Slope Stability Analysis Based on Analytical and Numerical Solutions

Abstract

Infiltration into soil slopes is a fundamental concern in civil engineering. Rainfall infiltration leads to changes in pore-water pressure and reduces matric suction in soils, making it one of the main triggers of slope failure (Ali et al. in Comput Geotech 61:341–354, 2014; Wu et al. in Hydro-mechanical analysis of rainfall-induced landslides. Springer, 2020). Slope instabilities caused by water infiltration are called rainfall-induced landslides (Xu and Zhang in Landslides 7:149–156, 2010; Wu et al. in Hydro-mechanical analysis of rainfall-induced landslides. Springer, 2020).

5.1 Introduction

Infiltration into soil slopes is a fundamental concern in civil engineering. Rainfall infiltration leads to changes in pore-water pressure and reduces matric suction in soils, making it one of the main triggers of slope failure (Rahimi et al. 2010; Ali et al. 2014; Wu et al. 2020). Slope instabilities caused by water infiltration are called rainfall-induced landslides (Xu and Zhang 2010; Wu et al. 2020).

There are three methods that can be taken to investigate the effect of rainfall infiltration on pore-water pressure or pressure head profile and hence on unsaturated soil slope stabilities. The approaches include numerical simulation, field monitoring, and analytical analysis (Zhan et al. 2013). A number of numerical studies have been conducted to investigate the hydrodynamic behaviors of soil slopes due to rainfall infiltration and to estimate the influence on the slope stability (Ng and Shi 1998; Iverson 2000; Collins and Znidarcic, 2004; Kim et al. 2004; Zhang et al. 2005; Garcia et al. 2011; Ali et al. 2014). Factors affecting the soil slope stability due to rainwater infiltration comprise the rainfall characteristics, the saturated permeability coefficient, the geometry of the slope, and the boundary and initial soil moisture conditions (Ali et al. 2014). Laboratory and field experiments have been performed to investigate variations in matric suction due to rainfall infiltration to improve the understanding of the mechanism of rainfall-induced soil slope failures (Wu et al. 2015, 2017, 2020). Field monitoring is helpful in the study of the effects of rainfall infiltration on the slope stability, but it is a costly procedure (Zhan et al. 2013).

The analytical methods involve theoretical infiltration and approximate infiltration models (Wu et al. 2022). In a theoretical infiltration model, the partial differential equation describing water infiltration in soils (i.e., the Richards’ equation) is proposed based on continuum mechanics, and the solution can be obtained through integral transformation or numerical methods (Srivastava and Yeh 1991; Zhu et al. 2020, 2022).

The analytical method requires that certain assumptions be made regarding the closed-form equations that are derived. If the assumptions can be shown to be reasonable, then the analytical methodology becomes a simple and practical tool for studying possible pressure head distribution and the factor of safety of the slope.

Coupled poromechanical approaches incorporating the unsaturated soils have been employed to analytically and numerically examine the hydrodynamic behaviors of partially or completely saturated slopes due to rainfall (Chen et al. 2005). These coupled poromechanical studies are in view of various constitutive models, which represent the mechanical response of the unsaturated soils using nonlinear elastic stress–strain relationships (Cho and Lee 2001; Zhang et al. 2005) or elastoviscoplastic and elastoplastic models. The coupled hydromechanical approach has been combined with a slope stability analysis according to finite element methods to examine the response of an unsaturated soil slope subjected to a rainstorm and to consider the variability in the soil properties (Zhang et al. 2005). Consequently, using the improved method proposed in the previous chapters, the infiltration equation of the unsaturated soil slope is solved, and the numerical and analytical solutions are applied to the slope model to study the effect of rainfall infiltration on the slope stability.

The practical infiltration process is very complicated and affected by many factors such as soil profile and rainfall conditions and becomes difficult to be described accurately with theoretical formulations (D’Aniello et al. 2019; Srivastava et al. 2020).

