Fractal Evolution Characteristics on the Three-Dimensional Fractures in Coal Induced by CO 2 Phase Transition Fracturing

: To analyze the transformed effect of three-dimensional (3D) fracture in coal by CO 2 phase transition fracturing (CO 2 -PTF), the CO 2 -PTF experiment under a fracturing pressure of 185 MPa was carried out. Computed Tomography (CT) scanning and fractal theory were used to analyze the 3D fracture structure parameters. The fractal evolution characteristics of the 3D fractures in coal induced by CO 2 -PTF were analyzed. The results indicate that the CO 2 phase transition fracturing coal has the fracture generation effect and fracture expansion-transformation effect, causing the maximum fracture length, fracture number, fracture volume and fracture surface area to be increased by 71.25%, 161.94%, 3970.88% and 1330.03%. The fractal dimension ( D N ) for fracture number increases from 2.3523 to 2.3668, and the fractal dimension ( D V ) for fracture volume increases from 2.8440 to 2.9040. The early dynamic high-pressure gas jet stage of CO 2 -PTF coal influences the fracture generation effect and promotes the generation of 3D fractures with a length greater than 140 µ m. The subsequent quasi-static high-pressure gas stage influences the fracture expansion-transformation effect, which promotes the expansion transformation of 3D fractures with a length of less than 140 µ m. The 140 µ m is the critical value for the fracture expansion-transformation effect and fracture generation effect. Five indicators are proposed to evaluate the 3D fracture evolution in coal caused by CO 2 -PTF, which can provide theoretical and methodological references for the study of fracture evolution characteristics of other unconventional natural gas reservoirs and their reservoir stimulation.


Introduction
As an important unconventional natural gas energy, the efficient development and utilization of coalbed methane (CBM) cannot only provide clean fuel, but also help reduce coal and gas outburst disasters, and alleviate the greenhouse effect [1][2][3].However, low permeability is one of the key factors restricting the efficient development of CBM [4][5][6].Therefore, a variety of technologies that enhance gas drainage and eliminate coal and gas outbursts have been developed, but different technologies are suitable for coal seams and mining areas with different gas geological conditions [7,8].The research and development of safe, effective, economical and highly applicable permeability-enhanced technology is one of the main research contents and major needs in the field of coal and gas comining research.
CO 2 phase transition fracturing (CO 2 -PTF) technology originated from the CARDOX technology invented in the United States in 1914 [9].The CO 2 -PTF technology or CARDOX technology is a non-explosive physical blasting technology that uses the high-pressure gas expansion energy generated by the heating and expansion of liquid CO 2 to act on the coal or rock, causing the coal or rock to fracture [10].The energy analysis indicates that CO 2 Fractal Fract.2024, 8, 273 2 of 13 undergoes a transformation from the liquid to the supercritical and gaseous state, 1 kg of liquid CO 2 undergoes phase change and the produced energy is equivalent to 397 g of TNT explosive [11].The field applications in coalmines show that CO 2 -PTF is a new reservoir stimulation method for enhancing permeability and eliminating gas outbursts, which has unique technical advantages, such as high safety performance and avoiding the negative effects of hydraulic measures [12][13][14].
Coal is a multi-scale pore-fracture media, and its pore-fracture structure characteristics and connectivity directly control the CBM desorption, diffusion and seepage [15][16][17].In recent years, the transformation effect of CO 2 -PTF on the pore-fracture structure (gas migration channel) of coal has attracted the attention of scholars.In terms of pore transformation by CO 2 -PTF, Bai et al. found that the coal pore volume with pore diameter greater than 1 µm increases after CO 2 -PTF [18].The study by Xia et al. showed that the pore volume and average pore diameter of coal increased, and the pore fractal dimension decreased after CO 2 -PTF [19].Recent studies have indicated that CO 2 -PTF mainly affects pores with pore diameters larger than 2 nm, and the mesopore fractal dimension is reduced [20].In terms of fracture transformation by CO 2 -PTF, Cao et al. found that the macroscopic fractures with a length of 1-3 m were produced by on-site monitoring after CO 2 -PTF [21], and the fractured coal sample by CO 2 -PTF is further observed by Scanning Electron Microscope (SEM), which found a large number of tri-radial wing micro-fractures and damage marks [22].Liu and Liao et al. used the fracture fractal theory to calculate the fractal dimension of the two-dimensional (2D) fracture observed by SEM after CO 2 -PTF, and the fractal dimension for 2D fracture increased [23,24].
In summary, at this stage, the pore structure scale of coal transformed by CO 2 -PTF has been revealed, and the fractal theory is widely used to characterize the complexity of the pore-fracture structure in coal.However, the related research on the fracture structure induced by CO 2 phase transition fracturing coal (CO 2 -PTF coal) mainly focuses on the 2D fracture by macroscopic monitoring and SEM observation.The distribution of fractures in coal has obvious three-dimensional (3D) structural characteristics of length, aperture and height, and fracture volume comprehensively reflects fracture length, aperture and height [25][26][27].The fracture surface area is the total area of the internal surface of the 3D fractures in coal [28,29].Therefore, the number, length, volume and surface area of fractures are key parameters that reflect the 3D fractures [29][30][31].The current research on the 2D fracture structure of CO 2 -PTF coal is not enough to fully reveal the 3D fracture evolution characteristics of CO 2 -PTF coal.CT scanning is a non-destructive technology, and can accurately characterize 3D fracture in coal [32,33], which provides technical support for analyzing the 3D fracture evolution of CO 2 -PTF coal.
Therefore, this paper carried out the CO 2 -PTF coal experiment, the CT scanning and fractal theory were applied to research the 3D fracture evolution characteristics of CO 2 -PTF coal, and the transformation effect of CO 2 -PTF on the 3D fracture was analyzed.This study further reveals the mechanism of coal seam transformation caused by CO 2 -PTF, provides new evaluation and analysis methods for studying the effect of CO 2 -PTF, and is conducive to further improving the reservoir stimulation theory of CO 2 -PTF coal.

