Calculation Method and Treatment Scheme for Critical Safety Rock Pillar Thickness Based on Catastrophe Theory

Abstract

To investigate the safety risks associated with gas tunnel coal uncovering, a physical and mechanical model of the critical safety rock pillar is proposed through a combination of theoretical analysis, numerical simulation, and field testing. Based on the principles of energy conservation and catastrophe theory, an expression for calculating the critical safety for rock pillar thickness is derived. The effects of tunnel radius, burial depth, axial stress, coal seam dip angle, and gas pressure on the critical thickness are systematically analyzed. The results indicate that the critical safety of rock pillar thickness increases with the tunnel radius, burial depth, gas pressure, and axial stress. Moreover, as the tunnel radius increases, the growth rate of the critical thickness also increases. Conversely, as the burial depth increases, the growth rate of the critical thickness decreases. For gas pressure and axial stress, the growth rate remains relatively constant. Using a tunnel project in Hunan as a case study, theoretical analysis yields a critical safety rock pillar thickness of 3.95 m. A corresponding numerical model is developed to simulate this scenario, and the simulation results align well with the theoretical predictions. Based on these findings, a combined treatment scheme of “advanced small-pipe grouting + gas drainage and pressure relief” is proposed for excavation upon reaching the critical rock pillar thickness. This scheme successfully ensures safe tunnel passage through the coal seam.

1. Introduction

With the rapid advancement of tunnel engineering, encountering coal seams during excavation has become increasingly common. Tunneling through coal seams presents significant challenges, particularly due to the instability of the tunnel face caused by the coal seam, which becomes prone to failure [1]. Additionally, the risk of coal and gas outburst and seepage behavior in the process of tunneling increases the complexity of construction [2,3,4]. To address these challenges, many researchers both domestically and internationally have conducted extensive studies on tunneling through coal seams. By combining physical similarity simulation with numerical modeling, Xu et al. [5] determined the safe thickness of reserved protective rock pillars, the stress concentration zones within them, and the required pillar thickness for safe tunnel advancement through coal seams. Similarly, based on theoretical analysis, numerical simulation, and field monitoring, Sun et al. [6] studied the additional compressive stress field model of surrounding rock with bolt support, and revealed the influence of support parameters on the spatial distribution of the compressive stress field. Kang et al. [7] identified the initial deformation zones in inclined coal-bearing strata. Yuan et al. [8] proposed a honeycomb gas flow network channel structure based on the distribution morphology of the borehole’s surrounding rock to address gas outburst. Through numerical simulation and field observation, Singh et al. [9] revealed the mechanism of rock pillar spalling due to the existence of weak surfaces in high stress environments. Zi et al. [10] simulated the influence of accumulated damage on the stability of the surrounding rock through experiments. Li et al. [11] explored the influence of coal seam inclination on roof deflection under the same mechanical state. Du et al. [12] experimentally obtained the energy dissipation characteristics during rock failure. Pan [13] analyzed the dynamic mechanical properties and fracture mechanisms of rocks through experimental and simulation methods. Du et al. [14] obtained the roadway deformation mechanism and support technology through field investigations and experimental research. Ma et al. [15] proposed control technologies and methods of gob-side entry through grouting modification and high-strength support for deep soft rock roadways. Sun et al. [16] adopted a stress relief technique to maintain the stability of high-pressure mines. Sakhno et al. [17] analyzed the stability of safe rock pillars in underground coal gasification through numerical simulation, indicating that maintaining a certain width of safe rock pillars plays a key role in controlling surface subsidence and preventing rock rupture.

