The Fractal Dimension, Structure Characteristics, and Damage Effects of Multi-Scale Cracks on Sandstone Under Triaxial Compression

Abstract

To study the influence of the spatial distribution and structure of multi-scale cracks on the mechanical behavior of rocks, triaxial compression tests and cyclic triaxial complete loading and unloading tests were conducted on sandstone, with real-time wave velocity monitoring and CT scan testing. The quantitative classification criteria for multi-scale cracks on sandstone were established, and the constraint effect of confining pressure was analyzed. The crack with a length less than 0.1 mm is considered a small-scale crack, 0.1–1 mm is a medium-scale crack, and larger than 1 mm is a large-scale crack. As the confining pressure increases, the spatial fractal dimension of large-scale cracks decreases, while that of medium-scale cracks increases, and that of small-scale cracks remains stable. The respective nonlinear models of the aspect ratio were established with the length and density of multi-scale cracks. The results indicate significant differences in the effects of cracks of different scales on rock damage. The distribution density of medium-scale cracks in the failed specimen is higher, which is the main reason to produce damage. The small-scale cracks mainly originate from relatively uniform initial cracks in rocks, mainly distributed in medium-density and low-density areas. The results of this research provide important insights into how to quantitatively evaluate the damage of rocks.

1. Introduction

The multi-scale failure behavior of rock mass is the fundamental cause of dynamic disasters in underground mining engineering [1,2]. Rocks are typical porous materials with complex pores, fractures, and skeletal structures. The damage and fracture of rocks are dynamic processes from micro-scale to macro-scale, and their failure exhibits significant scale effects [3,4,5]. Previous studies have shown that the size and location of cracks have a significant impact on the mechanical properties of rocks [6,7]. The quantitative study of multi-scale characteristics of cracks and damage is of great significance for the evaluation of rock mechanical behavior [8].

The evolution process of cracks with different scales can usually represent the instability mechanism of rocks [9]. To capture the evolution process of cracks, scholars have adopted various monitoring or testing methods [10,11]. Acoustic emission monitoring technology can achieve three-dimensional spatial localization of the fracture source, and the size of cracks can be determined by the energy level and distribution density [12,13]. Zheng et al. analyzed the fracture modes of small-scale and large-scale cracks using acoustic emission technology, representing shear fracture and tensile fracture, respectively [14]. Electron microscopy scanning technology can directly observe the distribution of microcracks on the fracture surface of failed rocks. The microscopic damage constitutive model can be established by counting the size and distribution characteristics of cracks [15,16,17]. Nuclear magnetic resonance technology can detect the volume of pores and cracks in materials [18]. Qin et al. compared the changes in large, medium, and small pores before and after rock failure and found that the connectivity of medium and large pores was significantly enhanced [19]. Ren et al. found the pores of 0–1 μm play an important role in the compaction stage, while the cracks with a pore size larger than 100 μm determine the change in porosity in the yield stage [20]. Lu et al. believed that pores with a diameter less than 0.1 μm are micro-pores, 0.1–1 μm are meso-pores, and those larger than 1 μm are macro-pores, according to the characteristics of the NMR T2 spectrum of slate [21,22]. The volume, spatial location, and structure of cracks can be detected using CT scanning technology [23]. Daniel et al. discovered a positive correlation between the acoustic emission count and the crack volume of rock using real-time CT scanning and acoustic emission monitoring technology [24]. Charalampidou et al. studied the multi-scale development of particles and cracks during the formation of shear failure zones in porous sandstone [25]. Yang et al. divided the complexity of the micro structure of coal specimens based on the stages of the porosity curve [26]. Wang et al. quantitatively characterized the complexity of cracks on shale using fractal dimension and concentration values [27]. Wang et al. analyzed the volume contribution of cracks with a different aperture and studied the distribution pattern, morphology, and structural differences of cracks [28].

The distribution of cracks is used to describe the continuity of rock structure but cannot directly characterize changes in mechanical properties. For example, previous studies have shown that the failure of rock fracture structures exhibits a continuous evolution characteristic, but the damage rate exhibits a mutation phenomenon [29]. This indicates that the distribution and propagation of cracks cannot directly characterize the damage degree. Current research indicates that the size effect of cracks on rock strength is not only determined by their volume, but also significantly influenced by the crack morphology [30]. The influence of crack structural parameters on the rock mechanical properties cannot be ignored, except for the volume and distribution of cracks [31,32,33]. Alessio et al. calculated the length, width, and other parameters of 2D microcracks on rock fractures and found that the crack aspect ratio on failed rock is mainly concentrated below 0.1 [34]. Benoît et al. found that the crack structure and potential cracking path have a significant impact on the macro mechanical behavior of rocks [35]. Li et al. studied the nonlinear relationship between the aspect ratio and porosity of various pores [36]. In addition, under the same porosity conditions, changing the pore size and the crack shape can also have a significant impact on the elastic modulus and strength of rocks [37,38].

