Effect of Rainfall, Runoff and Infiltration Processes on The

water

Article Effect of Rainfall, Runoff and Infiltration Processes on the Stability of Footslopes

Hung-En Chen 1,*, Yen-Yu Chiu 2, Tung-Lin Tsai 3 and Jinn-Chuang Yang 1

1 Disaster Prevention & Water Environment Research Center, National Chiao Tung University, Hsinchu 30010, Taiwan; [email protected] 2 Department of Soil and Water Conservation, National Chung Hsing University, Taichung 402204, Taiwan; [email protected] 3 Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi 60004, Taiwan; [email protected] * Correspondence: [email protected]

Received: 27 March 2020; Accepted: 23 April 2020; Published: 25 April 2020

Abstract: To analyze the effect of runoff on shallow landslides, a model coupling one-dimensional rainfall–runoff and two-dimensional infiltration was established to simulate rainfall, infiltration, and runoff processes. Based on Bishop’s limit equilibrium method, the slope failure of a hypothetical footslope was studied. First, conditions with and without inflow were compared. The results reveal a remarkable difference in factors of safety (FS) between the two conditions, suggesting that considering the effect of runoff is crucial for landslide modeling. In terms of a series of tests of the various magnitudes, durations, lag-time, and peak position of the hydrograph, analyses show that larger inflow leads to more accumulated infiltration and triggers landslides earlier. A long-term duration inflow decreases the stability more than short intensive inflow does. With subsequent surface inflow, slope failure may occur after rainfalls stop, owing to the inflow, and the shape of inflow hydrographs could slightly affect the variance in FS. Results also indicate the necessity of considering the surface runoff when using a numerical model to analyze landslide, particularly on a footslope.

Keywords: shallow landslide; rainfall–runoﬀ; numerical model; slope stability

1. Introduction Landslides occur in numerous locations worldwide and present serious threats to lives and the economy. There is a general consensus that rainfall is the most common external factor that stimulates shallow landslides. Therefore, the mechanism underlying rainfall-induced landslides has been widely studied and discussed in recent decades [1,2]. A relationship between water and landslides, known as “landslide hydrology,” has been proposed [2]. In the hydrological cycle, water can originate from many sources, such as precipitation, snowmelt, surface water and groundwater. Water infiltrating soil surfaces leads to low matric suction and a reduction in the shear strength of soil, thereby increasing the instability. The groundwater seepage has been broadly and deeply analyzed using numerical models, wherein different soil conditions and facilities have been considered [3–7]. However, it is difficult to completely reproduce the mechanism underlying rainfall-triggered shallow landslides using a numerical model. With the mechanism greatly simplified in the earliest studies, landslide models have gradually progressed in recent decades and currently include more mechanisms. For example, Iverson [8], Baum et al. [9], and Tsai and Yang [10] considered pore-pressure diffusion as a near-saturation response. Tarantino and Bosco [11], Collins and Znidarcic [12], and Tsai et al. [13] used the Richards equation and the extended Mohr–Coulomb failure criterion [14] to develop shallow landslide models for both saturated and unsaturated soils.

Water 2020, 12, 1229; doi:10.3390/w12051229 www.mdpi.com/journal/water Water 2020, 12, x FOR PEER REVIEW 2 of 19 extended Mohr–Coulomb failure criterion [14] to develop shallow landslide models for both saturated and unsaturated soils. Based on previous research, there has been progress in the clarificationWater 2020, 12 ,of 1229 landslide behavior, and the improved modeling capabilities allow researchers2 ofto 19 explore details. The influence of rainfall infiltration on landslide has been studied using numerical models. Tsai Based on previous research, there has been progress in the clarification of landslide behavior, and the [15] used a modified version of Iverson’s one-dimensional (1D) model to assess the influence of improved modeling capabilities allow researchers to explore details. different rainfall patterns on the groundwater table and slope stability in near-saturated soils. Tsai The influence of rainfall infiltration on landslide has been studied using numerical models. and Wang [16] demonstrated the effect of different patterns of rainfall on shallow landslides in Tsai [15] used a modified version of Iverson’s one-dimensional (1D) model to assess the influence of unsaturated soils. Chen et al. [17] further investigated the effects of the rainfall duration, rainfall different rainfall patterns on the groundwater table and slope stability in near-saturated soils. Tsai and amount, and lateral flow-induced slope failures using a vertical two-dimensional (2D) numerical Wang [16] demonstrated the effect of different patterns of rainfall on shallow landslides in unsaturated landslide model. The relationship between rainfall infiltration and slope stability has been discussed soils. Chen et al. [17] further investigated the effects of the rainfall duration, rainfall amount, and extensively. lateral flow-induced slope failures using a vertical two-dimensional (2D) numerical landslide model. Recently, hydrological and landslide susceptibility models were combined [18], and the results The relationship between rainfall infiltration and slope stability has been discussed extensively. revealed that the runoff depth may be an appropriate analysis factor, instead of the rainfall depth or Recently, hydrological and landslide susceptibility models were combined [18], and the results maximum rainfall intensity. Chiu et al. [19] combined a landslide model with a surface flow revealed that the runoff depth may be an appropriate analysis factor, instead of the rainfall depth or simulation of a small catchment and observed that the accumulated surface water increases the maximum rainfall intensity. Chiu et al. [19] combined a landslide model with a surface flow simulation possibility of landslide, especially in the downstream slope. The latter study emphasized the of a small catchment and observed that the accumulated surface water increases the possibility importance of considering the effect of runoff on the slope stability during the analysis. However, the of landslide, especially in the downstream slope. The latter study emphasized the importance of simulation was based on a regional watershed, and it was difficult to control the runoff conditions at considering the effect of runoff on the slope stability during the analysis. However, the simulation was an affected slope. Therefore, different surface flow conditions have not been studied yet. based on a regional watershed, and it was difficult to control the runoff conditions at an affected slope. The objective of this study is to analyze the effect of runoff on the slope failure under different Therefore, different surface flow conditions have not been studied yet. conditions including magnitudes, durations, lag time, and peak positions of hydrographs. This study The objective of this study is to analyze the effect of runoff on the slope failure under different aimed to analyze the variety in footslope stability with respect to time. A footslope refers to a slope conditions including magnitudes, durations, lag time, and peak positions of hydrographs. This study near the toe of a hill (Figure 1). The geographical position allows the converged water from the upper aimed to analyze the variety in footslope stability with respect to time. A footslope refers to a slope near areas of the catchment to flow through the footslope; therefore, compared to other slopes, a footslope the toe of a hill (Figure1). The geographical position allows the converged water from the upper areas is more likely to be affected by runoff. A physical-based model was developed to simulate the of the catchment to flow through the footslope; therefore, compared to other slopes, a footslope is more rainfall–runoff and slope instability based on the limit equilibrium method. The model is governed likely to be affected by runoff. A physical-based model was developed to simulate the rainfall–runoff by the coupling of the 1D kinematic wave equation and 2D Richards equation. A case study and slope instability based on the limit equilibrium method. The model is governed by the coupling of demonstrating the different results of conditions with and without surface inflow on the slope the 1D kinematic wave equation and 2D Richards equation. A case study demonstrating the different stability of shallow landslides is presented, and various surface inflow hydrographs have been results of conditions with and without surface inflow on the slope stability of shallow landslides is adopted to test the slope stability in different conditions. presented, and various surface inflow hydrographs have been adopted to test the slope stability in different conditions.