Currently, many scholars apply catastrophe theory to analyze the minimum safe thickness of protective rock pillars and the key influencing factors when tunneling through adverse geological conditions. Lignos et al. [18] provided a unified classification and accurate solution framework for the nonlinear buckling behavior of multi-parameter discrete systems by applying catastrophe theory, especially focusing on the universal solution of the cusp catastrophe potential energy function. Zhang and Yin et al. [19,20], through theoretical calculations and numerical simulations, summarized approaches for evaluating the stability of surrounding rock in coal-bearing strata. Taking into account gas outburst pressure, Li et al. [21] utilized a cusp catastrophe model to derive a criterion for identifying the instability of critical safety rock pillars in outburst-prone coal seam tunnels. Lei et al. [22] established the early warning model of fractured sandstone with the cusp catastrophe theory. Yuan et al. [23] constructed an evaluation model that can provide assistance for tunnel construction risk assessment. Furthermore, Kou, Guo and Lin et al. [24,25,26] studied the failure mechanisms of tunnel faces in water-rich karst strata and the minimum rock pillar thickness required under such conditions, providing useful engineering references. Sun et al. [27] discussed the size of safe rock pillars according to the unified strength theory. Wen et al. [28] analyzed the stability of stope pillars based on cusp catastrophe theory. Chen et al. [29] studied the safety thickness of a roof in goaf based on cusp catastrophe theory. Zhu et al. [30] proposed a criterion for the critical thickness of safe rock pillars in deep-buried tunnels. Yuan et al. [31,32] analyzed and studied the characteristics of the plastic zone of surrounding rock in deep tunnels and the mechanical properties of rock masses with different lithology.

These previous studies on tunneling through coal seams and karst caves have yielded substantial insights. Building on this foundation, the present study aims to more accurately determine the critical safety rock pillar thickness and its key influencing factors. Specifically, we investigate the effects of tunnel radius, burial depth, axial stress, coal seam dip angle, and gas pressure on the critical thickness. A cusp catastrophe model is constructed based on the mechanical behavior of the safety rock pillar, from which a criterion for critical thickness is derived and its influencing factors analyzed. Additionally, numerical simulations are employed to examine the evolution of the plastic zone during tunnel excavation, thereby validating the theoretical criterion. The findings are then applied to practical engineering projects to provide guidance for safe tunnel construction through coal seams.

2. Theoretical Calculation of the Critical Safety Rock Pillar

2.1. Catastrophe Theory: Fundamental Concepts

Catastrophe Theory [33] examines discontinuous phenomena and sudden state transitions within systems. Rooted in mathematical disciplines such as topology, singularity theory, and dynamical systems, it offers a theoretical framework for analyzing how systems shift between stable states as control parameters evolve, thereby modeling abrupt changes triggered by the crossing of critical thresholds. Xia [34] demonstrated that the cusp catastrophe model enables the quantitative characterization of complex geological engineering problems through mathematical modeling or empirical calibration.

There are three primary catastrophe models: the cusp, swallowtail, and butterfly catastrophes. Among these, the cusp catastrophe model is most widely used in geotechnical engineering due to its analytically manageable critical manifold, which facilitates practical application. As shown in Figure 1, the cusp catastrophe model provides a geometric representation of the bifurcation behavior of systems governed by two control parameters.

The potential function of the cusp catastrophe model is given as follows [35]:

$$V\left(x\right)={\displaystyle \frac{1}{4}}{x}^4+{\displaystyle \frac{1}{2}}\alpha x^2+\beta x$$

(1)

The equilibrium surface differential equation and the singular point set equation are given by

$$\left\{\begin{array}{c}{V}^{\prime }\left(x\right)=x^3+\alpha x+\beta =0\\ V^{″}\left(x\right)=3x^2+\alpha =0\end{array}\right.$$

(2)

By solving Equations (1) and (2) simultaneously, the bifurcation set equation is obtained:

$$\Delta =4{\alpha }^3+27{\beta }^2$$

(3)

The equilibrium surface defines the manifold of the system’s critical points. The failure of the safety rock pillar is determined by the bifurcation set crossing criterion (Δ = 0). When the control variables α and β satisfy the bifurcation set equation, the system reaches the critical state of instability, thereby defining the condition for catastrophic failure.

2.2. Potential Energy Function of the Safety Rock Pillar

2.2.1. Simplified Mechanical Representation of the Rock Pillar

Given the complex geological conditions of the actual tunnel project, the following assumptions are made for the instability analysis of the reserved safety rock pillar:

(1)

The combined effects of friction between the coal seam and the safety pillar, coal seam gas pressure, and intermediate principal stress are simplified as a uniformly distributed force Q acting perpendicular to the coal seam at the coal–rock interface;

(2)

The rock pillar is subjected to a vertical load P₁ on its top and an axial load P₂ along its axis;

(3)

The influence of groundwater, karst formations, and similar geological factors is neglected;

(4)

The pressure distribution on both sides of the rock pillar is assumed to be symmetrical.

