Heavy rainfall in extreme climates often causes natural disasters such as floods, landslides, and debris flows. Rainfall-induced slope instabilities are major geological natural disasters (Glade 1998; Dai et al. 1999; Iverson 2000; Lee and Pradhan 2007; Li et al. 2016a, b; Wu et al. 2020) that can result in considerable loss of life and damage to infrastructure. Extreme events such as storms, which are becoming more severe because of climate change, can trigger fatal landslides. Storm-induced slope failures frequently occur because of rainfall infiltration, particularly in tropical areas (Fourie 1996; Cevasco et al. 2014). Global climate change in many mountainous areas could lead to more severe fluctuations in rainfall, and trigger of soil slope deformations and even slope instability because of the alteration of intensity, frequency, and quantity of rainfall (Dixon and Brook 2007; Jeong et al. 2008). The influence of climate change on rainfall characteristics has the potential to alter the stability of unsaturated soil slopes. Rainfall infiltration causes a decrease in matric suction and an increase in moisture content and hydraulic conductivity in unsaturated soils. The rainfall intensity and duration, initial water table, and hydraulic conductivity are the parameters that significantly affect slope stability (Ng and Shi 1998). An increase in pore-water pressure can reduce the effective stress and thereby weaken the shear strength of slopes. Complex geological environment and human engineering activities are also significant factors of slope instability under rainfall conditions.

Rainfall-induced slope failures have been examined based on experimental modeling, analytical and numerical methods (Ng and Shi 1998; Iverson 2000; Chen et al. 2005; Wu et al. 2009, 2016, 2020; Zhu et al. 2019). Laboratory and field experiments have been carried out to examine the infiltration mechanisms associated with rainfall-induced slope failures (e.g., Lee et al. 2011; Wu et al. 2015, 2017, 2018). Many numerical and analytical studies have investigated the hydraulic responses of slopes to rainfall infiltration and the stability of slopes under such conditions (Iverson 2000; Cai and Ugai 2004; Wiles 2006; Ali et al. 2014; Zhu et al. 2022). Numerical analysis solves for the matric suction or pressure head distribution in a soil slope with varying permeability, and considering different surface conditions of a soil. Many cases demonstrate that change in rainfall patterns may lead to slope failure due to infiltration. The results can provide an indication of the potential influence of climate change on shallow landslides in many mountainous areas (Kim et al. 2012).

Slope stability problems are commonly encountered in engineering projects. Many slope failures are attributed to water infiltration (Cho and Lee 2002; Cai and Ugai 2004). Matric suction is crucial to the stability of soil slopes because dissipation of matric suction leads to decrease in shear strength of unsaturated soils. Slope failure is closely related to the rainfall-induced transient infiltration of slopes in unsaturated soils (Fredlund and Rahardjo 1993; Wu et al. 2020). Shallow landslides are related to periods of intense rainfall. Engineering activities can result in severe geological and environmental issues. Slope failures induced by engineering activities may occur and may progress into landslides. The internal mechanics of slope movements are stress redistribution and the consequent changes in engineering–geological conditions (Marschalko et al. 2012).

Landslide forecasting should take into account rainfall infiltration into soil slopes. The models in predicting the timing and location of landslides are related to dynamic water infiltration in soil slope. The coupled process in an unsaturated soil is of major interest because of its implications for disaster prevention and environmental issues. Analytical approaches have been developed to provide a basic understanding of unsaturated infiltration in terms of the coupling effect (Wu et al. 2020). Meanwhile, numerical approaches provide a powerful tool for solving complex, nonlinear infiltration into unsaturated soils. These numerical models effectively investigate the coupled hydro-mechanical problem involved in unsaturated rooted slope stability issues (Oka et al. 2010; Wu et al. 2020; Zhu et al. 2022).

