Deformation evolution characterization of shallow-buried asymmetrically loaded twin-arch tunnels under long-term rainfall

Abstract

The rapid development of global transportation infrastructure has led to increasing applications of large-span shallow-buried tunnels in complex geological conditions. Shallow-buried twin-arch tunnels with asymmetric overburden are prone to deformation and instability under long-term rainfall, particularly in soft clayey strata. This study investigates the rainfall-induced deformation behavior of such tunnels through a combined physical modeling and numerical simulation approach, based on the Wulongshan Tunnel in Nanjing. Results show that rainfall primarily affects shallow slopes within the first 3 d, with delayed seepage responses and infiltration depths up to twice the tunnel diameter. Under sustained torrential rainfall, crown settlement increases by 50% compared to dry conditions, while surface deformation exceeds twice the dry-state value and surpasses crown settlement. A coupled seepage–deformation model incorporating strength softening captures a “slow–rapid–slow” settlement pattern with increasing rainfall and highlights elevated deep-seated sliding risk under extreme conditions. The findings clarify the deformation mechanisms of twin-arch tunnels under rainfall and provide a basis for support design, construction timing, and risk control in similar geotechnical environments.

Graphical abstract

Keywords

deformation evolution / physical model test / twin-arch tunnel / silty clay / shallow-buried asymmetrical loading

1 Introduction

The widespread application of shallow-buried asymmetrically loaded twin-arch tunnels in complex geological environments has heightened concerns regarding their long-term stability, particularly under persistent rainfall conditions. In weak soils such as silty clay, rainfall infiltration leads to increased bulk density, reduced matric suction, and a significant decline in shear strength, which jointly accelerate stress redistribution and deformation accumulation around tunnel structures. These effects pose substantial risks to both construction safety and long-term service performance.

Previous studies have addressed various aspects of twin-arch tunnel behavior under asymmetric loading. Tian et al. [1] used physical and numerical models to analyze asymmetric stress release and proposed optimized material ratios for surrounding rock and support structures. Li et al. [2] proposed grouting techniques to reduce construction-induced impacts. Zhang et al. [3] established relationships between stress and displacement under different overburden thicknesses through theoretical–numerical hybrid approaches. Wang et al. [4] conducted finite element analysis of ultra-shallow, large-span twin-arch tunnels under expressways. Hong et al. [5] explored displacement fields and pressure arch development in loess, while Sui et al. [6] employed distributed optical fiber monitoring over a 12-month period and developed an inverse analysis method based on curved beam theory. Qiu et al. [7] further demonstrated through fluid–solid coupling that stratigraphic inclination causes intensified deformation on deep-side walls and tunnel crowns.

Concurrently, significant progress has been made in understanding slope–rainfall interactions. Deng et al. [8] examined infiltration behaviors in weathered sandstone. Calvello et al. [9], Cai et al. [10], and Zhang et al. [11] developed predictive models for rainfall-induced landslides, incorporating saturated seepage and probabilistic frameworks. Yang and Zhang [12] applied Bayesian inversion for unsaturated hydraulic parameters. Morbidelli et al. [13] synthesized slope–rainfall interaction mechanisms, while Liu et al. [14] analyzed the influence of rainfall intensity, soil type, and slope angle on shallow landslides. Wang et al. [15] implemented Material Point Method (MPM) for rainfall-induced slope failure modeling. Wang et al. [16] used the Material Point Method and unsaturated slope modeling to simulate failure progression and quantify post-rainfall safety factors. Huang et al. [17] developed physical models for red clay slope stability under excavation and rainfall conditions, and Na et al. [18] incorporated temporal rainfall patterns into humidity-index-based stability assessments. With respect to tunnels under rainfall, Xu and Li [19] identified progressive degradation of tunnel performance parameters, while Chen et al. [20] analyzed failure modes and reinforcement strategies for portal collapses under rainfall.

Despite these contributions, existing studies primarily focus on either slope failure mechanisms or isolated tunnel response under dry or simplified saturated conditions. Few have examined the coupled evolution of deformation and seepage in shallow-buried, asymmetrically loaded twin-arch tunnels under long-term rainfall infiltration, particularly in unsaturated silty clay formations typical of floodplain regions. Important issues such as time-dependent crown settlement, slope–tunnel interaction, and rainfall-induced strength softening remain insufficiently explored.

To address these gaps, this study adopts a hybrid approach combining physical model tests with artificial rainfall and numerical simulations. Using the Wulongshan Tunnel in Nanjing as a case study, the research investigates rainfall infiltration patterns, deformation evolution, and earth pressure responses. The findings elucidate the deformation mechanisms of shallow-buried asymmetrical twin-arch tunnels under long-term rainfall and provide scientific guidance for tunnel design, support optimization, and construction risk control in similar geological settings.

2 Model experiment design

2.1 Engineering and geology overview

The Wulongshan Tunnel is situated on the south bank of the Xianxin Road River-Crossing Channel in Qixia District, Nanjing, with a total length of 247 m. Designed as a large-span shallow-buried twin-arch tunnel, it features a maximum cross-sectional span of 33.3 m and a clearance height of 10.9 m, with the maximum overburden depth reaching 35 m. The overlying strata primarily consist of silty clay with thickness varying from 5 to 30 m. These soils exhibit medium to high compressibility and low shear strength. The tunnel foundation is underlain by strongly to completely weathered argillaceous siltstone, while the main structure is entirely embedded within the silty clay layer. Detailed longitudinal geological profiles and cross-sectional designs are presented in Fig. 1 and Fig. 2, respectively.