Due to the nonlinearity of differential equations, numerical software packages are frequently needed to solve the complex problem of infiltration for predicting moisture movement and pore-water pressure change in unsaturated soils (Masoudian et al. 2019; Yang et al. 2018; Zhang et al. 2016).

Numerical approaches often suffer from convergence and mass balance problems and also are inefficient and expensive in many conditions. To provide simplified solutions to the complex infiltration issue, a number of theoretical models based on the wetting front have been proposed (Iverson 2000; Conte and Troncone, 2008). A series of analytical approaches (Srivastava and Yeh 1991; Iverson 2000; Wu et al. 2022) was developed to calculate the pressure head change during rainfall. Unfortunately, these models can only be applied under the assumption that the rainfall intensity is a constant. However, in practice, there are often cases where the rainfall intensity is a variable function depending on duration.

The objective of this chapter is to study the application of improved numerical methods to the slope stability assessment and to discuss the influence of various factors on the slope stability. Combined with the monitoring data of slope pore-water pressure in the Tung Chung area of Hong Kong, the book carried out a comparative study to verify the accuracy and practicability of the proposed improved method. The Xiaoba landslide in Guizhou, China, a typical rainfall-induced landslide, was investigated using numerical methods.

5.2 Application of Unsaturated Soil Slope Stability Under Rainfall

5.2.1 Application of Chebyshev Spectral Method in Slope Stability Analysis

For unsaturated soil slopes (Fig. 5.1), the governing equations are modified as follows:

$$\frac{\partial }{\partial z}\left[ {K_{z} \left( h \right)\left( {\frac{\partial h}{{\partial z}} + \cos \beta } \right)} \right] = \frac{\partial \theta }{{\partial t}}$$

(5.1)

Fig. 5.1

Schematic diagram of a homogeneous unsaturated slope

where $\beta$ is the slope angle.

According to modified Eq. (2.34), the governing equation considering the hydromechanical coupling in soil slopes can be obtained:

$$\frac{\partial }{{\partial z^{ * } }}\left[ {K(h)\frac{\partial h}{{\partial z^{ * } }}} \right] + \frac{\partial K(h)}{{\partial z^{ * } }}\cos \beta = C(h)\frac{\partial h}{{\partial t}} + \frac{{\alpha_{{\text{c}}} \gamma_{{\text{w}}} (\theta - \theta_{{\text{r}}} )}}{{(\theta_{{\text{s}}} - \theta_{{\text{r}}} )F}}\frac{\partial h}{{\partial t}}$$

(5.2)

The analytical solution of the infiltration equation along the z-axis can be expressed as:

$$h_{{\text{t}}}^{*} \left( {z,t} \right) = \frac{{2\left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)}}{Lc}{\text{e}}^{{\alpha \cos \beta \left( {L - z} \right)/2}} \sum\limits_{m = 1}^{\infty } {\left( { - \,1} \right)^{m} } \left( {\frac{{\lambda_{{\text{m}}} }}{{\mu_{{\text{m}}} }}} \right)\sin \left( {\lambda_{{\text{m}}} z} \right){\text{e}}^{{ - \mu_{{\text{m}}} t}}$$

(5.3)

$$h_{{\text{s}}}^{*} \left( z \right) = \left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)\frac{{1 - {\text{e}}^{ - \alpha \cos \beta z} }}{{1 - {\text{e}}^{ - \alpha \cos \beta L} }}$$

(5.4)

This application example simulated transient infiltration in an unsaturated slope using the Chebyshev spectral method (CSM). The slope thickness (L) and angle ($\beta$) are assumed to be 10 m and 33°, respectively (Fig. 5.1). The initial condition was unchanged at h(z, t = 0) = −10 m. The upper and lower boundary conditions of the soil can be expressed as h(z = 0) = − 10 m and h(z = L) = 0.

For this example, this chapter selected sandy soil and silty loam. The experimental data (Brooks and Corey 1964; Lu and Likos 2004) and fitting curve of the two soils are shown in Fig. 5.2. The fitted parameters including k_s, θ_s, θ_r, and $\alpha$ are listed in Table 5.1. The total simulation time was 2 h and N = 40.

