Failure Mode of Tunnel Face Under Transient Unsaturated Seepage with Temperature Influence

Abstract

The seepage caused by heavy rainfall and storm runoff is not a static phenomenon. On the contrary, it is a dynamic process known as unsaturated transient seepage. Under the condition, the spatiotemporal variations in suction stress cannot be overlooked. With the development of tunnel mechanics, there has been an emergence of tunnels affected by high ground temperatures or temperature influences, highlighting the necessity of incorporating temperature effects into the analysis. This article proposes a new framework for the spatiotemporal response of tunnel face safety to temperature-affected and unsaturated transient seepage conditions. A one-dimensional transient seepage assumption is used to develop an analytical model describing unsaturated transient seepage, which is then integrated centered on suction stress strength theory for unsaturated soils to acquire suction stress variations with depth and time. The temperature impact on the unsaturated soil shear strength is incorporated, applying a temperature-sensitive effective stress model in conjunction with the soil–water characteristic curve to quantitatively analyze temperature-induced apparent cohesion changes. The 3D logarithmic spiral failure model is used to analyze the tunnel face stability. The validity of the proposed failure model is demonstrated through an engineering calculation. The rates of internal dissipation and external work are calculated, and a kinematic approach related to strength reduction is used to determine the safety factor of the tunnel face with zero support pressure. The results show that considering transient unsaturated seepage and temperature effects can increase the safety factor. The influence of these effects mainly depends on the soil type, tunnel geometric parameters, and seepage conditions. This work explores the influence of variations in a series of parameters on the failure mode of tunnel faces under temperature effects, taking into account unsaturated transient seepage, thereby providing valuable references for the design and construction of tunnels in the future.

1. Introduction

Tunnel engineering has been extensively utilized in the construction of urban subways, large river-crossing passages, and other transportation infrastructure projects. The instability and failure of a tunnel face can lead to ground deformation or even surface collapse, making the tunnel face’s stability a key focus throughout the excavation process. Consequently, significant efforts have been devoted to investigating this issue. Existing methods can be categorized into model testing [1,2,3,4,5,6] and numerical simulation [7,8]. In addition to model testing and numerical simulation methods, scholars have made numerous attempts to develop simplified theoretical or analytical models to enable effective and reliable calculations to evaluate the tunnel’s stability [9,10,11]. The limit equilibrium method and the upper-bound limit analysis, both grounded in geotechnical plasticity mechanics, are two primary theoretical approaches for studying tunnel face stability. Both methods heavily rely on assumed failure mechanisms to describe potential failure zones. The establishment of failure mechanisms typically involves combining existing geometric shapes, such as cones, cylinders, and logarithmic spirals. Experimental and numerical studies provide insights into the failure characteristics of tunnel faces, offering guidance for constructing failure mechanisms. Based on the assumed failure mechanisms, the limit equilibrium method and the upper-bound limit analysis method formulate force equilibrium and energy dissipation equilibrium equations, respectively, to analyze the limit state.

In the limit equilibrium method, the most classic failure mechanism is the wedge–prism model put forward by Horn. In this model, a horizontal wedge part and a prism part are included, with the former located in front of the excavation face and the latter positioned above it. Anagnostou and Kovári [12] introduced this wedge–prism model to analyze the tunnel faces’ stability. Subsequently, Oreste and Dias [13] replaced the wedge with a pyramidal shape, establishing a new model that combines a triangular base with a prismatic structure. The results from this model were found to be in close agreement with those from three-dimensional numerical analyses. The wedge–prism model was further refined by Anagnostou and Perazzelli [14], who applied the slice method to the wedge, removing the necessity to hypothesize the vertical stress distribution within it. The improved model can also handle non-uniform ground conditions and non-uniform support at the excavation face. The limit equilibrium method is widely used for analyzing the stability of geotechnical structures. Typically, this method requires an initial assumption of the failure surface and the stress distribution along it. However, under complex conditions, determining a reasonable stress distribution on the failure surface can be challenging, which limits the applicability of the limit equilibrium method.

Compared to the limit equilibrium method, the limit analysis method [15,16,17], based on classical plasticity theory, offers advantages such as clear conceptualization and rigorous theoretical foundations, making it a powerful tool for stability analysis in geotechnical engineering. Limit analysis is categorized into the upper-bound and lower-bound methods. The lower-bound method demands constructing a statically acceptable stress field, a process usually more complicated, so it is less often used. On the flip side, the upper-bound method involves predefining a kinematically admissible failure mechanism and then applying the principle of energy balance for calculations. This approach is relatively simpler, as it avoids the need to solve complex partial differential equations, and is, therefore, widely adopted.

