1 Introduction

Owing to the increase in underground space exploitation, a large number of foundation pit projects have been carried out by deep excavation technology in unsaturated soils [32]. Once rainfall occurs during excavation, variation in the hydraulic properties of unsaturated soils may lead to a potential risk for both foundation pits and pre-existing infrastructures [30, 51]. For instance, the torrential rainfall in Henan Province of China, in July of 2021, caused 142 foundation pits under construction, of a total of 604, to collapse with damage to nearby infrastructures [22]. To ensure engineering safety, it is essential to investigate the responses of foundation pits to rainfall and excavation in unsaturated soils.

In the past decades, several studies have been carried out in order to study the excavation-induced performance of foundation pits [9, 18, 20, 27, 57]. Only a little attention has been paid to the rainfall-induced performance of foundation pits in unsaturated soils. In fact, rainfall infiltration could significantly affect the performance of foundation pits, since it varies the hydraulic properties of unsaturated soils [26, 33, 45,46,47]. To reveal the rainfall-related issues, Shahrokhabadi et al. [36] and Huang et al. [21] carried out analytical methods to assess the effect of rainfall on supporting structures in unsaturated soil. Wang et al. [42] established a 2D numerical model with fixed hydraulic conductivities of soils, while Gao et al. [14] proposed a 2D numerical model that included a simplification of the hydraulic gradient reverse change for equivalent unsaturated conditions. However, all these studies ignored the effect of post-rainfall excavation. Based on Jiang and Tan [22], 22.4% of 170 rainfall-related failures of foundation pit cases took place after the rainfall, indicating that there is still a high risk of failures in post-rainfall excavation construction. Overall, the rainfall-related performance of foundation pits could be treated as a consequence of the combined effect of rainfall, excavation construction, and geological conditions.

As recognized in the literature [48, 49], excavation will change the entire seepage field by changing both seepage paths and boundaries. Meanwhile, rainfall infiltration will change the stress field by changing the pore-water pressure and the hydraulic properties of unsaturated soil [41, 43, 48]. The hydro-mechanical coupling effect that is induced by the above factors inherently governs the excavation-rainfall-excavation performance [6, 8, 52, 56]. In the majority of the previous studies, the seepage and stress are usually treated separately in two steps: (1) analyzing the stress field first and (2) imposing rainfall on the seepage boundary, without considering the post-rainfall excavation [14, 24, 39, 42, 48]. Zhang et al. [54] conducted a preliminary study on the excavation-rainfall-excavation performance through a 2D numerical model. However, the hydro-mechanical coupling response of the foundation pit to rainfall and excavation is still unclear.

In the present work, numerical simulations of a case study of the braced foundation pit were conducted to investigate its performance in unsaturated soil under rainfall and excavation. A 3D numerical model was developed, by considering the hydro-mechanical coupling effect, as well as the variation in the hydraulic properties of unsaturated soils. Based on the numerical model, the performances of the foundation pit can be discussed in detail.

2 Methodology

To investigate the performance of foundation pits under the combined effect of rainfall and excavation in unsaturated soil, the Biot consolidation theory was applied to consider the hydro-mechanical interaction. The groundwater flow was described by Darcy’s law. Regarding the constitutive relationship, the van Genuchten model was employed to describe the variation in the hydraulic properties of unsaturated soils. The Hardening Soil (HS) model was used to describe the stress and strain behavior of the soils.

2.1 Hydro-mechanical coupling method

2.1.1 Continuity equation

The water’s flow in saturated soils is commonly described by Darcy’s law, which has been used to describe fluid transport. Similarly, in unsaturated soils, transient seepage conditions can also be described by Darcy’s law [7, 12]. Based on conservation of mass and Darcy’s law, the governing equation for the saturated–unsaturated groundwater flow can be formulated as follows:

$$ \left\{ {\frac{\partial }{\partial x}\left[ {k_{x} \frac{\partial H}{{\partial x}}} \right] + \frac{\partial }{\partial y}\left[ {k_{y} \frac{\partial H}{{\partial y}}} \right] + \frac{\partial }{\partial z}\left[ {k_{z} \left( {\frac{\partial H}{{\partial z}} + 1} \right)} \right]} \right\} = \left[ {C\left( h \right) + S \cdot S_{{\text{s}}} } \right]\frac{\partial H}{{\partial t}} $$
(1)

where \(H\) is the total head; kx, ky, and kz are the permeability of soils; \(C\left( h \right) = \left( {\partial \theta /\partial h} \right)\) is the specific moisture capacity; \(\theta\) is the volumetric moisture content; \(S\) is the degree of saturation; and \(S_{{\text{s}}}\) is the specific storage. For simplification, the compressibility of soil particles can be neglected, leading to the following:

$$ S_{{\text{s}}} = \left( {n\rho_{{\text{w}}} g/K_{{\text{w}}} } \right) $$
(2)

where \(n\) is the ratio of the volume of voids to the total volume, \(\rho_{{\text{w}}}\) is the water density, \(g\) is the gravitational acceleration, and \(K_{{\text{w}}}\) is the compression modulus of the air–water mixture.

