Stability Analysis of the Surrounding Rock-Lining Structure in Deep-Buried Hydraulic Tunnels Having Seepage Effect

Abstract

To clarify the factors affecting the stability of deep-buried hydraulic tunnels containing pore water, the elastoplastic theory and the Mogi-Coulomb strength criterion were used to derive the analytical solutions of stress on the surrounding rock-lining structure, tunnel wall displacement, and plastic zone radius in surrounding rock under different operating conditions. During this process, the seepage effect and surrounding rock-lining interaction were considered. The influencing rules of seepage action, intermediate principal stress coefficient, lining permeability coefficient, and lining thickness on the stability of the surrounding rock-lining structure were investigated in depth. The results show that the seepage effect significantly changed the stress distributions in the lining structure and surrounding rock, reduced the bearing reaction force, and lowered the tunnel stability. The bearing reaction force was decreased considerably from the intermediate principal stress, and the plastic zone radius in the surrounding rock and the tunnel wall displacement was obviously reduced. Moreover, the bearing reaction force was reduced, and the plastic zone radius in the surrounding rock and the tunnel wall displacement was increased with the decrease of the lining permeability coefficient. With increasing the lining thickness, the bearing reaction force was enhanced, and an apparent restriction on the development of plastic zone in the surrounding rock appeared at the beginning, but the restriction effect weakened subsequently. This research can theoretically provide references for analyzing the stability of hydraulic tunnels containing pore water.

1. Introduction

With the development of the economy and society, many circular hydraulic tunnels have been constructed in lofty mountains and steep hills around the world. After excavating and lining tunnels, the surrounding rock will experience multiple stress redistributions under the seepage, stress-coupled action, and the surrounding rock-lining interaction [1,2]. It is a problem that cannot be ignored in tunnel construction. During the solution based on classical mechanics of elasticity, the stress solution of surrounding rock under uniformly distributed pressure can be obtained by treating rock mass as a pure elastomer [3,4]. However, the surrounding rock undergoes changes from the original elastic state to the elastic-plastic state after tunnel excavation [5,6,7]. Therefore, the rock mass should be considered an elastic-plastic body in analysis.

Several scholars have mainly employed theoretical derivation from the last century to perform elastic/plastic analysis on tunnel surrounding rocks. For example, Reza M regarded gravity load as the radial force exerted on the ground medium and derived a closed-form analytical solution to the deep-buried tunnel in the elastic-plastic rock mass by using the Mohr-Coulomb (M-C) criterion and the potential plastic function [8]. Based on the D-P criterion, Jiang A N et al. established a stress-seepage-damage model of surrounding rock in water-rich tunnel areas and the multi-field coupled solution iteration method [9]. Zareifard M R et al. carried out the elastic-brittle-plastic analysis of circular tunnels in infinite H-B media considering steady-state seepage force according to the generalized effective stress law [10]. Zhang B Q et al. treated the seepage force generated in non-Darcy flow as the body volume and added it to the stress field to derive the elastic-plastic analytical solution based on the unified yield criterion [11]. Wang L et al. introduced a dimensionless method to improve the generalized Hoek–Brown (H-B) or M-C criterion considering the seepage effect and concluded the stress/strain incremental calculation method for the elastic-plastic zone of surrounding rock [12]. Du J M et al. considered the interaction between the surrounding rock and the lining structure. They proposed a mechanical model of a circular lined tunnel with variable mechanical properties under the action of hydrostatic pressure and the surface pressure in the lining structure [13]. Zareifard M R et al. considered the stable-state flow of seepage force and the fluid mechanical pressure between adjacent regions and designed an analytical method to calculate the seepage stress and displacement in the underwater lining circular pressure tunnel [14]. Sun X M et al. employed 3 Dimension Distinct Element Code (3D-EC) and simulated the variation characteristics of the tunnel surrounding rock and the wall of the well before and after anchoring support [15]. Yang F J et al. validated the reasonability of on-site reinforcement schemes by comparing field monitoring data and numerical simulation results and evaluated the safety of the lining structure at run time [16,17]. Han J H et al. established a pore water-porous rock-well wall interaction model, examined the volume change under pore water pressure through ANSYS simulation, and revealed the mechanisms of shaft lining fracture induced by high-pressurized pore water [18].