The above research undoubtedly enriches the scientific connotation of rock crack and damage distribution characteristics. However, the current research has technical and methodological limitations. The spatial distribution and degree of damage of cracks cannot be evaluated using electron microscopy scanning technology [39,40]. The key method for the characterizing of damage is to analyze the spatial distribution of multi-scale cracks. Nuclear magnetic resonance technology can accurately detect pores with a diameter of less than 1 μm, but its detection accuracy is lower for larger cracks, and it cannot directly detect the spatial position of cracks with different scales [41]. In addition, there are few reports on the classification of multi-scale cracks with clear mechanical meanings. There is also a lack of research on characterizing the damage degree through statistical crack distribution.

In this work, triaxial compression tests and cyclic triaxial complete loading and unloading tests were conducted on sandstone, with real-time wave velocity monitoring and CT scan testing. Firstly, a quantitative classification of large-scale, medium-scale, and small-scale cracks was proposed, by the crack volume characteristics. Secondly, the fractal dimensions of 3D cracks with various scales and their mechanical effects were revealed. Then, the compaction effect of initial cracks and the damage effect of new cracks were studied through ultrasonic velocity and real-time porosity inversion. Finally, the relationship between crack aspect ratio and crack length and density was analyzed, and the damage distribution characteristic of multi-scale cracks was discussed.

2. Research Methods

2.1. Sandstone Physical Property and Testing System

The sandstone tested in this research was sourced from Ganqing Tunnel of the Xining-Chengdu Railway, located in Qinghai Province, China. All processed specimens have a height of 100 mm and a diameter of 50 mm, in accordance with the recommendations of the International Society for Rock Mechanics and Rock Engineering, as shown in Figure 1a. The maximum allowable error for the non-parallelism of the upper and lower end faces of specimens is ±0.02 mm, and the maximum allowable error for the diameter is ±0.2 mm. As shown in Figure 1b, the main mineral components of this sandstone are quartz, plagioclase, mica, etc., accounting for about 78.58%, 6.66%, and 5.37%, respectively. In the naturally dry state, the density of sandstone is 1, the initial elastic wave velocity is 2.36 g/cm³, and the average initial elastic wave velocity is about 2250 m/s. The porosity of sandstone obtained through mercury intrusion testing is 19.87%. Figure 1c shows the microscopic fracture photos of sandstone using an electron microscope scanner under 1000× magnification conditions. The composition mode of the sandstone mineral skeleton structure is mainly point-plane detachment, with interconnected pores between particles and a relatively loose structure [42].

In this research, all rock mechanics tests were conducted using the MTS-815, which produced by MTS Systems Corporation, Eden Prairie, MN, USA, as shown in the Figure 2a. The wave velocity testing of rocks was completed in an integrated acoustic emission and velocity testing system, which is attached to the MTS-815 system. The deformation of rocks was obtained through testing with axial and circumferential strain extensometers fixed in the middle of the specimen. The testing accuracy of the extensometer is ±0.001 mm, which meets the testing requirements. The real-time wave velocity of rocks is calculated by the propagation time of ultrasonic signals between the transmitting and receiving ends, which attached to both ends of the specimen. The frequency of the ultrasonic pulse signal is 300 kHz, and the time interval between two sets of pulse waves is 50 s [43]. A prepared sandstone specimen with extensometers and probes is shown in Figure 2b.

2.2. Test Procedure

Under traditional triaxial compression tests, the mechanical behavior of rocks involves pressure effects. The structure of rocks not only undergoes damage and fracture, but cracks are also compacted and closed under stress loading. This phenomenon can lead to some physical parameters such as wave velocity, and porosity cannot accurately characterize the mechanical state of rocks. Therefore, two types of rock mechanics tests were conducted in this study. The first type is the conventional triaxial compression test of rocks (TC test), aimed at studying the rock properties and the distribution characteristics of cracks with different scales. The second type is rock cyclic loading and unloading tests under triaxial compression (CTCLU test), aimed at detecting the structural changes inside the rock under stress-free conditions, and obtaining the compaction and damage characteristics of the rock. In both types of tests, the confining pressure was set to 0 MPa (representative of the uniaxial compression test), 2 MPa, 4 MPa, 8 MPa, 12 MPa, and 16 MPa [43].

Figure 3a shows the stress path of the TC test. Firstly, the confining pressure is loaded to the set value at a speed of 0.1 MPa/s. Then, we maintain the constant confining pressure and apply axial force at a speed of 0.5 kN/s. When the axial stress is higher than the damage strength of the rock, the loading mode of the axial load switches to circumferential strain control, with a speed of 0.04 mm/min.

Figure 3b shows the stress path of the CTCLU test; the loading and unloading process are shown below. Firstly, the confining pressure is loaded to the present value at a speed of 0.1 MPa/s. Then, the confining pressure remains constant, and deviatoric stress is loaded to the present value. Subsequently, the deviatoric stress is unloaded to 0 MPa. Finally, the confining pressure is unloaded to 0.1 MPa at a speed of 0.1 MPa/s. In all triaxial tests, when the axial stress is lower than the damage strength of sandstone, the deviatoric stress is loaded with a speed of 0.5 kN/s. When the stress is higher than the damage strength, the deviatoric stress is applied by the circumferential displacement control mode with a speed of 0.04 mm/min. The unloading speed of deviatoric stress is 0.5 kN/s. Under the CTCLU tests, the upper limit of axial stress is sequentially superimposed at 10% of the peak strength until the specimen is in failure.

