# The Key Parameters Involved in a Rainfall-Triggered Landslide

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### Numerical Model

_{x}and k

_{y}are the hydraulic conductivities in directions x and y, respectively, and θ

_{w}is the volumetric moisture content. Both the permeability coefficient and volumetric water content versus suction are important entry parameters when infiltration is modeled. Q is the precipitation intensity applied.

_{w}is the suction and mw can be considered constant for a given time during a transient process. Substituting Equation 2 into Equation 1 gives the differential equation governing the flow of water in unsaturated media:

_{w}is the slope of the soil retention curve, which can be determined experimentally [6].

_{s}for saturated), four shape variables are needed for correct modeling: a

_{f}, n

_{f}, m

_{f}and h

_{r}. Each one is responsible for a different shape, which also affects the air entry value (AEV) of the curve. The AEV indicates the maximum suction value at which the soil is still saturated. The four parameters are detailed as follows:

_{f}= parameter related to the air entry value of the soil;

_{f}= parameter function of the rate of water extraction from the soil once the AEV has been exceeded;

_{f}= parameter function of the residual water content;

_{r}= suction value at residual water content (kPa).

_{f}, m

_{f}, n

_{f}and h

_{r}), other mandatory entry parameters are needed to feed the numerical model. These parameters include slope geometry (declivity) and mechanical material properties, such as friction angle, unsaturated friction angle, cohesion and specific weight. The initial pore water pressure condition is also required, as well as boundary conditions related to the soil–air interface (precipitation) and related to lateral model limits.

_{b}), as the matric suction in unsaturated soil affects the shear strength.

_{a}and u

_{w}represent pore air and pore water pressure, respectively, ϕ′ is the effective frictional angle and ϕ

_{b}is the angle that defines the increase in shear strength due to negative pore water pressure. The angle ϕ

_{b}is a material property, and was also assessed in the parametric/sensitivity study. It can range from zero to ϕ′ depending on the material type. A commonly used value for ϕ

_{b}is ϕ′/2 [51].

## 3. Results

#### 3.1. Parametric Analysis

#### 3.1.1. Slope Geometry

#### 3.1.2. Initial Pore Water Pressure Distribution

^{3}). However, for the reference analysis, it was considered that the suction values could reach up to 40 kPa above the water table level and then return to zero. This behavior is shown in the figure below with the “Maximum 40 kPa—Basic Run” series. The behavior is comparable to that observed in the literature for partially saturated slopes [52,54].

#### 3.1.3. Saturated Volumetric Water Content

^{−6}m/day was applied. The permeability function is not affected by the variation in the volumetric water content. Three values of saturated volumetric water content (θ

_{sat}) were studied. Higher values of θ

_{sat}represent greater slope stability. However, their variation is slightly perceptible, which causes us to assume that its variation does not significantly influence the slope stability.

#### 3.1.4. Hydraulic Fitting Fredlund and Xing Parameters

_{f}[40] parameter, values equal to 20, 100 and 1000 were adopted (Figure 7a). In Figure 8a, the relationship between the a

_{f}value and F.S. shows that there is a downward trend as the value of a

_{f}increases. However, the variation is too small to confirm that any important correlation exists. As in all other situations, the analyses were performed under the condition that all other parameters associated with the curve remain constant. The AEV is a function of the a

_{f}value of the curve and, therefore, it also changes in each analysis. As per definition, it is the suction value above which the air begins to enter the larger voids in the soil. The AEV is directly proportional to a

_{f}, so that its increase also causes a drop in the factor of safety. Analyses performed with different a

_{f}values and other soil properties as well as those adopted in this paper have already demonstrated different behavior from that observed here. This emphasizes the importance of evaluating a wider range of possibilities with respect to the variation in this parameter, whose nature is logarithmic, i.e., its variation can be observed only in a magnitude order oscillation.