The tunnel portal section (K5 + 920 to K5 + 960) within approximately 40 m constitutes a shallow-buried asymmetrically loaded zone. Characterized by steep terrain, the minimum overburden thickness is merely 4 m. The crown depth disparity between left and right sides exceeds 200%, with severely compromised self-arching capacity of surrounding rock and significant asymmetrical loading, leading to poor construction stability in the portal section.

The construction site lies within a subtropical humid monsoon climate zone. As depicted in Fig. 3, the annual average rainfall at the tunnel site reaches approximately 1183.8 mm, with over 60% of precipitation concentrated between May and September. Long-term rainfall during the wet season induces notable deterioration of shear strength in the overlying soil layer, exacerbating surrounding rock destabilization during tunnel excavation.

2.2 Model test equipment

This study investigates the mechanical responses, deformation behavior, and seepage characteristics during the construction of large-span shallow-buried asymmetrically loaded twin-arch tunnels in weak soil mass under rainfall conditions. To this end, a specialized model testing apparatus for simulating rainfall effects during tunnel excavation was developed. The experimental setup comprises four core subsystems: a model box system, an artificial rainfall system, a tunnel excavation system, and a monitoring and data acquisition system. The integrated system configuration is illustrated in Fig. 4(a).

Model box system. The model chamber measures 120 cm × 100 cm × 60 cm (length × height × width). Constructed with a steel frame, its front and rear panels are composed of acrylic glass plates for observational clarity. Drainage holes at the base are filled with gravel-sand mixtures to prevent soil loss while maintaining permeability. To ensure the accuracy of model test results and eliminate potential boundary effects, the chamber dimensions were determined based on Saint-Venant’s Principle (Xu et al. [21]). The effective distance between the region of interest (tunnel and adjacent slope) and model boundaries exceeds five times the tunnel diameter, which effectively ensures that the influence of chamber boundaries on the local stress and deformation fields is negligible. A schematic of the chamber is shown in Fig. 4(b).

Artificial rainfall system. A programmable nozzle-based rainfall simulator utilizes 12 misting sprinklers (arranged in 4 columns × 3 rows) mounted on angle steel frames above the chamber. Each nozzle achieves a maximum precipitation coverage diameter of 60 cm, enabling uniform rainfall simulation across the entire chamber. The system replicates rainfall intensities ranging from 0 to 130 mm/h. Figure 4(c) illustrates the rainfall apparatus.

Tunnel Model. Prefabricated tunnel linings employ 304-grade waterproof stainless-steel strips. Based on equivalent bending stiffness principles for thin-walled cylindrical structures, the lining thickness is set to 0.3 mm.

Monitoring and data acquisition system. The monitoring equipment includes high-precision displacement meter, waterproof dial indicator, micro soil pressure gauge, strain test element, micro pore water stress gauge, soil moisture monitors and data acquisition instrument. The specific range and accuracy are shown in Table 1.

2.3 Parameter and working condition selection

According to the First Similarity Theorem, where subscripts p and m denote prototype and model physical quantities respectively, and C represents similarity constants, this study implements a controlled relaxation of gravitational similitude requirements due to technical limitations in artificial mass compensation. This decoupling strategy enables independent selection of elastic modulus ratio, density ratio, and geometric scaling ratio. The experimental setup employs remolded site-specific silty clay in stratified compaction layers, maintaining identical soil reconstitution protocols with established rainfall-tunnel interaction studies. Currently, there is no uniform criterion for setting the similarity ratio of rainfall intensity. This test mainly refers to the setting of rainfall similarity relationship by Fang et al. [22], namely C_{\mathrm{q}} = C_{\mathrm{t}}= \sqrt{C_{\mathrm{L}}}. Key physical similitude ratios are systematically tabulated as follows.

According to the general classification method of the meteorological department, the rainfall levels can be divided into six grades as shown in Table 2 according to the height of rain falling to the ground within 24 h (mm/24 h).

The experimental program simulates a 7-d construction cycle, with tunnel excavation divided into 7 sequential rounds following each rainfall event. Based on 24-h rainfall intensity classification (Table 3) and operational feasibility considerations, a standardized 6-h effective rainfall duration per day was adopted. Through precise modulation of rainfall nozzles, slight adjustments to rainfall duration and intensity were implemented while maintaining constant cumulative precipitation. All test conditions are detailed in Table 4.

2.4 Measurement point arrangement and test process

2.4.1 Measurement point layout

Three monitoring cross-sections are established along the tunnel model: Portal section (I-I cross-section); Central section (II-II cross-section); Exit section (III-III cross-section). Subsequent sections detail the instrumentation layout and monitoring apparatus configuration within each cross-section.

1) Tunnel instrumentation layout

At each of the three monitoring cross-sections, the following devices are installed symmetrically on both left and right sides: Crown area. Dial indicators to measure tunnel settlement deformation; earth pressure cells to monitor surrounding rock pressure; strain gauges mounted on the extrados of lining crown for internal force analysis. Sidewall area. Earth pressure cells at exterior sidewalls for lateral earth pressure monitoring; strain gauges on lining sidewall extrados to capture axial force variations. Invert area. Strain gauges on lining invert extrados to assess uplift resistance characteristics; The comprehensive layout is illustrated in Fig. 5.

