Experiment and DEM simulation study on mechanical behaviors of shale under triaxial cyclic loading and unloading conditions

Abstract

During shale gas exploration, natural fractures in shale reservoirs may be induced by cyclic loads frequently encountered in different geological processes, including tectonic movements, seismic actions, and artificial construction interference (vertical and horizontal wells drilling and cyclic hydraulic fracturing). In this paper, experiments and PFC^2D simulations are conducted to investigate the mechanical response and mechanism of shale specimens under cyclic loading and unloading. Using the experimental and simulation results, the strength and deformation characteristics and strain energy and damage evolutions during the cyclic loading-unloading process are quantified and analyzed. The damage variable characterized by plastic deformation, deformation modulus, and dissipated energy is thoroughly analyzed. Based on the PFC^2D simulation, the micro-crack distribution and evolution are further studied, and the results can reflect the experiments well and reveal the damage mechanism under cyclic loading. Furthermore, the experiments and simulations indicate that the degree of fatigue damage is closely related to the number of cycles the specimens undergo, and a small number of cycles may not have a distinct effect on the strength.

Highlights

• 1. Cyclic loading induces fatigue damage on the shale specimen. However, the degree of damage is closely related to the number of cycles the specimens undergo. A small number of cycles may not distinctively weaken the strength.

• 2. Cyclic loading leads to more complex cracks than monotonic loading, and the energy dissipation rate of each cycle first decreases and then increases with the increase in the number of cycles.

• 3. The effect of cyclic loading is weakened when the bedding plane dominates the failure mode of the shale specimens and when the bedding inclination is 60°.

1 Introduction

Shale gas is now an important and rapidly developing energy over the world. To exploit shale gas deep underground, artificial vertical and horizontal wells are often drilled and extended to the shale reservoir. To facilitate gas release from the shale matrix, complex fracture networks are expected by connecting the natural fractures with hydraulic fractures. Natural fractures in shale reservoirs affect the permeation behavior, hydraulic fracture propagation, and anisotropic mechanical behavior and can be induced by cyclic loads, which are frequently encountered in the diverse geological processes involving tectonic movements and seismic actions (Liu and Dai 2021). On the other hand, some rock engineering applications, such as drilling vertical and horizontal wells and cyclic hydraulic fracturing technics (Zhuang et al. 2016, 2019; Ji et al. 2021), can induce fatigue damage in shales. As shown in Fig. 1, such actions cause the shale reservoir to show different mechanical properties from that under monotonic stress conditions. The effects of cyclic loading on shale rock are manifested as the progressively deteriorating properties of the damaged rock. The damage evolution and failure characteristics can differ from that of monotonic loading. The strength, deformation, and failure characteristics of shale strata with bedding structures under cyclic stress in shale gas wells deserve further study.

Fig. 1

Shale reservoir may undergo cyclic loads induced by diverse geological processes, including tectonic movements, seismic actions, and cyclic hydraulic fracturing technics

Since cyclic loading is a common stress condition for rocks underground, researchers have conducted cyclic loading tests on different types of rocks under different experimental conditions. According to the loading path, cyclic loading in laboratory tests can be classified into the following types (Cerfontaine and Collin 2018; Liu and Dai 2021): (a) regular triangular, sinusoidal, and actual/ideal square waveforms, (b) waveforms with constant and stepwise increasing amplitudes, (c) waveforms with multi-level cyclic amplitudes and those with increasing average stresses and constant amplitudes, and (d) random cyclic loading. Based on these loading paths, Xiao et al. (2009, 2010) studied the fatigue damage evolution under different numbers of cycles. Song et al. (2013, 2016) conducted uniaxial cyclic loading of different amplitudes on sandstone specimens and studied the damage evolution process and crack development. Momeni et al. (2015) conducted uniaxial cyclic loading tests on granite specimens and studied the effects of maximum load, amplitude, and frequency on fatigue behavior. Jia et al. (2018) performed a series of triaxial cyclic compression tests with constant-amplitude cyclic loading and increasing-amplitude cyclic loading. Yang et al. (2018a) studied the damage evolution of sandstone under two types of cyclic loading. Yang et al. (2018b) studied the fatigue characteristics of limestone subjected to multi-level amplitude cyclic loading. Peng et al. (2019) studied the effects of the lower stress limit during cyclic loading and unloading on the deformation characteristics. The above experimental research suggests that the number of cycles (fatigue degree), stress amplitude, and the upper and lower limits of cyclic stress are the key factors affecting the mechanical behavior of rock materials.

Damage evolution has been the focus of many studies, and many indicators or parameters have been used as damage variables, such as elastic modulus, irretrievable deformation, AE event, and dissipated energy. Moreover, these parameters can be easily obtained during cyclic loading. In each loading cycle, the damage can be estimated based on these parameters, thus facilitating the evaluation of the damage evolution in the entire failure process. Yang et al. (2015, 2017) studied the mechanical damage characteristics of sandstone and crystalline marble under triaxial cyclic loading. Liu et al. (2016) developed a damage constitutive model for rocks under cyclic loading, which revealed the relationship between the strength and deformation of rocks under cyclic loading conditions and fitted the cyclic loading test results well. In addition to the damage characteristics, the effects of confining pressure on the mechanical behavior under cyclic loading were investigated (Zhang et al. 2017), and the compressive strength (Fuenkajorn and Phueakphum 2010; Yang et al. 2020a), Young's modulus and Poisson's ratio (Ma et al. 2013), deformation anisotropy (Feng et al. 2020), crack evolution (Akesson et al. 2004; Chen et al. 2011), acoustic emission characteristics and energy evolution (Meng et al. 2016; Song et al. 2019; Duan et al. 2021), permeability and porosity evolution (Wang et al. 2017; Jiang et al. 2017), and fatigue behavior (Wang et al. 2013) subjected to cyclic loading were studied and evaluated. These researches revealed distinct differences between the mechanical behaviors of cyclic and monotonic loading.

