A Modified Method for Evaluating the Stability of the Finite Slope during Intense Rainfall

Abstract

The Green–Ampt (GA) model is a widely used analytical method to calculate the depth of the wetting front during intense rainfall. However, it neglects the existence of the transition layer and the seepage parallel to the slope surface. Therefore, a modified stratified Green–Ampt (MSGA) model is proposed. A process to assess the stability of the finite slope during a rainfall event is demonstrated by combining the MSGA model and the limit equilibrium method. In the case of the Liangshuijing landslide, the factor of safety presents a negative correlation with the depth of the wetting front. The factor of safety obtained by the stratified Green–Ampt (SGA) model is smaller than that calculated by the MSGA model, and the gap between the factor of safety based on the two methods widens with time. The moving speed of the wetting front accelerates with the increase in the length of the slope surface, and the size effect becomes apparent when the length is short. In the initial stage of infiltration, the effect of the seepage parallel to the slope surface is small. The effect of the seepage cannot be neglected at the latter stage. The result calculated by the MSGA model agrees well with the measured result in the test.

1. Introduction

Landslides, one of the most serious and widespread geological disasters, pose a severe threat to the safety of the public. The main induced factor of the landslide is heavy rainfall [1,2,3]. During the rainfall infiltration process, the gravity of the soils and the seepage force will increase continuously with the development of the wetting front. In addition, the internal friction angle and cohesion of the soils will decrease. Numerous studies have shown that the factor of safety of the slope reduces gradually with rainfall infiltration [4,5,6]. Rainwater infiltration can greatly affect the stability of the slope. Therefore, it is essential to simulate the rainfall infiltration process precisely.

Richards’ equation model and GA are the two most popular methods used to simulate the process of rainfall infiltration [7,8]. Richards’ equation is a numerical model derived from the mass conservation law and Darcy’s law, which has been used widely [9,10,11]. However, using Richards’ equation to simulate the infiltration process requires many implicit parameters, which are sometimes unavailable [12]. Moreover, the numerical solution requires constant iteration and a large amount of calculation [13]. The GA model is an analytic method, another popular method used to simulate rainfall infiltration [14,15]. Since the concept of the GA model is clear and concise, the GA model is widely used [16,17].

The GA model was initially proposed to simulate the rainfall infiltration process in the plane. Then, it was extended to the slope by Chen and Young [18]. Wang [19] improved the GA model by considering the seepage parallel to the slope surface. The original GA model assumes that the soil above the wetting front is saturated without considering the existence of the transition layer. Yao [20] modified the GA model by using an elliptic curve to describe the water content of the transition layer. Jiang and Liu [21] proposed an improved Green–Ampt model considering the heterogeneity of the soil. However, previous research did not consider both the transition layer and seepage parallel to the slope surface at the same time. Therefore, a modified stratified Green–Ampt (MSGA) model is proposed that takes the transition layer and the seepage parallel to the slope surface into account at the same time. A stability evaluation method considering the seepage force for a finite slope during a rainfall event is given.

In this paper, the MSGA model is proposed to simulate rainfall infiltration in a finite slope considering both the transition layer and the seepage parallel to the slope surface. A process to evaluate the time-dependent stability of the finite slope is demonstrated based on the MSGA model and limit equilibrium method. This paper is organized as follows: The background of the GA model and one-dimensional Richards’ equation is introduced in Section 2. The proposed MSGA model and time-dependent method for evaluating the finite slope stability are demonstrated in Section 3. Section 4 gives an example of using the modified method for evaluating the stability of a finite slope during intense rainfall. Several discussions and the validation of the MSGA model are included in Section 5. The main conclusions are summarized in Section 6: Conclusions.

2. Theoretical Background

2.1. GA Model

Green and Ampt [7] proposed the GA model based on Darcy’s law to describe the infiltration process. The GA model assumes that the soil profile is homogeneous and that the initial water content is uniformly distributed. It also assumes that there is a wetting front in the profile. The soil above the wetting front is saturated, while the soil below the wetting front is unsaturated. The water content of the unsaturated soil is equal to the initial water content value. Figure 1a shows the original GA model. However, the original GA model did not consider the influence of the slope dip angle. Chen and Yong [18] modified the GA model by extending it from the plane to the slope. The modified model is shown in Figure 1b. When β is 0 degrees, the modified model degenerates into the original model. The original model is a special case of the modified model. In this paper, the infiltration equation of the GA model has considered the effect of the dip angle of the slope surface, which can be expressed by Equation (1).

$$i=K_s{\displaystyle \frac{Z_f{\cos }^2\beta +H\cos \beta +S_f}{Z_f\cos \beta }}$$

(1)

where i represents the infiltration rate for ponded infiltration (m/h), K_s represents the saturated hydraulic conductivity (m/h), Z_f represents the depth of the wetting front in the vertical direction (m), H represents the depth of the ponding water in the vertical direction (m), S_f represents the suction head of the wetting front (m), and β represents the dip angle of the slope surface (°).