The permeability of soils changes with the degree of saturation [40], and their relationship is usually empirically determined and described by an appropriate mathematical model. There are many mathematical models that describe the above hydraulic behavior of unsaturated soils [4, 13, 15], where the most widespread in the groundwater literature is van Genuchten model [10, 15], which is a more general form of Mualem’s model [29]. Therefore, the van Genuchten model was used to describe the hydraulic behavior of unsaturated soils. Following van Genuchten model, a closed-form expression of the permeability is expressed as follows:

$$ k\left( S \right) = k_{{{\text{sat}}}} \left( {S_{{{\text{eff}}}} } \right)^{{g_{l} }} \left\{ {1 - \left[ {1 - S_{{{\text{eff}}}}^{{\left( {\frac{{g_{n} }}{{g_{n} - 1}}} \right)}} } \right]^{{\left( {\frac{{g_{n} - 1}}{{g_{n} }}} \right)}} } \right\}^{2} $$
(3)

where \(k_{{{\text{sat}}}}\) is the saturated permeability; \(g_{l}\) and \(g_{n}\) are fitting parameters; \(S_{{{\text{eff}}}}\) is the relative saturation defined as \(S_{{{\text{eff}}}} = \left( {S - S_{{{\text{res}}}} } \right){/}\left( {S_{{{\text{sat}}}} - S_{{{\text{res}}}} } \right)\); \(S_{{{\text{sat}}}}\) and \(S_{{{\text{res}}}}\) are the degree of saturated and residual saturation, respectively.

2.1.2 Equilibrium equation

The partial differential equation of overall static equilibrium is written as follows:

$$ \left[ L \right]\sigma_{ij} + b_{ij} = 0 $$
(4)

where \(\sigma_{ij}\) is the total stress tensor; \(b_{ij}\) is the unit body force; and \(\left[ L \right]\) is the differential operator corresponding to the definition of engineering strains.

In the fully hydro-mechanical coupling analysis, Bishop’s effective stress concept was used to detect the failure of soil in the constitutive model [2]. The effective stress equation is expressed as follows:

$$ \sigma_{ij}^{{\prime }} = \sigma_{ij} - u_{{\text{a}}} \delta_{ij} + \chi \left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)\delta_{ij} $$
(5)

where \(\sigma_{ij}^{{\prime }}\) is the effective stress tensor; \(\chi\) is the effective stress parameter relating to the degree of saturation; ua and uw are the pore-air and pore-water pressure, respectively; and \(\delta_{ij}\) is the Kronecker delta.

2.1.3 Coupling method

The hydro-mechanical interaction was considered by using the coupling method. Based on the Biot consolidation theory [1], the coupling behavior can be represented by both the continuity equation and equilibrium equation. The pore-water pressure and the displacement of the solid skeleton were chosen as basic variables of the coupling method, which is expressed as follows:

$$ \left[ K \right]\frac{{{\text{d}}v}}{{{\text{d}}t}} + \left[ Q \right]\frac{{{\text{d}}P_{{\text{w}}} }}{{{\text{d}}t}} = \frac{{{\text{d}}f_{u} }}{{{\text{d}}t}} $$
(6)
$$ \left[ C \right]\frac{{{\text{d}}v}}{{{\text{d}}t}} - \left[ S \right]\frac{{{\text{d}}P_{{\text{w}}} }}{{{\text{d}}t}} = \left[ H \right]P_{{\text{w}}} + G + q_{p} $$
(7)

where \(\left[ K \right]\), \(\left[ Q \right]\), \(\left[ C \right]\), \(\left[ S \right]\), and \(\left[ H \right]\) are the stiffness matrix, the coupling matrix, the saturation matrix, the compressibility matrix, and the permeability matrix, respectively; \(v\) is the displacement of the solid skeleton; \(P_{w}\) is the pore-water pressure; \(f_{u}\) is the load vector; \(G\) is the gravity vector; and \(q_{p}\) is the flux on boundaries.

2.2 Constitutive relationship of soil

2.2.1 Soil–water characteristic curve

According to Bishop’s effective stress concept, the terms \((\sigma_{ij} - u_{{\text{a}}} \delta_{ij} )\) and \(\left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)\) are referred as the net stress and the matric suction, respectively, and together they determine the effective stress of soils. Moreover, the value of matric suction is mainly determined by the degree of saturation [53], and their relationship is described by the Soil–Water Characteristic Curve (SWCC), which reflects the capacity of soils to keep water at different stresses [17].