_{f}values. Moreover, as expected in this case, both shapes are influenced by the n

_{f}value [40]. Again, three values of n

_{f}were evaluated: 0.4 (Basic Run), 0.2 and 4.0. Figure 8b shows that despite the small change, an increase in n

_{f}represents a larger F.S. The F.S. = 1 line was not crossed, even when other trials were performed. According to [44], an increase in the value of n

_{f}indicates a more uniform ground. This means that, for the studied soil slope, the more uniform the soil, the more stable it tends to be.

_{f}equal to 1.5 (Basic Run), 0.5 and 4.0 were adopted (Figure 7c). The increase in this factor, associated with the greater verticalization of the curve and a lower residual volumetric water content [44], also indicates an increase in the factor of safety (Figure 8c). The residual volumetric water content is the moisture content from which a large variation in suction is required to remove more water from the soil; this denotes the moisture value above which increases in suction do not produce significant variations in moisture content. Again, the variation observed is minor, but clearly demonstrates the positive relationship between the two variables. In this case, the point where the soil slope crosses the stability line (m

_{f}~ 0.75) can be seen.

_{r}equal to 10,000 (Basic Run), 1 and 1,000,000 were adopted (Figure 7d). Figure 8d shows that, even considering a log-scale variation for the h

_{r}parameter, the F.S. was not affected. No influence was exerted by this parameter for the situation studied.

_{f}, n

_{f}and m

_{f}for the Fredlund and Xing model, have significant effects on the risk of landslides for the reservoir water’s drawdown. The saturated permeability coefficient of the soil and the velocity of water-level fluctuation also were key potential properties controlling stability. Their response can also be linked to the drainage conditions of the soils composing the reservoir area.

#### 3.1.5. Saturated Permeability Coefficient

_{sat}values assessed in the parametric study, plotted against the associated factor of safety. It is evident that the higher the saturated permeability coefficient of the soil, the lower the factor of safety. This occurs because, with the same standard rain condition considered, higher k

_{sat}values allow more water to be absorbed by the soil and, therefore, a more positive pore pressure is generated, leading the slope towards a more unstable condition and, then, failure.

_{sat}adopted. A higher k

_{sat}value (Figure 10a) led to positive pore water pressures (blue line) at higher depths, as well as caused the initial water level to rise. This suction condition at the second day of rainfall is responsible for the failure. On the other hand, the smallest k

_{sat}value (Figure 10c) insignificantly changes the initial water level and absorbs water only in the first meter of the soil slope. The situation is not sufficient to lead to failure when considering the critical slip surface into the soil layer.

_{sat}, of soil has a unique consequence on the stability of both good and poorly drained soil slopes. Their observation agrees with the findings presented above.

#### 3.1.6. Cohesion

#### 3.1.7. Friction Angle

_{b}parameter in Equation (6) is used to quantify the rate of increase in shear strength relative to suction. According to [51], when the ϕ

_{b}value is unknown, a ϕ

_{b}equal to 15° can be used in the slope stability study to evaluate the influence of matric suction on the F.S. A ϕ

_{b}value of zero can also be used, signifying that the effect of matric suction is neglected. The author affirms that if the air entry value (AEV) of the soil is smaller than 1 kPa, the effect of matric suction on soil slope stability is trivial and the ϕ

_{b}value can assumed to be zero. In this study, the Basic Run value of AEV is equal to 0.86 kPa. Thus, the unsaturated friction angle (Figure 12b) could be decreased to zero in order to show the variation in the factor of safety, generating the slope failure. Nonetheless, according to [51], if the air entry value (AEV) of the soil is between 20 and 200 kPa, an assumed ϕ

_{b}value of 15° provides a reasonable estimation of the effects of unsaturated shear strength in most cases, and for soils with an AEV greater than 200 kPa, ϕ

_{b}can generally be assumed to be equal to the effective angle of internal friction, ϕ′. Table 3 shows the ϕ

_{b}values adopted based on these considerations.

