Mining activities induce sudden changes in stress states, which are the primary cause of fault-slip events (Li et al. 2016, 2021; Dong et al. 2023). Cao et al. (2023) explored the total strain energy at the working face under various fault slip conditions, delineating the risk of coal outbursts under different longwall-fault distances. Fault coseismic slip, a mechanism for releasing accumulated elastic strain energy, plays a pivotal role in fracture initiation, fault surface sliding, and seismic wave generation (Noda et al. 2013; Uenishi 2017; Dong and Luo 2022; Li 2024b; Li et al. 2024b). Wei et al. (2020, 2021) conducted research on the magnitude of energy released during fault activation, particularly as the longwall face progressed in the direction of the fault. A portion of this released energy is assessed using seismic moment and static stress drop, derived from seismic or geodetic observation records (e.g., Venkataraman and Kanamori 2004). As coal mining progressively extends into deeper and geologically more complex regions, the incidence of coal bursts has become more pronounced, posing significant challenges to ensuring safety and efficiency in coal mine operations (Dai et al. 2021; Li et al. 2023; He et al. 2018; Agrawal et al. 2023; Zhu et al. 2024). Fault structures significantly threaten coal mining safety (Li 2024a). For instance, a fault-related coal burst at the Qianqiu coal mine claimed 10 lives (Liu et al. 2019). The Rudna Mine, situated in southwestern Poland, plunged over 1150 m into the earth, where a mining-induced fault-slip released 2.84 × 10⁹ J, causing a 4.2 magnitude induced earthquake (Lizurek et al. 2015).

However, understanding the total strain energy release is more crucial than just focusing on the available energy for a comprehensive understanding of the energy balance in fault rupture, as emphasized in the works of Kostrov (1974) and Matsu’ura (2024). Traditionally, strain energy release assessments have focused on the cumulative volume integral within the elastic medium, often overlooking spatial variations (Steketee 1958; Savage 1969). To understand mining-induced earthquake and coal burst generation in terms of energy accumulation and release during mining, it is well worth evaluating the spatial variation of strain energy changes in the fault-mining zone. To evaluate the strain energy changes associated with fault coseismic slip, we need not only coseismic stress changes but also background crustal stress and mining induced additional stress as known information (Saito et al. 2018; Noda et al. 2020). For the background stress, we can measure both of the orientation and absolute value of in situ stresses via observation techniques in boreholes such as the hydraulic fracturing technique, but they are limited in depth, location, and accuracy (e.g., Charsley et al. 2003; Ljunggren et al. 2003; Peng 2023). The challenge of determining absolute background stress values hinders the evaluation of spatial shear strain energy variations.

In this study, we assess and visualize the two-dimensional (2-D) shear strain energy changes associated with the F16 reverse fault at Yuejin coal mine. Initially, we introduce an innovative ‘0 Model’ concept, a hypothetical fault, considering shear strain energy as the foundational shear strain energy under the conditions of background stress and mining-induced additional stress. Our methodology involves estimating the coseismic fault slip distribution, accounting for variations in mining distances and fault types. By integrating the static stress changes from the fault coseismic slip with the baseline ‘0 Model’ data, we analyze coseismic shear strain energy variations. Finally, we conduct a comparative study of the shear strain energy fluctuations due to mining-induced coseismic fault slip and the bursting shear strain energy during the period of July 18, 2010, to September 14, 2012, corresponding to the operation of Longwall (LW) 25110.

We represent the stress (τ) and strain tensors (ε), denoted as τ_ij and ε_ij respectively. The elastic strain energy density, henceforth referred to simply as ‘elastic strain energy,’ stored in rock, particularly for isotropic rocks, E, can be generalized through certain equations reorganized from Eqs. 5.148–5.151 of Jaeger et al. (2007)

where E is the elastic strain energy; E_v is the volumetric strain energy; E_s is the distortional strain energy (shear strain energy); τ_m is the mean normal stress; {s₁, s₂, s₃} are principal deviatoric stress; K is bulk modulus; G is shear modulus.

In the elastic range of mining induced fault slip, the mean stress τ_m controls the volumetric change of a rock, whereas the deviatoric stresss controls the distortion. Moreover, many of the criteria for failure are concerned primarily with distortion, in which case these criteria are most conveniently expressed in terms of the invariants of the stress deviation. In an arbitrary coordinate system, these invariants take the form algebraic manipulation of the previous equations leads to the following alternative forms for J₂, which is the invariant that appears most often in failure criteria:

To calculate the total shear strain energy stored in a mining-fault system, one must integrate the shear strain energy across the entire system. This involves a comprehensive consideration of how mining activities, particularly near faults, impact the shear strain energy within the rock.