Samples
The experimental samples are collected from the Ji 15 coal seam in Pingdingshan No. 8 Coalmine, Henan province.The coal sample is bright, showing layered and massive structures, the original band structure is well preserved and the bedding is visible.The dry ash-free volatile content of the sample is 24.91%, and the maximum vitrinite reflectance is 1.22%.After the coal samples are collected, they are sealed, packaged and transported to the laboratory, where a linear cutting machine is used to prepare the coal pillar with a diameter (Φ) of 50 mm and length of 75 mm for the CO 2 -PTF experiment.

CO 2 Phase Change Fracturing Experiment
The experimental system for CO 2 -PTF coal is shown in Figure 1.The experimental system mainly includes the CO 2 -PTF device composed of liquid CO 2 , a heater, a pressurecontrolled bursting disc, an external test pipe of the CO 2 -PTF device and a test holder (the test holder is facing the discharge port of the CO 2 -PTF).The main experiment process of the CO 2 -PTF coal is as follows: (1) select the pressure-controlled bursting disc of 185 MPa, assemble the CO 2 -PTF device and fill it with liquid CO 2 ; (2) fix the external test pipe of the CO 2 -PTF device on the bracket in safety experimental chamber; (3) put the CO 2 -PTF device into the external test pipe, adjust the discharge port of the CO 2 -PTF so that it faces the test holder; (4) connect the wire of CO 2 -PTF device, close the external test pipe, close the safety experimental chamber, start the heater and then conduct CO 2 -PTF coal experiment; and (5) remove the test holder and take out the coal sample.The coal is labeled PR before CO 2 -PTF, and the coal is labeled PF after CO 2 -PTF.

CO2 Phase Change Fracturing Experiment
The experimental system for CO2-PTF coal is shown in Figure 1.The experimenta system mainly includes the CO2-PTF device composed of liquid CO2, a heater, a pressure controlled bursting disc, an external test pipe of the CO2-PTF device and a test holder (the test holder is facing the discharge port of the CO2-PTF).The main experiment process o the CO2-PTF coal is as follows: (1) select the pressure-controlled bursting disc of 185MPa assemble the CO2-PTF device and fill it with liquid CO2; (2) fix the external test pipe of the CO2-PTF device on the bracket in safety experimental chamber; (3) put the CO2-PTF device into the external test pipe, adjust the discharge port of the CO2-PTF so that it faces the tes holder; (4) connect the wire of CO2-PTF device, close the external test pipe, close the safety experimental chamber, start the heater and then conduct CO2-PTF coal experiment; and (5) remove the test holder and take out the coal sample.The coal is labeled PR before CO2 PTF, and the coal is labeled PF after CO2-PTF.

CT Scanning Measurement
CT scanning technology can obtain the internal structure information of coal or rock without damaging the internal structure.The instrument used in this CT scan is GE's Phoenix v|tome|xs, and the coal sample CT scanning process is shown in Figure 2. The main parameters of CT scanning are a resolution of 30.4 µm, scanning voltage of 200 kV scanning current of 170 µA, and exposure time of 500 ms, and 2467 slices of 1611 × 1611 pixels are obtained for each scanning.Here, to facilitate our analysis of the 3D fracture evolution characteristics of coal in duced by CO2-PTF.Before CO2-PTF, CT scanning was performed on the coal sample used for the CO2-PTF experiment to obtain the original fracture structure.After CO2-PTF, the CT scanning was performed on the coal sample again to obtain the internal fracture struc ture.