Figure 1. Schematic showing ﬁve hillslope positions: summit, shoulder, back slope, footslope, and toe Figureslope 1. (modiﬁed Schematic after showing [20,21]). five hillslope positions: summit, shoulder, back slope, footslope, and toe slope (modified after [20,21]). 2. Method and Materials

2.2.1. Method Mathematical and Materials Basis Water 2020The, 12 proposed, x; doi: FOR model PEER REVIEW can be divided into two parts: (1) hydrological modulewww.mdpi.com/j and (2)ournal/water soil failure module. The former is used to simulate the rainfall, infiltration, and runoff, whereas the latter is used to Water 2020, 12, 1229 3 of 19 calculate the slope stability. These two modules are physical-process-based to determine the physical mechanisms triggering landslides. The theory underlying 1D rainfall–runoff, 2D infiltration equation, and slope stability analysis is briefly described in this section.

2.1.1. Hydrological Module The hydrological module mainly simulates the rainfall infiltration and overland flow processes. Surface flow is obtained from kinematic wave approximation, ignoring inertia and pressure terms. The infiltration simulation considers both saturated and unsaturated conditions to compute the pore-water pressures by solving the 2D Richards equation. The kinematic wave and infiltration equations are briefly described below.

Kinematic Wave Equation The actual physical flow processes may be quite complex, but a simplified hydrodynamic model is sufficient for practical conditions (Henderson [22], Gunaratnam and Perkins [23], and Miller [24]). Kinematic wave equations [25] are often used to simulate the rainfall–runoff process in small- and average-sized basins with steep slopes. The details and the derivation of these equations are provided by Chow et al. [26] and Henderson [22]. The equation governing flow over a plane under the assumption of a kinematic wave is

∂q ∂h + s = i f , (1) ∂X ∂t − where hs denotes the depth of the slope surface flow; i is the rainfall intensity; f is the infiltration rate; t is the time; X is the distance downslope; and q is the flow rate per unit width and can be obtained as follows √S q = 0 h5/3, (2) n × s where n is Manning’s roughness coefficient, and S0 is the bed slope.

Infiltration Equation Infiltration flow is generally considered to be an essential triggering mechanism that must be addressed when studying the slope failure following a period of heavy rainfall. In this study, both saturated and unsaturated infiltration flows were considered when evaluating the pore-water pressure by solving the 2D Richards equation. The equation that can be used to calculate the groundwater flow in response to rainfall infiltration on a hillslope with a local rectangular Cartesian coordinate system is [27] " !# " !# ∂ψ ∂θ ∂ ∂ψ ∂ ∂ψ = Kx(ψ) + Kz(ψ) + 1 , (3) ∂t ∂ψ ∂x ∂x ∂z ∂z where θ is the moisture content; ψ is the pore-water pressure head; and Kx and Kz are the hydraulic conductivities in the x and z directions, respectively, a function of the soil properties presented in Equation (8). Equation (3) can be solved by using the appropriate initial and boundary conditions. Regarding an initial steady state with a water table of dZ in the vertical direction, the initial condition in terms of the pressure head can be expressed as

ψ = Z dZ, (4) − where Z is the elevation. In addition, the surface of the hillslope subjected to the rainfall yields ! ! ∂ψ ∂ψ f = Kx(ψ) sin α Kz(ψ) 1 cos α, (5) − ∂x − ∂z − Water 2020, 12, 1229 4 of 19 where α is the slope angle. When the ground surface is saturated, the pressure head ψ of the ground surface is equal to the depth of ﬂow; thus, the boundary condition can be expressed as

ψ = hs (6)

Solving Equations (3)–(6) requires a relationship among pressure, moisture content, and hydraulic conductivity. The water retention curve proposed by van Genuchten [28] was used in this study for the hydraulic conductivity K = Kx = Kz for isotropic soils

 M θ θr  1  S = − =   (7) θs θr  + ( )N  − 1 ξψ  2 1/2 " 1/M#M K(θ) θ θr  θ θr  = − 1 1 −  , (8) Ks θs θr  − − θs θr  − − where S denotes the degree of saturation; Ks is the saturated hydraulic conductivity; θs is the saturated moisture content; θr is the residual moisture content; and ξ, N, and M are the ﬁtting parameters (M = 1 1/N). The calculated pore-water pressures were used to evaluate the factor of safety (FS) of − a hillslope.

2.1.2. Soil Failure Module (Slope Stability Analysis) The slope stability is commonly analyzed using the limit equilibrium method of slices [29] and based on the consideration of the moment equilibrium of the soil mass. The limit equilibrium methods assume that the potential failure surface is governed by Mohr–Coulomb relationships between shear strength and the normal stress on the failure surface. The soil mass can be subdivided into a number of vertical slices that exhibit a width b above an assumed circular slip surface (Figure2). The maximum shear stress τ of the unsaturated soil acting at the lower boundary of a slice is related to the total normal stress σ, according to the Mohr–Coulomb failure criterion [30], and can be expressed as follows

b τ = c0 + (σ uα) tan φ0 + (uα uw) tan φ , (9) − − b where c0 denotes the eﬀective cohesion; φ0 and φ are the shearing resistance and friction angles with respect to the matric suction, respectively, and uα and uw are the pore-air and pore-water pressures, respectively. In Bishop’s method [29], the force transmitted between adjacent slices is assumed to be strictly horizontal. The equilibrium of moments relative to the center of the slip circle can be expressed by equating the sum of the moments of the weight of each slice relative to the center of the circle to the sum of the moments of the shearing forces at the bottom of the slices. The depth-averaged unit weight γ in a slice can be expressed as Z 1 h γ = [(1 θ) γw Gs + θ γw]dh, (10) h 0 − · · · where Gs denotes the speciﬁc gravity of a solid; h is the height of each slice; and γw is the unit weight of water. Water 2020, 12, x FOR PEER REVIEW 4 of 19 where 𝛼 is the slope angle. When the ground surface is saturated, the pressure head 𝜓 of the ground surface is equal to the depth of flow; thus, the boundary condition can be expressed as

𝜓=ℎ (6) Solving Equations (3)–(6) requires a relationship among pressure, moisture content, and hydraulic conductivity. The water retention curve proposed by van Genuchten [28] was used in this study for the hydraulic conductivity 𝐾=𝐾 =𝐾 for isotropic soils 𝜃−𝜃 1 𝑆= = (7) 𝜃 −𝜃 1+(𝜉𝜓)

() / / = 1 − 1 − , (8) where 𝑆 denotes the degree of saturation; 𝐾 is the saturated hydraulic conductivity; 𝜃 is the saturated moisture content; 𝜃 is the residual moisture content; and 𝜉, 𝑁, and 𝑀 are the fitting parameters (𝑀=1−1/𝑁). The calculated pore-water pressures were used to evaluate the factor of safety (FS) of a hillslope.