The GA model is by far one of the best models available to describe infiltration during rainfall events [22]. Since the GA model is concise and simple, it has been widely modified and applied to slopes for evaluating the stability of landslide masses [23].

2.2. Distribution of Water Content Based on the SGA Model

The GA model assumes that there are only saturated and natural layers in the water content profile. However, this is not consistent with observations. According to the observation from Colman and Bodman [24], there is a transition layer between the saturated layer and the natural layer. The water content of the soil in the transition layer gradually changes from saturated to the initial water content. According to the experiments and theoretical analysis, it was found that the thickness of the saturated layer approximates to half of the depth of the wetting front [25,26]. Additionally, Wang [25] proposed to use the ellipse function to describe the transition layer, which has been used and validated by other researchers [20,27].

Considering the existence of the transition layer, Yao [20] proposed the SGA model to describe the infiltration of rainwater. The SGA model assumes that there are three layers in the soil water content profile, and the thickness of the saturated layer equals half of the depth of the wetting front. In addition, it assumes that the soil water content varies with depth as an ellipse function. The distribution of the water content in the soil profile can be defined by Equations (2) and (3):

$$\theta (Z)=\{\begin{array}{c}{\theta }_s,0\le Z\le Z_s\\ {\theta }_i+({\theta }_s-{\theta }_i)\sqrt{1-{({\displaystyle \frac{Z-Z_s}{Z_s}})}^2},Z_s\le Z\le Z_f\\ {\theta }_i,Z\ge Z_f\end{array}$$

(2)

$$Z_s={\displaystyle \frac{Z_f}{2}}$$

(3)

where θ denotes water content, θ_i denotes initial water content, θ_s denotes saturated water content, Z_s denotes the thickness of the saturated layer (m), and Z denotes the depth (m). According to the SGA model, the water content distribution is shown in the Figure 2:

2.3. One-Dimensional Richards’ Equation

The one-dimensional Richards’ equation is widely used to simulate the infiltration of rainwater in unsaturated soil, and the equation is solved by the numerical method. In the case of one-dimensional vertical infiltration on the plane, Richards’ equation can be expressed as follows:

$${\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{\partial }{\partial z}}(D{\displaystyle \frac{\partial \theta }{\partial z}})-{\displaystyle \frac{dK}{d\theta }}{\displaystyle \frac{\partial \theta }{\partial z}}$$

(4)

where z denotes the vertical spatial coordinate (m), D denotes soil water diffusivity (m²/h), K denotes the permeability coefficient (m/h), and t denotes time (h). According to Chen and Young [18], when the one-dimensional Richards’ equation is used on the slope, it can be expressed as follows:

$${\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{\partial }{\partial z*}}(D{\displaystyle \frac{\partial \theta }{\partial z*}})-{\displaystyle \frac{dK}{d\theta }}{\displaystyle \frac{\partial \theta }{\partial z*}}\cos \beta$$

(5)

where z* denotes the coordinate in the normal direction to the slope surface (m), and β denotes the dip angle of the slope (°). The diagram of the coordinates is shown in Figure 3.

3. Methodology

3.1. MSGA Model for a Finite Slope

The water exchange among the slope consists of the recharge from the rainfall infiltration and the discharge of the seepage parallel to the slope surface. For the infinite slope, since the volume of the recharge from the infiltration of rainwater is much greater than the discharge of the seepage parallel to the slope surface, it is reasonable to neglect the influence of seepage parallel to the slope surface. However, the seepage parallel to the slope surface should be considered for the finite slope. Few expansions of the GA model have considered both the existence of the transition layer and the seepage parallel to the slope surface for the finite slope. Therefore, the MSGA model is proposed. While soil parameters are widely recognized to exhibit spatial variability, most of the modified GA models assume that soil is homogeneous for simplicity [19,28,29]. The MSGA model assumes that soil is homogeneous. The MSGA model considers the seepage parallel to the slope based on the SGA model. The schematic diagram of the MSGA model for a finite slope during a rainfall event is shown in Figure 4. Usually, the ponding water head is small compared to the suction head of the wetting front, so the value of the ponding water head is assumed to be equal to 0 m.