In the context of a 2-D plane shear strain condition, which is a specific scenario often encountered in geological and mining engineering, further details and calculations are provided in text S1.

We assume a linear relationship between coseismic stress changes Δσ_ij and shear strain changes Δε_kl: Δσ_ij = C_ijklΔε_kl, where C_ijkl represents the stiffness tensor (Saito et al. 2018; Noda et al. 2020). The shear strain energy is expressed as follows:

where E_s0 is the initial state of the shear strain energy.

The shear strain energy change (E_s change) is

The change in total shear strain energy is

In this study, we introduce a hypothetical fault (without slip, i.e., with an infinite friction coefficient) as the ‘0 Model’. The shear strain energy, under the conditions of background stress and additional stress induced by mining, is considered the baseline shear strain energy, denoted as E_s0. By using this ‘0 Model’, we can systematically analyze the spatial variations in strain energy caused by fault slip.

The F16 fault is an east-west trending compression-shear reverse fault that spans 45 km through several coal mines, including Changcun, Yuejin, Qianqiu, Gengcun, and Yangcun, in the Yima coalfield, China (Wang et al. 2020). It has dip angles ranging from 75° near the surface to 15°-35° at greater depths. This fault exists in a high horizontal tectonic stress environment and combined with mining activities, it frequently reactivates, leading to rock bursts and seismic events that significantly impact mining safety and operations (Cao et al. 2023; Cai et al. 2021).

Located in the Yima coalfield of Yima City, Henan Province, China, the Yuejin coal mine is adjacent to the significant F16 reverse-thrust fault to its south. At the Yuejin coal mine, the 25110 longwall face (LW25110) extends to depths between 800 and 1200 m. The seam measures 7.4 to 13.8 m in thickness and inclines at 12°. Adjacent to the panel’s northern boundary is the LW25090, while the F16 reverse-thrust fault lies to the south, oriented east-west with a dip angle progressing from 30° to 50° in the vicinity of the LW25110.

3.1 Engineering background and PyLith simulation

On August 11, 2010, the LW25110 suffered a devastating coal burst during its early mining stage, releasing 9 × 10⁷ J of energy and causing a seismic event measured at 2.7 in magnitude. The disaster occurred close to the lower roadway, extending damage over 362 m, and happened when the mining face was less than 100 m from the F16 fault. Recognizing the critical role of fault slip in such bursts, research by Cai et al. (2015, 2021) and Wang et al. (2020) has been pivotal. Following this, the microseismic monitoring system set up on May 15, 2009, has been crucial for monitoring seismic activity and measuring strain energy in the mines. It has particularly highlighted the build-up of energy near the F16 fault, a high-risk area, which was evident in the significant seismic events recorded in August and March of 2010.

Building on the research conducted by Li et al. (2024a) on mining-induced fault failure and coseismic slip across different geological conditions, our investigation examines the shear strain energy fluctuations due to coseismic slip in more depth (Fig. 1), employing the parameters identified in their studies (Tables 1 and 2). The local coordinate (red line) L is set along the fault, being the origin is at the center of the coal seam line (white dashed line), and the positive direction of L is taken upward. D_m is the mining distance measured from the origin of the local coordinate L to the working face, taken positive at the hanging wall side. The static friction model produces shear traction proportional to the fault normal traction plus a cohesive stress,

where C is the cohesive stress on the fault; τ and σ_n are the shear and normal stress on the fault, and µ is the friction parameter, respectively (e.g., Li et al. 2024a; Sainoki et al. 2020; Wen et al. 2024).

Our analysis is based on the mechanical parameters listed in Tables 1 and 2, and the simulations were conducted using PyLith 4.0.0 (Aagaard et al. 2013, 2023). This finite element software is renowned for its capability to model crustal deformation across a wide range of spatial scales, from a few meters to several hundred kilometers. It is particularly adept at both quasi-static and dynamic earthquake fault modeling, as noted by Aagaard et al. (2023). The software version we use has undergone extensive validation through tests by the Southern California Earthquake Center (SCEC). We apply PyLith 4.0.0 specifically to model the spatial variations in shear strain energy resulting from coseismic slip in faults affected by mining activities.

