# Coupled Infiltration and Kinematic-Wave Runoff Simulation in Slopes: Implications for Slope Stability

^{*}

## Abstract

**:**

## 1. Introduction

^{3}), respectively. The stability of the slope decreases as the thickness of saturation $z$ increases. The degree of stability is measured by the factor of safety (FS) of the slope soil, which equals the ratio of the forces resisting sliding to the forces driving sliding. A slope with an FS value larger than 1 is stable, it is at limiting equilibrium when FS = 1, and it fails if FS < 1. It is known (see, e.g., [7]) that the FS of the long slope in Figure 1 associated with effective stress analysis is given by the following equation:

## 2. The Generic Layout of a Slope Subjected to Rainfall that May Undergo Translational Sliding

## 3. Kinematic-Wave Runoff Affected by Rainfall and Infiltration

^{3}/s per unit width of slope) at location $x$ and time $t$ is given by Manning’s equation:

## 4. Infiltration on a Slope

#### 4.1. The Elapsed Time of Rainfall Required to Initiate Runoff on a Slope

#### 4.2. Infiltration after Runoff Formation

## 5. Numerical Solution of the Coupled Infiltration and Runoff Equations

#### 5.1. Finite-Difference Discretization of the Runoff Equation

^{3}/s)/m) :

#### 5.2. Finite-Difference Discretization of the Infiltration Equation

#### 5.3. Explicit Solution Approach to the Runoff and Infiltration Equations

## 6. Translational Stability Analysis of Long Slopes

## 7. Results and Discussion

#### 7.1. Data Input for Runoff Simulation

#### 7.2. Data Input for Simulation of Infiltration

#### 7.3. The Elapsed Time (${t}_{P}$) of Rainfall Required to Initiate Runoff on the Slope

#### 7.4. Calculated Depth of Runoff and Depth of Infiltration

#### 7.5. Slope Stability Analysis

## Author Contributions

## Conflicts of Interest

## References

- Cedegreen, H.R. Seepage, Drainage, and Flow Nets; John Wiley & Sons: New York, NY, USA, 1989. [Google Scholar]
- Anderson, S.A.; Sitar, N. Analysis of rainfall-induced debris flows. J. Geotech. Eng.
**1995**, 121, 544–552. [Google Scholar] [CrossRef] - Iverson, R.M. Landslide triggering by rain infiltration. Water Resour. Res.
**2000**, 36, 1897–1910. [Google Scholar] [CrossRef] - Wang, G.; Sassa, K. Factors affecting rainfall-induced flowslides in laboratory flumes tests. Géotechnique
**2001**, 51, 587–599. [Google Scholar] [CrossRef] - Duncan, J.M.; Wright, S.G.; Brandon, T.L. Soil Strength and Slope Stability; John Wiley & Sons: Hoboken, NJ, USA, 2014. [Google Scholar]
- Loáiciga, H.A. Steady-state phreatic surfaces in sloping aquifers. Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef] - Loáiciga, H.A. Groundwater and Earthquakes: Screening Analyses for Slope Stability. Eng. Geol.
**2015**, 193, 276–287. [Google Scholar] [CrossRef] - Morgenstern, N.R.; Price, V.E. The analysis of the stability of general slip surfaces. Géotechnique
**1965**, 15, 79–93. [Google Scholar] [CrossRef] - Spencer, E. A method of analysis of the stability of embankments assuming parallel inter-slice forces. Géotechnique
**1967**, 17, 11–26. [Google Scholar] [CrossRef] - Griffiths, D.V.; Marquez, R.M. Three-dimensional slope stability by elasto-plastic finite elements. Géotechnique
**2007**, 57, 537–546. [Google Scholar] [CrossRef] - Green, W.H.; Ampt, G.A. Studies on soil physics, part I: The flow of air and water through soils. J. Agric. Sci.
**1911**, 4, 1–24. [Google Scholar] - Loáiciga, H.A.; Huang, A. Ponding analysis with Green-and-Ampt infiltration. J. Hydrol. Eng.
**2007**, 12, 109–112. [Google Scholar] [CrossRef] - Chow, V.T. Open-Channel Flow; McGraw-Hill Kogakusha Ltd.: Tokyo, Japan, 1959. [Google Scholar]
- Cunge, J.A.; Holly, F.M., Jr.; Verwey, A. Practical Aspects of Computational River Hydraulics; Pitman Publishing Ltd.: London, UK, 1980. [Google Scholar]
- Chaudry, H.C. Open-Channel Hydraulics; Prentice Hall: Upper Saddle River, NJ, USA, 1993. [Google Scholar]
- Rawls, W.J.; Brakensiek, D.L. A procedure to predict Green and Ampt infiltration parameters. In Advances in Infiltration; America Society of Agricultural Engineering: St. Joseph, MI, USA, 1983; pp. 102–112. [Google Scholar]
- Rawls, W.J.; Ahuja, L.R.; Bakensiek, D.L.; Shirmohammadi, A. Chapter 5: Infiltration and soil water movement. In Handbook of Hydrology; Maidment, D.R., Ed.; McGraw-Hill: New York, NY, USA, 1992. [Google Scholar]
- Dingman, S.L. Physical Hydrology; Waveland Press: Long Grove, IL, USA, 2015. [Google Scholar]
- Hydrologic Engineering Center. Hydrologic Modeling System HEC-HMS: Technical Reference Manual; United States Corps of Engineers: Davis, CA, USA, 2000. [Google Scholar]
- Holtz, R.D.; Kovacs, W.D.; Sheahan, T.C. An Introduction to Geotechnical Engineering; Pearson Education Inc.: Upper Saddle River, NJ, USA, 2011. [Google Scholar]