2) Slope instrumentation layout

An earth pressure box is placed at the top of each of the three sections to monitor the earth pressure at the top of the slope; a displacement meter is placed at the top and middle of each of the three sections to monitor the settlement of the top and slope. A pore water stress gauge is placed every 9 cm from bottom to top on the soil above the partition wall of the tunnel in the three sections, and 2, 3, and 4 are arranged in sections I-I, II-II, and III-III, respectively. A moisture meter is arranged every tunnel diameter (9 cm) from the top of the slope (2 cm) along the height direction from top to bottom to analyze the rainfall infiltration law at different depths. The layout is shown in Fig. 6.

2.4.2 Test process

This experimental study investigates tunnel excavation under rainfall conditions through a scaled physical model, integrating two sequential phases: artificial rainfall simulation and tunnel construction. The procedure initiates with precise calibration of rainfall nozzles to achieve target precipitation intensity. Following each 30-min rainfall cycle, manual tunnel excavation is executed incrementally at a rate of 8.5 cm per ring, accompanied by photogrammetric displacement documentation and real-time data acquisition. After confirming stabilization of both seepage field and stress field, the cycle repeats until completing seven excavation rings. Figure 7 shows the model test process.

3 Experimental results and analysis

3.1 Seepage evolution on tunnel slopes

3.1.1 Soil saturation variability

Figure 8 shows the development curve of soil saturation under rainfall, and the results show that the soil saturation of the biased slope is about 40% under the initial condition. Under Torrential rain conditions, the monitoring point at the slope surface (d = 2 cm) responded rapidly, and the saturation increased to more than 90% in the first 1.2 h, at a rate of about 33.3%/h, after which the growth rate slowed down to an average of more than 95% saturation in the first 1.5 h, and then reached about 99% at the end of the rainfall. For the soil below the surface layer of 1 time the diameter of the hole (d = 11 cm), the initial saturation change is small, after 1 h with the surface layer of soil close to saturation, the excess rainwater gradually infiltration, resulting in an increase in the rate of increase in saturation, the end of the rainfall saturation is about 60%. The saturation of the soil 20 cm away from the slope surface remained between 40% and 45% throughout the test, and the effect of rainfall infiltration on it was negligible.

3.1.2 Soil pore water stress pattern

Figure 9 shows the development curve of soil pore water stress under torrential rain. Before rainfall, initial pore water stress at all monitoring points was within 0.05 kPa. During the first 30 min, except for surface gauges A, C, and F, other points remained nearly unchanged, indicating rainfall was mainly absorbed by the upper soil layer with limited infiltration depth. From 0.5 to 1 h, surface pore water stress increased rapidly, with central point B, D, and G also showing significant rise. After about 1.2 h, surface points tended to saturate, while points E, H, and I, located twice the borehole diameter from the slope surface, continued to increase at a slightly faster rate, indicating deeper infiltration. Throughout the test, points farther than twice the borehole diameter (e.g., E, H, I) showed minimal changes, suggesting limited rainfall impact below this depth during tunnel excavation.

Under torrential rain, pore water stress in the surface layer of the asymmetric slope was highest, averaging 2.93 kPa with a rising rate of about 2.42 kPa/h. Increments at points beyond twice the borehole diameter (e.g., A, E, I) were within 0.6 kPa. Overall, pore water stress is influenced by rainfall intensity and location: higher intensity and longer duration lead to greater stress and faster increase; at the same rainfall, soil farther from the infiltration surface responds slower with smaller changes. During the 7-d construction period, infiltration depth under torrential rain approached but did not exceed twice the borehole diameter.

As shown in Fig. 10, pore water stress at shallow measuring points (e.g., A, C, F) increases rapidly during the early stage of rainfall and approaches a quasi-steady-state at approximately 1.7 h. In contrast, deeper points (e.g., E, H, I) exhibit delayed but continuous growth, reflecting the progressive downward movement of the wetting front under unsaturated infiltration. To better capture this process, stress–depth profiles at 0.8 h and 1.7 h were extracted and fitted using power-law models, as shown in Fig. 10. The results indicate that at 0.8 h, the stress is concentrated in the surface zone with limited penetration, while by 1.7 h, the stress distribution becomes more uniform, indicating deeper infiltration and a gradual extension of rainfall effects. The selection of these two moments allows for the representation of both the initial and transitional phases of the infiltration process.

Based on the fitted parameters, an empirical model was developed to describe the coupled relationship among pore water stress (σ), depth (d), and rainfall duration (t):

(1)\sigma (d,t)={\left(\frac{d}{2.23t+1.47}\right)}^{\frac{1}{0.120t+0.883}}\left(\mathrm{k}\mathrm{P}\mathrm{a}\right).

This model effectively captures the nonlinear attenuation of pore water stress with depth and its amplification over time during the transient stage of unsaturated infiltration. However, as the model is derived from data before full stabilization of surface stress, it is most applicable when rainfall is ongoing, and the wetting front is still advancing. Once the shallow soil reaches full saturation and stress stabilizes, the vertical stress distribution may deviate from the power-law pattern. Therefore, this relationship is primarily suitable for analyzing the early to mid-stages of infiltration and for estimating stress propagation during active rainfall conditions.

