Numerical analysis of multiple slope failure due to rainfall: Based on laboratory experiments

Catena 150 (2017) 173–191

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Numerical analysis of multiple slope failure due to rainfall: Based on laboratory experiments Ram Krishna Regmi a, Kwansue Jung b,⁎, Hajime Nakagawa c, Xuan Khanh Do d, Binaya Kumar Mishra e a

Environment & Resource Management Consultant, Kathmandu, Nepal Department of Civil Engineering, Chungnam National University, Daejeon, 305-764, Republic of Korea Ujigawa Hydraulics Laboratory, Disaster Prevention Research Institute, Kyoto University, Kyoto, 612-8235, Japan d Department of Civil Engineering, Chungnam National University, Daejeon, 305-764, Republic of Korea e Institute for the Advanced Study of Sustainability, United Nations University, Tokyo, 150-8925, Japan b c

a r t i c l e

i n f o

Article history: Received 26 December 2015 Received in revised form 30 October 2016 Accepted 4 November 2016 Available online xxxx Keywords: Numerical analysis Multiple slope failure Dynamic programming Failure surface Sliding block model Experimental data

a b s t r a c t The investigation of mechanisms of multiple slope failure and the displacement of the resulting failure mass is highly important for the safety of the mountainous environments of the world. This study attempts to investigate such phenomena through numerical analysis. A one-dimensional (1D) surface flow and erosion/deposition model, a two-dimensional (2D) seepage flow model, a 2D slope stability model (the Spencer method of slope stability analysis), and a 1D sliding block model were combined as a single unit such that the developed model can also successfully analyze the surface water flow and erosion/deposition on the model slope soil surface, seepageflow phenomena within the soil domain, and stability of the model slope during the movement of the sliding mass by updating the shape of the model slope according to the new position of the sliding mass. The Spencer method of slope stability analysis was incorporated into dynamic programming to predict the time of a slope failure and the shape of the failure surface. The data obtained from the numerical simulation results were compared with the experimental data obtained from Regmi et al. (2014) for validation. The application of the model in the real field would have significant impact for appropriate mitigation measures against probable disasters that may be caused by rainfall-induced landslides and slope failures. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Extreme rainfall events have caused landslides and slope failures in mountainous environments worldwide. Increased pore pressure and seepage flow during periods of intense rainfall cause landslides and slope failures in general (Terzaghi, 1950; Sidle and Swanston, 1982; Sitar et al., 1992; Anderson and Sitar, 1995; Wang and Sassa, 2003). During rainfall, a wetting front moves downward into the soil, resulting in an increase in water content and an increase in pore pressure. The increase in pore pressure results in a decrease in effective stress, reducing the shear strength of the soil and ultimately resulting in landslide/slope failure (Brand, 1981; Brenner et al., 1985). Various methodological approaches have been developed for the investigation of landslide and slope failure processes. A series of flume tests were conducted by Sassa (1972, 1974), and it was concluded that the changes in rigidity of sand and the upper yield strain within a slope were essential to slope stability analyses. Fukuzono (1987) ⁎ Corresponding author. E-mail addresses: [email protected] (R.K. Regmi), [email protected] (K. Jung), [email protected] (H. Nakagawa), [email protected] (X.K. Do), [email protected] (B.K. Mishra).

http://dx.doi.org/10.1016/j.catena.2016.11.007 0341-8162/© 2016 Elsevier B.V. All rights reserved.

conducted an experiment using near-actual-scale slope models providing heavy rainfall to examine the conditions leading to slope failure. A rainfall-based landslide-triggering model was developed from landslide episodes in Wellington, New Zealand, termed the ‘Antecedent Water Status Model,’ to predict landslide occurrence by providing a 24-h forecast (Crozier, 1999). A physical model was developed using the complete Richards' equation, which measures the effect of the slope angle (Tsai et al., 2008), and the extended Mohr-Coulomb failure criterion of Fredlund et al. (1978) was also adopted to describe the unsaturated shear strength. A numerical model was developed to estimate the extent of rain-water infiltration into an unsaturated slope, the formation of a saturated zone, and the change in slope stability (Mukhlisin and Taha, 2009). Then, the model was used to analyze the effects of soil thickness on the occurrence of slope failure. Numerical simulations and flume experiments were performed by Regmi et al. (2012) to investigate the mechanism of slope failure due to rainfall events. However, these works are not applicable to multiple slope failures. Tsutsumi and Fujita (2008) investigated several landslide sites and used physical experiments and numerical simulations with a combination of rainwater infiltration for their slope stability analysis, which was applicable to represent multi-stage failure. After a failure, the failure mass slides slowly down along a well-defined slip surface as long as

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the slope gradient is steep enough, and it stops sliding when it arrives at a flatter area. This mass also supports the remaining soil mass such that the remaining mass does not immediately lose its stability. However, in their analysis, they did not consider the presence of such a failure mass. In the stability analysis, they used Janbu's simplified method, which satisfies only the force equilibrium. However, it would be better to use a method that satisfies both force and moment equilibria to obtain more accurate results. Regmi et al. (2012) performed numerical simulations and flume experiments to investigate the mechanism of slope failure due to rainfall events. A three-dimensional (3D) seepage-flow numerical model was coupled with a 2D surface-flow and erosion/deposition model for seepage analysis. Janbu's simplified 3D method of slope-stability analysis, as well as the extended Spencer method of slope-stability analysis, was incorporated into dynamic programming to locate the critical slip surface of a single failure event. However, these works did not examine the movement of the failure mass along the failure surface that may occur on the slope and probable multiple failures. Regmi et al. (2014) performed slope failure experiments consisting of a series of successive failures, with particular emphasis on the time of failure; the shape, size and position of the slip surface; and the final shape of the model slope after the displacement of the failure masses. However, numerical analysis of these phenomena is still lacking. Several researchers have proposed physically based models, which correlate the movement of landslide masses along slip surfaces to one or more variables controlling the slide behavior. Hong et al. (2005) used a statistical approach to correlate the relationship between intense rainfall and landslide movement. Leroueil (2001) proposed a physically based model to take into account the complexity of the hydrological and mechanical responses of the soil for the movement of landslide masses

along pre-existing slip surfaces due to rainfall-triggered pore pressure fluctuations. Calvello et al. (2008, 2009) developed a numerical model to predict the movement of landslide masses along pre-existing slip surfaces. The model is comprised of the following: a transient seepage finite-element analysis to compute the variations of pore-water pressures from rainfall; a limit equilibrium stability analysis to compute the factors of safety along the slip surface associated with transient pore pressure conditions; an empirical relationship between the factor of safety and the rate of displacement of the slide along the slip surface; and an optimization algorithm for the calibration of analyses and relationships based on the available monitoring data. Although the other researchers, besides Hong et al. (2005), used physically based model to evaluate the movement of landslide masses along pre-existing slip surfaces, their works are also not applicable for multiple failure. If a landslide occurs due to the rise of the saturated zone, at least in the neighborhood of the slip surface, the void-rich soil structure will be destroyed, forming a liquefied layer (Takahashi, 2007). Suwa et al. (1985) emphasized the role of the liquefied layer beneath the landslide mass, categorizing it as a significant cause of its movement. Many authors (e.g., Iverson et al., 1997; Sassa, 1997; Hutchinson, 1988; Okura et al., 2002; Wang and Sassa, 2002; van Asch et al., 2006) discussed the importance of liquefaction within the saturated zone as the cause of the rapid motion of the landslide mass. The nature of the movement of landslide masses is essentially viscous (Savage and Chleborad, 1982; Leroueil and Marques, 1996; Corominas et al., 2005), and the displacement rate is related to the changing saturated zones, which affect the shear stress level along the slip surface. In such cases, Vulliet and Hutter (1988) stated that “the slope is neither still nor ruptured but simply moves.” They proposed several phenomenological relationships between the displacement rate and the shear stress acting along the slip

Geometry of the model slope, and initial and boundary conditions

Seepage flow model Pore water pressure, Moisture content

Model parameters

Degree of saturation, Infiltration rate Surface flow and erosion/deposition model

Erosion/deposition thickness

Next time step

Model parameters

Surface water head

New geometry of the model slope

Model parameters

Slope stability model

No

Existing failure surface

Yes

Safety factor for a critical surface, Fs > 1

No New failure surface

Yes Sliding block model New geometry of the model slope

New geometry and position of the failure mass

Fig. 1. General flow chart of the coupled numerical simulation model.

