1 Introduction

Since the “carbon peak, carbon neutral” dual-carbon goal was put forward, China’s energy pattern has undergone significant changes, and clean and efficient energy is the development trend of China’s future.Compared to conventional natural gas, shale gas, an unconventional natural gas resource, offers advantages such as cleanliness, efficiency, a longer extraction life, and an extended production cycle (Philip 2009). Therefore, under the new development trend, shale gas is expected to replace conventional natural gas in the future (Lisjak et al. 2014; Shuai et al. 2015; Zhonghu et al. 2017; Xuhui et al. 2020; ** et al. 2021). According to the exploration results, China’s recoverable shale gas reserves have reached 36.1 × 1012m³, ranking first in the world (John 2002; Daniel et al. 2007), however, because of the low porosity and poor permeability of shale strata, resulting in the shortcomings of great difficulty and high cost of extraction (Zou et al. 2016; Hengxing et al. 2018; Wu et al. 2018; Yuewu et al. 9) (Hengtao et al. 2020).

Fig. 9
figure 9

Image after threshold segmentation

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Considering the non-uniformity of the shale, the Weibull statistical distribution (Weibull 1951) was used to determine each mechanical parameter of the shale model microcells in RFPA3D:

$$\phi \left( \alpha \right) = \frac{m}{{\alpha _{0} }} \cdot \left( {\frac{\alpha }{{\alpha _{0} }}} \right)^{{m - 1}} \cdot {\text{ }}e^{{ - \left( {\frac{\alpha }{{\alpha _{0} }}} \right)_{m} }}$$
(1)

In Eq. (1): α represents the mechanical property parameters (modulus of elasticity, strength, Poisson's ratio, etc.) of the fine-scale unit of the rock medium; α0 represents the average value of the mechanical property parameters of the fine-scale unit; m is the uniformity coefficient of the material, reflecting the homogeneity of the material, and the larger the coefficient of homogeneity, the more homogeneous the material is (Lou et al. 2020; Wu et al. 2020), each mechanical parameter of shale is detailed in Table 1.

Table 1 Mechanical parameters of shale matrix
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In order to study the influence of the inclination angle of natural cracks on the fracture process, mechanical properties and acoustic emission behavior of shale at the microscopic scale, seven groups of models with different inclination angles of natural cracks were reconstructed, the 3D numerical models are developed with inclination angles of 0°, 15°, 30°, 45°, 60°, 75°, and 90°, respectively, and numerical simulation experiments were carried out in uniaxial compression by RFPA3D, and the loading model is shown in Fig. 10, and the current experiment was loaded by displacement control volume with a single incremental step of 0.001 mm.

Fig. 10
figure 10

Schematic diagram of model loading

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4 Experimental findings and analysis

4.1 Mechanical properties

Figure 11 illustrates the stress-strain curves obtained from the uniaxial compression tests conducted on shale specimens containing natural cracks at various inclination angles. As shown in the figure, different natural crack inclination angles have a significant effect on the mechanical properties of shale, and the stress-strain curves are roughly divided into four stages, namely, the elastic deformation stage, the yielding stage of inelastic deformation, the destruction stage and the post-peak stage, and present obvious brittle rock characteristics.

As shown in Fig. 11, the damage curves of shale under each natural crack angle are roughly divided into two categories, namely, 0–45° and 60–90°, which are more typical brittle damage, but the stagnation time of shale with low natural crack angle in the damage stage is obviously higher than the latter.

Taking 0° as an example, firstly, the elastic deformation stage, at this time, with the loading, the stress-strain curve is linearly increasing. Next is the yield stage, continue loading, the stress began to increase nonlinearly, the microcracks inside the shale began to be compacted, and the stress-strain curve slowed down; the third stage is the destructive stage, at which the shale began to be destroyed, reaching the peak stress, but under the influence of the natural cracks of 0°, the shale did not lose its bearing capacity within a certain period of time. did not lose the bearing capacity within a certain period of time, but produced a period of stagnation, always maintaining the peak stress, in which the natural cracks inside the shale were continued to be compacted until the ultimate loss of bearing capacity; the fourth stage is the post-peak stage, in which the shale gradually loses all the bearing capacity, and the stress-strain curves fall down directly.