3. Multi-Scale Fracture Characteristics

3.1. Macro Mechanical Behavior on Sandstone

The variation curves of axial strain and circumferential strain with axial stress on sandstone are shown in Figure 4, with different confining pressures under TC tests and CTCLU tests.

Table 1. Results of TC and CTCLU tests on sandstone.

Test	No.	${\mathit{\sigma }}_3$ /MPa	Ρ /(g/cm3)	${\mathit{v}}_0$ /(m/s)	${\mathit{\sigma }}_{\mathbf{P}}$ /(MPa)	${\mathit{\epsilon }}_1^{\mathbf{P}}$ /(%)	${\mathit{\epsilon }}_3^{\mathbf{P}}$ /(%)	${\mathit{\sigma }}_{\mathbf{cd}}$ /(MPa)	${\mathit{\sigma }}_{\mathbf{cd}}$ $/{\mathit{\sigma }}_{\mathbf{P}}$
TC test	TC-0	0	2.12	2203	35.13	0.668	−0.642	15.79	0.449
	TC-2	2	2.16	2305	53.21	0.786	−0.474	30.55	0.574
	TC-4	4	2.16	2228	76.30	0.913	−0.308	53.04	0.695
	TC-8	8	2.18	2316	93.93	1.064	−0.363	66.39	0.707
	TC-12	12	2.17	2296	112.15	1.232	−0.203	85.33	0.761
	TC-16	16	2.18	2250	138.27	1.610	−0.286	106.59	0.771
CTCLU test	CTCLU-0	0	2.13	2251	37.61	0.817	−0.635	17.84	0.474
	CTCLU-2	2	2.12	2237	55.25	0.939	−0.551	30.08	0.544
	CTCLU-4	4	2.12	2288	70.42	1.112	−0.462	42.16	0.599
	CTCLU-8	8	2.12	2218	92.14	1.264	−0.346	61.91	0.672
	CTCLU-12	12	2.13	2245	112.13	1.376	−0.221	80.01	0.714
	CTCLU-16	16	2.13	2253	134.83	1.553	−0.224	104.01	0.771

shows the main mechanical parameters on sandstone under the TC tests and CTCLU tests. The ${\sigma }_{\mathrm{P}}$ is the peak stress of sandstone. The ${\epsilon }_1^{\mathrm{P}}$ and ${\epsilon }_3^{\mathrm{P}}$ are the axial peak strain and circumferential peak strain, respectively. The ${\sigma }_{\mathrm{cd}}$ is the damage strength, which is determined by the inflection point from the compression to expansion of the sandstone specimen.

Under the CTCLU tests, the outer envelope of the stress–strain curve is very close to that of the TC test. Under different confining pressure, the peak stress varies by 0.02% to 7.71%, the axial peak strain varies by 3.54% to 18.24%, and the circumferential peak strain varies by 1.10% to 8.87%, compared to the CTCLU tests and TC tests. Under the TC test, the damage strength of sandstone is 15.79, 30.55, 53.04, 66.39, 85.33, and 106.59 MPa, and the normalized damage strength relative to peak strength is 0.449, 0.574, 0.695, 0.707, 0.761, and 0.771, respectively. Under the CTCLU test, the damage strength of sandstone is 17.84, 30.08, 42.16, 61.91, 80.01, and 104.01 MPa, and the normalized damage strength relative to the peak strength is 00.474, 0.544, 0.599, 0.672, 0.714, and 0.771, respectively. The normalized damage strength of sandstone in the two types of tests differs by 0.07% to 13.88%. When the confining pressure is lower than 8 MPa, the failed specimen shows the tensile shear composite model, accompanied by many unconnected tension cracks. Under high confining pressure conditions (when the confining pressure is greater than 8 MPa), the failed sandstone showed the through type shear model, and no secondary cracks can be observed.

Considering that rocks are typical porous media materials, the discreteness of the sandstone specimen may lead to differences in strength and deformation. However, compared to the TC and CTCLU tests, the differences in peak strength, damage strength, and peak strain are almost negligible. The above macro mechanical parameters and failure modes indicate that the limited does not significantly affect the mechanical properties of rocks under quasi-static loading conditions. Although there are differences in stress paths between the TC and CTCLU tests, the failure mechanism of rocks can be considered consistent. This test result ensures the scientific validity of multi-scale crack distribution and compaction damage effects, which were characterized by CTCLU tests.

3.2. Multi-Scale Crack Classification Criteria

The quantitative characterization of multi-scale cracks is a prerequisite for establishing damage variables. It is generally believed that macro cracks can be directly observed. The distribution and propagation process of micro and macro cracks can be monitored using techniques such as acoustic emission, CT, and nuclear magnetic resonance. However, there is no relevant research on methods for quantitatively distinguishing micro cracks and macro cracks.