R.K. Regmi et al. / Catena 150 (2017) 173–191

surface. Using the same approach, Ferlisi (2004) developed a mechanical model in which the movements along the sliding surface, due to imposed water-level fluctuations, are assumed to be of a viscoplastic type. Further, other researchers have suggested different approaches to take into account the viscosity of the soil. If the earth mass is dense (porosity of less than approximately 0.33), the liquefied layer is hardly formed (Fleming et al., 1989). However, if the earth mass is sparse, it is suddenly liquefied, and the motion is accelerated (Iverson et al., 2000). Although the porosity of the considered soil mass is 0.43 (N0.33), for simplicity, this study assumes that no liquefied layer is produced in the slip surface along the saturated soil layer and simply treats a frictional shear stress along the slip surface instead of viscous shear stress. Although some studies have been conducted to investigate the movement of landslide masses along pre-existing slip surfaces using physically based model, a study related to multiple failure of a slope is still lacking. It is worth noting that multiple slope failure in mountainous areas is common due to rainfall. Therefore, this study attempts to develop a numerical simulation model to investigate the multiple failure of a slope due to rainfall events. For slope stability analysis, the limit equilibrium method of slices is the most popular and widely used method due to its simplicity and accuracy. In this approach, the soil mass above the slip surface is discretized into vertical slices, and the equilibrium of the slices is considered (Abramson et al., 2002). The ordinary method, the Bishop simplified method, Janbu's simplified method, Janbu's generalized procedures of slices, the Spencer method, the Morgenstern-Price method, the Sarma method, the Lowe and Karafiath method, and the Corps of Engineers method are the commonly used slice methods. A detailed review of equilibrium methods of slope stability analysis is presented by Duncan (1996). Most of the methods are very similar in their basic formulations with only minor differences in the assumptions on the interslice shear forces. The limit equilibrium method can be broadly classified into two main categories: simplified methods and rigorous methods. For the simplified methods, either force or moment equilibrium can be satisfied but not both at the same time. For the rigorous methods, both force and moment equilibrium can be satisfied, but the analysis is usually more tedious and may sometimes experience nonconvergence problems. The Spencer method, Morgenstern-Price method and Sarma method are the rigorous methods, and the others are simplified methods. Many researchers (e.g., Regmi et al., 2010; Wei et al., 2010; Agama et al., 2016; etc.) prefer the Spencer method among the other rigorous methods because it has a simple inter-slice force function.

175

This study aims to analyze the multiple failure of a slope due to a rainfall event. A 2D seepage-flow model, a 1D surface-flow and erosion/deposition model, a 2D slope stability model (the Spencer method of slope stability analysis), and a 1D sliding block model were combined as a single unit so that the developed model can also successfully analyze the seepage-flow phenomena within the soil domain, the surface water flow and erosion/deposition on the model's slope soil surface, and the stability of the model slope during the movement of the sliding mass by updating the shape of the model slope, according to the new position of the sliding mass, as well. The 1D sliding block model analyzes the motion of the failure mass as a rigid body earth block. The 2D seepage flow model computes the degree of saturation, rate of infiltration, pore-water pressure and moisture content within the body of the considered model slope. The 1D surface-flow and erosion/ deposition model computes the surface water flow and erosion/deposition on the model slope soil surface. The pore-water pressure, moisturecontent, and surface-water head data obtained by these models were used to analyze the stability of the model slope. The Spencer method of slope-stability analysis was incorporated into dynamic programming to locate the critical slip surface and the corresponding factor of safety simultaneously. The computation flow chart of the combined model is presented in Fig. 1. The experimental study conducted by Regmi et al. (2014) was composed of detailed observations of the slope failure process and consisted of a series of successive failures, with particular emphasis on the time of failure; the shape, size and position of the slip surface, and the final shape of the model slope after down-slope displacement of the successive failure mass. Moisture profiles at different points within the model slope soil domain and the surface-water forefront profiles were also observed. The study also noticed a strong correlation between rainfall intensity, sliding initiation time and the position of its slip surface head. Regmi and Jung (2016) applied a dynamic programming technique to locate the critical failure surface in the slope stability analysis for experimental cases of Regmi et al. (2014). The dynamic programming was employed in conjunction with limit-equilibrium techniques. Their article clearly discusses the boundary conditions for such a slope stability problem and describes the calculation procedure for the dynamic programming technique step-by-step. However, the article does not include the physical phenomena, such as the seepage flow, surface runoff, slope stability analysis and movement of the sliding mass, to model the slope failure induced by a rainfall event. The present study clearly illustrates these essential physical phenomena included in the developed numerical simulation model with verification through the data obtained from laboratory flume experiments conducted by Regmi

O

Z

P QR

e

X

W

x 1, z 1

e

xn+1,zn+1 xn , zn

x 2, z 2 ith slice xi, zi xi+1, zi+1 General slip surface

a) Sliding mass and vertically divided slices

QL N

b) Forces acting on a slice

Fig. 2. Two-dimensional general slip surface and forces acting on a typical slice.

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R.K. Regmi et al. / Catena 150 (2017) 173–191

i

of the failure mass was analyzed by the 1D sliding block model, just like that of the rigid body. These numerical models were combined into a single unit (Fig. 1). The simulation model computes the shape of the soil domain at each time step. Hence, it also analyzes the seepage flow and the stability of the soil domain during the movement of the failure mass by updating the shape of the soil domain, according to its position, as well. Furthermore, the simulation model predicts the new slope failure masses with their slip surfaces and time of occurrence in series.