Shale is a typical anisotropic rock material, and its mechanical characteristics under conventional loading conditions have been extensively investigated (Yang et al. 2020b, 2021). However, only a few studied anisotropic rock materials under cyclic loading. Gatelier et al. (2002) investigated the effect of structural anisotropy on the mechanical behavior of sandstone specimens under cyclic compression and considered different confining pressures and loading orientations. Their cyclic loading test results indicated two inelastic mechanisms in the sandstone, namely, compaction and microcracking, with distinct mechanical effects. Wang et al. (2019) carried out triaxial cyclic loading and unloading experiments on different backfill specimens and studied the effect of layering structure on the mechanical properties and failure modes of backfill. Li et al. (2020) conducted uniaxial fatigue loading experiments on shale specimens to study the effects of bedding layer orientation on fatigue characteristics. The above research discussed the effects of cyclic loading or fatigue on the mechanical characteristics of layered rock materials and contributed to understanding the anisotropic mechanical behavior under cyclic loading. However, more relevant research is needed, and the anisotropic effects should be considered and studied in depth.

To further study the effects of cyclic loading or fatigue on the mechanical behavior of rock materials, some numerical simulation studies have been conducted. Fu et al. (2020) investigated the crack development and mechanical behavior of marble specimens under three different cyclic loading paths and simulated the experiments with RFPA^2D, the results of which agreed well with the experimental results. Song et al. (2019) simulated the fatigue characteristics of concrete specimens under cyclic loading using the nonlinear parallel-bonded stress corrosion (NPSC) model. The simulated fatigue tests replicated the main mechanical features of concrete specimens under cyclic loading observed in the laboratory. Sinaie et al. (2018) developed a multi-phase implementation of the discrete element method (DEM) to simulate concrete under cyclic loading and yielded sufficient results illustrating the capability of the model in predicting the cyclic loading properties of concrete. Xu et al. (2021) proposed a numerical mesoscale approach modeling the damage and fracturing of sandstone under cyclic loading and studied the effects of maximum stress, stress amplitude, and specimen homogeneity on the fatigue behavior of sandstone. To investigate the mesoscopic mechanism by which cyclic loading affects mechanical behavior and damage evolution, further simulation studies on the effects of cyclic loading on damage evolution and failure are needed.

This research aims to investigate the mechanical behavior of laminated shale specimens under cyclic loading and unloading conditions and establish an anisotropic DEM model to analyze the mesoscopic mechanism of the failure process. The type of cyclic loading in this research is designed as damage-controlled using increasing mean and amplitude stress, and the shale specimens with three different bedding inclinations of 0°, 60°, and 90° are tested. The strength and failure characteristics and damage evolution process are analyzed and compared with those under monotonic loading. Based on the experimental results, the shale DEM model is established in PFC^2D and tested to reveal the mesoscopic mechanism of damage evolution in cyclic loading. By combining the experiment analysis and numerical simulation, the distinct and deep-laying mechanical characteristics and damage evolution mechanism are presented. The conclusions could help comprehend the anisotropic mechanical behavior of shales comprehensively.

2 Experimental methodology and procedures

2.1 Specimens

The shale specimens tested in this study were collected from Changning Country on the southern edge of the Sichuan Basin, the most important place for shale gas exploration in China (Yang et al. 2020b). The shale specimens were extracted from the lower Silurian Longmaxi formation, Paleozoic of the middle and upper Yangtze regions. X-ray diffraction test showed the mineral composition and proportion of the shale specimens as follows: quartz (36.3%), calcite (27.6%), Fe-muscovite (18.7%), clay (14.1%), and other minerals such as plagioclase, potash feldspar, and pyrite in minimal amounts (Yang et al. 2020b).

2.2 Experiment program and procedure

Most of the shale mass in the reservoir had horizontal dip angles, and the bedding inclinations of the shale mass in the drilled vertical and horizontal wells were 90° and 0°, respectively. Therefore, the shale specimens prepared for the cyclic loading and unloading tests in this research were at the bedding inclinations of 90° and 0°. The shale specimens at bedding inclinations of 0° and 90° corresponded to the engineering conditions of shale rock around the horizontal and vertical wells, respectively. To compare with the case of medium bedding inclination, specimens at the bedding inclination of 60° were also tested.

The mechanical properties of the shale specimens under conventional triaxial compression have been studied by Yang et al. (2020b). Based on the strength and deformation parameters from conventional triaxial compression tests, the experimental parameters for the cyclic loading and unloading test were determined. Three confining pressures of 5 MPa, 10 MPa, and 20 MPa were set for the tests, which were loaded at the rate of 8 MPa/min, and the displacement loading mode at the rate of 0.05 mm/min was selected for axial stress loading. The stress loading mode was selected for axial stress unloading, and the unloading rate was 100 Bar/min (the loading equipment used in the tests adopted the stress unit of Bar, which was converted into International System Units in subsequent data processing). Figure 2 illustrates the loading and unloading programs. During the loading process, the strain increment Δε₁ in each loading stage is constant at 0.001 and can be converted to 0.1 mm for the specimens. The test procedures are as follows:

• (1) The specimen was installed, and the confining pressure was adjusted to the set value according to the set procedure;

• (2) In the first stage of loading, the axial loading displacement for the first stage was set. The loading stopped automatically as the set loading displacement was reached. Meanwhile, the loading mode was immediately changed to stress loading, and the axial pressure was unloaded to 0 MPa according to the set unloading rate;

• (3) In the second stage of loading, the same process was followed as the first stage, except that the set loading displacement was increased by 0.1 mm compared with the first stage, and the loading displacement of each subsequent stage was increased by 0.1 mm compared with the previous stage;

• (4) The above loading and unloading steps were repeated until the specimen reached the residual strength or completely lost its bearing capacity. At that point, the test stopped, the remaining axial pressure was removed, and the remaining confining pressure was removed;

• (5) As all the loading pressures were removed, the hydraulic oil was drained, the specimen was removed, and the damaged specimen was photographed.

Fig. 2

Schematic of triaxial cyclic loading procedure on shale specimen

3 Results of triaxial cyclic loading and unloading tests on shale

3.1 Stress-strain curve

Figure 3a–i show the stress-strain curves of shale specimens at bedding inclinations of 0°, 60°, and 90° under confining pressures of 5 MPa, 10 MPa, and 20 MPa, respectively, and their comparisons with those of specimens under monotonic loading, where the peak stress point of each cyclic loading is marked.

