A Case Study on Tunnel Excavation Stability of Columnar Jointed Rock Masses with Different Dip Angles in the Baihetan Diversion Tunnel

Abstract

Columnar jointed rock mass (CJRM) formed by intact rock divided by special symmetrical columnar joints is a special type of rock with poor mechanical properties, strong anisotropy, and weak self-supporting ability, severely affecting the excavation safety and stability of underground tunnels. In this study, taking the Baihetan hydropower station as the engineering background, CJRM geological numerical models with different dip angles that combined well with the natural CJRM were generated based on the geological statistical parameters of the engineering site and were verified to have high rationality and accuracy. Tunnel excavation and overloading tests were carried out on these numerical models, and the results showed that the stress and displacement distributions after excavation exhibited strong anisotropic characteristics under different dip angles, and the positions where engineering safety problems are most likely to occur are the side walls, which are prone to stress-structure-controlled failure mode. The self-supporting ability at different dip angles after excavation from weak to strong are 45°, 60°, 75°, 90°, 30°, 0°, and 15°. The safety factors assessed by overloading for CJRM with dip angles of 0–90° degrees were 2.5, 2.6, 2.6, 1.8, 2.1, and 2.2, respectively, providing a valuable reference for the construction safety and support measures of CJRM excavation.

1. Introduction

Columnar jointed rock mass (CJRM) is a kind of magmatic rock with a special symmetrical joint network, which is widely considered to be formed by the cooling of erupted magma and is mainly developed in basalt [1,2]. Intact basalt rock has excellent engineering properties. However, due to the existence of columnar joints, CJRM shows strong and complex anisotropy, discontinuity, and inhomogeneity [3,4,5], which poses tremendous challenges for the design and construction of engineering projects [6,7,8]. The complex mechanical characteristics and special symmetrical joints result in extremely weak self-supporting ability after excavation, easily causing engineering accidents including rockfall and the cracking of side walls [9,10]. Currently, there are many engineering projects around the world that involve CJRM, such as the Baihetan, Tongjiazi, and Xiluodu hydropower stations. Therefore, it is crucial to fully understand the anisotropic mechanical properties of CJRM for the stable excavation of tunnels.

The Baihetan hydropower station located in southwest China is of large scale; CJRM basalt is widely presented in the structural components of the hydropower station, such as the tailrace tunnel, diversion tunnel, dam foundation, and underground powerhouse, and its brittle mechanical properties and complex anisotropy pose a severe threat to construction and the long-term stable operation of the project [11]. In the context of this engineering project, many scholars have conducted extensive research on the geometric characteristics, mechanical anisotropy, seepage characteristics, and other aspects of columnar joints, using research methods mainly divided into three categories: in situ testing, laboratory testing, and numerical simulation.

Throughout the entire process of planning, construction, and operation of the Baihetan hydropower station, a large number of in situ tests have been conducted at the project site, providing crucial references for the basalt CJRM engineering project. Shan et al. [12] studied the mechanical properties and failure characteristics of CJRM through rigid bearing plate tests. Fan et al. [13] conducted in situ triaxial tests to investigate the unloading and relaxation characteristics of CJRM and proposed comprehensive reinforcement measures for the dam foundation. Sun et al. [14] carried out an in situ drill test to study the relationship between the excavation damage zone of CJRM surrounding rock and drilling conditions, and proposed a method for determining the excavation damage zone under confined pressure. Lin et al. [15] revealed the typical unloading, cracking, and relaxation characteristics of CJRM during foundation excavation through field investigations and acoustic wave tests.

In terms of laboratory tests, small-scale similar material physical model tests were mainly conducted. Physical model tests have the advantages of low production cost and the ability to simulate multiple working conditions that are not limited by engineering realities, making them an important means of studying CJRM. Lu et al. [16,17] prepared regular hexagonal CJRM specimens with different dip angles and conducted uniaxial, conventional triaxial, and true triaxial compression tests to study the strength and deformation anisotropy characteristics of CJRM, proposing an anisotropic constitutive model for CJRM anisotropy. Que et al. [18,19,20,21,22] proposed a pentagonal cross-section and an irregular cross-section that more realistically simulates the CJRM cross-section, and prepared pentagonal, hexagonal, quadrangular, and irregular cross-section specimens for multi-axial testing, revealing the effects of different cross-sectional shapes on CJRM strength and deformation anisotropy, and proposed a failure strength criterion for CJRM. Niu et al. [23,24] conducted true triaxial tests on a CJRM physical model under hydraulic coupling, revealing CJRM hydraulic characteristics. These laboratory tests and results show the enormous effect of dip angles on mechanical properties and provide important references for the selection of mechanical parameters and strength estimation towards CJRM engineering projects.