Assuming the coal seam dip angle is θ, the idealized stress state of the safety rock pillar under these assumptions is shown in Figure 2.

2.2.2. Expression for the Total Potential Energy of the Safety Rock Pillar

Under the combined action of vertical pressure, axial stress, and gas pressure, the tunnel face undergoes compressive buckling deformation, forming an outward bulge in the direction of excavation, as schematically illustrated in Figure 2b.

According to [36], the axial deflection curve of the rock pillar can be expressed as

$$y(s)=u\mathit{sin}{\displaystyle \frac{\pi s}{R}}$$

(4)

where u is the deflection at the midpoint of the rock beam; s is the distance from the end of the beam; and R is the diameter of the tunnel.

Based on the mechanical models in Figure 2, the total potential energy of the rock beam consists of the bending strain energy and the work exerted by external forces, including the contributions from vertical pressure, axial stress, and gas pressure:

$$V=U+W_1+W_2+W_3$$

(5)

where U is the strain energy of the rock pillar; W₁ is work exerted by vertical pressure P₁; W₂ is work exerted by axial stress P₂; and W₃ is work exerted by gas pressure Q.

According to elastic theory, the strain energy U of the rock pillar is given by

$$U={\displaystyle \frac{1}{2}}{\int }_0^{R}M\left(s\right)d\phi$$

(6)

where M(x) is the bending moment at a distance x from the tunnel face, and dφ is the angular change in the rock pillar’s deflection curve. Here, M(x) and dφ can, respectively, be expressed as:

$$M\left(s\right)=EIk{\left(s\right)}^2$$

(7)

$$K\left(s\right)=y^{″2}{\left(1-y^{\prime 2}\right)}^{-\frac{1}{2}}$$

(8)

Substituting Equations (7) and (8) into Equation (6) yields

$$U={\displaystyle \frac{1}{2}}{\int }_0^{R}M\left(s\right)d\phi ={\displaystyle \frac{1}{2}}{\int }_0^{R}EIk{\left(s\right)}^{2}ds={\displaystyle \frac{1}{2}}{\int }_0^{R}EI{\left(y^{″}\right)}^2{\left[1-{\left(y^{\prime }\right)}^2\right]}^{-1}ds$$

(9)

where E is the elastic modulus of the rock pillar; I is the moment of inertia of the rock pillar; and y′ and y″ are the first and second derivatives of the deflection curve y, respectively.

Applying a Taylor expansion to Equation (9), the strain energy of the rock pillar becomes

$$U={\displaystyle \frac{1}{2}}{\int }_0^{R}EI{\left(y^{″}\right)}^2\sqrt{1-{\left(y^{\prime }\right)}^2}={\displaystyle \frac{EI{\pi }^6}{16R^5}}{u}^4+{\displaystyle \frac{EI{\pi }^4}{4H^3}}{u}^2$$

(10)

The work exerted by vertical stress W₂ is expressed as

$$W_1=-{\displaystyle \frac{1}{2}}{P}_1{{\int }_0^R\left(y^{\prime }\right)}^{2}dx=-P_1{\displaystyle \frac{{\pi }^2}{4R}}{u}^2$$

(11)

where P₁ = γH, with γ = 0.025 and H denoting the tunnel burial depth.

The work exerted by axial stress W₃ is expressed as

$$W_2=-P_2{\int }_0^{R}y\left(s\right)ds=-{\displaystyle \frac{2P_{2}R}{\pi }}u$$

(12)

From Figure 2a, the gas pressure is decomposed into vertical and horizontal components. Assuming a coal seam dip angle of θ, the work exerted by gas pressure on the safety rock pillar W₄ is the sum of the work exerted by its vertical and axial components:

$$W_3=-{\displaystyle \frac{1}{2}}Q\mathit{cos}\theta {{\int }_0^R\left(y^{\prime }\right)}^{2}dx-Q\mathit{sin}\theta {\int }_0^{R}y\left(s\right)d=-{\displaystyle \frac{{\pi }^2}{4R}}Q\mathit{cos}\theta u^2-{\displaystyle \frac{2H}{\pi }}Q\mathit{sin}\theta u$$

(13)

