Numerical Analysis on the Effect of Geometric Parameters of Reverse Fault on Tunnel Mechanical Response

Abstract

Permanent displacements caused by active faults can lead to the severe deformation of tunnel liners. To investigate the effect of fault fracture deformation patterns on the deformation of tunnel liners under fault dislocation, this paper categorized three fault-zone fracture deformation patterns and conducted numerical simulations for tunnel’s surrounding rock-liner systems under different fracture deformation patterns. Furthermore, the longitudinal displacement, relative deformation, axial stress, and shear stress of the tunnel liner were measured to characterize the mechanical response of the tunnel, and the effects of fault geometric parameters on the mechanical response of the tunnel liner were explored. The results showed that fracture deformation patterns were broadly categorized into uniform fracture deformation, linear fracture deformation, and nonlinear fracture deformation patterns. The distribution patterns of tunnel liner stress and deformation under these fracture deformation patterns were similar, but the magnitude of the peaks and the intensity of their effects differed. Under reverse fault dislocation, the peak values of tunnel liner deformation and shear stress occurred at the rupture plane. In contrast, the maximum axial stress was observed at the interface between soft and hard rock masses. When the core width of the fault zone decreased and the fault dip direction increased, the intensity of the mechanical response of the tunnel liner increased. With the fault dip decreased, the axial stress in the tunnel liner transitions from tensile-compressive stress to compressive stress, the shear stress decreases, and the intensity of the relative deformation of the tunnel liner increases. These research results can provide significant guidelines for tunnel design crossing the reverse fault.

1. Introduction

In recent years, with the rapid development of infrastructure construction in western China, numerous long-distance underground linear projects, such as hydraulic tunnels, are either under construction or in the planning stage. However, the complex geological structure and intense regional seismic activity in this area mean that these tunnels often have to cross active faults. The potential threat of fault dislocation to tunnel structures has been vividly demonstrated by many earthquake-related disasters. For instance, during the 1906 San Francisco earthquake, the lining of the Wright tunnel crossing the San Andeans fault suffered different degrees of permanent dislocations [1]. In the 1995 Hanshin earthquake in Japan, the Rokaka Tunnel experienced severe shear cracks and peeling [2] in its vault when passing through the fault section. The 2008 Wenchuan earthquake caused the Longxi tunnel across the fault to undergo serious misalignment and rotation deformation [3], and the Daliang tunnel, during the 2022 Menyuan earthquake in Qinghai province, suffered significant shear dislocation and non-uniform displacement after passing through the Lenglongling fault [4], as shown in Figure 1. These real cases highlight the severe damage that fault dislocation can inflict on underground linear engineering.

Scholars have made efforts to explore the deformation characteristics of tunnel lining structures affected by dislocated faults. Physical model experiments have become a common research method. Cui et al. [5] developed a test device capable of simulating the complex fault dislocation mechanism under high-ground-stress environments and found that deep-buried tunnels mainly experience large compressive deformations, while shallow-buried tunnels are prone to shear failure at the fault location. Liu et al. [6,7] conducted experimental research on fault dislocation models with different angles and concluded that the failure of tunnel linings is a combination of bending tension and direct shear failures. Zhou et al. [8] carried out small-scale physical model tests and discovered that the hinged tunnel under faults presents an “S” shape, with the orthogonal angle being the optimal one for crossing the fault zone. Liu et al. [9] conducted a deep-tunnel model test across an active fault and found that the creep slip dislocation deformation mode of the fault is “S”-shaped, and significant slip displacement occurs at the fracture surface when the fault momentum is large.

However, physical model experiments are time-consuming, and the monitoring means are relatively limited. Thus, some scholars have turned to numerical simulation methods to study the mechanical characteristics of tunnel linings. Sun et al. [10] assumed that fault dislocation occurs only at the fault plane and numerically simulated the surrounding-rock lining under normal fault dislocation, identifying the lining at the fault plane as the most severely affected area. Yan et al. [11] considered the mechanical property differences between the hanging and foot walls and used a three-dimensional discrete-continuous coupling method to analyze the influence of the fault-zone width on the lining’s mechanical response under fault dislocation. Zhang et al. [12] utilized a linear fault dislocation deformation model to simulate fault dislocation and analyzed the effects of different fault tendencies, dip angles, and widths on the displacement, relative deformation, and stress distribution of tunnel structures. Additionally, many scholars have established 3D models of cross-fault tunnels with various engineering backgrounds, considering different fault dislocation deformation modes to research tunnel mechanical responses [13,14,15,16].