Compared to the two research methods mentioned above, numerical simulation has the advantages of repeatability, simulation of various complex conditions, and no need for experimental costs, which provides an effective approach for studying CJRM. Di et al. [25] used 3DEC software to conduct numerical compression tests on different sizes of basalt CJRM specimens under confinement, proposed a characterization of the size of the element, and studied the macroscopic mechanical properties and equivalent strength parameters of columnar jointed rock masses. Feng et al. [26] used numerical methods to simulate the excavation of the Baihetan diversion tunnel and proposed a new support design method that uses rock cracking indicators to suppress the development of surrounding rock cracking. Zhao et al. [27] established a numerical model of the excavation process of a pressure-regulating pool cavity, analyzed the mechanical properties, failure modes, and acoustic emission characteristics of the pressure-regulating pool cavity, and proposed an evaluation method for the stability of large-scale columnar jointed rock masses and interlayer shear weak zones based on the BP neural network.

From a review of the above literature, numerical simulation, in situ tests, and laboratory experiments have provided valuable references for a detailed understanding of the mechanics and failure characteristics of CJRM. However, there are still some shortcomings in studying the potential dangers caused by the weak self-supporting capacity of CJRM after excavation due to the existence of special symmetric joints. These shortcomings mainly include the following aspects: (1) the dip angle is considered to be the most influential factor on the engineering mechanical properties of CJRM [28,29], but in the stability studies of CJRM tunnel excavation, only the typical dip angle of 75° has been studied, and the stability of CJRM tunnel excavation under different dip angles has not been systematically studied; (2) the cross-sectional shape of CJRM has an important impact on the mechanical anisotropy of jointed rock mass, but the current numerical simulations of CJRM tunnel excavation and foundation pit support have not fully utilized geotechnical parameters for generating a numerical model to better combine with engineering conditions; and (3) a safety factor for the CJRM tunnel and cavern excavation under different dip angles still needs to be studied for engineering applications.

In respect of the above issues, in this study, the Baihetan basalt CJRM was taken as the engineering background, the irregular CJRM geometric model reconstruction method was used to generate CJRM numerical models with different dip angles that were well combined with geological statistical parameters to better simulate real geological conditions, and the effectiveness and accuracy of the models were verified. For the CJRM geological numerical models with different dip angles, numerical simulations of tunnel excavation and overloading tests were carried out. The characteristics of stress and displacement after tunnel excavation were studied. The self-supporting abilities of CJRM with different dip angles after excavation were analyzed, and the safety factors of CJRM with different dip angles were assessed.

Numerical simulations of tunnel excavation tests were conducted on the different inclined CJRM geological models to investigate the stress–strain variation rules of CJRM tunnel excavation and analyze the self-supporting capacity of CJRM after excavation for different inclinations. Numerical simulations of tunnel overload tests were conducted on the basis of tunnel excavation simulation tests to calculate the overload safety coefficients of CJRM rocks for different inclinations, providing a valuable reference for CJRM tunnel excavation safety and support design.

2. Geological Overview of the Baihetan Hydropower Station

2.1. General Project Information

The Baihetan hydropower station is located in the Jinsha River Canyon, at the junction of Sichuan Province and Yunnan Province in southwest China, which is a major tributary of the Yangtze River. Connected to the Wudongde hydropower station and adjacent to the Xiluodu hydropower station, the station is situated in an area where the terrain is low in the south and high in the north. On the left bank of the river is the Daliangshan Mountain range at an elevation of 2600 m, while on the right bank is the Yaoshan Mountain range at an elevation of over 3000 m. With a total installed capacity of 14,000 MW and a guaranteed output of 5500 MW, the Baihetan hydropower station is the world’s second-largest hydropower station, with an average annual power generation of 64.095 billion kW·h. The main components of the hydropower station system include arch dams, cofferdams, diversion tunnels, and power generation systems, as shown in Figure 1.