After the rock mechanics test, a Nano Voxel CT scanner was used to scan the multi-scale cracks in failed specimen TC-0. The device can recognize discontinuous structures with a minimum size of 50 μm. The scanning interval for each slice is 400 μm, with a total of 2500 slices per specimen. The Vgdef algorithm was used to reconstruct the 3D cracks in all slices, and the structural parameters such as volume, surface area, length, and spatial coordinates of the cracks were calculated. The 3D cracks of sandstone under uniaxial compression are shown in Figure 5.

According to CT scanning and calculation, the total volume of cracks in the specimen is 8269.92 mm³, and the smallest identifiable crack volume is 4.31 × 10⁵ μm³, and its surface area is 1.72 × 10⁵ μm². Figure 6 shows the crack volume of different sizes. When the crack volume is larger than 1 mm³, the volume of all cracks is 7783.351 mm³. When the crack volume is between 0.001 mm³ and 0.008 mm³, the total crack volume is 242.99 mm³. When the crack volume is between 0.008 mm³ and 0.027 mm³, the total crack volume is 48.42 mm³. When the crack volume is between 0.027 mm³ and 0.064 mm³, the total crack volume is 10.53 mm³. When the crack volume is between 0.064 mm³ and 0.125 mm³, the total crack volume is 5.78 mm³. When the crack volume is between 0.125 mm³ and 1 mm³, the total crack volume is 2.21 mm³. When the crack volume is between 0.000125 mm³ and 0.000512 mm³, the total crack volume is 105.44 mm³. When the crack volume is between 0.000512 mm³ and 0.001 mm³, the total crack volume is 71.19 mm³.

Based on the above analysis, the crack scale can be divided into three categories: large scale, medium scale, and small scale. The volume of large-scale cracks is higher than 1 mm³, the volume of medium-scale cracks is between 0.001 mm³ and 1 mm³, and the volume of small-scale cracks is less than 0.001 mm³. The crack length can more directly represent its size than its volume and surface area. If an irregular crack is equivalent to a cube with the same volume, the length of the cube can be considered as the equivalent crack length $L_{\mathrm{crack}}$. Therefore, the length of large-scale cracks is greater than 1 mm, the length of medium-scale cracks is between 0.1 mm and 1 mm, and the length of small-scale cracks is less than 0.1 mm.

Current research indicates that the spatial distribution of cracks with different scales exhibits self-similarity characteristics. The self-similarity of crack distribution can be used as an indicator to measure the accuracy of multi-scale crack classification. Large-scale cracks show obvious continuity, indicating that crack propagation follows a self-similar law. Medium-scale and small-scale cracks cannot be directly observed by their spatial distribution. Therefore, in this study, the scientific and mechanical significance of multi-scale crack classification standards is defined by calculating the fractal dimension of cracks.

To verify the spatial distribution self-similarity of medium- and small-scale cracks, in this study, the specimen is divided into 10 equal parts based on height (z), with each part having the radius of 2.5 mm and the height of 10 mm. We project the spatial centroid coordinates of the crack in the cylindrical element onto the xy plane and calculate the distance from the circle center $R_i$ is as shown in Figure 7a. According to the calculation theory of fractal dimension based on the coverage method, the relationship between the covered cracks number $N_r$ by a circular ring, and its radius $r$ is as follows:

$$N_r=C{r}^D$$

(1)

where $C$ is a constant.

By taking the logarithm of Equation (1), it can be obtained:

$$\mathit{lg}{N}_r=\mathit{lg}C+D\mathit{lg}r$$

(2)

The corresponding $N_r$ can be obtained for a given radius of $r$. By plotting $\left(\mathit{lg}r,\mathit{lg}{N}_r\right)$ in the coordinate system and linearly fitting, fractal dimension is the slope of the linear fitting.

Figure 7b,c show the coordinates projections of medium-scale and small-scale cracks on the xy plane, when the height is between −5 mm and 5 mm, respectively. Figure 8a shows the fractal dimension calculation results of the medium-scale and small-scale cracks in sandstone. The agreement between the test data and the linear fitting results are all above 0.950, indicating the validity of the calculated results.

As shown in Figure 8b, the fractal dimension curves of medium- and small-scale cracks first increase and then decrease with the height, and are generally symmetric around the axis of z = 0. This indicates that the distribution characteristics of cracks in the upper and lower parts of the sample are similar. In addition, the fractal dimension of small-scale cracks has always been greater than that of medium-scale cracks. This indicates that the medium-scale cracks are more concentrated, while the small-scale cracks are more complex. These calculation results indicate that medium- and small-scale cracks show self-similarity, and the multi-scale crack classification method proposed in this work is scientifically reasonable.

3.3. Mechanical Effects of Different Scales Cracks

Table 2. The 3D spatial distribution of multi-scale cracks on sandstone specimens.