Despite these previous studies, there are still some crucial gaps in the existing research. First, the simulation methods used in previous studies vary widely, which makes it difficult to draw unified and accurate conclusions. For example, different assumptions about the fault dislocation mechanism and boundary conditions lead to inconsistent results, and there is a lack of a systematic comparison and integration of these methods. Second, most studies only focus on a few fault parameters, and the comprehensive influence of multiple fault parameters on the mechanical response of tunnel linings has not been fully explored. Under different engineering background conditions, the parameters of active fault zones vary greatly, and understanding how these parameter changes interact and affect tunnel mechanical responses is essential for practical engineering applications.

Based on a tunnel project crossing a reverse fault in western China, this study aims to fill these research gaps. We summarize the existing numerical simulation models of fault dislocation deformation, establish numerical models of tunnels crossing active faults under different dislocation-deformation modes, and conduct numerical calculations of reverse-fault dislocation. By comprehensively considering the influence of multiple fault parameters, including fault core width, total fault zone width, fault dip angle, and fault dip direction, on the mechanical behaviors of tunnel linings, this research provides a more in-depth understanding of the mechanical response of tunnels crossing reverse faults. The results offer more accurate and practical guidance for tunnel design in fault-prone areas, thus advancing the field of tunnel engineering in seismically active regions.

2. Methodology

2.1. Numerical Modeling

This paper calculates the prototype based on a reference of a water diversion tunnel project in the west, which penetrates an active fault zone. The total width is 180 m, the width of the core part of the fault zone is 50 m, the dip angle is 88°, and the dip direction is 216°. The horizontal displacement value of the defense level is considered to be 2 m. The above parameters are derived from the geological investigation data. A three-dimensional numerical model with a length of 600 m and a width of 100 m is constructed based on FLAC3D software V5.0 (Figure 2). The Y-axis represents the tunnel axis direction, the radius of the circular tunnel section is 7.0 m, and the concrete lining is combined and considered, with a thickness of 1.0 m. The model consists of six parts, namely, the core part, the influence zone, the fracture surface, the tunnel lining, and the stable hanging and foot walls, and all adopt the strain softening model. The maximum mesh size is set to 0.5 m.

The material parameters in the model correspond to the lining material parameters of a tunnel project in western China and the physical-mechanical parameters of the surrounding rock obtained from geological surveys and are derived from the recommended parameters in the Chinese code [17] (GB/T50218-2014 Standard for engineering classification of rock mass), with the specific parameters listed in

Table 1. Physical and mechanical parameters of the surrounding rock and liner.

Materials Name	Wall Rock Grade	Density (kg/m3)	Elastic Modulus (GPa)	Poisson’s Cohesion	Friction Ratio	Angle (°)
Hanging wall	III	2600	6.0	0.29	3.0	37
Affected zone	IV	2250	4.3	0.31	0.5	31
Core of fault zone	IV	1650	1.0	0.35	0.2	23
Lining	-	2500	30.0	0.20	2.0	25

. Since this study primarily focuses on creep slip displacement, the fault zone displacement is treated as static loading applied to the hanging wall boundary. Therefore, the dynamic damping setting was not used in the current simulation.