2.2. Engineering Geological Conditions

The terrain on both sides of the hydropower station is mainly composed of steeply inclined slopes. According to the geological survey reports, the rock strata at the dam site belong to the Permian Emeishan Formation (${\mathrm{P}}_2\mathsf{\beta }$) basalt. Columnar joints are mainly developed in several layers, including ${\mathrm{P}}_2{\mathsf{\beta }}_2^2$, ${\mathrm{P}}_2{\mathsf{\beta }}_2^3$, ${\mathrm{P}}_2{\mathsf{\beta }}_3^2$, and ${\mathrm{P}}_2{\mathsf{\beta }}_3^3$. The strike direction of the rock strata is N30°~E50°, as shown in Figure 2 [30]. The columnar basalts are densely developed in ${\mathrm{P}}_2{\mathsf{\beta }}_2^3$ and ${\mathrm{P}}_2{\mathsf{\beta }}_3^3$, where the integrity of the columnar basalts is poor, and the rock quality designation (RQD) value is generally low, showing typical discontinuity and anisotropy, which has a significant adverse effect on the stability of the dam foundation, slopes, and underground caverns of the hydropower station.

Based on the field investigation and geometric characteristics of the drilled rock mass (Figure 3), the CJRM in each rock layer at the dam site of the hydropower station can be classified into three types [31]: The first type of CJRM has slender columns generally between 2 and 3 m long, with an average diameter of 13–25 cm, and a high development density of internal fractures. The rock mass is not completely cut and presents a mosaic structure. The second type of CJRM has column lengths between 0.5 and 2m and a diameter of 25–40 cm, with irregular development between the rock blocks. The third type of CJRM has good integrity, a low joint density, irregular joint development, thick columns between 1.5 and 5 m long, and a diameter of 0.5–2.5 m. This type of rock mass is less affected by weathering and has good quality, and is often not the main focus of engineering research. The first type of rock mass, with poor integrity of the columnar jointed rock mass, is easily damaged after weathering and is an important research object in engineering. Therefore, in this paper, the first type of columnar jointed rock mass was selected as the research object.

2.3. Geological Conditions of the Diversion Tunnels in the Baihetan Station

Based on the engineering survey, the rock strata in which the left bank diversion tunnel is located belong to the monocline strata. The degree of joint development in the ${\mathrm{P}}_2{\mathsf{\beta }}_3$ layer, where the diversion tunnel is located, varies. According to the classification standard, the rock mass in the ${\mathrm{P}}_2{\mathsf{\beta }}_2^2$ layer is classified as the second type of CJRM, while the rock mass in the ${\mathrm{P}}_2{\mathsf{\beta }}_2^3$ layer is classified as the first type of CJRM. The position of the inlet of the first diversion tunnel on the left bank is between 450 and 850 m upstream of the arch dam. The thickness of the overlying rock in the upper part of the first diversion tunnel ranges from 30 to 397 m, and the horizontal depth of the tunnel ranges from 0 to 705 m. The thickness of the overlying rock in the upper part of the second diversion tunnel ranges from 50 to 380 m, and the horizontal depth of the tunnel ranges from 0 to 765 m. The thickness of the overlying rock in the upper part of the third diversion tunnel ranges from 38 to 358 m, and the horizontal depth of the tunnel ranges from 0 to 825 m. The exposed rock strata in the right bank diversion tunnel are ${\mathrm{P}}_2{\mathsf{\beta }}_2^2$, ${\mathrm{P}}_2{\mathsf{\beta }}_2^3$, ${\mathrm{P}}_2{\mathsf{\beta }}_3$, and ${\mathrm{P}}_2{\mathsf{\beta }}_4$, and the degree of rock mass development varies in each layer. According to the classification standard, the rock mass in the ${\mathrm{P}}_2{\mathsf{\beta }}_3^2$ and ${\mathrm{P}}_2{\mathsf{\beta }}_3^3$ layers is classified as the first type of columnar jointed rock mass. The position of the inlet of the right bank diversion tunnel is between 340 and 570 m upstream of the Dazhai Gully. The thickness of the overlying rock in the upper part of the fourth diversion tunnel ranges from 17 to 459 m, and the horizontal depth of the tunnel ranges from 0 to 290 m. The thickness of the overlying rock in the upper part of the fifth diversion tunnel ranges from 14 to 518 m, and the horizontal depth of the tunnel ranges from 0 to 360 m, as shown in Figure 2.

The length of the left bank diversion tunnel passing through the CJRM rock mass is 429–486 m, while the length of the right bank diversion tunnel is 412–455 m. Compared with the right bank diversion tunnel, the length of the first type CJRM rock mass in the rock layer passed through by the left bank diversion tunnel is smaller, indicating that the integrity of the surrounding rock mass of the right bank diversion tunnel is slightly better than that of the left bank diversion tunnel.