Therefore, the potential energy functional V of the rock pillar is given by

$$V={\displaystyle \frac{EI{\pi }^6}{16R^5}}{u}^4+{\displaystyle \frac{{\pi }^2}{4R}}\left[{\displaystyle \frac{EI{\pi }^2}{R^2}}-Q\mathit{cos}\theta -P_1\right]u^2-{\displaystyle \frac{2R}{\pi }}\left(Q\mathit{sin}\theta +P_2\right)u$$

(14)

2.3. Safety Criterion for Rock Pillar Thickness

According to the canonical form of cusp catastrophe theory in Equation (1), the total potential energy functional of the rock pillar in Equation (14) can be transformed into its standard form through variable substitution. We can define the following:

$$m={\left(\sqrt[4]{{\displaystyle \frac{4R^5}{EI{\pi }^6}}}\right)}^{-1}u$$

(15)

Here, the moment of inertia I of the rock pillar is defined as

$$I={\displaystyle \frac{L^3}{3}}$$

(16)

Substituting Equations (15) and (16) into Equation (14) yields the expression for the total potential energy functional:

$$V={\displaystyle \frac{1}{4}}{m}^4+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{3R^3}{E{L}^3{\pi }^2}}}\left[{\displaystyle \frac{E{L}^3{\pi }^2}{3R^2}}-Q\mathit{cos}\theta -P_1\right]m^2+{\displaystyle \frac{2R}{\pi }}\left(Q\mathit{sin}\theta +P_2\right)\sqrt[4]{{\displaystyle \frac{12R^5}{E{L}^3{\pi }^6}}}m$$

(17)

By comparing Equation (1) with Equation (17), the control parameters α and β of the potential energy functional can be identified as

$$\left\{\begin{array}{l}\alpha =\sqrt[2]{{\displaystyle \frac{3R^3}{E{L}^3{\pi }^2}}}\left[{\displaystyle \frac{E{L}^3{\pi }^2}{3R^2}}-Q\mathit{cos}\theta -P_1\right]\\ \beta ={\displaystyle \frac{2R}{\pi }}\left(Q\mathit{sin}\theta +P_2\right)\sqrt[4]{{\displaystyle \frac{12R^5}{E{L}^3{\pi }^6}}}\end{array}\right.$$

(18)

Substituting Equation (18) into the bifurcation set Equation (3) leads to the critical stability criterion for the safety rock pillar:

$$\Delta =32(\frac{6r^3}{E{L}^3{\pi }^2}{)}^{\frac{3}{2}}{\left[{\displaystyle \frac{2E{L}^3{\pi }^2}{12r^2}}-Q\mathit{cos}\theta -P_1\right]}^3+3456(Q\mathit{sin}\mathsf{\theta }+P_2{)}^2\sqrt[2]{{\displaystyle \frac{6r^9}{E{L}^3{\pi }^{10}}}}$$

(19)

where r represents the tunnel radius, R = 2r.

According to cusp catastrophe theory, when the bifurcation set function Δ > 0, the rock pillar remains in a stable state; when Δ < 0, the pillar becomes unstable; and when Δ = 0, the system reaches a critical point at which even minor disturbances can induce instability. Therefore, the rock pillar thickness that satisfies Δ = 0 defines the critical safety thickness.

3. Investigation of Influencing Factors on Critical Safety Thickness of Rock Pillars

Equation (19) indicates that the critical safety thickness of rock pillars is influenced by tunnel radius, overburden depth, axial compressive stress, gas pressure, and the dip angle of geological structures. A parametric sensitivity analysis of these governing factors is presented below.

3.1. Influence of Tunnel Radius and Overburden Depth on the Critical Safety Thickness of Rock Pillars

To investigate the influence of tunnel radius and overburden depth on the critical safety thickness of rock pillars, based on Equation (19), the following parameters are set: axial stress P₂ = 5 MPa, gas pressure Q = 1 MPa, and coal seam dip angle θ = 30°. The relationship between the critical safety rock pillar thickness L and tunnel radius R (ranging from 2 m to 6 m) under different tunnel depths H (50 m to 400 m) is illustrated in Figure 3.

Figure 3 shows how tunnel depth and radius jointly influence the critical safety rock pillar thickness. It can be observed that both tunnel radius and depth are positively correlated with the critical rock pillar thickness. As tunnel radius and depth increase, the critical thickness also increases. This is because greater tunnel depth leads to higher energy concentration within the rock pillar, increasing its likelihood of failure.