The infiltration process during intense rainfall can be divided into two stages depending on the relation between rainfall intensity and infiltration capacity. At stage 1, the rainfall intensity is less than the infiltration capacity. At stage 2, the rainfall intensity is greater than the infiltration capacity. The infiltration ability of the slope can be expressed as follows:

$$i_a=K_s{\displaystyle \frac{Z_f{\cos }^2\beta +S_f}{Z_f\cos \beta }}$$

(6)

where i_a denotes infiltration ability (m/h).

Since the permeability coefficient has an exponential relation to the water content, the permeability coefficient in the saturated state is much greater than that in the unsaturated state. Therefore, the MSGA model only considers the seepage in the saturated layer at stage 2.

At stage 1, which represents the initial stage of the rainfall infiltration, the shallow soil of the slope is unsaturated. Rainwater infiltrates into the soil entirely because the infiltration capacity is greater than the rainfall intensity. Therefore, the infiltration rate is dependent on the rainfall intensity, which can be calculated by:

$$i=q\cos \beta$$

(7)

in which q denotes the rainfall intensity (m/h).

With the infiltration of rainwater, the state of shallow soil changes from unsaturated to saturated. When the wetting front forms, this is called the critical time. The infiltration ability (i_a) is equal to the projection of the rainfall intensity on the slope surface (qcosβ) at the critical time. At this time, the depth of the wetting front can be calculated by combing Equations (6) and (7), which is expressed as follows:

$$Z_{fc}={\displaystyle \frac{S_f}{(q/K_s-1){\cos }^2\beta }}$$

(8)

where Z_fc is the depth of the wetting front at the critical time (cm). Based on Equations (2) and (3), the cumulative infiltration depth is defined by:

$$\begin{array}{c}I=({\theta }_s-{\theta }_i)Z_s\cos \beta +{\displaystyle \underset{Z_s}{\overset{Z_f}{\int }}(\theta -{\theta }_i)\cos \beta dz}\\ =({\theta }_s-{\theta }_i){\displaystyle \frac{Z_f}{2}}\cos \beta +{\displaystyle \underset{Z_f/2}{\overset{Z_f}{\int }}({\theta }_s-{\theta }_i)\cos \beta \sqrt{1-{(\frac{z-Z_f/2}{Z_f/2})}^2}dz}\\ ={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)}{8}}\cos \beta Z_f\end{array}$$

(9)

where I is the cumulative infiltration depth (m). The cumulative infiltration depth at the critical time and the critical time can be calculated as follows:

$$I_c={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)}{8}}\cos \beta Z_{fc}={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)S_f}{8(q/K_s-1)\cos \beta }}$$

(10)

$$t_{\mathrm{c}}={\displaystyle \frac{I_c}{q\cos \beta }}={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)S_f}{8q(q/K_s-1){\cos }^2\beta }}$$

(11)

where I_c is the cumulative infiltration depth at the critical time (m), and t_c is the critical time (h).

Stage 1 changes to stage 2 when the critical time is reached. At stage 2, the shallow soil is saturated. The moving speed of the wetting front is controlled by the rainfall infiltration and the seepage parallel to the slope surface. The effect of the rainfall infiltration on the moving speed of the wetting front can be expressed as follows:

$$({\displaystyle \frac{\mathrm{d}{Z}_f}{dt}}{)}_1={\displaystyle \frac{8}{({\theta }_s-{\theta }_i)(4+\pi )\cos \beta }}{({\displaystyle \frac{dI}{dt}})}_1={\displaystyle \frac{8K_s(Z_f{\cos }^2\beta +S_f)}{({\theta }_s-{\theta }_i)(4+\pi )Z_f{\cos }^2\beta }}$$

(12)

According to the law of conservation of mass and Darcy’s law, the following equation can be obtained:

$$K_s\sin \beta Z_s\cos \beta dt={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)}{8}}Ld{Z}_f\cos \beta$$

(13)

where L is the length of the surface for the finite slope (m). Therefore, the effect of the seepage on the moving speed of the wetting front can be expressed as follows:

$${({\displaystyle \frac{d{Z}_f}{dt}})}_2={\displaystyle \frac{4K_s{Z}_f\sin \beta }{({\theta }_s-{\theta }_i)(4+\pi )L}}$$

(14)

