A determination method of rainfall type based on rainfall-induced slope instability

This work presents a determination method of rainfall types based on rainfall-induced slope instability to provide an effective approach for early warning of different patterns of rainfall-induced slope failures/landslides. Firstly, 5808 scenarios of slope instability are simulated with 11 kinds of soil properties under 528 designed intensity-duration (I−D) conditions to get the threshold curves of slope failure under different rainfall conditions and soil properties. Then through analysis of the threshold curves when slope failure occurred, rainfall is classified into two types: short-duration—high-intensity (SH) type and long-duration—low-intensity (LL) type. According to the analysis results, it indicates that rainfall types affect the initiation of slope failure, i.e., different I−D conditions will affect the slope failure initiation under LL type rainfall, while the slope failure initiation will not be affected by the change of I−D conditions under SH type rainfall. In addition, the results show that the classification of rainfall types does not depend on the soil shear strength parameters (effective cohesion and effective internal friction angle), although the change of soil shear strength parameters will cause the shift of threshold curve of slope failure in the I−D conditions two-dimensional (2D) plane. The findings in this study benefit to understanding the effect of rainfall type on the mechanism of slope failure initiation, which will promote the development of an early warning system of slope failure in the future by considering the identification of rainfall types.


Introduction
Rainfall is considered as a key factor for triggering slope failure, and more than thousands of rainfall-induced slope failures occur around the world every year (Rahardjo et al. 2005;Zhang et al. 2014;Van Asch et al. 2018;Zhu et al. 2020a). Particularly, the different rainfall conditions/types cause slope failure to occur in different patterns, for example, heavy rainfall like rainstorms or torrential rain will make the surface layer of the soil saturated in a short time (Zhu et al. 2020b). The pore air between the saturated surface layer and the phreatic layer is enclosed in the soil, thereby preventing the downward infiltration of water. In this case, the water that does not infiltrate into the soil produces runoff on the ground surface, which will cause erosions of the ground surface and shallow slope failure/ landslides (Larsen and Simon 1993). On the other hand, during light rain, the weak rainfall intensity will not quickly saturate the surface layer of the soil. This makes it difficult to create the closed region of pore air. The infiltrated water replaces the pore air in the soil, and the pore air is expelled from the soil through the soil surface (Kuang et al. 2013). Accordingly, water can continuously infiltrate into the deep soil layer, resulting in the continuous increase of the water content in the deep soil layer, thereby reducing the matric suction and subsequently inducing the occurrence of deep-seated slope failure/landslide (Larsen and Simon 1993). Therefore, it cannot be ignored that the type of rainfall has a significant influence on the infiltration behavior of soil moisture and the failure pattern of the slope.