Furthermore, as tunnel depth increases, the rate of growth in critical pillar thickness also increases. This suggests that the sensitivity of the pillar thickness to tunnel depth becomes more pronounced, resulting in an exponential growth trend in the required thickness of the safety rock pillar with increasing tunnel depth.

3.2. Influence of Axial Stress, Gas Pressure, and Coal Seam Dip Angle on the Critical Safety Rock Pillar Thickness

To explore the influence of axial stress, gas pressure, and coal seam dip angle on the critical thickness of the safety rock pillar, the tunnel radius R = 4 m and tunnel depth H = 300 m are fixed, as per Equation (19). The variation in critical rock pillar thickness L with gas pressure Q (0.8 MPa to 1.6 MPa) and coal seam dip angle (20° to 40°) under different axial stresses P₂ (3 MPa to 7 MPa) is shown in Figure 4.

Figure 4 illustrates how the critical safety rock pillar thickness varies with axial stress and gas pressure at different coal seam dip angles. It can be seen that the coal seam dip angle is negatively correlated with the critical pillar thickness, i.e., as the dip angle increases, the critical thickness decreases. Similarly, with increasing axial stress, the critical thickness also gradually decreases.

As shown in Figure 4a, with increasing axial stress, the rate at which the critical thickness decreases becomes slower. This occurs because the stress state of the rock pillar transitions from uniaxial to biaxial compression. The energy-bearing capacity of the pillar increases under this condition, and its sensitivity to further axial stress decreases. Consequently, the rate of decrease in critical thickness slows as axial stress increases.

As shown in Figure 4b, gas pressure generates stress components in both the vertical and axial directions. Since the pillar’s sensitivity to axial stress is lower than that to vertical stress, an increase in gas pressure leads to an increase in the required thickness of the critical safety rock pillar.

4. Case Analysis of a Gas Tunnel Project

4.1. Project Profile

Taking a tunnel project in Hunan Province as an example, the tunnel adopts a single-center curved wall cross-section for its internal contour. The section is 6 m high and 8 m wide above the design line and intersects a single coal seam layer, with an average burial depth of 300 m. The coal seam has a dip angle of approximately 30° and a thickness of about 1 m, and the tunnel adopts a separated double-hole structure for the left and right lines. The left tunnel section from K70 + 470 to K73 + 165 and the right tunnel section from Y9K70 + 470 to Y9K73 + 165 pass through the coal seam. This study focuses on the left tunnel, where the section K72 + 180 to K72 + 430 crosses the coal seam and is identified as a relatively high-risk segment. Due to the fractured nature of the surrounding rock in this section and its close proximity to the coal seam, tunnel excavation is accompanied by risks such as gas emissions and instability of the surrounding rock. The geological map of the left tunnel section is shown in Figure 5.

4.2. Theoretical Analysis

According to Equation (19), the actual project parameters are substituted as follows: tunnel radius R = 4 m, burial depth H = 300 m, axial stress P₂ = 5 MPa, gas pressure Q = 1.2 MPa, and dip angle θ = 30°. The resulting relationship between the bifurcation set value Δ and the thickness of the safety rock pillar is illustrated in Figure 6.

According to catastrophe theory, when Δ < 0, the system is unstable; when Δ > 0, the system remains stable; and when Δ = 0, the system is at a critical state where even small disturbances can trigger instability. From Figure 6, it can be observed that when the thickness of the safety rock pillar is less than 3.95 m, the bifurcation set Δ remains less than 0, indicating that the pillar is highly unstable and prone to failure. When the thickness exceeds 3.95 m, Δ remains greater than 0, signifying a stable state. When the thickness is exactly 3.95 m, Δ = 0, identifying the critical safety thickness required for this tunnel section.

4.3. Numerical Simulation Analysis

4.3.1. Establishment of Tunnel Model Crossing a Coal Seam

According to actual geological survey data, the rock layers near the coal seam consist of shale and mudstone. Siltstone is present near the shale layer, and limestone lies adjacent to the mudstone. For simplification, the model assumes a coal seam thickness of 1 m, a shale layer of 49 m, and a mudstone layer of 50 m. The model dimensions are set with a tunnel length of 100 m and a tunnel cross-sectional radius of 4 m. The simulation domain is 40 m in both length and width. The established model is shown in Figure 7.