A i+1 1 2 3

1 2 3

j k

1 2 3 4

B

2.1. Seepage flow model

Fig. 3. Typical schematic sample of stages, states and slip surface in a 2D slope.

et al. (2014). They conducted their experiments using three rainfall conditions (50-, 60- and 72-l/min discharges from a water-supply pump). The measured average rainfall inside the flume was 42.12, 55.78 and 108.45 mm/h in 50, 60 and 72-l-per-minute supply pump discharge, respectively. 2. Numerical method A runoff was provided in the flume, upstream of the model slope, to develop a water-saturated zone in the bottom layer of the model slope, which is essential for understanding the slope failure phenomenon. On the surface of a soil slope, it is necessary to identify if surface runoff has occurred during rainfall events or not while analyzing the seepage and stability of the slope. The runoff produced may result in erosion/deposition on the surface. At the same time, the erosion/deposition rate affects the runoff rate and its depth. The depth and position of the runoff has a significant role in the seepage process within the soil domain and, ultimately, in the stability of the slope. In the present study, a 1D surface flow and erosion/deposition model, a 2D seepage flow model, a 2D slope stability model, and a 1D sliding block model were combined as a single unit. In the slope-stability analysis, the Spencer method was coupled with dynamic programming to simultaneously locate the critical slip surface and the corresponding factor of safety. Dynamic programming is a very powerful algorithm design in which an optimization problem is solved by combining sub-problem solutions. Its essential characteristic is the multistage nature of the optimization procedure. It is applicable when the sub-problems are not independent. In such problems there can be many possible solutions. Each solution has a value, and the goal is to find a solution with the optimal value. After a slope failure, the failure mass slides slowly down along a well-defined slip surface as long as the slope gradient is steep enough, and it stops sliding when it arrives at a flatter area. This sliding mass also supports the remaining soil mass so that the remaining mass does not lose its stability immediately. Treating it as a solid block, the motion

The following 2D pressure-based Richards' equation, valid for both saturated and unsaturated soil, was used in the seepage-flow model to calculate the change in pore-water pressure inside the soil domain.

ðC þ Sw Ss Þ

∂hw ∂ ¼ ∂x ∂t




 ∂hw ∂ ∂hw þ Kx Kz þ1 ∂z ∂x ∂z

where x is the horizontal spatial coordinate; z is the vertical spatial coordinate, taken as positive upwards; Kx, and Kz are the hydraulic conductivity in the x and z directions, respectively; C = ∂ θw/∂hw is the specific moisture capacity; θw is the soil volumetric moisture content; hw is the pore-water pressure head; Sw is the saturation ratio; Ss is the specific storage; and t is time. Ss depends on the compressibility of the solid matrix and fluid, so it approaches zero in an unsaturated, unconfined porous medium. To solve Eq. (1), the following constitutive relationships proposed by van Genuchten (1980) were used:  η −m Se ¼ 1 þ jαhw j

ð2Þ

Se ¼

θw −θr θs −θr

C¼

ðno −θr Þηmα jαhw j  η mþ1 1 þ jαhw j

ð3Þ η−1

ð4Þ

h  m i2 K w ¼ K ws Se 0:5 1− 1−Se 1=m

ð5Þ

where Se is the effective saturation; α and η are empirical parameters; θs is the saturated volumetric moisture content; θr is the residual volumetric moisture content; Kws is the saturated hydraulic conductivity; no is the porosity of soil; and m = 1–1/η. Eq. (1) was numerically solved by a line-successive over-relaxation (LSOR) scheme with an implicit iterative finite-difference scheme as used by Freeze (1971, 1978).

a)

b) Unsaturated

ub

Saturated us Slip surface

ð1Þ

θx Xt

X Fig. 4. a) Earth block above the failure surface as an assembly of vertical strips and b) Geometric parameters in a typical strip.

Distance (m)

Pump discharge = 50 ltrs/min Average Rainfall = 42.12mm/hr PR3

0.2

PR2

177

240 225 210 195 180 165 150 135 120 105 90 75 60 45 30 15 0

R.K. Regmi et al. / Catena 150 (2017) 173–191

Side A

PR1

0

4.7

5.2

3.7

4.2

3.2

2.7

2.2

1.7 Side B

Distance (m)

Distance (m) Pump discharge = 60 ltrs/min Average Rainfall = 55.78mm/hr 0.2

PR3

60

PR2

Side A

PR1

0

4.7

5.2

3.7

4.2

3.2

2.7

2.2

1.7

Pump discharge = 72 ltrs/min Average Rainfall = 108.45mm/hr PR3

0.2

PR2

1.2

Side B

Distance (m) Distance (m)

1.2

Side A

PR1

0

4.7

5.2

4.2

3.7

3.2

2.7

Model slope position

2.2

1.7

Distance (m)

1.2

Side B

Position of d/s 4m length flume at horizontal condition Fig. 5. Distribution of rainfall intensity (mm/h) over the rainfall zone (d/s of 4 m in length) of the flume (Regmi et al., 2014).

The continuity equation of the total volume is

2.2. Surface-flow and erosion/deposition model The mathematical model developed by Takahashi and Nakagawa (1994) was used to investigate the surface flow and erosion/deposition on the surface of the model slope. The depth-wise averaged 1D momentum equation for the x-wise (down valley) direction is

∂h ∂M þ ¼ ib fc þ ð1−c Þsb g þ R−I ∂t ∂x

ð7Þ

The continuity equation of the particle fraction is ∂M ∂ðuMÞ ∂ðh þ zb Þ τ þβ ¼ gh sinθb −gh cosθb − b ρT ∂t ∂x ∂x

ð6Þ

∂ðchÞ ∂ðcMÞ þ ¼ ib c ∂t ∂x

ð8Þ

VC3

Rainfall Simulator

Filter mat

Wooden plate

PR 1 PR 2 VC2

PR 3 PC

VC1

Flume

Fig. 6. Experimental setup in Ujigawa Open Laboratory, DPRI, Kyoto University, Japan (Regmi et al., 2014).

Data Logger

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R.K. Regmi et al. / Catena 150 (2017) 173–191

74.64

37.87

13.3 8.23

40º

53.46

35.64

Rods for measuring surface-water forefront 4 3 2 1 10.010.010.010.0

13.3 13.90

35.64

13.3 10.0 9.97

31.27

SR 6

SR 1

27º

SR 5

SR 9

Medium-grained silica sand All dimensions are in cm

Fig. 7. Model slope with the arrangement of SRs and surface water forefront measurement rods (Regmi et al., 2014).

The equation for the change of the bed surface elevation is ∂zb ¼ −ib ∂t

ð9Þ

where M (=uh) is the flow discharge per unit width in the x direction; u is the depth-averaged velocity in the x direction; h is the water depth; g is the gravitational acceleration; β is the momentum correction factor equal to 1.25 for stony-type debris flow (Takahashi et al., 1992) and 1.0 for both an immature debris flow and a turbulent flow; θb is the bed slope; zb is the erosion or deposition thickness measured from the original bed elevation; τb is the bottom shear stresses in the x direction; ρT is the mixture density [ρT = c · σ + (1 − c)ρ]; R is the rainfall intensity; I is the infiltration rate; sb is the degree of saturation in the bed; ib. is the rate of hydraulic erosion or deposition from the flowing water; c is the sediment concentration in the flow; c⁎ is the maximum sediment

concentration in the bed; ρ is the density of water; and σ is the density of the sediment particles. Using the following relationships, s b and I were obtained through seepage analysis: sb = θ w/θs for the top cell. If h N 0, then I = Kws(∂ hw/∂z + 1) of the top cell; otherwise, I = R. The erosion and deposition rates given by Takahashi et al. (1992) are expressed in Eqs. (10) and (11), respectively.



 i σ −ρT tanφ tanφ h pbffiffiffiffiffiffi ¼ K e sinhiθb 1− c −1 hi  −1 ðc∞ −cÞ ρT tanθ tanθ dm gh ð10Þ ib ¼ δd

ðc∞ −cÞ u c

ð11Þ

100

100

Experimental

Matric potential (-m)

80 70

Percent finer

Best fit curve

10

90

60 50 D50=0.326mm 40 30

1

0.1

0.01

20 10

0.001

0

0 0.1

0.1

0.2

0.3

0.4

1

Particle diameter (mm) Fig. 8. Grain size distributions of the sediment (Regmi et al., 2014).