2.2. Characterization of Geometric Parameter Variation in Fault Zones

The mechanical responses of tunnels crossing an active fault under different engineering conditions vary due to fault displacement. Hence, it is essential to investigate the effect of fault zone geometric parameters on the mechanical behavior of tunnel linings. While obtaining mechanical parameters of fault zone rock masses (e.g., elastic modulus, strength, friction coefficient) is challenging and typically relies on complex laboratory tests or field drilling, with high uncertainty and spatial variability, the geometric parameters of fault zones are comparatively easier to acquire and quantify. This study aims to analyze the geometric parameters of fault zones that affect the mechanical behaviors of tunnels crossing active faults. (1) Fault zone displacement deformation mode: Spatial alterations in fault zone rock masses caused by displacement, slip, tension, or compression. (2) Fault zone width: The length perpendicular to the strike direction. (3) Core width of fault zone: In practical engineering, faults often exhibit complex internal structures. This study defines the core region as the area adjacent to potential rupture surfaces. (4) Fault Dip Angle: The angle between the true inclination line of the fault and its horizontal projection, representing the inclination of the rupture surface relative to the horizontal plane in a vertical cross-section perpendicular. (5) Fault Dip Direction: The direction of the true inclination line projected onto the horizontal plane, pointing toward the hanging wall. On the same horizontal projection plane, the dip direction is perpendicular to the strike line.

2.3. Boundary Condition

Current numerical models for simulating fault deformation can be categorized into the following three approaches [18,19,20,21]: (1) The model consists of the lining, hanging wall, footwall, and fracture zone. The rupture surface is positioned at the center of the fracture zone. The surrounding rock on the footwall side of the rupture surface is fixed, while a uniform displacement is applied to the hanging wall side (Figure 3a). (2) The model includes a lining, hanging wall, footwall, and fracture zone, with the rupture surface located at the center of the fracture zone. The footwall rock mass is fixed, and a uniform displacement is applied to the hanging wall. Additionally, a linear displacement is imposed within the fault zone (Figure 3b). (3) Similar to the above structure, the rupture surface is placed at the center of the fracture zone. The footwall is fixed, a uniform displacement is applied to the hanging wall, and a nonlinear displacement is introduced within the fault zone (Figure 3c).

Table 2. Comparative table of the three approaches in boundary conditions.

Boundary Conditions Name	Model Composition	Rupture Surface Position	Fixed Area	Displacement Application
				Hanging Wall	Fault Zone
Type I	Lining, hanging wall, footwall, and fracture zone	Center of the fracture zone	Footwall	Uniform displacement	/
Type II	Lining, hanging wall, footwall, and fracture zone	Center of the fracture zone	Footwall	Uniform displacement	Linear displacement
Type III	Lining, hanging wall, footwall, and fracture zone	Center of the fracture zone	Footwall	Uniform displacement	Nonlinear displacement

presents the comparison of the three approaches.

The deformation in the fault zone is analogous to the deflection curve of a fixed-end beam subjected to differential settlement at its supports, both exhibiting an S-shaped deformation pattern. Consequently, an S-shaped fault dislocation deformation model is proposed based on the deflection curve assumption of a “fixed-end beam with differential settlement”. Figure 4 shows the schematic diagram of structural deformation for a beam experiencing differential settlement at its fixed ends.

According to the theory of materials mechanics, the deflection curve equation of a beam with ends fixed under uneven support settlement is as follows:

$$w={\displaystyle \frac{2a{x}^3}{l^3}}+{\displaystyle \frac{3a{x}^2}{l^2}}$$

(1)

where w is the deflection, a = ΔAB is the vertical displacement, l is the beam length, and x is the point position under the coordinate system.

3. Impact Analysis of Deformation Patterns Caused by Reverse Fault Displacement

3.1. Lining Deformation Response

Figure 5 presents the vertical displacement curves of the tunnel lining crown under different dislocation deformation models. As shown in Figure 5, the vertical displacements of the lining within both the fault core and damage zone vary remarkably across different deformation models. Specifically, for the uniform dislocation model, fault core displacement is 0.1934 m (accounting for 96.7% of total displacement); for the linear dislocation model, fault core displacement is 0.0814 m (accounting for 40.7% of total displacement); for the nonlinear dislocation model, fault core displacement is 0.1121 m (accounting for 56.1% of total displacement); and for the transfer-type dislocation model, fault core displacement is 0.1498 m (accounting for 74.9% of total displacement). These results indicate that the majority of fault displacement (60.7–96.7%) is concentrated within the narrow fault core, while the remaining displacement occurs in the more spatially extensive damage zone. This demonstrates that dislocation deformation models critically govern the distribution patterns of the lining displacements across fault zones.