According to the site data of a coal mine in Hunan, the coal seam dip angle is 30°, and the main surrounding rock types are mudstone, sandstone, and coal. The physical and mechanical properties of the rock strata used in the model are listed in

Table 1. Physical properties of rock strata.

	Properties	Density /kg·m−3	Bulk Modulus /GPa	Shear Modulus /GPa	Cohesion /MPa	Tensile Strength /MPa	Internal Friction Angle /°
Rock
Shale		2660	2.00	1.5	1.2	1.00	28
Coal		1400	1.00	1.4	0.4	0.30	20
Mustone		2500	2.04	1.5	1.2	0.72	32

.

The model comprises 487,865 elements and 388,934 nodes. Fixed boundary conditions are applied at both ends, the front, back, and bottom of the model, while the top is treated as a free boundary. Given the average burial depth of 300 m, a vertical stress of 7.5 MPa is applied at the top of the model. The ratio of vertical to horizontal tectonic stress is set to 0.5, and the gas pressure is 1.2 MPa. A full-section excavation method is simulated, with an excavation increment of 1 m per step.

4.3.2. Plastic Zone Analysis

To more intuitively illustrate the extent of damage to the safety rock pillar caused by external stress and excavation disturbance during tunnel advancement, the simulation retains the sections of the tunnel lining and surrounding rock. Sequential cross-sections of the tunnel face are selected to observe the distribution of the plastic zone when the minimum normal distance from the coal seam ranges from 2 m to 9 m. The evolution of the plastic zone distribution is shown in Figure 8.

As shown in Figure 8, with ongoing tunnel excavation, the plastic zone within the safety rock pillar gradually approaches the coal seam. When the tunnel face is 9 m from the coal seam, only the safety rock pillar exhibits a limited plastic zone, indicating a stable state. At a distance of 8 m, a small plastic zone begins to appear within the coal seam. When the tunnel is between 5 m and 8 m from the coal seam, the plastic zone in the coal seam expands significantly, though the plastic zone in the rock pillar does not yet fully penetrate it, implying that the pillar retains some load-bearing capacity. When the tunnel is 4 m from the coal seam, the plastic zone fully penetrates the rock pillar, indicating that it has reached its ultimate limit state. At 3 m, the plastic zone occupies a large volume of the pillar, showing a substantial loss in load-bearing capacity. When the tunnel is only 2 m from the coal seam, both the rock pillar and the coal seam exhibit extensive plastic zones that converge, signifying the complete instability and failure of the rock pillar. Therefore, the 8 m distance—when the plastic zone begins to appear in the coal seam—is identified as the early warning threshold for the rock pillar. When the plastic zone fully penetrates the pillar at 4 m, this is defined as the critical safety thickness of the rock pillar.

4.3.3. Stress Analysis

During tunnel excavation, an advance stress field forms in the rock pillar behind the tunnel face. When the peak of this advance stress reaches the coal seam, the pillar becomes highly unstable due to the elevated stress concentration. At this moment, the corresponding rock pillar thickness is considered the critical safety thickness. To analyze the evolution of advance stress in the rock pillar during tunnel excavation and determine its critical safety thickness, the vertical stress distributions were extracted for tunnel face positions ranging from 3 m to 7 m from the coal seam. The corresponding stress evolution curves are presented in Figure 9.

As illustrated in Figure 9, with tunnel advancement, the peak of the vertical stress behind the tunnel face shifts progressively closer to the coal seam. When the tunnel is excavated to 4 m from the coal seam, the vertical stress peak coincides with the coal seam and reaches its maximum magnitude. With further excavation, the stress peak remains within the coal seam, indicating that at 4 m, the stability of the safety rock pillar has reached its critical limit. These simulation results align with the theoretical analysis and confirm the validity and reliability of the previously derived critical safety rock pillar thickness of 3.95 m using the catastrophe discriminant equation.

5. Safety Rock Pillar Treatment Scheme for Gas Tunnel

5.1. Instability Mechanism of Safety Rock Pillars

A tunnel in Hunan Province intersects a coal seam and is identified as a gas tunnel. Due to the presence of gas, tunnel excavation through the coal seam may lead to instability and the failure of the safety rock pillar. According to the field test, when the tunnel is excavated, the distance between the plastic zone and the stress peak and the coal seam will be as shown in Figure 10. Based on a comprehensive analysis of the geological and mechanical environment of the tunnel and the lithological characteristics of the safety rock pillar, the primary causes of instability and failure are as follows.