The SWCC is divided into three sections according to the values of the air-entry suction and residual suction. When the value of the matric suction is smaller than that of the air-entry suction, the degree of saturation is assumed to be Ssat, but when the value of matric suction is larger than that of the residual suction, the degree of saturation is assumed to be Sres. When the value of matric suction is between the values of the air-entry suction and residual suction, the degree of saturation is computed in the van Genuchten model as follows:

$$ S = S_{{{\text{res}}}} + \left( {S_{{{\text{sat}}}} - S_{{{\text{res}}}} } \right)\left[ {1 + \left( {g_{a} \psi } \right)^{{g_{n} }} } \right]^{{g_{c} }} $$
(8)

where \(\psi = u_{a} - u_{w}\) is the matric suction; and ga, gn, and gc are fitting parameters. According to the Mualem model [29], the following assumption can be made: gc = (1-gn)/gn. Thus, the van Genuchten model is converted to a two-parameter equation, leading to the van Genuchten–Mualem model.

2.2.2 Constitutive model

The HS model is an advanced double-yield-surface soil model, which can be adapted to all the types of soil [19, 35]. Previous studies have indicated that the HS model is suitable for the numerical simulation of foundation pits, because of its different stress-dependent stiffnesses for unloading/reloading [16, 55]. Zhang et al. [55] applied the HS model to study the influence of groundwater drawdown on the responses of foundation pits. The results were in agreement with the measured responses. Therefore, the HS model was adopted in this study.

The HS model is derived from the hyperbolic model, accounting for a double-isotropic-hardening rule, i.e., volumetric and shear hardening, which is governed by volumetric and deviatoric plastic strains, respectively. The soil stiffness in the HS model is defined by parameters \(E_{50}^{{{\text{ref}}}}\), \(E_{{{\text{oed}}}}^{{{\text{ref}}}}\), which control the volumetric behavior, and \(E_{{{\text{ur}}}}^{{{\text{ref}}}}\), which states the unloading–reloading modulus. The parameter m rules the hyperbolic stress dependency of the stiffness to the reference stiffness. In addition, parameters c′, φ′, and υur denote cohesion, friction angle, and elastic Poisson’s ratio for unloading/reloading, respectively.

The hyperbolic function for standard drained triaxial tests can be described as:

$$ - \varepsilon_{1} = \frac{1}{{E_{i} }}\frac{d}{{1 - \frac{d}{{d_{{\text{a}}} }}}}\;for\;q < q_{f} $$
(9)

where \(\varepsilon_{1}\) is the vertical strain; \(d\) is the deviatoric stress; \(d_{{\text{a}}}\) is the asymptotic value of the shear strength; Ei is the initial stiffness; and qf is derived from the Mohr–Coulomb failure criterion. The yield function for the triaxial tests has the following formulation:

$$ f = \overline{f} - \gamma^{P} $$
(10)

where \(\overline{f}\) is a function of the stress, and \(\gamma^{P}\) is a function of the plastic strain, which are expressed as follows:

$$ \overline{f} = \frac{2}{{E_{i} }}\frac{d}{{1 - d/d_{{\text{a}}} }} - \frac{2d}{{E_{{{\text{ur}}}} }} $$
(11)
$$ \gamma^{P} = 2\varepsilon_{1}^{p} - \varepsilon_{v}^{p} $$
(12)

where \(E_{{{\text{ur}}}}\) is the stress-dependent stiffness modulus; \(\varepsilon_{1}^{p}\) is plastic strain; and \(\varepsilon_{v}^{p}\) is the plastic volume change.

3 Case study description

3.1 General description

The investigated braced foundation pit is situated in the downtown area of Xiamen, China. It was constructed for a part of a highway tunnel, named as Second West Tunnel of Xiamen. The braced foundation pit was conducted using the cut-and-cover method, with a maximum excavation depth of 12.8 m, and it was ladder-shaped with a width of 32.8 m. The lengths of the long and short sides were 66 m and 45 m, respectively, as shown in Fig. 1. The foundation pit crossed over two tunnels of Xiamen Metro Line 1, whose cover depth was about 20 m. Thus, the minimum distance between the final excavation surface and the crown of the tunnels was only 7.2 m. Moreover, an operating road named Jiahe Road was located on the west side of the foundation pit. The minimum distance between the roadbed and the foundation pit was only 5.0 m, while the average height of the roadbed was about 5.0 m. Figure 2 shows a typical profile of the investigated foundation pit, where the location of the profile is shown in Fig. 1.

