The Influence of Horizontal Variability of Hydraulic Conductivity on Slope

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Article The Inﬂuence of Horizontal Variability of Hydraulic Conductivity on Slope Stability under Heavy Rainfall

Tongchun Han 1,2,* , Liqiao Liu 1 and Gen Li 1

1 Research Center of Costal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China; [email protected] (L.L.); [email protected] (G.L.) 2 Key Laboratory of Soft Soils and Geoenvironmental Engineering, Ministry of Education, Zhejiang University, Hangzhou 310058, China * Correspondence: [email protected]

Received: 5 August 2020; Accepted: 10 September 2020; Published: 15 September 2020

Abstract: Due to the natural variability of the soil, hydraulic conductivity has significant spatial variability. In the paper, the variability of the hydraulic conductivity is described by assuming that it follows a lognormal distribution. Based on the improved Green–Ampt (GA) model of rainwater infiltration, the analytical expressions of rainwater infiltration into soil with depth and time under heavy rainfall conditions is obtained. The theoretical derivation of rainfall infiltration is verified by numerical simulation, and is used to quantitatively analyze the effect of horizontal variability of the hydraulic conductivity on slope stability. The results show that the variability of the hydraulic conductivity has a significant impact on rainwater infiltration and slope stability. The smaller the coefficient of variation, the more concentrated is the rainwater infiltration at the beginning of rainfall. Accordingly, the wetting front is more obvious, and the safety factor is smaller. At the same time, the higher coefficient of variation has a negative impact on the cumulative infiltration of rainwater. The larger the coefficient of variation, the lower the cumulative rainwater infiltration. The conclusions reveal the influence of the horizontal variation of hydraulic conductivity on rainwater infiltration, and then the influence on slope stability.

Keywords: hydraulic conductivity; variability; slope stability; rainwater inﬁltration

1. Introduction Hydraulic conductivity is an important factor affecting rainwater infiltration, and then inducing landslides. Rainwater infiltration into soil causes the increase in soil’s saturation and the corresponding decrease in matrix suction. On the other hand, rainwater infiltration also will lead to an increase in soil weight. All these adversely affect slope stability [1]. Due to the differences in material composition, settlement conditions, stress history, and geological effects, etc., slope soil has significant spatial variability of hydraulic conductivity [2–4]. In fact, among all the soil parameters, the hydraulic conductivity shows the strongest variability, and the resulting impact on rainwater infiltration has to be considered in the slope stability analysis. The coefficient of variation (CoV) of some soil parameters is shown in Table1.

Water 2020, 12, 2567; doi:10.3390/w12092567 www.mdpi.com/journal/water Water 2020, 12, 2567 2 of 23

Table 1. Variation coeﬃcient of diﬀerent soil parameters.

Property Soil Classification CoV (%) Source Unit weight (γ) 3–7 [5] Natural water content (wn) Clay and silt 8–30 Liquid limit (wL) Clay and silt 6–30 [6] Plasticity index (PI) Clay and silt 10–40 Effective stress friction angle (◦) 2–13 [7] Undrained shear strength (su) 1 Clay 10–40 Undrained shear strength (su) 2 Clay 7–25 [6] Saturated clay 68–90 [5] Hydraulic conductivity Partly saturated clay 130–240 [8] - 200–300 [9] 1 Unconfined compression test; 2 Unconsolidated-undrained triaxial compression test.

Table1 shows that the degree of variation of the saturated hydraulic conductivity is the strongest. Therefore, scholars around the world have extensively discussed the influence of the variability of the hydraulic conductivity on infiltration, and the impact on slope stability. Santoso et al. [10] regarded the saturated hydraulic conductivity as the random field of the static lognormal distribution, discussed the influence of the spatial variability of the saturated hydraulic conductivity along the depth direction on the infiltration of unsaturated soil by using the subset simulation method, and proposed probabilistic framework for evaluating unsaturated slope stability. Jiang et al. [11] proposed a non-intrusive stochastic finite element method for stability analysis of unsaturated soil slopes based on Latin hypercube sampling, considering the spatial variability of multiple soil parameters, and carried out reliability analysis of unsaturated soil slopes under steady seepage conditions. The Monte Carlo method is often used in random field simulation. Based on the Monte Carlo method, Qin et al. [12] established a model for bedrock layer slope stability considering the variation of hydraulic conductivity, and analyzed the effect of the variation coefficient of hydraulic conductivity and rainfall intensity on the failure probability of landslides. Zhu et al. [3] used the fast Fourier transform technique to generate the random field of saturated permeability coefficient, and then studied the influence of the variation of the hydraulic conductivity on the slope stability under rainfall conditions based on the Monte Carlo method. Zhang et al. [13] proposed a probabilistic method for predicting the occurrence time of rainfall-induced landslides considering the variability of hydraulic conductivity based on the physical infiltration model and the Monte Carlo method. Dou et al. [14] regarded the saturated permeability coefficient as a non-stationary random field, and considered the decreasing trend of soil hydraulic conductivity with depth. Based on the Monte Carlo method the one-dimensional non-stationary random field model was established. Cho [15] developed a one-dimensional random field model based on KL (Karhunen–Loeve) to establish the spatial variability of saturated permeability coefficient, and discussed the failure mechanism of shallow weathered residual soil slope under rainfall conditions. Li et al. [16] constructed a geological disaster meteorological risk early warning model based on the information method, and took a certain area of Honghe Prefecture as an example to provide early warning of geological disasters such as rainfall-induced landslides. Zhu et al. [17] undertook stochastic infinite slope analysis by assuming saturated hydraulic conductivity as the only random variable to assess the effect of depth on slope stability. The spatial variability of the saturated hydraulic conductivity considering the depth dependency is generated using the generic random field theory. The results show that if the depth dependency is not considered, the probability of failure is incorrectly estimated. The above studies are almost based on the numerical simulation of the random field with hydraulic conductivity. In general, the Monte Carlo method is used to generate the random field of hydraulic conductivity. Then, slope stability analysis is carried out for these random fields, and the failure probability of the slope is obtained. All these methods require a large number of random fields to be Water 2020, 12, 2567 3 of 23 simulated. On the one hand, the calculation efficiency is low. Secondly, we can only qualitatively study the effect of the horizontal variability of hydraulic conductivity on slope stability. In this paper, we focus on the quantitative analysis of the influence of horizontal variability of hydraulic conductivity on slope stability. The structure of the paper is as follows. First, hydraulic conductivity is considered as a random variable, and a random variable model is used to describe the variability of hydraulic conductivity. Second, based on the improved Green–Ampt (GA) model of rainwater infiltration, the analytical solution of rainwater infiltration with depth and time under heavy rainfall conditions are obtained. Third, the safety factors of infinite slope under heavy rainfall conditions are calculated based on the theory of unsaturated soil strength. In the end, the quantitative analysis of the effect of the horizontal variability of hydraulic conductivity on slope stability is implemented.