In this study, to clarify the factors affecting the stability of deep-buried hydraulic tunnels containing pore water, the elastoplastic theory and Mogi-Coulomb strength criterion were used to derive the analytical solutions of stress on the surrounding rock-lining structure, tunnel wall displacement, and plastic zone radius in surrounding rock under different operating conditions. The seepage effect and the surrounding rock-lining interaction were considered during this process. The permeability coefficients of lining under different operating conditions were calculated and analyzed based on a hydraulic tunnel in Xinjiang. Finally, we also clarified the effects of lining thickness on the hydraulic seepage stress in the surrounding rock-lining structure, the bearing reaction force, the plastic zone radius in the surrounding rock, and the tunnel wall displacement. The research results have an essential reference and engineering significance for the stability analysis of tunnel surrounding rock and the design of lining structure for deep-buried hydraulic tunnels having seepage effect.

2. Theory and Methodology

2.1. Mechanical Model of Tunnel

For the convenience of calculation, the following assumptions are made. (1) The water-bearing rocks are regarded as uniformly isotropic two-phase media that satisfy the Darcy law. (2) The lining structure is regarded as the elastomer. (3) Since the size of the tunnel cross-section is far smaller than the tunnel length, it is simplified as the plane strain problem for analysis. (4) The pressure stress is positive.

Figure 1 shows the mechanical model of a circular tunnel during the construction period. The figure shows that the surrounding rock-lining structure is divided into lining zone, plastic zone, and elastic zone. The rocks in the elastic zone are complete rocks unaffected by tunnel excavation. However, under seepage and crustal stress, the rock stress near the excavation face exceeds the ultimate rock strength and damages the rock mass, forming the elastic zone. To ensure both stability and operating safety in the tunnel, the lining structure should be applied to form the lining zone.

In Figure 1, r represents the distance between the tunnel center and the infinity, r₀ represents the net tunnel radius after lining, r₁ represents the tunnel excavating radius, r_p represents the length from the center of the tunnel to the external boundary of the plastic zone, p₀ represents the crustal pressure. It is because underground water in rock pores forms the seepage field in the rock, and the region with a radius of over R is the original seepage field unaffected by tunnel excavation. P_i denotes the hydraulic pressure outside the original seepage field, and h₀ represents the underground water head at infinity.

2.2. Mogi-Coulomb (MO) Strength Criterion

Previous research showed that the intermediate principal stress could realize the full strength potential of rocklike materials, resulting in much closer calculation results to actual values. However, when performing research on surrounding rock in deep-buried tunnels, compared with the M-C criterion that does not consider the intermediate principal stress, engineers are inclined to use the unified strength theory, or the Mogi-Coulomb strength criterion considers intermediate principal stress, for solution [19,20]. Since there have been a lot of theoretical derivation results based on the unified strength theory, this study uses the MO strength criterion for the solution.

Mogi et al. conducted a great deal of true tri-axial experiments on rocks and concluded the general formula of Mogi empirical strength [21]:

$$\left\{\begin{array}{l}{\tau }_{\mathit{oct}}=\frac{1}{3}\sqrt{{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2}\\ {\sigma }_{\mathrm{m}}=\frac{{\sigma }_1+{\sigma }_3}{2}\end{array}\right.$$

(1)

where τ_oct denotes the shear stress, MPa; σ₁ denotes the maximum principal stress, MPa; σ₂ denotes the intermediate principal stress, MPa; σ₃ denotes the minimum principal stress, MPa; σ_m denotes the mean principal stress, MPa.

In combination with the criterion proposed by Mogi and the Mohr-Coulomb yield criterion, Al-Ajmi A M et al. established the MO strength criterion [22,23]:

$${\tau }_{\mathit{oct}}=n+k{\sigma }_{\mathrm{m}}$$

(2)

where n and k are MO strength criterion parameters, $n=2\sqrt{2}c\cos \phi /3$ and $k=2\sqrt{2}\sin \phi /3$; c denotes the rock cohesive force, MPa; $\phi$ denotes the internal friction angle of rocks (°).

During the calculation of the strength of surrounding rocks, the intermediate principal stress coefficient b, which is used for characterizing the relative magnitudes of σ₁, σ₂ and σ₃, can be written as [24]:

$$b=\frac{{\sigma }_2-{\sigma }_3}{{\sigma }_1-{\sigma }_3}$$

(3)

Substituting Equations (1) and (3) into Equation (2), the MO strength criterion under the plane strain state can be written as:

$${\sigma }_1=A{\sigma }_3+B$$

(4)

where $A=\frac{2\sqrt{2(b^2-b+1)}+3k}{2\sqrt{2(b^2-b+1)}-3k}$; and $B=\frac{6n}{2\sqrt{2(b^2-b+1)-3k}}$.