**Figure 1.**Long slope (elevation view) showing some elements of a translational slide. Not drawn to scale.

**Figure 2.**Generic elevation view of a slope subjected to rainfall and infiltration susceptible to translational sliding. Not drawn to scale.

**Figure 3.**Schematic of the infiltration process. The pressure head at point $x$ is approximately equal to $d=y\xb7{\mathrm{cos}}^{2}\beta $. Elevation view not drawn to scale.

**Figure 5.**Diagram of a landslide caused by an advancing wetting front. Elevation view not drawn to scale.

**Figure 9.**Calculated runoff depth ($y$) as a function of slope station ($k$ ) and time ${t}^{\prime}=t-{t}_{P}$.

**Figure 10.**Calculated infiltration depth (${z}_{f}$) as a function of slope station ($k$ ) and time ${t}^{\prime}=t-{t}_{P}$.

Slope ($\mathit{S}$) | Roughness Csdoeff. ($\mathit{N}$) | Length ($\mathit{L}$, m) | Width ($\mathit{b}$, m) | Rainfall ($\mathit{w}$, m s^{−1}) |
---|---|---|---|---|

1/3 ($\beta =18.43\xb0)$ | 0.20 | 270 | 50 | Variable with time (see Figure 7) |

Porosity ($\mathit{n}$) | Volumetric Water Content (${\mathit{v}}_{0}$) | Wetting-Front Water Tension (${\mathit{\psi}}_{\mathit{f}}$, m) | Hydraulic Conductivity (${\mathit{K}}_{\mathit{s}\mathit{a}\mathit{t}}$, m s^{−1}) |
---|---|---|---|

0.30 | 0.10 | 0.10 | 1.39 × 10^{−}^{6} |

Unit Weight ($\mathit{\gamma}$, kN/m^{3}) | Unit Weight (${\mathit{\gamma}}_{\mathit{s}\mathit{a}\mathit{t}}$, kN/m^{3}) | Slope Angle ($\mathit{\beta}$, $\xb0$) | Friction Angle (${\mathit{\varphi}}^{\prime}$, $\xb0$) | Antecedent Cohesion * (${\mathit{c}}_{\mathit{a}\mathit{c}}^{\text{'}}$, kN/m^{2}) | Infiltration-Reduced Cohesion * (${\mathit{c}}_{\mathit{r}}^{\text{'}}$, kN/m^{2}) |
---|---|---|---|---|---|

19 | 20 | 18.43 | 16 | 9.8 | 0.49 |

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**MDPI and ACS Style**

Johnson, J.M.; Loáiciga, H.A.
Coupled Infiltration and Kinematic-Wave Runoff Simulation in Slopes: Implications for Slope Stability. *Water* **2017**, *9*, 327.
https://doi.org/10.3390/w9050327

**AMA Style**

Johnson JM, Loáiciga HA.
Coupled Infiltration and Kinematic-Wave Runoff Simulation in Slopes: Implications for Slope Stability. *Water*. 2017; 9(5):327.
https://doi.org/10.3390/w9050327

**Chicago/Turabian Style**

Johnson, J. Michael, and Hugo A. Loáiciga.
2017. "Coupled Infiltration and Kinematic-Wave Runoff Simulation in Slopes: Implications for Slope Stability" *Water* 9, no. 5: 327.
https://doi.org/10.3390/w9050327