3.2 Characterization of tunnel-slope deformation

3.2.1 Characterization of tunnel crown settlement

Crown settlement is a key indicator for tunnel engineering monitoring, which can reflect the safety and stability of the tunnel and surrounding rock and guide the adjustment of construction and support strategies. This model test focuses on the crown settlement of the initial support mold during tunnel excavation, and three monitoring sections, I-I, II-II, and III-III, are set up according to the depth of burial and the degree of bias, and the monitoring results are shown in Fig. 11. In the figure, a negative settlement value indicates downward deformation, and a positive value indicates upward deformation. The test procedure is divided into seven groups, each containing three steps, corresponding to one artificial rainfall, one excavation of the left hole (shallow-buried side) and the right hole (deep-buried side) of the tunnel, and step 0 represents the initial state.

As can be seen from Fig. 11, the settlement curves of each monitoring point show a “step” change with the excavation. Taking no rainfall condition as an example, the deformation law of tunnel excavation is analyzed. At the initial stage, when the palm face is far away from the monitoring surface, the settlement increases slowly. When the palm face is close to and passes through the monitoring surface, the crown settlement increases sharply due to the concentration of stress release around the tunnel as the excavation progresses. The surrounding soil experiences rapid relaxation as the excavation face passes, leading to a sudden increase in settlement. After the palm face passes through, the stress redistribution effect weakens, and the settlement curve’s slope decreases, as the system stabilizes and the rate of settlement slows, and the growth slows down, until the palm face is far away from the monitoring surface after two times of the hole diameter distance, and then the settlement tends to be stabilized. And because the left side is shallow-buried side and the right side is deep-buried side, the increase of crown settlement on the right side and the final settlement are larger than that on the left side. When the excavation reaches the monitoring section I-I, II-II, III-III, the arch deformation generated by the current excavation step can reach more than 60% of the total deformation.

After artificial rainfall, although the trend of crown settlement is similar to that when there is no rainfall, the infiltration of rainwater leads to the increase of soil bulk weight, the increase of load at the top of the tunnel and the degradation of peripheral rock properties, which in turn significantly improves the cumulative settlement value of the crown on the lining molds, and the intensity of rainfall is positively correlated with the increase of settlement. For example, in the monitoring section III-III, the crown settlement values of the right cavern were 2.10 and 3.16 mm under the two working conditions of no rainfall and Torrential rain, i.e., under Torrential rain conditions, the right crown settlement was increased by 50.6% compared with that of no rainfall, respectively.

As can be seen in Fig. 12, the average crown settlement value of the right side of the three sections is about 25.0% more than that of the left side due to the bias effect of the soil in the upper part of the tunnel in the case of no rainfall. In Torrential rain case, the average crown settlement value of the right side of the three sections increases by about 50.8% compared to the no-rainfall case. This indicates that rainfall infiltration will increase the soil bulk weight and weaken the surrounding rock properties, thus increasing the crown deformation.

3.2.2 Characterization of surface settlement of overlying soils

The evolution of slope surface settlement with rainfall and excavation progress was plotted from the monitoring data, where a negative value of settlement indicates downward settlement and a positive value indicates upward bulging. The excavation steps are the same as in the previous subsection.

From the analysis in Fig. 13, the monitoring area shows an overall subsidence trend with tunnel excavation and rainfall. Under the condition of no rainfall, the vertical settlement at each monitoring point is between 1.25 and 1.65 mm, which is relatively small compared to the crown settlement.

In the case of no rainfall, the deformation of the slope during tunnel excavation is mainly due to the disturbance of the soil body during excavation, which changes the original distribution of the ground stress, and then leads to the concentration or release of the stress, which leads to the deformation of the slope surface. Rainfall significantly exacerbated the vertical deformation of the slope surface, especially under Torrential rain conditions, which is different from the time when there is no rainfall, when the softening of the slope body is the main cause of the deformation of the slope body, and the settlement value of each monitoring point is more than twice that of the time when there is no rainfall, and it exceeds the average crown settlement of the tunnel. This shows that the increase in rainfall intensity directly leads to an increase in slope surface settlement, and during Torrential rain, the slope surface settlement even exceeds the tunnel crown settlement deformation, so it is necessary to focus on preventing the risk of slope surface collapse.

As can be seen from Fig. 14, the average slope top settlement for the three sections is about 10.9% more than that in the middle of the slope under the no-rainfall condition due to the redistribution of stresses in the slope caused by the tunnel excavation. The average cumulative settlement value of the top of the slope in the three sections under Torrential rain condition is twice as much as the increase in the no-rainfall condition. In the Torrential rain condition, the settlement was mainly concentrated in the initial rainfall, attributed to the low water content of the soil body at the initial stage, and the rapid infiltration of rainwater led to a surge in the downward force of the slope, a sharp reduction in the shear capacity, and accelerated deformation. Subsequently, the soil body is close to saturation and the settlement rate slows down. In addition, under Torrential rain conditions, the settlement of the top of the slope in section III-III exceeded 0.8 mm during the 6th rainfall, which was presumably caused by the puddle formed by continuous infiltration and scouring.