3.2 Model setup

The model in this study is based on the research conducted by Li et al. (2024a). We use a 2-D plane-strain model as shown in Fig. 1. The computational volume is 2400 m in length along the mining direction (x-direction), 1600 m in the vertical direction (y-direction), and 1 m in width (z-direction). The coordinate origin is situated at the bottom left corner. Contraction is taken negative in the present computations. We introduce the displacement boundary conditions,

where u_i is ith-component of the displacement.

Next, we set the following stress condition,

where σ_v(h) and σ_h(h) are vertical and horizontal far field principal stresses at depth of h. σ_v, representing the self-gravity, is defined as ρgh, where ρ and g are density and gravitational acceleration. We introduced a parameter r_b, which is the background stress ratio.

The mechanical parameters forming the basis of our analysis are detailed in Table 1. It should be noted that D_m represents the distance between the fault and the mining face, and it is considered positive when the mining face is located on the hanging wall side. As shown in Fig. 1a, we introduced a local coordinate denoted as L along the fault. The origin of L was set at the intersection of the fault and the center of the coal seam layer, taking upward as positive and downward as negative.

In our model, the mesh size is set to 0.2 m near the fault and working face to ensure accuracy. Away from the target locations, the mesh size gradually increases by a factor of 1.02, which balances precision and computational resource requirements (Table S1).

A schematic illustration of the 2-D plane strain model, including the coal mining working face and the fault ahead of the working face. a The xyz-coordinate system is set as shown in the bottom left of the figure. The origin is located at the bottom left corner of the modeling region. σ_v is vertical principal stress and σ_h is horizontal principal stress in the x-direction. On the left and bottom sides of the model, the displacement boundary condition is applied as described in the text. φ stands for fault dip angle; b Cohesive cells with no thickness regulate fault slip through the movement difference between vertices on the fault’s opposite sides

We propose a method to numerically calculate the spatial variation of coseismic slip and particle displacement under fixed mining distances by altering fault friction characteristics. Our foundational assumption begins with a ‘0’ model as a baseline, meaning the displacement at the working face and under background stress. Subsequently, faults with varying friction properties are introduced. This allows for the analysis of surrounding rock particle displacement caused by fault slip, thereby examining fault slip variation due to friction properties.

4.1 Slip distribution with variation in friction properties

Our investigation into the fault slip behavior, as informed by the data illustrated in Fig. 2, elucidates the influence of frictional properties on fault slip patterns, highlighting the critical role of friction in modulating the fault’s response to mining-induced stresses.

Figure 2 shows fault slip occurrences under a specific scenario where the D_m is ± 60 m, r_b is set at 2.0, and µ is varied. This setup provides a detailed examination of how changes in µ influence the spatial distribution of fault slip and, consequently, the shear strain energy changes in the vicinity of mining operations. The fault slip distribution demonstrates a clear dependency on µ, with significant spatial variations observed as µ changes. Figures 2a and b and Fig. S1 show the spatial distribution of horizontal (x-axis) and vertical (y-axis) displacement components within the surrounding rock mass. It is evident from Fig. 2c that as fault slip occurs, the displacement region within the rock mass can directly extend to the working face, especially the distribution of the vertical displacement component near the fault side. The distribution of displacement components also indicates the presence of a reverse fault, suggesting that fault slip occurs not only on the side of the working face but also on both the hanging wall and footwall sides. The distribution of displacement components under varying fault friction characteristics is depicted in Fig. 2c and d demonstrate that with lower values of µ, fault slip is more pronounced, signifying that decreased frictional resistance leads to increased displacement along the fault. This behavior is pivotal for understanding the mechanics of seismic energy release and accumulation in the context of mining-induced earthquake. Specifically, the conditions under which the fault slip transitions from stable to dynamic can be closely tied to variations in µ, reflecting the critical threshold for slip initiation and propagation. By analyzing the fault slip under these varied frictional conditions, we observed that as µ decreases, the extent of fault slip increases, suggesting that lower µ is conducive to more extensive fault slip. This observation is critical for predicting the potential for seismic events, as larger fault slips could lead to higher energy releases, thereby increasing the likelihood of mining-induced earthquake and rockburst.

