Parametric Analysis of Rainfall-Induced Loess Soil Slope Due to the Rainwater Infiltration

Abstract

Hydraulic properties (such as soil–water characteristic curves (SWCC) and hydraulic conductivity function (HCF)) play an important role in evaluating the stability of unsaturated soil slopes. Loess soils are widely distributed in Gansu Province in China, and most of them are in unsaturated conditions due to the deep groundwater table (G.W.T). In this study, twenty-eight sets of data published in the literature were analyzed to develop the upper and lower bounds of the SWCC for loess soil in Gansu. The variation of HCF for the loess soil was estimated from the upper and lower bounds curve developed in this study. Subsequently, numerical analyses incorporating scenarios considering different SWCCs, HCFs, and rainfall conditions were conducted for investigating the effects of those factors on the rainfall-induced slope stability. The results of analyses indicate that the infiltration plays an important role in the rainfall-induced slope stability. Higher permeable soil leads to a larger infiltration amount, which, in turn, results in a lower safety factor. In addition, the effect of the hydraulic property on the rainfall-induced slope stability decreases with the increase in slope angle.

1. Introduction

Loess soils are widely distributed in northwest China, and most of them are situated in unsaturated conditions due to the deep ground water table. Numerous steep slopes are stable in nature. However, the stability of these slopes is affected by climatic conditions. Rahardjo et al. [1] indicate that rainfall is commonly considered as the major factor resulting in shallow landslides. Brand [2], Kristo et al. [3], Rahardjo et al. [4], and Rahardjo et al. [1] indicate that the infiltration of rainwater is commonly recognized as the main factor leading to rainfall-induced slope failure. The works of Crosta [5], Basile et al. [6], and Ost et al. [7] indicate that several factors such as climatic conditions, geological features, topography, vegetation, or a combination of these factors, could result in the failure of slopes under rainfall conditions. Fredlund et al. [8] indicate that the shear strength function (SSF) of unsaturated soil is a function of soil suction that is sensitive to the infiltrated rainwater. Climatic conditions are gradually changing due to global warming. From a sustainability perspective, it is good to understand the effects of factors, such as those mentioned above, on the variation of the stability of the loess soil slopes. Therefore, in this paper, a parametric study is conducted to investigate the effects of factors such as variation in soil properties (considering different soil–water characteristic curves or SWCCs, hydraulic conductivity functions or HCFs, and SSFs), geometric profile, and variation in rainfall conditions on the stability of loess soil slopes. The safety factor (F_s) for loess soil slopes with different slope angles, soil properties, and rainfall conditions were primarily computed. Subsequently, the infiltration rates near the ground surfaces of the slopes were computed. Consequently, the normalized F_s was re-produced from the computed F_s. It is observed that the F_s for the high permeable soil drops drastically during rainfall. The rate of dropping in F_s decreases with an increase in slope angle.

2. Mathematical Models for SWCC, HFC, and SSF

Various continuous mathematical models, such as Gardner [9], Brooks and Corey [10], van Genuchten [11], and Fredlund and Xing [12] have been proposed for the representation of the SWCC. Fredlund and Xing’s equation [12], as illustrated in Equation (1), is one of most popular models, widely used by geotechnical engineers. Therefore, this study adopts Fredlund and Xing’s equation [12] for the representation of the SWCC of loess soil.

$$\theta =C(\psi )\frac{{\theta }_s}{{\left\{\ln \left[e+{\left(\frac{\psi }{a_f}\right)}^{n_f}\right]\right\}}^{m_f}}=\left[1-\frac{\ln \left(1+\frac{\psi }{C_r}\right)}{\ln \left(1+\frac{{10}^6}{C_r}\right)}\right]\frac{{\theta }_s}{{\left\{\ln \left[e+{\left(\frac{\psi }{a_f}\right)}^{n_f}\right]\right\}}^{m_f}}$$

(1)

where a_f, n_f, and m_f are fitting parameters, and C_r is the input parameter for the rough estimation of the residual suction. Zhai and Rahardjo [13] and Zhai et al. [14] recommend that C_r = 1500 kPa for most cases. The work from Wang et al. [15] indicates that C_r should be larger than the air-entry value (AEV) for the soil with AEV greater than 1500 kPa; θs is the saturated volume water content; ψ is soil suction; θ is the volumetric water content.