3.3 Tunnel-slope earth pressure evolution analysis

There are significant differences in the surrounding rock stresses in various regions of the shallow-buried asymmetrically loaded twin-arch tunnel, especially near the arch roof and sidewalls. Therefore, this subsection monitors and analyzes the surrounding rock earth pressure at the tunnel crown and outer sidewalls for the whole test, and the specific results are shown in Fig. 15.

According to the data in Fig. 15, the data of the measurement points on the deeply buried side of each section are generally higher than those on the shallowly buried side, but there is a significant difference in the development pattern of the crown and the side wall. For the crown perimeter rock pressure, the earth pressure at the monitoring points shows a serrate pattern of gentle development to rapid decline and then to gentle change in the absence of rainfall, in which the earth pressure in the left and right crown areas of the I-I, II-II, and III-III sections decreases significantly in specific excavation steps (0–2, 6–8, and 12–14), which is attributed to the stress release after tunnel excavation. stress release after tunnel excavation. The increase in burial depth resulted in a relative increase in the percentage of stress release in the surrounding rock, with decreases ranging from 58% to 78% on the shallow-buried side and more than 80% on the deeper buried side. Under rainfall conditions, the earth pressure in the crown perimeter rock rises due to the increase in the bulk weight of the soil, and the effect of increasing rainfall intensity is enhanced. For example, under Torrential rain condition, the earth pressure in the left hole (shallow-buried side) of I-I monitoring surface increased by 25%. At the same time, the rainfall weakened the unloading effect during excavation, and the proportion of stress relief decreased under Torrential rain condition (e.g., from 58.9% to 47.3%) compared with that in the absence of rainfall, which was attributed to the softening of geotechnical properties by the rain.

The earth pressure at the arch waist of the tunnel is smaller than that at the arch top, and it tends to increase with the excavation, and the increase is especially significant in the left hole of the shallow-buried side, which is due to the stress concentration under the bias load caused by the leftward shift of the tunnel chamber after the excavation. Torrential rain exacerbates this phenomenon, and the earth pressure at the arch waist of the left cavern increases by 108.9% under Torrential rain condition, which is much higher than that of 67.2% when there is no rainfall. Therefore, the shallow-buried asymmetrically loaded twin-arch tunnel excavation is prone to lead to stress concentration in the arch girdle on the shallow-buried side, and rainfall exacerbates this effect.

4 Numerical calculation method for soft soil tunnel considering rainfall softening effect

4.1 Numerical modeling method for tunnel excavation considering rainfall softening effect

4.1.1 Flow-consolidation coupled computational model considering the effect of saturation on the weakening of soil strength parameters

When considering seepage from slopes and tunnel envelope, the geotechnical body can be regarded as a pore-continuous medium. Therefore, the change of soil stress due to rainfall can be expressed by the theory of Terzaghi base:

(2){\sigma }^{\prime }=\sigma -p_{\mathrm{w}},

where σ is the total stress, σ′ is the effective stress and p_w is the pore water stress.

For unsaturated soil, the effective stress has the following expression:

(3){\sigma }^{\prime }=\sigma -S_{\mathrm{r}}\cdot p_{\mathrm{w}},

where S_r is the degree of saturation of the soil.

The unsaturated soil seepage field is described by the Richards equation for rainfall-induced moisture transport:

(4)\rho \left(\frac{C}{\rho g}+S_{\mathrm{r}}{S}_{\mathrm{S}}\right)\frac{\mathrm{\partial }p}{\mathrm{\partial }t}-\mathrm{\nabla }\cdot \rho \left[\frac{K_{\mathrm{S}}{k}_{\mathrm{r}}}{\rho g}\left(\mathrm{\nabla }p+\rho g\mathrm{\nabla }D\right)\right]=Q_{\mathrm{m}},

where \rho is the fluid density, C is specific water coefficient, g is gravitational acceleration, \mathrm{\nabla }D is the hydraulic gradient caused by gravity head, S_S is the water storage rate, k_r and K_S are the relative permeability coefficients of unsaturated soil and saturated soil, respectively, and Q_m is the mass source term.

The degree of saturation is closely related to soil seepage, and the unsaturated seepage of soil is calculated using the V-G [23] model of Eq. (2).

(5)\theta ={\theta }_{\mathrm{r}}+\frac{{\theta }_{\mathrm{S}}-{\theta }_{\mathrm{r}}}{{\left[1+{\left(\psi /\alpha \right)}^n\right]}^m},

where {\theta }_{\mathrm{r}} is the residual liquid volume water content, {\theta }_{\mathrm{S}} is the saturated volume water content, \psi is matrix suction, α, m, and n are fitting parameters.

After combining the above equation with the theory of Mualem [24] the relationship equation for the permeability coefficient of the soil can be obtained:

(6)K=K_{\mathrm{S}}\frac{{\left[1-\left(\alpha {\psi }^{n-1}\right)\left(1+{\left(\alpha {\psi }^n\right)}^{-m}\right)\right]}^2}{\left({\left(1+\alpha {\psi }^n\right)}^{\frac{m}{2}}\right)}.