Fig. 1
figure 1

Plane view of the deep foundation pit and pre-existing tunnels

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Fig. 2
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Profiles of the deep excavation and the existing tunnels

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3.2 Geotechnical and climatic conditions

According to the geological investigation report, soils at the site were Quaternary deposits, with the typical geological profile shown in Fig. 2. The groundwater level was 17 m below the ground surface. There were six soil layers with a total depth of up to 50 m in this project site, of which the third soil layer was the main one encountered within the excavation depth. The third layer was granite residual soil (GRS), with a thickness ranging from 5 to 60 m. The GRS was characterized by a strong structure in terms of bonding and fabric.

The studied area was situated in a subtropical zone, with a subtropical marine monsoon climate. The average and maximum annual precipitation are 1335.8 mm and 1707.0 mm, respectively. Monsoons and typhoons significantly impact this region from May and August, which can represent more than 60% of the annual rainfall. Torrential rainfall may cause a significant adverse impact on the performance of the deep foundation pit, especially in the case that the unsaturated soil is the main bearing layer.

3.3 Supporting system

The supporting system of the excavation included retaining structures, internal struts, and columns. The cast-in-situ bored retaining piles were constructed to retain the soils, being supported by two levels of struts. Most of the retaining piles were 23 m in length, with a diameter of 1 m, and axial spacing of 1.2 m, whereas the piles above the tunnels were 17 m long, due to the small buried depth of the tunnels. The first level strut was reinforced concrete struts, with cross-sectional dimensions of 900 mm in width, and 900 mm in height. The remaining level employed steel pipe struts, with 609 mm in diameter, and 16 mm in thickness. All the struts were vertically supported by steel lattice columns. During excavation, shotcreting with mesh was constructed inside the retaining piles, to cut off the seepage of water flow.

3.4 Construction procedures

Prior to excavation, the construction of retaining piles was already finished. The excavation started on March 27, 2017, and ended on July 18, 2017, after the base slabs were cured. The excavation was divided into four layers, and the thicknesses of each layer were 1.0 m, 6.0 m, 3.0 m, and 2.8 m. Considering the effect of constructing struts and slabs, the construction of the foundation pit was divided into seven stages, which are summarized in Fig. 3. A torrential rainfall occurred in stage 5 when the excavation reached the level of 10 m below the ground surface. The maximum daily rainfall was over 200 mm, and the average daily rainfall was about 100 mm. The total rainfall accounted for 75% of the average annual rainfall, and the daily distribution of precipitation is depicted in Fig. 4. Note that the soils between piles above the excavation surface were exposed during the torrential rainfall.

Fig. 3
figure 3

Timetable of the excavation

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Fig. 4
figure 4

Distributions of daily precipitation in the rainfall event

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4 Numerical simulation

4.1 Model description

In this study, a 3D numerical model was developed in PLAXIS 3D [3], a finite element analysis software, to study the performances of the braced foundation pit, under the combined effect of rainfall and excavation. The 3D numerical model comprised the foundation pit, the underlying tunnels, and adjacent Jiahe Road, as shown in Fig. 5. The distance between the edges of the foundation pit and model boundaries was more than five times the maximum excavation depth, which could cover the influence zone of the excavation [23]. Thus, the plan dimension of the model was set to be 190 m × 170 m, with a depth of 80 m to eliminate the influence of the bottom boundary of the model. Note that the soil layer distribution was modeled considering the undulation according to the geological investigation report, as illustrated in Fig. 5.

Fig. 5
figure 5

Geometry and mesh of the numerical model

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Boundary conditions included the mechanical and hydraulic boundaries. As for the mechanical boundary, soil movements were set to be free at the top boundary, and restrained in all directions at the bottom boundary. The soil movements of the four vertical boundaries of the model were constrained in their normal directions. For the hydraulic boundary, the bottom boundary and the four vertical boundaries were set to be impervious. The top boundary of the model and the bottom of the foundation pit were set to be a time-dependent hydraulic boundary. Under conditions without rainfall, there is no flux over the time-dependent hydraulic boundary, while under rainfall, the precipitation rate for steady-state seepage calculation was applied to the time-dependent hydraulic boundary to simulate the rainfall infiltration. In the simulation, the precipitation rate was set to be consistent with the real rainfall event shown in Fig. 4. The value of flux over the hydraulic boundary depended on the precipitation rate, as well as the capacity of soils. If the precipitation rate exceeds this capacity, ponding takes place and the boundary condition switches from inflow to prescribed head. As soon as the soil capacity meets the precipitation rate, the condition switches back. Therefore, the time-dependent hydraulic boundary is a complex mixed boundary condition, which is expressed as follows:

$$ \left\{ {\begin{array}{*{20}c} {H = y + h_{{{\text{max}}}} } & {{\text{if}}\; {\text{ponding}}} \\ { - q_{w} = q_{x} n_{x} + q_{y} n_{y} + q_{z} n_{z} } & {\begin{array}{*{20}c} { } \\ { {\text{if}}\; h_{{{\text{min}}}} < h < h_{{{\text{max}}}} } \\ \end{array} } \\ \end{array} } \right. $$
(13)

where \(q_{w}\) is the value of inflow; \(q_{x}\), \(q_{y}\), and \(q_{z}\) are the vector components of the total discharge; nx, ny, and nz are the outward pointing normal vector components on the boundary; \(H\) is the total head; y is the elevation head; h is the hydraulic head; \(h_{{{\text{max}}}}\) is the maximum pore pressure head, related to the elevation of the boundary; and \(h_{{{\text{min}}}}\) is the minimum pore pressure head, related to the elevation of the boundary.