Taking the numerical simulation study of the non-uniform faulting deformation mode as an example, Figure 6 shows the relative deformation curves of the lining in different tunnel diameter directions. The distribution of relative deformation in the tunnel lining is quite complex, with the most significant relative deformation occurring at the rupture surface. The relative deformation of the tunnel lining is more pronounced in the relatively small fault core region. With increases in the distance from the rupture surface axis, the relative deformation of the tunnel lining shows a decreasing trend until it becomes negligible after extending a certain distance into the upper and lower blocks. The direction of the fault slip is consistent with the 0–180° tunnel diameter direction and is perpendicular to the 90–270° tunnel diameter direction. The smaller the angle between the tunnel diameter and the direction of fault slip, the greater the relative deformation of the lining, with the maximum relative deformation direction aligning with the direction of fault slip.

Figure 7 presents the relative deformation curves of the lining at the tunnel crown and tunnel invert under different faulting deformation modes. From Figure 7, it is evident that in the uniform faulting deformation mode, the maximum relative deformation of the lining is 0.0081 m, with a second peak of relative deformation occurring at the boundary between the fault core and the influence zone of the fault, where the second peak value is −0.0004 m, approximately 4.9% of the maximum value. In the linear faulting deformation mode, the maximum relative deformation of the lining is 0.00139 m, and a second peak of relative deformation appears within the influence zone of the fault, with the second peak value being 0.0009 m, approximately 64.7% of the maximum value. In the nonlinear faulting deformation mode, the maximum relative deformation of the lining is 0.00148 m, with a second peak of relative deformation occurring at the boundary between the influence zone of the fault and the upper and lower blocks, where the second peak value is 0.00012 m, approximately 8.1% of the maximum value. In the faulting deformation mode IV, the maximum relative deformation of the lining is 0.0021 m, with a second peak of relative deformation appearing at the boundary during the fault core and influence zone of the fault, where the second peak value is 0.0011 m, approximately 52.4% of the maximum value. Additionally, a third peak of relative deformation occurs at the boundary between the influence zone of the fault and the upper and lower blocks, with the third peak value being 0.00075 m, approximately 3.6% of the maximum value. Due to fault dislocation, the maximum relative deformation will occur at the fracture plane, and the relative deformation will decrease as the axial measurement point moves away from the fracture plane. Due to the connection between the fault zone core and the influence zone, and the fault zone and the soft-hard gradient section of the upper and lower plates, the relative deformation of the lining will produce a local peak value.

In summary, the intensity of lining deformation at the fracture surface is as follows: uniform dislocation deformation mode > nonlinear dislocation deformation mode > linear dislocation deformation mode; the intensity of the deformation of the lining at the interface is as follows: linear dislocation deformation mode > nonlinear dislocation deformation mode > uniform dislocation deformation mode; the influence range of lining deformation is as follows: nonlinear dislocation deformation mode > linear dislocation deformation mode > uniform dislocation deformation mode.

3.2. Axial Tensile and Compressive Stress Analysis of Lining

Figure 8 presents the axial stress curves of the tunnel liner under different fracture deformation patterns. Under fault dislocation, the axial stress at the top and bottom of the tunnel presents an antisymmetric distribution on both sides of the fracture surface, with one side being strained and the other side being compressed. At the same measuring point, the stress direction of the roof and the bottom of the cave is the opposite; that is, at the same lining section, when the roof is stretched, the bottom of the cave is pressured. The peak axial stress of the top and bottom of the tunnel occurs at the junction of the core and the influence zone of the fault zone. The axial stress of the left wall and the axial stress of the right wall are mainly compressive stresses. The axial stress of the lining changes significantly in the core, and the changes in vertical shear stress affecting the region and the gradient section are also more intense. Under different dislocation deformation modes, the peak of axial stress and the intensity of change are different. In the uniformly distributed dislocation deformation mode, the maximum axial tensile stress of the lining reaches 29.5 MPa, and the maximum axial compressive stress of the lining reaches 33.9 MPa. In the linear dislocation deformation mode, the maximum axial tensile stress of the lining reaches 8.0 MPa, and the maximum axial compressive stress of the lining reaches 11.2 MPa. In the nonlinear dislocation deformation mode, the maximum axial tensile stress of the lining reaches 11.7 MPa, and the maximum axial compressive stress of the lining reaches 16.0 MPa. In the dislocation deformation mode Ⅳ, the maximum axial tensile stress of the lining reaches 19.8 MPa, and the maximum axial compressive stress of the lining reaches 25.1 MPa.