Many models have been proposed for the estimation of the HCF for the unsaturated soil. Mualem [16] categorized the models into three groups: empirical (such as Richards [17], Winds [18], and Gardner [9]), macroscopic (such as Averjanov [19] and Yuster [20]), and statistical (such as Childs and Collis-George [21] and Mualem [16]) models. Recently, Fredlund et al. [12] and Zhai and Rahardjo [22] proposed integration form and summation form for the statistical model, respectively. D.G. Fredlund and M.D. Fredlund [23] indicate that Zhai and Rahardjo’s equation [22] is workable using Microsoft Excel, which provides convenience for the engineers. Therefore, Zhai and Rahardjo’s [22] equation (Equation (2)) is adopted for predicting the HCF of the loess soil in this study.

$$k\left({\psi }_k\right)=\frac{k\left({\psi }_{ref}\right){\displaystyle \sum _{i=k}^N\left\{{\left[S\left({\psi }_k\right)-S\left({\psi }_{i+1}\right)\right]}^2-{\left[S\left({\psi }_k\right)-S\left({\psi }_i\right)\right]}^2\right\}{r}_i^2}}{{\displaystyle \sum _{i=1}^N\left\{{\left[1-S\left({\psi }_{i+1}\right)\right]}^2-{\left[1-S\left({\psi }_i\right)\right]}^2\right\}{r}_i^2}}$$

(2)

where S(ψ_k) is the saturation of soil when the soil suction is equal to ψ_k,; S(ψ_i) is the saturation of soil when the soil suction is equal to ψ_i; k(ψ_k) is the permeability coefficient when the soil suction is equal to ψ_k; and k(ψ_ref) is the permeability coefficient when the soil suction at the reference point is equal to ψref.

The correlation between S(ψ) and ψ is defined using Fredlund and Xing’s equation [12] in this study. Fredlund et al. [8] proposed the model for estimating the SSF for the unsaturated soil with two independent variables. Vanapalli et al. [24] and Zhai et al. [25] further improved this model. Zhai et al. [26] conducted stress analysis on the soil structure where the meniscus is created between soil particles. It is observed that meniscus results in two types of action on the soil, including isotropic compression to the soil structure and capillary bonding between soil particles. In addition, Zhai et al. [25] also indicate that hydroscopic water attached to the individual soil particle does not result in any action on the soil structure. The hydroscopic water should not be included in the estimation of the SSF for the unsaturated soil. Zhai et al. [26] indicate that the fourth term in the original Zhai et al. [25] equation has an insignificant effect on the estimated results, and this term can be ignored for most cases. As a result, the simplified equation from Zhai et al. [25], as shown in Equation (3), is adopted to estimate SSF for the loess soil in this paper.

$$\tau =c\prime +\left(\sigma -u_a\right)\tan \varphi \prime +\frac{S-S\prime }{1-S\prime }\left(u_a-u_w\right)\tan \varphi \prime$$

(3)

where S is saturation; S′ is the saturation corresponding to 3100 kPa soil suction; and S(ψ_i) is the saturation corresponding to soil suction ψ_i.

3. Hydraulic Properties of the Loess

Loess soils are widely distributed in northwest China. A large number of high and steep loess slopes maintain long-term stability under natural conditions. Twenty-eight sets of SWCC data for the loess soil were collected from different sources in the published literature, and are illustrated in

Table 1. Index properties of loess soil collected from the different literature.

