2.1 Preparation of shale disc samples with a central hole (SDCH) specimens

The shale specimens used in this study were all obtained from Lushan Mountain, Jiujiang City, Jiangxi Province. These shale reservoirs show a uniform bedding structure and are characterized by good integrity. The specimens were processed into cylindrical discs with a diameter of 50 mm and a thickness of 25 mm. The nonperpendicularity and nonuniformity of both ends of the processed specimens did not exceed 0.02 mm. The specimen preparation and testing procedures were conducted in strict adherence to the guidelines of the ISRM (ISRM.Isrm, 1978) and CSEB (Blasting SOEA, CSEBC, 2018).

As shown in Figure 1a, the bedding angle was defined as the angle between the loading direction and the bedding plane. Five kinds of specimens were prepared with bedding angles of 0°, 30°, 45°, 60°, and 90°. A high-precision drilling machine was used to fabricate central holes in the discs, with diameters of 4, 6, 8, and 10 mm. In total, 180 specimens were prepared, as shown in Figure 1b.

2.2 Dynamic test system

The dynamic Brazilian splitting test system is shown in Figure 2. Using the m-SHPB system, HS photography system, combined with 3D-DIC technology, the tensile strength and dynamic fracture propagation characteristics of SDCH under different impact pressures were carried out.

The m-SHPB equipment mainly includes the following devices: control equipment, nitrogen cylinders, impact bullets (500 mm), incident rods (3000 mm), transmission rods (3000 mm), buffer rods (1200 mm), buffer equipment, bullet velocity collectors, and strain signal collectors. The rods are made of silicon–manganese spring steel, and the key physical parameters are shown in Table 1 (Luo et al., 2023).

2.3 Principles of 3D-DIC technology

As shown in Figure 3a, the HS photography system integrates a HS camera and a high-intensity fill light, serving as the hardware foundation for 3D-DIC technology. Given the brittle nature of shale, which results in the sudden and transient propagation of fractures, the frame rate was specifically set to 100 000 fps to capture these rapid events effectively. Figure 3b,c depicts schematic diagrams of the shale impact loading setup, providing a clear illustration of the experimental configuration and the forces applied during the loading process (Li, Luo, et al., 2024).

The basic principle of 3D-DIC technology is to combine the principle of binocular stereo vision (as shown in Figure 4a) with the digital image correlation matching technology to restore the three-dimensional spatial coordinates of each point on the surface of the measured object before and after deformation, and then obtain the surface topography and three-dimensional deformation information of the object, as shown in Figure 4b (Blaber et al., 2015).

3.1 Theoretical calculation of tensile strength

3.2 Statistics of tensile strength

According to the theoretical formula, the tensile stress of SDCH was calculated by taking 0.2 MPa as an example, and its variation law is shown in Figure 5. Due to the existence of pores and fissures within the shale, the specimens initially entered the compaction stage, and then rapidly transformed into the linear elastic deformation stage. The microcracks were gradually initiated by the progressive accumulation of failure strain, leading the specimens to gradually attain the peak tensile strength. The propagation of microcracks led to macroscopic failures, causing a rapid decline in tensile stress. The stress–time curve might show a bimodal phenomenon due to the influence of the central hole, consistent with the findings of Wu et al. (2020).

In order to further explore the influence of the bedding angle (0°, 30°, 45°, 60°, 90°) and the central aperture (4, 6, 8, 10 mm) on the dynamic tensile strength under different impact pressures (0.2, 0.3, 0.4 MPa), a total of 60 groups of experiments were carried out. To reduce the discreteness of the experimental results as much as possible, each group of experiments was performed three times, and the average of the experimental results was obtained. The statistical results of the dynamic tensile strength are shown in Table 2.

Table 2. Statistics of experimental results of the dynamic tensile strength.

3.3 Qualitative analysis of the effects of the tensile strength

The results of the tensile strength under different conditions, as shown in Figure 6, showed a clear relationship with the bedding angle and the central aperture. The minimum tensile strength always occurred when the aperture was 10 mm and the bedding angle was 0°, while the maximum tensile strength was observed when the aperture was 4 mm and the bedding angle was 90°. Specifically, the minimum tensile strengths were 13.48, 17.12, and 18.13 MPa and the maximum tensile strengths were 22.00, 27.85, and 34.68 MPa under the condition of 0.2, 0.3, and 0.4 MPa, respectively. These findings indicated that an increase in impact pressure significantly enhanced the tensile strength. When the impact pressure and the central aperture remained constant, the tensile strength increased with an increase of the bedding angle. However, under the same impact pressure and bedding angle, the tensile strength decreased significantly as the central aperture increased.