According to the different dislocation deformation modes, the intensity of axial stress of thelining is as follows: uniform dislocation deformation mode > nonlinear dislocation deformation mode > linear dislocation deformation mode. The width of the influence range of the axial stress is as follows: linear dislocation deformation mode > nonlinear dislocation deformation mode > uniform dislocation deformation mode.

3.3. Analysis of Vertical Shear Stress of Lining

Figure 9 presents the vertical shear stress curves of the tunnel lining under different fault dislocation deformation modes. The vertical shear stress distributions at both the vault and the invert exhibit remarkable consistency under fault dislocation. The vertical shear stress reaches its maximum at the rupture surface within the fault zone. In addition to the global peak value, local stress peaks are observed at the interface between the fault core and the damage zone, as well as within the soft-to-hard transition segments connecting the fault zone with the hanging wall and footwall. The most dramatic changes in vertical shear stress occur near the rupture surface of the fault zone, while significant stress variations are also evident in both the damage zone and the soft-hard transition segments between the fault zone and surrounding rock masses. Under different fault dislocation deformation modes, the peak values and variation intensity of vertical shear stress exhibit distinct characteristics. In the uniform dislocation deformation mode, the maximum vertical shear stress in the lining reaches 21.9 MPa; under linear dislocation deformation mode, this value decreases to 7.6 MPa; in the nonlinear dislocation deformation mode, the peak stress increases to 9.5 MPa; while under the fault dislocation deformation mode IV, the maximum vertical shear stress reached 15.9 MPa. According to different dislocation deformation modes, the intensity of lining shear stress is as follows: uniform dislocation deformation mode > nonlinear dislocation deformation mode > linear dislocation deformation mode. The width of the influence range of the axial stress of the lining is in the following order: linear dislocation deformation mode > nonlinear dislocation deformation mode > uniform dislocation deformation mode.

Among the three dislocation deformation modes, the distribution of deformation and stress of the tunnel lining is basically the same, and the intensity of the deformation and stress and the width of the influence range are different. The intensity of deformation and stress is in the following order: uniform dislocation deformation mode > nonlinear dislocation deformation mode > linear dislocation deformation mode. The intensity of the influence range width is in the following order: nonlinear dislocation deformation mode > linear dislocation deformation mode > uniform dislocation deformation mode. In general, the deformation and stress in the uniformly distributed dislocation deformation mode are mainly concentrated near the fracture plane, which cannot reflect the mechanical response. The linear dislocation deformation model can reflect the gradual deformation and stress in the entire fault, whereas due to its fixed dislocation deformation mode, it lacks adaptability when considering the fault zone with large scale and complex internal structure. The nonlinear dislocation deformation model takes into account the influence of the complex structure inside the fault on the simulated dislocation. The subsequent analysis considers the use of the nonlinear dislocation deformation model.