When analyzing the stress distribution of the elasto-plastic zone in the tunnel surrounding rock during the construction period under the plane strain condition, if the lateral stress coefficient λ = 1, the tangential stress σ_θ is the maximum principal stress, while the radial stress σ_r is the minimum principal stress. Since the axial stresses σ_z, σ_θ, and σ_r are mutually orthogonal, σ_z can be treated as the intermediate principal stress, i.e., σ_z = σ₂ [25]. Then, Equation (4) can be rewritten as:

$${\sigma }_{\theta }=A{\sigma }_r+B$$

(5)

2.3. Analysis of the Surrounding Rock-Lining Interaction

As shown in Figure 2, the lining structure is generally processed with concretes after excavating the tunnel to avoid great deformation in the excavation face. Under the action of p₀, stress will be re-distributed in the rock at a certain range from the excavation face. At the surrounding rock internal boundary, uniformly distributed pressure q is generated and applied to the lining structure, and uniformly distributed pressure p is generated at the external boundary and applied to the surrounding rock. Accordingly, q and p are a pair of interaction forces. When considering the effects of seepage and p₀, due to the lining permeability coefficient being far lower than the surrounding rock, p decreases gradually with the increase of the hydraulic pressure at the surrounding rock-lining interface until the surrounding rock-lining structure reaches dynamic stability. It will result in the expansion of the plastic zone in the surrounding rock. Under the combined action of crustal stress and seepage, the surrounding rock and lining structure maintain a synergistic deformation. Based on the theory of deformation compatibility, the surrounding rock radial displacement equals that of the lining structure at the interface.

Among various parameters that affect the bearing reaction force, the intermediate principal stress coefficient, seepage action, lining permeability coefficient, and lining thickness were investigated in depth. These factors can directly or indirectly affect p, and the variation of p will affect the stress field in the surrounding rock, thereby affecting the distribution of plastic zone in the surrounding rock.

2.4. Calculation of the Seepage Field

Due to the permeability of the surrounding rock and lining structure, after the tunnel excavation and lining for a period, a stable seepage field is formed in the surrounding rock and lining structure under external water pressure. For the axial-symmetry planar stable seepage field, the differential equation can be written as [26,27,28]:

$$\frac{d^2{P}_w}{d{r}^2}+\frac{1}{r}\frac{d{P}_w}{dr}=0$$

(6)

where P_w denotes the seepage pressure (MPa).

Substituting the boundary conditions ${\left(P_w=0\right)}_{r=r_0}$, ${\left(P_w=P_{w1}\right)}_{r=r_1}$, and ${\left(P_w=P_i\right)}_{r=R}$ into Equation (6), we can obtain:

$$\left\{\begin{array}{ll}{P}_w=\frac{P_{w1}\ln \frac{r}{r_0}}{\ln \frac{r_1}{r_0}}& \left(r_0\le r\le r_1\right)\\ P_w=\frac{P_i\ln \frac{r}{r_1}+P_{w1}\ln \frac{R}{r}}{\ln \frac{R}{r_1}}& \left(r_1\le r\le R\right)\end{array}\right.$$

(7)

where P_w1 denotes the seepage pressure at the surrounding rock-lining interface (Mpa).

The seepage velocities in the lining structure and surrounding rock, denoted as v_c and v_d, can be written as [26]:

$$\left\{\begin{array}{l}{v}_c=-\frac{k_{c}d{P}_w}{{\gamma }_{w}dr}\left(r_0\le r\le r_1\right)\\ v_d=-\frac{k_{d}d{P}_w}{{\gamma }_{w}dr}\left(r_1\le r\le R\right)\end{array}\right.$$

(8)

where k_c and k_d are the permeability coefficients of the lining structure and the surrounding rock, respectively, m/s; γ_w denotes the bulk density of water (kN/m³).