By observing the failure patterns and velocity field characteristics of tunnel excavation faces at the ultimate limit state in model tests, researchers have proposed numerous failure mechanisms based on the upper-bound limit analysis method. Common two-dimensional failure mechanisms for tunnel faces include the 2D rotational failure mechanism based on spatial discretization techniques [18]. The 2D failure mechanisms simplify the tunnel excavation face as a plane strain problem, neglecting the actual three-dimensional soil arching effect, which can lead to an underestimation of the face stability. In contrast, the construction of 3D failure mechanisms, though more complex, better reflects the actual conditions of the problem. Typical 3D failure mechanisms for tunnel faces include the conical failure mechanism [19], the multi-block failure mechanism [7], the horn failure mechanism [20], and the 3D rotational failure mechanism based on spatial discretization techniques [21]. The spatial discretization technique for generating failure mechanisms proposed by Mollon [21] has been successfully developed and applied to various complex conditions, including random fields. Therefore, this study will also adopt this failure mechanism for evaluating the stability of tunnel faces under transient unsaturated seepage and temperature effect conditions.

The classical Rankine or Coulomb earth pressure theory has been extended to unsaturated conditions in some studies to assess the function of suction [22,23]. In recent studies, the suction stress concept introduced by Lu and Likos [24,25] has been utilized to depict the stress characteristics of unsaturated soils, and approaches to evaluating the tunnel face stability under steady unsaturated seepage conditions have been devised [26,27,28]. Although these studies may have provided in-depth insights into the effect of matric suction on the stability of tunnel faces, they have exclusively considered steady-state unsaturated seepage conditions. This study employs the assumption of one-dimensional transient seepage to establish an analytical model describing unsaturated transient seepage, upon which the stability analysis of the tunnel excavation face is then conducted. The spatiotemporal response of the dynamic infiltration process can be simulated using a closed-form equation of the one-dimensional form of Richards’ equation [29,30].

Historically, the impact of temperature on tunnel excavation stability has frequently been overlooked. In recent years, however, as numerous tunnels globally have been subjected to temperature-related effects, researchers have initiated studies to address the challenges associated with temperature-influenced tunnel conditions. Liu et al. [31] revealed the temperature variation patterns during high-temperature tunnel construction through field experiments. Vahedifard et al. [32] demonstrated that thermally-induced variations in the soil moisture content serve as the primary mechanism governing temperature-dependent alterations in the tunnel face stability, subsequently modifying soil matric suction and apparent cohesion characteristics. Thoat and Vahedifard [33] experimentally simulated the undrained behavior of unsaturated soils under elevated temperatures, revealing a monotonic reduction in matric suction with rising thermal gradients. Xiao et al. [34] further advanced this understanding by developing a temperature-dependent matric suction equation coupled with a modified soil–water characteristic curve (SWCC) model, demonstrating that increased temperatures reduce saturation levels under constant matric suction. Collectively, these investigations—spanning diverse analytical models, experimental protocols, thermal regimes, and stress states—establish temperature as a critical determinant of the unsaturated soil strength. In summary, temperature variations can induce water migration and alter the soil water content, and such thermally induced water migration can change the matric suction and apparent cohesion characteristics, thereby influencing the tunnel face stability. Consequently, thermal effects must be explicitly integrated into geomechanical stability assessments for tunnel face excavations.

Transient unsaturated seepage, a dynamic process closer to real conditions, is of great importance in tunnel construction. The groundwater seepage state is dynamic, and the water migration and stress distribution in unsaturated soils change accordingly. Traditional static analysis cannot capture the mechanical behavior under such dynamic changes. Matric suction in unsaturated soils is crucial for tunnel face stability. It reduces the surrounding rock pressure on the face, and when it equals the air-entry value, the pressure curve inflects. Under transient seepage, matric suction changes the soil strength and deformation, impacting the face stability. Transient seepage also dynamically changes the seepage field in the surrounding rock, altering the pore water pressure and thus the soil mechanical properties.

In this study, the unsaturated transient seepage, a dynamic process that more closely aligns with real-world conditions, is innovatively introduced. The impact of temperature on the tunnel face stability under unsaturated transient seepage conditions is explored by analyzing the relationship between the soil matric suction, shear strength, and temperature within the unsaturated soil mechanics framework. Then, a work rate balance equation matching the external power with internal energy dissipation is proposed to evaluate the tunnel face stability. The safety factor of the tunnel face, affected by temperature and unsaturated transient seepage, is calculated using the strength reduction method [35], defined as the ratio of the soil’s initial strength to its strength at the critical failure point. Relevant parameters are analyzed and investigated, which provides a foundation and basis for the subsequent stability analysis of tunnel faces under temperature-influenced unsaturated transient seepage conditions.