Fig. 3

Stress-strain curve comparison between shale specimens under monotonic loading and cyclic loading

As shown in Fig. 3, due to the brittleness of the shale specimens, the stress drops rapidly after reaching the peak strength and losing the bearing capacity, and the specimens cannot continue to undergo cyclic loading and unloading. Therefore, this study mainly focused on the mechanical characteristics before the peak value of cyclic loading and unloading.

According to Fig. 3, when considering the effect of specimen dispersion, the failure strength of the shale specimens under cyclic loading and unloading is not significantly different from that under monotonic loading. For instance, as shown in Fig. 3a–d, i, the failure strength of these specimens under the two loading modes is similar, while the failure strength of the other specimens under the two loading modes is different but within the permissible range of dispersion and shows no specific differences. On the other hand, the stress-strain curves under the two loading modes are similar shape-wise both in the axial deformation and lateral deformation, and the curves still maintain a high degree of volume strain curve similarity. The endpoints of each stress loading stage are on the monotonic loading stress-strain curve, indicating a similar deformation modulus under the two loading modes. It can be observed that the deformation characteristics of the specimens under the two loading modes are also highly consistent. In addition, the hysteretic loops of the cyclic loading and unloading curve suggest that the spacing between each hysteretic loop is very small before reaching the peak strength. Thus, the plastic deformation of the specimens during unloading is very small, further indicating the high brittleness of the shale specimens in the tests. When the loading reached the peak stress under cyclic loading and unloading, the specimens lost their bearing capacity rapidly due to their high brittleness. Meanwhile, the stress dropped to a very low level rapidly, leaving no residual strength, and the specimens failed to continue undergoing cyclic loading and unloading. Therefore, few hysteresis loops can be observed in the post-peak stage.

3.2 Analysis of strength and deformation characteristics

To further analyze the strength characteristics of shale specimens under cyclic loading and unloading conditions, the relationship between peak strength and confining pressure is plotted in Fig. 4, where the monotonic loading (ML) and cyclic loading (CL) test results are compared.

Fig. 4

Peak strength comparison between shale specimens under monotonic loading (ML) and cyclic loading (CL)

Figure 4 presents the monotonic and cyclic loading test results of shale specimens with three sets of bedding inclinations, and the dashed line is the linear fitting line of the monotonic loading test results. It can be observed that the failure strength points under cyclic loading are in the range of the monotonic loading test and near the fitting line, indicating little difference between the failure strength under the two loading conditions. High-frequency cyclic loading, also called fatigue loading, can cause the rock material to fail even under low compression stress, and greater stress amplitude may require fewer cycles to reach failure. Thus, the number of cyclic loadings can affect the failure strength, but the effect is mainly observed under high-frequency cyclic loadings. Since the cyclic loading stress is increased stepwise at a frequency far from high-frequency loading, the effect on failure strength is limited. Therefore, the failure strength obtained from the cyclic loading tests is very close to the results under monotonic loading. The same conclusion was reached by a previous study concerning multistep loading on brown coal by Taheri et al. (2016), i.e., the multistep loading tests yielded failure strength values similar to those obtained from the monotonic loading tests.

To further analyze the deformation parameters and damage evolution of the shale specimens in the cyclic loading tests, the stress-strain curve characteristics of every single cycle are illustrated and discussed. Figure 5a presents the typical stress-strain curves of the shale specimens under cyclic loading, and Fig. 5b presents the schematic diagram of one single cycle curve in Fig. 5a. As shown in Fig. 5b, the initial point of the nth loading cycle curve (ε_n) does not overlap with the endpoint of the unloading cycle curve (ε_n+1, also the initial point of the next loading cycle curve) since the plastic strain generated in the loading process is unrecoverable. The difference between ε_n and ε_n+1 is the plastic strain ε_pⁿ of the nth loading cycle. On the other hand, the elastic modulus of the loading curve is E_L, calculated by the slope of the approximate linear segment in the loading curve. The elastic modulus of the unloading curve is E_u, calculated by the slope of the approximate linear segment in the unloading curve.

Fig. 5

Completed loading and unloading cycle curve and single cycle curve

Figure 6 shows the evolution of plastic strain ε_pⁿ in one single cycle of the complete cyclic loading test. The abscissa is the strain corresponding to the maximum stress of one single cycle of the loading curve. Due to the high brittleness of the specimens, almost no post-peak cyclic curve is observed during the loading process. Thus, all the plastic strains presented in Fig. 6 are pre-peak values. As shown in Fig. 6a, the plastic strain evolution of the specimens at the bedding inclination of 0° is consistent under different confining pressures. With the increase of cycles, the plastic strain generally increases. Specifically, the plastic strain increases slowly in the first few cycles but significantly as the load approaches the peak and most rapidly before failure. In Fig. 6b, the plastic strain evolution trend of the specimens at the bedding inclination of 60° under different confining pressures is different. Specifically, at a low confining pressure of 5 MPa, the plastic strain of one single cyclic loading decreases first and then increases before the peak, showing a "V" shaped variation trend. As the confining pressure increases to 10 MPa and 20 MPa, the plastic strain increases first and then decreases suddenly before increasing again. As shown in Fig. 6c, the plastic strain evolution trend of the specimens at the bedding inclination of 90° is similar to that at the bedding inclination of 60°. Under the confining pressures of 5 MPa and 10 MPa, the plastic strain of one single loading cycle decreases first and then increases before the peak, showing a "U" shaped trend. When the confining pressure is 20 MPa, the plastic strain increases first and then decreases slowly before continuing to increase.

Fig. 6

Plastic strain evolution in the cyclic loading test

Based on the above analysis, the plastic strain evolution trends of one single loading cycle can be summarized and discussed as follows:

• (1) The plastic strain increases slowly at the early loading stage and rapidly as the load approaches the peak. At the early loading stage, the specimen is dominated by elastic deformation with little internal damage, and the increment of plastic strain is small. However, when the load approaches the peak, the specimens accumulate much more internal damage, leading to the acceleration of plastic deformation increment.