Furthermore, the spatial patterns of fault slip provide insights into the zones of increased risk near the mining face. Areas adjacent to the fault, particularly those experiencing significant slip, represent zones of heightened seismic potential. In summary, the detailed analysis of fault slip variation due to friction properties, as depicted in Fig. 2, underscores the complexity of fault mechanics under mining-induced stresses. By highlighting the significant impact of frictional properties on fault behavior, our study contributes to a more nuanced understanding of the mechanisms driving seismic events in mining environments.

Fault slip under the conditions of D_m = -60 m, r_b = 2.0, and µ = 0.6. a x-component of displacement; b y-component of displacement; c Fault slip with various µ under the conditions of D_m = -60 m, r_b = 2.0; d Fault slip with various µ under the conditions of D_m = + 60 m, r_b = 2.0

4.2 Spatial variations of total shear strain energy in fault-mining zone

Based on Eqs. (6) and (7), it becomes evident that the variation in the spatial pattern of shear strain energy is determined by the weighted sum of the interaction term. The presence of positive values indicates that the changes in coseismic stress align with the direction of the background stress field, while negative values signify a misalignment. Figure 3 illustrates the two-dimensional spatial variation in coseismic changes of shear strain energy related to the mining-induced faulting sequence across five scenarios with varying frictional coefficients (µ) of 0.6, 0.7, 0.8, 0.9, and 1.0. For a detailed illustration of hangwall mining under varying friction coefficients please refer to Fig. S2.

Spatial variation of coseismic slip changes in E_s associated with various µ. a to e Correspond to µ values of 0.6, 0.7, 0.8, 0.9, and 1.0, respectively

The spatial distribution of E_s under the combined effects of additional stress at the working face and background stress is depicted in Fig. S3. In the context of footwall mining with D_m = -60 m, Fig. 2d displays the fault coseismic slip distribution across various µ scenarios. The result reveals that fault slip predominantly occurs above the mining horizon, aligning with the spatial distribution of E_s change depicted in Fig. 3. The depiction in blue indicates regions where E_s change shows a decrease, illustrating a direct correlation between the magnitude of fault coseismic slip and the extent of E_s change. Importantly, near the working face adjacent to the fault, there is a notable transition where E_s change assumes positive values, indicative of energy amplification zones. This phenomenon becomes more pronounced as the frictional coefficient diminishes, leading to a significant increase in the positive alteration of E_s.

Exploring the relationship between the total coseismic shear strain energy changes and the frictional coefficient µ represents a compelling avenue of inquiry. The total change in coseismic shear strain energy, denoted as \(\:\Delta\:{E}_\text{total}={\int}_{\text{m}\text{i}\text{n}\text{i}\text{n}\text{g}-\text{f}\text{a}\text{u}\text{l}\text{t}}^{}\Delta\:{E}_\text{s}dV\), can be expressed as the weighted sum of the interaction term, with µ serving as the relative weight. This integration of E_s change was conducted around the mining-fault area, as illustrated in Fig. 3. Figure 4 presents the calculated ΔE_s values for µ = 0.6, 0.7, 0.8, 0.9, and 1.0, depicted through vertical profiles. Here, red and blue bars signify the increase and decrease in shear strain energy across each 5-m-thick horizontal layer, respectively. The overall negative ΔE_s indicates that mining-induced shear faulting acts to dissipate the shear strain energy accumulated in the surrounding rock. Notably, as µ decreases from 1.0 to 0.6, the total released E_s escalates from 0.39 × 10⁸ J to 2.31 × 10⁸ J, underscoring the significant impact of the frictional coefficient on energy dynamics. For the total released E_s under footwall and hanging wall mining with different D_m, please refer to Figs. S4, S5 and Table S2.

The overall distribution of coseismic shear strain energy changes across each 5-m-thick horizontal stratum. The red bars represent increases, while the blue bars signify decreases. a–e Correspond to µ values of 0.6, 0.7, 0.8, 0.9, and 1.0, respectively

4.3 Fault slip and shear strain energy changes

Building on the critical analysis of Fig. 5, our investigation delves into the complex relationship between coseismic slip and changes in E_s within the surrounding rock affected by mining activities. This detailed study provides essential insights into the induced seismic associated with mining by delineating the E_s variations across layers of 5-m thickness. Such variations, meticulously represented through numerical data, illuminate the crucial impact of coseismic slip on seismic events within mining zones. The visual representation using green bars to denote coseismic slip and its direct correlation with both increases (red bars) and decreases (blue bars) in E_s offers a vivid illustration of the seismic energy redistribution occurring due to slip disturbances induced by fault activity.