No.	Soil Type	Liquid Limit LL (%)	Plastic Limit PL (%)	Density ρ (Mg/m3)	Water Content w (%)	Void Ratio e	References
1	Lanzhou loess	29.5	9.4	-	14.8	-	Li et al. [27]
2	Xi’an loess	30.9	17.6	-	3.3	-	Zhang et al.[28]
3	Xining loess	23.2	14.7	-	-	-	Chen et al. [29]
4	Tongchuan loess	34.8	23.5	1.49	16.7	1.12	Jiang [30]
5	Linxia loess	-	-	1.28	15	/	Liu [31]
6	Qingyang loess	31.2	17.8	1.34	18	1.03	Li et al. [32]
7	Gansu loess	-	-	-	-	-	Huang et al. [33]
8	Shaanxi loess	31.33	15	-	12.0	-	Li et al. [34]
9	Lanzhou loess	27.8	17.7	1.50	7.4	-	Jiang et al. [35]
10	Lanzhou loess	27.5	18.3	1.43	8.3	1.06	Xie et al. [36]
11	Yan’an loess	28.21	17.15	1.35	5–25	-	Wang et al. [37]
12	Qingyang loess	31.2	17.8	1.34	37.8	1.03	Li et al. [38]
13	Ningxia loess	-	-	1.29	2.6	1.099	Sun et al. [39]
14	Yulin loess	24	12.8	-	12	-	Cai et al. [40]
15	Weinan loess	28.6	19	1.28	15.2	-	Hu et al. [41]
16	Yan’an loess	28.9	16.1	-	14.11	-	Nie et al. [42]
17	Yan’an loess	28.9	16.1	-	14.11	-	Wang et al. [43]
18	Yili loess	-	-	1.37	6.67	-	Wang et al. [44]
19	Xi’an loess	-	-	1.52	14.1	0.82	Zhang et al. [45]
20	Xi’an loess	34.2	18.6	1.4	21	-	Zheng et al. [46]
21	Yan’an loess	-	-	-	13	0.786	Fu et al. [47]
22	Luoyang loess	30.1	17.6	-	-	-	Xing et al. [48]
23	Luoyang loess	29.8	17.2	-	-	-	Xing et al. [48]
24	Badong loess	-	-	-	-	-	Jian et al. [49]
25	Xi’an loess	30.9	19.8	-	-	-	Zhang et al. [50]
26	Xi’an loess	35.7	16.2	1.52	16.6	-	Mu et al. [51]
27	Xianyang loess	30.5	18.6	1.30	-	1.085	Xing et al. [52]
28	Yuncheng loess	30.1	18	-	12.8	-	Wang et al. [53]

. Some of the index properties for the soils were not reported from the referring literature and are marked as “-“ in Table 1. Both upper and lower bounds for the determined SWCC of loess soil are estimated from the collected data based on the confidence level of 95%, as shown in Figure 1. Subsequently, the HCF were estimated from the best-fitted curve, upper and lower bounds, and shown in Figure 2.

4. Evaluation of the Rainfall-Induced Slope Stability

The Seep/W is formulated based on the assumption that water flow in the soil (either saturated or unsaturated soil) follows Darcy’s law. Richard [17]’s equation, as shown in Equation (4), is commonly used to solve the problems related to the seepage in unsaturated soil.

$$\frac{\partial }{\partial x}\left(k_x\frac{\partial H}{\partial x}\right)+\frac{\partial }{\partial y}\left(k_y\frac{\partial H}{\partial y}\right)+Q=m_w{\gamma }_w\frac{\partial H}{\partial t}$$

(4)

where H is the total water head; k_x and k_y are the hydraulic conductivity in x and y directions, respectively; m_w is the storage capacity of soil; γ_w is the unit weight of water; and Q is the applied boundary condition.

The seepage analyses were conducted using Seep/W, and the results from the seepage analyses were subsequently incorporated in the slope stability analyses using Slope/w. Equation (3) was adopted for estimating SSF of the loess soil from different SWCCs. The schematic diagram of the seepage model and the boundary conditions are illustrated in Figure 3. Various scenarios considering different soil properties, geometric profile, and rainfall conditions are illustrated in

Table 2. Shear strength properties of soil.

