**1. Introduction**

Rainfall is one of the main factors causing slope failure in tropical areas. The effects of rainfall properties on slopes have been studied in large amounts of research that have analyzed slope stability by both investigation and simulated models. By the simulation of different conditions of slopes and rainfall boundaries, previous research has indicated the changes of soil unsaturated properties, such as the reduction of suction, shear strength, and the increasing of hydraulic conductivity and pore-water pressure [1–20]. Finally, these changes cause slope failure. Before rain, the slope area above the groundwater table is considered as being in a partly unsaturated and dry state near the surface [1–4]. The unsaturated area reduces by rainfall water infiltration during and after the rainfall event. The change of the area from unsaturated to saturated is caused by the advancement of the wetting front from the surface [4,5,9,13,21–23] and groundwater table from depth [6–8,10–20]. Those processes were found to be the primary factors controlling the instability of slopes due to rainfall and were greatly affected by rainfall intensity (RI) and soil properties, especially by unsaturated

soil hydraulic conductivity [4,17–36]. The unsaturated soil hydraulic conductivity (HC) controls the transient seepage, the depth of rainfall infiltration, the changes in pore pressure during the rainfall event, and finally, affects the FOS. The effects of soil hydraulic conductivity on slope stability in the rainy season are usually assessed from the point of view of three topics: (1) the effects of changes in the value of soil hydraulic conductivity and rainfall intensity [17–30]; (2) the soil anisotropic HC and hydraulic hysteresis [31–35]; (3) the failure delay phenomenon due to the defense of HC and RI [36]. This research focuses on the overview and analysis of the first of these topics.

The effects of the HC on the FOS were specifically simulated by [25,26]. Those research works reported that high hydraulic conductivity led to rapid saturation. Additionally, infiltration causes the wetting front to quickly shift downward. This shift causes water to contact the underlying impermeable soil, leading to a rapid rise in pore-water pressure and the formation of a perched water table. The slope reaches full saturation and experiences a reduction in soil resistance. Consequently, the factor of safety rapidly decreases [26]. High-intensity rainfall has large effects on the slope if the soil slope has high hydraulic conductivity (Ks ≥ 10−<sup>4</sup> m/s) and when the slope has poorly drained soils (i.e., Ks ≤ 10−<sup>6</sup> m/s) [25].

The simulated analyses were also carried out with different boundaries of rainfall. When building the relationship of rainfall intensity and unsaturated hydraulic conductivity by the ratio of RI/HC, the previous research analyzed the RI = HC [5], RI ≥ HC [19,22]; and RI < HC [8,17,21,23,26,29] cases. Setting up a boundary with RI = HC or RI ≥ HC and no pond [5,19,22], the development of a wetting front from the crest of the slope and the reduction of soil suction during rainfall were found to dominantly affect the slope stability. The development of the defense of the wetting front under increased rainfall intensity shortens the time required for the wetting front to reach the pore-water pressure and moisture content sensors. The growing of groundwater has not been mentioned much. The same trend was also found in [17,21,23], which had the boundary of RI < HC. The larger the coefficient of permeability is, the greater the depth of the wetting front is. The opposite mechanism is seen in [8,29] with a boundary of small RI and high initial saturation and hydraulic conductivity, and the pore-water pressure increases gradually from the deep part to the crest of the slope [8,29]. For a slope with a larger HC, the slope failures possibly take place under rainfall with a shorter duration and a greater intensity [8]. Those research works also concluded that when the rainfall intensity is greater than the saturated hydraulic conductivity of the surface soils, a runoff occurs along the slope surface [8,22].

Although those previous research works had unobvious objectives focusing on building the effects of hydraulic conductivity, the results clearly showed that the slope stability was significantly affected by the soil hydraulic conductivity and the rainfall intensity boundary. The ratios of RI/HC were divided into three cases; however, each previous research work only concentrated on one of those cases. Moreover, there was a difference regarding the development of the pore-water pressure by the rising groundwater table [8,29] or by a wetting front from the surface [21,23] when those analyses had the same initial conditions. Further research should cover all three cases of different RI/HC ratios.

A riverbank is a special slope that relates closely to hydraulic dynamics, not only from rainfall infiltration from the surface concerning the space above the slope, but also significantly from the fluctuation of the river water level and groundwater table [27,37–43]. Past research works assessed the effects of rainfall on riverbank stability using the indirect process of transient seepage due to water level changes and assumed that rainfall has no effect on the riverbank surface.

By reviewing previous papers, it was obvious that the hydraulic conductivity has grea<sup>t</sup> effects on slope stability during rainfall events. However, some limitations were found in the reviewed research. The effects of the different RI/HC values were not specific to a researcher, and there was a difference in the previous discussion on seepage mechanics, such as the changes in the wetting front and the groundwater table. Moreover, research on the effects of hydraulic conductivity on riverbank slope stability was performed in only a few papers. Therefore, the objective of this paper is to build the mechanics of riverbank failure with different rainfall infiltration and soil hydraulic conductivity models. The numerical analyses by the GeoSlope program with both SEEP/W and SLOPE/W moduli are applied in this study.