In the past few decades, many scholars have made great efforts to establish the link between rainfall and slope instability (e.g., Larsen and Simon 1993;Guzzetti et al. 2008;Saito et al. 2010;Segoni et al. 2014;Chen et al. 2015;Guo et al. 2016;Abraham et al. 2020;Marin et al. 2021;Kim et al. 2021). A considerable part of their research focuses on a widely used method, namely intensity-duration (I−D) threshold models, since the I-D threshold is considered as an effective indicator for rainfall-induced slope failure/landslides (Kim et al. 2021). Among them, the most famous one is the power function of rainfall intensity as rainfall duration (i.e., I = aD b , I is rainfall intensity; D is rainfall duration; a and b are fitting parameters). For example, Marin et al. (2021) evaluated the applicability of the two methods (physically-based model and observed landslides) to define rainfall I-D thresholds in individual basins and emphasized the merits of both methods. Abraham et al. (2020) pondered the effect of the selection of rainfall parameters for developing a regional scale rainfall threshold in a data-sparse region and pointed out that the approach selecting the rain gauge based on the most extreme rainfall parameters performed better than the other approaches. However, the main focus of discussion in these studies is the method of the determination of the rainfall intensity, i.e., the average rainfall intensity (Saito et al. 2010), the peak rainfall total (Jarosińska 2018), or the antecedent rainfall (Kim et al. 2021).
For the rainfall types, the concepts of short-duration-high-intensity (SH) type and long-duration-low-intensity (LL) type have been mentioned in many studies (Chen et al. 2015;Perera et al. 2017;Deng et al. 2018;Jin et al. 2021;Marin et al. 2021;Kim et al. 2021). However, in these studies, a clear judgment theory or standard of rainfall type is not given. In addition, the criteria for classifying rainfall types are not uniform. For example, Jin et al. (2021) defined the rainfall event (I = 25 mm/h and D = 8 h) as SH type rainfall and the rainfall event (I = 0.4 mm/h and D = 168 h) as LL type rainfall. Chen et al. (2015) set the criteria of SH type rainfall as short-duration (< 2 h) and high-intensity (> 16.1 mm/h) and the criteria for LL type rainfall as long-duration (> 71 h) and low-intensity (< 8.8 mm/h). While Perera et al. (2017) set the criteria of SH type rainfall as short-duration (< 2 h) and high-intensity (> 54 mm/h) and the criteria for LL type rainfall as long-duration (> 8 h) and low-intensity (< 25 mm/h). Apart from the above studies, Saito et al. (2010) examined rainfall I-D conditions of 1174 shallow landslides that occurred during 2006−2008 in Japan and classified the rainfall type into SH type and LL type based on statistical results as shown in Fig. 1. However, from Fig. 1b, it can be seen that the criterion for classifying rainfall types proposed by Saito et al. (2010) is too small. The main reason may be that the object of investigation in the study of Saito et al. (2010) is shallow landslides, which are usually induced by heavy rainfall or the erosion of the surface caused by long-term light rain.
It can be seen from the above that SH type rainfall and LL type rainfall will trigger slope failure/landslides with different patterns (shallow/deep-seated). Therefore, the determination of the rainfall type is a key point to determine slope failure patterns. However, at present, in the field of early warning of rainfall-induced geological disasters, the determination of SH type rainfall and LL type rainfall is not consistent. The well-established or reasonable determination method of rainfall types is not reported. Therefore, this study attempts to present a determination method of rainfall type based on rainfall-induced slope instability. To avoid unreliable results due to the insufficient number of simulations and discuss the effects of soil shear strength parameters on the classification of rainfall types, through the analysis of the I−D conditions corresponding to 5808 scenarios of slope instability with different soil shear strength parameters, and subsequently, according to the effect of rainfall types on the initiation of slope failure, a determination method of rainfall type (SH type and LL type) based on rainfall-induced slope instability is proposed, which provides an effective approach for early warning of different types of rainfall-induced slope failures/landslides.