Therefore, to accurately understand the mechanical characteristics and provide a basis for engineering construction, this paper selected the left bank diversion tunnel with poor engineering properties of the Baihetan hydropower station as the engineering background, established a geological numerical model of irregular CJRM with different dip angles, and carried out numerical model experiments on tunnel excavation and overloading, providing a valuable reference for the safe excavation of the tunnel and surrounding rock support.

3. Basalt CJRM Geological Numerical Model Establishment

3.1. Construction of CJRM Geometric Model with Geological Parameters

Combined with field survey data, it was found that the CJRM distributed through the diversion tunnel is mainly in the shape of quadrilaterals, pentagons, and hexagons, and they are non-uniformly distributed. Different cross-sections of CJRM show different mechanical characteristics, and the irregular arrangement of rock masses in the diversion tunnel makes their anisotropic mechanical characteristics even more complex. To fully utilize the geological statistical parameters and establish a geometric model that better simulates the CJRM geological features for tunnel excavation simulation, this paper adopted the CJRM geometric model reconstruction method proposed by Zhu et al. [32] to generate the geometric model. The geological statistical geometric parameters of the left bank diversion tunnel [26,33,34] are shown in

Table 1. Geometric parameters of the CJRM geometric model with different dip angles.

Size (m)	Average Prism Diameter (m)	Irregular Factor	Dip Angle	Strike Angle	Transverse Joint Connectivity	Transverse Joint Spacing (m)
20 × 10 × 30	0.21	38.42%	0° 15° 30° 45° 60° 75° 90°	S89°E27°	0	1.5

. Since there are fewer transverse joints compared to the densely distributed columnar jointed network in the Baihetan basalts, which has little effect, and accurate geological statistical parameters are difficult to obtain, the transverse joint connectivity rate was set to 0.

Firstly, a single-random movement Voronoi diagram of uniform seed points [32] was used to generate cross-sections that conform to the average prism diameter and the irregular factor (defined by the variation coefficient of the cross-sectional polygon area) using MATLAB [35], as shown in Figure 4a. Then, the two-dimensional cross-section was stretched into a solid using the graphical software Rhino, and the solid was cut according to a cube with the target geometric size, dip angle, and strike angle, as shown in Figure 4b. Finally, a geological geometric model of CJRM that better simulates irregular CJRM rock masses using geological statistical parameters was generated, as shown in Figure 4c.

3.2. Establishment of Numerical Model

3DEC is a computational analysis program based on the discrete element theory, which describes the mechanical behavior of discrete media. The program divides rock joints and fractures into many discontinuous rock blocks, treating each block as a continuum. This is in line with the CJRM, where the intact rock mass is divided into cylindrical columns and intervening joints, forming a binary medium. Therefore, 3DEC is widely used in CJRM numerical simulation analysis [36]. In the columnar prism-intervening joint binary medium, the intact columns are modeled as deformable bodies and assigned to the Mohr–Coulomb constitutive model. The Coulomb slip joint constitutive model based upon elastic stiffness, frictional, cohesive and tensile strength properties, and dilation characteristics common to rock joints simulating displacement-weakening of the joint by loss of cohesive and tensile strength at the onset of shear or tensile failure can better reflect the sliding failure and tension of joint surfaces, providing a linear representation of joint stiffness and yield limit, which is very suitable for complex rock masses with multiple joint surfaces such as CJRM. Therefore, this joint model [37,38] was used in this paper, and the schematic diagram of the contact surface constitutive model is shown in Figure 5. According to the geological survey report and related numerical simulation studies of CJRM [25,39], the numerical model parameters are listed in

Table 2. Prism parameters of the CJRM geological numerical model.

Bulk Density (kg/m3)	Friction Angle (°)	Cohesion (MPa)	Tensile Strength (MPa)	Elastic Modulus (GPa)	Poisson’s Ratio
2780	56.13	12.4	5.6	60.4	0.21

and

Table 3. Columnar joint parameters of the CJRM geological numerical model.

Normal Stiffness (GPa/m)	Shear Stiffness (GPa/m)	Friction Angle (°)	Cohesion (MPa)	Tensile Strength (MPa)
100.8	52.3	36.5	0.62	0

.

The irregular CJRM geometric models with different dip angles established in Section 3.2 were imported into the 3DEC software to generate geological numerical models of CJRM. The generated numerical models of basalt CJRM with different dip angles are shown in Figure 6.