Initial conditions include hydraulic, stress, and displacement conditions. As for the initial hydraulic condition, a steady-state seepage calculation was done with a small precipitation rate, which was donated to the daily rainfall. The initial groundwater level was set to be − 17 m. The initial stress condition was generated by gravity loading. Regarding the initial displacement, it was set as zero.

Structural elements of the foundation pit in the numerical model included retaining piles and internal struts. The retaining piles and structures of underlying tunnels were modeled using plate elements. The internal struts were modeled using node-to-node anchors, as shown in Fig. 6.

Fig. 6
figure 6

Geometry of the structures

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4.2 Soil parameter calibration

HS model parameters used for numerical simulation were calibrated by simulating laboratory triaxial and oedometer tests of GRS. As shown in Fig. 7, the laboratory tests were simulated in a 2D finite element axisymmetric model, with the soil samples being set as weightless material [38]. The deformations along the boundaries for the triaxial model (lines AC, and CD), and the oedometer model (lines EG, GH, and FH) were kept free to allow a smooth movement along the axes of symmetry, while the deformations perpendicular to the boundaries were fixed. In relation to the boundaries AB, BD, and EF, these were free to move. The type of load was simulated as a uniform linear displacement (0.4%/min) for the triaxial model. The applied confining pressure was simulated as a distributed load system \(\sigma_{3}\) for the triaxial model, while the applied normal effective stress was simulated as a distributed load system \(\sigma_{v}^{{\prime }}\) for the oedometer model. The test results and numerical results are presented in the form of εl-q, log \(\sigma_{v}^{{\prime }}\)-e, and p′-q curves for comparison, as shown in Fig. 8. The numerical results revealed good agreement with the test curves for GRS. Therefore, the HS model could describe the mechanical behavior and volumetric behavior of GRS.

Fig. 7
figure 7

Simplified geometries of a triaxial test and b oedometer test infinite element models

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Fig. 8
figure 8

GRS parameters calibrations a εl-q curves, b log \(\sigma_{v}^{{\prime }}\)-e curve, and c p'-q curves

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Table 1 lists the input parameters for the soils in this study. SWCC parameters for the second and third soil layers were obtained from laboratory tests conducted by Chen and Gong [5], as illustrated in Fig. 9. Due to the lack of accurate test data, the international soil classification system (USDA) series was used as the hydraulic model of the other soil layers. Tables 2 and 3 list the input parameters of the structural elements for numerical simulation. The stiffness of the supporting structure was assumed to be 80% of the normal value, due to the possible bending moment-induced cracking [25].

Table 1 Soil parameters used in numerical analysis
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Fig. 9
figure 9

The Soil–Water character curve of residual soil in Xiamen

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Table 2 Physical parameters and dimensions of struts used in numerical analysis
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Table 3 Physical parameters and dimensions of enclosure structures used in numerical analysis
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4.3 Validation of the numerical model

As shown in Fig. 10, calculated lateral displacements of the retaining structure at section I-I (see Fig. 1) were compared to field data. At the depth of 0 m, there was negative deflection for the short pile, and positive deflection for the long pile, due to the surcharge from Jiahe Road. For the long pile, the calculated maximum lateral displacement was smaller than field data in stage 3 and the position of maximum lateral displacement was higher than field data in stage 7. Regarding short pile, the calculated lateral displacement was slightly smaller than the field data in stage 7. These discrepancies might be attributed to the following reasons: (1) the surcharge from Jiahe Road and undulation of strata could not be accurately simulated; (2) several scattered boulders were found in the excavation process, while they were not considered in the numerical simulation; (3) the HS model was unable to consider the strain-softening behavior of soils, which affected the accuracy of numerical simulation. The overall good agreement between the calculated results and field data proved the feasibility of the numerical model for the simulation of the performances of the braced foundation pit, to rainfall and excavation.