2.1.2. Soil Failure Module (Slope Stability Analysis) The slope stability is commonly analyzed using the limit equilibrium method of slices [29] and based on the consideration of the moment equilibrium of the soil mass. The limit equilibrium methods assume that the potential failure surface is governed by Mohr–Coulomb relationships between shear strength and the normal stress on the failure surface. The soil mass can be subdivided into a number of vertical slices that exhibit a width b above an assumed circular slip surface (Figure 2). The maximum shear stress 𝜏 of the unsaturated soil acting at the lower boundary of a slice is Waterrelated2020 to, 12 the, 1229 total normal stress 𝜎, according to the Mohr–Coulomb failure criterion [30], and5 ofcan 19 be expressed as follows

Rsinαn R C

γ∙h R

b τ σ A

Figure 2. Slope of the Bishop slip circle.

When using Equation (10) togetherFigure 2. with Slope the of shear the Bishop strength slip of circle. unsaturated soil given by Equation (9) and assuming that the pore-air pressure is atmospheric, the Factor of safety (FS) for the equilibrium of moments based on Bishop’s𝜏=𝑐 method+ ( [𝜎−𝑢29], which) tan 𝜙′was + derived(𝑢 −𝑢 by) tan Chen 𝜙 , [17], can be expressed as (9) where c’ denotes the effective cohesion; 𝜙′ and 𝜙 are the shearingb resistance and friction angles (c +γ h tanφ γw ψc tanφ γwψp tan φ ) P 0 · 0− · 0− with respect to the matric suction, respectively,cosα(1 +andtanα tan𝑢φ 0and/FS) 𝑢 are the pore-air and pore-water FS = P , (11) pressures, respectively. In Bishop’s method [29], γtheh sin forceα transmitted between adjacent slices is assumed to be strictly horizontal. The equilibrium of· moments relative to the center of the slip circle wherecan be theexpressed soil is unsaturated by equating when the sum the of groundwater the moments pressure of the weight head becomes of each negativeslice relative and toψ ctheequals centerψ, whichof the circle can be to obtainedthe sum of using the Equationmoments (3),of the whereas shearingψp forcesis zero. at Conversely,the bottom of when the slices. the groundwater The depth- averagedpressure headunit weight becomes 𝛾 positive, in a sliceψ canp is be identical expressed to ψ as, ψ c is zero, and the soil is saturated. When FS is less than one, it means the slope is unstable. Equation (11) indicates that slope failure occurs not only in saturated soil due to an increase in the positive groundwater pressure head, but also in unsaturated Watersoil due 2020, to 12, a x; decrease doi: FOR PEER in the REVIEW negative groundwater pressure head, indicatingwww.mdpi.com/j the dissipationournal/water of the matric suction. As shown in Figure2, Bishop slip circle is to be determined by a center and radius which has the overall minimum factor of safety calculated by Equation (11).

2.1.3. Computational Procedure The numerical model is solved using the finite difference method; the discretization of the kinematic wave equation is computed using the Preissmann four-point finite difference scheme [30], whereas the Richards equation is computed with a fully implicit scheme [19,31]. An iterative procedure (Figure3) is adopted to solve Equations (1) to (11). The iterative approach alternates the solutions of the infiltration and overland flow equations at each time step until convergence is reached. The groundwater pressure head of a hillslope is obtained by assuming that the infiltration rate equals the rainfall intensity, as presented in Equation (5), and the surface runoff is solved in succession using Equations (1) and (2). If the pressure head on the slope surface is less than or equal to zero, surface runoff does not occur, and the calculation progress to the next time step. However, if the calculated pressure head or water depth on the slope surface is larger than zero, that is, ψ0 > 0 or hs > 0, and the error between the pressure head and the water depth is assessed. If the error is greater than the minimum permissible error, (ψ0 + hs)/2 transfer into a boundary condition of infiltration for the recalculation within the same time step. Finally, the calculated pore-water pressures at the slopes are substituted into Equation (11), and FS is obtained. The physical-based method for the landslide simulation has a high accuracy under field and laboratory conditions. However, it ignores other factors, such as the vegetation. Water 2020, 12, 1229 6 of 19

Water 2020, 12, x FOR PEER REVIEW 6 of 19

Input rainfall & inflow hydrogragh

Simulate infiltration equation from Equations (3)-(5)

Simulate kinematic wave equation from Equation (1)

surface ψ0 ≤ 0 & hs = 0 Simulate infiltration equation by Surface check B.C. Equation (6)

surface ψ0 > 0 or hs > 0

e=|(hs-ψ0 )/(hs+ψ0 )|

-4 e>10 , ψ0=(hs+ψ0 )/2 Check Error e

e≤ 10-4

Calculate safety factor from Equation (11)

Move forward to the next time step

FigureFigure 3. 3.FlowFlow chart chart of of the the model model developed developed in in this this study. study.

2.2.2.2. Description Description of ofthe the Study Study Case Case AA hypothetical hypothetical slope slope is is used used for for demonstration. demonstration. Fi Figuregure 44 presents presents the the geometry geometry of of the the slope slope with with anan angle angle of of 31° 31 ◦andand height height of of 6 6m. m. The The red red dash dash line line in in Figure Figure 44 is is a aschematic schematic line line of of potential potential circular circular slipslip that that has has the the overall overall minimum minimum factor factor of of safety. safety. Th Thee slope slope is is set set as as a adeep deep slope, slope, where where the the gradient gradient isis larger larger than than the the friction friction angle. angle. The The boundary boundary conditions conditions are are as as follows: follows: abab isis an an impervious impervious bottom bottom (rock(rock mass); mass); aeae and bdbd areare thethe Dirichlet Dirichlet boundaries boundaries with with total total water water heads; heads; boundary boundarydf hasdf has no flux;no flux;and and the groundthe ground surface surfaceef is subjectedef is subjected to rainfall. to rainfall. The initial The groundwaterinitial groundwater table cd tablewas 4cd m was below 4 m the belowground the surface ground of thesurface hillslope. of the The hillslope. soil is assumed The soil to is be assumed sandy loam to be [28 sandy], and theloam parameters [28], and are the as 2 parametersfollows: θ sare= 0.47as follows:; θr = 0.17 𝜃 ;=0.47Ks = ;0.031 𝜃 =0.17m/h;; N 𝐾= =2; 0.031M = m/h0.5; ;ξ =𝑁0.01 = 2;; c𝑀=0.50 = 500; N 𝜉/m =; 0.01φ0; =𝑐′26 =◦; b 500φ N/m= 13◦;; and 𝜙′G =s =26°; 2.65𝜙 .= All 13° symbols; and 𝐺 refer=2.65 have. All been symbols defined refer in Appendixhave beenA defined. This hypothetical in Appendix slope A. Thisis adopted hypothetical to all theslope tests is adopted presented to inall the the study tests presented with various in the surface study inflow with conditions.various surface inflow conditions.

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FigureFigure 4. 4.Geometry Geometry of of the the hypothetical hypothetical slope. slope.