#### 3.1.8. Specific Weight

#### 3.2. Sensitivity Analysis

_{i}found for each analysis i studied in the proposed range is related to the respective central F.S.

_{central}, to calculate the percentage variation in the factor of safety (ΔF.S.):

_{f}fitting parameter of F&X, saturated volumetric water content, unsaturated friction angle, n

_{f}fitting parameter of F&X, saturated hydraulic conductivity, hr and a

_{f}F&X fitting parameter.

## 4. Discussion and Conclusions

_{b}had an effect of only 7%.

_{b}can lead to a factor of safety that is only 7% affected.

_{b}is related to the effect of matric suction, which, even contributing to the shear strength in unsaturated soils, plays a secondary role if compared to the friction angle itself.

_{sat}variation exerts a low influence on slope stability. However, the idea of magnitude order for the saturated permeability coefficient should be taken into account. If we do so, one magnitude order of variation can represent an even greater influence on stability than the other mechanical parameters discussed previously. The soil’s infiltration capacity was reached in all analyses, for a better comparison.

_{f}, n

_{f}and a

_{f}. While m

_{f}had 7% influence and n

_{f}2%, a

_{f}had only 0.2%. The small effect observed is probably due to the parameter’s nature. The values here studied varied based on a log scale. This means that when analyzing the parameter percentage variation against the F.S. percentage variation, the ratio found was necessarily small. This issue could be bypassed if a different methodology, taking the AEV value into consideration, was adopted.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Model geometry, boundary conditions for infiltration process, finite element mesh and pore water control section locations (AA, BB, CC and DD).

**Figure 7.**SWCC curves and permeability functions for different Fredlund and Xing parameter values: (

**a**) a

_{f}; (

**b**) n

_{f}; (

**c**) m

_{f}; (

**d**) hr.

**Figure 8.**Factor of Safety variation versus Fredlund and Xing parameters (

**a**) a

_{f}; (

**b**) n

_{f}; (

**c**) m

_{f}; (

**d**) h

_{r.}

**Figure 10.**Factor of safety variation for different permeability coefficients. (

**a**) k

_{sat}= 1.71 × 10

^{−6}m/s, (

**b**) k

_{sat}= 1.71 × 10

^{−7}m/s (Basic Run), (

**c**) k

_{sat}= 1.71 × 10

^{−8}m/s.

**Figure 12.**Factor of safety variation versus saturated friction angle (

**a**) and unsaturated friction angle (

**b**).

Parameter | Reference Value | Reference | |
---|---|---|---|

Slope declivity (degrees) | 40 | [30,47] | |

Maximum suction on unsaturated zone—ZAM ^{a} (kPa) | 40 | [52,54] | |

θ_{sat} (porosity) | 0.36 | [53] ^{b} | |

Fredlund and Xing fitting parameters | a_{f} (kPa) | 20 | [53] ^{b} |

n_{f} | 0.4 | [53] ^{b} | |

m_{f} | 1.5 | [53] ^{b} | |

h_{r} (kPa) | 10,000 | [53] ^{b} | |

k_{sat} (m/s) | 1.71 × 10^{−7} | [39] | |

c′ (kPa) | 2 | [39] | |

ϕ′ (degrees) | 34 | [39] | |

ϕ_{b} (ϕ′/2) | 17 | [51] | |

γ natural wet specific weight (kN/m^{3}) | 16.20 | [39] |

^{a}Refers to Zero Above a Maximum predetermined value; the values refer to the maximum allowed.

^{b}Field data were adjusted to [40] model in order to estimate the fitting parameters.