4. Fault Parameters Influence Analysis

4.1. Fault Zone Core Width

To investigate the impact of core width on the mechanical response of the tunnel linings, this research sets three different total widths for the fault zone, 40 m, 50 m, and 60 m, while keeping other relevant conditions unchanged. By analyzing the crown displacement, the relative deformation between the crown and invert, the axial stress at the crown, and the vertical stress on the east wall, this study explores the specific effects of variations in the core width on the mechanical response of the tunnel lining. Figure 10a presents the displacement patterns of the tunnel lining at the crown under different core widths. The distribution patterns of lining deformation were similar. However, the core width does have a certain effect on the displacement of the tunnel lining. The displacement values of the tunnel lining located within the core increase with a widening of the fault core. This is because the fault zone, with its relatively weak mechanical properties, is more prone to displacement, leading to an expansion of the area susceptible to displacement, which, in turn, increases the amount of displacement in that region. Figure 10b illustrates the relative deformation patterns between the crown and the invert of the tunnel lining under different core widths. The distribution patterns of relative deformation of tunnel lining are generally similar. However, the maximum relative deformation of the tunnel lining located within thefault zone core decreases as the core width increases, with the maximum relative deformation decreasing from 0.0049 m to 0.0021 m, a decrease of 57.1%. Conversely, the relative deformation of the tunnel lining located within the influence zone increases with a widening of the total fault zone width. As the area prone to displacement expands, the intensity of deformation in that region decreases. Figure 10c presents the patterns of axial stress at the crown of the tunnel lining under different core widths. The distribution patterns of axial stress at the crown of the tunnel lining are generally similar. The maximum axial stress at the crown, located at the boundary between the zone core and the influence zone, decreases as the core width increases. Specifically, the maximum axial tensile stress decreases from 26.42 MPa to 18.55 MPa, a reduction of 29.8%, while the maximum axial compressive stress decreases from 33.33 MPa to 23.14 MPa, a reduction of 30.6%. With the total width increasing, the area experiencing relative deformation also expands. Figure 10d depicts the pattern of vertical shear stress on the east sidewall of the tunnel lining under varying core widths. The distribution pattern of vertical shear stress on the east sidewall remains fundamentally consistent. Notably, the maximum vertical shear stress decreases as the core width increases. Conversely, the vertical shear stress of tunnel lining situated within the influence zone increases with the expansion of the total fault zone width. Specifically, the maximum shear stress reduces from 20.63 MPa to 15.51 MPa, marking a significant decrease of 24.8%. In summary, when the fault displacement is constant, a reduction in the core width of the fault zone leads to an intensification of the mechanical response of the tunnel lining, e.g., due to stress redistribution over a larger volume.

4.2. Total Width of Fault Zone

This research sets three different total widths of the fault zone, 180 m, 200 m, and 220 m, while keeping other relevant conditions unchanged. By analyzing the displacement of the tunnel crown, the relative deformation between the tunnel crown and the tunnel bottom, the axial stress at the tunnel crown, and the vertical shear stress of the east wall, this study explores the specific effects of changes in the total width on the mechanical response of tunnel linings. Figure 11a presents the displacement patterns of the tunnel lining at the crown under different total widths. The distribution patterns of the tunnel lining deformation are generally similar. However, the total width has a certain impact on the displacement of the tunnel lining. The displacement of the tunnel lining located within the influence zone of the fault zone increases with the increase in the total width of the fault zone. This is because with a constant amount of displacement, the area influenced by the fault zone increases, leading to a larger displacement in that region (2.6 mm to 4 mm). Figure 11b illustrates the relative deformation patterns between the tunnel crown and tunnel bottom under different total widths. The relative deformation of the tunnel lining is generally similar. However, the maximum relative deformation of the tunnel lining located at the core decreases as the total width increases. The maximum relative deformation decreases from 0.0039 m to 0.0026 m, representing a reduction of 33.3%. As the total width of the fault zone increases, the area experiencing relative deformation also expands. Figure 11c presents the axial stress patterns at the crown of the tunnel lining under different total widths. The distribution patterns of the axial stress at the tunnel crown are generally similar. However, the maximum axial stress at the crown, located at the boundary between the core of the fault zone and the influence zone, decreases as the total width increases. The maximum axial tensile stress decreases from 23.14 MPa to 19.12 MPa, representing a reduction of 17.4%, while the maximum axial compressive stress also decreases from 23.14 MPa to 19.12 MPa, showing a reduction of 17.4%. Figure 11d shows the vertical shear stress patterns of the east wall of the tunnel lining under different total widths. The distribution patterns of the vertical shear stress on the east wall of the tunnel lining are generally similar. However, the total width has a certain effect on the vertical shear stress of the east wall. The maximum vertical shear stress at the east wall located at the core decreases as the total width increases, while the relative deformation located in the influence zone increases as the total width increases. The maximum shear stress decreases from 18.54 MPa to 15.52 MPa, representing a reduction of 16.3%. In general, when the amount of fault displacement is constant, a decrease in the total width will lead to an increase in the intensity of the mechanical response of the tunnel lining.