3

3

Water content (m /m ) Fig. 9. Water retention curve for silica sand S6 (Regmi et al., 2014).

0.5

R.K. Regmi et al. / Catena 150 (2017) 173–191

179

Table 1 Experimental program (Regmi et al., 2014). Exp. no.

Case

Exp. type

Pump supply

Avg. rainfall inside flume

Rainfall duration

Measurement

1 2 3 4 5 6

1

Slope stability Seepage Slope stability Seepage Slope stability Seepage

50 l/min

42.12 mm/h

4380 s

60 l/min

55.78 mm/h

3120 s

72 l/min

108.45 mm/h

1180 s

Slip surface, position of failure mass Moisture profiles, surface-water forefront Slip surface, position of failure mass Moisture profiles, surface-water forefront Slip surface, position of failure mass Moisture profiles, surface-water forefront

2 3

where Ke is the parameter of the rate of erosion; θb is the bed slope; ø is the internal friction angle of the bed; tanθ is the gradient of the energy slope; δd is a constant; and c∞ is the equilibrium solids concentration. The finite difference forms of Eqs. (6) to (8) were obtained from the solution methods developed by Nakagawa (1989) using the leapfrog scheme. 2.3. Slope stability model The limit equilibrium method of slices was used for the slope stability analysis. The Spencer method has been incorporated into an effective minimization procedure based on dynamic programming by which the minimal factor of safety and the corresponding critical non-circular slip surface are determined simultaneously. The analysis of the stability of a slope using the limiting equilibrium method of slices necessitates the determination of the critical slip surface, which yields the minimal factor of safety. The method of slices requires the failure soil mass to be divided into n vertical slices, while the slip surface is represented by n + 1 vertices, with the respective coordinates of (x1,z1),···,(xi,zi),···,(xn + 1,zn + 1) (Fig. 2). The mobilized shear force T and the total normal force N on the base of the slice can be expressed as T¼

  1 ce l þ N−up l tanφ F

ð12Þ

  1 N ¼ ðW þ P Þ cosδ− ce l−up l tanφ sinðα e −δÞ =mα F

ð13Þ

where
 tanðα e −δÞ tanφ mα ¼ cosðα e −δÞ 1 þ F

ð14Þ

(a)

F is the safety factor for the given slip surface; ce is the effective cohesion; l is the length of the base of the slice; up is the average pore water pressure on the base of the slice; W is the weight of the slice; P is the vertical external force, i.e., the surface water weight, acting on the top of the slice; αe is the inclination of the base to the horizontal force; and δ is the inclination of inter-slice forces (QL and QR) to the horizontal. The factor of safety expressions for the Spencer method (Bardet and Kapuskar, 1989) is as follows: X   ce l þ ðW þ P Þ cosα e −up l tanφ =mα X Ff ¼ ðW þ P Þ sinα e =mα

Fm

X   ce l þ ðW þ P Þ cosα e −up l tanφ D cosðθe −δÞ=mα X ¼ ðW þ P Þ sinα e D cosðθe −δÞ=mα

ð15Þ

ð16Þ

where Ff and Fm are the safety factors with respect to the force and moment equilibria, respectively (to calculate the value of mα from Eq. (1)4, set F = Ff for Eq. (15) and Fm for Eq. (16)); Ff and Fm are the safety factors with respect to the force and moment equilibria, respectively; D is the distance from the base center of the slice to an arbitrary reference point; and θe is the angle between the vertical direction and the D direction in the xz plane. Ff and Fm can be computed separately from Eqs. (15) and (16) for several appropriately given values of δδ. Then, two curves showing the relationships of Ff − δ and Fm − δ can be plotted so that the intersection of these two curves leads to a required δo value and the corresponding safety factor Fs, satisfying both the force and moment equilibria.

(b) 1 2 3 4 3 2 1

Profile probes Surface-water forefront measuring rods Fig. 10. Pictures showing the flume setup for measuring moisture profiles and surface-water forefront; (a) view from downstream - front view, (b) view from upstream (Regmi et al., 2014).

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R.K. Regmi et al. / Catena 150 (2017) 173–191

Eq. (15) can be represented by the following expression:

Table 2 Some parameter values of the sediment considered (Regmi et al., 2014). Sediment type

S6

Saturated volumetric moisture content, θs Residual volumetric moisture content, θr van Genuchten parameter, α van Genuchten parameter, η Saturated hydraulic conductivity, Ks (cm/s) Specific gravity, Gs Mean grain size, D50 (mm) Angle of repose, ør Porosity, no

0.43 0.03 4.85 5.92 0.03518 2.65 0.326 34° 0.43

The dynamic programming search scheme coupled with Eqs. (15) and (16) for determining the critical 2D surface and minimum values of Ff and Fm is illustrated as follows.

N X

Ff ¼

i¼1 N X

Rfi ð17Þ T fi

i¼1

where,    Rfi ¼ ce l þ ðW þ P Þ cosα e −up l tanφ =mα

ð18Þ

T fi ¼ ðW þ P Þ sinα e =mα

ð19Þ

View from the front

View from side B

Side B

2

1

Time = 2,734 sec 2

Side A Side B

3

Time = 2,767 sec

3

Side A Side B

1

4 4

1

Time = 2,859 sec

Side A Side B

1, 5 5

1

Time = 2,915 sec

Side A Fig. 11. Photos showing a series of slides, Case 1 (contd.) (Regmi et al., 2014). Photos showing a series of slides, Case 1 (Regmi et al., 2014).

R.K. Regmi et al. / Catena 150 (2017) 173–191

View from the front

181

View from side B

Side B

6

Time = 3,292 sec 6

Side A Side B

7

Time = 3,748 sec

7

Side A Side B 8 8

Time = 4,072 sec

Side A Fig. 11 (continued).

When applying dynamic programming, the minimization of G is carried out over all admissible slip surfaces:

Eq. (16) can be represented by the following expression: N X

Fm ¼

i¼1 N X

Rmi ð20Þ

Gs ¼ minG ¼ min

N N X X ½Rmi − F m T mi  ¼ min DGi ð j; kÞ i¼1

T mi

i¼1

where,    Rmi ¼ ce l þ ðW þ P Þ cosα e −up l tanφ D cosðθe −δÞ=mα

ð21Þ

T mi ¼ ðW þ P Þ sinα e D cosðθe −δÞ=mα

ð22Þ

where Gs is the minimum value of function G, yielding the minimum value of Fm. A value of Fm in Eq. (23) is initially assumed and is updated by iteration until a convergence criterion is satisfied. Fig. 3 illustrates the stage-state system for a 2D slope. An arbitrary line jk which connects points (i, j) and (i + 1,k) is considered as a part of an assumed slip surface. Rmi and Tmi on the surface jk are obtained from Eqs. (21) and (22), and the return function is calculated using Eq. (25). G¼

N X

½Rmi −F m T mi  ¼

i¼1

Dynamic programming is applicable only to additive functions (Baker, 1980). To change Fm, defined in Eq. (20) in an additive function, the return function (or auxiliary function) G is defined as follows (Baker, 1980):

G¼

N X i¼1

ð24Þ

i¼1

½Rmi −F m T mi  ¼

N X

DGi ð j; kÞ

ð23Þ

N X

DGi ð j; kÞ

ð25Þ

i¼1

If Hi(j), the optimal value function in dynamic programming, signifies the minimum value of G from point A (Fig. 3) to the point (i, j). Next, based on Bellman's principle of optimality, the minimum G value from A to (i + 1,k) is given by Eq. (26): H iþ1 ðkÞ ¼ min½H i ð jÞ þ DGi ðj; kÞ; i ¼ 1  N; j ¼ 1  Si ; k ¼ 1  Siþ1 ð26Þ

i¼1

The boundary conditions are where DGi(j, k) is the change of G on the base jk, corresponding to the return function.