#### **2. Materials and Methods**

The case study is a riverbank area in the Red River in Hanoi city, Vietnam. The Red River originates in Yun-nan, China, and flows through northwest Vietnam. The total length of the Red River in Hanoi is about 90 km with about 40 km of riverbank inside the urban area of Hanoi, where the population density and density of houses built near the natural riverbank are highest. These areas are located in high river terraces, outside the river dike, and near natural banks. In the annual rainy season, the destruction of local houses, roads, and land-use along the Red River bank has occurred.

The selected riverbanks are almost natural banks, and some are supported in the toe by vegetation. The riverbanks in these locations have either collapsed or have high failure potential due to fluctuations in the river water level, river water flow stress, rainfall, and human activities. Figure 1 shows the road near the riverbank (left), and the natural riverbank (right), which was broken in the rainy season of 2016 and 2017.

**Figure 1.** Some current problems near the riverbank in the study area.

#### *2.1. Field Investigations and Soil Properties*

The field investigation was performed during both the dry and the rainy seasons to describe the status of the riverbank and the river water level changes. Field data measurement and collection included bank geometry (i.e., height, slope), alluvial area, current river water level, and soil samples. The monitoring data, which included groundwater level, river water level fluctuation, and rainfall, were also collected from the National Meteorology Station.

The soil properties included soil physical properties such as water content, density, and grain size. The saturated hydraulic conductivity and shear strength were obtained in the geotechnical laboratory of VNU University of Science, Vietnam National University, Hanoi. The unsaturated soil properties, such as soil suction, were determined by using a pressure plate apparatus in the geotechnical laboratory of Ibaraki University, Japan, as shown in [42]. Based on the soil samples collected along the riverbank in the Hanoi area and the soil properties experiment, the riverbank in this area is quite homogeneous, with a silt or silty-clay layer in the bank layer and fine sand from the toe of the riverbank to the sediments. The silt layer is composed of less than 20% fine sand, 30–70% silt, and 10–30% clay. This paper selects one section of the soil bank to analyze riverbank stability, as shown in Table 1. Figures 2 and 3 present the suction and unsaturated hydraulic conductivity, respectively. The unsaturated shear strength and hydraulic conductivity were estimated by using the models of Vanapalli (1996) [44] and Van Genuchten (1980) [45] in the GeoSlope program [46,47].

**Figure 2.** Soil–water characteristic curves.

**Figure 3.** The unsaturated hydraulic conductivity curves.


**Table 1.** Soil properties used in riverbank stability analysis in Ba Vi area.


**Table 1.** *Cont*.

#### *2.2. Numerical Model Framework*

This paper uses the commercial GeoSlope program (GeoSlope International Ltd.) [46–49] as a numerical model to analyze riverbank stability. GeoSlope is one of the most useful and widely used programs in slope and riverbank stability analysis in many kinds of research [1–5,12,14,15,24–26]. The present paper uses a pair of analyses of transient seepage in SEEP/W and slope stability in SLOPE/W in the GeoSlope program.

In general, by using the SEEP/W, the mechanism describes specifically the transient seepage by rainfall infiltration. The obtained results from SEEP/W, which include pore-water pressure distribution and changes in soil properties, become the input data for the next modulus SLOPE/W for analyzing the riverbank stability. The result of the FOS indicates the riverbank stability when FOS is higher than one. The results and discussion focus on building the relationships between the FOS and the different conditions of initial saturation, rainfall intensity, and soil hydraulic conductivity.

The field investigation, laboratory testing, and monitoring of the support input data included three factor groups: the riverbank geometry, the soil properties, and the hydraulic conditions (such as the river water level and rainfall intensity). The riverbank stability analysis was performed for different initial saturation conditions, soil hydraulic conductivity, and rainfall intensity. The initial saturation conditions were set up by the initial river water level and the capillary height or the maximum negative head. In SEEP/W, the maximum negative head can be used to build the assumption of the predetermined negative pore pressure profile, and then the saturation condition can be set up. Other simulated factors such as the soil hydraulic conductivity and rainfall intensities were installed as the difference between the soil properties and boundaries. The conditions of the riverbank, hydraulics, and boundaries for different scenarios of the initial saturation and rainfall intensity are described in more detail below.

#### 2.2.1. Riverbank Geometry and Hydraulic Boundary Condition

This paper uses a riverbank configuration with the initial conditions shown in Figure 4. The riverbank has a slope angle of 52 degrees and a height of 10 m. The riverbank is homogeneous silty soil with a sand layer underneath. The analyzed soil properties are shown in Table 1 and Figures 2 and 3.