3.4 Quantitative analysis of multiple regression of tensile strength

To further quantify the differences in the influence of the impact pressure, the bedding angle, and the central aperture on the tensile strength, multivariate linear regression analysis was carried out to statistically analyze the experimental results (Table 2). As a scientific and effective analytical method, multivariate linear regression analysis (Ismail et al., 2023; Moosavi & Mohammadi, 2021) represents a complex model in regression analysis, capable of comprehensively considering the combined effects of multiple input variables on the output variables. Unlike univariate linear regression, it introduces additional factor variables to establish a more holistic system of relationships, thereby more accurately capturing the intricate influence of each factor on the output variables.

The nonstandardization coefficients of the impact pressure and the bedding angle were positive, suggesting a positive correlation between these variables and tensile strength. Conversely, an increase in the central aperture showed a negative effect on tensile strength. It is important to note that nonstandardized coefficients are influenced by factors such as variable scales and orders of magnitude, making them unsuitable for directly assessing the relative impact of independent variables on the dependent variable. In contrast, standardized regression coefficients enable a direct comparison of the effects of different variables. The standardized regression coefficients of the impact pressure, the bedding angle, and the central aperture were 0.674, 0.588, and −0.311, respectively. These values indicate that the influence of the three variables on tensile strength follows the order, from highest to lowest impact pressure, bedding angle, and central aperture. The stress fluctuations induced by impact pressure represent the most direct and significant factor, rapidly elevating local stress levels to the rock's fracture strength, thereby initiating and propagating cracks. Consequently, impact pressure exerts the greatest influence on tensile strength. The bedding angle plays a secondary role in determining tensile strength, primarily by influencing crack propagation pathways. At smaller bedding angles, cracks are more likely to propagate along the bedding planes, leading to reduced tensile strength. As the bedding angle increases, cracks may deviate from the bedding planes, altering their propagation direction and enhancing tensile strength. This variation in the bedding angle fundamentally changes the crack propagation mechanisms, thereby significantly affecting the rock's overall mechanical properties. The central aperture, on the other hand, predominantly affects tensile strength through localized stress concentrations, which diminish the rock's load-bearing capacity. Larger pores exacerbate stress concentrations in specific regions, promoting crack propagation. However, the influence of central aperture is more localized and indirect, resulting in a relatively weaker effect on tensile strength compared to the other two variables.

Under different impact pressures, with the bedding angle and the central aperture as independent variables, the key statistical characteristic coefficients for binary regression analysis are shown in Table 4. The regression analyses conducted under the three impact pressures all satisfied the conditions of normality, homogeneity of variance, and independence, confirming the validity of the statistical results.

As shown in Figure 8, the average tensile strengths of 0.2, 0.3, and 0.4 MPa were 17.02, 21.37, and 24.03 MPa, respectively. The standard mean deviations were 2.034, 2.817, and 4.298, respectively, indicating a progressive increase in tensile strength with increasing impact pressure. Notably, the regression analyses conducted under the three impact pressure conditions all satisfied the fundamental assumptions of normality, homogeneity of variance, and independence, confirming the validity and reliability of the regression model.

Surface fitting diagrams with error bars at different impact pressures are shown in Figure 9. At 0.2, 0.3, and 0.4 MPa, the fitting decrease rates of the tensile strength with respect to the central aperture were 0.366, 0.596, and 0.808 MPa/mm, respectively, and the increase rates with respect to the bedding angle were 0.059, 0.076, and 0.115 MPa/(°), respectively. These results demonstrated that an increase in the impact pressure could strengthen the effect of both the bedding angle and the central aperture on the tensile strength. Under high-impact pressures, the crack propagation path and mechanisms became more intricate. The interaction between the central aperture and the bedding angle resulted in a more pronounced influence on tensile strength. Specifically, as the impact pressure increased, the stress concentration induced by the central aperture interacted with the bedding effects, thereby exacerbating crack propagation and significantly altering the mechanical response of the rock. However, the fitting error increased, indicating that the experimental data showed greater dispersion and that the control effects of the bedding and aperture were weakened. The normalization coefficients of the bedding angle were 0.895, 0.829, and 0.821, while those for the central aperture were −0.412, −0.485, and −0.431, respectively. This indicated that, across the experimental range, the influence of the bedding angle on tensile strength was consistently greater than that of the central aperture. This conclusion was also evident from Figure 9, where the inclination angle of the fitted surface in the direction of the bedding angle was obviously greater than that in the direction of the central aperture.