2. Variability of Hydraulic Conductivity The variability of hydraulic conductivity is due to its heterogeneity in spatial distribution, and is affected by many factors. Field experiments conducted by Lei et al. [18] showed that the ratio of the maximum value to the minimum value of soil hydraulic conductivity measured on a 40 m 70 m plot could reach 1426, showing significant horizontal variability of hydraulic conductivity. × In addition, when the rainfall period is short and the intensity is high, rainfall-induced landslides are generally shallow landslides, and the landslide body is generally longer. In this case, the influence of the variability on rainwater infiltration cannot be ignored. Therefore, in slope design or landslide prediction, when hydraulic conductivity is taken as a constant value without considering its variability, rainwater infiltration into the landslide body will not be correctly reflected, and the impact of rainwater infiltration on the slope stability also cannot be correctly evaluated. Many people have studied the variability of hydraulic conductivity. In general, the saturated hydraulic conductivity Ks is considered to follow a lognormal distribution, and the mean µKs , variance σKs or CoV of the saturated hydraulic conductivity are used to describe its probability distribution. Correspondingly, ln Ks is a random variable that obeys a normal distribution, µln Ks and σln Ks are its mean and variance respectively. The relationship between the parameters is expressed as follows:   σ2 2  Ks  2 σ = ln1 +  = ln 1 + CoV , (1) ln Ks  µ2  Ks

1 2 µln K = ln(µK ) σ , (2) s s − 2 ln Ks

The probability density function of saturated hydraulic conductivity Ks is expressed as:   1  1 2 ( ) =   f Ks exp ln Ks µln Ks , (3) √2πσ K −2σ2 −  ln Ks s ln Ks

The cumulative inﬁltration distribution function is as follows:   ln[K (x)]  1  s µln Ks  P[Ks > Ks(x)] = erfc − , (4) 2  √  2σln Ks where erfc is the complementary error function.

3. Infiltration Model Rainfall enters the soil layers through the process of infiltration. Many infiltration models have been developed, and can simulate the rainwater infiltration well. We apply the Green–Ampt model in this paper to calculate water infiltration into soils. Thus, the following assumptions are made: First, as rain continues to fall and water infiltrates, the wetting front advances at the same rate with depth, Water 2020, 12, 2567 4 of 23 Water 2020, 12, x FOR PEER REVIEW 4 of 23 contentswhich produces remain aconstant well-defined above wetting and below front, the like wetti a pistonng front movement. as it advances. Second, Third, the volumetric the soil-water water suctioncontents immediately remain constant below above the wetting and below front the remains wetting constant front as with it advances. both time Third, and location the soil-water as the wettingsuction immediatelyfront advances. below Meanwhile, the wetting soil surface front remains becomes constant saturated with immediately both time after and locationheavy rainfall. as the wettingThe front driving advances. force that Meanwhile, induces soilthe surfacevertical becomesinfiltration saturated into soils immediately consists of after two heavy parts: rainfall. one is gravity,The the driving other force is the that matric induces flow the potential vertical infiltration Φ of the unsaturated into soils consists soil. For of two unsaturated parts: one issoils, gravity, the hydraulicthe other isconductivity the matric flowand potentialthe matricΦ flowof the potentia unsaturatedl are related soil. For with unsaturated the relative soils, volumetric the hydraulic water content.conductivity We adopt and the the matric Brook flowand potentialCorey model are relatedto determine with thethe relativerelationship volumetric between water the content.relative Wehydraulic adopt theconductivity Brook and and Corey the modelwater tosaturation, determine an thed the relationship relationship between between the the relative relative hydraulic matric flowconductivity potential and and the water water saturation, saturation, which and theare relationshipdefined as [19] between the relative matric flow potential ()==η /λ and water saturation, which are definedKsrs as [19 KK] / s , (5)

η/λ Kr(s) = K/Ks =ψ s , (5) Φ=Φ=()sK/ b , rs()1−η sη−1/λ (6) ψb Φr(s) = Φ/Ks = , (6) where s =−()θθ/ () θ − θ is the relative volumetric(1 waterη)sη 1content,/λ which is water saturation, θ rsr − − s iswhere the saturateds = (θ θvolumetricr)/(θs θ rwater) is the content, relative θ volumetric is the residual water volu content,metric which water is content, water saturation, λ is a constantθs is − − r the saturated volumetric water content, θr is the residual volumetricη water=+ content,λ λψis a constant fitting parameter reflecting the distribution index of soil pore size, 23, and b is the air ﬁtting parameter reﬂecting the distribution index of soil pore size, η = 2 + 3λ, and ψ is the air entry entry value of the pressure potential. b value of the pressure potential.

3.1. Spatial Spatial Averaging Averaging of the Infiltration Infiltration Model In order to consider the influenceinfluence of the spatial variability of the hydraulic conductivity on the infiltration,infiltration, thethe slope slope surface surface is evenlyis evenly divided divided into numerousinto numerous independent independent rectangular rectangular cube elements, cube elements,as shown inas Figureshown1 .in The Figure soil properties 1. The soil of properti the heterogeneouses of the heterogeneous field are assumed field to are be assumed uniform alongto be uniformthe vertical along direction the vertical of every direction infiltration of element,every infiltration but are allowed element, to varybut are in theallowed horizontal to vary directions. in the horizontalThe hydraulic directions. conductivity The hydraulic of each rectangularconductivity infiltration of each rectangular element is infiltration the same, element and the variationis the same, of andthe hydraulicthe variation conductivity of the hydraulic of different conductivity infiltration of elements different represents infiltration the elements variability represents of hydraulic the variabilityconductivity of inhydraulic the horizontal conductivity direction. in the horizontal direction. x

Infiltration element

y z

Figure 1. Schematic diagram of rectangular cube inﬁltration elements. Figure 1. Schematic diagram of rectangular cube infiltration elements.

Initially, we assumed the relative volumetric water content in the ﬁeld to be identically si Initially, we assumed the relative volumetric water content in the field to be identically s everywhere, i.e., s = si at time t = 0. At time t, under given boundary conditions, the water saturationi = everywhere,at the surface i.e., becomes s si satt, time and thet = 0. water At time saturation t, under belowgiven boundary the wetting conditions, front L is the still watersi based saturation on the Green–Ampt model. Take the water saturation s as the main state variable, then Darcy’s law along the at the surface becomes st , and the water saturation below the wetting front L is still si based on the vertical direction can be written as [20] Green–Ampt model. Take the water saturation s as the main state variable, then Darcy’s law along the vertical direction can be written as [20] ∂Φr[s(z, t)] qz(z, t) = Ks + K(z, t), (7) − ∂z

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Integration of Equation (7) from the soil surface to the wetting front L in vertical direction leads to the following depth-integrated Darcy’s law:

Z L Z L qz(z, t) dz = K(z, t)dz + Ks[Φr(st) Φr(si)], (8) 0 0 −

Based on the assumption that the water content profile is rectangular above the wetting front, and the water saturation below the wetting front is always kept at the initial state si, Equation (8) can be rewritten as: qz(0, t)L(t; Ks) = KsKr(s )L(t; Ks) + Ks[Φr(st) Φr(s )], (9) 0 − i where s0 is the water saturation at the surface. Considering only vertical flow along infiltration element, the continuity equation becomes:

∂ ∂qz(z, t) (θs θr)[s(z, t) s ] = , (10) ∂t − − i − ∂z Integration of Equation (10) from the soil surface to the wetting front brings about the following depth-integrated continuity equation:

∂ qz(0, t) = KsKr(s ) + [(θs θr)(st s )L(t; Ks)], (11) i ∂t − − i

3.2. Equation Solving Under heavy rain, the soil surface becomes saturated immediately, i.e., the surface water content s0 = 1.0 is a constant. The only unknown variable of the depth-integrated Darcy’s law Equation (9) and the depth-integrated continuity Equation (11) under this boundary condition is the wetting front L. The combination of Equations (9) and (11) gives the diﬀerential equation for the wetting front location L(t; K ): s " # d L(t; Ks) β = 1 + , (12) dt µKs L(t; Ks) where parameters β and µ are deﬁned by: " # " # Φr(s0) Φr(si) Kr(s0) Kr(si) β = − , µ = − Kr(s ) Kr(s ) (θs θr)(s s ) 0 − i − 0 − i

Equation (12) is separable, the initial condition is L(0; Ks) = 0 at t = 0, and its solution is expressed as: " # tKsµ L(t; Ks) L(t; Ks) = log 1 + , (13) β β − β

