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

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Method and Materials

#### 2.1. Mathematical Basis

#### 2.1.1. Hydrological Module

#### Kinematic Wave Equation

#### Infiltration Equation

#### 2.1.2. Soil Failure Module (Slope Stability Analysis)

#### 2.1.3. Computational Procedure

#### 2.2. Description of the Study Case

## 3. Results and Discussions

^{2}/s over 12 h and (2) the same rainfall conditions but without surface inflow. The results of FS, accumulated infiltration and pore pressures are presented in Figure 5 and Figure 6. In Figure 5a, when the inflow is considered, the FS decreases faster and a landslide is triggered at the 9th hour; however, in the condition wherein the inflow is not considered, the FS is always larger than 1.0. The lowest FS of the two cases are 1.03 and 0.73, respectively. The variation in accumulated infiltration is presented in Figure 5b, indicating that, in the case considering inflow, the infiltration is higher and the difference of the two cases increases with time. In the case without considering the inflow, the final accumulated infiltration is 34,985 cm

^{2}, however, in the case that considers the inflow, it is 48,769 cm

^{2}, i.e., 1.4 times the other case. Figure 6 presents the pore pressures of the two cases. The pore pressures in Figure 6c,d increase faster than those in Figure 6a,b, that is, they are more likely to cause landslide. The abovementioned results indicate that, firstly, the hillslope inflow may increase the slope instability, and the phenomenon is more likely to occur on footslopes than on upper slopes due to the possibility of runoff. In addition, from the numerical modeling perspective, the mechanism considering the runoff is crucial for these footslopes.

#### 3.1. Magnitude of Surface Inflow

^{2}/s, 0.00002 m

^{2}/s, 0.00004 m

^{2}/s, and 0.0001 m

^{2}/s) were selected in this study to analyze the change in the FS with time. These values are in a reasonable range as a result of estimating the length of upper slope are about tens to hundreds of meters. In addition, the slope is subjected to a uniform rainfall event (200 mm, 12 h). As presented in Figure 7a, a landslide is triggered in all cases, except for the one with the smallest inflow. The times at which landslides are triggered is 11.7, 9.5, and 9 h for the cases with 0.00002, 0.00004, and 0.0001 m

^{2}/s inflow, respectively. In addition, although these cases all exhibit that the lowest FS appears at the same time (12th h), the time at which the landslides are triggered differ, depending on the magnitude of the inflow. Figure 7b presents the variance in accumulated infiltration with time, and reveal that larger inflow leads to more accumulated infiltration. In the tests, the accumulated infiltrations are 30,638, 36,304, 45,383, and 48,390 cm

^{2}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 that the magnitude of inflow accelerates the wetting front propagating downward. Therefore, a larger inflow increases the probability of triggering a slope failure and may bring forward the trigger time.

#### 3.2. Duration

^{2}/s for 3 h, (ii) Scenario 2: 0.00008 m

^{2}/s for 6 h, and (iii) Scenario 3: 0.00004 m

^{2}/s for 12 h. The designs were used to observe whether an intense, short inflow affect the FS more, or a mild, longterm inflow does. We firstly analyzed the variance in FS with time, as presented in Figure 9a. In the first 3 h, the differences in FS among three scenarios are small; the values of FS are 1.13, 1.14, 1.15, respectively, for Scenario 1, 2, and 3, and Scenario 1 has the lowest FS. From 3 to 6 h, Scenario 2 yields the lowest FS. Similarly, because the surface flow increases the infiltration rate, Scenario 1 has a higher accumulated infiltration in the first 3 h (Figure 9b). However, after the runoff stops, for example, at the 4th h, the infiltration of Scenario 1 is still higher than that of Scenario 3, but the FS rebounds from 1.13 to 1.192 in a short time and exhibits a different trend than the infiltration, due to surface pressure dissipation. Figure 10 presents the results for the pore pressure that explains the rebound of FS. Between the 3rd h (Figure 10a) and the 4th h (Figure 10b), the surface pressure of Scenario 1 changes from 0 to −30 cm, whereas those of Scenarios 3 and 2 remain at 0 (Figure 10c,d). The results emphasize the effect of continuous inflow on the stability, and long-term duration is more likely to trigger a landslide.

#### 3.3. Delayed Surface Inflow

#### 3.4. Shape of Inflow

^{2}/s, Figure 15b indicates the possibility that only Type 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.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