2. Suction Stress and Apparent Cohesion Under Transient Unsaturated Seepage

2.1. Suction Stress and Apparent Cohesion in Unsaturated Soils

In the field of soil mechanics, the significance of effective stress in regulating the mechanical responses of unsaturated soils is broadly acknowledged. To conduct research on the tunnel face stability in unsaturated soils, a reliable and clear understanding of effective stress is essential. Lu and Likos [24,25] proposed a unified expression for effective stress:

$${\sigma }^{\prime }=\sigma -u_{\mathrm{a}}-{\sigma }^{\mathrm{s}}$$

(1)

where $\sigma$ is the total stress, ${\sigma }^{\prime }$ is the effective stress, ${\sigma }^{\mathrm{s}}$ is the suction stress, and $u_{\mathrm{a}}$ is the pore–air pressure. Suction stress is a function of matric suction $\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)$ and their relationship is as follows [36]:

$${\sigma }^{\mathrm{s}}=-\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)$$

(2a)

$${\sigma }^{\mathrm{s}}=-S_{\mathrm{e}}\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)$$

(2b)

where $u_{\mathrm{w}}$ represents the pore–water pressure. It should be noted that Equation (2a) is applicable to fully saturated soils, while Equation (2b) is suitable for unsaturated soils. In Equation (2b), the parameter $S_{\mathrm{e}}$ is the effective degree of saturation that can be expressed by:

$$S_{\mathrm{e}}={\displaystyle \frac{\theta -{\theta }_{\mathrm{r}}}{{\theta }_{\mathrm{s}}-{\theta }_{\mathrm{r}}}}$$

(3)

where $\theta$ is the volumetric water content, ${\theta }_r$ is the residual volumetric water content, and ${\theta }_s$ is the saturated volumetric content. Within the effective stress framework, the Mohr–Coulomb yield criterion is employed to define the strength of unsaturated soils [37]:

$${\tau }_f=c^{\prime }+{\sigma }^{\prime }\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }=c^{\prime }+(\sigma -u_{\mathrm{a}})\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }-{\sigma }^{\mathrm{s}}\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }$$

(4)

where ${\tau }_f$ is the soil strength, $c^{\prime }$ is effective cohesion, and ${\phi }^{\prime }$ is the effective internal friction angle. Due to the suction stress, the term $-{\sigma }^{\mathrm{s}}\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }$ in Equation (4) is considered as extra cohesion, known as apparent cohesion [37].

$$c_{\mathrm{a}\mathrm{p}\mathrm{p}}=-{\sigma }^{\mathrm{s}}\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }$$

(5)

2.2. Impact of Temperature

The temperature effect on the soil strength is mainly manifested through its effect on matric suction, which is defined as $\psi =\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)$. How matric suction varies with temperature can be expressed as [38]:

$$\psi ={\psi }_{T_{\mathrm{r}}}({\displaystyle \frac{\beta +T}{{\beta }_{T_{\mathrm{r}}}+T_{\mathrm{r}}}})$$

(6)

where $T$ represents the temperature in Kelvin. For clarity and ease of interpretation, while Kelvin is used in the calculations, Celsius will be employed in subsequent discussions. Here, ${\psi }_{T_{\mathrm{r}}}$ denotes the matric suction at the reference temperature $T_{\mathrm{r}}$ (0 °C), and ${\beta }_{T_{\mathrm{r}}}$ is the regression parameter at $T_{\mathrm{r}}$, which depends on factors such as the surface tension, specific immersion enthalpy, and the contact angle. $\beta$ and ${\beta }_{T_{\mathrm{r}}}$ are calculated as follows [38]:

$$\beta ={\displaystyle \frac{-∆h}{-∆h+(a+b{T}_{\mathrm{r}})\mathrm{c}\mathrm{o}\mathrm{s}\alpha }}$$

(7)

$${\beta }_{T_{\mathrm{r}}}=-{\displaystyle \frac{{∆h}_{T_{\mathrm{r}}}}{C_1}}$$

(8)

where $\alpha$ is the temperature-dependent soil–water contact angle, a and b are the fitting parameters that can be estimated as $a=0.11766 \mathrm{N}/\mathrm{m}$ and $b=0.0001535 \mathrm{N}/(\mathrm{m}·\mathrm{K})$ [39], and $C_1$ is a constant calculated by Equation (11). $∆h$ represents the immersion enthalpy per unit area, and the temperature-dependent expression for $∆h$ is:

$$∆h={∆h}_{T_{\mathrm{r}}}{\left({\displaystyle \frac{1-T_{\mathrm{r}}}{1-T}}\right)}^{0.38}$$

(9)

where ${∆h}_{T_{\mathrm{r}}}$ is the $∆h$ at the reference temperature. The parameters related to the soil–water contact angle are as follows [40]:

$$\mathrm{c}\mathrm{o}\mathrm{s}\alpha ={\displaystyle \frac{-∆h+T{C}_1}{a+bT}}$$

(10)

$$C_1={\displaystyle \frac{{∆h}_{T_{\mathrm{r}}}+(a+b{T}_{\mathrm{r}})\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0}{T_{\mathrm{r}}}}$$

(11)

where ${\alpha }_0$ is the soil–water contact angle at $T_{\mathrm{r}}$, and $\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0$ is set to the empirical value of 0.8.