Fig. 5.2

Fitting curves of the SWCC: a sandy soil and b silty loam

Table 5.1 Parameters of unsaturated soils for analysis

The pressure head calculated using the CSM was introduced into the infinite slope stability analysis, and the factor of safety (F_s) could be obtained (Iverson 2000; Liu et al. 2017):

$$F_{{\text{s}}} = \frac{{c^{\prime}}}{{z\gamma_{{\text{t}}} \cos \beta \sin \beta }} + \frac{{\tan \phi^{\prime}}}{\tan \beta } - \frac{{h\gamma_{{\text{w}}} \tan \phi^{\prime}}}{{z\gamma_{{\text{t}}} \cos \beta \sin \beta }}$$

(5.5)

where $c^{\prime}$ denotes the effective cohesion, $\phi^{\prime}$ denotes the effective friction angle, $\gamma_{{\text{t}}}$ denotes the unit weight of soil, $\gamma_{{\text{w}}}$ denotes the unit weight of water, and z denotes the soil thickness. It was assumed that the unit weight of soil was 19 kN/m³, and the effective cohesiveness and effective friction angle of sandy soil and silty loam were 0 kPa, 32° and 10 kPa, 20°, respectively (Lu and Likos 2004).

The revised F_s for analyzing slope stability can be obtained as (Liu et al. 2017):

$$F_{{\text{s}}} = \frac{{c^{\prime}}}{{z\gamma_{{\text{t}}} \cos \beta \sin \beta }} + \frac{{\tan \phi^{\prime}}}{\tan \beta } - \frac{{h\gamma_{{\text{w}}} \left( {\frac{{\theta - \theta_{{\text{r}}} }}{{\theta_{{\text{s}}} - \theta_{{\text{r}}} }}} \right)\tan \phi^{\prime}}}{{z\gamma_{{\text{t}}} \cos \beta \sin \beta }}$$

(5.6)

The profiles of the pressure head of the two soils over time were compared with the analytical solutions. Figure 5.3 shows that the computed results were in good agreement with the analytical solutions. The numerical findings for sandy soil and silty loam indicate that the accuracy of the absolute error can reach the order of 10⁻⁵ and 10⁻⁶, respectively (Fig. 5.4).

Fig. 5.3

Results comparison with the exact solution for unsaturated soil: a sandy soil and b silty loam

Fig. 5.4

Absolute error of the computed results with the analytical solutions for unsaturated soil: a sandy soil and b silty loam

Figure 5.5 demonstrates the computed results of F_s for different unsaturated soils. Additionally, the range of F_s for different soils is different, and F_s for silty loam is significantly smaller than that for sandy soil. Table 5.2 shows variations in F_s at different depths over the infiltration time for different soils. It can be seen that the factor of safety decreased with increasing depth and infiltration time. F_s for sandy soil was greater than 1.0 in Table 5.2, and F_s for silty loam was less than 1.0 at z = 9.263 m. Consequently, the stability of the slope of sandy soil was better than that of the slope of silty loam. The comparison of the computed results reveals that rainfall-induced landslides are closely related to soil types.

Fig. 5.5

Computed results of F_s for unsaturated soil: a sandy soil and b silty loam

Table 5.2 Variations in the factor of safety (F_s) at different depths over the infiltration time for different soils

5.2.2 Application of P-CSIM in the Stability Analysis of Shallow Rainfall Slope

The proposed P-CSIM is used to solve the Richards’ equation for the slope, then the numerical solution of the pressure head is substituted into the infinite slope model, and the stability of the slope is analyzed. The factor of safety of the slopes is calculated as Eqs. (5.5)–(5.6).