• (2) The plastic strain increases slowly at first, decreases in the mid-term, and then increases again. In this evolution trend, the specimens are compacted in the early loading stage, and the deformation is irreversible, increasing the plastic deformation. With the specimens compressed, the initial damage decrease and deformation extend to the elastic stage, and the plastic deformation begins to decrease. As the loading increases, the new damage accumulates, and the plastic deformation increases again. In this case, the plastic deformation of the specimens is mainly caused by the compaction of the internal micro-cracks in the early loading stage. With the loading increase, new damage appears at the bedding plane, and the plastic deformation is mainly caused by the shear slip along the bedding plane.

• (3) The plastic strain first decreases and then increases before the peak, showing "V" or "U" shaped variation trends. In this case, the plastic deformation is mainly caused by the compaction of the internal micro-cracks at the early loading stage. However, the shale specimens in this research are very tight, and the internal micro-cracks are compacted soon. Hence, the plastic deformation increment gradually decreases. The plastic deformation increment shows almost no increase in the elastic deformation stage and increases rapidly as the damage accumulates near the peak.

Figure 7 shows the evolution of loading modulus E_L and unloading modulus E_U in every single cycle of loading and unloading. The abscissa is the strain corresponding to the maximum stress of one single loading cycle curve. The unloading modulus in the last cycle is not calculated since no unloading is performed. As shown in Fig. 7, the unloading modulus E_U is greater than the loading modulus E_L because the plastic deformation is irreversible during the unloading process, which decreases the total unloading deformation and increases the unloading modulus. On the other hand, the variation trend of the two elastic moduli is consistent. At first, the moduli increase continuously, and the increment is very fast. As the cycles continue, the modulus increment slows down and gradually stabilizes. When the loading reaches the peak, the modulus begins to decrease. The deformation modulus evolution reflects the internal damage development of the shale specimens. At the early stage of the loading test, the micro-cracks and pores inside the specimens are gradually compacted, i.e., the initial damage decreases, the specimens harden, and the elastic modulus increases. When the loading reaches a particular stage, the specimens are compacted, and the initial damage is minimized. At that time, the specimens are dominated by elastic deformation, resulting in a relatively stable deformation modulus. As the loading continues, new damage begins to generate and accumulate in the specimens, the specimens begin to soften, and the deformation modulus begins to decrease. The two modulus evolution trends under cyclic loading are in close accordance with the results of Xiao et al. (2021). It should be noted that the specimens in this research have high brittleness and fail rapidly in many cases after a long elastic deformation stage, showing no significant plastic deformation. For instance, as shown in Fig. 7d–f, the deformation modulus of specimens at the bedding inclination of 60° shows an increasing trend and does not decrease at the peak. In addition, Figs. 7g–i show that the deformation modulus of specimens at the bedding inclination of 90° decreases very slowly before failure, indicating insignificant plastic deformation.

Fig. 7

Deformation modulus evolution during cyclic loading test

3.3 Analysis of strain energy and damage evolution

• (1) Strain energy evolution during cyclic loading tests

From the thermodynamics perspective, it is assumed that the testing machine and the rock specimens compose a closed system without heat exchange with the outside world. Only the pre-peak energy evolution under static loading is considered, while the transformation of strain energy into the kinetic energy of the broken rock blocks pre-peak and post-peak is ignored. Then, the total strain energy U obtained by the rock element under compression can be expressed as:

$$U={U}_{e}+{U}_{d}$$

(1)

where U_e is the elastic strain energy that can be released by the rock element, and U_d is the dissipated energy of the rock element. The dissipated energy U_d is the main factor causing damage and plastic deformation in rock elements and irreversibly changing the internal state of the rock material.

Figure 8 shows the strain energy evolution and calculation of one single cycle of loading and unloading. Figure 8a is the stress-strain curve of the loading cycle. The area in blue enclosed by the curve section and the strain axis (abscissa axis) is the total strain energy U obtained from the sample element. Figure 8b shows a complete stress-strain curve of one single loading and unloading cycle. The area in green enclosed by the unloading curve in red and the strain axis (abscissa axis) is the elastic strain energy U_e that can be released by the specimen element, and the area in yellow enclosed by the loading curve and unloading curve is the dissipated energy U_d of the rock element.

Fig. 8

Diagram of strain energy calculation for one single cycle

The tests in this research are carried out under conventional triaxial compression conditions, i.e., σ₁ > σ₂ = σ₃, and the strain energy of the rock element can be expressed as:

$$U={\int }_{0}^{{\varepsilon }_{1}}{\sigma }_{1}\,d{\varepsilon }_{1}+2{\int }_{0}^{{\varepsilon }_{3}}{\sigma }_{3}\,d{\varepsilon }_{3}$$

(2)

The confining pressure σ₃ remains constant during the loading process. According to Eqs. (1) and (2), the dissipated energy U_d can be calculated as:

$$ U_{d} = 2\sigma_{3} \varepsilon_{3}^{p} + \mathop \smallint \limits_{0}^{{\varepsilon_{i}^{L} }} \sigma_{1}^{L} d\varepsilon_{1}^{L} - \mathop \smallint \limits_{0}^{{\varepsilon_{i}^{U} }} \sigma_{1}^{U} d\varepsilon_{1}^{U} $$

(3)

where ε₃^p is the lateral plastic strain equal to the difference between the total strain and the elastic strain; σ₁^L and σ₁^U are the axial stresses under loading and unloading, respectively, and ε₁^L and ε₁^U are the axial strains under loading and unloading, respectively. The two definite integrations are the areas under the loading curve and the unloading curve, respectively.

Energy dissipation can reflect the evolution of internal damage and plastic deformation of rock materials. With the increase of cycles, the accumulation of internal damage and plastic deformation of the specimens inevitably lead to more strain energy dissipation. Therefore, to better reflect the degree of energy dissipation in the cyclic loading process, the energy dissipation rate R_U is defined, which is the ratio of dissipated energy U_d to the total strain energy U:

$${R}_{U}={U}_{d}/U$$

(4)

where the energy dissipation rate R_U is between 0 and 1. A larger R_U means a higher degree of energy dissipation and greater plastic deformation in the specimens. An R_U equaling 0 means no energy dissipation in the loading process, and the specimens only go through elastic deformation, while an R_U equaling 1 means that all the strain energy from the loading is dissipated, and the specimens only go through plastic deformation. According to the test data, the total strain energy U, elastic strain energy U_e, dissipated energy U_d, and energy dissipation rate R_U of the shale specimens at the bedding inclinations of 0°, 60°, and 90° before peak in the cyclic loading and unloading process can be calculated by using Eqs. (3) and (4).