3.3. Results Results and and Discussions Discussions Firstly,Firstly, two two cases cases were were selected selected for for comparison:comparison: (1)(1) a uniform rainfall rainfall event event with with an an intensity intensity of 2 of25 25 mm/h mm/ hand and a surface a surface inflow inflow of 0.0001 of 0.0001 m2/s m over/s over 12 h 12and h (2) and the (2) same the rainfall same rainfall conditions conditions but without but withoutsurface surface inflow. inflow. The results The resultsof FS, accumulated of FS, accumulated infiltration infiltration and pore and pressures pore pressures are presented are presented in Figures in Figures5–6. In5 Figureand6. 5a,In Figurewhen 5thea, wheninflow the is inflowconsidered, is considered, the FS decreases the FS decreases faster and faster a landslide and a landslide is triggered is triggeredat the 9th at hour; the 9thhowever, hour; however,in the condition in the conditionwherein the wherein inflow is the not inflow considered, is not considered,the FS is always the FSlarger is alwaysthan 1.0. larger The thanlowest 1.0. FS The of the lowest two FScases of theare two1.03 casesand 0.73, are 1.03respectively. and 0.73, The respectively. variation in The accumulated variation ininfiltration accumulated is presented infiltration in isFigure presented 5b, indicating in Figure that5b,, indicating in the case that, considering in the case inflow, considering the infiltration inflow, is thehigher infiltration and the is difference higher and of the the di twofference cases of increase the twos caseswith increasestime. In the with case time. without In the considering case without the 2 consideringinflow, the thefinal inflow, accumulated the final infiltration accumulated is 34,985 infiltration cm2, however, is 34,985 in the cm case, however, that considers in the the case inflow, that 2 considersit is 48,769 the cm inflow,2, i.e., 1.4 it is times 48,769 the cm other, i.e., case. 1.4 times Figure the 6 presents other case. the Figure pore pressures6 presents of the the pore two pressures cases. The ofpore the twopressures cases. Thein Figure pore pressures6c,d increase in Figure faster6 c,dthan increase those in faster Figure than 6a,b, those that in is, Figure they6 a,b,are more that is, likely they to arecause more landslide. likely to causeThe abovementioned landslide. The abovementioned results indicate results that, firstly, indicate the that, hillsl firstly,ope inflow the hillslope may increase inflow maythe increaseslope instability, the slope and instability, the phenomenon and the phenomenon is more likely is moreto occur likely on footslopes to occur on than footslopes on upper than slopes on upperdue to slopes the possibility due to the of possibility runoff. In addition, of runoff .from In addition, the numerical from themodeling numerical perspective, modeling the perspective, mechanism theconsidering mechanism the considering runoff is crucial the runo forff theseis crucial footslopes. for these footslopes.

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1.7

1.6 Inflow 0.0001 m2/s Without inflow 1.5 FS = 1 1.4 1.3 1.2

FS 1.1 1.0 0.9 0.8 0.7 0.6 04812162024 Time (hr)

(a) Factor of safety

60x103

) 3 2 50x10

40x103

30x103

20x103

3 2 Accumulate rainfall infiltration (cm infiltration rainfall Accumulate 10x10 Inflow 0.0001 m /s Without inflow

0 0 4 8 12162024 Time (hr)

(b) Accumulated infiltration

FigureFigure 5. 5.The The variancevariance inin ((aa)) FactorFactor ofof safetysafety andand ((bb)) AccumulatedAccumulated infiltrationinfiltration with with time time for for condition condition consideringconsidering inflow inflow and and without without inflow. inflow.

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10 10 12Unit hours : cm 12Unit hours : cm FS=1.263 FS=1.263 8 8 0 -10 0 0 6 -20 6 -10 00 00 Z(m) -1 Z(m) -1 4 0 4 0 100 100 2 2 200 200 300 300 0 0 0 2 4 6 8 10 0 2 4 6 8 10 X(m) X(m)

(a) 6th hour (b) 12th hour

10 10 12Unit hours : cm 12Unit hours : cm FS=1.263 FS=1.263 8 8 0 0 0 00 5 6 -1 6 0 0 10 -10 Z(m) 0 Z(m) 0 00 0 2 4 10 4 00 00 2 00 3 -1 2 300 2 400 400

0 0 0 2 4 6 8 10 0 2 4 6 8 10 X(m) X(m)

FigureFigure 6.6. PressurePressure headhead contourscontours for rainfall of 25 mmmm//hh overover 1212 hh andand nono inﬂowinflow atat thethe time:time: ((aa)) 6th6th hourhour andand ((bb)) 12th12th hour;hour; Pressure Pressure head head contours contours for for the the same same rainfall rainfall and and considered considered hillslope hillslope inﬂow inflow at theat the time: time: (c) ( 6thc) 6th hour hour and and (d) ( 12thd) 12th hour. hour.

AccordingAccording toto thethe previousprevious results,results, thethe eeffectffect ofof thethe inflowinflow onon thethe slopeslope stabilitystability isis notable.notable. In situ, thethe magnitude,magnitude, duration,duration, laglag time,time, andand peakpeak ofof thethe hydrograph [[32]32] maymay varyvary accordingaccording toto thethe stormstorm conditionsconditions andand catchmentcatchment characteristics.characteristics. Therefore, different different conditions of inflowinflow andand associatedassociated eeffectsffects onon thethe slopeslope stabilitystability areare discusseddiscussed inin thethe followingfollowing sections.sections.

3.1.3.1. Magnitude of Surface InflowInflow TheThe sizesize ofof thethe upperupper catchment catchment area area and and the the magnitude magnitude of of rainfall rainfall cause cause di ffdifferenterent magnitudes magnitudes of 2 2 2 surfaceof surface inflow. inflow. Four Four diff erentdifferent magnitudes magnitudes of surface of surface inflow inflow (0.00001 (0.00001 m /s, m 0.000022/s, 0.00002 m /s, m 0.000042/s, 0.00004 m /s, 2 andm2/s, 0.0001 and 0.0001 m /s) werem2/s) selected were selected in this studyin this to study analyze to theanalyz changee the in change the FS within the time. FS with These time. values These are invalues a reasonable are in a rangereasonable as a result range of as estimating a result of the estimating length of the upper length slope of areupper about slope tens are to about hundreds tens ofto meters.hundreds In of addition, meters. theIn addition, slope is subjected the slope to is asubjected uniform rainfallto a uniform event rainfall (200 mm, event 12 h).(200 As mm, presented 12 h). As in Figurepresented7a, a in landslide Figure 7a, is triggereda landslide in is all triggered cases, except in all forcases, the except one with for thethe smallestone with inflow. the smallest The times inflow. at 2 whichThe times landslides at which are landslides triggered isare 11.7, triggered 9.5, and is 9 11.7 h for, 9.5, the and cases 9 withh for 0.00002, the cases 0.00004, with 0.00002, and 0.0001 0.00004, m /s inflow,and 0.0001 respectively. m2/s inflow, In addition, respectively. although In addition, these cases although all exhibit these that cases the lowestall exhibit FS appears that the at lowest the same FS timeappears (12th at h),the the same time time at which (12th h), the the landslides time at which are triggered the landslides differ, depending are triggered on thediffer, magnitude depending of the on inflow.the magnitude Figure7 bof presents the inflow. the Figure variance 7b in presents accumulated the variance infiltration in accumulated with time, andinfiltration reveal that with larger time, inflowand reveal leads that to morelarger accumulated inflow leads infiltration. to more accu Inmulated the tests, infiltration. the accumulated In the infiltrationstests, the accumulated are 30,638, infiltrations are 30,638, 36,304, 45,383, and 48,390 cm2 to the inflow, from small to large. The pore pressure also reveals the effects of different magnitudes of inflow, as presented in Figure 8, indicating