In the subsequent excavation process, the dimensions of the tunnel are shown in Figure 7a (taking a dip angle of 75° as an example) using a straight-wall circular arch shape with a width of 4 m, a height of 6 m, and a circular radius of 2 m. In order to monitor the deformation release response and stress change trend of the surrounding rock during tunnel excavation, eight monitoring points (P1 on the vault, P2 on the left spandrel, P3 on the right spandrel, P4 on the middle left wall, P5 on the middle right wall, P6 on the left wall foot, P7 on the right wall foot, and P8 on the floor) are set at the cross-section, which is vertical to the axis of the middle tunnel, as shown in Figure 7b. This is to facilitate the subsequent analysis of the stress–strain variation laws at different positions during the excavation and overloading process of tunnels in the CJRM with different dip angles.

3.3. Validation of the Basalt Geological Numerical Model

Model validation is a critical step in numerical simulation to check the rationality and reliability of the model. Comparing the results of on-site engineering tests with those of numerical simulations is the most effective method for model validation. To prove the accuracy of numerical simulations in this study, the in situ rigid bearing plate test conducted by Shi et al. [40] was used to verify the established numerical model. This test used a rigid bearing plate with a length of 50 cm, width of 50 cm, and thickness of 6 cm to establish the relationship between the pressure on the bottom surface of the bearing plate and the deformation of the rock mass below the bearing plate to obtain the deformation modulus of the rock mass. The pressure on the rigid bearing plate was increased from 0 MPa to 10 MPa at a regular interval of 2 MPa in the diversion tunnel, and the displacement was measured using four dial indicators. To simulate the test exactly after the tunnel excavation was completed, the numerical rigid bearing plate (Figure 8) was generated with the same pressure following the same test scheme, and the displacement of the numerical rigid bearing plate was observed to calculate the deformation modulus of the numerical model.

The in situ rigid bearing plate experimental results and numerical simulation results are shown in

Table 4. Comparison of experimental data and numerical results.

Load Steps	0	1	2	3	4	5
Applied stress	0	2.0	4.0	6.0	8.0	10.0
Actual experiment data/mm	0.068	0.143	0.212	0.292	0.361	0.068
Numerical results/mm	0.077	0.135	0.236	0.279	0.370	0.077
Relative error/%	13.56	−5.72	11.31	−4.56	2.75	13.56

and Figure 9. The relative errors between the numerical simulation and the actual test range from 2.75% to 13.56%, which is deemed highly consistent. Therefore, the irregular basalt CJRM geological numerical model generated based on geological parameters is believed to be capable of simulating the mechanical characteristics of real basalt CJRM rock mass impeccably.

4. Numerical Simulation of Tunnel Excavation in CJRM with Different Dip Angles and Discussion

4.1. Simulation Scheme of Tunnel Excavation

In this numerical experiment, the seven irregular CJRM geological numerical models with different dip angles generated in the previous section were used and calculated in sequence using the 3DEC numerical software, with their geometric parameters shown in Table 1 and their mechanical parameters in Table 2 and Table 3.

Firstly, the boundary conditions were applied to the model by fixing the bottom of the model and applying initial geo-stress, and the initial geo-stress was referenced to the initial geo-stress on the left bank of the Baihetan hydropower station. The relationship between the initial geo-stress (MPa) and the burial depth h (m) is shown in Equation (1):

$$\left\{\begin{array}{l}{\sigma }_1=0.0304h+10.5\\ {\sigma }_2=0.0225h+7.6 ,\\ {\sigma }_3=0.028h\end{array}\right.$$

(1)

After the initial stresses were applied and balanced, the full-length excavation of the tunnel was carried out, and the model was calculated until it reached equilibrium and the unbalanced forces converged to 1e-5, achieving stability after the completion of tunnel excavation. The numerical tunnel excavation test was thus completed.

4.2. Analysis of Stress Characteristics during Excavation of Tunnels in CJRM with Different Dip Angles

Figure 10 and Figure 11 are XX-stress and YY-stress distributions of excavated geological numerical models with different dip angles, respectively. In order to facilitate the comparison of stress distribution contour maps of different dip angles, the range and interval were set to be the same. The figures show that tunnel excavation causes different types and sizes of stress redistribution in the surrounding rock in different directions. At the same time, with the change of the dip angle, strong asymmetry along the tunnel axis can be observed, and symmetry is only present when the dip angle is 0°and 90°. This reflects and corresponds to the strong anisotropy characteristics of the CJRM.