The actual infiltration speed of the wetting front is calculated by combining Equations (12) and (14), which is expressed as follows:

$${\displaystyle \frac{\mathrm{d}{Z}_f}{dt}}={\displaystyle \frac{8K_s}{({\theta }_s-{\theta }_i)(4+\pi )}}({\displaystyle \frac{Z_f{\cos }^2\beta +S_f}{Z_f{\cos }^2\beta }}-{\displaystyle \frac{Z_f\sin \beta }{2L}})$$

(15)

Since the expression for Z_f obtained by integrating Equation (15) is complicated, it is suggested to use the numerical method to calculate Z_f. The depth of the wetting front at stage 2 is calculated by the Runge-Kutta 4th-order method, with the initial condition that Z_f = Z_fc and t = t_c.

3.2. Stability Analysis for Various Sliding Surfaces

Slope failure induced by rainfall frequently occurs, and the shallow failure is always parallel to the slope surface [11,30]. The shallow failure may occur along the wetting front or the interface between the saturated layer and the transition layer. Besides shallow failure, failure also occurs along the surface of the bedrock.

For the slip parallel to the slope surface along the wetting front, the factor of safety can be calculated by:

$$F_s={\displaystyle \frac{(G_s+G_t)\cos \beta \tan {\phi }^{\prime }+(u_a-u_w)\tan {\phi }^{b}L+c^{\prime }L}{(G_s+G_t)\sin \beta +P}}$$

(16)

in which F_s denotes the factor of safety, G_s denotes the gravity of the soil in the saturated layer (N/m), G_t denotes the gravity of the soil in the transition layer (N/m), φ′ denotes the effective friction angle of the natural soil (°), φ^b is the angle that denotes the contribution of matrix suction against shear strength (°), c′ denotes the effective cohesion of the natural soil (Pa), u_a denotes pore air pressure (Pa), u_w denotes pore water pressure (Pa), and P denotes the seepage force (N/m).

According to Vanapalli [31], tanφ^b can be expressed as follows:

$$\tan {\phi }^b=\Theta \cdot \tan {\phi }^{\prime }$$

(17)

in which Θ = (θ − θ_r)/(θ_s − θ_r) denotes relative water content. For the slip parallel to the slope surface along the wetting front, combing Equations (16) and (17), the factor of safety can be calculated by:

$$F_s={\displaystyle \frac{\tan {\phi }^{\prime }\left[\left(G_s+G_t\right)\cos \beta +(u_a-u_w){\Theta }_{i}L\right]+c^{\prime }L}{(G_s+G_t)\sin \beta +P}}$$

(18)

in which Θ_i = (θ_i − θ_r)/(θ_s − θ_r).

For the slip parallel to the slope surface along the interface between the transition layer and saturated layer, the factor of safety can be calculated by:

$$F_s={\displaystyle \frac{G_s\cos \beta \tan {\phi }_s^{\prime }+c_s^{\prime }L}{G_s\sin \beta +P}}$$

(19)

in which φ_s′ represents the effective friction angle of the saturated soil (°), and c_s′ represents the effective cohesion of the saturated soil (Pa).

For the slip along the surface of the bedrock, the factor of safety can be calculated by:

$$F_s={\displaystyle \frac{\left(G_s+G_t+G_n\right)\cos \beta \tan {\phi }^{\prime }+c^{\prime }L}{(G_s+G_t+G_n)\sin \beta +P}}$$

(20)

in which G_s denotes the gravity of the soil in the natural layer (N/m).

3.3. Variation in F_s for a Natural Slope during a Rainfall Event

Natural slopes usually have irregularly curved surfaces and slip surfaces, which makes it difficult to calculate the depth of the wetting front for natural slopes during intense rainfall. Without the depth of the wetting front, the stability of the natural slope cannot be evaluated by the limit equilibrium method.

The slice method is adopted to calculate the depth of the wetting front. The sides of the slices are vertical. The curved slope surface and slip surface of each slice are replaced by straight lines. Each slice can be deemed as a finite slope. The simplified model of the natural slope is shown in Figure 5. Then, the MSGA model is used to calculate the depth of the wetting front of each slice.

According to Equation (15), the depth of the wetting front during intense rainfall is influenced by the slice’s dip angle and the slice’s length. Therefore, different slices have different depths of the wetting front. The depth of the wetting front should be continuous. This kind of discontinuity can be reduced by increasing the number of slices.