List of Symbols | |

b | Slice width |

${c}^{\prime}$ | Effective cohesion |

$f$ | Infiltrate rate |

${h}_{s}$ | Depth of slope surface flow |

$i$ | Rainfall intensity |

n | Manning’s roughness coefficient. |

${S}_{0}$ | Bed slope |

$t$ | Time |

u_{a} | Pore air pressure |

u_{w} | Pore water pressure |

${K}_{x}$ | Hydraulic conductivities in the x-direction |

${K}_{z}$ | Hydraulic conductivities in the z-direction |

M | Fitting parameter |

N | Fitting parameter |

$S$ | Degree of saturation |

$X$ | Distance downslope |

$\alpha $ | Slope angle |

$\gamma $ | unit weight of soil |

${\gamma}_{w}$ | unit weight of water |

$\xi $ | Fitting parameter |

$\sigma $ | Normal effective stress |

$\theta $ | Moisture content |

${\theta}_{r}$ | Residual moisture content |

${\theta}_{s}$ | Saturated moisture content |

${\varphi}^{\prime}$ | Effective friction angle |

${\varphi}^{b}$ | Friction angle with respect to the matric suction |

$\psi $ | Pore water pressure head |

${\psi}_{c}$ | Positive pore water pressure head |

${\psi}_{p}$ | Negative pore water pressure head |

## References

- Sidle, C.R.; Greco, R.; Bogaard, T. Overview of Landslide Hydrology. Water
**2019**, 11, 148. [Google Scholar] [CrossRef] [Green Version] - Bogaard, T.A.; Greco, R. Landslide Hydrology: From Hydrology to Pore Pressure. WIREs Water
**2016**, 3, 439–459. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Pujades, E.; Jurado, A.; Carrera, J.; Vázquez-Suñé, E.; Dassargues, A. Hydrogeological Assessment of Non-Linear Underground Enclosures. Eng. Geol.
**2016**, 207, 91–102. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - 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**. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Iverson, R.M. Landslide triggering by rain infiltration. Water Resour. Res.
**2000**, 36, 1897–1910. [Google Scholar] [CrossRef] [Green Version] - 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.
- Tsai, T.-L.; Yang, J.-C. Modeling of rainfall-triggered shallow landslide. Environ. Geol.
**2006**, 50, 525–534. [Google Scholar] [CrossRef] - 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. [Google Scholar]
- Collins, B.D.; Znidarcic, D. Stability Analyses of Rainfall Induced Landslides. J. Geotech. Geoenviron. Eng.
**2004**, 130, 362–372. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Fredlund, D.G.; Morgenstern, N.R.; Widger, R.A. The shear strength of unsaturated soils. Can. Geotech. J.
**1978**, 15, 313–321. [Google Scholar] [CrossRef] - Tsai, T.-L. The influence of rainstorm pattern on shallow landslide. Environ. Geol.
**2008**, 53, 1563–1569. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Chen, H.-E.; Tsai, T.-L.; Yang, J.-C. Threshold of slope instability induced by rainfall and lateral flow. Water
**2017**, 9, 722. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] [Green Version] - Wysocki, D.A.; Schoeneberger, P.J.; LaGarry, H.E. Geomorphology of soil landscapes. Handb. Soil Sci.
**2000**, 1, 315–321. [Google Scholar] - 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. [Google Scholar] [CrossRef] [Green Version] - Henderson, F.M. Open-Channel Flow; Macmillan: New York, NY, USA, 1966. [Google Scholar]
- 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. [Google Scholar]
- Miller, J.E. Basic Concepts of Kinematic-Wave Models (No. 1302); US Geological Survey: Reston, VA, USA, 1984.
- 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. [Google Scholar] [CrossRef] - Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill: New York, NY, USA, 1988. [Google Scholar]
- Richards, L.A. Capillary conduction of liquids through porous mediums. Physics
**1931**, 1, 318–333. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] [Green Version] - Bishop, A.W. The use of pore-pressure coefficients in practice. Geotechnique
**1954**, 4, 148–152. [Google Scholar] [CrossRef] - 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. [Google Scholar]
- 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. [Google Scholar] [CrossRef] - Subramanya, K. Engineering Hydrology, 4e; Tata McGraw-Hill Education: New York, NY, USA, 2013. [Google Scholar]

**Figure 5.**The variance in (

**a**) Factor of safety and (

**b**) Accumulated infiltration with time for condition considering inflow and without inflow.

**Figure 6.**Pressure head contours for rainfall of 25 mm/h over 12 h and no inflow at the time: (

**a**) 6th hour and (

**b**) 12th hour; Pressure head contours for the same rainfall and considered hillslope inflow at the time: (

**c**) 6th hour and (

**d**) 12th hour.

**Figure 8.**Pressure head contours at 12th hour with (

**a**) 0.00001 m

^{2}/s, (

**b**) 0.00002 m

^{2}/s, (

**c**) 0.00004 m

^{2}/s, and (

**d**) 0.0001 m

^{2}/s surface inflow rate.

**Figure 9.**The variance in: (

**a**) factor of safety, and (

**b**) accumulated infiltration, for different inflow durations.

**Figure 10.**Pressure head contours for various inflow durations: (

**a**) Scenario 1 at 3rd hour, (

**b**) Scenario 1 at 4th hour, (

**c**) Scenario 2 at 4th hour, (

**d**) Scenario 3 at 4th hour, (

**e**) Scenario 2 at 12th hour, and (

**f**) Scenario 3 at 12th hour.

**Figure 12.**The different development of factor of safety with time in the tests of different delay times.

**Figure 13.**Pressure head contours in the tests of different delay times: (

**a**) Delay of 4 h at 12th hour and (

**b**) delay of 4 h at 16th hour.

**Figure 15.**Factor of safety for various inflow hydrograph types. (

**a**) Maximum inflow rate of 0.0002 m

^{2}/s and (

**b**) maximum inflow rate of 0.00008 m

^{2}/s.

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**MDPI and ACS Style**

Chen, H.-E.; Chiu, Y.-Y.; Tsai, T.-L.; Yang, J.-C.
Effect of Rainfall, Runoff and Infiltration Processes on the Stability of Footslopes. *Water* **2020**, *12*, 1229.
https://doi.org/10.3390/w12051229

**AMA Style**

Chen H-E, Chiu Y-Y, Tsai T-L, Yang J-C.
Effect of Rainfall, Runoff and Infiltration Processes on the Stability of Footslopes. *Water*. 2020; 12(5):1229.
https://doi.org/10.3390/w12051229

**Chicago/Turabian Style**

Chen, Hung-En, Yen-Yu Chiu, Tung-Lin Tsai, and Jinn-Chuang Yang.
2020. "Effect of Rainfall, Runoff and Infiltration Processes on the Stability of Footslopes" *Water* 12, no. 5: 1229.
https://doi.org/10.3390/w12051229