The prediction model of dynamic tensile strength was established according to the regression analysis, and the differences between the measured and the regressed prediction values are shown in Figure 10. All dynamic tensile strength values were predicted within the 95% prediction interval, demonstrating high accuracy. The dispersion of the predicted scatters was the smallest at 0.2 MPa impact pressure, indicating that the regression model performed better in terms of prediction accuracy at lower impact pressures.

4.1 Fracture evolution characteristics

As shown in Figure 11, the fractures were divided into four typical fracture modes according to the cause of the fracture and the propagation path (excluding the small microcracks) (Luo et al., 2023; Wu et al., 2021): tensile fracture along the bedding plane (TS), tensile fracture cross the bedding plane (TC), shear fracture along the bedding plane (SA), and shear fracture cross the bedding plane (SC).

As illustrated in Figure 12, the final fracture characteristics of the SDCH under different bedding angles and central apertures at 0.2 MPa impact pressure were analyzed, with the fracture modes summarized in Figure 13. Through a detailed horizontal and vertical comparison, the following conclusions were drawn: Overall, the fracture characteristics showed increasing complexity as the bedding angle and the central aperture increased. Tensile fractures typically propagated in the direction of the impact loading before transitioning to shear fractures influenced by the bedding plane. Specifically, the initial fracture mode that originated from the disc's edge was the SA, while the SC likely evolved from the SA under the combined influence of the bedding surface and the central hole. Additionally, an increase in both the bedding angle and the central aperture showed a pronounced effect in promoting the formation of the SC.

The SDCH with a 0° bedding angle presented typical TA characteristics. As the central aperture increased, a limited amount of SC was generated. Interestingly, for all cases except the 0° bedding angle, the shear fracture forms were predominantly SA, indicating that the fracture propagation direction caused by shear fracture was consistent with the bedding direction. The bedding surface would obviously induce the fracture to expand along the bedding plane. The fracture characteristics of 30°, 45°, and 60° were mainly shear fractures. Notably, the occurrence frequency of SC increased significantly with an increase of the bedding angle, even under the same central aperture. For the SDCH with a 90° bedding angle, both tensile and shear fractures were evident simultaneously. Specifically, the SA and SC fractures were biased toward the central hole, while the TC fractures were located in the impact loading direction. The fracture propagation direction of the SA along the bedding direction generated from the edge of the disc and the SC extending from the SA almost always passed through the central hole. In contrast, the SC fractures extending from the TA or TC fractures deviated from the central hole. These findings suggested that the existence of the central hole only had an obvious guiding effect on fractures propagating along the bedding plane, but its influence was less significant on other fracture modes.

As illustrated in Figure 14, this investigation presented the fracture characteristics of the SDCH with a central aperture of 10 mm under the combined effects of impact pressure and bedding angle. The results demonstrated that as the impact pressure increased, the width of primary tensile fractures significantly increased, and there was an increased propensity for the formation of additional shear secondary fractures. While the overall fracture characteristics became more complex, they continued to show pronounced bedding-induced effects. Except for 0°, with the increase of the impact pressure, the SA generated on the side of the disc increased significantly and the bedding effect of shear fracture propagation was enhanced. The SC formed by the expansion of the SA still tended to be in the direction of the central hole. Notably, the SA originating at the central hole gradually evolved into the TC during its expansion process.

Figure 15 shows the fracture characteristics of the SDCH at a 90° bedding angle under the influence of the impact pressure and the central aperture. The SDCH specimens at 90° all formed two obvious main tensile fractures in the horizontal direction of the central hole. The increase in the impact pressure promoted the initiation and propagation of fractures near the central hole, resulting in a significant increase in the width of the tensile fracture. When the aperture was 4 mm, the increase of the impact pressure led to the formation of more shear fractures near the central hole, thereby making the fracture morphological characteristics more complex. Conversely, the fracture mode was the simplest when the aperture was 10 mm, but the fracture width was the largest. For apertures of 6 and 8 mm, the overall fracture mode became the most complex, with an increased generation of SC generated by the TC and SA. Importantly, the SC generated by SA all pointed toward the central hole, whereas the SC generated by TC deviated from it. This observation further demonstrated that the central hole only had a significant guiding effect on the SC generated by the SA.