Combining Equations (7) and (8) and ${\left(v_c=v_d\right)}_{r=r_1}$ together, the seepage pressure on the surrounding rock-lining structure during the construction period can be written as:

$$\left\{\begin{array}{l}{P}_w=P_i\frac{k_d\ln \frac{r}{r_0}}{k_c\ln \frac{R}{r_1}+k_d\ln \frac{r_1}{r_0}}\left(r_0\le r\le r_1\right)\\ P_w=P_i\frac{k_c\ln \frac{r}{r_1}+k_d\ln \frac{r_1}{r_0}}{k_c\ln \frac{R}{r_1}+k_d\ln \frac{r_1}{r_0}}\left(r_1\le r\le R\right)\end{array}\right.$$

(9)

3. Theoretical Analysis of the Surrounding Rock-Lining Structure

3.1. Mechanical Analysis of the Surrounding Rock-Lining Structure without the Consideration of Seepage

For the axisymmetric problem, if ignoring the effect of volume force, the plastic zone in the surrounding rock satisfies the following differential equation of equilibrium [29]:

$$\frac{d{\sigma }_r}{dr}+\frac{{\sigma }_r-{\sigma }_{\theta }}{r}=0$$

(10)

Combining Equations (5) and (10) and ${({{\sigma }_r}^p=P_1)}_{{\mathrm{r}=\mathrm{r}}_1}$, the ${{\sigma }_r}^p$ and ${{\sigma }_{\theta }}^p$ of the plastic zone in surrounding rock can be written as:

$$\left\{\begin{array}{l}{{\sigma }_r}^p=\frac{B}{1-A}+\left(P_1-\frac{B}{1-A}\right){\left(\frac{r_1}{r}\right)}^{1-A}\\ {{\sigma }_{\theta }}^p=\frac{AB}{1-A}+A\left(P_1-\frac{B}{1-A}\right){\left(\frac{r_1}{r}\right)}^{1-A}+B\end{array}\right.$$

(11)

Substituting ${({{\sigma }_r}^e=p_0)}_{r\to \infty }$, ${({{\sigma }_r}^p={{\sigma }_r}^e)}_{r=r_p}$, and ${({{\sigma }_{\theta }}^p={{\sigma }_{\theta }}^e)}_{r=r_p}$ into Equation (11), the ${{\sigma }_r}^e$ and ${{\sigma }_{\theta }}^e$ of the elastic zone in surrounding rock can be written as:

$$\left\{\begin{array}{l}{{\sigma }_r}^e\\ {{\sigma }_{\theta }}^e\end{array}\right.=p_0\pm \left[\frac{B}{1-A}+\left(P_1-\frac{B}{1-A}\right){\left(\frac{r_1}{r_p}\right)}^{1-A}-p_0\right]{\left(\frac{r_p}{r}\right)}^2$$

(12)

Combining Equations (11) and (12), and ${({{\sigma }_{\theta }}^p={{\sigma }_{\theta }}^e)}_{r=r_p}$, the plastic zone radius in surrounding rock can be derived as:

$$r_p=r_1{\left\{\frac{2\left[p_0-B/\left(1-A\right)\right]}{\left(A+1\right)\left[P_1-B/\left(1-A\right)\right]}\right\}}^{\frac{1}{A-1}}$$

(13)

The purely elastic lining structure can be regarded as a cylinder problem with finite thickness under uniformly distributed pressure. On that basis, the lining structure is only under uniformly distributed external pressure. This external pressure and P₁ constitute a pair of interactions. Therefore, the displacement and stress of the lining structure can be written as:

$$\left\{\begin{array}{l}{{\sigma }_r}^l=\frac{{r_1}^2(r^2-{r_0}^2)}{({r_1}^2-{r_0}^2)r^2}{P}_1\\ {{\sigma }_{\theta }}^l=\frac{{r_1}^2(r^2+{r_0}^2)}{({r_1}^2-{r_0}^2)r^2}{P}_1\\ u_l=\frac{\left(1+{\mu }_c\right){r_1}^2}{E_c\left({r_1}^2-{r_0}^2\right)r}\left[\left(1-2{\mu }_c\right)r^2+{r_0}^2\right]P_1\end{array}\right.$$

(14)

Assuming that the plastic zone volume in the surrounding rock remains unchanged, by combining Equations (12) and (13), and the thick-walled cylinder of finite length theory, the internal radius displacement of surrounding rock (denoted as u_d) can be derived as:

$$u_d=\frac{\sin \phi }{2G{r}_1}\left(p_0+c\cot \phi \right){r_p}^2$$

(15)