1.1 Rainfall Infiltration Equation

According to the modified Green–Ampt model (Mein and Larson 1973), the water infiltration process in soils during uniform precipitation can be divided into two stages: a stage controlled by rainfall intensity and a stage controlled by pressure head. The infiltration rate (fa) determined by rainfall intensity (qr) can be written as:

$$f_{{\text{a}}} = q_{{\text{r}}} \cos \beta$$
(1.1)

where qr is the rainfall intensity; and β is the slope angle.

The infiltration rate (fb) determined by the pressure head can be expressed as:

$$f_{{\text{b}}} = k_{{\text{s}}} \frac{{z_{{\text{f}}} \cos \beta + s_{{\text{f}}} }}{{z_{{\text{f}}} }}$$
(1.2)

where ks represents the permeability coefficient at saturation; sf represents the suction head of the wetting front; and zf represents the wetting front depth.

If fa = fb, the water ponding time (tp) can be obtained as:

$$t_{{\text{p}}} = \frac{{s_{{\text{f}}} (\theta_{{\text{s}}} - \theta_{{\text{i}}} )}}{{\cos^{2} \beta \left( {q_{{\text{r}}} /k_{{\text{s}}} - 1} \right)q_{{\text{r}}} }}$$
(1.3)

where θs and θi are the saturated moisture content and initial moisture content, respectively.

According to the water balance principle and Darcy law, wetting front movement during rainfall can be calculated as follows:

$$\left\{ {\begin{array}{*{20}l} {z_{{\text{f}}} = \left[ {q_{{\text{r}}} \cos \beta } \right]\frac{t}{{(\theta_{{\text{s}}} - \theta_{{\text{i}}} )}},} & {t < t_{{\text{p}}} } \\ {\frac{{{\text{d}}z_{{\text{f}}} }}{{{\text{d}}t}} = \frac{1}{{(\theta_{{\text{s}}} - \theta_{{\text{i}}} )}}\left[ {k_{{\text{s}}} \frac{{z_{{\text{f}}} \cos \beta + s_{{\text{f}}} }}{{z_{{\text{f}}} }}} \right],} & {t \ge t_{{\text{p}}} } \\ \end{array} } \right..$$
(1.4)

Equation (1.4) is the GA model suitable for slopes (Chen and Young 2006).

A penetration test of saturated sand layers was conducted and found a quantitative relationship between the infiltration velocity of water in the soil and the head loss, namely Darcy law:

$$q = - k_{{\text{s}}} \frac{{{\text{d}}H}}{{{\text{d}}L}}$$
(1.5)

where q is the flux (discharge per unit area, with units of length per time, m/s), H is the total water head, and L is the seepage length.

When soil mass is unsaturated, most scholars believe that Darcy law can also be used for analyzing water movement in unsaturated soils. Richards (1931) extended the Darcy law of saturated soil to the unsaturated infiltration and introduced the continuity equation to derive the equation of motion of the unsaturated soil–water flow, namely the Richards’ equation. The permeability coefficient (k) is expressed as a function of matric suction, and Darcy’s law can be expressed as:

$$q = k\nabla H$$
(1.6)

where \(\nabla H\) is a hydraulic gradient, including two components of gravity and suction.

Continuity equation:

$$\frac{\partial \theta }{{\partial t}} = -\nabla \cdot q$$
(1.7)

Then:

$$\nabla \cdot \left[ {k\nabla H} \right] = \frac{\partial \theta }{{\partial t}}$$
(1.8)

Substituting Eq. (1.6) into Eq. (1.8), one can obtain:

$$\nabla \cdot \left[ {k\left( h \right)\nabla \left( {h + z} \right)} \right] = \frac{\partial \theta }{{\partial t}}$$
(1.9)

As shown in Fig. 1.1, the Richards’ equation governing one-dimensional vertical infiltration in unsaturated soils can be written as:

$$\nabla \cdot \left[ {k\left( h \right)\nabla h} \right] + \frac{\partial k\left( h \right)}{{\partial z}} = \frac{\partial \theta }{{\partial t}}$$
(1.10)
Fig. 1.1
A diagram of a rectangular homogeneous unsaturated soil sample of height L with vertical infiltration from the top. At the top of the sample is h equals 0, and at the bottom of the sample is h equals h subscript d.