Through the analysis of the horizontal stress distribution under different dip angles, it can be found that the range of disturbance caused by the excavation of the rock mass is about 6 m in the horizontal direction, and the disturbance is most significant within 2 m around the tunnel. Moreover, with the increase in the dip angle, different degrees of asymmetry are presented. As the dip angle increases, the disturbance range on the right side of the tunnel increases continuously, while the disturbance range on the left side of the tunnel shows a trend of first increasing and then decreasing. It is worth noting that, except for the special angles of 0°and 90°, the disturbance range on the right side of the tunnel is larger than that on the left side. Combining the stress situation of the eight monitoring points set in advance, it can be found that the locations where the horizontal stress is greater are the side walls. Under each dip angle, the horizontal stress on the right wall is greater than that on the left wall, and the stress range on the right side under each dip angle is between 5.72 MPa and 8.53 MPa, with the average stress on the right wall being the highest at a dip angle of 45°. The fact that the stress is greater on the left and right walls means that this part is prone to stress-induced failure during the tunnel excavation process. Considering the special structure of the columnar joints, the rock mass in this part is susceptible to failure, and its failure mode is the stress-structure-controlled failure mode, which is also confirmed by the current physical model test results.

Through the analysis of the vertical stress contour maps under different dip angles, it can be found that, similar to the horizontal stress contour maps, strong asymmetry is exhibited, showing significant anisotropy characteristics of CJRM. Due to the excavation of the tunnel, the surrounding rock experienced varying degrees of stress relaxation at the face, resulting in different levels of tensile stress on the vault, floor, and side walls. Under different dip angles, the range of tensile stress on the floor ranges from 2.81 MPa to 4.46 MPa, with the largest average range occurring at 45°. However, it still does not exceed the tensile strength of the columnar basalt. It is worth noting that the possibility of floor heave deformation failure still exists due to overloading and disturbance caused by the progress of surrounding engineering work. In Figure 11, stress concentration occurs at the vault in all dip angles of the CJRM, with the largest stress range occurring at 60 °. In summary, it is necessary to provide timely support for the side walls, floor, and vault during excavation to prevent stress-induced failure and sliding failure of the columns.

4.3. Analysis of Displacement Characteristics during Excavation of Tunnels in CJRM with Different Dip Angles

Figure 12 and Figure 13 are the displacement distribution in the X direction and Y direction of excavated geological numerical models with dip angles, respectively. In order to facilitate the comparison of displacement distribution contour maps of different dip angles, the range and interval were also set to be the same. Similar to the features of stress distribution contour maps, the displacement distribution exhibits a strong non-symmetrical pattern with varying dip angles, with only two special angles (0° and 90°) presenting a symmetrical distribution along the centerline of the tunnel, reflecting the anisotropic characteristics of the strength and deformation of the CJRM.

In terms of vertical displacement, the locations with larger displacements are concentrated at the vault, followed by the wall feet at all dip angles, which corresponds to the stress concentration at the vault and the tensile stress borne by the wall feet in the vertical stress contour map. The settlement range at the vault in numerical models with various dip angles is 10.92 mm to 18.66 mm, and the maximum settlement occurs in the CJRM with a dip angle of 60°, which is 18.66 mm. The bulging range at the floor at various angles is 7.08 mm to 12.80 mm, and the maximum bulging occurs in the CJRM with an angle of 60°, which is 12.80 mm. Although the deformation value is relatively low compared to that at the side walls, the tunnel column fracture and the bottom bulging phenomenon of the CJRM still need to be given attention, considering the overloaded stress caused by the presence of the diversion tunnel intersection and adjacent areas in actual engineering processes.

Based on the eight monitoring points (P1–P8) set in advance, the following table (

Table 5. Displacement results of monitoring points (P1–P8).

Monitoring Point	P1	P2	P3	P4	P5	P6	P7	P8
Monitored Position	Vault	Left Spandrel	Right Spandrel	Middle Left Wall	Middle Right Wall	Left Wall Foot	Right Wall Foot	Floor
Displacement at the dip angle of 0°	12.24	21.60	23.60	19.52	21.35	9.46	9.36	8.85
Displacement at the dip angle of 15°	12.04	21.31	22.74	19.95	21.83	10.43	8.47	8.78
Displacement at the dip angle of 30°	11.14	21.50	23.57	19.15	22.39	7.65	9.92	7.65
Displacement at the dip angle of 45°	16.90	30.31	32.13	28.82	33.92	13.79	14.47	12.78
Displacement at the dip angle of 60°	18.66	26.27	29.33	27.10	30.59	8.91	13.68	12.80
Displacement at the dip angle of 75°	11.75	24.15	28.91	22.52	25.82	8.67	10.97	8.40
Displacement at the dip angle of 90°	10.92	24.02	26.67	23.29	25.65	9.46	9.55	7.08

) is listed.