Our analysis underscores the pivotal role that coseismic slip plays in either storing or releasing seismic energy, significantly influenced by the specific geological and mining conditions at mining sites. It is particularly noteworthy that areas with larger coseismic slips, indicated by more pronounced green bars, frequently correlate with considerable stress drop zones. These zones can either experience an amplification or reduction in E_s, respectively. This phenomenon is of paramount importance for the planning and management of mining operations, providing a quantifiable framework to assess the impact of fault slips induced by mining on the stability and energy dynamics of the surrounding rock mass.

By rigorously examining the interplay between coseismic slip and E_s changes across different strata, mining engineers and geologists are better equipped to forecast and counteract the induced seismic risks inherent to deep underground mining operations, such as rockbursts and the reactivation of faults. This exploration not only enriches the theoretical understanding of fault mechanics and energy distribution in response to mining activities but also emphasizes the critical need for a comprehensive grasp of energy accumulation and release mechanisms within the surround rock.

Correlation of E_s changes with coseismic slip across 5-m-thick layers. The green bars represent coseismic slip, the red bars indicate increases, and the blue bars signify decreases. a–e Correspond to µ values of 0.6, 0.7, 0.8, 0.9, and 1.0, respectively

We estimate the seismic moment (M₀) of mining-induced earthquakes under varying friction parameters, as initially defined by Aki (1967) and Kanamori and Anderson (1975). M₀ is defined as follows,

In this analysis, the shear modulus (G) of fault rocks is 6.5 GPa, referencing data from Table 1. The variables L_s and L_w represent the coseismic slip length and width along the fault, respectively, with D indicating the fault’s average coseismic slip. Given our study employs a 2-D plane strain model with a model width of 1 m, we consider L_w to be constant. To quantitatively analyze fault slip, we introduce the term ‘L_sD’. The proximity of a mining fault necessitates examining its slip and the resultant changes in stored strain energy (E_s) impacting the mining face. We focus on a 5 m by 5 m area near the fault at the mining face corner to assess its response.

As shown in Fig. 6, for footwall mining at a distance (D_m) of -60 m and varying friction coefficients between 0.6 and 1.0 in increments of 0.1, L_sD values decrease from 4.5 × 10³ m² to 7.5 × 10³ m². Correspondingly, the maximum slip reduces from 270 mm to 80 mm, and stored E_s energy decreases from 5.44 × 10⁷ J to 1.26 × 10⁷ J. In the case of hanging wall mining at a distance (D_m) of 60 m, with the same range of friction coefficients, L_sD values decrease from 2.5 × 10³ m² to 1.5 × 10³ m², and the maximum slip reduces from 180 mm to 23 mm. Here, stored E_s energy falls from 3.44 × 10⁷ J to 2.5 × 10⁶ J.

Notably, while the conditions for hanging wall mining mirror those of footwall mining in terms of friction parameters and absolute mining distance, the outcomes for the former are consistently lower. Li et al. (2024a) provide insights into this phenomenon, highlighting that fault slip increases shear strain energy storage around the mining face. This amplified shear strain energy concentration can potentially trigger or exacerbate rock (coal) bursts.

The relation between E_s change, maximum coseismic slip, and L_sD for different values of µ. a Footwall mining, and b Hanging wall mining

4.4 Comparing numerical simulations and field data on strain energy near the F16 fault

On March 1, 2011, at 10:09 AM, a rockburst event occurred at the LW 25110 working face of the Yuejin Coal Mine (China), injuring three miners. Cai et al. (2014) conducted a case study on the impact of rockbursts at the LW 25110 working face beneath the F16 reverse fault. By employing a microseismic monitoring system, they comprehensively monitored microseismic events and the magnitude and distribution of bursting strain energy during the mining at the LW 25110 working face. Between July 18, 2010, and September 14, 2012, they observed an increase in strain energy near the fault compared to areas farther away. A significant concentration of strain energy was detected in the mining area (fault-working face), as shown in Fig. S6. This region experienced an increased frequency of microseismic events, with over 30 events releasing energy exceeding 10⁶ J (Cai et al. 2015). These findings align with the numerical simulation results depicted in Fig. 3, indicating that fault slip leads to the accumulation of shear strain energy (E_s) near the fault zones at the working face. Furthermore, as illustrated in Fig. S3, considering the background stress and the stress induced by mining, this energy concentration is particularly noticeable on the side of the working face near the fault. This accumulation of E_s is considered a potential trigger for rock bursts.