1D rainfall infiltration model

Full size image

1.2 Infiltration Equation for Unsaturated Slopes

The 2D generalized RE for unsaturated infiltration is expressed as (Ku et al. 2017; Wu et al. 2020):

$$\frac{\partial }{\partial x}\left[ {K_{x} \left( h \right)\frac{\partial H}{{\partial x}}} \right] + \frac{\partial }{\partial z}\left[ {K_{z} \left( h \right)\frac{\partial H}{{\partial z}}} \right] = C\left( h \right)\frac{\partial H}{{\partial t}}$$
(1.11)

where \(K_{z} (h)\) and \(K_{x} (h)\) are the permeability coefficients along the vertical direction and lateral direction in unsaturated soils, respectively. To study the groundwater flow of the unsaturated slope (Fig. 1.2), the RE needs to be rotated. Total head can be written as:

$$H = h + E$$
(1.12)

and elevation head E can be described as:

$$E = x\sin \beta + z\cos \beta$$
(1.13)
Fig. 1.2
A diagram of the slope ground surface that makes an angle beta with the horizontal plane has a depth of L. Uneven mesh nodes are arranged along that depth with labeled top, initial, and bottom boundary conditions.

A slope infiltration model

Full size image

By substituting Eqs. (1.12) and (1.13) into Eq. (1.11), Eq. (1.11) can be re-expressed as:

$$\frac{\partial }{\partial x}\left[ {K_{x} \left( h \right)\left( {\frac{\partial h}{{\partial x}} - \sin \beta } \right)} \right] + \frac{\partial }{\partial z}\left[ {K_{z} \left( h \right)\left( {\frac{\partial h}{{\partial z}} + \cos \beta } \right)} \right] = C\left( h \right)\frac{\partial h}{{\partial t}}$$
(1.14)

According to Iverson’s model (2000), the modified RE only considers infiltration in the vertical direction, which can be given by:

$$\frac{\partial }{\partial z}\left[ {K_{z} \left( h \right)\left( {\frac{\partial h}{{\partial z}} + \cos \beta } \right)} \right] = \frac{\partial \theta }{{\partial t}}$$
(1.15)

1.3 Linearized Richards’ Equation

Combined with an exponential model, Eq. (1.15) is linearized. Here, a new parameter \(h^{*}\) is defined as:

$$h^{*} = {\text{e}}^{\alpha h} - \lambda$$
(1.16)

where \(\lambda\) is the key parameter for solving the linearized Richards’ equation in the conversion method, which can be defined as a constant \(\lambda = {\text{e}}^{{\alpha h_{{\text{d}}} }}\) (Tracy 2006; Liu et al. 2015); hd is the pressure head value when the soil is dry. The relative permeability coefficient is expressed as:

$$K\left( h \right) = \frac{{K_{z} \left( h \right)}}{{k_{{\text{s}}} }} = {\text{e}}^{\alpha h}$$
(1.17)

Taking the derivative of Eq. (1.16) with respect to z, one can obtain:

$$\frac{{\partial h^{*} }}{\partial z} = \alpha {\text{e}}^{\alpha h} \frac{\partial h}{{\partial z}}$$
(1.18)

Equation (1.18) can be re-stated as follows:

$$\frac{\partial h}{{\partial z}} = \frac{1}{\alpha }{\text{e}}^{ - \alpha h} \frac{{\partial h^{*} }}{\partial z}$$
(1.19)

Furthermore, substituting Eq. (1.16) into Eq. (1.19), one can obtain:

$$K\frac{\partial h}{{\partial z}} = {\text{e}}^{\alpha h} \left( {\frac{1}{\alpha }{\text{e}}^{ - \alpha h} \frac{{\partial h^{*} }}{\partial z}} \right)$$
(1.20)