Fig. 10
figure 10

The calculated and measured displacement of the retaining structure: a the long pile; b the short pile

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5 Analysis of the numerical results

5.1 Displacement of retaining piles

To interpret the deformation of the foundation pit induced by rainfall and excavation, two scenarios (i.e., with rainfall and without rainfall) were calculated for comparison. Figure 11 shows the calculated lateral displacements of the retaining structure at section I-I, in the two scenarios. In the scenario without rainfall, the final maximum displacement (δhm) of the long pile and short pile was 17.7 mm and 14 mm, respectively. The ratio of δhm to He was both less than 0.2%, where He is the final excavation depth. In the scenario with rainfall, the δhm of the long pile and short pile was 43.6 mm and 34.7 mm, respectively. And the ratio of δhm to He was 0.34% and 0.27%, respectively, which was consistent with the range of 0.2–0.5% found in the research performed by Ou et al. [34]. Comparing the two scenarios, the δhm of the retaining structure increased by 143% for the long pile, and 171% for the short pile due to rainfall.

Fig. 11
figure 11

The lateral displacement of the retaining structure in two scenarios

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The position of the maximum lateral displacement with rainfall was lower than the one without rainfall. In the scenario without rainfall, the positions of δhm of the long pile and short pile were located at 0.7 He and 0.78 He under the ground surface, respectively. In the scenario with rainfall, the movement pattern of the long pile was a convex deformation model, due to the limitation of internal struts and the embedment of bottom strata. The position of δhm of the long pile was located at 0.81 He under the ground surface, which was near the excavation level. Whereas for the short pile, a clear tendency of kick-in deformation could be observed after the post-rainfall excavation. The position of δhm of the short pile was located at 0.8 He under the ground surface. The short pile was more likely to kick than the long pile, which means that the embedded depth of the retaining structure had an important impact on the deformation of the braced foundation pit suffering rainfall.

The displacement of the foundation pit is greatly influenced by both the excavation depth, and plane shape. The maximum and minimum displacements generally occur in the middle of the long side and the corner of the foundation pit, respectively, which is known as the spatial effect of the foundation pit. Comparing the total displacement contours of the two scenarios shown in Fig. 12, the overall displacement increased, while the spatial effect decreased due to rainfall. This can be explained by the fact that the seepage of groundwater weakened the soil arching, which was the main cause of the spatial effect of the braced foundation pit [44].

Fig. 12
figure 12

Total displacement contours of the retaining structure a the scenario without rainfall and b the scenario with rainfall

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5.2 Earth pressure

To illustrate the force of the retaining structure, the earth pressure borne by short piles at section I-I is depicted in Fig. 13. The active and passive earth pressures under transient flow in unsaturated soils were calculated by the formulations proposed by Shahrokhabadi et al. [36] for comparisons. The earth pressure in the active zone can be considered as the external load acting on the retaining structure, and the earth pressure in the passive zone can be considered as the resistance acting on the retaining structure.

Fig. 13
figure 13

Earth pressure (EP) on the retaining structure in the scenario with rainfall

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In the active zone, the matric suction in the unsaturated soil contributed to the appearance of tensile stresses which caused the earth pressure near the ground surface to be zero. After excavation to − 10 m, the calculated distribution form of earth pressure was close to the active earth pressure under unsaturated conditions. The calculated and analytical depth of the tension zone was 5.1 m and 8.1 m, respectively. Those results showed that soils in the active zone have not reached the active state. After rainfall, the calculated distribution form of the earth pressure presented the “R” type, due to the surcharge from Jiahe Road. The earth pressure in the active zone increased but has not reached the active state after rainfall.

In the passive zone, the calculated earth pressure at the excavation surface almost reached the passive earth pressure after excavation to − 10 m. However, the earth pressure under the excavation surface was far from reaching the passive state, which is a phenomenon that might be attributed to the convex form of the retaining structure deformation. After rainfall, the calculated earth pressure changed into a linear distribution, while the earth pressure at the excavation surface was close to zero, due to the loss of matric suction. Soils in the passive zone have not reached the passive state after rainfall.

After the post-rainfall excavation, the calculated earth pressures in active and passive zones decreased, being consistent with the active and passive earth pressure under saturated conditions, respectively. The results showed that rainfall and excavation changed both the distribution and values of earth pressure.

5.3 Seepage responses

Figure 14 presents the distribution of pore-water pressure at section I-I, before and after rainfall, where a positive value represents the positive pore-water pressure, and a negative value represents the negative pore-water pressure. Before rainfall, the pore-water pressure is distributed uniformly along the horizontal direction. The maximum matric suction was located at the ground surface, with a magnitude of 240 kPa. In addition, the pore-water pressure changed linearly with the depth, becoming positive near the groundwater level. After rainfall, the pore-water pressure near the foundation pit was about 50 kPa and became negative far away from the foundation pit. This was due to the waterproof effect of retaining structures restricting the seepage path of groundwater around the foundation pit [50]. Therefore, the waterproof curtain will increase the earth pressure and cause an adverse impact on the force balance of the retaining structure under rainfall.