Note that the wetting front location L(t; Ks) in Equation (13) is an implicit function of time t and saturated hydraulic conductivity Ks. The wetting front location L(t; Ks) is a monotonically increasing function of saturated hydraulic conductivity Ks for a fixed time t, i.e., the greater the saturated hydraulic conductivity of the soil cube infiltration element, the greater the depth of the wetting front. Let us define Ksz(L, t) as the hydraulic conductivity Ks which satisfies Equation (13) with a given L(t; Ks) and a given time t. For any given z, t and Ks, L(t; Ks) < z is equivalent to Ks < Ksz(z, t), and L(t; Ks) > z is equivalent to Ks > Ksz(z, t). The function Ksz(z, t) is defined by: " !# β z z Ksz(z, t) = log 1 + , (14) µt β − β Water 2020, 12, 2567 6 of 23

3.3. Saturation at Any Depth

Integration of s(z, t; Ks(x)) over the horizontal area of interest gives the mean water saturation at vertical location z at time t:     1 1   s (z, t) = As(z, t; Ks(x))dx =  sidx + s0dx, (15) h i Ax Ax x  A1 A2 where A1 is the area occupied by soil of Ks(x) such 0 < Ks(x) < Ksz(z, t), and A2 is the area occupied by soil of Ks(x) such that Ksz(z, t) < Ks(x) < . ∞ When the probability density function or the frequency distribution function of Ks(x) over the area A is known, we have the following relations: Z 1 Ksz = ( ) = [ ( )] x A1 sidx si fKs Ks dKs siPr Ks < Ksz z, t , A 0 Z 1 0 = ( ) = [ ( )] x A2 s0dx s0 fKs Ks dKs s0Pr Ks > Ksz z, t , A Ksz

As mentioned earlier, the saturated hydraulic conductivity Ks obeys the lognormal distribution. The probability Pr[Ks > Ksz(z, t)] can be explicitly expressed in terms of the complementary error function according to Equation (4). Consequently, the analytical expression of time-space varying mean areal water saturation is obtained as:

s0 si s (z, t) = s + − erfc[u(Ksz(z, t))], (16) h i i 2 where the function u is deﬁned as: ln k µln K u(k) = − s √ 2σln Ks

4. Numerical Model The theoretical model of rainwater infiltration under heavy rainfall conditions is established above. Based on the different hydraulic conductivities of every soil infiltration elements, the rainwater infiltration calculation considering the horizontal variation of hydraulic conductivity is implemented. In order to verify the correctness of the theoretical infiltration model, a numerical simulation is done for comparison. A Richards equation solver named suGWFoam is used for numerical simulation, which is an open source 3D saturated-unsaturated groundwater flow solver based on OpenFOAM [21].

4.1. Model Establishment Zavattaro et al. [22] measured near-saturated hydraulic conductivity in a 8 8 m plot on a silt × loam soil at Säby, in east central Sweden, to investigate the spatial dependence of near-saturated hydraulic conductivity. In total, 37 measurement locations were located on the plot. The variability was represented by a single parameter, the scale factor δi, following the Miller and Miller similar media theory, based on which the saturated conductivity is scaled with the scale factor:

K = K δ2, (17) si ref × i where Kref is the reference value of the plot, which is equal to 30.9 cm/h in the plot, and i refers to each location. The contour map of the scale factor is shown in Figure2. WaterWater 2020 2020, ,12 12, ,x x FOR FOR PEER PEER REVIEW REVIEW 77 of of 23 23

K wherewhere Krefref is is the the reference reference value value of of the the plot, plot, which which is is equal equal to to 30.9 30.9 cm/h cm/h in in the the plot, plot, and and i irefers refers to to Watereacheach 2020location. location., 12, 2567 The The contour contour map map of of the the scale scale factor factor is is shown shown in in Figure Figure 2. 2. 7 of 23 Distance(m) Distance(m) 02 468 02 468 8 00 2462468 Distance(m)Distance(m) FigureFigureFigure 2.2. 2. Contour Contour map map of of scale scale factor factor δ δ.. .

TheThe plot plot surface surface waswas was divideddivided divided intointo into aa a gridgrid grid ofof of 1616 16 rowsrows rows andand and 1616 16 columns,columns, columns, eacheach each squaresquare square isis is 0.50.5 0.5 × × 0.5 0.5 m m in in × size,size, and and the the hydraulic hydraulic conductivity conductivity of of each each square square is is represented represented byby by thatthat that atat at the the midpoint midpoint of of the the small small square,square, whichwhich which isis is obtainedobtained obtained byby by interpolation.interpolation. interpolation. Thus,Thus, eacheach squaresquare corresponds corresponds to to a a soil soil column,column, column, and and thethe the thicknessthicknessthickness ofof of thethe the soilsoil soil layerlayer layer isis is 33 3 m.m. m. The The distribution distribution of of hydraulichydraulic conductivityconductivity inin ParaViewParaView is is shown shown in in FigureFigure3 3. .3.

FigureFigure 3. 3. The The distribution distribution ofof hydraulichyhydraulicdraulic conductivity. conductivity.

TheThe established established model model is is shown shown in in Figure Figure4 4. .4. ToTo To focusfocus focus onon on thethe the initialinitial initial infiltration,infiltration, infiltration, thethe the upperupper upper partpart part ofof of thethethe modelmodel model hashas has aa a finerfiner finer meshmesh mesh thanthan than lowerlower lower part.part. part. The The boundary boundary conditions conditions areare are specifiedspecified specified asas as follows:follows: follows: for for top top surface,surface, saturationsaturation saturation isis is setset set toto to 1.01.0 1.0 atat at tt t == = 0, 0, and and for for four four side side surfaces, surfaces, symmetry symmetry boundary boundary isis is set.set. set. The The initial initial water content θθiθis set to 0.12 for the whole soil body. waterwater content content i i is is set set to to 0.12 0.12 for for the the whole whole soil soil body. body.

Water 2020, 12, x FOR PEER REVIEW 8 of 23

WaterWater 20202020, 12, 12, x, 2567FOR PEER REVIEW 8 8of of 23 23

Figure 4. Numerical model for infiltration.

