1 Introduction

As a major global energy resource, coal is widely used in areas such as electricity generation, heating, industrial production, and transportation (Ishaq et al. 2022), with an annual consumption reaching as high as 160 EJ. Particularly in the field of electricity, approximately 30% of global power supply relies on coal as a fuel source (Ma et al. 2022). Coal plays a crucial role in the industrial and economic development of nations, especially in developing countries such as China, Brazil, and India. However, it is accompanied by a significant environmental pollution issue, including the release of pollutants and greenhouse gases during coal combustion, as well as contamination of soil and water sources (Ivanova et al. 2022; Munawer 2018). In recent years, countries and international organizations have been actively promoting the adoption of clean energy to reduce environmental impacts, mitigate climate change, and facilitate sustainable energy development (Jaiswal et al. 2022; Shahsavari and Akbari 2018; Vanegas Cantarero 2020). As a result, the crucial question for coal lies in how to continue its usage while implementing clean, efficient, and sustainable extraction methods to minimize environmental impacts.

Tar-rich coal is a type of coal resource with a tar yield exceeding 7% (Ju et al. 2023). Its main feature is the abundance of a hydrogen-rich structure in the coal, which can be pyrolyzed to generate oil and gas, as well as having a high tar yield. The in-situ pyrolysis technology can effectively extract coal-based oil and gas resources from tar-rich coal. It is a method of heating the coal seam to the pyrolysis temperature in the original stratum through various means (such as electric heating, high-temperature gas heating, etc.) to directly obtain oil and gas products under pyrolysis, achieving the extraction of oil and the carbon sequestration effect, without the need for conventional mining techniques. Compared to traditional mining techniques, in-situ pyrolysis technology, as it operates underground and occupies a smaller surface area, has the potential to reduce emissions of polluting gases and the risk of soil and water source contamination (Bhutto et al. 2013). Therefore, it is considered an economically, environmentally friendly, and sustainable novel mining technology.

X-ray computed tomography (CT) is a non-destructive testing technique that does not harm the object being examined, thereby enhancing the accuracy of research data. CT technology has significant advantages in studying the distribution of porous network structures, quantitatively characterizing structural properties, and the flow behavior of internal structures within objects (Mostaghimi et al. 2017). Yao et al. (2009) used X-ray scanning technology to extract and quantitatively characterize the pores of coal in Chinese coal samples. Shi et al. (2018) employed CT technology to study characterization parameters of micro-fractures in different coal grades and explored the relationship between fractal dimension and permeability. Wang et al. (2019a) conducted three-dimensional reconstruction of coal and studied the relationship between coal deformation behavior and permeability by CT imaging technology. Furthermore, many researchers have quantitatively characterized the pore-fracture structure and permeability of coal (Li et al. 2012; Wang et al. 2020a, b; Zhang et al. 2019; Zhao et al. 2018), but most of these studies have focused on characterizing and analyzing the pore structure of coal at normal temperature. To our knowledge, there has been relatively limited research on coal’s pore-fracture structure under high-temperature pyrolysis conditions (Shi et al. 2023a, b; Yu et al. 2012; Zhao et al. 2017), with even less research on tar-rich coal under high-temperature pyrolysis.

Therefore, this study will take the tar-rich coal in northern Shaanxi of China as the research object. The coal samples will be heated to different pyrolysis temperatures under a nitrogen atmosphere, and advanced Nikon CT scanner will be employed to scan the tar-rich coal samples after high-temperature pyrolysis. Pore-fracture structures of tar-rich coal under high-temperature pyrolysis will be extracted, and the evolution characteristics of coal’s pore and fracture structures will be studied separately, along with quantitative characterization. The fractal dimension of the pore and fissure structure of the coal was calculated. The development of pores and fractures, as well as the homogeneity, was analyzed at different temperatures. Finally, the impact of pores, fractures, and their coupling effects on permeability at different pyrolysis temperatures was analyzed. These research findings will enhance the theoretical foundation of tar-rich coal under high-temperature pyrolysis. They provide valuable information for in-situ pyrolysis extraction technology of tar-rich, contribute to improving pyrolysis and product recovery efficiency, and promote the sustainable development of tar-rich coal.

2 Experimental methods

2.1 Sample collection

The experimental samples were collected from Jurassic coal seams in the Shenfu Mining Area of northern Shaanxi, China, with a coal seam burial depth of approximately 400 m. Details of industrial and elemental analysis data are shown in Table 1, and the coal tar yield was measured at 9.67% using the Gray–King assay. To prevent internal oxidation, decomposition, and evaporation of coal chunks after extraction, it is transported to the laboratory after being strictly sealed. Subsequently, the original coal chunks are cut and polished with sandpaper to achieve a smooth surface, with dimensions of approximately 3.5 cm × 3.5 cm × 1.5 cm. About 2.5 g of fragmented coal chunks are collected as a control group for the heating experiments.