CT Scanning Measurement
CT scanning technology can obtain the internal structure information of coal or rock without damaging the internal structure.The instrument used in this CT scan is GE's Phoenix v|tome|xs, and the coal sample CT scanning process is shown in Figure 2. The main parameters of CT scanning are a resolution of 30.4 µm, scanning voltage of 200 kV, scanning current of 170 µA, and exposure time of 500 ms, and 2467 slices of 1611 × 1611 pixels are obtained for each scanning.

CO2 Phase Change Fracturing Experiment
The experimental system for CO2-PTF coal is shown in Figure 1.The experimental system mainly includes the CO2-PTF device composed of liquid CO2, a heater, a pressurecontrolled bursting disc, an external test pipe of the CO2-PTF device and a test holder (the test holder is facing the discharge port of the CO2-PTF).The main experiment process of the CO2-PTF coal is as follows: (1) select the pressure-controlled bursting disc of 185MPa, assemble the CO2-PTF device and fill it with liquid CO2; (2) fix the external test pipe of the CO2-PTF device on the bracket in safety experimental chamber; (3) put the CO2-PTF device into the external test pipe, adjust the discharge port of the CO2-PTF so that it faces the test holder; (4) connect the wire of CO2-PTF device, close the external test pipe, close the safety experimental chamber, start the heater and then conduct CO2-PTF coal experiment; and (5) remove the test holder and take out the coal sample.The coal is labeled PR before CO2-PTF, and the coal is labeled PF after CO2-PTF.

CT Scanning Measurement
CT scanning technology can obtain the internal structure information of coal or rock without damaging the internal structure.The instrument used in this CT scan is GE's Phoenix v|tome|xs, and the coal sample CT scanning process is shown in Figure 2. The main parameters of CT scanning are a resolution of 30.4 µm, scanning voltage of 200 kV, scanning current of 170 µA, and exposure time of 500 ms, and 2467 slices of 1611 × 1611 pixels are obtained for each scanning.Here, to facilitate our analysis of the 3D fracture evolution characteristics of coal induced by CO2-PTF.Before CO2-PTF, CT scanning was performed on the coal sample used for the CO2-PTF experiment to obtain the original fracture structure.After CO2-PTF, the CT scanning was performed on the coal sample again to obtain the internal fracture structure.Here, to facilitate our analysis of the 3D fracture evolution characteristics of coal induced by CO 2 -PTF.Before CO 2 -PTF, CT scanning was performed on the coal sample used for the CO 2 -PTF experiment to obtain the original fracture structure.After CO 2 -PTF, the CT scanning was performed on the coal sample again to obtain the internal fracture structure.

Fractal Dimension Calculation for 3D Fracture
The fractal dimension is usually used to quantitatively describe the complexity of complex and irregular structures [34][35][36].The fractal dimension D N for 3D fracture number and fractal dimension D V for 3D fracture volume in this research are applied to characterize the fracture complexity of CO 2 -PTF coal.

Fractal Dimension Calculation for 3D Fracture Number
The calculation principle of the fractal dimension D N for 3D fracture number is as follows: The number of fractures of the fracture length greater than L i is recorded as N(L i ), the fractal scaling relationship between N(L i ) and L i satisfies the Equation (1) [37,38].Taking the logarithm of N(L i ) and L i and drawing the relation between logN(L i ) and logL i , we obtain the absolute value of the slope of the line by the linear regression analysis, which is the fractal dimension D N for 3D fracture number, as shown the Equation (2).
where L i is the fracture length; N(L i ) is the number of fractures with the fracture length greater than L i ; D N is the fractal dimension for the fracture number; C 1 is a fitting constant.

Fractal Dimension Calculation for 3D Fracture Volume
The fracture volume V and surface area A are obtained based on CT scanning, and they satisfy the fractal scaling relationship of Equation ( 3) [39].Taking the logarithm of both sides of Equation ( 3), the fractal dimension D V for 3D fracture volume can be calculated by Equation (4).
where A is the surface area of the fracture; V is the volume of the fracture; D V is the fractal dimension for the 3D fracture volume; C 2 is a fitting constant.

2D Fractures
Figure 3 indicates the fractures in the 2D slice before and after CO 2 -PTF coal.Before CO 2 -PTF, there are no obvious large fractures inside the coal sample PR.The aperture of some original fractures is small and the connectivity of the fractures is poor (Figure 3a,b).After CO 2 -PTF, the obvious fracture network is generated in the coal sample PF, and the connectivity between fractures is enhanced.Furthermore, there are some new fracture structures formed from CO 2 -PTF coal.Large damage marks (DM) are formed and the four-radial-wing (FRW) fractures extending in four directions is generated around the large damage marks in the middle of the coal sample PF (Figure 3c,d).Some small damage marks and radial fractures are also formed in the coal sample PF (Figure 3c,d).Cao et al. also found that CO 2 -PTF promotes the formation of damage marks and tri-radial-wing (TRW) fractures (the three single fractures initiated from one point) by SEM observations [22].In this study, the radial fractures formed by CO 2 -PTF include not only tri-radial-wing (TRW) fractures, but also four-radial-wing (FRW) fractures (the four single fractures initiated from one point) (Figure 3c,d).