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36,304,Water 45,383, 2020, 12 and, x FOR 48,390 PEER REVIEW cm2 to the inflow, from small to large. The pore pressure also reveals10 of 19 the effects of different magnitudes of inflow, as presented in Figure8, indicating that the magnitude of that the magnitude of inflow accelerates the wetting front propagating downward. Therefore, a larger inflowinflow accelerates increases the the wetting probability front of propagating triggering a slope downward. failure and Therefore, may bring a forward larger inflow the trigger increases time. the probability of triggering a slope failure and may bring forward the trigger time.

1.8

0.00001 m2/s 1.6 0.00002 0.00004 0.0001 1.4 FS = 1

1.2 FS

1.0

0.8

0.6 024681012141618202224 Time (hr) (a) Factor of safety

50x103 ) 2 40x103

30x103

20x103

0.00001 m2/s 10x103 0.00002 Accumulate rainfall infiltration (cm infiltration rainfall Accumulate 0.00004 0.0001

0 04812162024 Time (hr)

(b) Accumulated infiltration

FigureFigure 7. Factor 7. Factor of safety of safety and and accumulated accumulated infiltration infiltration for different different hillslope hillslope inflow inflow rates. rates.

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10 10 12Unit hours : cm 12Unit hours : cm FS=1.263 FS=1.263 0

8 -20 8 0 -2 -50 -50 6 00 6 0 -2 0 0 20 10 -1 -

Z(m) - Z(m) 0 0 20 0 4 4 - 00 0 1 10 0 0 20 20 2 00 2 0 3 30 400 400 0 0 0 2 4 6 8 10 0 2 4 6 8 10 X(m) X(m)

(a) 0.00001 m2/s at 12th hour (b) 0.00002 m2/s at 12th hour

10 12Unit hours : cm FS=1.263 8 0 0 6 0 0 0 Z(m) 1 -50 4 200 300 2 400

0 0 2 4 6 8 10 X(m)

Figure 8. PressurePressure head contours at 12th hour with (a) 0.00001 m2/s,/s, ( b) 0.00002 m 2/s,/s, ( (cc)) 0.00004 0.00004 m m2/s,/s, and ( d) 0.0001 m 2/s/s surface surface inflow inﬂow rate.

3.2. Duration Duration Three scenariosscenarios with with di ffdifferenterent durations durations but thebut samethe same inflow inflow volume volume were considered: were considered: (i) Scenario (i) 2 2 2 1:Scenario 0.00016 1: m0.00016/s for m 32 h,/s for (ii) 3 Scenario h, (ii) Scenario 2: 0.00008 2: 0.00008 m /s for m2/s 6 h,for and 6 h, (iii)andScenario (iii) Scenario 3: 0.00004 3: 0.00004 m / sm for2/s 12for h.12 The h. The designs designs were were used used to observe to observe whether whether an intense, an intense, short short inflow inflow affect affect the FS the more, FS more, or a mild, or a mild,longterm longterm inflow inflow does. does. We firstly We firstly analyzed analyzed the variancethe variance in FS in with FS with time, time, as presented as presented in Figurein Figure9a. 9a.In theIn the first first 3 h, 3 the h, ditheff erencesdifferences in FS in among FS among three three scenarios scenarios are small;are small; the valuesthe values of FS of are FS 1.13, are 1.13, 1.14, 1.15,1.14, respectively,1.15, respectively, for Scenario for Scenario 1, 2, and 1, 3,2, andand Scenario3, and Scenario 1 has the 1 lowesthas the FS. lowest From FS. 3 to From 6 h, Scenario 3 to 6 h, 2 Scenarioyields the 2 lowest yields FS.the Similarly, lowest FS. because Similarly, the surface because flow the increases surface theflow infiltration increases rate,the infiltration Scenario 1 hasrate, a Scenariohigher accumulated 1 has a higher infiltration accumulated in the infiltration first 3 h (Figure in the9b). first However, 3 h (Figure after 9b). the However, runo ff stops, after for the example, runoff stops,at the for 4th example, h, the infiltration at the 4th of h, Scenario the infiltration 1 is still of higher Scenario than 1 is that still of higher Scenario than 3, that but of the Scenario FS rebounds 3, but thefrom FS 1.13 rebounds to 1.192 from in a 1.13 short to time1.192 and in a exhibitsshort time a di andfferent exhibits trend a thandifferent the infiltration,trend than the due infiltration, to surface duepressure to surface dissipation. pressure Figure dissipation. 10 presents Figure the 10 results presents for the the results pore pressure for the pore that pressure explains that the reboundexplains theof FS. rebound Between of FS. the Between 3rd h (Figure the 3rd 10 a)h (Figure and the 10a) 4th an h (Figured the 4th 10 hb), (Figure the surface 10b), the pressure surface of pressure Scenario of 1 changes from 0 to 30 cm, whereas those of Scenarios 3 and 2 remain at 0 (Figure 10c,d). The results Scenario 1 changes− from 0 to −30 cm, whereas those of Scenarios 3 and 2 remain at 0 (Figure 10c,d). Theemphasize results emphasize the effect of the continuous effect of continuous inflow on inflow the stability, on the andstability, long-term and long-term duration duration is more likely is more to likelytrigger to a trigger landslide. a landslide.

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1.7

1.6 0.00016 m2/s for 3 hr (Scenario 1) 2 1.5 0.00008 m /s for 6 hr (Scenario 2) 0.00004 m2/s for 12 hr 1.4 (Scenario 3) FS = 1 1.3 FS

1.2

1.1

1.0

0.9 024681012141618202224 Time (hr)

(a) Factor of safety

50000

40000 ) 2

30000

20000 0.00016 m2/s for 3 hr (Scenario 1) 0.00008 m2/s for 6 hr Accumulate infiltration (cm infiltration Accumulate 10000 (Scenario 2) 0.00004 m2/s for 12 hr (Scenario 3) 0 024681012141618202224 Time (hr)

(b) Accumulated infiltration

FigureFigure 9. 9.The The variance variance in: in: (a) factor(a) factor of safety,and of safety, (b and) accumulated (b) accumulated infiltration, infiltration, for different for inflow different durations. inflow durations.