The shales with high natural fracture angles were more similar during the experiments, but all of them fell straight down when they reached the peak stress and did not produce a point of stagnation of the peak stress, where the 90° specimen produced a single stress reduction at 50% of the stress level, which may be the reason why its peak stress was lower than that of the 75° specimen.

Fig. 11
figure 11

Stress-strain curve under uniaxial

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Fig. 12
figure 12

Peak stress and modulus of elasticity at different angles

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Figure 12 presents the correlation between peak stress and modulus of elasticity in shale with varying inclinations of natural cracks.  As shown in Fig. 12, peak stress and modulus of elasticity in shale tend to rise with the increasing inclination of natural cracks. This observation aligns with the findings reported in the literature (Hao et al. 2020), and the peak stress of shale in the specimens of 30° and 90° only decreases, while that of shale in the specimens of 15° and 45° decreases. The peak stress is 40.45 MPa at 0°, 42.50 MPa at 15°, 43.74 MPa at 30°, 46.75 MPa at 45°, 54.22 MPa at 60°, 57.94 MPa at 75°, and 56.15 MPa at 90°. Obviously, natural fractures from different angles have a significant impact on the anisotropy of uniaxial compression in shale. As the natural crack angle increases, there is a decreasing trend in the strength loss observed during uniaxial compression of the shale. Overall, as the natural crack angle is raised, the peak stress and modulus of elasticity of the shale are increasing and less strength loss occurs in the shale.

4.2 Rupture mode

Figure 13 depicts the rupture process as well as the acoustic emission images of shale containing natural fractures at different angles during the full course of uniaxial compression. The fracturing process of shale containing natural fractures at different angles is divided into three main stages: crack generation, crack extension, and final penetration of shale. The corresponding acoustic emission diagrams during the crack penetration stage reveal variations influenced by the different angles of natural cracks, leading to diverse initiation points, extension processes, and eventual rupture modes. They are mainly classified into the following four categories:

  1. (1)

    Sprouting along the natural crack initiation to the middle and outside of the shale, and eventually forming a similar “X”-type crack (0°): During uniaxial compression with a natural crack inclination angle of 0°, the natural crack situated in the middle of the shale undergoes gradual compression and densification as the displacement loading increases. Subsequently, tiny cracks emerge from the middle of the natural crack, propagating towards the outer layers until they penetrate the interior of the shale. This process is followed by the extension of crack propagation. The tiny cracks sprouted from the middle of the natural cracks to the outside until they penetrated into the middle of the shale, and then the cracks gradually expanded perpendicular to the upper and lower bottom surfaces, and gradually formed X-shaped cracks on the surface. In the final stage of failure, the shale did not completely rupture along the already formed vertical cracks, but two 45° damage surfaces were formed from the upper and lower bottom surfaces.

  2. (2)

    Sprouting along the natural cracks towards the middle and outside of the shale, and eventually forming inverted “Y” cracks (15°, 30°): In uniaxial compression at 15°, for example, the cracks sprang from the ends of the natural cracks, and no longer sprang along the natural cracks but perpendicularly towards the middle and outside of the shale, and then the initiated cracks expanded towards the upper and lower bottom surfaces. In the crack expansion stage, it can be observed that the upward expanding cracks are still in the shape of “I” to extend in a single direction, while the downward expanding cracks split into an inverted “V” shape in the lower part of the shale. It eventually extended along the upper right side of the shale where the lower part had been damaged, until it expanded into a through crack on the other side. During the extension phase, the upper part of the crack extends only partially, without forming a penetrating crack. The rupture of the 30° specimen proceeds similarly to that described above, except that the crack develops more densely. As loading continues, cracks continue to extend to form an inverted "Y" shaped penetration crack.