A slice is divided into three layers based on the MSGA model, which is shown in Figure 6a. The gravity of the ith slice can be expressed as follows:

$$G_i=G_{is}+G_{it}+G_{in}={\gamma }_s{A}_{is}+G_{it}+{\gamma }_i{A}_{in}$$

(21)

where G_i represents the gravity of the ith slice. G_is represents the gravity of the saturated layer of the ith slice (N/m), G_it represents the gravity of the transitional layer of the ith slice (N/m), and G_in represents the gravity of the natural layer of the ith slice (N/m). γ_s represents the unit weight of the saturated soils (N/m³), and γ_i represents the unit weight of the natural soils (N/m³). A_is represents the area of the saturated layer of the ith slice (m²), and A_in represents the area of the natural layer of the ith slice (m²).

According to Yao [20], G_it can be expressed as follows:

$$G_{it}=\{\begin{array}{c}\left[L_1{Z}_f-{\displaystyle \frac{3Z_f^2\sin {\beta }_1}{4}}-{\displaystyle \frac{3Z_f^2\cos {\beta }_1}{4\tan ({\alpha }_1-{\beta }_1)}}\right]\cos {\beta }_1{\gamma }_d\left[{\displaystyle \frac{4(1+{\theta }_i)+\pi ({\theta }_s-{\theta }_i)}{8}}\right],i=1\\ L_i\cos {\beta }_i{\gamma }_d{Z}_f\left[{\displaystyle \frac{4(1+{\theta }_i)+\pi ({\theta }_s-{\theta }_i)}{8}}\right],1<i<n\\ \left[L_n{Z}_f-{\displaystyle \frac{3Z_f^2\sin {\beta }_n}{4}}-{\displaystyle \frac{3Z_f^2\cos {\beta }_n}{4\tan ({\beta }_n-{\alpha }_n)}}\right]\cos {\beta }_n{\gamma }_d\left[{\displaystyle \frac{4(1+{\theta }_i)+\pi ({\theta }_s-{\theta }_i)}{8}}\right],i=n\end{array}$$

(22)

where L_i denotes the length of the slope surface of the ith slice (m), β_i denotes the dip angle of the slope surface of the ith slice (°), and α_i denotes the dip angle of the slip surface of the ith slice (°).

In the saturated layer, the seepage parallel to the slope surface would cause the seepage force parallel to the slope surface, which is expressed as follows:

$$P_i={\gamma }_w{A}_{is}\sin {\beta }_i$$

(23)

where P_i is the seepage force of the ith slice (N/m), and γ_w is the unit weight of water (N/m³). The force diagram of the ith soil slice is shown in Figure 6b.

The factor of safety can be calculated by:

$$F_s={\displaystyle \frac{\left[G_i\cos {\alpha }_i+P_i\sin ({\beta }_i-{\alpha }_i)+E_{i-1}\sin ({\alpha }_{i-1}-{\alpha }_i)\right]\tan {\phi }^{\prime }+E_i+c^{\prime }{l}_i}{G_i\sin {\alpha }_i+P_i\cos ({\beta }_i-{\alpha }_i)+E_{i-1}\cos ({\alpha }_{i-1}-{\alpha }_i)}}$$

(24)

where F_s represents the factor of safety, E_i represents the thrust between the ith slice and the (i + 1)th slice (N/m), and l_i represents the length of the slip surface (m). Equation (24) can be written as follows:

$$\begin{array}{c}{E}_i=F_s{G}_i\sin {\alpha }_i-(G_i\cos a_i\tan {\phi }^{\prime }+c^{\prime }{l}_i)+P_i\left[\cos ({\beta }_i-{\alpha }_i)-\sin ({\beta }_i-{\alpha }_i)\tan {\phi }^{\prime }\right],i=1\\ E_i=F_s{G}_i\sin {\alpha }_i+E_{i-1}\left[F_s\cos ({\alpha }_{i-1}-{\alpha }_i)-\sin ({\alpha }_{i-1}-{\alpha }_i)\tan {\phi }^{\prime }\right]-(G_i\cos a_i\tan {\phi }^{\prime }+c^{\prime }{l}_i)\\ +P_i\left[\cos ({\beta }_i-{\alpha }_i)-\sin ({\beta }_i-{\alpha }_i)\tan {\phi }^{\prime }\right],i\ge 2\end{array}$$

(25)

To be consistent with reality, it is assumed that there is no tensile stress between adjacent slices. Therefore, the value of E_i is replaced by 0 when the calculation result of E_i is smaller than 0, which can be expressed as follows:

$$E_i=\{\begin{array}{c}0,E_i<0\\ E_i,E_i\ge 0\end{array}$$

(26)

The process for calculating the stability of the natural slope is illustrated as follows: Firstly, an arbitrarily small value of F_s is input. Then, for i = 1 to i = n, the values of E_i can be calculated in sequence. If E_n is not equal to 0, then F_s will be reset with a greater value, and the value of E_i will also be recalculated. F_s needs to be iterated until E_n equals 0. When E_n equals 0, the corresponding F_s is the actual factor of safety of the slope. The process of the proposed method is shown in Figure 7.