To model rainwater infiltration,Figure 4.theNumerical VG equati modelon proposed for infiltration. by van Genuchten in 1980 is used Figure 4. Numerical model for infiltration. [23] with the expression as below To model rainwater infiltration, the VG equation proposed by van Genuchten in 1980 is used [23]  θ −θ withTo the model expression rainwater as below infiltration, theθ + VG equatis r on proposedif h <0 by van Genuchten in 1980 is used  r m [23] with the expression as belowθ () =  n h   1+ ()α h  , (18)  θs θr    θθr +−θ − n m i f h < 0 θ(h)θ=θ+ s [1+r (α h ) if]otherwiseh <0 , (18)  r s m| | θ ()h =  θ+s()α n  otherwise, θ  θ 1 h θ θ (18) where is the soil water content, r  is the soil residual water content, r = 0.067, s is the soil θ where θ is the soil water content,θ θriss the soil residualotherwise water content, θr = 0.067, θs is the soil saturated saturated water content, s = 0.45 ， h is soil water potential, α is a scale parameter inversely h α water content,θ θs = 0.45, is soil waterθ potential, is−1 a scale parameter inverselyθ proportionalθ to mean whereproportional is the to soil mean water pore content, diameter, r αis = the 0.49 soil cm residual, n and waterm are content,the shape rparameters= 0.067, s of is soil the watersoil pore diameter, α = 0.49 cm 1, n and m are the shape parameters of soil water characteristic, n = 1.578, characteristic, n = 1.578, mθ− = 1 − 1/n，. However, as expressed in Equations (5) and (6), the Brooks and saturatedm = 1 1 /watern. However, content, as expresseds = 0.45 inh Equations is soil water (5) and potential, (6), the Brooks α is anda scale Corey parameter model is usedinversely in the Corey− model is used in the above theoretical derivation.−1 The parameters used by the two models are proportionalabove theoretical to mean derivation. pore diameter, The parameters α = 0.49 cm used, n by and the m two are modelsthe shape are parameters not the same. of soil Hence, water the not the same. Hence, the parameters in Brooks and Corey are converted from those of the VG model characteristic,parameters in n Brooks = 1.578, and m = Corey 1 − 1/ aren. However, converted as from expressed those of in the Equations VG model (5) according and (6), the to the Brooks equivalent and according to the equivalent conversion method proposed by Hubert J. Morel-Seytoux [24]. Coreyconversion model method is used in proposed the above by theoretical Hubert J. Morel-Seytoux derivation. The [24 parameters]. used by the two models are not the same. Hence, the parameters in Brooks and Corey are converted from those of the VG model 4.2. Results Comparison according4.2. Results to Comparisonthe equivalent conversion method proposed by Hubert J. Morel-Seytoux [24]. TheThe simulationsimulation of rainwaterof rainwater infiltration infiltration is completed is completed by suGWFoam, by suGWFoam, and the spatial and distributionsthe spatial 4.2. Results Comparison ofdistributions water content of inwater the plot content at diff inerent the timesplot at are different obtained. times Figure are5 showsobtained. the waterFigure content 5 shows distribution the water atcontentt The= 0.5 distributionsimulation h displayed atof in tParaView. =rainwater 0.5 h displayed infiltration in ParaView. is completed by suGWFoam, and the spatial distributions of water content in the plot at different times are obtained. Figure 5 shows the water content distribution at t = 0.5 h displayed in ParaView.

Figure 5. Distribution of water content at t = 0.5 h. Figure 5. Distribution of water content at t = 0.5 h. Result comparisons between the theoretical calculation and numerical simulation at diﬀerent Result comparisons between the theoretical calculation and numerical simulation at different times are shown in Figure6Figure. 5. Distribution of water content at t = 0.5 h. times are shown in Figure 6. Result comparisons between the theoretical calculation and numerical simulation at different times are shown in Figure 6.

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(a) t = 0.2 h (b) t = 0.4 h

(e) t = 1.0 h

Figure 6. Result comparisons of theoretical and numerical solution at didifferentﬀerent times.times.

DueDue to the uncertainty of the hydraulic conductivityconductivity in the random ﬁeld,field, we performed multiple simulations.simulations. In each simulation, thethe original hydraulic conductivity was multiplied by a proportional coefficient ω, so that different probability distributions of Ks were verified. The calculation results are shown in Figure 7.

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Water 2020, 12, x FOR PEER REVIEW 10 of 23 coefficient ω, so that different probability distributions of Ks were verified. The calculation results are shown in Figure7.

(a) ω = 0.2

(b) ω = 0.4

Figure 7. Cont.

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(d) ω = 0.8

FigureFigure 7. 7. ResultResult comparisons comparisons of of theoretical theoretical and and numerical numerical solution solution for for different diﬀerent ωω. .

AsAs can can be be seen seen in in above above figures, figures, the the difference difference between between the the numerical numerical solution solution and and theoretical theoretical solutionsolution isis small,small, which which shows shows that that the previousthe previous theoretical theoretical derivation derivation can be usedcan be to calculateused to rainwatercalculate rainwaterinfiltration infiltration under heavy under rainfall heavy conditions. rainfall conditions.

5.5. Slope Slope Stability Stability Analysis Analysis TheThe rainfall-induced rainfall-induced landslides landslides [25,26] [25,26] generally generally take take the the form form of of shallow shallow (typically (typically 1–3 1–3 m m deep), deep), translationaltranslational slides, slides, which which form form parallel parallel to to the the original original slope slope surface. surface. A A great great deal deal of of research research shows shows

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that the potential slip surface of the infinite slope is almost always coincident with the surface of the that the potential slip surface of the inﬁnite slope is almost always coincident with the surface of the wetting front. In the paper, the infinite slope model is used to represent shallow landslides induced wetting front. In the paper, the inﬁnite slope model is used to represent shallow landslides induced by by rainfall, as shown in Figure 8. rainfall, as shown in Figure8.

Rainwater z

Unit

z w Failure surface τ

σn

Figure 8. Infinite slope. Figure 8. Infinite slope. The limit equilibrium method can be readily applied to calculate the factor of safety. The shear resistanceThe limit of the equilibrium soil slice can method be determined can be readily using the applied modified to calculate Mohr–Coulomb the factor failure of safety. criterion The [27 shear,28] forresistance unsaturated of the soil: soil slice can be determined using the modified Mohr–Coulomb failure criterion [27,28] for unsaturated soil: b τ f = c + (σn ua) tan ϕ + (ua uw) tan ϕ , (19) τσ=+0 ()() −− ϕϕ +0 −− b fnaawcu'tan'tan uu, b (19) where c0 is the effective cohesion, ϕ0 is the effective angle of friction, tan ϕ is the change rate of shear strengthwhere c’ with is the matrix effective suction, cohesion,σn is theφ’ is total the effective normal stress angle on of the friction, failure tan plane,ϕb isua theis the change pore-air rate pressure of shear on the failure plane, uw is the pore-waterσ pressure on the failure plane, (σn ua) is the net normal strength with matrix suction, n is the total normal stress on the failure plane,− ua isb the pore-air pressure on the failure plane, (ua uw) is the matric suction on the failure plane, and ϕ is the angle − ()σ − thatpressure defines on thethe shearfailure strength plane, increasesuw is the pore-water with an increase pressure in matric on thesuction. failure plane, nau is the net The factor of safety for homogeneous()− infinite soil slope can be written as: ϕ b normal pressure on the failure plane, uuaw is the matric suction on the failure plane, and is

the angle that defines the shear strength increases with an increase in matricb suction. τ f c + (σn ua) tan ϕ + (ua uw) tan ϕ The factor of safety Ffors = homogeneous= 0 infinite− soil slope0 can− be written, as: (20) τm zγ cos α sin α τ +−()()σ ϕϕ +− b f cu'na tan ' uu aw tan where γ is the unit weightF of soil,==z is the depth of the wetting front during, inﬁltration process, and α(20)is s τγααz cos sin the slope angle. m b whereVanapalli γ is the unit et al. weight [29] determined of soil, z is the the relationship depth of the between wetting ϕfrontand during the relative infiltration volumetric process, water and contentα is thes through slope angle. a micro-analysis of unsaturated soil. Vanapalli et al. [29] determined the relationship between ϕ b and the relative volumetric water tan ϕb = s tan ϕ , (21) content s through a micro-analysis of unsaturated soil.· 0 b We adopt the ﬁtting formula proposedtanϕϕ by=⋅s Brook tan and ' , Corey [30] to determine the relationship(21) between the water content and the matric suction: We adopt the fitting formula proposed by Brook and Corey [30] to determine the relationship 1/λ between the water content and the matricψ suction:(s) = ψ s− , b · ψψ()=⋅−1/λ s b s , Thus, Equation (20) can be rewritten as: Thus, Equation (20) can be rewritten as: 1 1 1 λ tan ϕ c ψbγ1w− s − tan ϕ Fs = ϕ 0 + −0ψγ − λ ϕ 0, (22) tantan α ' cs'tan'bwzγ sin α cos α F =+ , (22) s tanαγααz sin cos where γw is the unit weight of water. Note that the safety factor of slope is related with the depth of γ thewhere wetting w isfront the andunit the weight water of saturation, water. Note which that arethe functionssafety factor of the of slope saturated is related hydraulic withconductivity the depth of the wetting front and the water saturation, which are functions of the saturated hydraulic