5.2.2.1 Homogeneous Soil Slope

The soil slope thickness is 150 cm, the slope angle is 33°, the initial condition for removing the boundary point is h(z, t = 0) = − 100 cm, and the boundary conditions are h(z = 0) = − 100 cm and h(z = L) = 0. The homogeneous unsaturated soil is assumed to be a sandy soil with a saturated permeability coefficient k_s, residual volumetric water content θ_r, and saturated volumetric water content θ_s of 9 × 10⁻² cm/h, 0.08, and 0.43, respectively (Liu et al. 2017). The desaturation coefficient $\alpha$ is 1 × 10⁻². In addition, the unit weight, effective cohesiveness, and effective friction angle of the soil are 21.5 kN/m³, 4.6 kPa, and 30°, respectively (Lu and Likos 2004).

Figures 5.6 and 5.7 represent the computed profile of the factor of safety and the pressure head for the homogeneous soil slope. The suction (negative pressure head) in shallow parts of the slope decreases with the increase of rainfall time. In addition, it is found that F_s decreases with the increase of rainfall time at the shallow slope zone. The unstable slope is mostly 130–150 cm deep, which may be strength deterioration and softening of the shallow layer caused by rainfall infiltration (Zhuang et al. 2018).

Fig. 5.6

Computed profile of pressure head for homogeneous unsaturated soil slopes

Fig. 5.7

Computed profile of safety factor for homogeneous unsaturated soil slopes

5.2.2.2 Two-Layer Soil Slopes

For slopes of two-layer soils, the thickness of unsaturated Soil 1 (L₁) and Soil 2 (L₂) is assumed to be 130 cm and 20 cm, respectively (Fig. 5.8). The slope angle and boundary and initial conditions are the same as the slope for homogeneous soils. The physical parameter settings are listed in Table 5.3. It is assumed that Soil 1 is loess, and that the permeability coefficient of Soil 2 is greater than that of Soil 1.

Fig. 5.8

Schematic diagram of two-layer soil slope model

Table 5.3 Parameters used in two-layer soil slopes

The calculated results (Fig. 5.9) show that increase of the pressure head in layered unsaturated soils dominates slope stability. It can be found that the pressure head changes faster at the interface. Besides, because the permeability coefficient of the bottom layer is lower than that of the upper layer, the top layer (unsaturated Soil 2) may become unstable. Figure 5.10 demonstrates F_s profile of the two-layer unsaturated soil slope. It shows that unstable slope ranges from the interface to the soil surface after t = 10 h. Figure 5.11 indicates that F_s varies with duration at the interface (z = 130 cm) for unsaturated soil slopes. It is found that F_s is strongly affected by duration, and that the upper layer of slope becomes unstable after t = 8.2 h. Accordingly, the results illustrate that the pressure head caused by rainfall infiltration is closely related to the soil layer, and the interface of the soil layer plays a key role in the slope stability related to rainfall infiltration.

Fig. 5.9

Computed profile of pressure head for two-layer unsaturated soil slopes

Fig. 5.10

Computed profile of safety factor for two-layer unsaturated soil slopes

Fig. 5.11

Results of safety factor at the interface (z = 130 cm) for two-layer unsaturated soil slopes

5.2.3 Application of Improved Picard Method into Unsaturated Slopes

In this example, this chapter investigates 1D transient infiltration into the two-layer soil slopes. The van Genuchten model is adopted here. In Fig. 5.8, the thickness of soil layer 1 (L₁) and soil layer 2 (L₂) is 3 m and 2 m, respectively, and the slope angle ($\beta$) is 33°. The governing equation is shown in Eq. (5.1). The VGM model parameters of the two soils are listed in Table 5.4 (Zha et al. 2013). The bottom boundary condition is groundwater table, defined as h(z = 0, t) = 0. The top boundary condition is given by (Wu et al. 2020):

$$\left[ {K\frac{\partial h}{{\partial z}} + K} \right]_{{z = L_{1} + L_{2} }} = q_{{\text{t}}} ,\quad t < t_{{\text{I}}}$$

(5.7)

Table 5.4 Parameter of van Genuchten model

$$h_{{\left| {z = L_{1} + L_{2} } \right.}} = h_{{\text{t}}} ,\quad t \ge t_{{\text{I}}}$$

(5.8)

where $t_{{\text{I}}}$ represents ponding time; $q_{{\text{t}}}$ denotes the rainfall flux at the soil surface when the rainfall time is less than the ponding time; and $h_{{\text{t}}}$ denotes constant pressure head after the ponding time. Here, $q_{{\text{t}}}$ is assumed to be $K_{{{\text{s}}2}} /4$ and $h_{{\text{t}}} = 0$. The initial condition is given by $h(z,t = 0) = - z \times 10$ m.