4. Case Study and Results

4.1. Background

The Liangshuijing landslide is selected as an example to assess the stability of a natural slope during a rainfall event. The Liangshuijing landslide is located in the Three Gorges reservoir area of Chongqing, adjacent to Guling Town. The location of the landslide is shown in Figure 8b. The Liangshuijing landslide was first discovered in 2003. In early April 2009, it accelerated [32]. The landslide has been monitored for many years, which provides a rich dataset [33]. The sliding body consists of a Quaternary soil layer (Q₄^del). The bedrock is comprised of interbedded sandstone and mudstones (J₂s).

The height of the landslide front is about 100 m, the height of the rear edge is about 319.5 m, the relative height difference is about 221.5 m, the longitudinal length of the plane is about 434 m, and the transverse width is 358 m. The total area of the Liangshuijing landslide is 11.82 × 10⁴ m², and the total volume is 407.79 × 10⁴ m³. The average thickness of the sliding body is about 34.5 m. The topographic map of the Liangshuijing landslide is shown in Figure 8a.

The Liangshuijing landslide is located in a subtropical monsoon humid climate area. Rainfall in the Liangshuijing landslide area is mainly distributed from March to August each year, and the total rainfall in these six months can account for more than 65% of the total annual rainfall. The average annual rainfall is 1436.5 mm, with the maximum daily rainfall reaching 189.6 mm.

4.2. Application of the MSGA Model

The 1-1′ profile is chosen as an example to study the process of calculating the safety factor of a natural slope during a rainfall event. Figure 9 shows the 1-1′ profile. According to the boundary and geometry of the slide body, the slide body is divided into eight pieces. From top to bottom, the numbers of the slices are from 1st to 8th.

The physical and mechanical parameters of the soil of the sliding body in the Liangshuijing landslide are shown in

Table 1. Soil parameters of the Liangshuijing landslide.

Parameter	Value
Saturated hydraulic conductivity Ks	0.011 m/h
Saturated water content θs	0.330
Natural water content θi	0.285
Suction at the wetting front Sf	6 m
Natural unit weight of the soil γi	23 kN/m3
Saturated unit weight of the soil γs	23.8 kN/m3
Effective cohesion c′	19.17 kPa
Effective friction angle φ′	25.03°

. The geometric elements and infiltration parameters of each slice in the 1-1′ profile are listed in

Table 2. Geometric elements and infiltration parameters of each slice in 1-1′ profile.

Number of the Slice	α (°)	β (°)	L (m)	Zfc (m)	tc (h)
1st	35	20	61	2.63	2.64
2nd	40	32	43	3.23	3.24
3rd	34	31	77	3.16	3.17
4th	26	15	27	2.49	2.50
5th	23	9	31	2.38	2.39
6th	18	22	56	2.71	2.71
7th	14	26	81	2.87	2.89
8th	10	38	50	3.74	3.76

. The rainfall intensity is set to 0.04 m/h. The MSGA model and SGA model are used to calculate the factor of safety of the slope. The variation in the factor of safety of the slope with time is shown in Figure 10.

The 5th slice has the smallest dip angle of the slope surface among the slices, and it reaches the critical state earliest. The difference between the MSGA model and the SGA model appears at t_c^5th = 2.39 h.

The slope of the curve of the average depth of the wetting front represents the average moving speed of the wetting front. The infiltration rate of the wetting front calculated by the MSGA model is fast at first, but it becomes slower and slower with time. At last, the depth of the wetting calculated by the MSGA model tends to be a constant. The explanation for the tendency might be that it reaches a state of balance. In the state of balance, the recharge from the rainwater infiltration just compensates for the discharge of the seepage parallel to the slope surface. The advancing speed of the wetting front calculated by the SGA model also decreases constantly. The gap between the speed calculated by the two methods widens with time. The factor of safety shows a negative correlation with the average depth of infiltration. The factors of safety calculated by the two models change from 1.195 at the initial time to 1.141 and 1.037 at 40 h, respectively.