A large number of studies have shown that changes in saturation caused by seepage in the soil have a significant effect on the strength of the soil (Al and Bandini [25], Fan et al. [26], Gu et al. [27]). According to the related research of Wu [28], the relationship between the strength parameters of pulverized clay and saturation is expressed as follows:

The data collected from the shear test of silty clay were analyzed by linear regression after taking the logarithm of cohesion as shown in Fig. 16. The relationship between cohesion and saturation is:

(7)\mathrm{l}\mathrm{g}c=1.779-0.005\times S_{\mathrm{r}},

(8)c=60.117\times 10^{-0.005S_{\mathrm{r}}}.

The relationship between the friction angle of soil under different saturation can be expressed as follows:

(9)\phi =26.710-0.116\times S_{\mathrm{r}}.

The compression modulus and elastic modulus of the samples at different saturation levels are linearly fitted, and the relationship equation is:

(10)E=\left(-0.561S_{\mathrm{r}}+73.656\right).

4.1.2 Calculation method of tunnel rainfall considering soil seepage-softening effect

Since numerical calculation cannot directly couple the transient calculation of rainfall infiltration with the steady-state calculation required for tunnel excavation, the transient rainfall-steady-state excavation deformation step-by-step iteration method is used for calculation. The specific calculation steps are as follows. 1) First, a steady-state calculation is performed on the initial stress state of the model. 2) The first round of rainfall calculation is performed, and the soil saturation changes through rainfall infiltration to calculate the slope deformation caused by rainfall. 3) The seepage field, stress field, and deformation field generated by the first round of rainfall are introduced into the steady-state calculation, so that the calculation results of the previous step become the initial boundary conditions of the steady-state calculation. Then, the tunnel excavation calculation is performed to obtain the deformation caused by tunnel excavation under a certain rainfall state. 4) The stress field and deformation field obtained by the steady-state calculation are iterated into the transient calculation of the next round of rainfall. 5) Steps 2) to 4) are repeated until the calculation is completed. The specific calculation process is shown in Fig. 17.

In addition, to reduce the computational deviation, the convergence tolerance for displacement and stress was strictly controlled, and the number of calculation steps was increased appropriately. In particular, smaller time steps were applied during key stages such as peak rainfall periods and tunnel face advancement, to improve temporal resolution and ensure better numerical accuracy.

4.2 Model calculation parameters and working condition selection

According to Saint-Venant’s principle, considering the excavation boundary effect, the model size is set to 150 m in the x direction, 72 m in the y direction, 60 m on the left boundary and 110 m on the right boundary in the z direction. The single span of the main tunnel is about 14 m, the span of the middle partition wall is about 5 m, and the groundwater level is 10 m below the tunnel. Free tetrahedral meshing is adopted. The schematic diagram of the model is shown in Fig. 18. There are 14 steps and seven rings in total. First, continuous rainfall for 7 d, each ring includes one left tunnel excavation and one right tunnel excavation.

According to the site conditions, the overall depth of the tunnel inlet section is below 2 times the tunnel diameter, so the soil layer overlying the tunnel is generalized as a slope with 45° inclination, and the depth of the shallow side crown at the inlet is set to be 0.5 times the tunnel diameter, and the depth of the shallow side crown at the outlet is set to be 2 times the tunnel diameter, and the depth of each section is increased along the tunnel boring direction, and the tunnel height-to-span ratio is 0.8. In terms of the size of the rainfall, we refer to the common division method of the meteorological department and the model test to set up four groups of rainfall conditions, namely no rainfall, light rain, heavy rain and heavy rain, and set up a group of heavy rain to impose a retaining wall. In terms of rainfall size, referring to the general classification method of the meteorological department and the model test conditions, four groups of rainfall conditions are set up, namely, no rainfall, light rain, heavy rain and Torrential rain, and one group of Torrential rain is applied to the retaining wall, and the angle of inclination of the ground surface is 45°. The specific working conditions are shown in Table 5.

This calculation model mainly contains two major calculation interfaces of solid mechanics and Richard’s equation, so the model parameters include two parts of physico-mechanical parameters and unsaturated seepage related parameters, and the physico-mechanical parameters are shown in Table 6. The shear strength and stiffness index of the soil body must consider the softening effect in the process of humidification, with reference to the calculation of Eqs. (6) and (9) in Section 4.1, this calculation selects the Van.Genuchten model with strong applicability for seepage field calculation, with reference to the suggested values of the field geological investigation report and the existing references, and the values of the relevant hydraulic parameters are shown in Table 7.

4.3 Model validation

Taking working condition 1 as an example, the deformation trend of the tunnel excavation process is analyzed. The model test and numerical simulation data are dimensionless processed by Eq. (11). The comparison between numerical simulation and model test is shown in Fig. 19.

(11)x_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=\frac{x-x_{\mathrm{m}\mathrm{i}\mathrm{n}}}{x_{\mathrm{m}\mathrm{a}\mathrm{x}}-x_{\mathrm{m}\mathrm{i}\mathrm{n}}}.

Through the normalized comparative analysis of the numerical simulation and model test results, it was found that the two showed a high consistency in the displacement trend of the tunnel crown. Under the conditions of the portal tunnel, central tunnel and exit tunnel, the displacement of the left and right crowns showed a gradual attenuation characteristic with the advancement of the construction sequence. The numerical simulation curve was consistent with the test data curve trend, which verified the reliability of the numerical model; however, the test value in the middle tunnel area (the third to fifth rings) was about 8%–12% higher than the simulation value, which may be related to the sensitivity of the surrounding rock contact surface parameters or the simplification of the test boundary conditions. In general, the numerical simulation results are relatively accurate, and the model verification is correct.