Table 1 Physical properties of tar-rich coal
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2.2 Experimental equipment and process

The samples are subjected to high-temperature heating in the OTF-1200X tube furnace, with a rated voltage of 220 V, a rated power of 1200 W, and a maximum temperature of up to 1200 °C. The instrument’s temperature control accuracy is ± 1 °C. To ensure uniform heating of the samples and simulate the actual conditions of high-temperature nitrogen convection heating for tar-rich coal, nitrogen gas (N2) is injected into the high-temperature furnace, and final temperature settings are established at 300 °C, 400 °C, 500 °C, and 600 °C, with a heating rate of 20 °C/min. Once the samples reach the set temperature, they are kept in a constant-temperature state for 2 h to ensure uniform heating. Afterward, the heating device is turned off, and the samples are placed in a drying oven to naturally cool to room temperature. X-ray computed tomography (CT imaging) technology is utilized to analyze the evolution of pore and fracture structures in tar-rich coal after high-temperature pyrolysis. Firstly, they are shaped into small cubic specimens, and finally, these specimens are polished to create 7 mm cubic blocks. They are then placed in an oven for drying treatment.

2.3 CT image processing

2.3.1 CT scan technical parameters

In this study, a Nikon XTH 320 X-ray CT scanner is utilized. The test samples are 7 mm cubic specimens. The scanner operates in a cone-beam scanning mode with a working voltage of 119 kV and a current of 120 μA. The image resolution of a CT scanner differs from the two-dimensional pixel concept commonly associated with electronic products. It employs the concept of spatial resolution to represent the size of the smallest detail features that CT images can identify (Wang and Fleischmann 2018). Based on the previous studies of some scholars, the pore and fracture structures of coal at high temperatures have larger scales (Wang et al. 2021; Yang et al. 2019). The focus of this study is on the structural development characteristics of coal under high-temperature conditions (300–600 °C). At the same time, the CT instrument is affected by factors like spatial resolution and noise, which are inevitable. In light of these considerations, this study determines the spatial resolution of the CT scanner to be 3.9 μm, with an exposure time of 1 s per projection. A total of 1755 slices with a cross-sectional area of 1790 × 1790 voxels and a thickness of 3.9 μm for each slice were obtained. To compare and analyze the structural development characteristics of tar-rich coal at normal temperature and high temperatures, this study conducted CT scans of 5 sets of samples corresponding to normal temperature (36 °C), 300 °C, 400 °C, 500 °C, and 600 °C.

2.3.2 Pre-processing of CT image

For CT image processing, 16-bit depth grayscale slice was chosen because they provide 65,536 different grayscale levels, which is helpful in capturing the fine details of the evolving internal structure of coal when performing threshold segmentation for substance extraction. Typically, in porous media like rocks and coal, higher grayscale values indicate higher material density. Furthermore, higher grayscale values correspond to brighter pixels, whereas lower grayscale values correspond to darker pixels (0 represents black, and 65,535 represents white). Figure 1 presents the grayscale slices of the five sets for CT samples. Darker pixel regions are recognized as pores or fractures, brighter pixel regions are identified as minerals, and areas with grayscale values falling between the two are classified as coal matrix regions.

Fig. 1
figure 1

Gray-scale slice of the sample after high-temperature heating

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Due to inevitable factors such as hardware and materials affecting the industrial CT imaging process, CT images may exhibit noise, appearing as speckled bright and dark spots or artifacts (Diwakar and Kumar 2018). These noises can potentially introduce errors in three-dimensional reconstruction and material extraction during threshold segmentation. This study employed the commercial visualization software Avizo developed by Thermo Fisher Scientific for noise reduction. To effectively control image noise and prevent secondary image noise from affecting the images, choosing the appropriate noise reduction algorithm is essential. Currently, the main filtering algorithms used in visual analysis processing include mean filtering, median filtering, Gaussian filtering, and so on (Maity et al. 2015). This study tried multiple algorithms and discovered that median filtering can effectively decrease speckle noise in CT images. The main features of this algorithm include: (1) It sorts the values in the pixel neighborhood and selects the middle value as the current pixel’s value. (2) Additionally, the algorithm can effectively remove salt-and-pepper noise caused by sensors and preserve edge information effectively (Erkan et al. 2018). The substance extraction from grayscale images is based on the threshold segmentation method in visual analysis. This method segments substances by selecting groups of different grayscale values to achieve the goal of substance extraction. To evaluate the effectiveness of denoising, the segmentation results before and after denoising can be compared. The 3D modeling process is illustrated with Fig. 2 as an example.

Fig. 2
figure 2

Three-dimensional modeling process for tar-rich coal sample

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2.3.3 Selection of representative element volume (REV)

This study aims to explore the pore and fracture evolution characteristics and the underlying mechanisms of permeability of tar-rich coal after high-temperature pyrolysis. But analyzing all slices of each sample in software would require substantial computational power and make data statistical analysis laborious. Therefore, REV in different samples was selected for study. Theoretically, when the area of the study unit exceeds a specific critical value, the structure, composition, and mechanical properties of the coal body tend to stabilize. This critical value is known as the REV of the coal body (Wang et al. 2020a, b). The steps for selecting REV are as follows: 1. In the central region of each group of CT images, a unit cell of size 50 × 50 × 50 voxels is selected, and the coal’s pore-fracture porosity is calculated using Volume Fraction algorithm in Avizo; 2. Subsequently, a new cell with an increment of 50 voxels in side length was obtained, and the porosity of the pore-fracture in the new cell was recalculated using the volume fraction algorithm in Avizo. By repeating the above steps, the relationship between cell size and porosity was ultimately obtained, as depicted in Fig. 3.