5. Discussion

5.1. The Applicability of the MSGA Model

In order to verify the accuracy of the MSGA model, it is used to calculate the depth of the wetting front under the M-3 test condition proposed by Lin [34]. The results calculated by the MSGA model are compared with the measured data. In addition, the GA model and SGA model are also used to compare with the MSGA model.

According to Ma [35], the results of the Richards equation solved by HYDRUS-1D are not very consistent with the measured results of the experiments on the flat surface. The HYDRUS-1D code severely underestimates the advancing depth of the wetting front. However, Richards’ equation is widely used to simulate the infiltration of the rainwater for the slope. Therefore, it is necessary to study the applicability of Richards’ equation to the slope. The Runge-Kutta 4th-order method is used to solve the Richards equation under the M-3 test condition. The results derived from Richards’ equation are compared with the MSGA model, GA model, and SGA model.

The soil water retention curve of the silt is described by the Van Genuchten model, which is expressed as follows [36]:

$$\{\begin{array}{c}{\displaystyle \frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}}={(1+{\left|\alpha h\right|}^n)}^{-m},h>0\\ \theta ={\theta }_s,h\le 0\end{array}$$

(27)

in which θ_r represents the residual water content, and h represents the soil water pressure head (m). a, m, and n represent empirical parameters (m = 1 − 1/n). The relation between the unsaturated hydraulic conductivity (K) and soil water pressure head (h) is expressed as follows [37]:

$$K(h)={\displaystyle \frac{K_s{\left\{1-{(\alpha h)}^{mn}{\left[1+{(\alpha h)}^n\right]}^{-m}\right\}}^2}{{\left[1+{(\alpha h)}^n\right]}^{ml}}}$$

(28)

in which l is an empirical parameter equal to 0.5 for most soils.

The M-3 test model conducted by Lin is shown in Figure 11. In the M-3 test, three tensiometers are installed in the model. Tensiometers 1, 2, and 3 are at a depth of 0.4 m, 0.1 m, and 0.167 m, respectively. The slope angle of the model is 33.7°. The rainfall intensity equals 0.04 m/h.

Table 3. Parameters of silt.

ks (m/h)	θs	θr	θi	Sf (m)	a (1/m)	n	m
0.016	0.405	0.1	0.1	0.09	2.020	1.587	0.3699

shows the soil parameters.

The depths of the wetting front calculated by different methods and measured by the tensiometers are shown in Figure 12. The times of the wetting front arriving at a depth of 0.1 m, 0.167 m, and 0.4 m are recorded in

Table 4. Comparison of the arrival time.

Method	Arriving Time at a Certain Depth (h)
	0.1 m	0.167 m	0.4 m
Tensiometer	0.68	1.26	4.06
GA	0.77	1.40	4.41
SGA	0.68	1.26	3.94
MSGA	0.68	1.26	4.04
Richards’ equation	0.80	1.95	7.02

.

The depths of the wetting front calculated by the MSGA model and SGA model are consistent with the measured results. The accuracy of the GA model is worse than that of the MSGA model and SGA model. The difference between the results calculated by the MSGA model and the SGA model widens over time.

At the first 0.7 h, the depth calculated by Richards’ equation is close to the results calculated by different methods and measured by the tensiometer. Then, the depth calculated by Richards’ equation diverges greatly from the measured results. The Richards equation solved by the Runge-Kutta 4th-order method severely underestimates the moving speed of the wetting front, which is in accordance with the results calculated by HYDRUS-1D in the plane [35].

5.2. Effects of the Factors for Infiltration

Besides the properties of the soil, the rainwater infiltration process based on the MSGA model is affected by the length of the slope surface and the dip of the slope surface. A series of parametric studies are performed to assess the effect and relative contribution of the controlling factors on infiltration. The parameters of the soil and the rainfall intensity in the parametric studies are set as follows: θ_s = 0.45, θ_i = 0.15, K_s = 0.016 m/h, S_f = 0.09 m, and q = 0.04 m/h.

Four lengths of the slope surface, L (1, 3, 5, and 10 m with an inclination of 45° from the horizon), are used for comparison. The result is shown in Figure 13. The depth of infiltration is positively correlated with the length of the slope surface. The size effect becomes apparent when L is short. When L is large enough, the MSGA model degenerates into the SGA model.

Three dip angles of the slope surface, β (15, 30, and 45° with a length of the slope surface of 2 m), are used for comparison. The result is shown in Figure 14. As mentioned above, for the MSGA model, the moving speed of the wetting front is controlled by the recharge of the rainfall infiltration and the discharge of the seepage parallel to the slope surface.