2.3. Analytical Solution to Transient Unsaturated Seepage

The transient unsaturated seepage in the vertical direction is governed by the one-dimensional form of the Richard’s equation [29,40,41]:

$${\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{\partial }{\partial z}}\left[k{\displaystyle \frac{\partial ({\psi }^{\prime }+z)}{\partial z}}\right]$$

(12)

where $t$ represents time, $z$ denotes the vertical coordinate (with positive values indicating the upward direction and zero representing the groundwater level), and ${\psi }^{\prime }$ signifies the suction head, which can be expressed as ${\psi }^{\prime }=-{\displaystyle \frac{\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)}{{\gamma }_w}}$. According to the Gardner model [42], the water content $\theta$ and hydraulic conductivity $k$ can be expressed as exponential functions of the suction head, namely:

$$k=k_{\mathrm{s}}{e}^{\alpha \psi \prime }$$

(13)

$$\theta ={\theta }_{\mathrm{r}}+({\theta }_{\mathrm{s}}-{\theta }_{\mathrm{r}})e^{\alpha \psi \prime }$$

(14)

where $\alpha$ and $k_s$ represent the reciprocal of the vertical height of the capillary fringe and the saturated hydraulic conductivity, respectively. Substituting Equations (13) and (14) into Equation (12) yields:

$${\displaystyle \frac{\alpha ({\theta }_{\mathrm{s}}-{\theta }_{\mathrm{r}})}{k_{\mathrm{s}}}}{\displaystyle \frac{\partial k}{\partial t}}={\displaystyle \frac{{\partial }^{2}k}{\partial z^2}}+\alpha {\displaystyle \frac{\partial k}{\partial z}}$$

(15)

To solve Equation (15), two boundary conditions and one initial condition are required. In this study, a special case is considered where the groundwater level is stable and the surface flow rate is constant. The conditions for the transient unsaturated seepage under investigation can be predefined as follows: (i) the initial surface flow rate is zero, meaning the initial suction head is linearly distributed within the soil; (ii) the suction head at the groundwater level remains zero at all times; and (iii) the infiltration rate at the surface, denoted by q, is assumed to be constant. For ease of expression, the following dimensionless parameters are defined:

$$\dot{K}={\displaystyle \frac{k}{k_{\mathrm{s}}}}, \dot{Z}=\alpha z, \dot{Q}={\displaystyle \frac{q}{k_{\mathrm{s}}}}, \dot{L}=\alpha l, \dot{T}={\displaystyle \frac{\alpha k_{\mathrm{s}}t}{{\theta }_{\mathrm{s}}-{\theta }_{\mathrm{r}}}}$$

where $l$ represents the vertical distance between the ground surface and the groundwater table. By substituting these five dimensionless parameters into Equation (15), it can be expressed as:

$${\displaystyle \frac{\partial \dot{K}}{\partial \dot{T}}}={\displaystyle \frac{{\partial }^2\dot{K}}{\partial {\dot{Z}}^2}}+{\displaystyle \frac{\partial \dot{K}}{\partial \dot{Z}}}$$

(16)

Additionally, the boundary and initial conditions can be expressed in the following forms:

$$\dot{K}(\dot{Z}, 0)=e^{-\dot{Z}}={\dot{K}}_0\dot{\left(Z\right)}$$

(17a)

$$\dot{K}(0, \dot{T})=1$$

(17b)

$${\left[{\displaystyle \frac{\partial \dot{K}}{\partial \dot{Z}}}+\dot{K}\right]}_{\dot{Z}=\dot{L}}=\dot{Q}$$

(17c)

By employing the Laplace transform technique and incorporating the assumed initial and boundary conditions, the solution to the partial differential Equation (16) can be obtained [30,40,43]. The derivation yields the dimensionless hydraulic conductivity $\dot{K}$ as:

$$\dot{K}=\dot{Q}-(\dot{Q}-1)e^{-\dot{Z}}-4\dot{Q}{e}^{{\displaystyle \frac{\dot{L}-\dot{Z}}{2}}}{e}^{-{\displaystyle \frac{\dot{T}}{4}}}\sum _{n=1}^{\infty }{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}(\dot{{\lambda }_n}\dot{Z})\mathrm{s}\mathrm{i}\mathrm{n}(\dot{{\lambda }_n}\dot{L})e^{-\dot{{\lambda }_n^2}\dot{T}}}{1+\frac{\dot{L}}{2}+2\dot{{\lambda }_n^2}\dot{L}}}$$

(18)

where $\dot{{\lambda }_n}$ represents the nth positive root of the following pseudo-periodic characteristic equation:

$$\mathrm{t}\mathrm{a}\mathrm{n}(\dot{{\lambda }_n}\dot{L)}+2\dot{{\lambda }_n}=0$$

(19)

Subsequently, the suction head and matric suction can be derived from Equation (13) as:

$${\psi }^{\prime }={\displaystyle \frac{\mathrm{l}\mathrm{n}\dot{K}}{\alpha }}$$

(20a)

$${\psi }_{T_{\mathrm{r}}}={\displaystyle \frac{{\gamma }_w\mathrm{l}\mathrm{n}\dot{K}}{\alpha }}$$