4.3. Fault Dip Angle

To investigate the impact of the relationship between fault dip angles and tunnel dip angles (the angle between the tunnel axis and the horizontal plane) on the mechanical response of tunnel linings, this research sets three different fault dip angles, 70°, 80°, and 88°, while keeping other relevant conditions constant. By analyzing the crown displacement of the tunnel, the relative deformation between the crown and the invert, the axial stress at the crown, and the vertical shear stress of the east wall, the specific effects of the variation in the core width on the mechanical response are explored. Figure 12a explores the displacement patterns of the tunnel lining at the crown under different fault dip angles. The distribution patterns of the tunnel lining displacement are essentially the same, indicating that the fault dip angle has almost no effect on the displacement of the tunnel lining. Figure 12b presents the relative deformation patterns of the tunnel lining between the crown and the invert under different fault dip angles. The distribution patterns of the relative deformation are essentially the same. The maximum relative deformation located in the core of the fault zone decreases as the fault dip angle increases, and the relative deformation located in the influence zone also decreases with an increasing in fault dip angle. The maximum relative deformation decreases from 0.0084 m to 0.0034 m, representing a reduction of 59.6%. The decrease in the fault dip angle leads to an increased range of influence of the relative deformation in the core. Figure 12c illustrates the axial stress patterns at the crown of the tunnel lining under different fault dip angles. The distribution patterns of the axial stress at the crown of the tunnel lining are essentially the same. The maximum axial tensile stress at the junction between the core and influence zone increases as the fault dip angle increases, while the maximum axial compressive stress at the same junction decreases with an increasing fault dip angle. The maximum axial tensile stress decreases from 22.69 MPa to 2.21 MPa, representing a reduction of 90.3%, while the maximum axial compressive stress decreases from 42.45 MPa to 26.30 MPa, a reduction of 38.5%. The smaller of dip angle, the larger of area of compressive stress experienced by the tunnel lining (45 MPa to 30 MPa). Figure 12d shows the vertical shear stress patterns on the east wall of the tunnel lining under different fault dip angles. The distribution patterns of the vertical shear stress on the east wall are essentially the same. The maximum vertical shear stress located in the core increases as the fault dip angle increases, while the shear stress of the tunnel lining in the influence zone of the fault also increases with an increase in the fault dip angle. The maximum shear stress increases from 13.83 MPa to 17.47 MPa, representing an increase of 26.4%.

Overall, when the fault displacement is constant, a smaller fault dip angle leads to a larger horizontal component of the fault displacement, leading to a transition of the axial stress in the tunnel lining from tensile-compressive stress to predominantly compressive stress. Additionally, a smaller vertical component of fault displacement leads to lower shear stress, while the relative deformation becomes more pronounced.