Soil Layer	Saturation Unit Weight γ (kN/m3)	Cohesion c′(kPa)	Friction Angle φ′(°)	Water Content w (%)
Loess	15.1	16	27.2	13.5
Weathered mudstone	18.5	20	30	13.2

.

Azarafza et al. [54] reviewed discontinuous rock slope stability analysis using the limit equilibrium approaches. Chen et al. [55] conducted numerical analyses on the slope stability by considering the interaction of rainfall and earthquakes. Zhang et al. [56] conducted slope stability analyses for cracked soil under the rainfall conditions. Deliveris et al. [57] investigated the effect of the rainwater infiltration on the stability of the deep excavation. Azadi et al. [58] conducted slope stability analyses under rainfall, considering the rapid drawdown conditions. In this paper, the parametric study on the rainfall-induced slope stability was conducted by considering the variation in the hydraulic properties of soil, geometrical profile, and rainfall intensity. The pore-water pressure profile was computed from the Seep/w, and subsequently coupled into the Slope/w for the stability analysis. The F_s was computed using the Morgenstern and Price [59] method, which satisfies both force equilibrium and moment equilibrium, as illustrated in Equations (5) and (6), respectively.

$$F_m=\frac{{\displaystyle \sum \left\{c\prime lR+\left(P-u_{w}l\frac{\tan {\varphi }^b}{\tan \varphi \prime }\right)R\tan \varphi \prime \right\}}}{{\displaystyle \sum Wx-{\displaystyle \sum Pf}}}$$

(5)

$$F_f=\frac{{\displaystyle \sum \left\{c\prime l\cos \alpha +\left(P-u_{w}l\frac{\tan {\varphi }^b}{\tan \varphi \prime }\right)\tan \varphi \prime \cos \alpha \right\}}}{{\displaystyle \sum P\sin \alpha }}$$

(6)

where F_f is the factor of safety that satisfies the force equilibrium and F_m is the factor of safety that satisfies the moment equilibrium; c’ is the effective cohesion; φ’ is the effective internal friction angle; l is the base length of the slice; and α is the angle between the base and the horizontal line; u_w is the pore-water pressure; P is the component of weight of slice that is perpendicular to the base; tan(φ^b) defines the rate increase in shear strength of unsaturated soil with an increase in soil suction; W is the summation of the weight of slices; x and f are the arm of force W and P with respect to the rotation point O, respectively; and R is the radius of the arc for the rotation.

Three slope heights Hs = 10 m was selected in the parametric studies. Four slope angles α (15°, 30°, 45°, and 60°) were utilized in the parametric studies. The initial depth H_w and the inclination of the groundwater table (G.W.T) at the slope toe is 4 m and 7°, respectively. Three different soil types were adopted in the parametric studies, namely S1, S2, and S3, illustrated in

Table 3. Fitting parameters of SWCC and saturated hydraulic conductivity.

Soil Type	Fitting Parameters			Saturated Hydraulic Conductivity
	af (kPa)	nf	mf	ks (m/s)
Upper bound loess S1	45.00	1.80	0.38	10−7
Fitting loess S2	12.64	1.99	0.51	10−6
Lower bound loess S3	5.70	1.60	0.95	10−5

. Three types of rainfall intensity including I_r = 8.45 mm/h, 80 mm/h, and k_s with duration of 24 h were set in the study, as shown in

Table 4. Parameter design of influencing factors of slope stability under constant rainfall.

No.	Soil Type	Slope Angle α (°)	Rainfall Intensity Ir (mm/h)	Slope Height Hs (m)	Total
A	S1 S2 S3	15	8.45 80 ks	10	36
		30
		45
		60

. A series of numerical analyses were carried out to evaluate the influence of soil properties on the coefficient of saturated permeability, and the influence of slope angle on the stability of soil slope under different rainfall conditions.