Figure 9 shows the energy dissipation rate evolution of shale specimens with different bedding inclinations under cyclic loading and unloading. As shown in Fig. 9, with the increase of axial strain, the energy dissipation rate of each cycle presents a "U" shape variation trend that decreases first and then increases. Therefore, the internal damage of the specimens decreases first and then increases. In the early loading stage, the energy dissipation is dominated by the compaction of the specimen. As the loading increases, the specimen is compacted, and the strain energy dissipation decreases. In the elastic deformation stage, the dissipation maintains a low level. When loading approaches the peak, damage gradually accumulates, and the dissipation increases. The energy dissipation rate maximizes at the peak of loading. Comparison between the energy dissipation evolution in Fig. 9 and the stress-strain curves in Fig. 3 shows that the point where the energy dissipation rate begins to increase corresponds to the inflection point of the volumic-strain curve, further indicating that the energy dissipation reflects the damage evolution in the specimens. In addition, the energy dissipation rate increases slowly within a small range near the peak point, indicating that the specimens still have a large amount of elastic energy before the peak, and the energy dissipation is not obvious. In particular, when the specimens are at the bedding inclination of 60°, as shown in Fig. 9b, the energy dissipation rate of the specimens barely increases before the peak. Therefore, the specimens are barely damaged before failure, showing significant brittle failures.

• (2) Damage evolution during cyclic loading tests

Fig. 9

Dissipated energy evolution in cyclic loading tests

Analysis of the plastic strain and deformation modulus evolution under cyclic loading suggests that the irreversible damage inside the rock material is the main factor of its plastic deformation. Meanwhile, the damage accumulation softens the rock material until failure. Generally, the damage variable D is used to quantitatively and intuitively describe the damage evolution in the failure process of rock materials and establish a relationship with other mechanical effects in the failure process. Essentially a thermodynamic variable, the damage variable cannot be measured directly via experiments. Since the damage evolution of rock materials is often accompanied by the variation of some physical and mechanical properties, the damage variable can be indirectly characterized by measuring the changes in such physical and mechanical variables. In general, the damage variable D is defined as:

$$D=1-\frac{A}{{A}_{0}}$$

(5)

where D is the damage variable, and A and A₀ represent the current and damage-free physical or mechanical parameters, respectively. For example, in the widely adopted Kachnnov method, A and A₀ represent the current effective area and the initial complete area of the rock, respectively. Other common methods to define the damage variable mainly include the elastic modulus method, ultrasonic wave velocity method, density method, energy method, strain method, CT number method, and acoustic emission cumulative number method. According to the basic principles of thermodynamics, the different descriptions of these damage variables are equivalent, i.e., these damage variables measure and define a specific damage state from different views, indicating a specific relationship between the definitions of various damage variable forms. Thus, the definition of damage variables can be varied.

Based on the analysis of plastic deformation and deformation modulus evolution above, two parameters can be used to characterize the damage variable. Damage variables D_E and D_ε are defined as follows:

$${D}_{E}=1-\frac{E}{{E}_{0}}$$

(6)

$${D}_{\varepsilon }=\frac{{\varepsilon }_{P}}{\varepsilon }=1-\frac{{\varepsilon }_{e}}{\varepsilon }$$

(7)

where E is the loading modulus in the cyclic loading and unloading of each stage, E₀ is the loading modulus of the damage-free specimen, ε is the total strain of each loading cycle, and ε_p and ε_e are the plastic strain and elastic strain of each loading cycle, respectively. In Eq. (6), it is important to determine E₀. As the initial damage in the rock specimens cannot be ignored, the loading modulus of initial loading cannot be considered to be E₀. Therefore, according to the stress-strain curves of each specimen in Fig. 3, the elastic modulus of the loading curve in the elastic deformation stage is determined as E₀.

According to the calculation results of Eqs. (6) and (7), the evolution trends of damage variables D_E and D_ε in the loading processes with different confining pressures are plotted in Figs. 10 and 11. The damage variable of the shale specimens at the three bedding inclinations decreases rapidly with the increase of axial strain, stabilizes, and then slowly increases, showing a "U" shaped trend. In the early loading stage, the compaction of initial micro-cracks in the rock matrix decreases the damage variables. As the loading increases, the rock matrix enters the elastic deformation stage, and little new damage occurs. Thus, the damage variable stays at a low level. As the loading continues, new internal damage accumulates. When the loading reaches the failure point, the damage variable increases rapidly, and the specimens fail. It can be concluded that the damage variables of most specimens show no significant increase from the elastic stage to the failure stage, indicating that the energy dissipation before the failure is not large, and most of the strain energy is rapidly released at the peak, resulting in significant brittle failure of the specimens.

Fig. 10

Evolution of damage variable D_E under cyclic loading

Fig. 11

Evolution of damage variable D_ε under cyclic loading

It should be noted that the energy dissipation evolution in Fig. 9 and the damage variable evolution in Figs. 10 and 11 show similar patterns, indicating a certain connection between them. The strain energy applied to the rock specimens can be generally divided into the elastic energy and the dissipated energy. The elastic energy is stored by the elastic deformation, and the dissipated energy is stored by the plastic deformation, which is caused by the damage accumulation. Therefore, the evolution of the dissipated energy and the damage variable is positively correlated.

3.4 Analysis of failure mode

Figure 12 shows the failure modes of shale specimens at bedding inclinations of 0°, 60°, and 90° under cyclic loading and unloading and different confining pressures. Figure 12a depicts the shale specimens at the bedding inclination of 0°. No distinct oblique shear crack is observed in the specimens under a low confining pressure of 5 MPa, and a single shear crack occurs when the confining pressure increases to 10 MPa and 20 MPa. Figure 12b shows the failure mode of shale specimens at the bedding inclination of 60°. In this case, all the specimens show single shear failure, and the shear crack is along the bedding plane. As shown in Fig. 12c, no distinct shear crack confining pressure increases to 10 MPa and 20 MPa, shear failure occurs across the bedding planes, and the dip angle of the shear plane is large.