3D reconstruction of Fractures
The 2D slices obtained from the CT scanning were filtered and denoised, and subjected to threshold segmentation.The VG Studio MAX 3.0 image processing software from the Volume Graphics company (Heidelberg, Germany) was used to visualize the 3D reconstruction of the processed 2D slices of coal samples.The pore/inclusion analysis module was used to extract and analyze the defects or fractures in coal samples.The detected fractures were color-coded and visualized, and the fracture structural parameters were calculated, mainly including fracture number, length, volume and surface area.
Figure 4 shows the 3D fracture of the coal sample from 3D reconstruction before and after CO2-PTF.Before CO2-PTF, the fracture rendering color inside the coal sample PR is mainly green and blue (Figure 4a,b), the fracture volume is small and the connectivity between fractures is poor.After CO2-PTF, the complex fracture network is formed inside the coal sample PF, and the fracture rendering color is mainly red (Figure 4c,d).The fracture volume increases, and the connectivity between the fractures is enhanced.Furthermore, compared with the 3D fracture in coal sample PR, the 3D fractures in coal sample PF after CO2-PTF have obvious longitudinal continuity.This is mainly due to the highpressure CO2 flow generated by CO2-PTF that acts on the top of the coal sample, prompting the fractures in the coal sample to expand longitudinally, forming the complex fracture network.The fractures in coal have obvious 3D characteristics [40][41][42].To further truly reflect the fracture evolution of CO 2 -PTF coal, it is necessary to perform 3D reconstruction of the 2D slices.

3D Reconstruction of Fractures
The 2D slices obtained from the CT scanning were filtered and denoised, and subjected to threshold segmentation.The VG Studio MAX 3.0 image processing software from the Volume Graphics company (Heidelberg, Germany) was used to visualize the 3D reconstruction of the processed 2D slices of coal samples.The pore/inclusion analysis module was used to extract and analyze the defects or fractures in coal samples.The detected fractures were color-coded and visualized, and the fracture structural parameters were calculated, mainly including fracture number, length, volume and surface area.
Figure 4 shows the 3D fracture of the coal sample from 3D reconstruction before and after CO 2 -PTF.Before CO 2 -PTF, the fracture rendering color inside the coal sample PR is mainly green and blue (Figure 4a,b), the fracture volume is small and the connectivity between fractures is poor.After CO 2 -PTF, the complex fracture network is formed inside the coal sample PF, and the fracture rendering color is mainly red (Figure 4c,d).The fracture volume increases, and the connectivity between the fractures is enhanced.Furthermore, compared with the 3D fracture in coal sample PR, the 3D fractures in coal sample PF after CO 2 -PTF have obvious longitudinal continuity.This is mainly due to the high-pressure CO 2 flow generated by CO 2 -PTF that acts on the top of the coal sample, prompting the fractures in the coal sample to expand longitudinally, forming the complex fracture network.Figure 5 shows that logN (Li) and logLi have an obvious linear relationship, and the correlation coefficient R 2 is greater than 0.99, indicating that the number distribution of the 3D fractures inside the coal sample has obvious fractal characteristics.Before CO2-PTF, the fractal dimension DN for 3D fracture number is 2.3523.After CO2-PTF, the fractal dimension DN for the 3D fracture number is 2.3668.

Estimation of the Fractal Dimension DV for 3D Fracture Volume
The fracture surface area A and volume V are fitted by Equation ( 4) to obtain the fractal dimension DV for 3D fracture volume.Figure 6 shows that logA and logV have an obvious linear relationship, and the correlation coefficient R 2 is greater than 0.98, reflecting that 3D fracture volume distribution inside the coal sample has obvious fractal characteristics.After CO2-PTF, the fractal dimension DV increases from 2.8440 to 2.9040.Figure 5 shows that logN(Li) and logLi have an obvious linear relationship, and the correlation coefficient R 2 is greater than 0.99, indicating that the number distribution of the 3D fractures inside the coal sample has obvious fractal characteristics.Before CO2-PTF, the fractal dimension DN for 3D fracture number is 2.3523.After CO2-PTF, the fractal dimension DN for the 3D fracture number is 2.3668.

Estimation of the Fractal Dimension DV for 3D Fracture Volume
The fracture surface area A and volume V are fitted by Equation ( 4) to obtain the fractal dimension DV for 3D fracture volume.Figure 6 shows that logA and logV have an obvious linear relationship, and the correlation coefficient R 2 is greater than 0.98, reflecting that 3D fracture volume distribution inside the coal sample has obvious fractal characteristics.After CO2-PTF, the fractal dimension DV increases from 2.8440 to 2.9040.