  3. (3)

    During uniaxial compression at a natural crack inclination of 45°, the cracks continue to develop mainly along the natural cracks, ultimately forming a "Y"-type crack, sprout to the upper and lower bottom surfaces, and damage occurs at the junction of the upper bottom surface and the side surface, followed by “V”-type damage on the upper bottom surface. V” type damage on the upper bottom surface. In the crack expansion stage, it can be seen that the crack gradually expands downward along the side, and eventually the crack formed on the upper bottom surface also gradually expands from the front along the 45° diagonal until the crack passes through, forming a side “Y”-type mainly through the crack.

  4. (4)

    The cracks do not follow the natural crack initiation, eventually forming “S” type cracks (60°, 75° and 90°): During the uniaxial compression at a natural crack inclination of 60°, it is evident that crack generation does not originate from the natural cracks, but rather sprouts in the middle of the natural cracks in the directions of the vertical and upper bottom surface, the side, and the front, resulting in three directions in the same stage of the shale, the upper bottom surface, the side, and the front. The crack initiation stage of the 75° and 90° specimens is similar to the above, however, the cracks mainly expand in different locations, resulting in three different angles of “S” type cracks throughout the final cracks.

Fig. 13
figure 13

Fracturing process of microstructure of shale

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In summary, the damage patterns of shale containing natural fractures with different dip angles can be roughly classified into four categories at the microscopic scale. The first category is to sprout along the natural fractures to the outer side of the shale, and ultimately form a crack similar to the “X”-type crack (0º); the second category is to sprout along the natural fractures to the middle and outer side of the shale, and ultimately form an inverted “Y” type crack ( 15°, 30°); the third type is the cracking along the sides of the natural cracks, eventually forming a “Y” type crack (45°); the fourth type is the cracking not along the natural cracks, eventually forming an “S” type crack (60°, 75° and 90°).

. In the acoustic emission diagram, the red color characterizes the units that have shear damage at the current loading step, and the blue color characterizes the units that have tensile damage at the current loading step. In uniaxial compression, when the inclination angles of natural cracks are 0°, 15°, 30° and 45°, it can be found that most of the units in the main cracks are tensile damaged during the rupture process, and a small amount of compression shear failure occurred in the remaining units; in uniaxial compression, when the inclination angles of natural cracks are 60° and 90°, it can be found that the main cracks are mostly tensile damaged during the rupture process, but some of the compression-shear damage occurs in units that have not yet formed cracks. During uniaxial compression at a natural crack inclination angle of 75°, the primary crack unit in the rupture process experiences both tensile and compressive-shear damage.

4.3 Stress distribution characteristics

Figure 14 illustrates the principal stress distribution characteristics of shale microstructures with varying inclination angles of natural cracks under uniaxial compression. Initially, at a natural crack inclination angle of 0°, 15°, or 30°, there is a noticeable stress concentration effect on the surfaces adjacent to the two ends of the natural cracks, while the front and back surfaces parallel to the natural cracks have less stress; at a natural crack inclination angle of 45°, it can be clearly seen that the stresses are mainly concentrated on the upper bottom surface; at a natural crack inclination angle of 60°, 75°, 90°, the stresses are also concentrated on the upper bottom surface and the side of the shale. At a natural crack inclination angle of 60°, 75° and 90°, the stress is also concentrated on the upper bottom surface and side of the sample, but different from the natural crack with low angle, the stress is more concentrated on a certain point rather than distributed along a single path, and with the increasing of the angle the point of concentration of the stress is moving from the side to the upper bottom surface, so that the first damaged surface is also moving with it. In the rupture stage, 15°, 30°, 45°, 60° specimens from the natural crack extension face, that is, around the side, the top and bottom of the expansion path of cracking, the stress is still concentrated in these surfaces, and finally occurred mainly in the face parallel to xz destruction; 0°, 75°, 90° specimens also occurred on the front side of the phenomenon of obvious stress concentration, indicating that at this time, natural cracks on shale impact has been restricted, the natural fracture. The location of the initiation of cracks occurs on the lateral or upper top surface, but eventually the cracks will extend to the front surface, and finally extend to the body damage of the whole specimen.