H 1 ð jÞ ¼ 0; j ¼ 1  S1

ð27Þ

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R.K. Regmi et al. / Catena 150 (2017) 173–191

Gm ¼ minG ¼ min½H nþ1 ð jÞ; j ¼ 1  Siþ1

ð28Þ

The difference between the value of Fm calculated by Eq. (20) after this procedure and the initially assumed value of Fm should be within tolerance; therefore, iteration is required to obtain the exact value of Fm along the slip surface. From Eq. (17), the minimum value for Ff along the slip surface can be obtained.

in the lower part of the earth block, even if it is saturated by water, the equation of motion of a strip is ∂us ¼ g sinθx −μ k g cosθx ∂t The x-wise motion of the sliding mass is X ub ¼

2.4. Sliding block model The motion of a sliding mass was analyzed as a rigid body earth block by using a Lagrangian treatment. The moving earth block was considered an assembly of vertical strips (Fig. 4). A viscous shear stress exists at the liquefied layer of the slip surface of any soil type and even in the slip surface of clayey soil as well. The soil used in this study was medium-grained silica sand. Assuming no liquefied layer was produced

View from the front

ð29Þ

M s us cosθx X Ms

ð30Þ

where θx is the gradient of the strip bottom; us is the strip velocity in the direction parallel to the strip bottom gradient; μk is the kinetic friction coefficient; ub is the x-wise motion of the sliding mass; and Ms. is the mass of the strip. Ms. is described as follows: Ms ¼ ρT X t ðα 1 hsb þ hs Þ

ð31Þ

View from side B

Side B

1 1

2

Time = 2,179 sec

Side A Side B

1 1

3

Time = 2,202 sec

Side A Side B

1 1

4

Time = 2,216 sec

Side A Side B

1 1

5 5

Time = 2,261 sec

Side A Fig. 12. Photos showing a series of slides, Case 2 (contd.) (Regmi et al., 2014). Photos showing a series of slides, Case 2 (Regmi et al., 2014).

R.K. Regmi et al. / Catena 150 (2017) 173–191

View from the front

183

View from side B

Side B

1 6

1, 6

Time = 2,333 sec

Side A Side B

7 7

Time = 2,393 sec

Side A Side B 8 8

Time = 2,533 sec

Side A Side B 9 9

Time = 2,898 sec

Side A Fig. 12 (continued).

where Xt is the strip thickness; hs is the saturated height of the strip; hsb is the unsaturated height of the strip; α1 = {c σ + (1 − c)ρ · sbs}/{c · σ + (1 − c)ρ}; and sbs is the degree of saturation of hsb. The time forward differencing approximation was used for the solution of Eq. (29). 3. Materials and methods In this study, experimental results obtained from Regmi et al. (2014) were used to validate the numerical simulation model. They had carried out their experiments on a 5-m-long, 30-cm-wide and 50-cm-deep rectangular flume set at a 27° slope. The experiments were conducted using three rainfall conditions (50-, 60- and 72-l/min discharges from a water-supply pump). The measured average rainfall inside the flume was 42.12, 55.78 and 108.45 mm/h in 50-, 60- and 72-l-per-minute supply pump discharges, respectively. Fig. 5 presents the contour maps showing the rainfall distribution over the flume for each rainfall condition. The value of rainfall intensity averaged in the lateral direction was applied for the numerical simulation as an input.

Two video cameras were used to capture the activities associated with slope failure. Another video camera was positioned above the flume to capture the surface-water forefront propagation. Three profile probes were used to measure the moisture profile at different locations inside the soil domain. A schematic diagram of the flume, including instrumentation and the data acquisition system, is shown in Figs. 6 and 7. presents the shape and size of the model slope with the arrangement of measurement accessories. The sediment used in the study was medium-grained silica sand produced by crushing the silica stone. It mainly consists of quartz produced by the weathering of granite. Because of the crushed stone, the shape of the sand was angular rather than round. Some parameter values of the sediment used in the experiments are listed in Table 2. The grain-size distributions of the sediment are shown in Figs. 8 and 9. shows the water retention curve for the sediment. Six experiments were carried out at an initial volumetric moisture content of 0.04 for the sediment used. The inclination of the sloping face of the model slope was set to 40° to trigger sudden sliding. The slope of the

184

R.K. Regmi et al. / Catena 150 (2017) 173–191

lower section (toe) was set to 27° (parallel to the flume-slope) to observe the deposition after the sliding. A wooden plate was fixed at the downstream face of the flume to support the soil mass. The experimental program is outlined in Table 1. Four measuring rods (Figs. 7 and 10) were placed in a vertical position on the top surface of the model slope in its central longitudinal section at 10-cm spacing to measure the forefront of the propagated surface-water flow. 4. Results and discussion In the seepage-flow model, the numerical simulation was conducted at 0.02-s time steps, 2.0-cm space steps in the x (horizontal) direction, and 1.0-cm space steps in the z (vertical) direction. In the surfacewater flow and erosion/deposition model, 0.01-s time steps and 2.0/ cosθb-cm space steps in the x direction (parallel to the assumed bed slope of the model slope) were used. Time steps of 1.0 s and space steps of 6 cm in the x direction were used in the slope-stability model.

View from the front

Ridge line

In the sliding block model, 0.02-s time steps and 2.0-cm space steps in the x direction were used. The parameters of the numerical simulation were: Kwx = Kwz = Kw; Kws = 0.0003518 m/s (Regmi et al., 2014); g = 9.81 m/s2; ρow = 1000.0 kg/m3; Ss = 1 × 10−7 m−1 (assumed for the saturated condition of the sediment); c⁎ = 1.0 - no = 0.57; Ke = 0.06 (assumed); δd = 0.03 (assumed); σ = 2650 kg/m3; n = 0.03 (assumed); c = 0.0 (non-cohesive soil); ø = ør = 34°; and μk = 0.541(assumed). Figs. 11 to 13 show the pictures of the observed sliding slope failure in different experimental cases (Regmi et al., 2014). A number of slidings occurred, and their times of occurrence were captured through video cameras. Fig. 14 shows the plots of the slip surfaces observed in experiments by Regmi et al. (2014) and computed by the simulation. The plotted experimental slip surfaces were observed from flume side B. The full-saturation and partial-saturation zones within the model slope at the time of failure and the position of the corresponding failure surface, as per the numerical computation, are presented in Fig. 15.

View from side B

Side B 1

2 2

Time = 1,106 sec Erosion-deposition phenomenon Side A Side B 3

3

Time = 1,126 sec Erosion-deposition phenomenon

1 2

Side A Side B 4 2, 4 1

Time = 1,143 sec Erosion-deposition phenomenon

3 Side A

Side B 5

Time = 1,164 sec Erosion-deposition phenomenon

5

Side A Fig. 13. Photos showing a series of slides, Case 3 (Regmi et al., 2014).