Estimation of the Fractal Dimension D V for 3D Fracture Volume
The fracture surface area A and volume V are fitted by Equation ( 4) to obtain the fractal dimension D V for 3D fracture volume.Figure 6 shows that logA and logV have an obvious linear relationship, and the correlation coefficient R 2 is greater than 0.98, reflecting that 3D fracture volume distribution inside the coal sample has obvious fractal characteristics.After CO 2 -PTF, the fractal dimension D V increases from 2.8440 to 2.9040.

3D fracture Structure Variation Induced by CO2-PTF Coal
According to the 3D fracture reconstruction from CT scanning in Section 4.2, the 3D fracture structure parameters before and after CO2-PTF are shown in Figure 7.After CO2-PTF, the maximum fracture length in the coal sample increases from 48.90 mm to 83.74 mm, with an increased rate of 71.25% (Figure 7a).The fracture number increases from 35,183 to 92,159, with an increased rate of 161.94% (Figure 7b).The fracture volume V increases from 158.12 mm 3 to 6436.87 mm 3 , with an increased rate of 3970.88% (Figure 7c).The fracture surface area A increases from 7733.05 mm 2 to 110,585.06 mm 2 , an increase of 1330.03% (Figure 7d).

3D fracture Evolution Characteristics of Coal Induced by CO2-PTF
Figure 8 shows that after CO2-PTF, the fracture volume and surface area with a fracture length less than 0.14 mm decreased, and the fracture volume and surface area with a

3D Fracture Structure Variation Induced by CO 2 -PTF Coal
According to the 3D fracture reconstruction from CT scanning in Section 4.2, the 3D fracture structure parameters before and after CO 2 -PTF are shown in Figure 7.After CO 2 -PTF, the maximum fracture length in the coal sample increases from 48.90 mm to 83.74 mm, with an increased rate of 71.25% (Figure 7a).The fracture number increases from 35,183 to 92,159, with an increased rate of 161.94% (Figure 7b).The fracture volume V increases from 158.12 mm 3 to 6436.87 mm 3 , with an increased rate of 3970.88% (Figure 7c).The fracture surface area A increases from 7733.05 mm 2 to 110,585.06 mm 2 , an increase of 1330.03% (Figure 7d).

3D fracture Structure Variation Induced by CO2-PTF Coal
According to the 3D fracture reconstruction from CT scanning in Section 4.2, the 3D fracture structure parameters before and after CO2-PTF are shown in Figure 7.After CO2-PTF, the maximum fracture length in the coal sample increases from 48.90 mm to 83.74 mm, with an increased rate of 71.25% (Figure 7a).The fracture number increases from 35,183 to 92,159, with an increased rate of 161.94% (Figure 7b).The fracture volume V increases from 158.12 mm 3 to 6436.87 mm 3 , with an increased rate of 3970.88% (Figure 7c).The fracture surface area A increases from 7733.05 mm 2 to 110,585.06 mm 2 , an increase of 1330.03% (Figure 7d).

3D fracture Evolution Characteristics of Coal Induced by CO2-PTF
Figure 8 shows that after CO2-PTF, the fracture volume and surface area with a fracture length less than 0.14 mm decreased, and the fracture volume and surface area with a