Governing equation of seepage flow
The governing equation for two-dimensional seepage flow is as given by Richards (1931) and Childs and Collins-George (1950), where ρ w is density of water (kg/m 3 ); D v is diffusion coefficient of water vapor through soil ((kg m)/(kN s)); P v is vapor pressure of soil moisture (kPa); y is elevation (m); P is pressure (kPa); k x , k y is hydraulic conductivity in the x-direction and y-direction (m/s), respectively; g is gravity acceleration (9.81 m/s 2 ); Q is boundary flux (1/s); m w is slope of the soil water characteristic curve, SWCC (1/kPa); γ w is unit weight of water (kN/m 3 ), and t is time (s). (1)

Shear strength of unsaturated soil
The shear strength of unsaturated soil is expressed based on Bishop's effective stress principle as given by Vanapalli et al. (1996), where τ is the shear strength of unsaturated soil (kPa); c′ is the effective cohesion (kPa); ϕ′ is the effective angle of internal friction (°); u a is the pore air pressure (kPa); u w is the pore water pressure (kPa); σ n is net total stress (kPa); θ w is volumetric water content (m 3 /m 3 ); θ s is saturated volumetric water content (m 3 /m 3 ), and θ r is residual volumetric water content (m 3 /m 3 ).

Slope stability analyzed by limit equilibrium method
Slope stability has been analyzed using the limit equilibrium method given by Morgenstern and Price (1965), where W is the total weight of a slice as shown in Fig. 2 (kN); N is the total normal force on the base of the slice (kN); x is the horizontal distance from the centerline of each slice to the center of rotation (m); f is the perpendicular offset of the normal force from the center of rotation (m); R is the radius of a circular slip surface (m); l is the base length of each slice (m); χ is a parameter related to the degree of saturation, χ = (θ w − θ r )/(θ s − θ r ). The Morgenstern and Price Method (Fig. 2) considers both shear (X) and normal (E) interslice forces and satisfies both moment and force equilibrium (Morgenstern and Price 1965).
Slope stability was analyzed by the limit equilibrium method given by Morgenstern and Price (1965)

Numerical model and materials
A 2D homogeneous conceptual slopes model is established to simulate slope stability under different I-D conditions by using the general software package, GeoStudio (2007). Figure 3 shows the size of the numerical model and the applied boundary conditions. During the simulation, rainfall data are applied on the soil surface (ab, bc, cd in Fig. 3), and no flow boundary is applied on the left side (af in Fig. 3), right side (de in Fig. 3), and bottom side (ef in Fig. 3) of the model. The initial groundwater level (GWL) is set as − 3 m (blue dash line in Fig. 3), and the stable GWL is the simulated GWL after the initial steady-state analysis (red dash line in Fig. 3). The slope height (H) is assumed as 10 m, and the slope angle (α) is assumed as 45°.
There are large land areas of Japan are covered with volcanic soils, which have been produced since the Quaternary. Likewise, in Hokkaido, volcanic soils are widely distributed over more than 40% of the total area (Siva Subramanian 2017). The soil properties are given as a kind of volcanic soil and Toyoura sand in Hokkaido, Japan as listed in Table 1. Dry density (ρ d ), porosity (n), effective cohesion (c′), effective internal friction angle (ϕ′), and saturated hydraulic conductivity (k s ) were measured by Sato et al. (2017) and Lin (2020). Figure 4 shows the volcanic soil grain size curve, soil-water characteristic curve  1 3 (SWCC), and hydraulic conductivity of the soil. The SWCC and hydraulic conductivity of the volcanic soil and Toyoura sand are estimated from the grain size distribution curve by referring to Fredlund et al. (2002). There are 22 different rainfall intensities and 24 different rainfall duration are designed as listed in Table 2. Therefore, a total of 528 kinds of I-D conditions have been designed. The minimum rainfall intensity is set to 10 mm/h, and the maximum rainfall intensity is set to 60 mm/h. The intensity of the rest rainfall increases in increments of 2 mm/h. It is worth noting that the rainfall intensity in this study means the average rainfall intensity, i.e., within the rainfall duration, the design intensity is a constant value.

Slope instability under different I-D conditions
Two different approaches can be selected to simulate the gradual progress of the slope from stable to unstable. One way is to keep the rainfall duration constant and gradually increase the rainfall intensity, while the other way is to keep the rainfall intensity constant and gradually extend the rainfall duration. Due to under the same rainfall intensity, with the increase in the rainfall duration, the stability of the slope will continue to decline, i.e., the factor of safety (FOS) of the slope will gradually decrease. This makes it much easier to capture the process of slope from stable to failure, so the second approach is used in this study. Figure 5 shows part of the simulation results (c′ = 3 kPa, ϕ′ = 35°) of the slope stability under different I-D conditions. Figure 5 suggests that under low rainfall intensity, the stability of the slope drops very smoothly. However, it is worth noting that in the case of weak rainfall, as the rainfall intensity increases, the FOS curve will become steeper. While, when the rainfall intensity increases to a certain degree, even if the rainfall intensity further increases, the stability of the slope does not change significantly as shown from Fig. 5g-i.
To discuss the effect of hydrological properties of volcanic soil and Toyoura sand on slope stability, the same value (c′ = 3 kPa, ϕ′ = 35°) of the shear strength parameters was assumed for both kinds of materials. The I-D conditions when the slope failure occurs  Fig. 6. From Fig. 6, it can be recognized that the threshold curve formed by the simulation results conforms to the rule in the wide-used early warning system that the rainfall intensity is the power function of the rainfall duration. This implies that the calculation results are reliable and effective. However, it should be noted that a very interesting phenomenon was discovered as described in the previous paragraph, the increase in rainfall intensity will advance the time of slope failure, but when the rainfall intensity exceeds 32 mm/h (volcanic soil, in the case discussed in this study) and 50 mm/h (Toyoura sand, in the case discussed in this study), the slope failure time will no longer change even if the rainfall intensity continues to increase.