Overall, the natural cracks from 0º to 60 ° determine the stress concentration location of the specimen, and have a greater influence on the path of crack initiation at the crack initiation stage, where the natural cracks from 0° to 30° determine the internal cracks to be initiated to the side, and the natural cracks at 45° and 60° determine the internal cracks to be initiated to the upper bottom surface. And when the natural cracks are 75° and 90° have less influence on the path of the crack when it sprouts. At the stage of rupture, the specimen formed a weak surface in the plane parallel to the natural cracks from 30° to 60° to incur damage, while the overall damage occurred at 0°, 75°, 90° (Fig. 14).

Fig. 14
figure 14

Stress distribution of microstructure of shale

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4.4 Evolutionary characteristics of acoustic emission distribution

Fig. 15
figure 15

Relationship among stress, AE count, cumulative AE count and loading steps under different angle

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Figure 15 depicts the correlation between stress, strain, acoustic emission and cumulative acoustic emission counts for shale specimens subjected to seven sets of natural cracks with varying inclinations. In RFPA3D, microfracture (strength degradation) of a unit is regarded as a single acoustic emission event, and the cumulative acoustic emission counts, which are the sum of acoustic emission events, represent the total number of unit damages in the overall specimen (quoted). As shown in the figure, the stress-strain curve shows good agreement with the acoustic emission strain curve. The phase changes of the acoustic emission and the stress-strain changes basically coincide with each other and reflect a clear angular influence.

In the seven groups of experiments in this experiment, the acoustic emission data have similar trends when the natural crack angle is 0°, 15°, 30° and 45°. Taking the natural crack angle of 45° as an example, the initial cumulative acoustic emission performance is relatively flat, and only when it reaches the middle and late elasticity stage, a very few acoustic emission events appear, and during the gradual loading process, with the stress continuing to increase, the shale The internal units began to continuously damage, the acoustic emission events continued to accumulate, the cumulative acoustic emission showed an accelerated growth trend, at this time the stress state is gradually approaching the peak stress of the specimen; the subsequent cumulative acoustic emission showed a sudden change, as the loading step continued to increase, the stress reached the peak, at this time the number of single acoustic emission events began to rapidly increase, the cracks in the shale began to accelerate the expansion of the entire damage stage The number of single acoustic emission events is very high and reaches a maximum value of 27,020 at approximately 50% of the peak stress level (29.42 MPa), and the shale is completely destroyed during continued loading.

In the three sets of experiments with natural crack inclination angles of 60°, 75°, and 90°, the acoustic emission curves behaved more similarly to those at low natural crack inclination angles, and the differentiation of the stages was consistent, but in the destruction stage, the destruction of the shale with high natural crack inclination angles was more drastic, which was manifested by the decrease in the number of acoustic emission events, but the level of acoustic emission events in the destruction stage was higher, which could be up to 61,788, 46, 605, respectively, 94,315, reflecting that the shale with high natural crack inclination is more brittle. 90° inclination shale specimen produces a stress drop, at this time, through the acoustic emission spatial distribution map of this loading step can be found, the upper surface of the specimen and the natural crack intersection position suddenly produces an extended tensile crack, resulting in the redistribution of stress, which may also be the shale specimen with 90° natural crack inclination final strength of the shale specimen with 90° natural crack inclination is lower than that of the shale specimen with 75° natural crack inclination.

The information carried by the acoustic emission signals is more consistent with the above changes in the mechanical properties and rupture process of shale with natural cracks of different inclinations, so, the acoustic emission signals can represent the stress-strain trends and rupture processes of naturally fractured shales with different dip angles on a microscopic scale.