3D Fracture Evolution Characteristics of Coal Induced by CO 2 -PTF
Figure 8 shows that after CO 2 -PTF, the fracture volume and surface area with a fracture length less than 0.14 mm decreased, and the fracture volume and surface area with a length greater than 0.14 mm increased.Figure 9 further indicates that after CO 2 -PTF, the fracture volume with a fracture length less than 0.14 mm decreases from 5.78 mm 3 to 5.13 mm 3 , a decrease of 11.24% (Figure 9a).The fracture volume with a fracture length greater than 0.14 mm increases from 152.34 mm 3 to 6431.74 mm 3 , an increase of 4121.96% (Figure 9b).After CO 2 -PTF, the fracture surface area with a fracture length of less than 0.14 mm decreases from 650.29 mm 2 to 593.06 mm 2 , a decrease of 8.80% (Figure 9c).The fracture surface area with a fracture length greater than 0.14 mm increases from 7082.76 mm 2 to 109,992.00 mm 3 , an increase of 1452.95% (Figure 9d).
Fractal Fract.2024, 8, x FOR PEER REVIEW 8 of 13 length greater than 0.14 mm increased.Figure 9 further indicates that after CO2-PTF, the fracture volume with a fracture length less than 0.14 mm decreases from 5.78 mm 3 to 5.13 mm 3 , a decrease of 11.24% (Figure 9a).The fracture volume with a fracture length greater than 0.14 mm increases from 152.34 mm 3 to 6431.74 mm 3 , an increase of 4121.96% (Figure 9b).After CO2-PTF, the fracture surface area with a fracture length of less than 0.14 mm decreases from 650.29 mm 2 to 593.06 mm 2 , a decrease of 8.80% (Figure 9c).The fracture surface area with a fracture length greater than 0.14 mm increases from 7082.76 mm 2 to 109,992.00 mm 3 , an increase of 1452.95% (Figure 9d).The above fracture structure variation indicates that CO2-PTF promotes the expansion transformation of fractures with a length less than 0.14 mm into fractures with a length greater than 0.14 mm.In addition to the expansion transformation from the fractures with a length less than 0.14 mm, it is mainly due to the CO2-PTF generating a large number of new fractures with a length greater than 0.14 mm in coal.Therefore, the fracture evolution caused by CO2-PTF coal has dual effects, which are the fracture generation effect and fracture expansion-transformation effect, and the boundary between the dual effects of CO2-PTF coal under the fracturing pressure of 185 MPa is 0.14 mm (140 µm).Fractal Fract.2024, 8, x FOR PEER REVIEW 8 of 13 length greater than 0.14 mm increased.Figure 9 further indicates that after CO2-PTF, the fracture volume with a fracture length less than 0.14 mm decreases from 5.78 mm 3 to 5.13 mm 3 , a decrease of 11.24% (Figure 9a).The fracture volume with a fracture length greater than 0.14 mm increases from 152.34 mm 3 to 6431.74 mm 3 , an increase of 4121.96% (Figure 9b).After CO2-PTF, the fracture surface area with a fracture length of less than 0.14 mm decreases from 650.29 mm 2 to 593.06 mm 2 , a decrease of 8.80% (Figure 9c).The fracture surface area with a fracture length greater than 0.14 mm increases from 7082.76 mm 2 to 109,992.00 mm 3 , an increase of 1452.95% (Figure 9d).The above fracture structure variation indicates that CO2-PTF promotes the expansion transformation of fractures with a length less than 0.14 mm into fractures with a length greater than 0.14 mm.In addition to the expansion transformation from the fractures with a length less than 0.14 mm, it is mainly due to the CO2-PTF generating a large number of new fractures with a length greater than 0.14 mm in coal.Therefore, the fracture evolution caused by CO2-PTF coal has dual effects, which are the fracture generation effect and fracture expansion-transformation effect, and the boundary between the dual effects of CO2-PTF coal under the fracturing pressure of 185 MPa is 0.14 mm (140 µm).The above fracture structure variation indicates that CO 2 -PTF promotes the expansion transformation of fractures with a length less than 0.14 mm into fractures with a length greater than 0.14 mm.In addition to the expansion transformation from the fractures with a length less than 0.14 mm, it is mainly due to the CO 2 -PTF generating a large number of new fractures with a length greater than 0.14 mm in coal.Therefore, the fracture evolution caused by CO 2 -PTF coal has dual effects, which are the fracture generation effect and fracture expansion-transformation effect, and the boundary between the dual effects of CO 2 -PTF coal under the fracturing pressure of 185 MPa is 0.14 mm (140 µm).
The fracture structure evolution induced by CO 2 -PTF coal is closely related to the process of CO 2 -PTF coal (Figure 10).The process of CO 2 -PTF coal is mainly divided into two stages: the early dynamic high-pressure gas jet stage and the subsequent quasi-static high-pressure gas stage [43][44][45].In the early dynamic high-pressure gas jet stage, the high-pressure CO 2 gas jet is ejected from the discharge port and acts on the surrounding coal, forming a large number of new fractures.This stage corresponds to the fracture generation effect.The fracture structure evolution induced by CO2-PTF coal is closely related to the process of CO2-PTF coal (Figure 10).The process of CO2-PTF coal is mainly divided into two stages: the early dynamic high-pressure gas jet stage and the subsequent quasi-static high-pressure gas stage [43][44][45].In the early dynamic high-pressure gas jet stage, the highpressure CO2 gas jet is ejected from the discharge port and acts on the surrounding coal, forming a large number of new fractures.This stage corresponds to the fracture generation effect.The high-pressure CO2 gas jet migrates along the fracture in coal and the CO2 gas pressure decays, entering the subsequent quasi-static high-pressure gas stage.The CO2 gas pressure is generally 5-8 MPa, which is much greater than the tensile strength of coal (0.5-1.5 MPa) [46], the "gas wedge effect" of high-pressure CO2 gas will cause micro-fracture to expand and transform [47,48], resulting in the fracture expansion-transformation effect.
The fracture dual evolution effect (the fracture generation effect and fracture expansion-transformation effect) induced by CO2-PTF coal, especially the fracture generation effect, promotes a significant increase in the fracture number, volume and surface area, which causes the enhanced connectivity between fractures, and the increase in fractal dimensions for the fracture number and volume.In summary, CO2-PTF coal enhances the connectivity between fractures and provides an effective channel for gas seepage, which is conceived in the CBM drainage.