Slope instability under different soil properties
To investigate whether the finding in Sect. 4.1 is an accidental phenomenon caused by an unreasonable model setting or an objective law that prevails in reality, 11 kinds of soil shear strength parameters with different effective cohesion (0 kPa, 1 kPa, 2 kPa, 3 kPa, 4 kPa, and 5 kPa) and effective internal friction angle (33°, 34°, 35°, 36°, 37°, and 40°) as listed in Table 3 were discussed. In the simulation, the controlled variable method is used, that is, when the effective cohesion changes, the effective internal friction angle remains constant (ϕ′ = 35°), and when the effective internal friction angle changes, the effective cohesion remains constant (c′ = 0 kPa). From Fig. 6, it suggests that the hydrological properties also are key parameters affecting the threshold curve of slope failure. It contains Total number 11 kinds of soil properties different saturated hydraulic conductivities and SWCCs. The changes of these two parameters (saturated hydraulic conductivity and SWCC) will become very complicated with different soil grain size curves, so they are not considered in this study, which is the limitation of this study. Therefore, more discussions need to be carried out in future related studies. In this section, the focus is on the discussion of the effects of soil shear strength parameters on the threshold curve of volcanic soil slope failure. A total of 5808 scenarios of slope instability are simulated with 11 kinds of soil properties under 528 designed intensity-duration (I−D) conditions. Figures 7 and 8 show the threshold curves formed by the occurrence points of slope failure (FOS is less than 1.0 for the first time) under the variation of effective cohesion and the variation of effective internal friction angle, respectively. From Fig. 7, it shows that the change of effective cohesion will cause the obvious rightward shift of the threshold curve of slope failure in the I−D conditions 2D plane no matter under relatively heavy rainfall or relatively weak rainfall. The difference is that under relatively heavy rainfall conditions, the threshold curve of slope failure moves to the right nearly parallel to the x-axis in the I−D conditions 2D plane, while under relatively weak rainfall conditions, the threshold curve of slope failure moves to the upper right at a certain angle to the x-axis in the I−D conditions 2D plane. In addition, Fig. 8 shows the effect of the change of effective internal friction angle on the threshold curve of slope failure. From Fig. 8, it can be identified that under relatively heavy rainfall conditions, there is no obvious movement of the threshold curve of slope failure that was observed in the I−D conditions 2D plane, while under relatively weak rainfall conditions, although the magnitude of the movement is not as obvious as in Fig. 7, the trend of the threshold curve of slope failure moving toward the upper right at a certain angle to the x-axis in the I−D conditions 2D plane still can be recognized. From the overall analysis of Figs. 7 and 8, the results show that no matter under different effective cohesion conditions or different effective internal friction angle conditions, when the rainfall intensity exceeds a certain value, i.e., critical intensity (I c ), the occurrence time of slope failure will no longer be advanced even if the rainfall intensity continues to increase. Furthermore, the critical intensity (I c ) is not affected by the variation in effective cohesion and effective internal friction angle, for example, in all 11 kinds of soil properties, the value of the critical intensity (I c ) is about 32 mm/h (in the case discussed in this study).