The total simulation time and time step are taken to be 2 h and 0.01 h, respectively. Let the number of nodes be N₁ = 60 and N₂ = 40. Furthermore, NTG-PI is applied to solve Eq. (5.1), where $\mu = 1$ is adopted. The unit weight of the soil, the effective cohesion, and friction angle are 19.5 kN/m³, 4.6 kPa, and 30°, respectively.

Figure 5.12 shows that the pressure head increases over duration, while the pressure head at the interface has a greater increase. Compared with NTG-PI, the pressure head obtained by traditional method PI has a greater increase at the interface as the rainfall time increases. This may underestimate the safety factor of the interface for two-layer soil slopes (Fig. 5.13). Moreover, the results illustrate that the soil layer structure will affect the distribution of the pressure head in the two-layer soil slopes during rainfall. When the hydraulic conductivity of the lower soil is lower than that of the upper one, it is easy to form ponding at the interface and induce landslides.

Fig. 5.12

Computed profile of pressure head for two-layer soil slopes

Fig. 5.13

Computed profile of F_s for two-layer soil slopes

5.2.4 Parameter Sensitivity Analysis of Slope Stability Under Rainfall

The test uses the Gardner model, as described by Eqs. (1.13)–(1.15). The slope soil layer is assumed to be homogeneous soil, the mathematical model is depicted in Fig. 5.1, its thickness is assumed to be 2 m, and the saturated and residual water contents are set to 0.46 and 0.1. The distribution characteristics of saturated permeability coefficient, desaturation coefficient $\alpha$, slope angle, and rainfall q on pore-water pressure are examined, and their influence on soil slope stability is discussed. The parameter settings of different conditions are shown in Table 5.5. For the slope stability analysis, the soil unit weight, effective cohesion, and effective internal friction angle are 19.9 kN/m³, 10 kPa, and 26°, respectively. The upper boundary is set as the flow boundary, the lower boundary is the groundwater level, its pressure head is set to 0, and the initial condition for removing the boundary points is h(z, t = 0) = −z m.

Table 5.5 Parameters for different cases

5.2.4.1 Influence of Saturated Permeability Coefficient on Slope Stability

Figure 5.14 depicts the effect of three different saturated permeability coefficients on the pressure head distribution in Case 1. With the increase of the saturated permeability coefficient of the soil, the distribution of the pressure head moves faster, and with the increase of the rainfall time, the change of the pressure head is large. The numerical results illustrate that in the numerical simulation of rainfall infiltration for soil slope, the size of the saturated permeability coefficient can directly affect the speed of the pressure head distribution. Figure 5.15 indicates that the increase of the saturated permeability coefficient has a great influence on the stability of the shallow soil slope.

Fig. 5.14

Influence of different saturated permeability coefficients on pressure head distribution (Case 1)

Fig. 5.15

Relationship between slope safety factor (z = 1 m) and saturated permeability coefficient

5.2.4.2 Influence of Desaturation Coefficient on Slope Stability

Figure 5.16 depicts the effect of different desaturation coefficients on the pressure head distribution in Case 2. As the desaturation coefficient increases, the pressure head distribution lags behind. At the same time, the curvature of the pressure head distribution becomes obvious with the increase of the desaturation coefficient. Figure 5.17 demonstrates that the slope safety factor decreases as the desaturation factor decreases.