**Table 2.**Parametric analysis matrix (Adapted from [55]).

Test | Declivity ( ^{o}) | Maximum Suction Allowed (kPa) | SWCC (Fredlund and Xing Fitting Parameters) | k_{sat}(m/s) | c′ (kPa) | ϕ′ ( ^{o}) | ϕ_{b} (^{o}) | γ (kN/m^{3}) | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

θ_{sat} | a_{f} | n_{f} | m_{f} | h_{r} (kPa) | |||||||||

Basic Run | 40 | ZAM * | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_01 | 30 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_02 | 50 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_03 | 40 | ZAM | 30 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_04 | 40 | ZAM | 50 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_05 | 40 | ZAM | 40 | 0.25 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_06 | 40 | ZAM | 40 | 0.45 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_07 | 40 | ZAM | 40 | 0.36 | 100 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_08 | 40 | ZAM | 40 | 0.36 | 1000 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_09 | 40 | ZAM | 40 | 0.36 | 20 | 0.20 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_10 | 40 | ZAM | 40 | 0.36 | 20 | 4.00 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_11 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 0.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_12 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 4.0 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_13 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 1 | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_14 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 1 × 10^{6} | 1.71 × 10^{−7} | 2 | 34 | 17 | 16.2 |

S_15 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−6} | 2 | 34 | 17 | 16.2 |

S_16 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−8} | 2 | 34 | 17 | 16.2 |

S_17 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 1 | 34 | 17 | 16.2 |

S_18 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 10 | 34 | 17 | 16.2 |

S_19 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 30 | 17 | 16.2 |

S_20 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 36 | 17 | 16.2 |

S_21 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 0 | 16.2 |

S_22 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 18 | 16.2 |

S_23 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 15 |

S_24 | 40 | ZAM | 40 | 0.36 | 20 | 0.40 | 1.5 | 10,000 | 1.71 × 10^{−7} | 2 | 34 | 17 | 24 |

a_{f} | n_{f} | m_{f} | h_{r} | AEV | ϕ′b |
---|---|---|---|---|---|

20 | 0.4 | 4.0 | 10,000 | 0.25 | 0 |

20 | 0.4 | 1.5 | 1 | 0.47 | 0 |

20 | 0.4 | 1.5 | 1,000,000 | 0.84 | 0 |

20 | 0.4 | 1.5 | 10,000 | 0.86 | 0 |

1000 | 0.4 | 1.5 | 10,000 | 3.55 | 0 |

20 | 4.0 | 1.5 | 10,000 | 13.06 | 15 |

20 | 0.2 | 1.5 | 10,000 | 27.58 | 15 |

100 | 0.4 | 1.5 | 10,000 | 76.28 | 15 |

20 | 0.4 | 0.5 | 10,000 | 237.19 | ϕ′ |

Parameter | ΔF.S./Δparameter (%) ^{a} |
---|---|

Slope inclination | 146 |

Initial suction condition | 38 |

θ_{sat} | 7 |

a_{f} | 0.2 |

n_{f} | 2 |

m_{f} | 8 |

h_{r} | 0.7 |

k_{sat} | 0.9 |

Cohesion | 13 |

Friction angle | 93 |

Unsaturated friction angle | 7 |

Specific weight | 22 |

^{a}Average of the absolute value.

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## Share and Cite

**MDPI and ACS Style**

Oliveira, E.d.P.; Acevedo, A.M.G.; Moreira, V.S.; Faro, V.P.; Kormann, A.C.M.
The Key Parameters Involved in a Rainfall-Triggered Landslide. *Water* **2022**, *14*, 3561.
https://doi.org/10.3390/w14213561

**AMA Style**

Oliveira EdP, Acevedo AMG, Moreira VS, Faro VP, Kormann ACM.
The Key Parameters Involved in a Rainfall-Triggered Landslide. *Water*. 2022; 14(21):3561.
https://doi.org/10.3390/w14213561

**Chicago/Turabian Style**

Oliveira, Elisangela do Prado, Andrés Miguel González Acevedo, Virnei Silva Moreira, Vitor Pereira Faro, and Alessander Christopher Morales Kormann.
2022. "The Key Parameters Involved in a Rainfall-Triggered Landslide" *Water* 14, no. 21: 3561.
https://doi.org/10.3390/w14213561