Water 2020, 12, 2567 13 of 23

Water 2020, 12, x FOR PEER REVIEW 13 of 23 of soils and the rainfall duration. Thus, the safety factor of the slope varies with time during rainfall, and isconductivity aﬀected by of the soils hydraulic and the rainfall conductivity duration. of Thus soils., the safety factor of the slope varies with time duringIn addition, rainfall, owing and is affected to rainwater by the inﬁltrationhydraulic conductivity into the soil, of soils. the weight of the soil also increases, which is expressedIn addition, as: owing to rainwater infiltration into the soil, the weight of the soil also increases, which is expressed as: γt = γ + ∆θ γw, (23) γγ=+Δ⋅θ γ · tw, (23) where γt is the unit weight of soil at time t, and ∆θ is the increase in volumetric water content. γ Δθ Substitutewhere thet is expression the unit weight of the of increase soil at time in volumetric t, and wateris the increase content in∆θ volumetric= θi + (θ waters θr )(content.st si ) into Δ+−−θθ()() θ θ − − EquationSubstitute (23), the the expression unit weight of the of theincrease soil canin volumetric be rewritten water as: content = isrtis s into Equation (23), the unit weight of the soil can be rewritten as: = + [ + ( )( )] γγγγθt =+γ γ ⋅w +θ()()i θθ −θs θ −r st si , (24) twisrti· −s s ,− (24) ThroughoutThroughout the the preceding preceding discussion, discussion, the safety safety factor factor of ofthe the slope slope can can be evaluated, be evaluated, and the and the correspondingcorresponding solution solution procedure procedure is is shown shown inin Figure 99..

Start

Input parameters

Input number of sub-layers N

Set loop variable i = 0

i = i + 1

Calculating the coefficient Ksz by Equation (14)

Calculating the saturation s by Equation (16)

Calculating the unit weight γt by Equation (24) Loop for every sub-layers every for Loop Calculating the safety factor Fs by Equation (22)

Y i < N

End

FigureFigure 9. 9.Flow Flow chartchart for for safety safety factor factor calculations. calculations.

Water 2020, 12, 2567 14 of 23

6. Case Study A hypothetical natural unsaturated soil slope, whose surface was covered by a layer of 1.0 m thick sandy clay loam, was adopted as an example, and the slope angle is equal to 30 degrees. The soil is uniformly dry, and the soil surface was held to be saturating at the beginning of the inﬁltration. Some of parameters are summarized in Table2[ 31]. Among them, only the saturated hydraulic conductivity was treated as a random variable obeying lognormal distribution, and the mean µKs of the saturated hydraulic conductivity was ﬁxed.

Table 2. Parameters for the study case.

Parameters Deﬁnition Value

Ks Saturated hydraulic conductivity µKs = 0.43 cm/h θs Saturated volumetric water content 0.398 θr Residual volumetric water content 0.068 θi Initial volumetric water content 0.148 ψb Air entry value 0.2809 m λ Pore distribution index 0.25 3 γd Unit dry weight of soil 16.5 kN/m c0 Eﬀective cohesion 3.0 kPa ϕ0 Eﬀective angle of internal friction 18◦

6.1. Rainfall Duration Infiltration is a process in which rainwater falling to the ground surface enters the soil. In general, the longer the infiltration time, the more rainwater seeps into the soil. At the same time, the infiltration is affected by the variability of hydraulic conductivity, and is not distributed evenly in the whole field. In this section, the influence of rainfall duration on rainwater infiltration and the corresponding safety factor under the condition of variation of hydraulic conductivity is discussed. To this end, for a specific soil hydraulic conductivity mean and coefficient of variation, the changes of water content in the soil and the corresponding safety factor are calculated with time increase. Many factors affect the variability of soil saturated hydraulic conductivity, which varies in a larger range. The in-situ test data collected by Duncan [5] shows that the coefficient of variation of the saturated hydraulic conductivity ranges from 68–90%, as shown in Table1. By analyzing a lot of soil samples of a loess slope from the South Jingyang Plateau, northwest China, Wei Wang [32] concluded that Kh and Kv measurements of the loess samples collected in the trench and adit all show log-normal distribution, and the hydraulic conductivity varied between 53% and 114%. In the case study, the coefficient of the variation of the saturated hydraulic conductivity (CoV) takes 60%. Changes in saturation (denoted by s) can intuitively reflect the infiltration process of rainwater. The saturation profiles at different times are plotted in Figure 10, and clearly show the process of rainwater infiltration. At time t = 2.0 h, the infiltration distance is about 0.1 m. At time t = 4.0 h, the infiltration distance is about 0.15 m. At time t = 8 h, the infiltration distance is about 0.25 m. At time t = 16.0 h, the infiltration distance is about 0.5 m. It is obvious that the infiltration distance increases with time, and the infiltration rate decreases significantly over time. On the other hand, it also can be seen that the saturation profiles have an extended leading portion at deeper locations, and as time increases, the leading edge of the saturation profiles is longer. This is because the rainwater moves faster at those elements of larger Ks value, which results in the differences between different infiltration elements, and the differences will grow bigger with time. The distribution of the wetting fronts of all infiltration elements determines the shape of the saturation. Water 20202020,, 1212,, 2567x FOR PEER REVIEW 15 of 23 Water 2020, 12, x FOR PEER REVIEW 15 of 23

Figure 10. Saturation profiles proﬁles at didifferentﬀerent times.times. Figure 10. Saturation profiles at different times.

Figure 1111 showsshows thethe safety safety factor factor (denoted (denoted by byFs F)s profiles) profiles at at di ffdifferenterent times. times. It can It can be seenbe seen that that the Figure 11 shows the safety factor (denoted by Fs) profiles at different times. It can be seen that safetythe safety factors factors decrease decrease with with depth depth in the in shallower the shallower ground, ground, then increasethen increase below below the wetting the wetting fronts. the safety factors decrease with depth in the shallower ground, then increase below the wetting Therefore,fronts. Therefore, there is there a minimum is a minimu safetym safety factor factor near thenear surface the surface at di ffaterent different times, times, which which explains explains the fronts. Therefore, there is a minimum safety factor near the surface at different times, which explains reasonthe reason why why the shallow the shallow landslide landslide occurs occurs during during rainfall. rainfall. This is becauseThis is because the water the content water decrease content the reason why the shallow landslide occurs during rainfall. This is because the water content withdecrease rainfall with infiltration rainfall infiltration above the above wetting the front,wetting which front, induces which ainduces decrease a decrease in matrix in suction, matrix suction, and the decrease with rainfall infiltration above the wetting front, which induces a decrease in matrix suction, correspondingand the corresponding decrease decrease in safety factors,in safety and factor belows, and the wettingbelow the front wetting the water front content the water remains content the and the corresponding decrease in safety factors, and below the wetting front the water content sameremains as thethe initialsame value.as the initial Subsequently, value. Subsequently, as depth increases, as depth the increases, soil saturation the soil does saturation not change does much. not remains the same as the initial value. Subsequently, as depth increases, the soil saturation does not Accordingly,change much. the Accordingly, contribution the of contribution the matrix suction of the ma to thetrix safetysuction factor to the also safety does factor not changealso does much. not change much. Accordingly, the contribution of the matrix suction to the safety factor also does not Meanwhile,change much. owing Meanwhile, to the increase owing into soilthe weight,increase the in safetysoil weight, factor the continues safety tofactor show continues a decreasing to show trend a change much. Meanwhile, owing to the increase in soil weight, the safety factor continues to show a withdecreasing depth increase,trend with as showndepth increase, in Equation as (22).shown Minimum in Equation safety (22). factor Minimum value is reachedsafety factor at the value bottom. is decreasing trend with depth increase, as shown in Equation (22). Minimum safety factor value is Itreached also shows at the that bottom. the potential It also shows slide plane that the is not potent alwaysial slide coincident plane withis not the always wetting coincident front, and with sliding the reached at the bottom. It also shows that the potential slide plane is not always coincident with the maywetting also front, occur and at the sliding bottom may along also the occur bedrock at the plane. bottom The along wetting the front bedrock and theplane. bedrock The planewetting are front two wetting front, and sliding may also occur at the bottom along the bedrock plane. The wetting front potentialand the bedrock sliding planesplane are that two need potential to be taken sliding into planes account, that and need Ma to et be al. taken [33] discusses into account, the problem and Ma for et and the bedrock plane are two potential sliding planes that need to be taken into account, and Ma et anal. infinite[33] discusses slope. the problem for an infinite slope. al. [33] discusses the problem for an infinite slope.