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10 10 12Unit hours : cm 12Unit hours : cm FS=1.263 FS=1.263 30 50 - 8 - 8 0 -30 0 0 -50 -1 0 00 50 00 -5 -2 0 - 6 -2 6 -20 00 0 -1 -10 Z(m) Z(m) 0 0 0 4 100 4 10 0 0 20 20 0 00 2 30 2 3 0 00 40 4 0 0 0 2 4 6 8 10 0 2 4 6 8 10 X(m) X(m)

(a) Scenario 1 at 3rd hour (b) Scenario 1 at 4th hour

10 10 12Unit hours : cm 12Unit hours : cm FS=1.263 FS=1.263 50 8 - 8 -50 0 0 0 0 10 20 - 0 -50 - 0 20 20 50 6 - 0 6 - 0 - -10 -10 Z(m) 0 Z(m) 0 0 0 4 10 4 10 0 0 20 20 00 00 2 3 2 3 0 0 40 40

0 0 0 2 4 6 8 10 0 2 4 6 8 10 X(m) X(m)

10 10 12Unit hours : cm 12Unit hours : cm FS=1.263 FS=1.263 8 8 0 30 - 0 -30 6 -50 6 -50 0 0 00 Z(m) Z(m) 1 0 0 -3 100 -5 4 4 200 200 0 300 30 2 2 0 400 40

0 0 0 2 4 6 8 10 0 2 4 6 8 10 X(m) X(m)

(e) Scenario 2 at 12th hour (f) Scenario 3 at 12th hour

Figure 10. PressurePressure head contours for various inflow inﬂow durations: ( a) Scenario Scenario 1 at at 3rd 3rd hour, hour, ( b) Scenario Scenario 1 at 4th hour, hour, ( (cc)) Scenario Scenario 2 2 at at 4th 4th hour, hour, (d (d) )Scenario Scenario 3 3at at 4th 4th hour, hour, (e) ( eScenario) Scenario 2 at 2 12th at 12th hour, hour, and and (f) (Scenariof) Scenario 3 at 3 12th at 12th hour. hour.

3.3. Delayed Surface Inflow In this section, the start and end times of the inflow of all scenarios differ by 2 h (Figure 11). This arrangement was used to analyze whether or not landslides may occur after the rain stops, attributed

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Water3.3. Delayed2020, 12, x Surface FOR PEER Inflow REVIEW 14 of 19 In this section, the start and end times of the inflow of all scenarios differ by 2 h (Figure 11). This to the subsequent runoff. The slope is subjected to a uniform rainfall event (200 mm, 12 h), the same arrangement was used to analyze whether or not landslides may occur after the rain stops, attributed as the previous sections. to the subsequent runoff. The slope is subjected to a uniform rainfall event (200 mm, 12 h), the same as the previous sections.

FigureFigure 11. 11. TheThe input input inflow inﬂow hydrograph hydrograph and and rainfall rainfall hyet hyetographograph in in the the different diﬀerent delay times tests.

InIn scenarios withwith delayeddelayed surface surface inflow inflow (delay: (delay: 0, 0, 2, 2, and and 4 h),4 h), landslides landslides are are triggered triggered during during the therainfall rainfall event event (within (within 12 h), 12 as h), shown as shown in Figure in 12Fi.gure However, 12. However, in the other in the three other scenarios, three landslidesscenarios, landslidesare not triggered are not during triggered the rainfall during event the butrainfall rather event when but the rather slope receiveswhen the the subsequentslope receives surface the subsequentflow. The pore-pressure surface flow. variation The pore-pressure in Figure 13 variation indicates in that Figure the pressure13 indicates under that the the surface pressure continues under theto increase surface continues after the rainfallto increase stops, after which the rainfall is due st toops, surface which water, is due and to surface the wetting water, front and the propagates wetting frontdownward. propagates The resultsdownward. indicate The that results landslides indicate can that occur landslides after the can rain occur stops after and the delayed rain stops runo andff is delayedmore likely runoff to cause is more landslide. likely to cause landslide.

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1.81.8 DelayDelay 0 0 hr hr Delay 2 hr 1.6 Delay 2 hr 1.6 DelayDelay 4 4 hr hr DelayDelay 6 6 hr hr DelayDelay 8 8 hr hr 1.41.4 DelayDelay 10 10 hr hr FSFS = = 1 1 1.2 FS 1.2 FS

1.01.0

0.80.8

0.60.6 024681012141618202224024681012141618202224 TimeTime (hr) (hr)

FigureFigure 12. 12. The The different di differentﬀerent development development of of factor factor of of safety safety with with time time in in the the tests tests of of different di differentﬀerent delay delay times. times.

10 10 10 12Unit hours : cm 10 12Unit hours : cm FS=1.26312Unit hours : cm FS=1.26312Unit hours : cm FS=1.263 FS=1.263 8 8 8 8 00 00 50 6 - 50 6 1000 6 -000 6 10 -110 Z(m) - 0 Z(m) Z(m) 0 Z(m) 00 4 00 4 2200 4 1100 4 2000 00 20 3300 3000 0 2 30 2 4000 2 00 2 4 4400

00 00 00 22 44 66 88 1010 00 22 44 66 88 1010 X(m)X(m) X(m)X(m)

(a(a) ) (b(b) )

FigureFigure 13. 13. Pressure Pressure headhead head contourscontours contours in in in the the the tests tests tests of of diof ffdifferent differenterent delay delay delay times: times: times: (a) ( Delaya(a) )Delay Delay of 4of hof 4 at 4 h 12thh at at 12th hour12th hour andhour (andband) delay ( b(b) )delay delay of 4 hof of at 4 4 16thh h at at 16th hour. 16th hour. hour. 3.4. Shape of Inflow 3.4.3.4. Shape Shape of of Inflow Inflow Different basin shapes may lead to different types of hydrograph [32]. In this study, the inflow DifferentDifferent basin basin shapes shapes may may lead lead to to different different types types of of hydrograph hydrograph [32]. [32]. In In this this study, study, the the inflow inflow hydrograph is simplified into three types based on the peak position of the hydrograph, as shown in hydrographhydrograph is is simplified simplified into into thr threeee types types based based on on the the peak peak position position of of the the hydrograph, hydrograph, as as shown shown in in Figure 14: (1) peak appears at the beginning, (2) peak appears at middle, and (3) peak appears at the FigureFigure 14: 14: (1) (1) peak peak appears appears at at the the beginning, beginning, (2) (2) peak peak appears appears at at middle, middle, and and (3) (3) peak peak appears appears at at the the end. Figure 15a shows that the differences among the three types gradually expand until about the end.end. Figure Figure 15a 15a shows shows that that the the differences differences among among the the three three types types gradually gradually expand expand until until about about the the 2.6 h, the FS of Type 1, Type 2, and Type 3 are 1.20, 1.23 and 1.26, respectively. Moreover, the times at 2.62.6 h, h, the the FS FS of of Type Type 1, 1, Type Type 2, 2, and and Type Type 3 3 are are 1.20, 1.20, 1.23 1.23 and and 1.26, 1.26, respec respectively.tively. Moreover, Moreover, the the times times which landslides occurred are 9.3, 9.7, and 10.0 h. The FS of the Type 1 decreases faster than that of atat which which landslides landslides occurred occurred are are 9.3, 9.3, 9.7, 9.7, and and 10.0 10.0 h. h. The The FS FS of of the the Type Type 1 1 decreases decreases faster faster than than that that theof the others others and and triggers triggers the the landslide landslide slightly slightly earlier earlier than than the the others. others. However,However, ifif thethe peakpeak value of of the others and triggers the landslide slightly2 earlier than the others. However, if the peak value of the inflow is a lower value, such as 0.00008 m2 /s, Figure 15b indicates the possibility that only Type 3 thethe inflow inflow is is a a lower lower value, value, such such as as 0.00008 0.00008 m m/s,2/s, Figure Figure 15b 15b indicates indicates the the possibility possibility that that only only Type Type 3 3 triggers a landslide. However, compared with the tests in the previous sections, only slight differences can be found in the FS trends among the three types.