σ3/MPa	3D Reconstruction	Large-Scale Cracks	Medium- and Small-Scale Cracks	Medium-Scale Cracks	Small-Scale Cracks
4 MPa
8 MPa
16 MPa

shows the 3D spatial distribution of various scale cracks on sandstone specimens with different confining pressures. The confinement effect of confining pressure on sandstone is mainly reflected in the number and morphology of tensile cracks. Under the low confining pressure conditions, large-scale cracks in sandstone show the “N” shape. Under high confining pressure conditions, large-scale cracks penetrate through sandstone specimens with the shear failure model. When ${\sigma }_3=4 \mathrm{MPa}$, medium- and small-scale cracks are concentrated at one end of the specimen and distributed with a conical shape surrounded by large-scale cracks. When ${\sigma }_3=8 \mathrm{MPa}$, medium- and small-scale cracks are distributed on both sides of large-scale cracks as the veins. The distribution of small-scale cracks is relatively uniform, making it difficult to distinguish the crack propagation trajectory. However, the distribution of medium-scale shows significant continuity, which can be used to determine the direction and location of large-scale cracks.

Figure 9 shows the volume distribution of medium-scale and small-scale cracks with different crack length. When ${\sigma }_3=0 \mathrm{MPa}$, the volume of small-scale cracks and medium-scale cracks is 176.63 mm³ and 309.94 mm³, respectively, and the proportion of small-scale cracks is 36.30%. When ${\sigma }_3=4 \mathrm{MPa}$, the volume of small-scale cracks and medium-scale cracks is 2.28 mm³ and 171.09 mm³, respectively, and the proportion of small-scale cracks is 1.32%. When ${\sigma }_3=8 \mathrm{MPa}$, the volume of small-scale cracks and medium-scale cracks is 1.49 mm³ and 234.85 mm³, respectively, and the proportion of total small-scale cracks volume is 0.063%. When ${\sigma }_3=16 \mathrm{MPa}$, the volume of small-scale cracks and medium-scale cracks is 2.31 mm³ and 115.66 mm³, respectively, and the proportion of total small-scale cracks volume is 1.96%. Under triaxial compression, small-scale cracks are significantly constrained, with the proportion decreasing from 36.30% to below 2%. Under uniaxial compression, small-scale cracks in sandstone are mainly initial cracks, which have not been compacted. Under triaxial compression, many initial small-scale cracks are compressed. With the confining pressure increasing, small-scale cracks cannot further propagate into medium-scale cracks.

With the crack length increasing, the medium-scale and small-scale cracks volume shows an “S shaped” growth curve, as shown in Figure 9 using the red dashed line. This evolutionary characteristic conforms to the logistic curve growth function. This function can be shown in Equation (3), and the fitting curves are shown by the blue dashed line in Figure 9.

$$\Omega =A_1-{\displaystyle \frac{A_1}{1+{\left(L_{\mathrm{crack}}/B_1\right)}^{C_1}}}$$

(3)

where $A_1$, $B_1$, and $C_1$ are the parameters of the logistic curve growth model.

Figure 10 shows the relationship between the spatial fractal dimension of cracks with different scales and confining pressure. With the confining pressure increasing, the spatial fractal dimension of small-scale cracks remains fluctuating within the range of 2.4 to 2.6. The volume of small-scale cracks is affected by confining pressure and deviatoric stress, but the distribution characteristics are almost unchanged due to the initial pore. When the confining pressure increases, the fractal dimension of medium-scale crack increases, while the large-scale cracks decrease. These results indicate that under low confining pressure conditions, medium-scale cracks are prone to propagate and form large-scale cracks, causing the complex distribution of large-scale cracks. Under high confining pressure conditions, the size of micro cracks is difficult to increase, and the spatial distribution is more dispersed and uniform.

The rock failure process is the small-scale cracks converging to produce medium-scale cracks, which then propagate to form large-scale cracks. Under uniaxial compression, small-scale and medium-scale cracks can converge as much as possible to form macro cracks. Under triaxial compression, the number of small-scale cracks significantly reduced because of circumferential constraints. From these results, it can be found that the confining pressure compaction effect is mainly reflected in the closure of initial and new small-scale cracks, while the rock damage effect is mainly reflected in the propagation of medium-scale and large-scale cracks.

4. Crack Structure and Damage Effects

4.1. Real-Time Porosity Inversion and Compaction–Damage Process

During the triaxial loading process, new cracks in the rock may remain closed under stress. The wave velocity of rocks under stress constraints cannot truly reflect the damage in rocks. The wave velocity of the rock can only truly reflect the damage state after the external load of the rock is completely unloaded. Figure 11 shows the relationship between wave velocity and stress. The pre-peak stress level can be calculated by the ratio of the upper-limit stress to peak stress, and the post-peak stress level can be calculated by Equation (4).

$$l_{\sigma }=1+\left[1-\left({\sigma }_{\mathrm{upp}}-{\sigma }_3\right)/\left({\sigma }_{\mathrm{p}}-{\sigma }_3\right)\right]$$

(4)

where $l_{\sigma }$ is the stress level, and ${\sigma }_{\mathrm{upp}}$ is the upper-limit stress.