Fig. 12

Ultimate failure mode of the shale specimens after triaxial cyclic loading and unloading tests

4 DEM simulation results of triaxial cyclic loading and unloading test on shale

4.1 Simulation model and procedure

The variations of strength deformation parameters and the evolutions of strain energy and damage characteristics of shale specimens under cyclic loading and unloading are analyzed in the laboratory tests. To comprehensively reveal the damage evolution mechanism and failure process of shale specimens in the cyclic loading and unloading process, the DEM numerical model (Cundall 1971, 1979) established based on the Particle Flow Code (PFC) was employed to simulate the cyclic loading and unloading tests. The PFC programs (PFC^2D and PFC^3D) provide a general-purpose, distinct-element modeling framework with both a computational engine and a graphical user interface. The PFC model can simulate the movement and interaction of many finite-sized particles, which are rigid bodies with finite mass that can both translate and rotate independently (Itasca Consulting Group 2016).

To establish the shale numerical model in PFC^2D, the parallel bond (PB) model and smooth joint (SJ) model are used to simulate the particle contact in the shale matrix and bedding plane (Cho et al. 2007; Yang et al. 2019b), respectively. In Fig. 13, Fig. 13a shows the parallel bedding planes of a shale specimen in the laboratory test. Figure 13b, c show the established PFC^2D shale models with PB and SJ, where the bedding planes are generated based on the features of the shale specimen shown in Fig. 13a. The loading stage is conducted with the displacement mode at a loading rate of 0.05 m/s, and the unloading stage is conducted with the stress mode at an unloading rate of 1 MPa/100 steps. The meanings of these two rates are different from those in the actual physical world (Cho et al. 2007). The PFC program can guarantee that the specimen is under quasi-static loading conditions. It should be noted that the strain increment of each loading cycle is 1 × 10⁻³ before the plastic deformation stage and 0.5 × 10⁻³ when the loading increases to the plastic deformation stage. Due to the little damage in the rock specimen before plastic deformation, the strain increment of each loading cycle is set to 1 × 10⁻³ to improve the computing efficiency. However, the damage accumulates and becomes significant in the plastic deformation stage. Thus, the strain increment of each cyclic loading is decreased to 0.5 × 10⁻³ to obtain more and better damage data.

Fig. 13

PFC^2D model of the shale specimen in this research

In the simulation, the numerical model of shale can be established at any bedding inclinations. Therefore, shale models at the bedding inclinations of 0°, 15°, 30°, 45°, 60°, 75°, and 90° are simulated to comprehensively study their mechanical properties. The micro-parameters of the simulation model are calibrated by comparing the simulation and experiment results of specimens at the bedding inclinations of 0°, 60°, and 90°. In this paper, the "trial and error" method is applied to calibrate the micro parameters of the shale specimen numerical model (Yang et al. 2019a). In the calibration test, the simulation results of micro parameters in each group were compared with the laboratory test results, and the ones best reflecting the laboratory test results of the set of mesoscopic parameters were selected as the model specimen calibration parameters.

Using the "trial and error" method for micro parameters calibration, a group of micro parameters is determined for the shale numerical model. The simulation results can reflect the laboratory test results under different conditions. The main micro parameters of the shale numerical model are listed in Table 1.

Table 1 PFC^2D micro-parameters of the shale specimen model after calibration

4.2 Comparison of simulation and experiment results

Figure 14 compares the stress-strain curves of the simulation and laboratory tests under cyclic loading and unloading. Overall, the simulated stress-strain curves are similar to the experimental results. It can be observed that the peak strength of the laboratory test and the simulation correspond well, despite a certain difference at the bedding inclination of 90°. In the laboratory test curves, certain differences between the ML peak strength and CL peak strength can be observed. In the simulation curves, however, the differences between the curve slope before the peak, the peak strength, and the curve shape after the peak are very small. Thus, the numerical model has less discreteness and could better reflect the effects of the single variable on the test results. Further observation of the simulation curves indicates that the peak strength of the CL curve is lower than or slightly lower than the peak strength of the ML curves. Thus, the cyclic loading and unloading path could reduce the strength of the shale specimens compared with the ML conditions. The reason is that the CL path could produce certain fatigue damage, which weakens the strength of the specimens. On the other hand, compared with the experimental curves, the distinct hysteretic loops appear post-peak in the simulated CL curves, while the experimental CL curves have few hysteretic loops. Due to the high brittleness of the shale specimens, as the loading approaches the peak, the stress rapidly drops to a very low level, rendering it impossible to continue the cyclic loading and unloading. In the PFC simulation tests, the numerical models are built using circular particles with PB bonds. When the model is loaded to the peak strength and fails, the particles within maintain a certain degree of interlocking and mutual movement. Thus, the bearing capacity does not disappear immediately, and the specimens can withstand a certain number of cyclic loading and unloading after the peak.

Fig. 14

Comparison of monotonic loading stress-strain curves and cyclic loading and unloading stress-strain curves between simulation and experiment

Figure 15 compares the experimental and simulated strength envelopes of the specimens at the bedding inclinations of 0°, 60°, and 90° under ML and CL conditions. Figure 15a, b show specimens under the bedding inclinations of 0° and 60°, where the simulated strength under different confining pressures is similar to the strength envelope of the laboratory tests. Thus, the simulation results reflect the laboratory test results well. Although the simulation results of the specimens at the bedding inclination of 90° in Fig. 15c are lower than the experimental values, the fitting line slope of the simulated CL strength versus the confining pressure increases and is close to the slope of the experimental fitting line, which also reflects the effectiveness of the simulated tests.