Fig. 3
figure 3

Changes in pore-fracture porosity within the REV

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As shown in Fig. 3, when the unit cell is increased to 400 voxels, the pore-fracture porosity of the unit cell tends to stabilize. Therefore, the 500 × 500 × 500 pixels were chosen for subsequent quantitative analysis on samples subjected to different pyrolysis temperatures.

2.4 Separation of pore and fracture

In order to analyze the evolution processes of pore and fracture structures under high-temperature pyrolysis, the 3D shape factor is chosen as a quantitative parameter tool for separating pore and fracture structures, with its mathematical expression given in Eq. 1 (Herlinger and Vidal 2022). When the shape factor equals 1, it indicates that the labeled object is an ideal sphere, while a shape factor greater than 1 indicates a more complex shape of the labeled object

$$S_{f} { = }\frac{{A^{3} }}{{36 \times \pi \times V^{2} }}$$
(1)

where \(A\) is the surface area of each pore or fracture, \(V\) is the volume of each pore or fracture.

The extraction method for pores and fractures is as follows: 1. Firstly, all pores and fractures of coal samples are labeled through threshold segmentation. Then, the 3D shape factor of all labeled objects is calculated using the Label Analysis module in Avizo. Subsequently, based on the recognition that the number of internal pores in coal under high-temperature pyrolysis conditions far exceeds the number of fractures (Yang et al. 2019; Yu et al. 2012), traversal distribution statistics of the shape factor for all labeled objects are conducted with an increment of 5, and the quantitative distribution statistics results are shown in Fig. 4. This Fig. 4 illustrates that when the shape factor is less than 5, the number of labeled entities is 45,625. When the shape factor ranges from 5 to 10, the number of labeled objects is 1029, and when it ranges from 35 to 40, the number is 102. The shape factors of the labeled objects in this range are all close to 1, indicating a tendency towards spherical morphology. In this range, the labeled objects mainly represent some relatively large or morphologically complex pore structures existing inside the coal sample, as shown in Fig. 5a. As the shape factor exceeds 40, the number of labeled objects drastically decreases to single digits. Based on the previously mentioned recognitions, this shape factor value is utilized as the segmentation threshold for pores and fractures. The Extraction result of pores and fractures at room temperature are illustrated in Fig. 5b. All subsequent pore-fracture segmentations are based on this method.

Fig. 4
figure 4

The quantitative distribution of 3D shape factor for a sample at 36 °C

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Fig. 5
figure 5

Extraction of pores or fractures from a sample at 36 °C. a Large or morphologically complex pore structures. b Extraction result of pores and fractures

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3 Results and discussion

3.1 Porosity of pore-fracture

In the process of oil and gas migration, the connectivity of pore-fracture is a crucial factor that directly impacts the efficiency of product recovery (Wang et al. 2019b); Moreover, isolated pore-fracture can provide storage space for gases such as methane (Zou and Rezaee 2016). Under high-temperature conditions, the existence of isolated pores and fractures can have a significant influence on the connectivity and morphological development of the pore and fracture network. Therefore, studying the connected pore-fracture and isolated pore-fracture in coal after high-temperature pyrolysis and analyzing the evolution of these pores-fractures holds substantial significance. Using the built-in Volume Fraction algorithm in Avizo, the pore-fracture porosity, connected pore-fracture porosity, and isolated pore-fracture porosity of five sets of samples are calculated, and the results are shown in Fig. 6. With the increase in temperature, the pore-fracture porosity gradually increases. At normal temperature, the pore-fracture porosity of tar-rich coal is 5.38%. When the temperature rises to 300–400 °C, there is a slight increase in pore-fracture porosity, with values of 8.53% at 300 °C and 11.31% at 400 °C. However, at 500–600 °C, the pore-fracture porosity of tar-rich coal increases significantly to 19.99% and 24.40%, respectively. The trend of connected pore-fracture porosity closely resembles that of pore-fracture porosity, increasing with the elevation of temperature and showing a significant jump between 400 °C and 500 °C. It is worth noting that as the temperature rises, isolated pore-fracture porosity gradually decreases, and when the temperature exceeds 500 °C, the isolated pore-facture porosity is only about 1%.

Fig. 6
figure 6

The relationship between temperature and pore-fracture porosity, connected pore-fracture porosity and isolated pore-fracture porosity

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3.2 Evolutionary development of pore and fracture structures

This section will explore and analyze the evolutionary development process of pore and fracture structures at different temperatures. The pore and fracture structures at different temperatures were segmented and extracted using the previous method, with the results shown in Fig. 7.