As shown in Figure 11, $v_a$ and $v_b$ represent the real-time wave velocity of sandstone after stress completely unloaded and loaded, respectively. After the first three cycles, the wave velocity of sandstone remains stable at around 2500 m/s, indicating that the initial cracks closed irreversibly. However, the elastic pores between rock crystal particles can still close under stress, and $v_b$ continues to increase with increasing stress levels. When the stress is higher than ${\sigma }_{\mathrm{cd}}$, the new cracks appear and $v_a$ decreases slightly, while $v_b$ tends to stabilize. After peak strength, medium-scale and small-scale cracks converge and form large-scale cracks, leading to $v_a$ rapidly decreasing. However, in the post-peak stage, $v_b$ can remain at 3500–3700 m/s, indicating that the new large-scale cracks are in a “pseudo closed state” under stress [43].

According to the wave velocity real-time monitoring results, the damage effect can be scientifically characterized by the wave velocity after stress is completely unloaded. When the stress is lower than ${\sigma }_{\mathrm{cd}}$, a nonlinear model of compaction stage wave velocity is established, as shown in Equation (5).

$${\left.v_a\right|}_{l_{\sigma }<l_{\mathrm{cd}}}=v_a^{\mathrm{close}}\left[1-\exp \left(l_{\sigma }\cdot \alpha \right)\right]+v_0\cdot \exp \left(l_{\sigma }\cdot \alpha \right)$$

(5)

where $v_a^{\mathrm{close}}$ is the wave velocity after the initial crack closure, and $\alpha$ is the model parameter.

When the stress is higher than ${\sigma }_{\mathrm{cd}}$, many new cracks rapidly propagate, leading to the wave velocity decreasing exponentially. Therefore, a wave velocity exponential attenuation model is established, characterizing the development of damage in the post-peak stage, as shown in Equation (6).

$${\left.v_a\right|}_{l_{\sigma }\ge l_{\mathrm{cd}}}=v_a^{\mathrm{close}}-v_a^{\mathrm{open}}\left[\beta l_{\sigma }-\exp \left(\beta \cdot l_{\sigma }\right)+1\right]$$

(6)

where $v_a^{\mathrm{open}}$ is the decrease in wave velocity caused by the damage, and $\beta$ is the model index.

Therefore, the variation in wave velocity caused by rock compaction and damage under stress application can be characterized, represented by Equation (7).

$$v_a=\left\{\begin{array}{l}{v}_a^{\mathrm{close}}-\left(v_a^{\mathrm{close}}-v_0\right)e^{l_{\sigma }\cdot \alpha }\begin{array}{c}\begin{array}{c}\end{array} \mathrm{when},l_{\sigma }<l_{\mathrm{cd}}\end{array}\\ v_a^{\mathrm{close}}-v_a^{\mathrm{open}}\left[\beta \left(l_{\sigma }-l_{\mathrm{cd}}\right)-e^{\beta \left(l_{\sigma }-l_{\mathrm{cd}}\right)}+1\right]\begin{array}{c} \mathrm{when},l_{\sigma }\ge l_{\mathrm{cd}}\end{array}\end{array}\right.$$

(7)

The real-time wave velocity of sandstone after stress is completely unloaded is fitted using Equation (7), and the fitting curves are shown in Figure 12. When ${\sigma }_3=4 \mathrm{MPa}$, $v_a^{\mathrm{close}}$ is 2458 m/s, $\alpha$ is 10.70, $v_a^{\mathrm{open}}$ is 758 m/s, and $\beta$ is 1.35, with a goodness fit of $R^2=0.969$. When ${\sigma }_3=8 \mathrm{MPa}$, $v_a^{\mathrm{close}}$ is 2483 m/s, $\alpha$ is 19.46, $v_a^{\mathrm{open}}$ is 1064 m/s, $\beta$ is 1.18, with a goodness fit of $R^2=0.978$. When ${\sigma }_3=16 \mathrm{MPa}$, $v_a^{\mathrm{close}}$ is 2505 m/s, $\alpha$ is 35.84, $v_a^{\mathrm{open}}$ is 1358 m/s, $\beta$ is 1.51, with a goodness fit of $R^2=0.947$. This wave velocity model can accurately describe the characteristics of wave velocity increasing first, then stabilizing, and finally decreasing nonlinearly, revealing the compaction and damage process on rock.

The variation in the number and structure of cracks is the fundamental reason for the 1alteration of mechanical properties. The variation of the porosity during crack propagation and closure leads to changes in the rock wave velocity. Therefore, it is possible to accurately reflect the changes in the micro structure in the rocks by inverting the real-time porosity. In this study, based on the wave velocity theory model, a modified Raymer model was used to invert the real-time porosity, as shown in Equation (8).

$$v_a={\left(1-\varphi \right)}^2{\gamma }_m{v}_m+\varphi v_{\varphi }$$

(8)

where $\varphi$ is the total porosity of rocks, $v_m$ is the wave velocity of the rock matrix, ${\gamma }_m$ is the correction coefficient for the rock matrix wave velocity, and $v_{\varphi }$ is the wave velocity of the pore.