Based on the above analysis and in combination with Table 5, it can be concluded that during the excavation of the CJRM tunnel at various dip angles, the deformation of each monitoring point from large to small is as follows: side walls, spandrels, vault, wall feet, floor. The maximum deformation value appears at the right wall followed by the right spandrel, which means that the most common engineering safety accident during tunnel excavation is the failure of the side wall pillar and the collapse of the surrounding rock, which corresponds to the most common engineering problems encountered in the construction of the Baihetan hydropower station diversion tunnel [41], as shown in Figure 14.

The failure mode at the intersection of the two side walls and the circular section is the typical stress-structure-controlled mode, which is a more complex and severe failure mode of a large-scale cavity. It is greatly affected by mechanical anisotropy and geometric anisotropy caused by different dip angles of CJRM. The magnitude of deformation at the most prone position to failure also reflects the weakening effect of different dip angles on the strength level of the surrounding rock and the self-supporting ability of the surrounding rock after excavation. The weakening effect of the surrounding rock strength level from strong to weak at different dip angles is as follows: 45°, 60°, 75°, 90°, 30°, 0°, and 15°, which is also correspondent to the self-supporting ability from weak to strong at different dip angles. Clarifying the weakening effect of different dip angles on the strength level of the surrounding rock will provide an important basis for the safe production and construction of tunnel excavation and support in the CJRM.

5. Numerical Simulation of Tunnel Overloading after Excavation in CJRM with Different Dip Angles and Discussion

5.1. Simulation Scheme of Tunnel Overloading after Excavation

In the tunnel overloading numerical test, calculation parameters and the constitutive relationship of the geological numerical CJRM model with different dip angles were the same as those in the numerical tunnel excavation test. Like the tunnel excavation test, the model boundary was first fixed, and the initial geo-stress was applied as shown in Equation (1). After the initial ground stress was balanced, the entire length of the tunnel was excavated once, and the model was calculated to reach equilibrium until the unbalanced forces converged and the stability of the excavated chamber was achieved. These steps were the same as those in the numerical model test plan for tunnel excavation. After stability was reached, the ground stress was increased by increments of 0.1 times the initial geo-stress, and the model was recalculated for equilibrium every time the geo-stress was increased. When the unbalanced forces converged and the model was stable, the geo-stress was increased again by 0.1 times the initial geo-stress until a significant increase in displacement was recorded on the monitoring positions, which also indicated the end of the tunnel overloading numerical test. The equation for setting the geo-stress is shown in the following Equation (2):

$$\left\{\begin{array}{l}{\sigma }_1=(1+n)(0.0304h+10.5)\\ {\sigma }_2=(1+n)(0.0225h+7.6) ,\\ {\sigma }_3=(1+n)(0.028h)\end{array}\right.$$

(2)

where n is the increment overloading step of the geo-stress, with an interval of 0.1, and the values are 0, 0.1, 0.2, 0.3, and so on.

5.2. Safety Factor of CJRM with Different Dip Angles Assessed by Overloading

In the research of deep underground tunnels and other underground engineering projects using the geological model test method, it is common practice to gradually increase the geo-stress after completion of the excavation until instability and failure occur on the structure of the underground engineering buildings. The ratio of increased geo-stress ${\sigma }_{\mathrm{overload}}$ at which local damage occurs in the underground engineering structure to the initial geo-stress ${\sigma }_{\mathrm{initial}}$ is defined as the overloading safety factor $K\mathrm{s}$ [42,43], as shown in Equation (3):

$$K\mathrm{s}=\frac{{\sigma }_{\mathrm{overload}}}{{\sigma }_{\mathrm{initial}}},$$

(3)