Figure 11. Safety factor proﬁles at diﬀerent times. Figure 11. Safety factor profiles at different times. Figure 11. Safety factor profiles at different times.

Water 2020, 12, x FOR PEER REVIEW 16 of 23 Water 2020, 12, 2567 16 of 23 When rainfall duration lasts a longer time, like time t = 16 h as shown in Figure 11, the decrease in safetyWhen factors rainfall atduration the wetting lasts front a longer is not time, obviou like times at thist = 16 time h as compared shown in Figurewith other 11, the times. decrease At the in safetydepth factorsof about at the0.5 wettingm, the local front minimum is not obvious safety at thisfactor time occurs. compared As the with depth other further times. increases, At the depth the ofcontribution about 0.5 m, of the matrix local minimumsuction to safetythe safety factor fact occurs.or gradually As the depth increases; further correspondingly, increases, the contribution the safety offactor matrix increases suction slowly. to the safetyThis is factor because gradually the water increases; content correspondingly, has a slow change the with safety depth, factor and increases matrix slowly.suction Thisalso ishas because a corresponding the water content process has with a slow depth. change To withillustrate depth, this and point, matrix the suction matrix also suction has a correspondingprofiles at different process times with are depth. plotted To illustratein Figure this 12, point,in which the it matrix can be suction seen that profiles the atmatrix different suction times is areabout plotted 800 kPa in Figure corresponding 12, in which to the it can initial be seen water that cont theent. matrix With suction rainfall is infiltration, about 800 kPa the correspondingwater content toof thesoil initial above water the wetting content. front With rainfallincreases, infiltration, and the thecorresponding water content matrix of soil suction above thedecreases. wetting For front a increases,shorter time and period, the corresponding the shorter the matrix time, suctionthe steeper decreases. the change For ain shorter the matrix time suction period, along the shorter depth, the as time,shown the in steeperFigure 12. the At change the time in thet = 16 matrix h, the suction change alongin the depth,matrix assuction shown with in Figuredepth is12 slower,. At the which time tis= consistent16 h, the change with the in saturation the matrix suctionprofile in with Figure depth 10, is and slower, induces which the is formation consistent withof the the safety saturation factor profileprofile inin FigureFigure 1011., and induces the formation of the safety factor profile in Figure 11.

Figure 12. Matrix suction profiles at different times. 6.2. Coefficient of Variation Figure 12. Matrix suction profiles at different times.

6.2. CoefficientThe coeffi cientof Variation of variation, also known as “relative variability,” is equal to the standard deviation of a distribution divided by its mean, and represents the spatial variability of soil saturated hydraulic conductivity.The coefficient A larger of coevariation,fficient ofalso variation known indicatesas “relative higher variability,” spatial variability is equal ofto thethe saturatedstandard hydraulicdeviation conductivity.of a distribution In thisdivided section, by its the mean, coeffi cientand represents of variation the was spatial chosen variability to be 20%, of 50%,soil saturated 80% and 150%,hydraulic respectively, conductivity in order. A larger to evaluate coefficient the influence of variation of the indicates variability higher of hydraulic spatial variability conductivity of the on rainwatersaturated infiltrationhydraulic conductivity. and the safety In factor this section, of the slope. the coefficient of variation was chosen to be 20%, 50%, 80% and 150%, respectively, in order to evaluate the influence of the variability of hydraulic 6.2.1.conductivity Saturation on rainwater infiltration and the safety factor of the slope. Rainfall infiltration into soil leads to an increase in saturation. In this section we will discuss how 6.2.1. Saturation the variability of hydraulic conductivity affects the change in soil saturation. FigureRainfall 13 infiltration shows the into saturation soil leads profiles to an increase under in di saturation.fferent variability In this section of hydraulic we will conductivity discuss how conditionsthe variability at di offf hydraulicerent times. conductivity For comparison, affects the coe changefficient in of soil variation saturation.CoV = 0 is also considered, representingFigure 13 a uniformshows the soil saturation layer. From profiles Figure 13under, it is seendifferent that thevariability smaller theof hydraulic coefficient conductivity of variation, theconditions more obvious at different the change times. in For saturation. comparison, For example, coefficient the saturationof variation profile CoV of= uniform0 is also soil considered, layer has arepresenting right angle ata uniform the wetting soil layer. front, From indicating Figure a steep13, it is decrease seen that in the saturation, smaller th ande coefficient the saturation of variation, profile withthe more a coe obviousfficient of the variation change ofin 1.5saturation. is the slowest For exam in changeple, the of saturation saturation profile along depth of uniform from thesoil top layer to thehas bottoma right ofangle the slope.at the Thiswetting is because front, indicating the smaller a thesteepCoV decreaseof the saturatedin saturation, hydraulic and the conductivity, saturation theprofile more with uniform a coefficient the soil inof thevariation horizontal of 1.5 layer is th is;e likewise,slowest in the change smaller of thesaturation difference along of the depth saturated from the top to the bottom of the slope. This is because the smaller the CoV of the saturated hydraulic

Water 2020, 12, x FOR PEER REVIEW 17 of 23 conductivity,Water 2020, 12, 2567 the more uniform the soil in the horizontal layer is; likewise, the smaller the difference17 of 23 of the saturated permeability coefficient is, the easier it is to achieve synchronous infiltration between differentpermeability soil cube coeffi elements.cient is, the This easier is consistent it is to achieve with the synchronous results of numerica infiltrationl simulations between di byfferent Chen soil et al.cube [20]. elements. This is consistent with the results of numerical simulations by Chen et al. [20].

s (-) s (-) 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

0.1 0.1

0.2 0.2

0.3 Cov = 0.0 0.3 Cov = 0.2 Cov = 0.0 0.4 Cov = 0.5 0.4 Cov = 0.2 Cov = 0.8 Cov = 0.5 0.5 Cov = 1.5 0.5 Cov = 0.8 Cov = 1.5 0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

(b) t = 4.0 h (a) t = 2.0 h

s (-) s (-) 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4 Cov = 0.0 0.5 Cov = 0.2 0.5 Cov = 0.5 0.6 0.6 Cov = 0.8 Cov = 0.0 Cov = 1.5 0.7 0.7 Cov = 0.2 Cov = 0.5 0.8 0.8 Cov = 0.8 Cov = 1.5 0.9 0.9

1.0 1.0

FigureFigure 13. SaturationSaturation profiles profiles under under different different variability variability of of hydraulic hydraulic conductivity conductivity conditions conditions at at differentdifferent times.