5 Slope-tunnel deformation evolution under rainfall action

5.1 Analysis of the seepage-strength evolution law of the slope body

5.1.1 Analysis of soil saturation law

Figure 20 is a cloud diagram of soil saturation distribution under the influence of different rainfall intensities. Under no rainfall conditions, the saturation in the slope decreases linearly from bottom to top along the water level line, and the initial saturation is about 50% lower when it reaches the surface. When the slope is affected by light rain, due to limited rainfall, the saturation only increases slightly above the surface. Currently, the soil on the slope surface is not fully saturated, so the infiltration effect of water in the soil is not obvious. The effect of rainfall on the increase of the sliding force on the slope is not obvious, and the weakening effect of soil strength has limited impact on the tunnel. When the rainfall intensity is heavy rain, the saturation of the top of the slope and the slope surface is further expanded, the rainfall penetration depth increases, and an approximately saturated zone is formed from the foot of the slope to the left of the shallow-buried side arch, and the strength of the surrounding rock of the shallow-buried tunnel is significantly weakened. When the rainfall intensity is heavy rain, the saturation at different depths in the slope increases, and the slope surface reaches saturation. The groundwater level rises slightly, resulting in a significant increase in the saturation deep in the slope. As a result, the soil strength at the tunnel location is significantly weakened under the combined effect of slope surface infiltration and groundwater rise, and the eccentric thrust is significantly increased, thereby reducing the stability of the tunnel.

To analyze the saturation evolution law of the tunnel surrounding rock under the influence of rainfall infiltration, three characteristic cross-sections: Portal section (adjacent to tunnel entrance); Central section (mid-span of tunnel alignment); Exit section (proximate to tunnel terminus), and the specific laws are shown in Fig. 21. Torrential rain significantly affects the moisture dynamics of the surrounding rock: under the condition of light rain (Fig. 21(a)), only the saturation degree of the left crown of the portal tunnel is increased by about 5%; under the condition of heavy rain (Fig. 21(b)), the saturation degree of the left crown of the exit tunnel reaches more than 90%, with an average growth rate of 3%/d, which indicates that the moisture content of the soil is positively correlated to the intensity of the rain; under the condition of Torrential rain (Fig. 21(c)), the saturation degree of the shallow side of the tunnel (buried depth < 1.25 times of the diameter of the tunnel) reaches 99.5% at the entrance and 99.5% at the exit of the tunnel, respectively. saturation reached 99.5% and 87.4%, respectively, while there was no significant change in saturation in the area with burial depth > 2 times the diameter of the hole. In conclusion, Torrential rain will induce rapid saturation of the shallow-buried surrounding rock and aggravate the risk of deformation, but the effect of rainfall is negligible when the burial depth is more than 2 times the diameter of the hole, so it is recommended to pay attention to the stability of the shallow-buried section during the construction period.

5.1.2 Analysis of soil pore water stress law

As a key component of the strength characteristics of soil under infiltration conditions, the changes in its distribution characteristics under different rainfall conditions have an important impact on disasters such as landslides, as shown in Fig. 22. Under non-rainfall conditions, the pore water stress in the slope body gradually decreases from bottom to top, with the groundwater level as the boundary, positive below the water level and negative above. Because the soil is saturated below the water level, and there is matrix suction between the unsaturated soil particles above, the boundary is obvious. Under the action of light rain, the pore water stress cloud map of the model changes little, only affecting the surface soil near the top of the slope, which is consistent with the distribution law of the saturation field. When the rainfall intensity is heavy rain, the rate and amplitude of the change of pore water stress will increase significantly, the saturation of the slope surface will increase rapidly, and the effect of matrix suction between soil particles will be weakened. Under the action of heavy rain, the change of pore water stress on the slope surface tends to be gentle, and due to the seepage of surface rainwater to the deep layer, the positive pore water stress of the saturated zone soil increases accordingly. In summary, when the rainstorm lasts for more than 7 d, the matrix suction of the soil on the entire slope surface is almost completely lost, and the pore water stress approaches 0, resulting in a significant decrease in shear strength and the slope is very likely to slip and become unstable.

As shown in Fig. 23, the pore water stress at each measuring point is negative, indicating that the surrounding rock is mostly in an unsaturated state, and its change pattern is affected by rainfall and burial depth. After 7 d of light rain, only the pore pressure of the left arch of the cave increased by about 35 kPa, and the other measuring points remained almost unchanged; under the action of heavy rain, the pore pressure of the shallow-buried side arch of the cave increased from –201 to –65 kPa, the left arch in the cave increased to –161 kPa, and the other measuring points changed by < 15 kPa; after 7 d of torrential rain, the pore pressure of the left arch of the cave was close to 0, the left arch in the cave dropped to –87 kPa (saturation > 90%), and the other measuring points changed by 10–25 kPa. The results show that the greater the rainfall intensity and the shallower the burial depth, the faster the pore pressure response and the greater the amplitude. Light rain has limited effect on the strength of surrounding rock with a burial depth of less than 0.5 times the tunnel diameter, while the pore pressure of shallowly buried surrounding rock approaches 0 after 7 d of torrential rain, the soil strength is significantly reduced, and the risk of construction instability is high; when the burial depth is greater than 2 times the tunnel diameter, the impact of rainfall can be ignored.