Based on these observations and monitoring data, we propose a fourth mechanism that may cause rock bursts or coal bursts. This mechanism involves the redistribution of energy generated by coseismic slip on the fault and the localized concentration of E_s near the working face. This dynamic can potentially trigger rock bursts or coal bursts in areas close to the fault. Importantly, this suggests that localized energy accumulation due to fault slip can trigger rock or coal bursts, even in the absence of dynamic fault rupture. Therefore, this aspect warrants significant attention in mining operations near faults, especially in areas prone to rock (coal) bursts.

5.1 Stress drop of mining induced fault coseismic slip

The stress drop plays a crucial role in predicting the intensity of ground motion, especially when it can be determined reliably. Fukuyama and Madariaga (1995) established a boundary integral equation linking slip to stress drop, aligning with earlier findings and showing that final stress drop values are consistent across static and dynamic scenarios.

In recent years, the disposal of wastewater generated from the development of unconventional hydrocarbons has led to a notable increase in seismic activity in the central United States (Ellsworth 2013). Oklahoma, in particular, has experienced numerous moderate earthquakes, such as the 2014 Cushing earthquake (e.g., McNamara et al. 2015) and the 2016 Pawnee earthquake (Chen and Nakata 2017). Some researchers have suggested that induced earthquakes exhibit systematically lower stress drops compared to natural earthquakes (Hough 2014; Sumy et al. 2017), while other studies (Huang et al. 2017) have found that the stress drops of earthquakes induced in the central United States are comparable to those of tectonic events in the same region. Accurate measurements of stress drops can enhance our understanding of the impact of induced earthquakes on underground structures, such as workings faces and tunnels.

The data from Fig. 7 reveals a trend where the stress drop varies with µ, spanning -10.5 MPa to 9.5 MPa for µ = 0.6, -9.8 MPa to 8.4 MPa for µ = 0.7, -8.2 MPa to 7.3 MPa for µ = 0.8, -6.2 MPa to 6.0 MPa for µ = 0.9, and finally, -4.2 MPa to 5.0 MPa for µ = 1.0. This pattern underscores that larger slips correspond to greater stress drops, highlighting the significant impact of slip magnitude on stress drop values.

To validate the stress drop distribution calculated with PyLith, we compared it against the DC3D model (Okada 1985, 1992), known for its fault slip and strain distribution modeling around faults. By employing DC3D’s subroutine, we were able to evaluate displacement and its spatial derivatives from uniform slip on a finite rectangular fault within an elastic half-space. Given the irregular fault slip due to mining, a segmented slip distribution at 5 m intervals was adopted to estimate the stress drop distribution.

Figure 7 shows the stress drop distributions across five fault types, with the DC3D model serving as a benchmark. The outcomes, indicated by the brown curve with empty circles, show PyLith results in harmony with those from DC3D. Despite this alignment, slight differences in stress distribution near the slip region’s bottom, proximate to the mining area, were noted. These discrepancies, associated with sudden changes in shear stress and the use of a 0.5 m finite element size in PyLith, suggest limitations in reaching theoretical precision. Reducing element size is proposed as a way to potentially align closer with DC3D’s theoretical outcomes.

Distribution of stress drop (top) and coseismic slip (bottom) for different µ with r_b = 2, D_m = -60 m, and φ = 30°. a–e Correspond to µ values of 0.6, 0.7, 0.8, 0.9, and 1.0, respectively

5.2 Stress drop and shear strain energy changes

The interrelation between stress drop, change in E_s change, and coseismic slip under specific conditions—µ, r_b = 2, D_m = -60 m, and φ = 30°—presents insightful perspectives into the seismic mechanics of faults. The graphical representation in Fig. 8, employing blue, orange, and green lines to illustrate stress drop, E_s change, and coseismic slip respectively, reveals critical dynamics in fault behavior under seismic stress.