Fig. 7
figure 7

Evolution of pore and fracture structures under high temperature conditions

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From Fig. 7, it can be observed that at a temperature of 36 °C, there are tiny pores and fine fracture structures inside the coal due to geological or tectonic processes. At 300 °C, due to the thermal stress, coal matrix undergoes thermal expansion and significant loss of moisture, resulting in the formation of numerous pore structures filling the entire analysis area. Additionally, fracture structures expand and develop on the original basis, intertwining with each other to form a small-scale fractures network. As the temperature rises to 400 °C, fracture structures further develop, forming a large fracture zone penetrating the entire analysis area. Moreover, the connections between different fractures are significantly strengthened. It is worth noting that at this stage, the number and size of pore structures are relatively smaller compared to 300 °C, which is due to the thermal expansion and thermal stress inside the crack zone causing the surrounding pores to shrink or close. When the temperature rises to 500 °C, a large number of circular or elliptical pores are formed in the central area due to pyrolysis reactions, while many isolated pores still exist on both sides of the analysis area. This phenomenon has also been observed in the study by Feng and Zhao (2015). At this stage, the number of fractures decreases relative to 400 °C, and the fractures are distributed on both sides of the analysis area, with a significant decrease in the connectivity between each fracture. At 600 °C, due to intense pyrolysis reactions, a large number of large pore structures are generated and fill the entire analysis area, resulting in a significant reduction in isolated pores in the entire analysis area. At the same time, due to the destruction or merging of the large fracture zone under the pyrolysis reaction, the number and size of fractures decrease sharply.

3.3 Microstructural parameters of pore-fracture

The Avizo built-in label analysis algorithm was used for quantitative analysis of the extracted pores and fractures from five sample sets. Pore characteristic parameters include equivalent diameter, 2D shape factor, and pore number. Fracture characteristic parameters include fracture width, fracture length, and fracture count. The mathematical expression of the equivalent diameter is shown in Eq. 2 (Wang et al. 2019c). The change curves of the parameters of pore or fracture with temperature are shown in Figs. 8 and 9, respectively

$$D_{eq} = \sqrt[3]{{\frac{6 \times V}{\pi }}}$$
(2)

where \(V\) is the volume of each pore.

Fig. 8
figure 8

Change curve of pore parameters with temperature

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Fig. 9
figure 9

Change curve of fracture parameters with temperature

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As shown in Fig. 8a, b, with the increase in temperature, the average pore size first decreases and then increases, primarily distributed between 12 and 20 μm. Within the temperature range of 36–400 °C, the maximum pore size fluctuates significantly. When the temperature rises to 500 °C, the maximum pore size increases significantly; simultaneously, the 2D shape factor of the pores shows a trend of first increasing, then decreasing, and then increasing again with the temperature. This is because at temperatures below 400 °C, the internal substance of coal primarily undergoes thermal cracking to form fracture structures (Fig. 7). The expansion within the fractures exerts pressure on the surrounding areas, leading to the contraction or closure of pores, thereby reducing the size and shape of the pores. When the temperature reaches 500 °C, due to intense pyrolysis reaction, the tiny pores and fractures inside the coal undergo fusion or merging, forming morphologically complex pore structures. Consequently, at this stage, the diameter and shape factor of the pores are at a higher level. It is noteworthy that at 300 °C, the 2D shape factor of pores is significantly higher than at 36 °C, 400 °C, and 500 °C. This is because, in the initial stage, due to thermal expansion and loss of moisture in the coal matrix, numerous micro-pores and micro-fractures of various shapes and sizes are formed inside the coal (Fig. 1). In comparison, at 400 °C or 500 °C, the shapes of pores tend to be elliptical or circular, while large fracture structures present more regular band-like structures (Fig. 7). In Fig. 8c, it can be observed that with the increase in temperature, the number of pores initially increases and then decreases. This trend is consistent with the previous evolution of pore structures. When the temperature reaches 300 °C or 400 °C, due to thermal expansion and loss of moisture in the coal matrix, numerous tiny pores are formed inside the coal. When the temperature rises to 500 °C or 600 °C, due to pyrolysis reaction, pores melt or merge, ultimately leading to a sharp decrease in the number of pores.

As shown in Fig. 9a, the maximum fracture width increases initially and then decreases with the increase in temperature, reaching its maximum of 125 μm at 400 °C for tar-rich coal. This is because tar-rich coal undergoes primarily thermal cracking from 300 to 400 °C, and 400 °C marks the most intense stage of thermal cracking, causing the fracture of internal coal substance and the formation of large fractures. In Fig. 9b, it can be observed that with the increase in temperature, the maximum fracture length initially increases and then decreases. From 400 to 500 °C, the maximum fracture length decreases from 2063 to 1950 μm, whereas from 500 to 600 °C, the maximum fracture length decreases from 1950 to 1267 μm. The reason for this lies in the fact that at a temperature of 500 °C, even though coal undergoes pyrolysis reactions, large fractures, serving as the primary migration pathways for oil and gas molecules due to their high permeability, readily conduct heat. As a result, the internal temperature within the large fractures increases more slowly, making them less prone to pyrolysis reactions. Conversely, smaller fractures may experience disruption or fusion due to limited heat conduction. Thus, the maximum fracture length remains relatively stable between 400 and 500 °C, with a slight decrease in average fracture length. When the temperature reaches 600 °C, the internal heat within the coal is sufficiently high, leading to the extensive pyrolysis of organic matter. As a result, large fractures are disrupted or merge with the large pore structures formed during pyrolysis, causing a decrease in fracture length.