The analyzed results for scenario A are illustrated in Figure 4 for the soil type S1, in Figure 5 for the soil type S2, and in Figure 6 for the soil type S3.

Figure 4, Figure 5 and Figure 6 show that the factors of safety (F_s) drops dramatically during the rainfall period, and decreases slowly after the rainfall stops. It is different from the observation from Rahardjo et al. [60]. The authors conducted a separate analysis, and observe that when the groundwater table is near the ground surface, which is the condition reported in Rahardjo et al. [60], the F_s gradually restores after the rainfall stops. When the initial groundwater table is deep, the F_s decreases slowly after the rainfall stops. Figure 4, Figure 5, and Figure 6indicate that higher rainfall intensity results in lower Fs. It is interesting to note low Fs results from the large amount of rainwater infiltration. It is commonly believed that the saturated coefficient of permeability k_s is the threshold of the infiltration rate. However, Figure 4, Figure 5, and Figure 6show that the infiltration rate can be higher than the k_s. Therefore, the flux rate around the ground surface at 24 h (when rainfall stops) is monitored and shown in Figure 7.

Figure 7 indicates under the same rainfall condition, higher HCF results in a higher infiltration rate. When the rainfall intensity is higher than the k_s of the slope soil, the infiltration rate is higher than k_s. When the rainfall intensity I_r is higher than k_s, the infiltration rate on the section near the crest is larger than that near the toe. For this case, the soil near the ground surface is saturated, water drains out of soil near the toe, and results in a lower infiltration rate near the toe. When the rainfall intensity I_r is lower than k_s, the infiltration rate on the section near the crest is less than that near the toe. For this case, the soil near the ground surface is still unsaturated, and the suction near the toe is lower than that near the crest. As a result, the infiltration rate on the section near the toe is larger than that near the crest for the scenario that I_r is lower than k_s.

For a better comparison, the normalized F_s, which defines the ratio between the factor of safety and the maximum factor of safety at the initial state, is adopted. The results in Figure 4, Figure 5 and Figure 6 were reproduced using the normalized F_s, and illustrated in Figure 8.

It is observed that F_s drops 1% to 5% for the slope with different slope angles under 10 days rainfall for S1; drops 2% to 10% for the slope with different slope angles under 10 days rainfall for S2; and drops 8% to 35% for the slope with different slope angles under 10 days rainfall for S3. The magnitude of dropping in F_s for the three types of soil agrees with the infiltration rates shown in Figure 7. A higher infiltration rate results in a large dropping in F_s. Therefore, the effect of the variation in the hydraulic properties of soil (i.e., the differences between S1, S2, and S3) on the rainfall-induced slope stability is significant. Figure 8 also indicates that the magnitude of the decrease in the normalized F_s decreases with an increase in slope angle. It indicates that the significance of the effect of hydraulic properties on the rainfall-induced slope stability deceases with an increase in the slope angle.

5. Conclusions and Recommendations

The rate of infiltration into the slope soil is much dependent on the hydraulic conductivity of the soil. Higher hydraulic conductivity leads to a larger amount of infiltration (e.g., S3 > S2 > S1 in this study). It is also observed that the slope stability of the higher permeable soil is more sensitive to the rainfall than that of the lower permeable soil. The normalized F_s for S3 drops 8% to 35% under a specific rainfall condition, while those for S1 and S2 drop 1% to 5% and 2% to 10%, respectively. It is also observed that the magnitude of dropping in normalized Fs is 35% for the slope angle of 15°, while it is 18% for the slope angle of 60%. It indicates that the significance of the effect of hydraulic properties on the rainfall-induced slope stability decreases with an increase in the slope angle. As the infiltration plays an important role in the rainfall-induced slope stability, field instruments are recommended to be installed in the loess soil slopes to monitor the variation of pore-water pressure under rainfall conditions.