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6.2.2.6.2.2. Safety Safety Factor Factor RainfallRainfall infiltration infiltration induces an increase in water content,content, whichwhich adverselyadversely aaffectsffects the the stability stability of of a aslope. slope. HowHow thethe variabilityvariability ofof hydraulichydraulic conductivityconductivity aaffectsffects the change in safety factor of of a a slope slope is is discusseddiscussed inin thisthis section. section. The The safety safety factor factor profiles profiles under diunderfferent different variability variability of hydraulic of conductivity hydraulic conductivityconditions at conditions different times at different are shown times in are Figure shown 14. in It canFigure be seen14. It thatcan thebe seen smaller that the the coe smallerfficient the of coefficientvariation ofofthe variation saturated of the hydraulic saturated conductivity, hydraulic theconductivity, smaller the the safety smaller factor the at safety the corresponding factor at the correspondingwetting front, and wetting the higher front, theand increase the higher in safety the increa factorse below in safety the wetting factor below front. the By contrast,wetting front. the larger By contrast,the coeffi cientthe larger of variation, the coefficien the smallert of variation, the safety factorthe smaller at the correspondingthe safety factor wetting at the front, corresponding and the less wettingobvious front, the increase and the in less the safetyobviou factors the belowincrease the in wetting the safety front. factor In particular, below the when wetting the coe front.fficient In particular,of variation when is 1.5, the the coefficient slope safety of factorvariation decreases is 1.5, the monotonously slope safety from factor the decreases top to the monotonously bottom of the from the top to the bottom of the slope, which is different from other coefficients of variation. This slope, which is different from other coefficients of variation. This corresponds to the distribution of corresponds to the distribution of saturation at different times. The smaller the coefficient of saturation at different times. The smaller the coefficient of variation, the higher the saturation above variation, the higher the saturation above the wetting front. The lower the corresponding matrix the wetting front. The lower the corresponding matrix suction of soil, and the smaller the safety factor suction of soil, and the smaller the safety factor of the slope at the wetting front. Similarly, the lower of the slope at the wetting front. Similarly, the lower the saturation below the wetting front, the higher the saturation below the wetting front, the higher the corresponding matrix suction of soil, and the the corresponding matrix suction of soil, and the greater the safety factor of the slope. On the other greater the safety factor of the slope. On the other hand, the larger the coefficient of variation, the hand, the larger the coefficient of variation, the smoother the curve of the saturation profile, and the smoother the curve of the saturation profile, and the lower the saturation above the wetting front, the lower the saturation above the wetting front, the higher the corresponding matrix suction, and the higher the corresponding matrix suction, and the higher the safety factor of the slope, so that below higher the safety factor of the slope, so that below the wetting front the change of saturation is not the wetting front the change of saturation is not obvious, the corresponding change of matric suction obvious, the corresponding change of matric suction is also not obvious, and the change in safety factor is also not obvious, and the change in safety factor of the slope is slow. of the slope is slow.

Fs 0 204060

0.1

0.2

0.3

0.4 Cov = 0.0 0.5 Cov = 0.2 Cov = 0.5 0.6 Cov = 0.8 Cov = 1.5 0.7

0.8

0.9

1.0

(a) t = 2.0 h (b) t = 4.0 h

Figure 14. Cont.

WaterWater 20202020, ,1212, ,x 2567 FOR PEER REVIEW 1919 of of 23 23

FigureFigure 14. 14. EvolutionEvolution of of safety safety factor factor profiles proﬁles over over time. time.

FromFrom the the perspective perspective of of slope slope stability, stability, It can It can be concluded be concluded that thatin the in early the earlystage stageof rainfall, of rainfall, such assuch time as t time = 2 th= or2 ht or= 4t h= 4in h this in this example, example, the the smaller smaller the the coefficient coefficient of of variation, variation, the the smaller smaller the the correspondingcorresponding safety factor,factor, and and the the higher higher the the slope slope instability instability probability. probability. In the In late the stage late ofstage rainfall, of rainfall,for example, for example, at time t at= 15time h in t = this 15 example,h in this example, the difference the difference in safety factor in safety is not factor significant is not forsignificant different forcoe differentfficients ofcoefficients variation, of because variation, at this because time theat this upper time part the of upper the soil part layer of the issaturated soil layer regardlessis saturated of regardlessthe coefficient of the of coefficient variation. of variation.

6.2.3.6.2.3. Cumulative Cumulative Infiltration Infiltration TheThe safety safety factor factor of of the the slope slope is is related related to to ante antecedentcedent accumulative accumulative infilt infiltration.ration. Figure Figure 15 15 shows shows thethe variation ofof cumulative cumulative infiltration infiltration with with time time under under different different coeffi cientscoefficients of variation. of variation. With increase With increasein time, thein time, cumulative the cumulative infiltration infiltration gradually grad increases.ually increases. At the same At the time, same the largertime, the the larger coefficient the coefficientof variation, of variation, the smaller the the smaller corresponding the corresponding cumulative cumulative infiltration. infiltration. When the When coeffi thecient coefficient of variation of variationis 0.2, the is cumulative 0.2, the cumulative infiltration infiltration is close to is the close cumulative to the cumulative infiltration infiltration under uniform under infiltration, uniform infiltration,i.e., variation i.e., coe variationfficient equalscoefficient zero. equals With thezero. increase With the of theincrease coeffi cientof the of coefficient variation, of the variation, difference the of differencecumulative of infiltration cumulative is infiltration also increasing. is also When increa thesing. coeffi Whencient ofthe variation coefficient reaches of variation 1.5, the reaches difference 1.5, in thecumulative difference infiltration in cumulative is significant. infiltration This is significant. is because whenThis is the because coeffi cientwhen of the variation coefficient becomes of variation larger, becomesthe difference larger, in the the difference coefficient in of hydraulicthe coefficient conductivity of hydraulic among conductivity the cube elements among alsothe cube becomes elements larger, alsowhich becomes affects thelarger, cumulative which infiltration.affects the cumulative Figure 16 shows infiltration. the curves Figure of the 16 probability shows the density curves function of the probabilityfor different density coefficients function of variation.for different It cancoefficients be clearly of seenvariation. from theIt can curves be clearly that when seen CoVfrom= the0.5, curvesthe average that when value CoV of saturated = 0.5, the hydraulic average value conductivity of saturated is about hydraulic 2.59 cm conductivity/h, and the PDF is about curve 2.59 is basically cm/h, andsymmetrical the PDF curve with theis basically average symmetrical value. That is,with the the number average of saturatedvalue. That hydraulic is, the number conductivity of saturated for the hydrauliccube infiltration conductivity elements for the above cube the infiltration average and elements below above the average the average is substantially and below equal, the average in which is substantiallycase the accumulative equal, in which infiltration case the is close accumulative to the cumulative infiltration infiltration is close to under the cumulative uniform soil infiltration condition. underAs the uniform coefficient soil of variationcondition. increases, As the coefficient the average of value variation shifts significantlyincreases, th toe average the left, meaningvalue shifts that significantlythe generated to saturated the left, hydraulicmeaning that conductivity the generate is muchd saturated less than hydraulic the average cond value,uctivity and is the much larger less the thancoeffi thecient average of variation, value, theand more the obviouslarger the this coeffici trendent is. Therefore,of variation, under the themore assumption obvious this of lognormal trend is. Therefore,distribution under of saturated the assumption hydraulic of lognormal conductivity, distribution the higher of coesatuffiratedcient hydraulic of variation conductivity, has a negative the higherimpact coefficient on the cumulative of variation infiltration. has a negative impact on the cumulative infiltration.

Water 2020, 12, x FOR PEER REVIEW 20 of 23 Water 2020, 12, 2567 20 of 23 Water 2020, 12, x FOR PEER REVIEW 20 of 23

Figure 15. The variation of accumulative infiltration over time. Figure 15. The variation of accumulative inﬁltration over time. Figure 15. The variation of accumulative infiltration over time.