4.2 Dynamic fracture process obtained by the 3D-DIC

To further reveal the fracture initiation and propagation mechanism of the SDCH under impact load, shale specimens with five varying bedding angles were subjected to additional supplement dynamic splitting experiments under the condition of 0.2 MPa and 10 mm. Additionally, the evolution process of the SDCH surface strain field was obtained by the DIC full-field analysis technology. Due to differences in the experimental objectives and the fundamental physical properties of the rock samples, variations in fragmentation levels between the two sets of experiments are indeed present. However, the overall fracture patterns in Figure 16 remain consistent with those observed in Figure 14, providing additional insights into the initiation, propagation, and characteristics of fractures.

As shown in Figure 16, the final fracture propagation morphology was basically consistent with the previous analysis, showing overall symmetry of the central hole in the loading direction. Specifically, Figure 16a illustrates that the fracture mode remained the typical type of TA at a bedding angle of 0°. For 30° and 60° bedding angles (Figure 16b,d), the fractures on both sides of the central hole propagated horizontally, but their subsequent expansion direction deviated to some extent from the horizontal direction. Under these two bedding angles, both TC and SC were observed. In addition, the failure mode at 30° was more complex, with the additional presence of SA. As shown in Figure 16c,e, the fracture modes at 45° and 90° showed the mixed fracture characteristics of the TC and SA.

Figure 17 presents the evolution process of the vertical strain field under different bedding angles. While the vertical strain field showed good symmetry on both sides of the central hole, there were notable differences in the appearance position and the evolution process of the strain concentration area. These differences indicated that the fracture initiation position and the propagation path were significantly influenced by the bedding angle.

As shown in Figure 17a, when the bedding angle was 0°, the strain concentration area initially appeared at the loading position on the left side of the SDCH and subsequently extended gradually toward the central hole. Meanwhile, on the right side, the strain concentration area first emerged near the central hole and then extended progressively toward the right impact loading position. This asymmetric evolution of strain concentration zones was closely tied to the direction of the impact loading, which influenced both the fracture initiation and propagation. Ultimately, a symmetrical strain concentration area was formed on both sides of the central hole in the horizontal direction, consistent with the macroscopic fracture distribution characteristics observed in Figure 16a. In Figure 17b, at a bedding angle of 30°, the strain concentration areas on both sides of the central hole appeared almost simultaneously and then extended toward both ends under continuous impact loading. However, the strain concentration zone on the right side showed a significantly larger extension range and velocity compared to the left side. Notably, on the lower right side, a strain concentration area formed with an extension direction closely aligned the bedding angle. This behavior was primarily due to the interaction of the main fracture with the bedding plane during propagation, causing a deflection in the extension path. When the bedding angle was 45° (Figure 17c), the cracking sequence and the propagation direction were largely reversed compared to the 0° case, and the strain concentration area displayed a certain angle relative to the horizontal line. At bedding angles of 60° and 90° (Figure 17d,e), the strain concentration area initially appeared near the central hole and then gradually extended toward the loading ends on both sides.

4.3 Variation law of the fracture area

Combined with the fracture morphological features, the fracture area can quantitatively characterize the fracture complexity under different conditions. Understanding the variation law of the fracture area under dynamic impact is crucial for optimizing the engineering application of methane in situ explosion technology. The image processing software ImageJ was used to segment the threshold of surface fractures, and the actual area was obtained by converting pixel values into real dimensions. To simplify computational and statistical analyses, the fracture morphological feature was displayed by taking 90°, 10 mm as examples, as shown in Figure 18. Relevant studies (Engelder & Zevenbergen, 2018; Wu et al., 2020) have demonstrated that tensile fractures in shale reservoirs typically show large widths under explosion loading, serving as dominant fractures extending deep into the reservoir and providing primary channels for methane migration. Meanwhile, shear fractures demonstrate a greater ability to connect natural fractures within the reservoir, thereby enhancing methane gas release. An ideal fracture network should consist of a balanced distribution of both tensile and shear fractures. Based on the classification principle outlined in the preceding section, the total fracture area was subdivided into tensile and shear fracture areas, with the tensile fracture ratio defined as the proportion of tensile fracture area relative to the total fracture area.