(1)

Fluctuations in gas pressure within the gas tunnel affect the stability of the surrounding rock strata. As the tunnel advances, imbalances in gas pressure distribution can induce instability and lead to rock pillar failure.

(2)

Due to the weak strength and poor deformation resistance of coal-bearing strata, the rock pillar is highly susceptible to excavation-induced disturbances. Large deformations can occur during tunnel advancement through the coal seam, resulting in progressive damage. As the tunnel face approaches the coal seam, a broader damage zone is generated, compromising the stability of the safety rock pillar.

(3)

The movement of coal and gas also influences rock pillar stability. Tunnel excavation inevitably causes the infiltration, migration, and diffusion of coal gas, altering the physical properties of the surrounding strata and increasing the likelihood of pillar instability.

(4)

Vibrations and impacts from tunnel construction, mining, and transportation activities disrupt the original stress balance of the rock mass. These disturbances create stress concentration zones before and after the coal seam, further compromising the stability of the safety rock pillar.

5.2. Safety Rock Pillar Management Principles

Based on the preceding research, tunnel excavation through a coal seam inevitably induces widespread instability and failure of the safety rock pillar. To ensure construction safety, the following management principles are proposed:

(1)

Prior to implementing treatment measures, the rock pillars within the tunnel should be regularly monitored to detect early signs of instability, such as displacement and cracking, enabling timely intervention.

(2)

Given the complex geological and mechanical conditions of the coal-bearing strata, rock pillars are prone to disturbance and deformation. Therefore, appropriate reinforcement techniques should be applied to enhance the elastic modulus and energy absorption capacity of the pillars, thereby minimizing deformation and ensuring stability during excavation.

(3)

Measures such as pressure relief drilling and ventilation should be adopted to reduce gas pressure within the coal seam. Lowering the gas pressure decreases the energy exerted on the safety rock pillar, limiting its deformation and reducing the risk of instability.

5.3. Treatment Measures for Safety Rock Pillars

(1)

Advance detection and forecasting

When the minimum normal distance between the tunnel excavation face and the coal seam reaches 3.95 m, monitoring systems should be installed at the tunnel face. Displacement sensors and early warning systems should be deployed to monitor changes in the rock pillar and provide alerts for potential instability.

(2)

Advance small-pipe grouting

To ensure sufficient strength and deformation resistance of the safety rock pillar before excavation, advanced small-pipe grouting using pipes of Φ42 mm × L4000 mm is applied. The grouting pressure should be no less than 1 MPa, with an overlap length of no less than 1.5 m. The layout of the grouting pipes is shown in Figure 11.

(3)

Gas drainage and pressure relief

To maintain rock pillar stability, gas drainage and pressure relief are conducted in the coal seam. A “joint borehole layout and integrated extraction” method is adopted to maximize gas extraction efficiency. The extraction zone extends at least 12 m beyond the tunnel’s outer contour. The gas drainage layout is shown in Figure 12.

5.4. Field Application

Based on theoretical calculations and numerical simulation analysis, the tunnel in Hunan was excavated up to a distance of 3.95 m from the normal line of the coal seam. The corresponding construction layout is shown in Figure 13.

During excavation, the displacement and stress of the safety rock pillar were continuously monitored to detect the development of fractures promptly. Advanced small-pipe grouting was then implemented to reinforce the pillar, while gas drainage and pressure relief reduced the internal gas pressure. After each excavation cycle, the pillar was treated accordingly until the tunnel successfully passed through the coal seam. Field results indicate that the tunnel passed safely through the coal seam, demonstrating that the proposed construction scheme effectively mitigates the risk of rock pillar instability and satisfies tunnel safety requirements.