(20b)

By combining Equation (14) with the aforementioned dimensionless parameters, the degree of saturation can be derived as:

$$S_{\mathrm{e}}=\dot{K}$$

(21)

By combining Equations (3)–(6), the suction stress ${\sigma }^{\mathrm{s}}$ and apparent cohesion $c_{\mathrm{a}\mathrm{p}\mathrm{p}}$ under transient unsaturated seepage and temperature conditions can be derived and expressed as functions of $\dot{K}$:

$${\sigma }^{\mathrm{s}}=-{\displaystyle \frac{{\gamma }_w\dot{K}\mathrm{l}\mathrm{n}\dot{K}}{\alpha }}({\displaystyle \frac{\beta +T}{{\beta }_{T_{\mathrm{r}}}+T_{\mathrm{r}}}})$$

(22)

$$c_{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{{\gamma }_w\dot{K}\mathrm{l}\mathrm{n}\dot{K}}{\alpha }}({\displaystyle \frac{\beta +T}{{\beta }_{T_{\mathrm{r}}}+T_{\mathrm{r}}}})\mathrm{t}\mathrm{a}\mathrm{n}\phi$$

(23)

3. Failure Mode of Tunnel Face Under Transient Unsaturated Seepage

3.1. Three-Dimensional Logarithmic Spiral Failure Mechanism

The failure mechanism adopted in this study is illustrated in Figure 1. This work establishes a tunnel with zero support, a diameter of D, and an overburden depth of C, within soil experiencing transient unsaturated seepage and temperature effects. A horizontal plane is positioned at a distance $z_0$ above the tunnel base, and a z-coordinate axis is defined with the positive direction vertically upward, based on this horizontal plane.

This study uses the kinematic limit analysis method, with priority given to the construction of a collapse mechanism that meets the specified conditions. The soil is assumed to obey the associative flow rule, which suggests the velocity vector direction should correspond to the effective internal friction angle at the failure surface. Therefore, the normal velocity $v_{\mathrm{n}}$ can be expressed as:

$$v_{\mathrm{n}}=v_{\mathrm{t}}\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }$$

(24)

In the equation, $v_{\mathrm{n}}$ and $v_{\mathrm{t}}$ represent the normal and tangential components of the velocity on the sliding surface, respectively. It is demonstrated that, during rigid rotation, the outline of the failure mechanism takes the shape of a logarithmic spiral, and the sliding surface is at a tangent to a cone with a vertex angle of ${2\phi }^{\prime }$. The use of a 3D logarithmic spiral failure mechanism to describe tunnel face collapse is shown to satisfy the aforementioned conditions.

In this study, the 3D logarithmic spiral failure mechanism is adopted, which rotates about the central point O with an angular velocity $\omega$. Consequently, a polar coordinate system $(\theta ,\rho )$ is established to determine the velocity at any point on the mechanism:

$$v=\omega \rho$$

(25)

The failure mechanism’s path is determined by two logarithmic spiral trajectories, namely the curve AE and curve BE, formulated as follows:

$$r_{\mathrm{A}\mathrm{E}}=r_{\mathrm{A}}{\mathrm{e}}^{-\left(\theta -{\theta }_{\mathrm{A}}\right)\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}$$

(26a)

$$r_{\mathrm{B}\mathrm{E}}=r_{\mathrm{B}}{\mathrm{e}}^{\left(\theta -{\theta }_{\mathrm{B}}\right)\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}$$

(26b)

where $r_{\mathrm{A}}$ and $r_{\mathrm{B}}$ can be expressed in terms of ${\theta }_{\mathrm{A}}$ and ${\theta }_{\mathrm{B}}$:

$$r_{\mathrm{A}}={\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{B}}}{\mathrm{s}\mathrm{i}\mathrm{n}{(\theta }_{\mathrm{B}}-{\theta }_{\mathrm{A}})}}D$$

(27a)

$$r_{\mathrm{B}}={\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_A}{\mathrm{s}\mathrm{i}\mathrm{n}({\theta }_{\mathrm{B}}-{\theta }_{\mathrm{A}})}}D$$

(27b)

By combining Equations (26a), (26b), (27a) and (27b), the following can be obtained:

$$r_{\mathrm{A}\mathrm{E}}={\displaystyle \frac{D\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{B}}}{\mathrm{s}\mathrm{i}\mathrm{n}{(\theta }_{\mathrm{B}}-{\theta }_{\mathrm{A}})}}{\mathrm{e}}^{-\left(\theta -{\theta }_{\mathrm{A}}\right)\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}$$

(28a)

$$r_{\mathrm{B}\mathrm{E}}={\displaystyle \frac{D\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{A}}}{\mathrm{s}\mathrm{i}\mathrm{n}{(\theta }_{\mathrm{B}}-{\theta }_{\mathrm{A}})}}{\mathrm{e}}^{\left(\theta -{\theta }_{\mathrm{B}}\right)\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}$$