R.K. Regmi et al. / Catena 150 (2017) 173–191

185

Experiment

Simulation

Original topography

Flume bed

Original topography

Slip surface-1 (2,685 sec)

Slip surface-2 (2,689 sec)

Flume bed

Slip surface-3 (2,740 sec)

Slip surface-4 (2,818 sec)

Slip surface-1 (2,795 sec)

Slip surface-5 (2,883 sec)

Slip surface-6 (3,219 sec)

Slip surface-7 (3,438 sec)

Slip surface-8 (3,991 sec)

Slip surface-2 (2,813 sec) Slip surface-3 (2,822 sec)

2

Case1

2.2 2

Case1

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1 5

4.8

4.6

4.4

4.2

4

3.8

3.6

3.4

1 5.4

5.2

5

4.8

Distance (m)

4.6

4.4

4.2

4

3.8

3.6

3.4

Distance (m) Experiment

Simulation

Original topography

Flume bed

Original topography

Flume bed

Slip surface-1 (2,134 sec)

Slip surface-5 (2,242 sec)

Slip surface-1 (2,212 sec)

Slip surface-2 (2,213 sec)

Slip surface-6 (2,314 sec)

Slip surface-7 (2,362 sec)

Slip surface-3 (2,214 sec)

Slip surface-4 (2,218 sec)

Slip surface-8 (2,488 sec)

Slip surface-9 (2,741 sec)

2.2

2.2 2

Case2

2

Case2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1 5.4

5.2

5

4.8

4.6

4.4

4.2

4

3.8

3.6

3.4

1 5.4

5.2

5

4.8

Distance (m)

4.6

4.4

4.2

4

3.8

3.6

3.4

Distance (m)

Experiment

Simulation

Original topography

Flume bed

Original topography

Slip surface-1 (1,064 sec)

Slip surface-2 (1,079 sec)

Flume bed

Slip surface-3 (1,116 sec)

Slip surface-4 (1,130 sec)

Slip surface-1 (992 sec)

Slip surface-5 (1,152 sec)

2.2

2.2

2

Case3

2

Case3

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2 1

1 5.4

5.2

5

4.8

4.6

4.4

4.2

Elevation (m)

5.2

4

3.8

3.6

3.4

Distance (m)

Elevation (m)

5.4

Elevation (m)

2.2

5.4

5.2

5

4.8

4.6

4.4

4.2

4

3.8

3.6

3.4

Distance (m)

Fig. 14. Plots showing experimental (Regmi et al., 2014) and simulated slip surfaces in the original topography.

Generally, the geometric shape of a slip surface is 3D, in which its length and depth on either side are shorter and shallower than the inner part. Although this study aimed to investigate the 2D slope failure mechanisms, the nature of the failure in the experiments was also 3D. It was clearly viewed at slides towards the toe in the pictures taken from the front of the flume (Figs. 11 and 12). In the experimental Case 2, even the slidings 2 and 3 were not extended up to flume sidewall B (Fig. 12). The reason behind it was the non-uniform distribution of the rainfall in a lateral direction (Fig. 5) and the effect of flume sidewall friction. The friction on the sidewall of the flume was ignored in the computation. Additionally, the plot of experimental slip surfaces (Fig. 14) was also

based only on observation from the flume side. Therefore, the simulated slip surfaces were longer and deeper than the experimental one. In all the experimental cases, the slip surface of the initial sliding reached the base of the model slope and appeared to extend up to the downstream face's fixed support (Figs. 11 to 14). Thereby, the entire failure mass above the slip surface was retained by the fixed support. As a result, only a small amount of displacement with very slow movement was observed towards its head. This displacement was merely due to the compaction of the failure mass towards the down-slope from its own weight (Regmi et al., 2014). In the present numerical simulation, the point of exit of a potential slip surface was limited at any point on

R.K. Regmi et al. / Catena 150 (2017) 173–191

1.8

Case 1

2

Time = 2,813sec

1.8

1.6

1.6

Partial saturation Failure surface

1.4

Full saturation Failure surface

1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Case 1

2

Case 2

2

Time = 2,822sec

1.8

Time = 2,212sec

1.8 1.6

1.6 Partial saturation

Partial saturation Full saturation Failure surface

1.4

Full saturation Failure surface

1.2

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Case 2

2

Case 2

2

Time = 2,213sec

1.8

Time = 2,214sec

1.8

1.6

1.6

Partial saturation Full saturation Failure surface

Partial saturation

1.4

Full saturation Failure surface

1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

2

Time = 2,218sec

1.8

1.2

2

Case 3 Time = 992sec

1.8

1.6

1.6

Partial saturation Failure surface

1.4

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Case 2

Full saturation

Elevation (m)

Full saturation

Partial saturation

Elevation (m)

2

Elevation (m)

Case 1 Time = 2,795sec

Partial saturation

1.4

Full saturation Failure surface

1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Distance (m)

1.4

Elevation (m)

186

1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Distance (m)

Fig. 15. Saturated and unsaturated zones at the time of failure and the position of the corresponding failure surface.

the surface of the model slope but not at the downstream face's fixed support. The sliding mass was also assumed as a rigid body earth block. Hence, the initial sliding surface, as per the experiments, was not obtained in the simulation. The present numerical simulation model can only compute the upslope propagation of the successive failures. Therefore, for Case 3, there was no down-slope propagation of the successive failures in the numerical computation. Although seven slidings (except the initial sliding) occurred in the Case 1 experiment, only three slidings were computed by the simulation (Figs. 14 and 15). The locations of simulated slidings 1, 2 and 3 are comparable with the locations of experimental slidings 3, 6 and 8, respectively, even though the position, depth, and size of the sliding mass are different between the experiment and the simulation. In the Case 2 experiment, eight slidings (except the initial sliding) occurred, but only four slidings were computed by the simulation (Figs. 14 and 15). The locations of simulated slidings 1, 2, 3 and 4 are comparable with the locations of experimental slidings 5, 6, 7 and 8, respectively, even though the position, depth, and size of the sliding

mass are different between the experiment and the simulation for Case 2 as well. In the experimental observation for Case 1, sliding numbers 4 and 7 were visualized as a successive localized shallow slide after sliding numbers 3 and 6, respectively. Similarly, in the Case 2 experiment, sliding number 9 was visualized as a successive localized shallow slide after sliding number 8. However, the dynamic programming approach yields the absolute minimum only, disregarding any local minima that may exist. Four slidings (except the initial sliding) occurred in the Case 3 experiment, but only one sliding was computed by the simulation (Figs. 14 and 15). The position of the simulated sliding is comparable with the position of experimental sliding 1. Only one sliding in the numerical computation for Case 3 was due to the limitation of the present numerical simulation model in that it is unable to compute the down-slope propagation of successive failures. Once failure was initiated at the slope toe, the failure of the toe area consequently reduced the lateral support to the up-slope mass. This results in unstable zone propagation to the up-slope portion of the model slope, and it is supposed that the slope failure is easily induced with a