For CJRM engineering projects, currently, there is no unified safety factor, especially with regard to the different dip angles. The overloading assessment method of the safety factor $K\mathrm{s}$ provides a good approach and was used as the safety factor of CJRM in this paper [44,45]. Taking the overloading simulation of the CJRM geological numerical model with a dip angle of 45° after excavation, for example, the displacement curves of each monitoring point during the numerical overloading test are shown in Figure 15. When the ratio of geo-stress to the initial geo-stress is between 1.0 and 1.8, the displacement of each measuring point increases steadily without obvious displacement surges. This indicates that the overloading geo-stress is still within the self-supporting capacity range of the CJRM with a dip angle of 45°. However, when the ratio of geo-stress to the initial geo-stress reaches 1.9, there is a significant increase in displacement in the monitoring points of the left and right walls, as well as the right and left spandrels, indicating that local damage has occurred in these areas. Therefore, the overloading safety factor $K\mathrm{s}$ of the CJRM with a dip angle of 45° is determined to be 1.8. Using the same calculation method, the safety factors for the remaining dip angles were computed; the safety factors for each dip angle are shown in

Table 6. Safety factors of the CJRM with different dip angles.

The Dip Angle of CJRM	0°	15°	30°	45°	60°	75°	90°
$\mathrm{Safety} \mathrm{factor} K\mathrm{s}$	2.5	2.6	2.6	1.8	2.1	2.2	2.3

.

Through Table 6, it can be observed that the overloading safety factor of CJRM increases in the following order of dip angles: 45°, 60°, 75°, 90°, 0°, 15°. This is consistent with the self-supporting ability obtained from the excavation numerical test. The smaller the overload safety factor is, the weaker the self-supporting ability of CJRM is, and the greater the weakening of the rock strength level due to the presence of columnar joints. A dip angle of 45° is the most dangerous dip angle during the construction of the CJRM engineering projects, and more attention is needed when encountering this dip angle. The safety factor of CJRM with different dip angles assessed by overloading will provide a valuable reference for the selection of excavation and support measures for CJRM tunnels at different dip angles.

6. Summary and Conclusions

In this paper, the excavation of the Baihetan hydropower station diversion tunnel was taken as the engineering background. By using the irregular CJRM geometric model reconstruction method and fully combining this with geological statistical parameters, the geological numerical models of basalt CJRM with different dip angles were established. Based on these numerical models, tests on the numerical simulation of tunnel excavation and tunnel overloading in the CJRM with different dip angles were carried out, and the stress and displacement characteristics during tunnel excavation were analyzed. The self-supporting abilities of CJRM with different dip angles after excavation were studied, and the safety factors of CJRM with different dip angles were calculated.

(1) To better establish a CJRM numerical model that is applicable to engineering practice, the irregular CJRM geometric model reconstruction method proposed by Zhu et al. [32] was adopted to establish irregular CJRM geological numerical models with good integration with engineering practice. In order to validate the model, a rigid bearing plate test was carried out, and the results were compared with actual engineering tests, which showed that the model has great rationality and accuracy. The numerical model constructed in this paper can be adjusted accordingly and applied to the deformation and failure theory research and engineering application of related jointed rock masses such as layered rock masses, demonstrating strong practicality.

(2) In the numerical simulation of tunnel excavation, the stress and displacement contour maps after tunnel excavation exhibit strong asymmetry with respect to dip angles, indicating strong strength and deformation anisotropy. The positions where large deformations occurred and engineering accidents are prone to happen in descending order are: side walls, spandrels, vault, wall feet, and floor. The sidewalls are prone to stress-structure-controlled failure mode, which becomes the most likely position for engineering accidents. The CJRM with a dip angle of 45° shows the largest tunnel deformation with a maximum deformation of 33.92 mm occurring in the right wall. For the CJRM with different dip angles, the self-supporting ability at different dip angles after excavation from weak to strong are 45°, 60°, 75°, 90°, 30°, 0°, and 15°. This provides an important reference for the safety of CJRM tunnel excavation and tunnel operation.

(3) In the numerical simulation of tunnel overloading after excavation, the overloading safety in deep surrounding rock analysis method was adopted to assess the safety factor of CJRM with different dip angles. The smaller the overload safety factor is, the weaker the self-supporting ability of CJRM is, and the greater the weakening of the rock strength level due to the presence of columnar joints. By calculating the safety factor of the CJRM with dip angles of 0°, 15°, 30°, 45°, 60°, and 75°, it was found that the safety factors were 2.5, 2.6, 2.6, 1.8, 2.1, and 2.2, respectively. The safety factors are consistent with the weakening effect of different inclination angles on the strength level of the CJRM surrounding rock, providing valuable engineering references for CJRM tunnel excavation construction and support.