The boundary conditions of river water level (RWL) and rainfall intensity were set up in the SEEP/W model. Using the riverbank configuration shown in Figure 4, the initial RWL was the boundary in the river site from the bottom to 3 m, and the function RWL–time was the boundary of the entire lateral riverbank. The rainfall was the boundary on the surface of the riverbank. Both scenarios with and without the function "potential seepage face review" were used with the rainfall boundary. Without the function "potential seepage face review", the rainwater would infiltrate into the soil as long as the rainwater exceeded the infiltration capacity; under these conditions, a pond would present above the riverbank surface when there was excess rainwater. This condition often occurs at some riverbank sites that have a pond or low areas near the riverbank and during floods. If the "potential seepage face review" was included, the pond would not exist, and the excess rainwater would run off.

**Figure 4.** The initial riverbank configuration.

The rainfall intensity was determined based on daily monitoring data. The rainy season is from mid-June to mid-September, and high rainfall often occurs in August. Figure 5 shows the variation in the rainfall intensities (RI, mm/h) of several rainy days in the rainy month (August 2016), and Figure 6 shows the changes in daily rainfall and RWL monitored during the rainy season from 1 July to 31 August where some sites in the Red River bank broke. It can be seen that the rainfall intensity ranged from 0–50 mm/h. To simulate the effect of rainfall intensity on riverbank stability, three rainfall intensities were used: RI = 10 mm/h; RI = 30 mm/h; and RI = 50 mm/h. Based on the RWL data, the initial RWL was set as 3 m.

**Figure 5.** The hourly rainfall with high rainfall intensity for some days in August 2016.

The initial pore-water pressure and saturation conditions were set up by using the function of the maximum negative pressure head in SEEP/W. This function was built with knowledge of the pore pressure in unsaturated soil. Soil regions above the groundwater table were divided into two sub-regions: a dry zone near the surface and a partly saturated zone near the groundwater table. The pore-water pressure graph was linear and negatively sloped from the groundwater table to the maximum negative head. This meant that the negative pore-water pressure near the surface may have become too high in the dry zone. In fact, the soil was never completely dry and always retained some amount of water. The small surface flux had the effect of changing the pore-water pressure profile. Figure 7 shows a pore-water pressure profile with a non-dry surface condition in which the negative pore-water pressure was negatively linearly sloped to a maximum negative pore-water pressure and remained constant at a value in response to soil water content. The magnitude of the maximum

negative pore-water pressure was dependent on the shape of the hydraulic conductivity function, and to a lesser extent on the rate of infiltration. In SEEP/W, setting the maximum negative head can indicate the field pore pressure profile. Based on the investigation and experimental data of the soil water content and soil suction curve, the value pore pressure or saturation degree could be determined.

**Figure 6.** Daily river water level (RWL) and daily rainfall from 1 July to 31 August 2016.

**Figure 7.** Calculation pore-pressure condition by saturation condition.

To simulate the effect of the initial pore pressure and saturation conditions, we set the different initial negative pore pressures to 15 kPa and 33 kPa, respectively. At those pressures, the relative soil water contents were 41% and 33%, and the saturation degrees were 87% and 70% respectively. Those were average experimental values in the beginning of the rainy season and in the rainy season.

The initial riverbank soil had a saturation hydraulic conductivity of Ks = 7.39 × 10−<sup>5</sup> cm/s (Table 1). To simulate the effects of soil hydraulic conductivity on the processes of rainfall infiltration and riverbank stability, three values of saturation hydraulic conductivity were used to represent the initial saturated hydraulic conductivity: Ks = 7.39 × 10−<sup>3</sup> cm/s, Ks = 7.39 × 10−<sup>4</sup> cm/s, and Ks = 7.39 × 10−<sup>5</sup> cm/s. A hydraulic conductivity of Ks = 7.39 × 10−<sup>3</sup> cm/s was considered to show high conductivity (cases Hi); Ks = 7.39 × 10−<sup>4</sup> cm/s was considered medium hydraulic conductivity (cases Mi); and Ks = 7.39 × 10−<sup>5</sup> cm/s was considered low hydraulic conductivity (cases Li). Figure 8 shows three unsaturated hydraulic conductivity curves that correspond to the three

hydraulic conductivity values listed above and to the same suction properties for silty soil in the Red River bank.

**Figure 8.** The different unsaturated hydraulic curves used in analyses.

#### 2.2.2. Cases Used in Analysis

Table 2 describes the numbers of cases we analyzed to simulate the riverbank stability in the different initial suction conditions or negative pressure, the rainfall intensity, and the hydraulic conductivity, in which cases H-1-1, H-1-2, H-1-3 were the names of case studies with high saturation hydraulic conductivity (Ks = 7.39 × 10−<sup>3</sup> cm/s) at the initial saturation degree of 70% at three rainfall intensities of 10 mm/h, 30 mm/h, and 50 mm/h, respectively. Using this labeling for cases, there were 18 analyzed cases, as shown in Table 2.


**Table 2.** Cases used to analyze Ba Vi riverbank.

Where rainfall intensities (RI) = 10 mm/h = 2.7 × 10−<sup>4</sup> cm/s; RI = 30 mm/h = 8.3 × 10−<sup>4</sup> cm/s; RI = 50 mm/h = 1.3 × 10−<sup>3</sup> cm/s.