As shown in Fig. 9c, with the increase in temperature, the number of fractures initially increases and then decreases. The increase in the number of fractures from 36 to 400 °C is primarily due to the thermal cracking of internal coal substance, leading to the formation of numerous large fractures. The number of large fractures increases from 400 to 500 °C, as even though pyrolysis reactions primarily occur at 500 °C, there is also simultaneous thermal cracking of internal material, resulting in a slight increase in the number of fractures at 500 °C. When the temperature rises to 600 °C, due to intensified pyrolysis reactions, a large number of fractures undergo fracture or merge. At this stage, the number of fractures is only 21.6% of that at 500 °C, indicating a significant decrease.

3.4 Fractal dimension of pore-fracture

In the 1960s, Mandelbrot first introduced the concept of fractal dimension to quantitatively describe complex fractal structures. Fractals can accurately depict the characteristics of irregular and complex structures. In 1987, Friesen and Mikula used fractal geometric methods to describe the pore size distribution of coal, demonstrating that fractal geometry is a useful descriptive tool in porous media. In 1995, Kang et al. (1995) discovered that fractures in coal display clear fractal characteristics. Recently, Guo et al. (2019) used fractal theory to quantitatively assess the scale features of pores and fractures. The methods for calculating fractal dimensions primarily comprise the box counting method, box dimension method, and area measurement method, among others (Fernández-Martínez 2016). In this study, the box counting method is utilized. This approach entails placing a series of boxes of varying sizes on the fractal object and determining the number of boxes by adjusting the box’s side length. Ultimately, the fractal dimension is calculated based on the power-law relationship between the number of boxes and the box size (Ai et al. 2014)

$$D = - \mathop {\lim }\limits_{r \to 0} \frac{\lg N(r)}{{\lg (r)}}$$
(3)

where \(D\) is the fractal dimension, \(r\) is the box side length, \(N(r)\) is the number of boxes.

The built-in fractal dimension algorithm in Avizo was used to perform three-dimensional fractal dimension calculations on the binary images of pores and fractures from five sets of samples, with the results shown in Fig. 10. In Fig. 10a, as the temperature increases, the 3D fractal dimension of pores-fractures exhibits a trend of initial increase followed by a decrease, primarily ranging from 2.4 to 2.6. Figure 10b and c indicate that within the temperature range of 36–300 °C, coal samples lose a substantial amount of moisture, leading to the generation of thermal stress within the coal, resulting in the fracture of the coal’s material structure and the formation of more complex fractures. At this point, due to the expansion, merging, or connection of microscopic pores within the coal, the morphology and structure of the pores become more complex and irregular. As the temperature increases from 300–400 °C, the fractures become more complex due to the intensified thermal cracking. The merging and expansion of small fractures result in increased width and quantity of fractures. The expansion and thermal stress within the large fractures exert pressure on the surrounding area, leading to the closure and reduction in size of nearby pores. As a result, the fractal dimension of pores decreases at 400 °C. When the temperature rises above 500 °C, the pyrolysis reaction becomes more intense, releasing a large amount of oil and gas molecules from the coal matrix. Fractures expand into large pores or cavity structures, causing a reduction in the number of large fractures and an increase in the number of small fractures. Migration of oil and gas molecules within the small pores leads to pore expansion and fusion, resulting in the formation of large pore structures. Ultimately, the fractal dimension of coal pores increases, while the fractal dimension of fractures decreases. When the temperature reaches 600 °C, the pyrolysis reaction leads to further fusion and expansion of pore walls, forming complex morphological large pore structures.

Fig. 10
figure 10

Relationship between 3D-fractal dimension of pore-fracture and temperature. a Fractal dimension of pore-fracture, b Fractal dimension of pore, c Fractal dimension of fracture

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It is noteworthy that, at high temperatures, the fractal dimension of coal’s pore structure is slightly larger than that of its fracture structure, and the morphology is more complex. This can be attributed to the following reasons: 1. Under high-temperature conditions, the pyrolysis reaction of coal releases a large amount of oil and gas molecules, leading to the rearrangement and expansion of pores (Fig. 7). Ultimately, this leads to the complexity and irregularity of pore morphology. While the evolution of fracture structures is primarily influenced by substance cracking, as well as the merging and fusion of pore structures, resulting in contraction or expansion (Feng and Zhao 2015; Shi et al. 2023b). 2. Under high temperature conditions, the migration and diffusion of oil and gas molecules inside coal lead to an increase in pressure within the pores, resulting in a more developed pore structure network and a more complex morphology, thereby increasing the fractal dimension of the pore structure. On the contrary, the morphology of fractures is mostly a relatively simple banded structure (Fig. 7).