Figure 16. PDF curves of saturated hydraulic conductivity. Figure 16. PDF curves of saturated hydraulic conductivity. As stated above, the difference in coefficient of variation leads to a difference in the accumulative As stated above, theFigure difference 16. PDF in coefficientcurves of saturated of variation hydraulic leads conductivity. to a difference in the accumulative infiltration during the same period of time, and this difference also leads to different slope stability. infiltration during the same period of time, and this difference also leads to different slope stability. The variationAs stated of above, safety the factors difference at the in wetting coefficient front of with variation time leads are shown to a difference in Figure in 17 the. It accumulative can be seen The variation of safety factors at the wetting front with time are shown in Figure 17. It can be seen thatinfiltration at the same during time, the the same larger period the coe of ffitime,cient and of variation,this difference the higher also leads the corresponding to different slope safety stability. factor, that at the same time, the larger the coefficient of variation, the higher the corresponding safety factor, whichThe variation is consistent of safety with factors the water at the content wetting distribution front with along time depth, are shown because in theFigure larger 17. the It can coe ffibecient seen which is consistent with the water content distribution along depth, because the larger the coefficient ofthat variation, at the same the smoothertime, the larger the curve the coefficient of the saturation of variation, profile. the By higher contrast, the corresponding the lower the coe safetyfficient factor, of of variation, the smoother the curve of the saturation profile. By contrast, the lower the coefficient of variation,which is consistent the higher with the saturation the water at content the wetting distribu front,tion and along the depth, lower thebecause matrix the suction, larger the which coefficient results variation, the higher the saturation at the wetting front, and the lower the matrix suction, which inof a variation, smaller safety the smoother factor. In the addition, curve of as the time saturation increases, profile. the safety By contrast, factor decreases, the lower and the the coefficient difference of results in a smaller safety factor. In addition, as time increases, the safety factor decreases, and the betweenvariation, the the safety higher factors the corresponding saturation at tothe a diwettingfferent coefront,fficient and ofthe variation lower the is also matrix smaller. suction, This iswhich due difference between the safety factors corresponding to a different coefficient of variation is also toresults the fact in thata smaller when safety the rainwater factor. In infiltrates addition, for as a time certain increases, period ofthe time, safety regardless factor decreases, of the coe ffiandcient the smaller. This is due to the fact that when the rainwater infiltrates for a certain period of time, ofdifference variation, between the upper the soil safety is almost factors completely corresponding saturated, to a asdifferent shown coefficient in Figure 13 ofd, variation resulting is in also an regardless of the coefficient of variation, the upper soil is almost completely saturated, as shown in increasinglysmaller. This small is due diff erenceto the infact safety that factors. when the rainwater infiltrates for a certain period of time, Figure 13d, resulting in an increasingly small difference in safety factors. regardless of the coefficient of variation, the upper soil is almost completely saturated, as shown in Figure 13d, resulting in an increasingly small difference in safety factors.

Water 2020, 12, 2567 21 of 23 Water 2020, 12, x FOR PEER REVIEW 21 of 23

Figure 17. The variation of safety factors over time. Figure 17. The variation of safety factors over time. 7. Conclusions 7. Conclusions By improving and simplifying the GA model, this paper deduces the analytical formula of the By improving and simplifying the GA model, this paper deduces the analytical formula of the failurefailure probability probability of the of inﬁnitethe infinite slope slope under under heavy heavy rainfall rainfall conditions, conditions, andand quantitativelyquantitatively analyzes analyzes the inﬂuencethe influence of the horizontal of the horizontal variability variability of saturated of saturated hydraulic hydraulic conductivity conductivity on theon the slope slope stability. stability. Finally, someFinally, conclusions some conclusions are summarized are summarized as follows. as follows.

(1) (1)For aFor certain a certain coe coefficientfficient of of variation, variation, therethere is a local minimum minimum at atthe the wetting wetting front front in the in initial the initial periodperiod of the of the rainfall rainfall and, and, as as time time increases, increases, the reduction reduction of of the the safety safety factor factor at the at wetting the wetting front front becomes insignificant until it disappears. becomes insignificant until it disappears. (2) At the initial stage of rainfall, the smaller the coefficient of variation, the more rainwater is (2) At theconcentrated initial stage in the of rainfall,upper part the of smallerthe soil thelayer, coe andffi cientthe smaller of variation, the safety the factor. more With rainwater the is concentratedincrease of in time, the upperthe influence part of of the thesoil change layer, in coefficient and the smaller of variation the safety on slope factor. stability With becomes the increase of time,less significant the influence in the of later the period change of rainfall. in coeffi cient of variation on slope stability becomes less (3)significant There are in two the laterpotential period slip ofsurfaces rainfall. due to the influence of matrix suction and the increase in (3) Theresoil are weight two potential with depth. slip surfacesIn general, due with to thethe influenceincrease of matrixdepth, the suction safety and factor the increasegradually in soil weightdecreases, with depth. and the In safety general, factor with at the bottom increase of ofthe depth, weathered the safety layer factoris the lowest. gradually Thus, decreases, the bottom is a potential sliding surface. The above discussions show that in the early stage of and the safety factor at the bottom of the weathered layer is the lowest. Thus, the bottom is a rainfall, there is a significant reduction of the safety factor at the wetting front. Therefore, the potential sliding surface. The above discussions show that in the early stage of rainfall, there is wetting front is also a potential sliding surface, which is caused by sudden changes in matrix a significantsuction induced reduction by rainfall of the infiltration. safety factor at the wetting front. Therefore, the wetting front is (4)also The a potential high coefficient sliding surface, of variation which of is hydrauli causedc by conductivity sudden changes has a innegative matrix suctionimpact on induced the by rainfallcumulative infiltration. infiltration of rainwater. The larger the coefficient of variation, the lower the (4) The highcumulative coefficient infiltration. of variation Correspondingly, of hydraulic the conductivity larger the coefficient has a negative of variation, impact the on thehigher cumulative the infiltrationsafety factor of rainwater. at the initial The stage larger of the rainfall. coeffi Meanwhile,cient of variation, with the the increase lower theof time, cumulative the difference infiltration. Correspondingly,between the safety the largerfactors thecorresponding coefficient to of coeffi variation,cients theof variation higher the is becoming safety factor increasingly at the initial small. stage of rainfall. Meanwhile, with the increase of time, the difference between the safety factors correspondingThe effect of to slope coeffi anglecients on of rainwater variation infiltrati is becomingon was increasinglyneglected. The small. effect is complicated by many confounding factors [34], and due to the current technology level and theory, the main Theinfluence effect is ofambiguous, slope angle which on may rainwater result in infiltration a contradictory was view. neglected. In addition, The etheffect results is complicated show that by manyhydraulic confounding conductivity factors [has34], an and impact due toon the rainwater current infiltration technology and level corresponding and theory, slope the main stability. influence is ambiguous,Thus, in slope which stability may result analysis, in a special contradictory attention view. should In be addition, paid to thethe resultsvariability show of thathydraulic hydraulic conductivityconductivity. has an impact on rainwater infiltration and corresponding slope stability. Thus, in slope stability analysis, special attention should be paid to the variability of hydraulic conductivity. Author Contributions: T.H., supervision, conceptualization, and methodology; L.L., writing–Original draft preparation; G.L., numerical simulation. All authors have read and agreed to the published version of the Author Contributions: T.H., supervision, conceptualization, and methodology; L.L., writing–Original draft manuscript. preparation; G.L., numerical simulation. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Zhejiang Natural Science Foundation (LY18E080006). Conflicts of Interest: The authors declare no conflict of interest. Water 2020, 12, 2567 22 of 23

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