Table 2 AE energy and fractal dimension Ds at different stress levels for different natural crack inclination angles
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Fig. 16
figure 16

Plot of stress level versus fractal dimension Ds for different natural crack inclination angles

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Fractal theory can describe the irregular and complex things in nature, including the evolution of rock rupture process, and the fractal dimension Ds, as a basic parameter of fractal theory, can quantitatively characterize the intensity of the rock in the process of destruction and the complexity of the rock at the final destruction. In this paper, the rock rupture images under different natural crack inclinations and different stress levels are imported into the MATLAB and obtain the corresponding fractal dimension values. The following equation shows the solution of fractal dimension Ds:

$${D_S}= - \mathop {\lim }\limits_{{\frac{1}{{\text{r}}} \to 0}} \frac{{\log N(r)}}{{\log \frac{1}{r}}}$$
(2)

Ds represents the fractal dimension of the pore structure within the damaged region, while r represents the side length of the square box. N(r) is equal to the number of square boxes of side length r needed to cover the damaged area.

Specific values of AE (acoustic emission) energy and Ds （fractal dimension） are given in Table 2 for shale specimens with various natural crack inclinations at different stress levels, and Fig. 16 shows the stress levels versus AE energy and fractal dimension for shale specimens with different natural crack inclinations. As depicted in the figure above, both the AE (acoustic emission) energy and the fractal dimension of shale specimens increased with the stress level across each natural crack inclination angle; the overall trend case is that the peak stress and the corresponding AE energy values for each stress level condition also increased with the increase of natural crack inclination angle, while a decrease was observed at a natural crack inclination angle of 90° due to the fact that the specimens at a natural crack inclination angle of 90° were This is due to the fact that at 90° natural crack inclination, a stress reduction occurs at 50% stress level, during which the upper unit of the natural crack undergoes tensile damage and microcracks are generated, thus affecting the mechanical properties of the overall specimen.

The graphs indicate that the fractal dimension of specimens with varying angles of natural cracks is not 0 at the 10% stress level, which is attributed to the presence of natural cracks. Taking the natural crack inclination angle of 90° as an example, at a stress level of 10%, the fractal dimension value is 2.02, which is due to the inhomogeneity of the natural cracks (the whole is slightly in the form of a wave pattern); and at a stress level of 60%, the fractal dimension value is 2.243, and the cracks suddenly expand. At this time, the cracks suddenly expanded, new tensile cracks were generated at the intersection of the upper and natural cracks, and tensile cracks also appeared in the whole rock interior, especially at the periphery of the natural cracks, but at this time, the tensile cracks were not connected, but the overall complexity level of the cracks was greatly increased; in the process of the stress level growing from 60 to 100%, the fractal dimension number showed a non-linear growth, which was due to the shale compression of cracks produced continuously generated and compacted, the natural cracks, although dominating the cracks, are also compressed during further compression, and the fractal dimension is determined by the more intense primary and secondary cracks, and the compaction of the natural cracks and the expansion of the nascent cracks jointly determine the size of the fractal dimension. The value of the fractal dimension is affectde by both together during the destruction of the shale at each natural fracture angle.

As the natural crack angle increases, the fractal dimension value of shale at peak stress generally increases. Specifically, at 75°, the fractal dimension reaches its peak value of 2.611, which is the maximum value observed across the entire range of angles tested from 0° to 90°, but in the middle and early stage of the damage, 90° fractal dimension is the highest, which is due to the fact that it is the first one to produce tensile cracks, and at the same time, the trend of fractal dimension at the final damage is the same as that of the AE energy. Meanwhile, the trend of fractal dimension at final damage is the same as the trendof AE energy, and 75° final damage has the highest acoustic emission energy and the number of acoustic emission events, so its fractal dimension is also the highest.