5.2 Analysis of tunnel-slope deformation evolution

Figures 24 and 25 show the deformation evolution curves of the tunnel under different intensities. When there is no rainfall, the settlement of the tunnel crown is the smallest, and the deformation of the deep-buried side is more serious than that of the shallow-buried side. The settlement is distributed in a zigzag shape with the excavation process, and the maximum settlement occurs at the crown of the exit section (48.57 mm). After 7 d of continuous light rain, the settlement of each section increased, and the cumulative settlement value was between 27 and 54 mm. Under heavy rain and torrential rain conditions, 7 d of continuous rainfall made the slope almost saturated, the soil gravity increased, and the surrounding rock deformation increased sharply. The maximum settlement reached 71.4 mm, which was 46.9% higher than that when there was no rainfall, exceeding the control value, and deformation space needed to be reserved, and support strengthened. Under different rainfall intensities, the maximum cumulative settlement values were 48.57, 53.35, 71.36, and 77.64 mm, respectively. Compared with no rainfall, the settlement of light rain, heavy rain, and torrential rain conditions increased by 11.1%, 46.9%, and 59.9%, respectively. Rainfall will intensify the settlement of the tunnel crown, and the settlement will accelerate with the increase of rainfall intensity. However, when the intensity exceeds 50 mm/d, the settlement growth rate slows down and the sensitivity of surrounding rock deformation to rainfall intensity decreases.

As shown in Fig. 26, under different rainfall conditions, the overall horizontal displacement of the soil is negative, indicating that the tunnel and the slope move to the left as a whole under the bias pressure, and the maximum horizontal deformation area is located in the surrounding rock and slope surface to the left of the outer wall of the shallow-buried side of the tunnel. As the rainfall intensity increases, the soil bulk density increases, the sliding force on the slope and the tunnel load intensify, and the shear strength and elastic modulus of the soil soften, resulting in a significant increase in the horizontal displacement of the soil, such as from about 5.9 cm when there is no rainfall to more than 10 cm after 7 d of torrential rain, an increase of nearly 100%. Comprehensive analysis shows that, therefore, under heavy rainfall conditions, the slope surface deformation is more serious during the construction of the shallow-buried bias pressure multi-arch tunnel, especially in heavy rain weather, the risk of horizontal extrusion and shear sliding damage of the tunnel and slope is higher than the risk of vertical roof collapse settlement damage.

6 Conclusions

This study systematically investigates the strength degradation patterns of silty clay strata under long-term rainfall conditions through an integrated methodology combining physical model tests and numerical simulations. The research further elucidates the mechanisms governing the deformation of surrounding rock in shallow-buried asymmetrically loaded twin-arch tunnels within silty clay strata under varying rainfall intensities. The key findings are as follows.

1) The impact of rainfall on shallow slopes is mainly within the first three days of rainfall. The soil saturation and water pressure inside the slope have a hysteresis effect on the response to rainfall, and the response time increases with the increase in depth. When the slope depth exceeds 1 time the tunnel diameter, the seepage is approximately linear with the increase in rainfall. The maximum infiltration depth of rainfall on the slope is about 2 times the tunnel diameter.

2) Due to the shallow overall cover thickness, the excavation of asymmetrically loaded twin-arch tunnels causes greater deformation in deep-buried measurements than in shallow-buried measurements. Under continuous torrential rain conditions, tunnel deformation increases by about 50% compared to no rainfall, while the deformation of the slope surface reaches more than twice that of no rainfall conditions. Moreover, the deformation of the slope surface after rainfall is greater than the deformation of the tunnel arch. Continuous heavy rainfall causes the dominant factor of slope stability to change from tunnel excavation to slope softening.

3) A deformation-seepage evolution calculation model considering the softening effect of soil strength under the influence of rainfall was established. Through numerical analysis, it was found that the tunnel crown settlement showed a trend of slowly increasing, then rapidly increasing, and then slowly increasing with the increase of rainfall. When the rainfall was heavy, the tunnel deformation increment was the largest. In addition, with the increase of rainfall, the horizontal deformation of the tunnel also increased significantly, especially the deformation at the slope angle was the most obvious. The slope was at risk of deep sliding under the influence of heavy rainfall.

This study provides a theoretical and practical basis for tunnel design in soft clay strata under long-term rainfall. The identified three-stage deformation pattern, pore water stress response, and asymmetric pressure redistribution offer guidance for support optimization, drainage planning, and excavation control. The integration of physical modeling with coupled seepage–deformation simulation, along with strength parameter updating based on matric suction, enhances modeling accuracy under unsaturated conditions. However, the study is limited by the use of scaled models, simplified boundary conditions, and reliance on a single tunnel case. Future research should incorporate multi-field monitoring data, explore more diverse geological and hydrological settings, and consider material anisotropy, rainfall variability, and time-dependent strength degradation. Expanding the coupling framework to include thermal, chemical, or creep effects may further improve the applicability of the model in complex engineering scenarios.

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