Fig. 5.16

Influence of different desaturation coefficients on pressure head distribution (Case 2)

Fig. 5.17

Relationship between the slope safety factor (z = 1 m) and the desaturation factor

5.2.4.3 Influence of Slope Angle on Slope Stability

Figure 5.18 represents the effect of different slope angles on the pressure head distribution in Case 3. As the slope angle increases, the pressure head distribution shifts gradually to the right. At the same time, the curvature of the pressure head distribution becomes obvious with the increase of the slope angle. Figure 5.19 shows that the slope angle has an inherent key effect on the slope safety coefficient, and the slope safety factor decreases continuously with the increase of the slope angle.

Fig. 5.18

Influence of different slope angles on pressure head distribution (Case 3)

Fig. 5.19

Relationship between the slope safety factor (z = 0 m) and the slope angle

5.2.4.4 Influence of Rainfall on Slope Stability

Figure 5.20 depicts the effect of different rainfall on the pressure head distribution in Case 4. With the increase of rainfall, the migration speed of the pressure head distribution increases, which means that the soil matric suction dissipates faster. At the same time, the curvature of the pressure head distribution becomes obvious with the increase of rainfall. Figure 5.21 verifies that the slope safety factor decreases with increasing rainfall.

Fig. 5.20

Influence of different rainfall on pressure head distribution (Case 4)

Fig. 5.21

Relationship between the slope safety factor (z = 1 m) and the rainfall

To sum up, in the analysis of rainfall slope stability, the slope angle has an inherent key role in the factor of safety, and the saturation permeability coefficient, desaturation coefficient, and rainfall all have a large influence on the distribution of pressure head. For the short-term rainfall period, when the desaturation coefficient is small and the saturated permeability coefficient is large, the larger the rainfall, the larger the rainwater infiltration depth, which will lead to the softening of the soil, the reduction of the shear strength, and the shallow soil slope stability. As a result, factors such as desaturation coefficient, rainfall, permeability coefficient, and slope angle all affect the stability of unsaturated soil slopes.

5.3 Rainfall Landslide Case Study—Tung Chung Landslide

According to the improved method, a comparative study is carried out with the monitoring data of the pore-water pressure of a slope in Tung Chung, Hong Kong, to verify the application effect of the proposed method. The long-term rainfall monitoring in this area provides systematic data for investigating rainfall-induced landslides. Many researchers have conducted extensive and in-depth studies on landslides in the Lantau area, including field experiments, statistical analysis, and remote sensing interpretation (Zhang et al. 2016). The mathematical model of this application case is shown in Fig. 5.1.

The measured data of rainfall and pressure head at monitoring point SP3 are shown in Fig. 5.22. The change of pressure head is basically consistent with the change of rainfall, which better reflects the change of infiltration boundary. The monitoring data can be divided into three periods according to the time of the rainfall peak. Among them, the calibration period is from 0 to 192 h (0:00 on June 8, 2001 to 0:00 on June 16, 2001), and the Bayesian stochastic inversion of soil hydraulic parameters can be performed according to this stage (Yang et al. 2018). The model parameters and permeability coefficients of the two soil–water characteristic curves are shown in Table 5.6. After calibrating the soil hydraulic parameters by random back analysis, the error between the model calculated and measured values is very small (Yang et al. 2018). Numerical simulations are carried out for two rainfall periods in the verification period and compared with the monitoring data. The first period is 5 h (from 4:00 on June 27 to 9:00 on June 27), and the average rainfall is 1.4 cm/h. The second period is 5 h (from 1:00 on July 7 to 6 on July 7), and the average rainfall is 2.2 cm/h. The third period is 3 h (23:00 on July 15 to 2:00 on July 16), and the average rainfall is 2.4 cm/h.