In summary, under the influence of natural cracks with different inclination angles, the AE energy of shale specimens increases with the stress level, peaking at the maximum stress. Conversely, the fractal dimension values exhibit a zigzag pattern, also increasing, which can be attributed to the compaction of natural cracks and the extension of nascent cracks, both influencing the fractal dimension magnitude, and the compaction phenomenon is obviously weakened with the elevation of the angle; with the increase in the angle of inclination of the natural cracks increases, the AE energy and fractal dimension increase in the final damage, and only the 90° test exists a decrease, this is due to the 90° specimen produced secondary cracks, the final damage stress partially along the secondary crack release expansion, the damage intensity is reduced, the extension of the other cracks is less, and the rupture form is relatively simple (Fig. 17).

4.5 Model validation

Fig. 17
figure 17

comparison of 60° model results

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Comparison of the results of the 60° uniaxial compression acoustic emission experiments with those of the previous experiments reveals that the acoustic emission patterns are basically the same, thus confirming the accuracy of the present model (Wang et al. 2021).

5 Conclusions

The shale samples used in this study are from the Lower Cambrian Niushutang Formation in the northern Qianbei region. These samples were observed using SEM and CT scanning. A natural crack was selected, and each section picture of shale containing natural cracks was obtained through relevant processing. Digital images of 7 groups of shale containing natural cracks with different inclinations were imported through RFPA3D. Three-dimensional numerical models were then established for each group, and numerical tests (uniaxial compression) were conductde. The study drew the following conclusions: 

  1. (1)

    The inclination angle of natural cracks plays a crucial role in determining the fracture mode of shale. The damage patterns of shale containing natural cracks with varying inclination angles can be broadly classified into four categories on a microscopic scale. The first category is to sprout along the natural crack to the outside of the shale, and ultimately form a similar “X”-type crack (0º); the second category is to sprout along the natural crack to the middle and outside of the shale, and ultimately form an inverted “Y”-type crack (15°, 30°); the third category is to sprout along the natural crack to the middle and outside of the shale, and ultimately form an inverted “Y” crack (15°, 30 The second type of cracking is along the natural cracks to the outside of the middle of the shale, resulting in inverted “Y” cracks (15°, 30°); the third type of cracking is along the sides of the natural cracks, resulting in “Y” cracks (45°); and the fourth type of cracking is not along the natural cracks, resulting in “S” cracks (60°, 75°, and 90°). In the low natural fracture dip shale model, tensile damage mainly occurs, with compression shear damage; in the high natural fracture dip shale model, both tensile and compression-shear damage account for a larger proportion of the fracture process.

  2. (2)

    The microstructure of shale significantly impacts the distribution of principal stress, the natural cracks from 0º to 60° determine the stress concentration location of the specimen, and have a greater influence on the direction of crack initiation at the stage of crack initiation, where the natural cracks from 0° to 30° determine the internal cracks to be initiated to the side, and the natural cracks at 45° and 60° determine the internal cracks to be initiated to the bottom surface. And when the natural cracks are 75° and 90° have less influence on the path of the crack when it sprouts. In the rupture stage, the specimen formed a weak surface in the plane parallel to the natural cracks from 30° to 60° to incur damage, while the overall damage occurred at 0°, 75°, and 90°.

  3. (3)

    Acoustic emission can indicate the changes in stress and the fracture process of shale with natural cracks at varying inclination angles on a microscopic scale. The evolution of acoustic emission signal distribution is characterized by two types, 0°-45° test and 60°-90° test, and the difference is mainly reflected in the destruction stage, the destruction of shale with high natural crack inclination is more intense, which is manifested by the reduction of the number of acoustic emission events, but the level of acoustic emission events at the stage of destruction is higher, which can reach 61,788, 46,605, and 94,315, and the brittleness of the shale with high natural crack inclination is stronger.

  4. (4)

    The presence of natural cracks significantly impacts the fractal dimension value. Under the influence of natural cracks with different inclination angles, the fractal dimension value zigzags up with the increase of stress level, and the cracks generated by the sudden change of stress will directly affect the size of the fractal dimension, and the compaction of natural cracks and the expansion of the nascent cracks jointly determine the size of the fractal dimension.