6. Discussion

Based on the catastrophe theory, energy conservation principle, and numerical simulation, this study puts forward the criterion of critical safe rock pillar thickness and analyzes its influencing factors, providing a theoretical basis and engineering guidance for the safe construction of gas tunnels uncovering coal. However, there are still some aspects worth further discussion and future improvement. Firstly, when establishing the mechanical model of the safe rock pillar, it is assumed that the two sides of the rock pillar are under the same pressure. This paper mainly focuses on the influence of vertical pressure (P1) and axial stress (P2). The influence of significant horizontal tectonic stress and its directional difference that may exist in the actual geological environment is not fully considered. The inhomogeneity of the horizontal stress field may change the stress distribution pattern of the rock column and induce asymmetric deformation or even local shear failure, which is not reflected in the simplified model of this study, and may bring some deviation to the accuracy of the critical thickness prediction, especially in areas with strong tectonic activity or complex in situ stress fields.

Secondly, in this study, gas pressure (Q) was treated as a uniformly distributed load applied to the coal–rock interface, and the key dynamic process of gas seepage was not discussed in depth. In fact, the seepage of gas in coal and rock mass is a process closely coupled with stress field and damage evolution. Excavation disturbance will change the permeability of the coal seam and accelerate gas desorption, migration and pressure redistribution. This seepage–stress-damage coupling effect not only continuously changes the magnitude and distribution of the gas load acting on rock pillars, but also dynamically reduces the stability of rock pillars by affecting the mechanical properties of coal and rock (such as strength weakening) and inducing pore pressure changes. Ignoring the time-varying and spatial non-uniformity of seepage may lead to an underestimation of the promoting effect of gas on the instability of rock pillars and its risk.

Finally, in the selection of the theoretical framework, this study mainly applies the cusp catastrophe theory to construct the instability criterion. Its advantages are that the model is simple, the critical surface is easy to construct and the physical meaning of the parameters is clear, which means it has good practicability in engineering applications. However, trying or comparing other types of mutation theoretical models (such as swallowtail mutation, butterfly mutation, etc.) is a direction worth considering. These models may contain more control variables, which can theoretically describe more complex system instability behaviors, especially when multiple strong coupling factors (such as seepage, time effect, and finer stress state) need to be considered. It may provide a more comprehensive description of the instability mechanism or a more accurate prediction of the critical state. Future research can consider introducing horizontal stress components, establishing a gas seepage–stress coupling model, and exploring the application potential of other catastrophe theories in multi-factor complex systems to further improve the completeness and applicability of the critical safety rock pillar thickness prediction model.

7. Conclusions

(1)

Based on the principles of material mechanics and structural mechanics, a physical model of the critical safety rock pillar is established. Using potential energy theory and catastrophe theory, a catastrophe instability model is constructed for determining the critical thickness of the reserved safety rock pillar when the tunnel passes through a coal seam. A discriminant formula for the critical safety rock pillar thickness is derived. According to the bifurcation set Δ, when Δ > 0, the rock pillar remains stable; when Δ < 0, it is unstable; and when Δ = 0, the rock pillar reaches its critical instability state, which defines the critical safety rock pillar thickness. The model has the capacity for multi-factor coupling analysis, which can systematically integrate the geometric parameters, mechanical parameters and geological conditions of the tunnel, and quantitatively reveal the sensitivity and interaction effect of each factor on the critical thickness, which is better than the existing models, considering only a single or a few factors.

(2)

The calculation results show that the critical safety rock pillar thickness increases with rising tunnel radius, vertical pressure, axial stress, and gas pressure. It decreases with increasing cohesion, internal friction angle, and coal seam dip angle. As the confining pressure ratio increases, the thickness first decreases and then increases, showing a nonlinear relationship.

(3)

Numerical simulation software is used to model the evolution of the plastic zone and the vertical stress field behind the tunnel face during excavation. The simulation results indicate that when the tunnel advances to within 4 m of the normal line of the coal seam, the plastic zone of the reserved safety rock pillar fully penetrates to the coal seam, and the peak of the vertical stress also reaches the coal seam. At this point, the safety rock pillar completely loses stability. These findings are consistent with the theoretical predictions, verifying the reliability of using catastrophe theory combined with energy analysis to determine the critical thickness of the safety rock pillar.

(4)

Based on theoretical analysis, when a tunnel in Hunan is excavated to within 3.95 m of the coal seam’s normal line, a joint control scheme is proposed—consisting of “advance detection and prediction + advanced small-pipe grouting + gas drainage and pressure relief”—to mitigate rock pillar instability. The field results show that the tunnel successfully traversed the coal seam, validating the effectiveness and rationality of the proposed joint control approach for ensuring rock pillar stability.