The blue line indicating stress drop demonstrates a variable pattern as a function of µ. This variability underscores the sensitivity of fault rupture dynamics to frictional properties. Stress drop, a measure of the stress release during an earthquake, directly influences the seismic wave energy and, by extension, the ground motion experienced during seismic events. The observed pattern suggests that as µ increases, the stress drop tends to decrease, indicating a potential relationship between the fault’s frictional resistance and the magnitude of stress release during slip events. The orange line representing E_s change across different µ values illustrates the transformation in stored elastic energy in the vicinity of the fault as it undergoes slip. The modification in shear strain energy is essential for understanding the energy budget of an earthquake—specifically, how much energy is released versus stored during and after rupture. Notably, the pattern of E_s change complements the stress drop trend, highlighting the interplay between stress release and energy transformation during seismic events. Illustrated by the green line, the coseismic slip variation with different µ values provides insights into the fault displacement during an earthquake. Coseismic slip, a critical factor in determining the earthquake’s potency and potential damage, shows a nuanced relationship with the µ, reflecting how frictional properties may influence slip distribution and, consequently, the seismic risk profile of a region.

The integrated view presented by Fig. 8 allows for a comprehensive understanding of the seismic process. The juxtaposition of stress drop, E_s change, and coseismic slip underlines the complexity of fault mechanics and the pivotal role of frictional properties in shaping seismic outcomes. This analysis not only enhances our grasp of seismic phenomena but also contributes to more accurate risk assessments and the development of mitigation strategies. The investigation into the distribution of stress drop, E_s change, and coseismic slip for varying µ under specified conditions sheds light on the intricate mechanics of seismic fault behavior. It emphasizes the importance of considering multiple parameters in seismic studies to accurately model and predict seismic activity. This multifaceted approach is vital for advancing our understanding of earthquakes and enhancing our ability to safeguard communities against seismic hazards.

Distribution of stress drop (represented by the blue line), E_s change (depicted by the orange line), and coseismic slip (indicated by the green line) for varying values of µ with r_b = 2, D_m = -60 m, and φ = 30°. a–e Correspond to µ values of 0.6, 0.7, 0.8, 0.9, and 1.0, respectively

5.3 Limitations

Our study on spatial variations in shear strain energy changes associated with mining-induced fault coseismic slip primarily focuses on variations in fault types, specifically changes in friction parameters. However, there are several limitations in our study that need to be addressed in future research. Firstly, while our model systematically studies friction parameters, it does not account for their heterogeneity in actual engineering applications. Future research should address this heterogeneity and its impact on shear strain energy distribution. Additionally, future work should incorporate insights into the mechanical properties of coal, particularly its nonlinear mechanical behavior and support methods (e.g., Hao et al. 2015; 2017), to improve the accuracy and applicability of numerical models predicting shear strain energy changes associated with mining-induced fault coseismic slip. Furthermore, although we acknowledge the impact of fault roughness on friction parameters and the rupture preparation process, the distribution of coseismic slip on faults with varying roughness levels due to mining remains unexamined. Investigating how different roughness levels affect fault coseismic slip and shear strain energy distribution is essential for future studies.

Our numerical model also lacks validation with field data, which is crucial for enhancing the model’s reliability. Future research should include the collection and comparison of field data to validate our findings.

This study rigorously examines the effects of mining-induced fault coseismic slip on shear strain energy within deep mining settings, focusing on retreating longwall operations. We have shown, through theoretical and numerical analyses, that mining locations and fault characteristics crucially influence shear strain energy adjustments. This impact is especially pronounced when fault slip occurs near the mining face, substantially increasing the risk of rockbursts.

Our results underscore the importance of including fault coseismic slip dynamics within the comprehensive framework for understanding rock (coal) bursts, thereby advancing the theoretical basis for mitigating geological hazards in deep mining. The observed spatial variability in shear strain energy highlights the complex interplay between mining operations and fault dynamics. Validation with observational data near the F16 fault zone not only lends credence to our findings but also identifies mining-induced fault coseismic slip as a key element in understanding rockburst triggers. This knowledge is vital for improving risk assessment and enhancing safety measures in mining, particularly in geologically hazardous areas.

Significantly, our research enriches the fields of rock mechanics and mining engineering by clarifying the complex relationship between mining-induced seismicity and shear strain energy changes. The findings lay a robust groundwork for future studies aimed at reducing deep mining risks. Emphasizing an interdisciplinary approach that merges geological insights with cutting-edge analytical and numerical methods, this work effectively addresses the challenges of mining-induced geological hazards.