Therefore, according to Equations (7) and (8), the real-time wave velocity calculation formula for sandstone is shown in Equation (9).

$$\varphi =\sqrt{{\displaystyle \frac{v_a-{\gamma }_m{v}_m}{{\gamma }_m{v}_m}}+{\left({\displaystyle \frac{v_{\varphi }-2{\gamma }_m{v}_m}{2{\gamma }_m{v}_m}}\right)}^2}-{\displaystyle \frac{v_{\varphi }-2{\gamma }_m{v}_m}{2{\gamma }_m{v}_m}}$$

(9)

The skeleton of sandstone is quartz, which the wave velocity is 4500 m/s. Usually, for the hard rocks with dense microstructure, the rock matrix wave velocity correction coefficient ${\gamma }_m$ is selected as 1. For the sandstone with loose structure in this study, ${\gamma }_m$ is selected as 0.75, based on the mass proportion of quartz. The medium of the pores is air, with a wave velocity of 340 m/s. By using Equation (9) and the above assumptions, the evolution of real-time porosity with stress levels can be calculated.

Figure 13 shows the real-time porosity model curves. The initial porosity range of sandstone is 19.52%~20.42%, which is very close to the porosity of 19.87% in the mercury intrusion test. When ${\sigma }_3=4 \mathrm{MPa}$, the porosity after the initial crack closed is 15.58%, the small-scale crack closure is 3.94%, and the porosity after the specimen failed is 32.31%. When ${\sigma }_3=8 \mathrm{MPa}$, the porosity is 15.11% while the initial crack closed, the small-scale crack closure is 5.05%, and the porosity after the specimen failed is 38.94%. When ${\sigma }_3=16 \mathrm{MPa}$, the porosity after the initial crack closed is 13.88%, the small-scale crack closure is 6.54%, and the porosity after the specimen failed is 40.60%. It can be inferred that the initial cracks are significantly compacted as the axial stress approaches ${\sigma }_{\mathrm{cd}}$. When the axial stress is higher than ${\sigma }_{\mathrm{cd}}$, the propagation and convergence of medium-scale and large-scale cracks cause rock failure.

4.2. Cracks Structure Characteristics and Damage Distribution

The rocks mechanical parameters are influenced by the number, location, and structure of cracks. The scale and spatial distribution of cracks were studied in the previous section, and the quantitative study will focus on the relationship between crack structure and quantity. Approximation cracks as circular ellipsoids is the most common method for characterizing cracks in porous media materials. The aspect ratio is the ratio of the short axis to the long axis, which is an important structural parameter for cracks. As shown in Figure 14a, $a$ and $b$ are semi-major axes of the ellipsoid, and $c$ is a semi-minor axis. The crack aspect ratio can be expressed as $\lambda =c/a$. Due to the irregular shape of the actual crack, the crack short axis is usually not counted in CT scans. Therefore, the crack short axis needs to be calculated by a standard ellipsoid with the volume equal to the actual crack, as shown in Equation (10).

$$c=\sqrt{{\displaystyle \frac{3V_{\mathrm{crack}}^{\mathrm{el}}}{4\pi a}}}=\sqrt{{\displaystyle \frac{3V_{\mathrm{crack}}}{4\pi a}}}$$

(10)

where $V_{\mathrm{crack}}^{\mathrm{el}}$ is the volume of an ellipsoid equivalent to the actual crack.

According to the definition of crack aspect ratio, when $\lambda$ is 1, the crack can be regarded as a sphere. The larger $\lambda$, the rounder the crack is, and the smaller $\lambda$, the flatter the crack is. Figure 14b shows the negative exponential function relationship between the aspect ratio and length of the crack when ${\sigma }_3=16 \mathrm{MPa}$. The model is shown in Equation (11): as the crack aspect ratio increases, the length decreases nonlinearly. This indicates that the larger the crack size on the failed rock is, the smaller its aspect ratio and the flatter the shape. The shape of small-scale cracks is approximately a sphere, and its aspect ratio is close to 1.

$$L_{\mathrm{crack}}=A_2+B_2{e}^{-\lambda /C_2}$$

(11)

where $A_2$, $B_2$, and $C_2$ are parameters for the aspect ratio model.

By substituting Equation (11) into Equation (3), the relationship between the aspect ratio and the cumulative crack volume can be obtained, as shown in Equation (12).

$$\Omega =A_1-{\displaystyle \frac{A_1}{1+{\left[\left(A_2+e^{-\lambda /B_2}\right)/B_1\right]}^{C_1}}}$$

(12)

Henyey and Pomfrey found that the compressibility of a crack is inversely proportional to its aspect ratio, meaning that the smaller the aspect ratio, the easier the crack is to be compressed [44,45]. This indicates that although the microcrack volume is small, it may still show strong compressibility. Therefore, the structural parameter that affects the material elastic mechanical properties is not only porosity but also a parameter that is positively correlated with the distorted strain field volume around the crack. Therefore, the crack density is defined as shown in Equation (13).

$$\Gamma ={\displaystyle \frac{S〈a^3〉}{V}}$$

(13)

where $\Gamma$ is the crack density, $S$ is the crack number, 〈 〉 represents the average value, and $V$ is the material volume.