(28b)

The mechanism consists of multiple circles, each centered at points along the rotational axis of the two logarithmic spiral curves. The path of the central point is described as:

$$r_{\mathrm{m}}={\displaystyle \frac{r+r^{\prime }}{2}}$$

(29)

The radius of the aforementioned circles can be calculated as:

$$R={\displaystyle \frac{r-r^{\prime }}{2}}$$

(30)

The intersection point of the two logarithmic spiral curves is located at point E, which can be identified using polar coordinates:

$${\theta }_{\mathrm{E}}={\displaystyle \frac{{\theta }_{\mathrm{A}}+{\theta }_{\mathrm{B}}}{2}}+{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{B}}/\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{A}}\right)}{2\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}}$$

(31a)

$$r_{\mathrm{E}}={\displaystyle \frac{\sqrt{\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{A}}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{B}}}}{\mathrm{s}\mathrm{i}\mathrm{n}{(\theta }_{\mathrm{B}}-{\theta }_{\mathrm{A}})}}{\mathrm{e}}^{{\displaystyle \frac{1}{2}}\left({\theta }_{\mathrm{A}}-{\theta }_{\mathrm{B}}\right)\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}D$$

(31b)

In shallow tunnels, the failure mechanism may outcrop, causing curves AE and BE to intersect the ground at points C and D. Here, the mechanism is defined by curves AC, BD, and the ground CD. The ground positions of C and D are determined by ${\theta }_{\mathrm{C}}$ and ${\theta }_{\mathrm{D}}$ as follows:

$$C=r_{\mathrm{B}}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{B}}-r_{\mathrm{A}}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{C}}{\mathrm{e}}^{-\left({\theta }_{\mathrm{C}}-{\theta }_{\mathrm{A}}\right)\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}$$

(32a)

$${C=r}_{\mathrm{B}}\mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\theta }}_{\mathrm{B}}-r_{\mathrm{B}}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{D}}{\mathrm{e}}^{\left({\theta }_{\mathrm{D}}-{\theta }_{\mathrm{B}}\right)\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}$$

(32b)

In this framework, ${\theta }_{\mathrm{A}}$ and ${\theta }_{\mathrm{B}}$ are selected as optimization variables to characterize the failure mechanism and must satisfy the following constraints:

$$0<{\theta }_{\mathrm{A}}<{\theta }_{\mathrm{B}}<{\displaystyle \frac{\mathsf{\pi }}{2}}$$

(33a)

$${\theta }_{\mathrm{B}}<{\theta }_{\mathrm{E}}<\mathsf{\pi }$$

(33b)

3.2. Strength Reduction Technique and Safety Factor

Within the framework of the limit analysis, the safety factor is quantified as the ratio between the soil’s inherent shear strength and its reduced strength at the critical failure state of the tunnel face. Based on this criterion, FS is mathematically formulated as:

$$FS={\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime }}{\mathrm{t}\mathrm{a}\mathrm{n}{\phi }_{\mathrm{d}}^{\prime }}}={\displaystyle \frac{c^{\prime }}{c_{\mathrm{d}}^{\prime }}}$$

(34)

where FS represents the safety factor, with ${\phi }_{\mathrm{d}}^{\prime }$ and $c_{\mathrm{d}}^{\prime }$ defined as the strength-reduced parameters at critical collapse conditions. During this iterative procedure, the soil strength is progressively adjusted by systematically varying FS until the supporting pressure at the tunnel face diminishes to equilibrium conditions (${\sigma }_{\mathrm{c}}=0$). The analytical expression for ${\sigma }_{\mathrm{c}}$ is determined as follows:

$${\sigma }_{\mathrm{c}}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\displaystyle \frac{{\dot{W}}_{\mathrm{e}}-{\dot{W}}_{\mathrm{i}}}{2\omega r_{\mathrm{A}}^2{\mathrm{s}\mathrm{i}\mathrm{n}}^2{\theta }_{\mathrm{A}}{\int }_{{\theta }_{\mathrm{A}}}^{{\theta }_{\mathrm{B}}}\sqrt{R^2-d_1^2}\mathrm{c}\mathrm{o}\mathrm{t}\theta {\mathrm{c}\mathrm{s}\mathrm{c}}^2\theta \mathrm{d}\theta }}\right\}$$

