (Chai et al. 2014;Zhang et al. 2022;Liu et al. 2016). For this study, due to the high content of clay minerals in the sample and less times of drying-wetting treatment, the expansion, shrinkage and dissolution of clay minerals in the drying-wetting process are the main reasons for the structural damage of samples. It leads to more pores and cracks in the sample and promotes the expansion and extension of existing pores and cracks. Guo et al. (2020a) found that the destruction of rock matrix structure during the drying-wetting process leads to the increase of the pore size of primary pores and the initiation of new pores, while the connection between pores and the expansion and shrinkage of clay minerals will promote the formation of secondary microfractures. To sum up, the development of pores and microcracks will significantly increase the spatial heterogeneity of claybearing sandstone structure. The thermal conductivity distribution varies with the damage degree of the spatial structure of clay-bearing sandstone. Specifically, when the damage degree is small, the thermal conductivity decreases less, while when the damage is serious, the thermal conductivity decreases obviously, which leads to the increase of the distribution difference of thermal conductivity. Moreover, the transient hot wire device used in this study is sensitive to the mineral distribution and structural damage on the measuring surface. Therefore, drying-wetting process leads to the decrease of the average thermal conductivity of clay-bearing sandstone, while the increase of thermal inhomogeneity. In addition, the distribution of minerals and the development of pores and microcracks may change with the change of the axial depth of the sample, which may also affect the distribution of thermal conductivity.
At present, the evaluation of thermal conductivity of engineering rock mass is usually obtained by measuring rock samples in the laboratory. However, our experimental results show that even on the sample scale, the thermal conductivity will change with the change of rock spatial structure, and the drying-wetting process will also influence on the distribution of thermal conductivity. Therefore, it is necessary to carry out heat conduction experiments of large-scale fractured rock mass in the future, in order to increase the accuracy of thermal conductivity evaluation of engineering rock mass.

Effect of pore structure on thermal conductivity during drying-wetting process
The damage of rock structure is essentially the result of the development of pore structure. Pore structure not only promotes the interaction between water and clay minerals, but also affects the heat conduction process of rock. And according to Popov et al. (2003), the pores have a dominate influence on the thermal heterogeneity. Therefore, it is necessary to continue the quantitative analysis of pore structure.

Pore diameter and pore surface areas
Image Pro Plus (IPP) is a graphic analysis software which integrates image acquisition, processing and measurement (Yang et al. 2018). Firstly, the spatial scale of the IPP software is calibrated according to the scale on the SEM image, and the calibration unit is determined to be μm. Then, the SEM image is binarized to obtain the rock pore characteristics, the black area represents the pore area ( Fig. 12(b)). Finally, the accuracy of pore recognition is further improved by eliminating the interference region and segmenting the image threshold (Fig. 12(c)). We select the SEM image with magnification of 2000 times to analyze the pore characteristics. Figure 13 shows the cumulative percentage curve of pore diameter and pore surface area. The pore diameter and surface area of sandstone show an obvious increasing trend during the drying-wetting process, which is consistent with the change of pore structure observed in Figs. 10 and 11. These changes will significantly increase the total porosity of the sample, which can further explain the decrease of thermal conductivity after drying-wetting treatment. From Figs. 8,9,10,11 and 13,it can be inferred that the drying-wetting process will inevitably lead to an increase in the porosity of the sample. When the thermal conductivity of the sample in both dry and water-saturated conditions is measured, the porosity of the corresponding position can be calculated according to the geometric average model (Pribnow and Sass 1995;Pribnow et al. 1996;Haffen et al. 2017):

Porosity
where, sat is the thermal conductivity of in water-saturated condition. dry is the thermal conductivity in dry condition.
mat is the MTC. wat and air are the thermal conductivity of water (0.6 W/mK) and air (0.026 W/mK), respectively. It should be noted that Eq. (6) the conditions of establishment are as follows: (1) The MTC is the only value in both dry and water-saturated conditions. (2) The measured thermal conductivity in water-saturated condition is higher than that of dry condition. Therefore, in order to calculate the porosity of the measured position in three times dry condition, it is necessary to assume that the MTC of remains unchanged, where, sat I i−i is the thermal conductivity of i − i measuring line in Saturation I condition. dry i−i is the thermal conductivity of i − i measuring line in Dry I, Dry II and Dry III conditions. Therefore, corresponding to each measuring line, a porosity value can be calculated. Figure 14 shows more clearly the inhomogeneity of porosity distribution. And the statistical results show that the porosity increases in the dry and wet process, compared with Dry I condition, the increasing range of porosity is between -1.52% and 9.05% in Dry II condition, and the increasing range of porosity is between -0.43% and 12.56% in Dry III condition. Combined with Fig. 5, the thermal conductivity distribution varies with the change of porosity distribution. Therefore, according to Eq. (3), during the drying-wetting process, max − min increases and mean decreases, it can be further explained that the drying-wetting process leads to the increase of thermal inhomogeneity factor.
In addition, the effective porosity of each pair of samples is further obtained by using the Eqs. (8) and (9): where, n is the effective porosity of each pair of samples, V p is the pore volume, V is the bulk rock volume, M sat I is the mass of each pair of samples in Saturation I condition, M dry is the mass of each pair of samples in in Dry I, Dry II and Dry III conditions, w is the water density. Table 3 shows the comparison between the porosity calculated by Eq. (6) and the effective porosity, and there is an error between them. The possible reasons are as follows: (1) Limited by the experimental device, the sample surface data measured in this study is less, and the calculation results only represent the local information of the sample surface.