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triggers a landslide. However, compared with the tests in the previous sections, only slight Water 2020, 12, x FOR PEER REVIEW 16 of 19 differences can be found in the FS trends among the three types. triggers a landslide. However, compared with the tests in the previous sections, only slight differences can be found in the FS trends among the three types. Water 2020, 12, 1229 0.00025 0 16 of 19

0.000250.00020 0 40

2060 /s) 2 0.000200.00015 4080

60100 /s) 2 0.000150.00010 80120 Inflow rate (m rate Inflow 100 140 (mm/hr) intensity Rain 0.00010 Rainfall Intensity 120 0.00005 TYPE 1 160 Inflow rate (m rate Inflow TYPE 2 140180 (mm/hr) intensity Rain TYPE 3 Rainfall Intensity 0.000050.00000 TYPE 1 160200 024681012TYPE 2 180 TYPE 3 Time (hr) 0.00000 200 Figure 14. Three024681012 inflow types and the rainfall intensity in the tests of hydrograph shapes. Time (hr)

FigureFigure 14. 14. ThreeThree inflow inﬂow types types and and the the rainfall rainfall intens intensityity in in the the tests tests of of hydrograph hydrograph shapes. 1.7

1.6 1.7 TYPE 1 TYPE 2 1.5 1.6 TYPE 3 TYPEFS = 1 1.4 TYPE 2 1.5 TYPE 3 1.3 FS = 1 FS 1.4

1.2 1.3 FS 1.1 1.2

1.0 1.1

0.9 1.0 024681012 0.9 Time (hr)

024681012 (a) Maximum inflow rate of 0.0002 m2/s. Time (hr) Figure 15. Cont. (a) Maximum inflow rate of 0.0002 m2/s.

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1.7

1.6 TYPE 1 1.5 TYPE 2 TYPE 3 FS=1 1.4

1.3 FS

1.2

1.1

1.0

0.9 024681012 Time (hr)

(b) Maximum inflow rate of 0.00008 m2/s.

FigureFigure 15. 15.Factor Factor of of safety safety for for various various inflow inflow hydrograph hydrograph types. types. (a) Maximum(a) Maximum inflow inflow rate ofrate 0.0002 of 0.0002 m2/s andm2/s (b and) maximum (b) maximum inflow inflow rate of rate 0.00008 of 0.00008 m2/s. m2/s.

4.4. ConclusionsConclusions ThisThis study study focused focused on on the the effect effect of surface of surface water wate on ther on slope the stability.slope stability. In this study, In this a 2Dstudy, landslide a 2D modellandslide coupling model thecoupling 1D rainfall–runo the 1D rainfall–runoffff and 2D infiltration and 2D infiltration equations equations was developed. was developed. Based on Based the physical-basedon the physical-based model, themodel, diff erencethe difference between betw consideringeen considering inflow andinflow not and considering not considering inflow oninflow the FSon wasthe elucidated.FS was elucidated. With inflow With consideration,inflow consideratio the FSn, decreasedthe FS decreased much faster much than faster with than no with inflow no consideration.inflow consideration. This revealed This revealed that the considerationthat the consideration of the runo offf isthe crucial runoff for is modeling, crucial for particularly modeling, whenparticularly footslopes when are footslopes analyzed. Inare addition, analyzed. based In a onddition, a series based of simulations on a series of theof hypotheticalsimulations slope,of the thehypothetical magnitude, slope, duration, the magnitude, and lag time duration, affect the and FS andlag timetime ofaffect landslide the FS occurrence, and timealthough of landslide the influenceoccurrence, of thealthough peak position the influence of hydrograph of the peak isminor. position The of magnitude hydrograph tests isminor. showed The that magnitude a larger inflow tests hasshowed a lower that FS a andlarger triggers inflow landslide has a lower earlier. FS and Long-term triggers durationlandslideinflow earlier. decreases Long-term the duration stability inflow more thandecreases short the intensive stability inflow more does, than inshort the intensive duration infl tests.ow Indoes, addition, in the duration the delay tests. surface In addition, inflow tests the indicatedelay surface that landslides inflow tests may indicate occur afterthat landslides the rainfall may stops, occur attributing after the to rainfall the lingering stops, attributing inflow. to the lingeringIn practice, inflow. the result of the study supports the usages of drainages and agrees that hillslope managementIn practice, should the contriveresult of tothe reduce study runosupportsff, because the usages uncontrolled of drainages runo ffandcan agrees cause slopethat hillslope failure. Besides,management when should using numericalcontrive to model reduce to runoff, analyze because the shallow uncontrolled landslide runoff on footslopes, can cause slope we strongly failure. suggestBesides, modelling when using the numerical surface flow model and to considering analyze the the shallow inflow landslide from upper on slopesfootslopes, as well, we andstrongly the esuggestffect of runomodellingff on the the boundary surface conditionflow and considering should be calculated the inflow in thefrom model, upper owing slopes to as the well, remarkable and the influenceeffect of runoff we exhibited on the boundary in the study. condition should be calculated in the model, owing to the remarkable Authorinfluence Contributions: we exhibitedConceptualization, in the study. T.-L.T. and J.-C.Y.; Formal analysis, H.-E.C.; Methodology, H.-E.C., Y.-Y.C. and T.-L.T.; Supervision, T.-L.T. and J.-C.Y.; Writing—original draft, H.-E.C.; Writing—review and editing, Y.-Y.C.Author All Contributions: authors have readConceptualization, and agreed to theT.-L.T. published and J.-C.Y.; version Formal of the analysis, manuscript. H.-E.C.; Methodology, H.-E.C., Funding:Y.-Y.C. andThis T.-L.T.; research Supervision, received no T.-L.T. external and funding. J.-C.Y.; Writing—original draft, H.-E.C.; Writing—review and editing, Y.-Y.C. All authors have read and agreed to the published version of the manuscript. Conflicts of Interest: The authors declare no conflict of interest. Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