Fig. 5.22

Measured data of rainfall and pressure head at monitoring point SP3

Table 5.6 Soil–water characteristic curve and saturated permeability coefficient

The water pressure meter SP3 is buried at a depth of 200 cm, and the groundwater table depth is 250 cm. The thickness of the soil layer (L) is 250 cm, and $\beta$ is assumed to be 35°. The initial conditions and boundary conditions are expressed as follows:

$$\left[ {K\frac{\partial h}{{\partial z}} + K\cos \beta } \right]_{z = L} = q$$

(5.9)

$$h_{{\left| {z = 0} \right.}} = 0$$

(5.10)

The space step is 2.5 cm, and the time step is 0.1 h. Figures 5.23 and 5.24 compare the results between the numerical solutions of the Richards’ equation of the VGM and Gardner models and the measured values in the first period. It can be found that the Richards’ equation described by the VGM model fits the measured values better. When t = 5 h, the relative error between the simulated and measured values is only 1.23%.

Fig. 5.23

Comparison between the numerical solution of Richards’ equation for the VGM model and the measured value in the first period

Fig. 5.24

Comparison between the numerical solution of Richards’ equation for the Gardner model and the measured value in the first period

Figures 5.25 and 5.26 represent the comparison results between the numerical solutions of the Richards’ equation of the VGM and Gardner models and the measured values in the second period. Compared with the Gardner model, the fitting effect of the Richards’ equation described by the VGM model and the measured values is still better. The relative error is 9.22% when t = 5 h.

Fig. 5.25

Comparison between the numerical solution of Richards’ equation for the VGM model and the measured value in the second period

Fig. 5.26

Comparison between the numerical solution of Richards’ equation for the Gardner model and the measured value in the second period

Figures 5.27 and 5.28 compare the numerical solutions of the Richards’ equation and the measured values of the VGM and Gardner models in the third period. When t = 3 h, the relative error between the simulated value and the measured value computed by the two models is 3.54% and 3.67%, respectively. The numerical results of Richards’ equation using the VGM model are closer to the measured values.

Fig. 5.27

Comparison between the numerical solution of Richards’ equation for the VGM model and the measured value in the third period

Fig. 5.28

Comparison between the numerical solution of Richards’ equation for the Gardner model and the measured value in the third period

As shown in Table 5.7, in the three time periods, the overall root mean square error (RSE) and relative error (RE) of the VGM model and the measured data are both smaller than the values obtained by the Gardner model, which further indicates that the Richards’ equation described by the VGM model has a better fitting effect with the measured data. Numerical results illustrate that the proposed method can well simulate the time-varying response of pressure head in the rainfall infiltration of unsaturated soil slopes and has a good application effect.

Table 5.7 Error between the simulated and the measured values during the validation period

5.4 Conclusions

This chapter studies the application of Chebyshev spectral method in slope stability analysis and the application of improved iterative methods P-CSIM and NTG-PI in shallow rainfall slope stability analysis. In order to verify the accuracy and practicability of the proposed improved method, a comparative study was carried out combining the monitoring data of slope in the Tung Chung area of Hong Kong and the Xiaoba landslide area. The conclusions are drawn as follows:

• (1) The numerical results obtained by the CSM method are highly consistent with the transient analytical solution, which indicates that the proposed method is sufficiently accurate to handle transient infiltration problems associated with rainfall-induced landslides. The numerical solution obtained by the CSM method is introduced into the slope stability analysis to effectively evaluate the slope stability under rainfall conditions. The numerical solutions of the pressure head obtained by the improved methods P-CSIM and NTG-PI can effectively analyze the shallow landslides caused by rainfall. Combined with the rainfall data, a mathematical model was established for the monitoring points, the evolution process of the pressure head of the slope was solved by an improved iterative method, and the stability analysis of the unsaturated soil slope was carried out. The results indicate that the numerical solution is consistent with the measured value, and the relative error is small, showing a good application prospect. The numerical solution to coupled water flow and deformation in two-layer unsaturated porous media is obtained using a finite element method.

• (2) A conceptual model of two-layer unsaturated soils is established to analyze the rainfall infiltration process under different conditions. A simplified analysis of an infinite slope is used to compute the factor of safety as a function of the depth of wetting front, and special attention is paid to the hydrological response at the interface between two layers of unsaturated porous media.