At the initial stage, with the same dip of the slope surface, the depths of the wetting front calculated by MSGA and SGA are similar. The explanation for this might be that the recharge of the rainfall infiltration dominates the infiltration rate of the wetting front. The effect of the seepage parallel to the slope surface, which is neglected in the SGA model, is small at the initial stage. Therefore, the difference between the two methods is very small. With the same dip angle of the slope surface, the depth of the wetting front calculated by the MSGA model is less than the result calculated by the SGA model. It indicates that seepage parallel to the slope inhibits the development of rainfall infiltration depth. The infiltration depth increases with the increase in slope inclination, which is consistent with the conclusion of Chen and Young [18].

At the latter stage, the depth of the wetting front calculated by the MSGA model decreases with the increase in slope inclination, because the seepage parallel to the slope surface dominates the moving speed of the wetting front at the latter stage. The volume of the seepage parallel to the slope surface increases with the increasing depth of the wetting front, according to Equation (15). Therefore, from the initial stage to the latter stage, the effect of the seepage parallel to the slope surface increases constantly.

5.3. Stability for Different Sliding Surfaces

To compare the factor of safety for different sliding surfaces, a series of studies are performed on a finite slope, which is shown in Figure 15. The parameters of the soil are set as y_d = 13.45 kN/m³, θ_s = 0.45, θ_i = 0.15, θ_r = 0.10, c′ = 19 kPa, c_s′ = 12 kPa, φ′ = 19.8°, φ_s′ = 13.1°, u_a-u_w = 5 kPa. Two dip angles of the slope surface, α (15° and 25°), are used for comparison. The length of the slope surface is 5 meters. The method to calculate the factor of safety for various sliding surfaces is illustrated in Section 3.2: Stability analysis for various sliding surfaces.

The variation in factors of safety based on different sliding surfaces is shown in Figure 15. Compared with the factors of safety for sliding along the interface and the wetting front, the factor of safety for sliding along the bedrock surface decreases slowly with the depth of the wetting front. The factors of safety for sliding along the bedrock surface and the wetting front are equal when the wetting front reaches the bedrock surface. At the initial stage, the bedrock surface is the most dangerous sliding surface. With the advancing of the wetting front, the interface becomes the most dangerous sliding surface. When α = 15°, the factor of safety for sliding along the interface drops below 1 firstly. When α = 25°, the factor of safety for sliding along the bedrock surface drops below 1 firstly. The sliding surface of a slope is largely determined by the dip angle of α. The results are shown in Figure 16.

6. Conclusions

This paper proposes the MSGA model to simulate rainfall infiltration. The MSGA model considers the existence of the transition layer and the seepage parallel to the slope surface. This paper uses the Liangshuijing landslide as an example to demonstrate how to calculate the factor of safety of a natural slope. Also, this paper validates the accuracy of the MSGA model and discusses the applicability of Richards’ equation. Finally, this paper discusses the effects of the factors for infiltration. According to the analyses conducted, the conclusions are obtained as follows:

(1) The factor of safety of the Liangshuijing landslide decreases constantly with time during an intense rainfall event. The factor of safety presents a negative correlation with the depth of the wetting front. The factor of safety calculated by the SGA model is smaller than that calculated by the MSGA model, and the gap between the results widens with time.

(2) The MSGA model can more accurately simulate the process of rainfall infiltration compared with the SGA model and Richards’ equation. Richards’ equation solved by the Runge-Kutta 4th-order method evidently underestimates the advancing speed of the wetting front.

(3) The length of the slope surface presents a positive correlation with the advancing speed of the wetting front, and the size effect becomes apparent when the length is short. The recharge of the rainwater infiltration and seepage parallel to the slope surface controls the moving speed of the wetting front. At the initial stage, the effect of the seepage parallel to the slope surface is small. As the infiltration depth increases, the volume of the seepage parallel to the slope surface increases. At the latter stage, the seepage parallel to the slope surface dominates the moving speed of the wetting front.

In this paper, it is assumed that the soil is homogeneous. The spatial variability and the heterogeneity of the soil parameters are not considered. In addition, the 2D plane strain model is used in this article and most of the previous research. However, few studies have focused on the application of the GA model to 3D slope stability analysis due to the complexity of the calculation, while actual slopes are usually 3D. This may affect the reliability of the result. Therefore, further studies are required to be undertaken in the future to solve the problem mentioned above.