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Appendix A

List of Symbols b Slice width c0 Effective cohesion f Infiltrate rate hs Depth of slope surface flow i Rainfall intensity n Manning’s roughness coefficient. S0 Bed slope t Time ua Pore air pressure uw Pore water pressure Kx Hydraulic conductivities in the x-direction Kz Hydraulic conductivities in the z-direction M Fitting parameter N Fitting parameter S Degree of saturation X Distance downslope α Slope angle γ unit weight of soil γw unit weight of water ξ Fitting parameter σ Normal effective stress θ Moisture content θr Residual moisture content θs Saturated moisture content φ0 Effective friction angle φb Friction angle with respect to the matric suction ψ Pore water pressure head ψc Positive pore water pressure head ψp Negative pore water pressure head

References

1. Sidle, C.R.; Greco, R.; Bogaard, T. Overview of Landslide Hydrology. Water 2019, 11, 148. [CrossRef] 2. Bogaard, T.A.; Greco, R. Landslide Hydrology: From Hydrology to Pore Pressure. WIREs Water 2016, 3, 439–459. [CrossRef] 3. Pujades, E.; DeSimone, S.; Carrera, J.; Vázquez-Suñé, E.; Jurado, A. Settlements around Pumping Wells: Analysis of Influential Factors and a Simple Calculation Procedure. J. Hydrol. 2017, 548, 225–236. [CrossRef] 4. Pujades, E.; Jurado, A.; Carrera, J.; Vázquez-Suñé, E.; Dassargues, A. Hydrogeological Assessment of Non-Linear Underground Enclosures. Eng. Geol. 2016, 207, 91–102. [CrossRef] 5. Wu, Y.-X.; Shen, S.-L.; Lyu, H.-M.; Zhou, A. Analyses of Leakage Effect of Waterproof Curtain during Excavation Dewatering. J. Hydrol. 2020, 583, 124582. [CrossRef] 6. Wu, Y.-X.; Lyu, H.-M.; Shen, S.-L.; Zhou, A. A Three-Dimensional Fluid-Solid Coupled Numerical Modeling of the Barrier Leakage below the Excavation Surface Due to Dewatering. Hydrogeol. J. 2020.[CrossRef] 7. Wu, H.-N.; Shen, S.-L.; Chen, R.-P.; Zhou, A. Three-Dimensional Numerical Modelling on Localised Leakage in Segmental Lining of Shield Tunnels. Comput. Geotech. 2020, 122, 103549. [CrossRef] 8. Iverson, R.M. Landslide triggering by rain infiltration. Water Resour. Res. 2000, 36, 1897–1910. [CrossRef] 9. Baum, R.L.; Savage, W.Z.; Godt, J.W. TRIGRS–A Fortran Program for Transient Rainfall Infiltration and Grid-Based Regional Slope-Stability Analysis, Version 2.0; US Geological Survey: Reston, VA, USA, 2008. 10. Tsai, T.-L.; Yang, J.-C. Modeling of rainfall-triggered shallow landslide. Environ. Geol. 2006, 50, 525–534. [CrossRef] Water 2020, 12, 1229 19 of 19

11. Tarantino, A.; Bosco, G. Role of soil suction in understanding the triggering mechanisms of flow slides associated with rainfall. In Debris-Flow Hazards Mitigation: Mechanics, Prediction, and Assessment, Proceedings of the Second International Conference on Debris-Flow Hazards Mitigation, Taipei, Taiwan, 16–18 August 2000; A.A. Balkema: Rotterdam, The Netherlands, 2000; pp. 81–88. 12. Collins, B.D.; Znidarcic, D. Stability Analyses of Rainfall Induced Landslides. J. Geotech. Geoenviron. Eng. 2004, 130, 362–372. [CrossRef] 13. Tsai, T.-L.; Chen, H.-E.; Yang, J.-C. Numerical modeling of rainstorm-induced shallow landslides in saturated and unsaturated soils. Environ. Geol. 2008, 55, 1269–1277. [CrossRef] 14. Fredlund, D.G.; Morgenstern, N.R.; Widger, R.A. The shear strength of unsaturated soils. Can. Geotech. J. 1978, 15, 313–321. [CrossRef] 15. Tsai, T.-L. The influence of rainstorm pattern on shallow landslide. Environ. Geol. 2008, 53, 1563–1569. [CrossRef] 16. Tsai, T.-L.; Wang, J.-K. Examination of influences of rainfall patterns on shallow landslides due to dissipation of matric suction. Environ. Earth Sci. 2011, 63, 65–75. [CrossRef] 17. Chen, H.-E.; Tsai, T.-L.; Yang, J.-C. Threshold of slope instability induced by rainfall and lateral flow. Water 2017, 9, 722. [CrossRef] 18. Chan, H.-C.; Chen, P.-A.; Lee, J.-T. Rainfall-induced landslide susceptibility using a rainfall–runoff model and logistic regression. Water 2018, 10, 1354. [CrossRef] 19. Chiu, Y.-Y.; Chen, H.-E.; Yeh, K.-C. Investigation of the influence of rainfall runoff on shallow landslides in unsaturated soil using a mathematical model. Water 2019, 11, 1178. [CrossRef] 20. Wysocki, D.A.; Schoeneberger, P.J.; LaGarry, H.E. Geomorphology of soil landscapes. Handb. Soil Sci. 2000, 1, 315–321. 21. Fazlollahi Mohammadi, M.; Jalali, S.G.H.; Kooch, Y.; Said-Pullicino, D. Slope Gradient and Shape Effects on Soil Profiles in the Northern Mountainous Forests of Iran. Eurasian Soil Sci. 2016, 49, 1366–1374. [CrossRef] 22. Henderson, F.M. Open-Channel Flow; Macmillan: New York, NY, USA, 1966. 23. Gunaratnam, D.J.; Perkins, F.E. Numerical Solution of Unsteady Flows in Open Channels; Report No. 127; Massachusetts Institute of Technology, Department of Civil Engineering, Hydrodynamics Laboratory: Cambridge, MA, USA, 1970; 216p. 24. Miller, J.E. Basic Concepts of Kinematic-Wave Models (No. 1302); US Geological Survey: Reston, VA, USA, 1984. 25. Nguyen, T.S.; Luong, T.A.; Luong, H.D.; Tran, H.T. A finite element one-dimensional kinematic wave rainfall-runoff model. Pac. Sci. Rev. A Nat. Sci. Eng. 2016, 18, 233–240. [CrossRef] 26. Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill: New York, NY, USA, 1988. 27. Richards, L.A. Capillary conduction of liquids through porous mediums. Physics 1931, 1, 318–333. [CrossRef] 28. Van Genuchten, M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 1980, 44, 892–898. [CrossRef] 29. Bishop, A.W. The use of pore-pressure coefficients in practice. Geotechnique 1954, 4, 148–152. [CrossRef] 30. Preissmann, A. Propagation of translatory waves in channels and rivers. In Proceedings of the 1st Congress of French Association for Computation, Grenoble, France, 14–16 September 1961; pp. 433–442. 31. Celia, M.A.; Bouloutas, E.T.; Zarba, R.L. A general mass conservative numerical solution for unsaturated flow equation. Water Resour. Res. 1990, 26, 14. [CrossRef] 32. Subramanya, K. Engineering Hydrology, 4e; Tata McGraw-Hill Education: New York, NY, USA, 2013.

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