Fig. 14 Porosity distribution map in dry condition
(2) The calculated porosity is limited to the surface area of the sample and cannot represent the interior of the sample, and Fig. 11 shows that the drying-wetting process also causes damage to the microstructure inside the sample. (3) For high clay mineral sandstone, the process of water-rock interaction is complex. During the drying-wetting process, the structure of the sample changes continuously, and the mass is continuously lost, thus affecting the calculation results of Eqs. (8) and (9).

Heat conduction of porous rock
The numerical model is used to further reveal the influence mechanism of pores on rock heat conduction. Figure 15 is the simulation result of rock heat conduction under different porosity (MTC = 5W/mK) (Albert et al. 2017a, b). As can be seen from the heat flux diagram, because the MTC is significantly higher than that of the thermal conductivity porosity fluid, the heat flow will give priority to the path with low thermal resistance, which is called the "thermal bridge effect" (red arrow). The pores also affect the matrix heat conduction along the heat flow direction (White circle). As the pore diameter increases, the area of the affected area increases, and when the distance between the pores is closer, the affected areas will be connected to each other, forming larger areas, and the heat transfer effect of these areas will become significantly worse (white arrow). Therefore, in the same temperature range (288.15-293.15 K), with the increase of porosity, the "thermal bridge effect" is significantly weakened and the heat flux density decreases, resulting in thermal conductivity decrease (porosity from 10 to 40%, the thermal conductivity under dry and water saturation decreased by 52.75% and 40.10%, respectively). In addition, from the temperature field distribution map, it can be seen that the existence of pores will also lead to the heterogeneous distribution of temperature field. Under the same porosity, the pore fluid will affect the distribution of the temperature field. When the pore fluid is water, some obvious fluctuations in the temperature field tend to smooth (white wireframe). Under the same temperature gradient, the fluctuation of the temperature field will be more obvious in the area with a large number of pores. Therefore, pores will weaken the heat transfer effect of rock and lead to the heterogeneous distribution of temperature field, which is helpful to further explain the experimental results of this paper to some extent. However, our numerical model simplifies the rock structure, and the actual rock structure is much more complex. A more accurate numerical model will be more helpful to fully understand the heat conduction process of rock, which needs further research in the future.

Conclusion
Using transient hot wire techniques combined with 3D scanning and SEM, this paper focuses on the effect on the thermal conductivity inhomogeneity of clay-bearing sandstone subjected to drying-wetting process. In addition, the influence mechanism of pores on rock heat conduction is further revealed by the numerical model. The research results can provide reference for the evaluation of thermal conductivity of rock mass in deep engineering. The main conclusions are as follows: (1) The drying-wetting process leads to the decrease of the average thermal conductivity of clay-bearing sandstone, compared with the Dry I condition, the thermal conductivity in both Dry II and Dry III conditions decreased on average by 6.23% and 12.97%, respectively. Moreover, the drying-wetting process caused the increase of thermal inhomogeneity factor, compared with the Dry I condition, the β value in both Dry II and Dry III conditions increased by an average of 26.86% and 67.70%, respectively. (2) During the drying-wetting process, the dissolution and disintegration of clay minerals in the interaction with water, as well as the expansion and shrinkage of clay minerals will lead to an increase in the number and diameter of pores and the formation of secondary microcracks in clay-bearing sandstone. This is the main reason why the average thermal conductivity decreased and the thermal inhomogeneity increased. (3) The numerical results show that the existence of pores can significantly weaken the heat transfer effect of rock. On the one hand, it leads to the decrease of thermal conductivity, on the other hand, it leads to the heterogeneous distribution of temperature field. In addition, a more accurate numerical model will be more helpful to fully understand the heat conduction process of rock, which needs further research in the future.