4.4. Fault Dip Direction

To investigate the impact of the angle between the fault dip direction and tunnel axis angle on the mechanical response of tunnel lining, this research sets three different fault dip directions at 200°, 215°, and 230°, while keeping other relevant conditions unchanged. By analyzing the crown displacement, the relative deformation between crown and invert, the axial stress at the crown, and the vertical shear stress on the east wall, the specific effects of changes in fault dip direction on the mechanical response are explored. Figure 13a presents the displacement patterns of the tunnel lining crown under different fault inclinations. The distribution pattern of the tunnel lining deformation is generally similar. However, the fault dip direction has a certain impact on the displacement of the tunnel lining, with the displacement values located at the core increasing as the fault dip direction increases. Figure 13b illustrates the relative deformation patterns between the tunnel crown and invert under different fault dip directions. The distribution pattern of the relative deformation of the tunnel lining is generally similar. However, the maximum relative deformation located at the core increases with the fault dip direction, rising from 0.0026 m to 0.0046 m, which represents an increase of 76.9%. Conversely, the relative deformation located in the influence zone decreases as the fault dip direction increases. Figure 13c presents the axial stress patterns at the crown of the tunnel lining under different fault dip directions. The distribution pattern of the axial stress at the crown of the tunnel lining is generally similar. The maximum axial stress at the crown of the tunnel lining, located at the boundary between the core and the influence zone, increases with the fault dip direction. The maximum axial tensile stress rises from 20.64 MPa to 24.18 MPa, representing a decrease of 17.3%, while the maximum axial compressive stress increases from 26.71 MPa to 28.98 MPa, indicating a decrease of 8.9%. Figure 13d illustrates the vertical shear stress patterns on the east wall of the tunnel lining under different fault dip directions. The distribution pattern of vertical shear stress on the east wall is generally similar. The fault dip direction has a certain impact on the relative deformation of the tunnel lining, with the maximum vertical shear stress increasing as the fault dip direction increases. Conversely, the shear stress located in the influence zone decreases as the fault angle increases. The maximum shear stress rises from 14.24 MPa to 19.99 MPa, representing an increase of 40.4%. In general, when the fault displacement is constant, an increase in the fault dip direction will lead to a greater intensity of the mechanical response.

4.5. Discussion

The parameter analysis reveals three key trends critical for reverse fault tunnel design: (1) Reducing the fault core width and total zone width amplifies mechanical response intensity, with core width being the most sensitive parameter due to localized stress concentration at the core-damage zone interface. (2) Decreasing the fault dip angle shifts the axial stress from tensile-compressive to predominantly compressive, while increasing the dip direction enhances shear stress, highlighting the need to prioritize fault orientation in structural design. (3) Nonlinear deformation modes, which account for complex fault zone structures, more realistically predict stress distribution compared to simplified uniform/linear models. These findings underscore the importance of integrating detailed fault geometric parameters into numerical simulations, providing a basis for targeted reinforcement strategies—such as enhancing lining ductility at core interfaces and adjusting joint flexibility for shallow-dipping faults—to mitigate high-risk deformation and stress concentrations in practical engineering.

5. Conclusions

(1)

Three fault dislocation deformation patterns (uniform, linear, nonlinear) were categorized, and their effects on tunnel lining deformation and stress were analyzed. The results show that although the distribution patterns of lining stress and deformation under different patterns are similar, the peak values and influence intensities differ. For example, the maximum relative deformation and shear stress of the lining occur at the rupture plane, while the maximum axial stress appears at the interface between soft and hard rock masses.

(2)

The influence of fault geometric parameters (core width, total zone width, dip angle, and dip direction) on the mechanical response of the tunnel linings was explored. Reducing the core width and the total zone width, or increasing the fault dip direction, intensifies the mechanical response. Decreasing the dip angle transitions the axial stress from tensile-compressive to compressive, reduces shear stress, and increases relative deformation intensity. Numerical evidence shows that a 33.3% reduction in core width increases relative deformation by 57.1%, highlighting the sensitivity of narrow fault cores to stress concentration.

(3)

This study provides a systematic analysis of reverse fault geometric parameters, filling a gap in the prior research tradition, which lacked comprehensive parameter sensitivity analysis. The classification of nonlinear deformation modes, considering complex fault internal structures, offers a more accurate prediction of stress distribution. These findings can guide tunnel design in active fault zones, such as reinforcing the interface between fault cores and influence zones and adjusting lining parameters based on the dip angle/direction.

(4)

Limitations include the focus on static creep slip without dynamic seismic loading or fluid pressure effects, and the simplified mechanical parameters of the surrounding rocks. Future research may integrate dynamic simulations and physical tests to explore coupled hydromechanical behaviors under seismic conditions. In addition to the geometric parameter changes of faults having an impact on the mechanical behavior of tunnels, the mechanical parameters of faults also have a significant influence on the mechanical behavior of tunnels. These works will be discussed in future studies.