Fig. 14
figure 14

Distribution of pore-water pressure in the scenario with rainfall a before rainfall and b after rainfall

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Figure 15 illustrates the development of pore-water pressure at six points (Point A–F) caused by rainfall, where the positions of points A–F are shown in Fig. 2. The development trend of the calculated pore-water pressure was consistent with the change in rainfall intensity. The pore-water pressure began to increase after about 0.5 days of the beginning of the rainfall. There was a clear delay in pore-water pressure response to rainfall infiltration. Though the HS model is generally suitable for all soils, it does not account for viscous effects. Therefore, the delayed response of pore-water pressure may be due to the permeability of soils [11]. The pore-water pressure was redistributed within a certain period after rainfall. The positive pore-water pressure decreased, and the matric suction recovered partially. These results were consistent with the field observation reported by Ng et al. [31].

Fig. 15
figure 15

Pore-water pressure development under rainfall with time

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The variation trend of pore-water pressure at different depths was similar. This phenomenon was due to the fact that the permeability of GRS increased rapidly with the increase of the degree of saturation. As a result, a larger permeability enabled rainfall to infiltrate into the deep soil layer, instead of forming the transient saturation area on the surface. Therefore, the rainwater could influence the shear strength of unsaturated soils in the bearing layer. Compared with soils with low permeability, the braced foundation pit in unsaturated GRS was more vulnerable to rainfall infiltration.

6 Discussion

To further reveal the combined effect of rainfall and excavation on the braced foundation pit, the soil effective stress paths of six stress points (Point A′–F′) were investigated, with the location of these points shown in Fig. 2. The stress path of stress points is plotted in the p′-q coordinate, where the average effective principal stress (p′) and the shear stress (q) are defined as p′ = (\(\sigma_{1}^{{\prime }}\) + \(\sigma_{2}^{{\prime }}\) + \(\sigma_{3}^{{\prime }}\))/3 and q = (\(\sigma_{1}^{{\prime }}\) − \(\sigma_{3}^{{\prime }}\))/2, respectively. \(\sigma_{1}^{{\prime }}\) is the major principal effective stress, \(\sigma_{2}^{{\prime }}\) is the intermediate principal effective stress, and \(\sigma_{3}^{{\prime }}\) is the minor principal effective stress. Under the assumption of isotropy and normal consolidation, the initial direction of the major principal effective stress \(\sigma_{1}^{{\prime }}\) is the direction of gravity, and \(\sigma_{2}^{{\prime }}\) and \(\sigma_{3}^{{\prime }}\) are horizontal and equal.

Figure 16 depicts the evolution of the effective stress path of points A′ and B′ in the active zone. p′ decreased and q increased before rainfall. This behavior happened because of the unloading effect, which caused the decrease of minor principal effective stress in the active zone. During rainfall, p′ decreased significantly with the decrease of both the corresponding q and shear strength. This is due to the decrease of matric suction, and the increase of pore water pressure caused by rainfall. In addition, q increased further due to post-rainfall excavation. However, the decrease of p′ was insignificant, due to the dissipation of pore-water pressure. Finally, the stress state of most soils in the active zone was close to the Mohr–Coulomb Failure Line (MCFL).

Fig. 16
figure 16

Effective stress paths of the soil in the active zone. (Note: S1–S7 denote the construction stages.)

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Figure 17 depicts the evolution of the effective stress path of points C′, D′, E′, and F′ in the passive zone, where points C′ and D′ are located in front of the retaining structure, and points E′ and F′ are located at the center of the excavation. Both p′ and q decreased at stages 1–5 due to the unloading effect, which decreased the vertical stress above the stress points. With the decrease of vertical stress, the direction of principal stress rotated, while the direction of the major principal effective stress changed from vertical to horizontal. As a result, p′ continued to decrease but q started to increase. p′ decreased significantly after rainfall while remaining inside the yield surface. However, the stress path of points C′ and E′ reached the MCFL after the post-rainfall excavation. This indicated that the soils in the passive zone lost most of their resistance after the post-rainfall excavation.

Fig. 17
figure 17

Effective stress paths of the soil in the passive zone. (Note: S1–S7 denote the construction stages.)