Combining Equations (14) and (15), the bearing reaction force P₁ of the lining structure to surrounding rock without considering the seepage effect can be written as:

$$P_1=\frac{2\left[p_0-B/\left(1-A\right)\right]}{A+1}{\left\{\frac{2G\left(1+{\mu }_c\right)\left[\left(1-2{\mu }_c\right){r_1}^2+{r_0}^2\right]P_1}{E_c\sin \phi \left(p_0+c\cot \phi \right)\left({r_1}^2-{r_0}^2\right)}\right\}}^{\frac{1-A}{2}}+\frac{B}{1-A}$$

(16)

3.2. Mechanical Analysis of the Surrounding Rock-lining Structure with the Consideration of Seepage

Taking the seepage into account, the plastic zone in the surrounding rock satisfies the following differential equation of equilibrium [30]:

$$\frac{d{{\sigma }_r}^w}{dr}+\beta \frac{d{P}_w}{dr}+\frac{{{\sigma }_r}^w-{{\sigma }_{\theta }}^w}{r}=0$$

(17)

where β denotes the area coefficient under seepage pressure (generally, β = 1 [31]). The superscript w represents that the surrounding rock-lining structure is under seepage effect.

Combining Equations (5), (9) and (17), the stress of the elasto-plastic zone in the surrounding rock can be written as:

$$\left\{\begin{array}{l}{{\sigma }_r}^{wp}=A_2+\left(P_2-A_2\right){\left(r/r_1\right)}^{A-1}\\ {{\sigma }_{\theta }}^{wp}=A{A}_2+A\left(P_2-A_2\right){\left(r/r_1\right)}^{A-1}+B\end{array}\right.$$

(18)

$$\left\{\begin{array}{l}{{\sigma }_r}^{we}\\ {{\sigma }_{\theta }}^{we}\end{array}\right.=p_0\pm \left[A_2+\left(P_2-A_2\right){\left(\frac{r_p}{r_1}\right)}^{A-1}-p_0\right]{\left(\frac{r_p}{r}\right)}^2$$

(19)

where $A_1=k_c\ln \frac{R}{r_1}+k_d\ln \frac{r_1}{r_0}$ and $A_2=\frac{\left(B-k_c{p}_i/A_1\right)}{\left(1-A\right)}$.

Combining Equations (18) and (19), the following equation can be derived:

$${r_p}^w=r_1{\left[\frac{2p_0-A_2-A{A}_2-B}{\left(A+1\right)\left(P_2-A_2\right)}\right]}^{\frac{1}{A-1}}$$

(20)

According to the thick-walled cylinder of finite length theory, the stress of the lining structure is as follows:

$$\left\{\begin{array}{l}{{\sigma }_r}^{lw}=\frac{{r_1}^2(r^2-{r_0}^2)}{({r_1}^2-{r_0}^2)r^2}{P}_2\\ {{\sigma }_{\theta }}^{lw}=\frac{{r_1}^2(r^2+{r_0}^2)}{({r_1}^2-{r_0}^2)r^2}{P}_2\end{array}\right.$$

(21)

According to the thick-walled cylinder of finite length theory and the physical equation and geometrical equation under plane strain conditions, the displacement of the lining structure is as follows:

$${u_l}^w=\frac{\left(1+{\mu }_c\right)r_1}{E_c}\left[\frac{\left(1-2{\mu }_c\right)k_d{p}_i}{2A_1}+\frac{k_d{p}_i{r_0}^2\ln \left(r_1/r_0\right)}{\left({r_1}^2-{r_0}^2\right)A_1}+\frac{{r_0}^2+\left(1-2{\mu }_c\right){r_1}^2}{{r_1}^2-{r_0}^2}{P}_2\right]$$

(22)

Assuming the plastic zone volume in the surrounding rock remains unchanged, by combining Equations (19) and (20), and the thick-walled cylinder of finite length theory, the internal radius displacement in surrounding rock is as follows:

$${u_d}^w=\frac{\sin \phi }{2G{r}_1}\left(p_0+c\cot \phi \right){r_p}^{w^2}$$

(23)

Combining Equations (22) and (23), the bearing reaction force P₂ of the lining structure to surrounding rock with the consideration of seepage, can be obtained as:

$$P_2=\frac{2p_0-A_2-A{A}_2-B}{A+1}{\left\{\frac{2G\left(1+{\mu }_c\right)}{E_c\sin \phi \left(p_0+c\cot \phi \right)}\left[\frac{\left(1-2{\mu }_c\right)k_d{p}_i}{2A_1}+\frac{k_d{p}_i{r_0}^2\ln \left(r_1/r_0\right)}{\left({r_1}^2-{r_0}^2\right)A_1}+\frac{{r_0}^2+\left(1-2{\mu }_c\right){r_1}^2}{{r_1}^2-{r_0}^2}{P}_2\right]\right\}}^{\frac{1-A}{2}}+A_2$$