Following the method of Wang et al. (2019c), the two-dimensional fractal dimension and its mean absolute deviation (Eq. 4) were introduced to quantitatively analyze the homogeneity of pore and fracture internal structures under pyrolysis conditions

$$MAD = \frac{1}{n} \times \sum\limits_{i = 1}^{n} {\left| {d_{i} - \overline{d} } \right|}$$
(4)

where \(d_{i}\) is 2D- fractal dimension of the layer by layer sliced pores or fractures, \(\overline{d}\) is the average 2D-fractal dimension of the pores or fractures.

As shown in Fig. 11a, the 2D-fractal dimension of pore structures at 400 °C exhibits significant fluctuations with distinct peaks. At the normal temperature of 36 °C, the 2D-fractal dimension of pore structures also displays high degree of dispersion. At 500 °C, the complexity of pore structures demonstrates a relatively stable level of variability. As shown in Fig. 11b, the fractal dimension of fracture structures at 600 °C fluctuates significantly and exhibits an unstable trend, while at 400 °C, the development of fractures remains relatively stable with consistent variability.

Fig. 11
figure 11

The 2D-fractal dimension fluctuation curve of pore or fracture structure. a represents the 2D-fractal dimension fluctuation curve of fracture structure; b represents the pore structure. The horizontal lines in the Figure represent the average lines of the 2D-fractal dimensions of each coal sample

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As shown in Fig. 12, the MAD is used to measure the homogeneity of pore and fracture structures at different temperatures. Pore structures exhibit the poorest homogeneity at 400 °C. The homogeneity of pore structures is highest at 300 °C. When the temperature reaches 600 °C, fracture structures exhibit a high degree of heterogeneity, and the structures become extremely complex. At 400 °C, fracture structures exhibit the highest homogeneity. Furthermore, the MAD of pore structures is smaller than that of fracture structures, indicating that the development of pore structures is more uniform and homogeneous than fracture structures. This is consistent with the previous analysis, as when the temperature rises above 500 °C, the coal sample primarily undergoes pyrolysis reactions, resulting in various morphological changes in fractures, including expansion, swelling, and fusion, forming complex and irregular curved or branched structures.

Fig. 12
figure 12

Mean absolute deviation of 2D-fractal dimension of pore or fracture at different temperatures

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Overall, as the temperature increases, the homogeneity of pore structures is higher at 300 °C and 500 °C. However, at 400 °C, the development of pore structures is the most uneven, showing lower homogeneity and the most complex structural morphology. In comparison, the development of fracture structures is relatively uniform at 400 °C, but at 600 °C, the fracture structures become extremely complex, exhibiting high heterogeneity.

3.5 Permeability of coal

3.5.1 Permeability of pore-fracture

Studying the changes in permeability of tar-rich coal at different temperatures is of significant importance for the efficient recovery of oil and gas products under pyrolysis condition (Liu et al. 2021; Pan et al. 2010). In this study, three 50 × 50 × 50 voxel regions were randomly selected within each REV area extracted from the previous five sets of CT samples and subjected to 3D reconstruction (directly importing the REV regions would require immense computational resources and time, and convergence would be challenging). The three regions selected are labeled as R1, R2, and R3. Following this, surface mesh models were created and imported into numerical simulation software for model permeability calculations. Permeability calculations were performed using Darcy’s law.

$$K = \frac{Q\mu L}{{A(P_{1} - P_{2} )}}$$
(5)

where, \(K\) is absolute permeability, mD; \(Q\) is the volume flow rate, \({\text{m}}^{3} /{\text{s}}\); \(\mu\) is the dynamic viscosity, Pa s; \(L\) is the vertical distance between fluid inlet and outlet, m; \(A\) is the outlet area, m2; \(P_{1}\) is the pressure intensity of the inlet, \(Pa\); \(P_{2}\) is the pressure intensity of the outlet,\(Pa\). The parameter values of the model are set to: \(\mu = 3.5 \times 10^{ - 5} \;{\text{Pa}}\;{\text{s}}\); \(P_{1} = 1.2\;{\text{MPa}}\); \(P_{2} = 0.1\;{\text{MPa}}\).

As shown in Fig. 13, the permeability gradually increases with temperature, with a significant leap in permeability in the temperature range of 500–600 °C. With increasing temperature, the average permeability also gradually increases. At 36 °C, the average permeability of the coal is 0.12 mD, this value close to the permeability value of Shaanxi tar-rich coal obtained using gas permeability in the study by Yanjun et al. (2021) (0.14 mD). In the range from 300 to 400 °C, the coal’s average permeability is 50.8 mD and 110.4 mD, respectively. The increase in average permeability of the coal is relatively slow. It’s worth noting that at 300 °C, the average permeability of the coal samples is 360 times higher than at normal temperature. When the temperature reaches 600 °C, the coal’s average permeability reaches 373.2 mD, which is a 128% increase compared to 500 °C and a remarkable 741% increase compared to 300 °C.