Potential Application in Evaluating the Effect of CO2-PTF Coal
According to this study, combining the fractal dimension DV for fracture volume with the fractal dimension DN for fracture number can more comprehensively reflect the connectivity evolution characteristics for the fracture structure of CO2-PTF coal.Therefore, we propose evaluation indicators to evaluate the 3D fracture evolution in coal caused by CO2-PTF from the perspectives of Euclidean geometry and fractal geometry, which mainly includes the maximum fracture length (Lmax), fracture volume (V), the fracture number (N), the fractal dimension (DV) for fracture volume and the fractal dimension (DN) for fracture number.The high-pressure CO 2 gas jet migrates along the fracture in coal and the CO 2 gas pressure decays, entering the subsequent quasi-static high-pressure gas stage.The CO 2 gas pressure is generally 5-8 MPa, which is much greater than the tensile strength of coal (0.5-1.5 MPa) [46], the "gas wedge effect" of high-pressure CO 2 gas will cause micro-fracture to expand and transform [47,48], resulting in the fracture expansion-transformation effect.
The fracture dual evolution effect (the fracture generation effect and fracture expansiontransformation effect) induced by CO 2 -PTF coal, especially the fracture generation effect, promotes a significant increase in the fracture number, volume and surface area, which causes the enhanced connectivity between fractures, and the increase in fractal dimensions for the fracture number and volume.In summary, CO 2 -PTF coal enhances the connectivity between fractures and provides an effective channel for gas seepage, which is conceived in the CBM drainage.

Potential Application in Evaluating the Effect of CO 2 -PTF Coal
According to this study, combining the fractal dimension D V for fracture volume with the fractal dimension D N for fracture number can more comprehensively reflect the connectivity evolution characteristics for the fracture structure of CO 2 -PTF coal.Therefore, we propose evaluation indicators to evaluate the 3D fracture evolution in coal caused by CO 2 -PTF from the perspectives of Euclidean geometry and fractal geometry, which mainly includes the maximum fracture length (L max ), fracture volume (V), the fracture number (N), the fractal dimension (D V ) for fracture volume and the fractal dimension (D N ) for fracture number.
The fracture length reflects the fracture expansion range, so the larger the maximum fracture length (L max ), the larger the fracture expansion range.The fracture volume (V), fracture number (N), fractal dimension (D V ) for fracture volume and fractal dimension (D N ) for fracture number can represent the fracture connectivity [39,[49][50][51][52].The larger these values, the better the connectivity between fractures.The comprehensive evaluation diagram of five indicators for evaluating the 3D fracture evolution of CO 2 -PTF coal is shown in Figure 11.
Fractal Fract.2024, 8, x FOR PEER REVIEW 10 of 13 The fracture length reflects the fracture expansion range, so the larger the maximum fracture length (Lmax), the larger the fracture expansion range.The fracture volume (V), fracture number (N), fractal dimension (DV) for fracture volume and fractal dimension (DN) for fracture number can represent the fracture connectivity [39,[49][50][51][52].The larger these values, the better the connectivity between fractures.The comprehensive evaluation diagram of five indicators for evaluating the 3D fracture evolution of CO2-PTF coal is shown in Figure 11.This research further reveals the fracture evolution mechanism by CO2-PTF coal and provides a new evaluation and analysis indicators for the effect of CO2-PTF.In addition, the above research method can provide theoretical and methodological references for the study of fracture evolution characteristics of other unconventional natural gas reservoirs and their reservoir stimulation.

Conclusions
According to the CT scanning technology and fractal theory, the variations in the length, number, volume and surface area of the 3D fractures in coal before and after CO2-PTF under the fracturing pressure of 185 MPa are analyzed.The fractal dimension (DN) for fracture number and fractal dimension (DV) for fracture volume before and after CO2-PTF are calculated.The fractal evolution characteristics of the 3D fracture structure of coal induced by CO2-PTF are revealed.The main conclusions are as follows: CO2 phase transition fracturing coal has the fracture generation effect and fracture expansion-transformation effect, causing the maximum fracture length to increase from 48.90 mm to 83.74 mm, the fracture number to increase from 35,183 to 92,159, the fracture volume to increase from 158.12 mm 3 to 6436.87 mm 3 , and the fracture surface area to increases from 7733.05 mm 2 to 110,585.06 mm 2 .
After CO2-PTF, the fractal dimension for fracture number increases from 2.3523 to 2.3668, and the fractal dimension for fracture volume increases from 2.8440 to 2.9040, which reflects the increased connectivity between three-dimensional fractures.
The volume and surface area of fractures with a length of less than 140 µm are reduced under the influence of the fracture expansion-transformation effect.The fracture expansion-transformation effect and fracture generation effect jointly promote a significant increase in the volume and surface area of fractures with a length greater than 140 µm.The 140 µm is the critical value for the fracture expansion-transformation effect and fracture generation effect.This research further reveals the fracture evolution mechanism by CO 2 -PTF coal and provides a new evaluation and analysis indicators for the effect of CO 2 -PTF.In addition, the above research method can provide theoretical and methodological references for the study of fracture evolution characteristics of other unconventional natural gas reservoirs and their reservoir stimulation.