Fig. 15

Comparison of cyclic loading and unloading strength envelope between simulation and experiments

Figure 16 depicts the shale specimen failure modes between simulation and experiment under cyclic loading and unloading. The shale specimens are at the bedding inclinations of 0°, 60°, and 90° and under the confining pressures of 5 MPa, 10 MPa, and 20 MPa. In the simulation, one micro-crack is generated in the specimen when the bond (PB or SJ) between two particles fails as the tensile or shear stress exceeds the bond force, which is updated in each calculating step. As the loading increases, the micro-cracks accumulate and eventually form macroscopic fracture zones. As shown in Fig. 16a, under the confining pressure of 5 MPa, both tensile splitting and shear fractures appear in the simulated specimens at the bedding inclination of 0°, shear fractures along the bedding plane appear in the simulated specimens at the bedding inclination of 60°, tensile splitting along the bedding plane and shear fractures across the bedding plane appear in the simulated specimens at the bedding inclination of 90°. These results are consistent with the failure modes in the laboratory tests. Under the confining pressures of 10 MPa and 20 MPa, oblique shear fractures appear in the simulated specimens with different bedding inclinations. The shear fractures expand along the bedding plane in specimens at the bedding inclination of 60° and across through the bedding plane in specimens at the bedding inclinations of 0° and 90°. These results are also consistent with the failure modes in the laboratory tests. Therefore, in terms of failure modes, the simulated results can well reflect the laboratory test results under the effect of anisotropy and confining pressure.

Fig. 16

Comparison of cyclic loading and unloading failure mode between simulation and experiment

4.3 Discussion on simulation results

Under monotonic loading, the specimens only undergo monotonic stress changes, while under cyclic loading and unloading, the stress within the specimens increases and decreases repeatedly. This complex stress-loading history has a certain effect on mechanical properties.

Figure 17 shows the variation of peak strength (σ_p) and peak strain (ε_1p) of the simulated specimens under ML and CL at different bedding inclinations. In general, the peak strength variation at different bedding inclinations is similar to the peak strain regardless of the confining pressure or the loading path. According to Fig. 17, the peak strength and peak strain show little change at the bedding inclinations of 0° to 30° and decrease rapidly as the bedding inclination increases to 75° before increasing again at the bedding inclination of 90°.

Fig. 17

Comparison of peak strength and peak strain between monotonic loading and cyclic loading and unloading

On the other hand, the σ_p and ε_1p of the specimens under the two loading paths show little difference at the same bedding inclination. Further observation indicates that the CL σ_p is larger than that of ML under some conditions (e.g., σ₃ = 0 MPa, β = 30°, and 75°) and smaller under other conditions (e.g., σ₃ = 5 MPa, β = 15°, and 60°). Moreover, ε_1p also shows a similar pattern. Generally, changes in the mechanical properties of specimens under cyclic loading and unloading are closely related to the number of cycles. With more cycles, the degree of fatigue damage within the material is greater, as is the impact on the mechanical properties of the specimens. In this research, the number of cycles before the peak is within the range of 6 to 10, which is not enough to cause significant fatigue damage inside the specimens. Thus, the effect on the overall mechanical properties of the specimens is limited.

To further investigate the different strength deformation characteristics between the two loading paths, the secant modulus E_s at the peak point of the stress-strain curve is studied, which reflects the pre-peak damage accumulation in the loading process to a certain extent. The damage increases the plastic strain, which, in turn, decreases the stiffness of the specimens, manifested as the decrease of the secant modulus at the peak point.

Figure 18 shows the E_s variation with bedding inclination under the two loading paths. It can be observed that the E_s decreases with the increase of bedding inclination under different confining pressures. On the other hand, although the E_s of the CL path is lower than that of the ML path in most working conditions, different patterns are observed in other working conditions, which are similar to those analyzed in Fig. 17. However, specimens under cyclic loading and unloading generally show a weakening trend, which is closely related to the number of loading and unloading cycles. Therefore, if the number of cycles is further increased, the weakening trend becomes more significant.

Fig. 18

Comparison of secant modulus at peak between monotonic loading and cyclic loading and unloading

To further study the internal damage of specimens under the two loading paths, the number of different micro-cracks is calculated in the simulated specimens with different bedding inclinations at the peak strength. It should be noted that the PB shear cracks are not listed in Fig. 19 because they are not generated in the failed specimens. Overall, the variations of all types of micro-cracks with the bedding inclination are similar under different confining pressures, indicating that the confining pressures have no distinct effect on the anisotropic variation of micro-cracks. Under the two loading paths, the PB tensile cracks show a trend of fluctuation at the bedding inclinations of 0° to 30°, decrease rapidly as the bedding inclination exceeds 30°, and increase again at the bedding inclination of 90°. SJ tensile cracks and SJ shear cracks increase first with the increase of bedding inclination and then decrease as the bedding inclination exceeds 60°. As shown in Fig. 19, the difference in the number of cracks between the two loading paths is mainly reflected in the PB tensile crack, and the difference in SJ cracks is small. The SJ model seems to be barely affected by the cyclic loading. Thus, fatigue damage occurs more easily in the rock matrix. The SJ model simulates the behavior of a planar interface with dilation regardless of the local particle contact orientations along the interface. The behavior of a frictional or bonded joint can be modeled by assigning SJ models to all contacts between particles that lie on the opposite sides of the joint. Therefore, once the damage is done to the SJ model in the loading stage, the specimens cannot be restored in the unloading stage.

Fig. 19

Comparison of micro-crack numbers at peak between monotonic loading and cyclic loading and unloading

To further study the failure mechanism of shale specimens in the cyclic loading and unloading process, the peak points of each loading stage are identified separately to depict a stress-strain curve. The number of all types of micro-cracks at the peak point of each loading stage is counted, and the distribution of micro-cracks in the specimen is depicted. Due to the limited space, this paper only presents the simulation results under uniaxial compression and confining pressure of 20 MPa at bedding inclination of 0°, 60°, and 90°. As shown in Figs. 20 and 21, each figure can be divided into three parts: the evolution of stress and micro-cracks with strain, the failure mode comparison between ML and CL, and the evolution diagram of micro-crack distribution and propagation in the numerical specimen. Figure 20 shows the micro-crack evolution of specimens at bedding inclinations of 0°, 60°, and 90° under uniaxial cyclic loading and unloading.