Figure 19 shows the variation of the fracture area and the tensile fracture ratio with the bedding angle and the central aperture under 0.2 MPa.

As shown in Figure 19a, the fracture area showed a trend of initially decreasing and then increasing with an increase of the bedding angle for aperture sizes other than 6 mm, with the maximum fracture area occurring at 90°. Notably, when the aperture was 6 mm, the fracture area was significantly larger than that of the other three apertures at 30°, 45°, and 60°, and reached the maximum at 60°. Combined with Figure 19c,d, it is evident that the SDCH mainly presented tensile failure at 0°, with the shear fracture area progressively increasing as the central aperture size increased. The tensile fracture ratio at 90° remained relatively stable, hovering around 60%, regardless of the central aperture. Figure 19b shows that the SDCH maintained a high fracture area across all four at 90°, while the fracture area at 30°, 45°, and 60° peaked at the 6 mm aperture size. Therefore, to achieve an optimal fracture network during methane in situ ignition operations, it is recommended to align the methane explosion loading direction with the bedding plane at 90° (Chai et al., 2023; Wang et al., 2024b).

Under the 90° bedding angle, the variation law of fracture area under different explosive pressures is illustrated in Figure 20. This finding is particularly relevant for explosion fracturing applications in horizontal wells (Li et al., 2018; Li, Liu, et al., 2024), where the peak explosion pressure significantly influences fracture propagation near the perforation channel and the wellbore.

As shown in Figures 20 and 15, the fracture area under all apertures showed an increasing trend with an increase of the impact pressure, and the morphological features became progressively more complex. Specifically, the fracture areas for central apertures of 4, 6, 8, and 10 mm achieved their maximum values at 0.4 MPa, reaching 182.224, 214.142, 234.966, and 231.657 mm2, respectively. Among the three impact pressure conditions, the fracture areas for 8 and 10 mm apertures were significantly larger than those of the other two sizes, and their tensile fracture ratios remained relatively uniform despite variations in impact pressure. These findings demonstrate that larger perforation channels in the reservoir facilitate the formation of a more complex fracture network under the influence of stress waves.

4.4 Engineering application discussion

The findings of this study hold substantial practical significance, particularly in the realm of methane in situ combustion fracturing technology. By systematically controlling the impact pressure, the bedding angle, and the central aperture, it is possible to optimize crack propagation and significantly improve the efficiency of rock fracturing processes. This optimization is critical for improving the effectiveness of fracturing operations, especially in the contexts of shale gas extraction and oil and gas reservoir stimulation.

Previous research has indicated that the dynamic tensile strength of rock is closely associated with the formation of tensile fractures (Wang et al., 2023a, 2024b). While an increase in the impact pressure is positively correlated with the dynamic tensile strength, the fracture area still shows a significant increase, suggesting that moderately enhancing the peak explosive load can effectively enlarge the fracture area and create more complex fracture patterns (Wang et al., 2024a). Although the influence of the central aperture on the distribution of tensile fracture areas remains relatively minor (remaining around 60%), the substantial difference in fracture areas between 8 and 10 mm apertures compared to 4 and 6 mm apertures underscores the importance of selecting appropriate pore sizes to maximize fracturing efficiency. Notably, under the same impact pressure of 0.2 MPa, the fracture area at the 90° bedding angle is notably larger than that at other angles, indicating that, under specific geological conditions, adjusting the orientation of perforations relative to the bedding planes of shale reservoirs may be particularly effective in achieving larger fracture areas (Wei et al., 2024). Additionally, this study demonstrates that larger bedding angles and central pore diameters contribute to the formation of shear fractures. As previously established (Liu et al., 2023; Wang et al., 2023b), tensile fractures primarily serve as the main pathways for gas migration, while shear fractures are more effective in connecting natural fractures within the reservoir, thereby promoting the development of a more intricate and interconnected fracture network. Furthermore, the presence of a central pore plays a pivotal role in guiding fracture propagation along bedding planes during reservoir stimulation. These insights highlight the importance of optimizing both perforation pore sizes and the orientation of perforations relative to reservoir bedding planes to enhance fracturing outcomes.

In conclusion, this study emphasizes the necessity of adopting a comprehensive approach to fracturing design, taking into account the complex interactions between the impact pressure, the bedding angle, and the central aperture. The ability to optimize these factors has the potential to enhance techniques used in unconventional resource extraction, thereby improving both the technical performance and economic outcomes of energy production.