(24)

4. Model Validation

4.1. Project Profile

The total length of the deep-buried hydraulic tunnel in Xinjiang is about 17.3 km, the mountain slope along the route is steep, and the terrain presents a trend of high in the west and low in the east. The buried depth of surrounding rock varies from 500 to 1500 m, and the buried depth of the granite section is about 1200 m at most, which belongs to the deep-buried tunnel. According to the in situ stress test and geological prospecting data, the surrounding rock of the granite cave section within 0.7 km at the tail is a completely hard and massive structure. In combination with the actual geological condition, 3 drilling holes were arranged. At each drilling hole, 3 measuring ranges were set. Then, the feature parameters of the 9 measuring ranges, including fracturing pressure, initial pressure, closing pressure, and reopening pressure, were obtained. The maximum and minimum principal stresses were calculated. According to the calculation results, in this tunnel segment, σ₁ = 19.9~21.6 MPa, σ₃ = 10.2~12 MPa, and the uniaxial compressive strength σ_c of the surrounding rock was 28.7 MPa. When σ_c/σ₁ < 2, it can be concluded that the surrounding rock is under high crustal stress [32].

To gain in-depth knowledge of the influencing factors of tunnel stability, two different operating conditions, i.e., with and without the consideration of seepage, were applied. The results under two different conditions were then compared in detail. Combined with field measurement results, the related parameters were set below. The tunnel excavation radius r₁ = 2 m, the elastic modulus of surrounding rock E_d = 6 GPa, the Poisson’s ratio μ_d = 0.3, the cohesive force c = 0.8 MPa, the internal friction angle φ = 27°, the permeability coefficient of the rock mass k_d = 10⁻⁶ m/s. The elastic modulus of concretes E_c = 30 GPa, the Poisson’s ratio μ_c = 0.167, the lining permeability coefficient k_c = 10⁻⁹ m/s, the crustal stress p₀ = 20 MPa, the external water pressure P_i = 5 MPa, and the radius of the seepage field R = 20 m [33].

4.2. Model Validation

To validate the accuracy of the derived formulas in this study, we comparatively analyzed them with the formulas in previous literature. When b = 0, the MO criterion becomes the M-C criterion. Using the above parameters of the surrounding rock and lining structure, the lining thickness of 100 mm was substituted to the derived formulas and the existing formulas in Ref. [34] to calculate and analyze the stress on the surrounding rock and the plastic zone radius. As shown in Figure 3, the computed stresses on the surrounding rock using two different formulas fitted well with each other, and the plastic zone radius was 3.18 m, validating the accuracy of the derived formulas.

5. Analysis of the Influencing Factors

5.1. Seepage Effect

The excavation and lining of the tunnel will change the boundary condition of the seepage field in the tunnel. During the construction period, the water pressure on the internal wall surface of the lining structure equaled zero, and the external water pressure unaffected by tunnel excavation remained unchanged. Figure 4 shows the calculation results when b = 0.5 and at a lining thickness of 100 mm. From the figure, when the overall structure reached dynamic stability, the seepage pressure at the surrounding rock-lining interface was 4.785 MPa, close to the external water pressure of 5 MPa. The seepage pressure in the surrounding rock varied slightly, while it varied significantly in the lining structure. Under the first operating condition (without the consideration of seepage), the bearing reaction force was 5.05 MPa; under the second operating condition (with the reference of seepage), the bearing reaction force was 2.77 MPa, reducing by 45.15% compared with the first operating condition.

According to the calculation results, under the first operating condition, the plastic zone radius in the surrounding rock and the tunnel wall displacement were 1.22r₁ and 0.0032r₁, respectively; under the second operating condition, the plastic zone radius in the surrounding rock and the tunnel wall displacement were 1.483r₁ and 0.0047r₁, respectively. Compared with the first operating condition, the plastic zone radius in the surrounding rock and the tunnel wall displacement under the second operating condition were enhanced by 21.56% and 46.88%, respectively.