(35)

where the external work rate ${\dot{W}}_{\mathrm{e}}$ and internal work rate $\dot{W}$ are calculated as follows:

$$\begin{gathered} {\dot{W}}_{\mathrm{e}}=2\omega \gamma \left[{\int }_{{\theta }_{\mathrm{A}}}^{{\theta }_{\mathrm{B}}}{\int }_{d_1}^R{\int }_0^{\sqrt{R^2-y^2}}{\left(r_{\mathrm{m}}+y\right)}^2\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{d}x\mathrm{d}y\mathrm{d}\theta \right. \\ +{\int }_{{\theta }_{\mathrm{B}}}^{{\theta }_{\mathrm{C}}}{\int }_{-R}^R{\int }_0^{\sqrt{R^2-y^2}}{\left(r_{\mathrm{m}}+y\right)}^2\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{d}x\mathrm{d}y\mathrm{d}\theta \\ +{\int }_{{\theta }_{\mathrm{C}}}^{{\theta }_{\mathrm{D}}}{\int }_{-R}^{d_2}{\int }_0^{\sqrt{R^2-y^2}}{\left(r_{\mathrm{m}}+y\right)}^2\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{d}x\mathrm{d}y\mathrm{d}\theta ] \end{gathered}$$

(36a)

$${\dot{W}}_{\mathrm{i}}=2\omega \left[{\int }_{{\theta }_{\mathrm{A}}}^{{\theta }_{\mathrm{B}}}{\int }_{d_1}^{R}c{\displaystyle \frac{R{\left(r_{\mathrm{m}}+y\right)}^2}{\sqrt{R^2-y^2}}}\mathrm{d}y\mathrm{d}\theta +{\int }_{{\theta }_{\mathrm{B}}}^{{\theta }_{\mathrm{C}}}{\int }_{-R}^{R}c{\displaystyle \frac{R{\left(r_m+y\right)}^2}{\sqrt{R^2-y^2}}}\mathrm{d}y\mathrm{d}\theta \right.\left.+{\int }_{{\theta }_{\mathrm{C}}}^{{\theta }_{\mathrm{D}}}{\int }_{-R}^{d_2}c{\displaystyle \frac{R{\left(r_{\mathrm{m}}+y\right)}^2}{\sqrt{R^2-y^2}}}\mathrm{d}y\mathrm{d}\theta \right]$$

(36b)

The parameters $d_1$ and $d_2$ are determined by subtracting the radius of the median line from the respective distances between the central point O and points on the tunnel face or ground surface, as defined by:

$$d_1=r_{\mathrm{A}}{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{A}}}{\mathrm{s}\mathrm{i}\mathrm{n}\theta }}-r_{\mathrm{m}}$$

(37a)

$$d_2=r_{\mathrm{C}}{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{C}}}{\mathrm{s}\mathrm{i}\mathrm{n}\theta }}-r_{\mathrm{m}}$$

(37b)

An iterative computational approach is utilized in this study, with the procedural workflow detailed in Figure 2. A stability assessment is governed by two energy balance criteria: (i) Unstable condition—when the external energy input rate (${\dot{W}}_{\mathrm{e}}$) exceeds the internal energy dissipation rate (${\dot{W}}_{\mathrm{i}}$), resulting in ${\sigma }_{\mathrm{c}}>0$, an additional stabilizing pressure must be applied. In this scenario, FS₂ is iteratively updated to the midpoint value of the current interval. (ii) Stable condition—conversely, if ${\dot{W}}_{\mathrm{e}}<{\dot{W}}_{\mathrm{i}}$ and ${\sigma }_{\mathrm{c}}<0$, the tunnel face is deemed stable, requiring no supplemental support. Here, FS₁ is replaced by the midpoint value.

The iterative process terminates when either the calculated ${\sigma }_{\mathrm{c}}$ falls below 0.01 kPa, or the absolute difference between FS₁ and FS₂ reduces to less than 0.001.

4. Parametric Analysis and Discussions

This section applies the established analytical framework to parametric studies of three representative unsaturated soils, investigating the spatiotemporal effects of unsaturated seepage. The hydraulic and geomechanical parameters for the hypothetical soils are summarized in

Table 1. Representative parameters of sand, silt, and clay.

Soil Type	${∆\mathit{h}}_{{\mathit{T}}_{\mathit{r}}}(\mathbf{J}/{\mathbf{m}}^2)$	$\mathit{\alpha }\left({\mathbf{m}}^{-1}\right)$	${\mathit{k}}_{\mathit{s}}(\mathbf{m}/\mathbf{s})$	$\phi \left(°\right)$	c′/γD	${\mathit{\theta }}_{\mathbf{s}}$	${\mathit{\theta }}_{\mathbf{r}}$
Sand	$-0.285$	0.7	$5\times {10}^{-6}$	35	0.04	0.41	0.05
Silt	$-0.516$	0.5	$9\times {10}^{-7}$	30	0.07	0.45	0.10
Clay	$-0.516$	0.13	$5\times {10}^{-8}$	25	0.10	0.58	0.05

. The infiltration rate is assumed equivalent to the saturated hydraulic conductivity ($\dot{Q}=1$), representing an extreme precipitation scenario. Since natural rainfall-induced infiltration rates typically remain below saturated conductivity levels, results under $\dot{Q}=1$ inherently provide conservative estimates. Additional computational parameters are configured as follows: the unit weight of water ${\gamma }_w=10 \mathrm{k}\mathrm{N}/{\mathrm{m}}^3$, soil unit weight $\gamma =20 \mathrm{k}\mathrm{N}/{\mathrm{m}}^3$, and the groundwater level is set at the tunnel base ($z_0=0$).