R.K. Regmi et al. / Catena 150 (2017) 173–191

Saturation (%)

series of successive failures. However, it did not propagate rapidly to the up-slope region. In the experiments, there was a large time difference between the successive failures in comparison to simulation data. This is because the entire region within the potential failure surface was not fully saturated (Fig. 15) and still had enough strength to resist added weight of the slope from the rainfall. The strength was still provided by the existence of apparent cohesion due to matric suction of the unsaturated region. However, in the numerical method of slope stability analysis, the increase in shear strength due to the apparent cohesion was not taken into account (Eqs. (15) and (16)). Fig. 16 shows the experimental (Regmi et al., 2014) and simulated moisture profiles at different SRs. The time, t = 0 s, in the figure represents the starting time of the rainfall. The moisture profiles in the simulations and experiments match satisfactorily. Table 3 compares the experimental (Regmi et al., 2014) and simulated surface-water forefront propagation time at different measuring rods. After a failure, the failure mass slides slowly down along a welldefined slip surface. After and during the movement of the failure mass, the geometry of the model slope changed. The profile probes within the failure mass were also displaced together with the failure mass, and their upright firm position was also diminished such that there might be a gap around their excess tube and the existing soil mass. Gaps around the tube will result in generally low readings and poor response to soil moisture changes. Therefore, in the experiments, a sudden drop in moisture profile was observed for the sensors positioned in such a failure soil mass (for SR5, SR6 and SR9 in all the cases; Fig. 16) (Regmi et al., 2014). Although the movement of the failure mass was also simulated by the simulation model, the position of the SRs within the failure was assumed to be stationary such that no sudden drop in moisture profile was obtained in simulation.

100

80

80 60 SIM-SR1 (Case 1) EXP-SR1 (Case 1) SIM-SR1 (Case 2) EXP-SR1 (Case 2) SIM-SR1 (Case 3) EXP-SR1 (Case 3)

40 20 0

SIM-SR5 (Case 1) EXP-SR5 (Case 1) SIM-SR5 (Case 2) EXP-SR5 (Case 2) SIM-SR5 (Case 3) EXP-SR5 (Case 3)

40 20 0

0

Saturation (%)

Additionally, for SR5 and SR6 in Case 3, there was no moisture profile after the sliding of the slope. This is because there was no soil mass at their position after the movement of the soil mass. In addition to the rainfall intensity and its duration, the other conditions were the same for all the considered cases. The higher the rainfall intensity, the greater the movement of the lateral inflow was in creating a water table from the head reach to the toe. Additionally, the rise in the water table was faster than a lower intensity case. Figs. 14 and 17 clearly show these phenomena (the color blue indicates the full saturation soil layer). Due to a rainfall event, the surface runoff and infiltration phenomena of a particular soil are directly connected to the physics of the soil-water movement of that particular soil. The permeability function and the moisture retention function for an unsaturated soil are related to the soil-water characteristic curve. Therefore, these functions can be estimated from the soil-water characteristic curve. Because the moisture retention function and hydraulic conductivity function (Eqs. (2) to (5)) are controlled by the soil-water retention curve (Fig. 9), the rate of increase in the movement of the lateral inflow was lower than the rate of increase in the rise of the water table due to lateral inflow, while the rainfall intensity increased (Figs. 15 and 17). Therefore, the rise in the water table in the head reach of the model slope was faster than the rise of the water table in its downstream reach area (towards the toe), while the rainfall intensity increased (Figs. 15 and 17). Consequently, the surface-water forefront propagation was directly related to the rainfall amount: the higher the rainfall intensity, the faster the surfacewater movement (Tables 1 and 3). For Case 1, the surface-water forefront did not even reach measuring rod 4. Alternately, in the experiment for Case 3, the surface-water forefront crossed the ridge line, and it produced erosion-deposition phenomenon on the steep slope just after the ridge line (Fig. 13). The erosion-deposition was observed just before the

100

60

187

1000

2000

3000

4000

0

100

100

80

80

1000

2000

3000

4000

60

60 SIM-SR6 (Case 1) EXP-SR6 (Case 1) SIM-SR6 (Case 2) EXP-SR6 (Case 2) SIM-SR6 (Case 3) EXP-SR6 (Case 3)

40 20

SIM-SR9 (Case 1) EXP-SR9 (Case 1) SIM-SR9 (Case 2) EXP-SR9 (Case 2) SIM-SR9 (Case 3) EXP-SR9 (Case 3)

40 20 0

0 0

1000

2000

Time (sec)

3000

4000

0

1000

2000

3000

4000

Time (sec)

Fig. 16. Comparison of experimental (Regmi et al., 2014) and simulated moisture profiles (plot for the same sensor in different experimental cases).

188

R.K. Regmi et al. / Catena 150 (2017) 173–191

Table 3 Surface water forefront propagated time to measuring rods. Case 1

Case 2

Case 3

Measuring rod

Experiment

Simulation

Experiment

Simulation

Experiment

Simulation

1 2 3 4

308 s 3197 s 3641 s –

338 s 3262 s 3737 s –

148 s 558 s 1789 s 2127 s

181 s 726 s 1964 s 2306 s

59 s 132 s 235 s 412 s

64 s 161 s 290 s 463 s

1.8

0

10

20

30

40

50

70

60

1.8

Time = 1,000sec

1.6

Experiment

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

1.2 3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

2

Case 2

2

Case 2

1.8

Time = 500sec

1.8

Time = 1,000sec 1.6

Experiment

1.6

Experiment

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

1.2

2

2

Case 3 1.8

Time = 500sec

1.8

Time = 1,000sec

1.6

1.6

Experiment

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Distance (m)

1.4

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Case 3

Experiment

1.4

Elevation (m)

Experiment

1.6

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Distance (m)

Fig. 17. Temporal variation of the degree of saturation (contd.). Temporal variation of the degree of saturation.

Elevation (m)

Time = 500sec

2

Case 1

Elevation (m)

2

Case 1

80

90

2) and was sufficient for the instability of the model slope (Figs. 14 and 15). Therefore, the position of the initial slip surface head was closer to the head reach of the model slope with its larger failure mass size in the case with higher rainfall intensity rather than in the case with lower rainfall intensity (Table 1 and Figs. 11 to 14). Obviously, failure time was also faster for the higher rainfall intensity experimental cases (Fig. 14). Shear failure along a slip surface is associated with a decrease in the shearing resistance. Therefore, the sliding mass moves at an accelerated rate during its first phase of sliding. The motion of the sliding mass continues as long as the slope gradient is steep enough and liquefaction at the boundary between the sliding earth mass and the ground continues (Takahashi, 2007). As the slide proceeds, the resistance force that tends to maintain the sliding movement decreases because the mass enters increasingly stable positions. Therefore, the accelerated movement

0

10

20

30

40

50

60

70

80

90

initiation of the sliding. Conversely, in the simulation, the surface-water forefront moved slightly slower than in the experiment (Table 3), and the failure was also faster than in the experiment (Fig. 14). Therefore, it did not cross the ridge line before the failure of the model slope. In the lower rainfall intensity cases (Cases 1 and 2), the full saturation region in the head reach was smaller, which was insufficient for the instability of the model slope. The instability occurred when the full saturation was created at the toe area. Therefore, the position of the initial slip surface head was closer to the toe area (Fig. 15). Additionally, the time of the failure was too late in the case with the least rainfall intensity (Fig. 14) because of the slower movement of the lateral inflow, which is necessary to create full saturation in the toe. In the higher rainfall intensity case (Case 3), the full saturation region in the head reach was higher than the lower rainfall intensity cases (Case 1 and Case