Conclusions
According to the CT scanning technology and fractal theory, the variations in the length, number, volume and surface area of the 3D fractures in coal before and after CO 2 -PTF under the fracturing pressure of 185 MPa are analyzed.The fractal dimension (D N ) for fracture number and fractal dimension (D V ) for fracture volume before and after CO 2 -PTF are calculated.The fractal evolution characteristics of the 3D fracture structure of coal induced by CO 2 -PTF are revealed.The main conclusions are as follows: CO 2 phase transition fracturing coal has the fracture generation effect and fracture expansion-transformation effect, causing the maximum fracture length to increase from 48.90 mm to 83.74 mm, the fracture number to increase from 35,183 to 92,159, the fracture volume to increase from 158.12 mm 3 to 6436.87 mm 3 , and the fracture surface area to increases from 7733.05 mm 2 to 110,585.06 mm 2 .
After CO 2 -PTF, the fractal dimension for fracture number increases from 2.3523 to 2.3668, and the fractal dimension for fracture volume increases from 2.8440 to 2.9040, which reflects the increased connectivity between three-dimensional fractures.
The volume and surface area of fractures with a length of less than 140 µm are reduced under the influence of the fracture expansion-transformation effect.The fracture expansion-transformation effect and fracture generation effect jointly promote a significant increase in the volume and surface area of fractures with a length greater than 140 µm.The

Figure 1 .
Figure 1.Experimental system of CO 2 phase transition fracturing.

Figure 3 .
Figure 3. 2D Fractures induced by CO2-PTF coal: (a) and (b) are the 2D fractures of coal PR, (c) and (d) are the 2D fractures of coal PF.

Figure 3 .
Figure 3. 2D Fractures induced by CO 2 -PTF coal: (a,b) are the 2D fractures of coal PR, (c,d) are the 2D fractures of coal PF.

Figure 3
Figure 3 further shows that the 2D fractures on different slices of the same sample are obviously different.Therefore, the 2D fracture can only reflect the local fracture evolution.The fractures in coal have obvious 3D characteristics[40][41][42].To further truly reflect the fracture evolution of CO 2 -PTF coal, it is necessary to perform 3D reconstruction of the 2D slices.

Fractal 13 Figure 4 .
Figure 4. 3D Fracture in coal before and after CO2-PTF: (a) and (b) are the 3D fractures of coal PR, (c) and (d) are the 3D fractures of coal PF.

Figure 4 .
Figure 4. 3D Fracture in coal before and after CO 2 -PTF: (a,b) are the 3D fractures of coal PR, (c,d) are the 3D fractures of coal PF.

4. 3 . 13 Figure 4 .
Figure5shows that logN(L i ) and logL i have an obvious linear relationship, and the correlation coefficient R 2 is greater than 0.99, indicating that the number distribution of the 3D fractures inside the coal sample has obvious fractal characteristics.Before CO 2 -PTF, the fractal dimension D N for 3D fracture number is 2.3523.After CO 2 -PTF, the fractal dimension D N for the 3D fracture number is 2.3668.

Figure 7 .
Figure 7. Variation of the 3D fracture structure parameters before and after CO 2 -PTF: (a) maximum fracture length, (b) fracture number, (c) fracture volume, (d) fracture surface area.

Figure 8 .
Figure 8. Evolution of 3D fracture volume and surface area: (a) cumulative volume of fracture, (b) cumulative surface area of fracture.

Figure 8 .
Figure 8. Evolution of 3D fracture volume and surface area: (a) cumulative volume of fracture, (b) cumulative surface area of fracture.

Figure 8 .
Figure 8. Evolution of 3D fracture volume and surface area: (a) cumulative volume of fracture, (b) cumulative surface area of fracture.

Figure 9 .
Figure 9.Comparison of fracture volume and surface area with the fracture length 0.14mm as the boundary: (a) fracture volume with fracture length less than 0.14 mm, (b) fracture volume with fracture length greater than 0.14 mm, (c) fracture surface area with fracture length less than 0.14 mm, (d) fracture surface area with fracture length greater than 0.14 mm.

Fractal 13 Figure 9 .
Figure 9.Comparison of fracture volume and surface area with the fracture length 0.14mm as the boundary: (a) fracture volume with fracture length less than 0.14 mm, (b) fracture volume with fracture length greater than 0.14 mm, (c) fracture surface area with fracture length less than 0.14 mm, (d) fracture surface area with fracture length greater than 0.14 mm.

Figure 11 .
Figure 11.Five indicators for evaluating the 3D fracture evolution of CO2-PTF coal.

Figure 11 .
Figure 11.Five indicators for evaluating the 3D fracture evolution of CO 2 -PTF coal.