Proposal of the determination method of rainfall types
Indeed, two rainfall intensities (high/low intensity) and two rainfall duration (short/long duration) can be combined into four rainfall types: short-duration-high-intensity (SH) type, short-duration-low-intensity (SL) type, long-duration-high-intensity (LH) type, and long-duration-low-intensity (LL) type. However, the stability of the slope will not be significantly reduced under SL type rainfall, as rainfall intensity is very low, and its duration is also short, i.e., the rainfall conditions are in a safe zone. While, under LH type rainfall, a large number of slope failures will occur with a high probability. In this case, the rainfall conditions are in a dangerous zone. The applicability of the early warning system in these two situations is very low, so the discussion is concentrated on SH type rainfall and LL type rainfall because in these two cases, the slope changes from stable to failure.
Combining the findings of Sects. 4.1 and 4.2, through analysis of 5808 scenarios of slope instability with 11 kinds of soil properties under 528 designed intensity-duration (I−D) conditions, using geometric methods, rainfall is classified into two types: shortduration-high-intensity (SH) type and long-duration-low-intensity (LL) type as shown in Fig. 9a and b. The red solid line in Fig. 9a and b is the outer boundary line of SH type rainfall and LL type rainfall, namely critical duration (D c ), and the blue solid line in Fig. 9a and b is the inner boundary line of SH type rainfall and LL type rainfall, that is the critical intensity (I c ) in Fig. 7 and 8. It is worth noting that the inner boundary line (purple solid line) of SH type rainfall and LL type rainfall proposed by Saito et al. (2010) is too small. Accordingly, a new determination method of rainfall types (SH type rainfall and LL type rainfall) is proposed as shown in Fig. 10. The expression is as follow, where D L is the lower limit of rainfall duration (h); D U is the upper limit of rainfall duration (h); β is a slope angle between 0 and 90°. That is if the average rainfall intensity (I) exceeds the critical intensity (I c ), the rainfall duration between D L and D U is SH type rainfall, i.e., rainfall conditions are located in the light green rectangle zone in Fig. 10. If the average rainfall intensity (I) does not exceed the critical intensity (I c ) and the rainfall duration is in the light blue diamond-shaped zone surrounded by critical duration (D c ) and critical intensity (I c ) in Fig. 10, it is LL type rainfall.  conditions, it is the soil volumetric water content (matric suction) that plays a key role in the stability of the slope. Meanwhile, the type of rainfall is the main factor affecting the change of soil volumetric water content in the slope.

Conclusions
This work simulated 5808 scenarios of slope instability with 11 kinds of soil properties under 528 designed intensity-duration (I−D) conditions. Then through analysis of the I−D conditions when slope failure occurred, a determination method of rainfall types (SH type rainfall and LL type rainfall) is proposed. According to the analysis results, it indicates that rainfall types affect the initiation of slope failure, that is in the case of LL type rainfall, the increase in rainfall intensity will advance the time of slope failure. While, in the case of SH type rainfall, i.e., when the rainfall intensity exceeds the critical intensity (I c ), even if the rainfall intensity further increases, the stability of the slope does not change significantly, i.e., the occurrence time of slope failure will no longer change.
In addition, under LL type rainfall, the change of effective cohesion and effective internal friction angle will make the threshold curve of slope failure to move toward the upper right at a certain angle to the x-axis in the I−D conditions 2D plane. While under SH type rainfall, the change of effective cohesion will make the threshold curve of slope failure to move toward the right nearly parallel to the x-axis in the I−D conditions 2D plane, but the effect of changes in effective internal friction angle is not obvious. Furthermore, the classification of rainfall types only depends on two relationships: 1. the relationships between the rainfall intensity (I) and critical intensity (I c ), and 2. the relationship between the rainfall duration (D) and critical duration (D c ). It does not depend on the soil shear strength parameters (effective cohesion and effective internal friction angle).
The findings in this study will benefit to understand the effect of rainfall type on the mechanism of slope failure initiation, which will promote the development of an early warning system of slope failure in the future by considering the identification of rainfall types.