Overall, the tunnel excavation and lining can change the seepage field distribution, and the decline in the seepage effect will reduce the lining stress field distribution, thereby leading to the decline in the bearing reaction force. Furthermore, with the decrease of bearing reaction force, the stress field distribution in the elastic-plastic zone in surrounding rock changes, resulting in the plastic zone radius increment in surrounding rock and the tunnel wall displacement. Therefore, in actual projects, the seepage effect cannot be ignored when performing the overall calculation, especially the tunnel excavation in water-rich areas.

5.2. Effect of Intermediate Principal Stress Coefficient

At a lining thickness of 100 mm, when b = 0, 0.25, 0.5, 0.75, and 1, the plastic zone radius in surrounding rock was 1.591r₁, 1.513r₁, 1.483r₁, 1.513r₁, and 1.591r₁ (Figure 5a), while the tunnel wall displacement was 0.00537r₁, 0.00486r₁, 0.00466r₁, 0.00486r₁, and 0.00537r₁ (Figure 5b), respectively. These two variation curves are the quadratic parabolic curves concerning b = 0.5. When b = 0 or 1, the values of u_d and r_p were the largest and equaled each other. The calculation results fitted well with the results using the M-C criterion. When b = 0, the plastic zone radius and the tunnel wall displacement at b = 0.5 reduced by 6.79% and 13.22%, respectively.

From Figure 6, the stress curves when b = 0.25 and b = 0 coincide with the curves when b = 0.75 and b = 1. The bearing reaction force at b = 0.25 was 3.08 MPa, reducing by 21.03% compared with the 3.9 MPa at b = 0. The bearing reaction force at b = 0.5 was 2.77 MPa, reducing by 28.97% compared with the value at b = 0. The peak tangential stress at b = 0.25 was 30.87 MPa, which was 3.59% higher than the 29.8 MPa at b = 0. The peak tangential stress at b = 0.5 was 31.3 MPa, which was 5.03% higher than that at b = 0.

Overall, due to the consideration of the reinforcement effect of intermediate principal stress on rock strength, compared with the M-C criterion that shows relatively conservative results, the MO criterion can better reflect the effect of intermediate principal stress and intervals on the rock strength and realize the full potential of rock strength. As a result, it enables a closer calculation result to actual values and ensures a safer and more economical tunnel design and construction.

5.3. Effect of the Permeability Coefficient of Lining

Figure 7 shows the relationship between seepage pressure and bearing reaction force when b = 0.5 and at a lining thickness of 100 mm. From Figure 7, as the permeability coefficient of the lining decreased from 1.0 × 10⁻⁷ m/s to 1.0 × 10⁻¹⁰ m/s, the seepage pressure at the surrounding rock-lining interface increased from 0.911 to 4.978 MPa, with increasing amplitude of 2.54, 1.334, and 0.193 MPa, respectively. On the other hand, the bearing reaction force dropped from 4.69 to 2.68 MPa with decreasing amplitude of 1.3, 0.62, and 0.09 MPa, respectively.

From Figure 8, when the lining permeability coefficient dropped from 1.0 × 10⁻⁷ m/s to 1.0 × 10⁻¹⁰ m/s, the plastic zone radius in surrounding rock was 1.284r₁, 1.413r₁, 1.483r₁, and 1.493r₁, and the tunnel wall displacement was 0.00349r₁, 0.00423r₁, 0.00466r₁, and 0.00473r₁, respectively. The lining permeability coefficient’s decline increased the plastic zone radius in the surrounding rock and the tunnel wall displacement. Despite a decreasing amplitude, it still adversely affects tunnel stability.

Overall, reducing the lining permeability coefficient can enhance the seepage pressure at the surrounding rock-lining interface, reduce the bearing reaction force, and increase the plastic zone radius in the surrounding rock and the tunnel wall displacement. Therefore, during the tunnel excavation and lining, the lining structure with quite low permeability coefficients should be used to avoid the adverse effect of the overflow of pore water in the rock into the tunnel on the project quality. Meanwhile, the negative effect of increasing seepage pressure at the interface on the lining structure should also be considered.

5.4. Effect of the Lining Thickness

As shown in Figure 9, when b = 0.5 (b is the intermediate principal stress coefficient), the seepage pressure at the surrounding rock-lining interface and the bearing reaction force increased with the lining thickness (d is the thickness of the lining). The seepage pressure at the interface increased slightly from 4.583 MPa to 4.942 MPa. It can be attributed to the lining permeability coefficient being far lower than the surrounding rock. Under the second operating condition, the bearing reaction force increased from 1.146 MPa to 8.69 MPa; under the first operating condition, the bearing reaction force increased from 3.286 MPa to 8.69 MPa. However, the stresses all grew at a decreasing rate.