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For further investigation, the changes in relative shear stress (\(\tau_{{{\text{rel}}}}\)) of the six stress points are shown in Figs. 18 and 19, respectively. The relative shear stress \(\tau_{{{\text{rel}}}}\) is defined as the ratio of the shear stress (q) and the maximum shear stress (\(\tau_{{{\text{max}}}}\)), which approximately measures the relative distance between the soil stress path and the MCFL. It is important to note that \(\tau_{max}\) is the maximum shear stress that the soil can bear under the corresponding stress level. It can be seen that the relative shear stress in the active zone increased gradually with the development of excavation and rainfall. The relative shear stress in the passive zone initially decreased, while increasing afterwards due to the rotation of the principal stress direction. Although rainfall further increased the relative shear stress of the soil, both inside and outside the excavation, the soil was far from reaching the failure state. The post-rainfall excavation caused the soil in the passive zone to reach the critical failure state. Therefore, the post-rainfall excavation became the direct factor, causing the soil to reach the failure state in this case.

Fig. 18
figure 18

The relative shear stress of the soil in the active area. (Note: S1–S7 denote the construction stages.)

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Fig. 19
figure 19

The relative shear stress of the soil in the passive area. (Note: S1–S7 denote the construction stages.)

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Rainfall significantly reduced the average effective principal stress p′ of soils, by reducing the matric suction. Also, excavation significantly reduced the net stress of soils. According to Mohyla et al. [28] and Shu et al. [37], the stiffness of soil decreased with the decrease of both suction and net stress. Therefore, rainfall and excavation cause soils to be prone to deformation, by reducing the stiffness of unsaturated soil. In addition, the same change value of p′ will cause a larger change of v, under a smaller p′ according to the classical v-p′ relationship. Therefore, the post-rainfall excavation will produce larger deformation of the foundation pit, under the assumption that the excavation produced the same unloading in both scenarios. Moreover, the post-rainfall excavation caused part of the soil to reach the MCFL in the scenario with rainfall, which further aggravated the deformation of the foundation pit.

By comparing the results of both scenarios, rainfall was the inducing factor for the increase in the deformation of the foundation pit. The post-rainfall excavation further increased the deformation, by significantly weakening the resistance in the passive zone. The resistance in the passive zone was the key factor to control the deformation, and therefore, prompt propping or timely casting of base slabs should be performed after rainfall, to improve the resistance in the passive zone. In addition, post-rainfall excavation construction should be avoided to compensate for the schedule. In this regard, a period before post-rainfall excavation is useful for the dissipation of pore-water pressure as well as the increase of effective stress of soils, improving the resistance in the passive zone.

7 Summary and conclusions

Through 3D numerical simulation using the hydro-mechanical coupling method, this study investigated the performances of foundation pit under the combined effect of rainfall and excavation. The feasibility of the numerical model was validated by comparing it with field data. With these results, the performance of the foundation pit in the two scenarios (i.e., with rainfall and without rainfall) was discussed in detail. Based on the numerical analyses, the following conclusions could be drawn:

  1. (1)

    The large deformation in the real case was caused by the rainfall event. Under the effect of rainfall, the maximum displacement of the retaining structure increased by 143% for the long pile, and 171% for the short pile. In addition, rainfall had the following effects on the braced foundation pit: (1) the position of the maximum lateral displacement of the retaining structure was reduced; (2) short piles were more prone to kicking; (3) the spatial effect of the foundation pit was weakened by rainfall.

  2. (2)

    The earth pressure in the active zone can be regarded as the external load acting on the retaining structure. The earth pressure increased since the rainfall caused the increase of the pore-water pressure. The variation of pore-water pressure at different depths was consistent with the rainfall intensity. This phenomenon was due to the fact that the permeability of unsaturated soils increased rapidly under rainfall. The larger the permeability, the more easily rainfall infiltrates into the deep soil layer, instead of forming the transient saturation area on the surface. Besides, the waterproof effect of retaining structures restricted the seepage path of groundwater, around the foundation pit.

  3. (3)

    The earth pressure in the passive zone can be regarded as the resistance acting on the retaining structure. The earth pressure in the passive zone decreased after rainfall and post-rainfall excavation, which was mainly attributed to the following two aspects: (1) the matric suction of the soil decreased due to rainfall; (2) the vertical stress of the soil in the passive zone decreased due to the post-rainfall excavation. Moreover, the earth pressure in the passive zone was close to the value of the passive earth pressure after the post-rainfall excavation.

  4. (4)

    Rainfall and excavation weakened the stiffness of the soil, by reducing the matric suction, as well as the net stress. Rainfall was the inducing factor for the increase in the deformation of the foundation pit. The post-rainfall excavation caused the soil in the passive zone to reach the critical failure state and therefore, the post-rainfall excavation had a more significant impact on the mechanical behavior of soils than that without rainfall.

  5. (5)

    For the foundation pit that suffers from rainfall, the resistance provided by the passive zone was the key factor to control the deformation. Therefore, prompt propping or timely casting of base slabs should be performed after rainfall. Post-rainfall excavation should be avoided to compensate for the schedule. A period before post-rainfall excavation for the dissipation of pore-water pressure and increase of effective stress could improve the resistance in the passive zone.