Therefore, the crack density can be calculated by the porosity and crack aspect ratio, as shown in Equation (14).

$$\varphi ={\displaystyle \frac{4\pi \lambda }{3}}\Gamma$$

(14)

Figure 15 shows the theoretical relationship between the cumulative volume and the aspect ratio of cracks, when ${\sigma }_3=16 \mathrm{MPa}$. If the damaged rock contains k sets of cracks with different aspect ratios, the total crack volume is $\Omega$, and the total porosity is ${\varphi }_k=\Omega /V$. When the aspect ratio is lower than ${\lambda }_i$, the cumulative crack volume is ${\Omega }_i$. When the aspect ratio is lower than ${\lambda }_{i+1}$, the cumulative crack volume is ${\Omega }_{i+1}$. Therefore, when the aspect ratio is between ${\lambda }_i$ and ${\lambda }_{i+1}$, the crack volume is $\Delta {\Omega }_i={\Omega }_{i+1}-{\Omega }_i$. When ${\lambda }_i$ and ${\lambda }_{i+1}$ are very close, it can be considered that the volume of this group of cracks with an aspect ratio of ${\lambda }_i$ is $\Delta \Omega$. The density of this group of cracks can be calculated according to Equation (14), as shown in Equation (15).

$$\Delta {\Gamma }_i={\displaystyle \frac{3\Delta {\Omega }_i}{4\pi {\lambda }_{i}V}}$$

(15)

where $\Delta {\Gamma }_i$ represents the density of the group of cracks with an aspect ratio of ${\lambda }_i$.

We assume that the crack aspect ratio is numerically continuously distributed and the functional relationship between the cumulative volume and aspect ratio is continuous. Therefore, Equations (14) and (15) are continuous and differentiable, and Equation (15) can be expressed as:

$$\zeta \left(\lambda \right)={\displaystyle \frac{3}{4\pi \lambda V}}{\Omega }^{\prime }\left(\lambda \right)$$

(16)

where $\zeta \left(\lambda \right)=d\Gamma \left(\lambda \right)/d\lambda$ represents the crack density distribution function. Among them, $\Gamma \left(\lambda \right)$ represents the cumulative density of all cracks when its aspect ratio is lower than $\lambda$.

Therefore, the distribution of crack density with aspect ratios can be calculated by Equation (16). As shown in Figure 16, the crack density of failed sandstone increases first and then decreases with the crack aspect ratio, approximately showing a normal distribution. The maximum crack density is around 0.85%, corresponding to the crack aspect ratio of approximately 0.2. According to the numerical value and distribution of crack density, cracks are classified into high-density cracks, medium-density cracks, and low-density cracks. When the crack density is higher than 0.5%, it is considered a high-density crack. When the crack density is between 0.3% and 0.5%, it is considered a medium-density crack. When the crack density is less than 0.3%, it is considered a low-density crack.

As shown in Figure 16, the aspect ratio of large-scale cracks is generally below 0.05, and the corresponding crack density is also below 0.01%. Under triaxial compression, large-scale cracks show a single-shear failure model, and its distribution is concentrated, resulting in a very low density. The aspect ratio of medium-scale cracks is approximately within the range of 0.05 to 0.25, and they are distributed in high-density, medium-density, and low-density areas. The aspect ratio of small-scale cracks is generally greater than 0.25, mainly distributed in medium- and low-density areas, with only a small portion distributed in high-density areas. Under triaxial compression, medium-scale cracks are effectively constrained by confining pressure, making it difficult for them to further propagate into large-scale cracks. Therefore, the distribution density of medium-scale cracks in the failed specimen is higher, which is the main reason to produce damage. The small-scale cracks mainly originate from relatively uniform initial cracks in rocks, mainly distributed in medium-density and low-density areas.

5. Conclusions

In this article, the classification criteria of multi-scale cracks were established, and he structural parameters and damage effects of cracks were discussed. The main conclusions obtained are as follows.

Based on the volume and self-similarity characteristics of the spatial distribution of cracks, they can be divided into three categories: a crack with a length greater than 1 mm is considered a large-scale crack, 0.1–1 mm is considered a medium-scale crack, and less than 0.1 mm is considered a small-scale crack. As the confining pressure increases, the fractal dimension of small-scale cracks remains, the medium-scale cracks increase, while the large-scale cracks decrease.

The real-time porosity inversion model during rock failure was established, and the theoretical formula for crack density was derived based on the porosity and crack aspect ratio. The results indicate that the crack density of failed rock increases first and then decreases with the crack aspect ratio, approximately showing a normal distribution. Large-scale cracks are mainly distributed in low-density areas, while medium-scale cracks are in high-density areas. Due to limitations in the detection accuracy of CT scanners, smaller cracks were not included in this study. The distribution characteristics of smaller cracks and larger joints, fissures, and even faults will be carried out in future work.