From Figure 10, as the lining thickness increased, the plastic zone radius in the surrounding rock dropped rapidly but at a decreasing rate. Under the first operating condition, after the addition of lining with a thickness of 244 mm, the plastic zone thickness in the surrounding rock dropped to 0. At that moment, the plastic zone radius under the second operating condition was only 1.125r₁, and the plastic zone thickness dropped to 0 after the addition of lining with a thickness of 113 mm. At the same lining thickness, the plastic zone radius in the surrounding rock under the second operating condition was always larger than that under the first operating condition.

Therefore, it can be concluded that increasing the lining thickness can increase the seepage pressure at the surrounding rock-lining interface, and enhance the bearing reaction force. In the beginning, increasing the lining thickness can restrict the expansion of the plastic zone in the surrounding rock; however, the restriction weakened with the increasing lining thickness. A too-thick lining will also result in high project costs. Therefore, increasing the lining thickness can significantly reduce the plastic zone thickness in the surrounding rock, and some other construction methods should be used simultaneously to limit the deformation of the surrounding rock.

6. Results and Discussion

Based on the MO strength criterion, this study took a circular tunnel in Xinjiang to derive the analytical solutions of stress and displacement considering the surrounding rock-lining interaction under two different operating conditions (with and without the consideration of seepage). The relationship between seepage pressure and bearing reaction force was analyzed, and the relationship of bearing reaction force with the plastic zone radius in surrounding rock and the tunnel wall displacement was revealed. In addition, the calculation results of the stress on the surrounding rock and the plastic zone radius were compared with the data calculated according to the formulas in Ref. [34], showing a favorable agreement.

Under the seepage effect, the bearing reaction force of the lining structure to the surrounding rock decreased by 45.15%, the plastic zone radius in the surrounding rock increased by 21.56%, and the tunnel wall displacement increased by 46.88%. As the lining thickness increased from 50 mm to the value corresponding to zero plastic zone thickness, the seepage pressure at the interface increased by 0.359 MPa. Under the first operating condition, the bearing reaction force increased by 5.404 MPa; under the second operating condition, the bearing reaction force increased by 7.544 MPa. Therefore, the lining structure with a thickness of 244 mm was added under the first operating condition, while the lining structure with a thickness of 357 mm was added under the second operating condition, indicating that the lining structure can significantly restrict the expansion of plastic zone in surrounding rock at the beginning, but the restriction effect weakened gradually. Therefore, the seepage effect should be considered in tunnel stability analysis.

The formula proposed in this paper also has a limitation. It only applies to the circular tunnel during the construction period when the lateral pressure coefficient of surrounding rock is 1, and the water-bearing rock mass is a uniform isotropic two-phase medium conforming to Darcy’s law. Therefore, the analytical formula of surrounding rock under the action of multi-field coupling needs to be further studied.

7. Conclusions

For clarifying the factors affecting the stability of deep-buried hydraulic tunnels containing pore water, this research explored the elastoplastic theory and Mogi-Coulomb strength criterion to derive the analytical solutions of the surrounding rock-lining structure. Based on the studying, the main conclusions can be drawn as follows.

(1)

From the MO criterion, the solution was a quadratic parabolic curve symmetrical at b = 0.5. Compared with the results at b = 0, the bearing reaction force at b = 0.5 was reduced by 28.97%, the plastic zone radius was reduced by 6.79%, and the tunnel wall displacement was reduced by 13.22%. This suggests that the intermediate principal stress can enhance rock strength, and the effect of intermediate principal stress should not be ignored in the calculation.

(2)

As the lining permeability coefficient decreased from 10⁻⁷ m/s to 10⁻¹⁰ m/s, the seepage pressure at the surrounding rock-lining interface increased by 4.067 MPa, the bearing reaction force decreased by 2.01 MPa, the plastic zone radius in surrounding rock increased by 0.209r₁, and the tunnel wall displacement increased by 0.00124r₁. All of these were unfavorable for tunnel stability.

(3)

The research results have important theoretical significance and practical engineering application values for the stability analysis of deep-buried hydraulic tunnels. In the future, the damage and fracture of deep-buried hydraulic tunnels under multi-field coupling effects, the zonal fracture mechanism of surrounding rock, and the control mechanism of supporting structure need to be further studied.