2

Case 1

2

Case 2 1.8

Time = 1,500sec

1.8

Time = 1,500sec 1.6

Experiment

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

1.2 3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

2

2

Case 2 1.8

Time = 2,000sec

1.8

Time = 2,000sec 1.6

Experiment

1.6

Experiment

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

2

Case 1

2

Case 2 1.8

Time = 2,500sec

1.8

Time = 2,500sec 1.6

Experiment

Elevation (m)

Case 1

1.4

1.6

Experiment

1.4 1.2

3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

1.4

Elevation (m)

Experiment

1.6

Elevation (m)

0

10

20

30

40

50

60

189

70

80

90

0

10

20

30

40

50

60

70

80

90

R.K. Regmi et al. / Catena 150 (2017) 173–191

1.2 3.469 5.3 3.669 5.1 3.869 4.7 4.269 4.5 4.469 4.3 4.669 4.1 4.869 3.9 5.069 3.7 5.269 3.5 4.9 4.069

Distance (m)

Distance (m) Fig. 17 (continued).

changes into a retarded one, and finally it stops. Fig. 18 shows the simulated sliding mass movement pattern from the initial time of a particular sliding to just the time at which it stops. The plots represent the movement pattern of slidings 3, 4 and 1 for Case 1, Case 2 and Case 3, respectively. Fig. 19 compares the final shape of the model shape observed from the experiments and computed from simulations. Although a liquefied layer is produced in the slip surface along the saturated soil layer, this study assumes no liquefied layer existence as in the slip surface along the unsaturated soil layer and simply addresses a frictional shear stress along the slip surface instead of viscous shear stress. However, the final shape in the simulations and experiments are comparable. Hence, the improvement of the present sliding block model necessitates introducing the viscous shear stress in the slip surface along the saturated soil layer that produces liquefaction. Fig. 19 also includes the position of the surface water computed by the numerical simulation at the end of rainfall. In the case with higher rainfall intensity, the size of slip surfaces was comparatively larger and had a longer steep gradient portion than the cases with lower rainfall intensity. Therefore, the position of the failure mass deposition was also more concentrated towards the flume at the downstream end of the reach in the higher rainfall intensity case in comparison to the lower rainfall intensity case. In Case 1, the forefront of the deposited failure mass was ahead of the downstream end of the model slope. In Case 2, the failure mass was just initiated to cause outflow from the

downstream end, whereas the failure mass had already initiated an outflow in Case 3. The study conducted by Tsutsumi and Fujita (2008) confirmed that landslide prediction methods must be improved by incorporating various important factors, such as subsurface geomorphology, occurrence of multiple slides, and rainfall characteristics, in simulation models. However, in their multiple slope failure simulation, they did not consider the presence of previous failure masses in the seepage and slope stability analyses while computing the next failure, which is not a reality in the actual cases. The experimental observations of Regmi et al. (2014) also clearly show the presence of previous failure masses at the time of next failure. However, the simulation results obtained in the present study indicate that the simulation model successfully analyzes the seepage flow and stability of the slope to compute multiple failures, updating the shape of the model slope. It is noteworthy that it also analyzes the surface flow, erosion/deposition, and movement of the failure masses, together with the seepage flow and slope stability analyses. Alternately, other studies related to rainfall-induced slope failure (e.g., Regmi et al., 2010, 2012) only computed the failure surface for a single failure. The studies also performed laboratory flume experiments and simulation to analyze the seepage flow and stability of the slope with surface flow and erosion/deposition. Similarly, the numerical models developed by Leroueil (2001) and Calvello et al. (2008, 2009) can only predict the movement of landslide masses along pre-existing

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2.2

Case 1

Sliding 3

2,823.00

Case1

2 1.8

2,822.60

Flume bed Original topography Final topography (Sim) Final topography (Exp) Surface water (Sim)

2,822.40 2,822.20

1.4 1.2 1

2,822.00

5.4 2,821.80 4.33

4.34

4.35

4.36

4.37

4.38

5.2

5

4.8

4.6

4.4

4.2

4

3.8

3.6

3.4

Distance (m)

4.39

2.2

Distance (m)

2

Case2

2,219.40

1.8

Sliding 4

2,219.00 2,218.80 2,218.60 2,218.40

5.4

2,218.20

5.2

5

4.8

4.6

4.4

4.2

1.6 Flume bed Original topography 1.4 Final topography (Sim) Final topography (Exp) 1.2 Surface water (Sim) 1 4 3.8 3.6 3.4

Distance (m)

2,218.00 2,217.80 4.03

2.2 4.04

4.05

4.06

4.07

4.08

4.09

4.1

4.11

2

Case3

Distance (m)

1.8 993.80

Case 3

993.60

Flume bed Original shape Final topography (Sim) Final topography (Exp) Surface water (Sim)

Sliding 1

993.40 993.20

Time (sec)

Elevation (m)

Case 2

2,219.20

Time (sec)

1.6

993.00

1.6 1.4

Elevation (m)

Time (sec)

2,822.80

Elevation (m)

2,823.20

1.2 1

992.80

5.4

992.60

5.2

5

4.8

4.6

4.4

4.2

4

3.8

3.6

3.4

Distance (m)

992.40 992.20

Fig. 19. Final shape of the model slopes after stopping the rainfall.

992.00 991.80 3.9

4

4.1

4.2

4.3

4.4

4.5

Distance (m) Fig. 18. Sliding mass movement pattern in simulation.

slip surfaces. Considering factors such as creep, fatigue, destructuration, partial saturation and infiltration, Leroueil (2001) emphasized the brittleness of soils and its practical implications for the progressive failure developing at the pre-failure stage and on the characteristics of postfailure movements. The model proposed by Calvello et al. (2008, 2009) consists of a physically based transient groundwater analysis of slope and a kinematic analysis of rate of movement and the inverse of the local factor of safety at a critical point along the slip surface. 5. Conclusions This study conducted an experimental and numerical analysis to investigate the mechanism of multiple slope failure due to rainfall events. A 2D seepage-flow model, a 1D surface-flow and erosion/deposition model, a 2D slope stability model (the Spencer method of slope stability analysis), and a 1D sliding block model were combined into a single unit

so that the developed model can also successfully analyze the surface water flow and erosion/deposition on the model slope soil surface, seepage-flow phenomena within the soil domain, and stability of the model slope during the movement of the sliding mass by updating the shape of the model slope, according to the new position of the sliding mass, as well. Furthermore, the model can predict the new sliding masses with their slip surfaces and corresponding times of occurrence in series. In the numerical simulation, the point of exit of a potential slip surface was limited at any point on the surface of the model slope but not at the downstream face's fixed support. Therefore, the initial sliding surface that reached the base of the model slope and appeared to extend up to the downstream face's fixed support in the experiments was not obtained in the simulation. Additionally, the numerical model cannot simulate the down-slope propagation of the successive failures and localized shallow slide. However, the results obtained from the numerical simulation are comparable to the data obtained from the experiments. Further studies based on 3D numerical analysis with a modified simulation model to simulate both the up-slope and down-slope propagation of the successive failures are necessary to more realistically reproduce the sliding failures of a slope in series.

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Acknowledgment This research was supported by a grant (11-TI-C06) from the Advanced Water Management Research Program funded by the Ministry of Land, Infrastructure and Transport of the Korean government.

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