Fig. 20

Micro-crack evolution at different loading stress under uniaxial compression

Fig. 21

Micro-crack evolution at different loading stage under confining pressure σ₃ = 20 MPa

As shown in Fig. 20a, the specimens at the bedding inclination of 0° mainly show PB tensile cracks, with few other types of micro-cracks. The comparison between ML failure mode and CL failure mode suggests that the micro-crack distribution in specimens under ML is more dispersed, eventually forming some axial macro fracture zones. On the other hand, the micro-crack distribution in specimens under CL is more concentrated, and the macro fracture zone is more distinct. The micro-crack evolution indicates that PB tensile cracks increase when the loading increases and approaches the peak. Initially, the micro-crack concentrates on the right side and at the left corner of the specimens, and the micro-cracks accumulate rapidly when the loading increases to the peak, forming a distinct macroscopic fracture zone.

According to Fig. 20b, c, the number of SJ tensile cracks and SJ shear cracks increase significantly as the bedding inclinations increase to 60° and 90°, and the initiation time of SJ shear cracks is earlier. Before the peak, the PB tensile cracks are fewer and dispersed. The number increases rapidly and expands to form a large fracture when the loading is about to reach the peak point and after the peak. Further observation of the SJ micro-crack distribution diagram indicates that the distribution of SJ shear cracks is more extensive than that of SJ tensile cracks, which appear on almost all bedding planes, and the number of SJ shear cracks slowly increases until covering the whole bedding plane. The SJ tensile cracks are concentrated on one or several bedding planes, forming macroscopic fracture planes on these bedding planes. Some of the SJ tensile cracks are connected with the PB cracks in the matrix, forming fractures through the bedding plane. The failure mode comparison diagram between ML and CL suggests that although the failure modes of the specimens under the two loading paths are similar, the failure degree of specimens under the CL path is significantly greater, indicating that the cyclic loading and unloading can strengthen the damage on shale specimens.

Figure 21 shows the micro-crack evolution of the specimens at different loading stages and the bedding inclinations of 0°, 60°, and 90° under the confining pressure of 20 MPa. Compared with the simulation results under uniaxial compression, the number of micro-cracks in the specimens under the confining pressure of 20 MPa is larger, and the distribution of micro-cracks in the specimens is more extensive. Eventually, the specimens fail into a single oblique shear fracture zone. The simulated failure modes at different bedding inclinations indicate that the failure modes under the ML path and CL path are similar, and the micro-crack distribution and the morphology of the macro fracture zone are highly similar. Thus, the confining pressure weakens the effect of CL on the damage degree. The effect of high confining pressure plays a dominant role in the failure process compared to the loading path. Similar to the crack evolution under uniaxial compression, as shown in Fig. 21b, c, plenty of SJ shear cracks and SJ tensile cracks appear in the bedding planes of specimens with bedding inclinations of 60° and 90°. Thus, the bedding planes simulated using the SJ model will produce more micro-cracks under different confining pressures. The reason is that the mechanical parameter values of the SJ model are much lower than the parameter values of PB parameters in the matrix. As the particles in the SJ model can slip through the SJ interface, SJ shear cracks are more likely to appear.

According to the crack evolution in Figs. 20 and 21, the PB cracks and SJ cracks show distinct evolution patterns under different bedding inclinations and confining pressures. The comparison of the results under the three bedding inclinations indicates that the PB tensile cracks are the main micro-cracks in the specimens, which increase rapidly when approaching the peak strength. However, the SJ shear cracks and SJ tensile cracks are only distinct at the bedding inclinations of 60° and 90°. Therefore, the damage in the rock matrix is dominant at the bedding inclination of 0°, and little damage appears in bedding planes. In specimens at the bedding inclinations of 60° and 90°, damage in the bedding planes increases, and the fracture appears more easily in the bedding plane. On the other hand, the high confining pressure increases the damage both in the rock matrix and bedding plane.

5 Conclusions

The cyclic loading and unloading on shale rocks should be considered, whether they are attributed to geological movements, seismic actions, or artificial construction interference, such as vertical and horizontal well drilling and cyclic hydraulic fracturing. To investigate the mechanical response and mechanism of shale rocks under cyclic loading, experiments and DEM simulations were conducted to quantify the strength, deformation, energy, and damage evolution during the cyclic loading–unloading process. The damage variable characterized by plastic deformation, deformation modulus, and dissipated energy was analyzed in depth. Through PFC^2D simulation, the micro-crack distribution and evolution were further studied, and the results could reflect the experiments well and reveal the damage mechanism under cyclic loading. The conclusions are as follows.

• (1) In general, cyclic loading, particularly fatigue loading, had a certain weakening effect on the strength of rock materials. In this research, the failure strength points of the shale specimens under cyclic loading and unloading fell within the strength envelope under monotonic loading. The PFC^2D simulation results further indicated that the fatigue damage degree was closely related to the number of cycles on the specimens, and a small number of cycles might not distinctly affect the strength. The unloading modulus E_U was greater than the loading modulus E_L in each cycle. The two deformation moduli E_U and E_L increased at the early loading stage due to the compaction of the rock matrix. As the loading increased, the deformation modulus gradually stabilized. When the loading reaches the peak, the deformation modulus begins to decrease. The energy dissipation rate of each cycle decreased first and then increased with the increasing number of cycles.

• (2) With the increase of loading, the damage variable of shale specimens with the three groups of bedding inclinations decreased rapidly first and then stabilized before slowly increasing again. The damage variable increment of most specimens was not large from the elastic stage to the failure stage, indicating that the energy dissipation before failure was not large and most of the strain energy was released at failure, resulting in the significant brittle failure of the specimens.

• (3) After cyclic loading and unloading, oblique shear fracture occurred in shale specimens at the bedding inclinations of 0° and 90° under a high confining pressure. The shale specimen at the bedding inclination of 60° showed a single shear fracture under different confining pressures, and the shear fracture extended along the bedding plane. In addition, the specimens at the bedding inclinations of 0° and 90° could produce more complex cracks after cyclic loading and unloading. Specimens at the bedding inclination of 60° showed shear fracture along the bedding plane under the two loading modes (ML and CL), indicating that the effect of other factors was weakened when the bedding plane dominated the failure mode of the shale specimens.