Fig. 13
figure 13

Relationship between permeability and temperature. R1, R2, and R3 represent the permeability calculated from arbitrarily selected three 50 × 50 × 50 voxel units in the REV analysis region of each coal sample

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3.5.2 The relationship between porosity and permeability

The coal’s pore-fracture porosity and permeability may have a strong relationship (Zhou et al. 2016), and previous researchers have studied this connection (Pan et al. 2019; Zhao et al. 2009). Zheng et al. (2018) used nuclear magnetic resonance theory and fractal theory to analyze the permeability of coal mines in the Qinshui Basin and Junggar Basin, China. They found a clear exponential relationship between permeability and porosity. Additionally, fractal dimension, as a powerful tool for describing the complex structure of porous media, has been explored by researchers in attempts to correlate it with permeability (Jafari and Babadagli 2013; Xu et al. 2013). Wang et al. (2019c) found a negative correlation between fractal dimension and permeability of coal. However, some scholars have found no clear relationship between the two (Cai et al. 2015). From Figs. 6, 10 and 13, it can be observed that both porosity and permeability gradually increase with temperature, but fractal dimension exhibits a trend of initially increasing, then decreasing, and then increasing again with temperature. It can be found that in this study, there is no clear relationship between the fractal dimension of coal samples and permeability, which may be due to the confounding influence of porosity (Xia et al. 2019). Therefore, this section will delve into the relationship between the pore-fracture porosity and permeability of tar-rich coal.

As shown in Fig. 14, with the increase in temperature, the pore-fracture porosity gradually increases. Pore porosity exhibits a trend of decreasing first and then increasing, while the fracture porosity shows a trend of increasing first and then decreasing. It is worth noting that within the temperature range of 300–400 °C, the pore porosity decreases from 5.26 to 2.92%, while the fracture porosity increases from 3.28 to 8.39%. However, within this temperature range, the permeability of coal increases from 50.39 to 110.4 mD. Clearly, during this stage, the impact of the fracture structure on permeability predominates, and the enlargement of fracture effectively enhances coal permeability. Similarly, when the temperature rises from 400 to 600 °C, the fracture porosity gradually decreases, becoming 8.39%, 4.80%, and 4.39%, respectively, while the pore porosity increases gradually from 2.92 to 20.00%. Within this stage, the coal’s permeability increases from 110.4 to 373.2 mD. Therefore, during this stage, the pore structure predominantly affects permeability.

Fig. 14
figure 14

The relationship between pore-fracture porosity, pore porosity and fracture porosity and temperature

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As shown in Fig. 15, there are two stages of the relationship between pore-fracture porosity \(\varphi\) and average permeability. In the first stage, when the coal temperature rises between 36 and 400 °C, the pore-facture porosity \(\varphi\) is 5.38%, 8.54%, and 11.31%, and the average permeability is 0.12 mD, 50.38 mD, and 110.4 mD, respectively. The growth rate during this stage is relatively slow. In the second stage, when the coal temperature rises between 400 and 600 °C, the pore-fracture porosity \(\varphi\) is 11.31%, 19.99%, and 24.39%, and the average permeability is 110.4 mD, 292.5 mD, and 373.2 mD, respectively. At this stage, the growth rate is faster.

Fig. 15
figure 15

Relationship between pore-fracture porosity and average permeability

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This indicates that in the first stage, as the temperature rises from 36 to 400 °C, the coal mainly undergoes a thermal cracking process, resulting in the formation of numerous fractures. The formation of these fractures significantly increases the coal’s permeability, albeit at a relatively slow rate. However, in the second stage, when the temperature rises from 400 to 600 °C, the coal primarily undergoes pyrolysis reaction, forming large pore structures. During this stage, the permeable pathways in the coal are mainly dominated by the large pore structures, leading to a rapid increase in average permeability. Therefore, this indicates that at different temperatures, the change in the average permeability of coal is influenced by two different mechanisms, thermal cracking and pyrolysis reaction. These mechanisms come into play at different stages, resulting in varying growth rates of average permeability. The average permeability \(k\) exhibits exponential variation with the pore-fracture porosity \(\varphi\), described by empirical Eq. 6, \(R^{2} = 0.9753\). This empirical formula can be used as an assessment equation for the permeability of coal after high-temperature pyrolysis

$$k = - 371.1 + 304.8 \times e^{(\varphi /27.29)}$$
(6)

where \(k\) is the average permeability, \(\varphi\) is the pore-fracture porosity.

In order to gain a deeper understanding of the impact of pore and fracture structures on permeability at different pyrolysis temperatures, a bivariate fit of pore porosity, fracture porosity, and permeability was performed using Matlab. The fitting result is expressed by Eq. 7, \(R^{2}\) = 0.9241. Where, \(\varphi_{p}\) is the pore porosity, \(\varphi_{f}\) is the fracture porosity.

$$k = - 90.57 + 52.88 \times e^{{(0.08 \times \varphi_{p} + 0.13 \times \varphi_{f} )}}$$
(7)

Equation 7 is generalized to get Eq. 8. Where, \(X = e^{{0.08 \times \varphi_{p} }}\), \(Y = e^{{0.13 \times \varphi_{f} }}\).

$$k = - 90.57 + 52.88 \times X \cdot Y$$
(8)