Figure 3, Figure 4 and Figure 5 illustrate the variations of the normalized safety factor with temperature for three hypothetical soils under two typical depth-to-diameter ratios, corresponding to shallow and deep tunnel conditions, respectively. The infiltration process is jointly controlled by the thickness of the unsaturated soil layer, soil type, and infiltration time. The shallower the soil layer, the faster the infiltration. Over time, transient seepage effects extend to the water table. With sufficient time, the area between the water table and ground surface approaches full saturation, reducing suction stress. Transient unsaturated seepage development varies with the soil type. Infiltration in sandy soils is completed in days, while silty and clayey soils take longer.

Comparing our results with those of previous studies [44,45], the trends are consistent and the inflection points are close.

The spatiotemporal impacts of transient unsaturated seepage vary significantly across different soil types. Among the three selected soils (clay, sand, and silt), the transient unsaturated seepage effect exerts the most pronounced influence on clay. Compared to sand and silt, clay exhibits a substantially greater improvement in the safety factor. Figure 3, Figure 4 and Figure 5 demonstrate that the safety factors of all soil types initially increase, then decrease over time. Concurrently, the transient unsaturated seepage effect during infiltration progressively diminishes with time. As shown in the figures, with the increase in temperature, the safety factor of the three kinds of soil decreases. Clay is more sensitive to temperature than sand and silt. And the temperature influence on the case with a larger depth-to-diameter ratio is greater than that with a smaller one. Key observations include the following:

(1)

Layer thickness dependency: Thinner unsaturated soil layers experience earlier turning points and more pronounced safety factor variations. For example,

(a)

At D = 10 m and C/D = 0.5:

(i)

Sand: $\mathrm{t}\approx 8$ days.

(ii)

Silt: $\mathrm{t}\approx 30$ days.

(iii)

Clay: $\mathrm{t}\approx 100$ days.

(b)

At D = 10 m and C/D = 2:

(i)

Sand: $\mathrm{t}\approx 16$ days.

(ii)

Silt: $\mathrm{t}\approx 90$ days.

(iii)

Clay: $\mathrm{t}\approx 800$ days.

(2)

Convergence behavior: the safety factors gradually approach no-suction conditions, with thinner soil layers achieving this equilibrium state faster.

(3)

Early infiltration phase: during initial infiltration stages, saturation degrees and suction stresses near the groundwater level remain relatively unaffected, resulting in minimal safety factor variations.

(4)

Temperature changes affect the safety factor of the inflection point in unsaturated transient seepage. An increased temperature leads to a lower safety factor, and this impact is more significant in tunnels with greater depth-to-diameter ratios.

5. Conclusions

This article proposes a framework for assessing tunnel face safety under transient unsaturated seepage and temperature effects. Using the generalized effective stress principle, it derives the suction stress profile for one-dimensional unsaturated seepage. The extended Mohr–Coulomb criterion describes the unsaturated soil strength, with temperature effects on matric suction also considered. Based on the logarithmic spiral failure mechanism, it calculates the rates of internal dissipation and external work, and uses a kinematic approach related to strength reduction to determine the safety factor of an open tunnel face. An iterative method finds the safety factor with a zero support pressure. The main findings are as follow:

(1)

Rainfall-induced unsaturated transient seepage and ground temperature changes can lead to significant variations in unsaturated effects and temperature responses in unsaturated soils, particularly in silt and clay.

(2)

As the seepage progresses over time, the safety factor first rises, then falls, and gradually nears the value under no-suction conditions. Thinner soil layers reach equilibrium faster. This implies that considering transient unsaturated seepage is beneficial for tunnel face stability during excavation.

(3)

As temperatures change, so does the inflection point of the safety factor in unsaturated transient seepage. Higher temperatures lower the safety factor in tunnels with larger depth-to-diameter ratios. And the sensitivity to temperature changes is in the order: clay > silt $\approx$ sand.

Innovatively incorporating the effects of transient seepage and temperature on unsaturated soils can yield more practical solutions for tunnel face stability and guide real-time support force adjustments during excavation. The failure model and analysis framework developed in this study account for the impacts of transient seepage on the strength of unsaturated soils under temperature effects, providing a strict upper bound solution for tunnel face stability with high computational efficiency to guide face support force adjustments during the construction of the tunnel.

Actually, the combined effect of transient unsaturated seepage and temperature is a complex multiphysics coupling problem. In practical engineering, the two factors often coexist and interact. For example, temperature changes can affect seepage characteristics, while the seepage process can alter the temperature field distribution. Future research can focus on this coupling problem and combine the upper-bound method of limit analysis introduced in this paper and numerical simulation methods (e.g., finite element and finite difference methods). This combination can more accurately simulate the transient unsaturated seepage process and the impact of temperature on unsaturated soils, providing a scientific basis for assessing the stability of tunnel faces.