Subsequently, the permeability of coal samples at normal temperature is taken as the initial permeability (Eq. 9), and the permeability equations at different temperatures are transformed into Eq. 10. Where, \(X_{0}\), \(Y_{0}\) represent the impact factors of pore porosity and fracture porosity at normal temperature on permeability, respectively; \(a\) and \(b\) represent the constants − 90.57 and 52.88, respectively. \(\Delta X\), \(\Delta Y\) represent the impact factors of the growth of pore porosity and fracture porosity at the pyrolysis temperature on permeability, respectively.

$$k_{0} = a + b \times X_{0} \cdot Y_{0}$$
(9)
$$k_{i} = a + b \times (X_{0} + \Delta X) \cdot (Y_{0} + \Delta Y)$$
(10)

Equation 10 is split and transformed into Eq. 11. From Eq. 12, it can be observed that the permeability at the pyrolysis temperature consists of three components: the growth of pore porosity, the growth of fracture porosity, and the influence of their coupling. Therefore, based on the following Eqs. 13, 14, and 15, the contributions of pore structure, fracture structure, and their coupling to permeability are calculated at different pyrolysis temperatures. Where, \(c_{pi}\) is the contribution of pore porosity to permeability, \(c_{fi}\) is the contribution of fracture porosity to permeability, \(c_{ci}\) is the contribution of coupling effect to permeability.

$$k_{i} = a + b \times X_{0} \cdot Y_{0} + b \times (\Delta X \cdot Y_{0} + X_{0} \cdot \Delta Y + \Delta X \cdot \Delta Y)$$
(11)
$$k_{i} = k_{0} + b \times (\Delta X \cdot Y_{0} + X_{0} \cdot \Delta Y + \Delta X \cdot \Delta Y)$$
(12)
$$c_{{p_{i} }} = \frac{{b \times (\Delta X \cdot Y_{0} )}}{{k_{i} }}$$
(13)
$$c_{fi} = \frac{{b \times (X_{0} \cdot \Delta Y)}}{{k_{i} }}$$
(14)
$$c_{ci} = \frac{b \times (\Delta X \cdot \Delta Y)}{{k_{i} }}$$
(15)

During the pyrolysis process from 300 to 600 °C, the contributions of pore porosity, fracture porosity and their coupling to permeability are shown in Fig. 16. The contribution of pore porosity to permeability shows a trend of decreasing and then increasing, while the contributions of fracture porosity and their coupling exhibit a trend of increasing and then decreasing. At a temperature of 300 °C, the contribution of pore structure to permeability is 38.4%, the contribution of fracture structure is 42.5%, and the contribution of their coupling is 19.1%. At this point, the contribution of fracture structure is slightly higher than that of pore structure. When the temperature rises to 400 °C, the contribution of fracture structure reaches 89.83%, while the contributions of pore structure and coupling are 4.31% and 5.76% respectively. This result is consistent with the previous analysis, indicating that large fractures formed inside the coal play a dominant role in permeability at 400 °C. When the temperature increases from 400 to 600 °C, the contributions of pore structure are 4.31%, 60.28%, and 68.92%, while the contribution of fracture structure drops sharply from 89.83 to 8.08%. This result is also consistent with the earlier analysis, indicating that the large pore structure formed during pyrolysis plays a dominant role in permeability. It is worth noting that when the temperature increases from 400 to 600 °C, the contribution of coupling increases from 5.76 to over 20%. Although at this stage, the fracture structure is merged or combined into large pore structures, leading to a decrease in fracture rate. However, the interconnectivity between pore and fracture structures is enhanced, and they work more synergistically. Therefore, the contribution of coupling to permeability increases.

Fig. 16
figure 16

Contribution of pore porosity, fracture porosity and coupling effect to permeability

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4 Conclusions

In this study, the evolution of pores and fractures in tar-rich coal from the Shenfu mining area of northern Shaanxi under high-temperature pyrolysis conditions was studied and discussed using μCT technology. Precise segmentation of pores and fractures has been achieved. Based on this, the evolution processes of pores and fractures at different temperatures are studied. The changes in permeability of coal after high-temperature pyrolysis are analyzed, and a predictive equation for permeability is provided. The main conclusions are as follows:

  1. (1)

    The evolution of pores and fractures in tar-rich coal can be divided into two stages as the temperature increases from 36 to 600 °C. In the first stage, from 36 to 400 °C, the internal substance structure of the coal undergoes mainly thermal cracking, forming large fracture bands, and a well-connected fracture network. In the second stage, from 400 to 600 °C, the coal undergoes intense pyrolysis reaction, leading to the formation of mainly circular or elliptical large pore structures inside the coal, while the number of fracture decreases.

  2. (2)

    The 3D-fractal dimension of pore-fracture structures exhibits a trend of initially increasing and then decreasing with rising temperature. During high-temperature pyrolysis, the homogeneity of pore structure is superior to that of fracture structure.

  3. (3)

    With increasing temperature, the coal’s permeability exhibits segmented growth, with a significant leap observed from 400 to 500 °C. Research indicates a strong exponential relationship between the pore-fracture porosity of tar-rich coal and permeability. When the temperature exceeds 500 °C, the coupled effect of pores and fractures contributes more than 20% to permeability, and its impact cannot be ignored.