Equation (1.20) can be rewritten as:

$$K\frac{\partial h}{{\partial z}} = \frac{1}{\alpha }\frac{{\partial h^{*} }}{\partial z}$$
(1.21)

Equation (1.17) can be derived from z again:

$$\frac{\partial K}{{\partial z}} = \alpha {\text{e}}^{\alpha h} \frac{\partial h}{{\partial z}}$$
(1.22)

Substituting Eq. (1.19) into Eq. (1.22), one can have:

$$\frac{\partial K}{{\partial z}} = \frac{{\partial h^{*} }}{\partial z}$$
(1.23)

An exponential model is employed to describe soil moisture (Gardner 1958):

$$\theta \left( h \right) = \theta_{{\text{r}}} + \left( {\theta_{{\text{s}}} - \theta_{{\text{r}}} } \right){\text{e}}^{\alpha h}$$
(1.24)

where \(\theta (h)\) is volumetric water content; \(\theta_{{\text{r}}}\) is the residual volumetric water content.

The derivation of both sides of Eq. (1.24) with respect to time t has the following relationship:

$$\frac{\partial \theta }{{\partial t}} = \left( {\theta_{{\text{s}}} - \theta_{{\text{r}}} } \right)\frac{\partial K}{{\partial t}} = \left( {\theta_{{\text{s}}} - \theta_{{\text{r}}} } \right)\frac{{\partial h^{*} }}{\partial t}$$
(1.25)

Substituting Eqs. (1.19), (1.23), and (1.24) into Eq. (1.10), the linearized Richards’ equation can be obtained:

$$\frac{{\partial^{2} h^{*} }}{{\partial z^{2} }} + \alpha \frac{{\partial h^{*} }}{\partial z} = c\frac{{\partial h^{*} }}{\partial t}$$
(1.26)

where \(c = \alpha (\theta_{{\text{s}}} - \theta_{{\text{r}}} )/k_{{\text{s}}}\).

Equation (1.26) is also expressed as:

$$K_{a} \frac{{\partial^{2} h^{*} }}{{\partial z^{2} }} + K_{\theta } \frac{{\partial h^{*} }}{\partial z} = \frac{{\partial h^{*} }}{\partial t}$$
(1.27)

where \(K_{\theta } = k_{{\text{s}}} /\left( {\theta_{{\text{s}}} - \theta_{{\text{r}}} } \right)\), \(K_{a} = K_{\theta } /\alpha\).

The finite difference format of the linearized RE (Eq. 1.26) can be expressed as:

$$K_{a} \frac{{h_{i + 1}^{*n} - 2h_{i}^{*n} + h_{i - 1}^{*n} }}{{\Delta z^{2} }} + K_{\theta } \cos \beta \frac{{h_{i + 1}^{*n} - h_{i - 1}^{*n} }}{2\Delta z} = \frac{{h_{i}^{*n} - h_{i}^{*n - 1} }}{\Delta t}$$
(1.28)

where \(i,\Delta z,\Delta t\), and n denote the nodal point, grid size, time step, and time level, respectively.

It can be seen from Eqs. (1.26) and (1.27) that the nonlinear partial differential equation (Eq. 1.26) has been transformed into a linear partial differential equation. Once the linear partial differential equation is solved to obtain a numerical solution, the actual pressure head can be written as:

$$h\left( {z,t} \right) = \frac{1}{\alpha }\ln \left( {h^{*} \left( {z,t} \right) + \lambda } \right)$$
(1.29)

1.4 Unsaturated Soil Slope Stability Under Rainfall

Shear strength is a fundamental material property that is required to address a variety of engineering problems including bearing capacity, slope stability, lateral earth pressure, pavement design, and foundation design. Recently, many researches have focused on the shear strength of unsaturated soils (Fredlund et al. 1996; Lu et al. 2010).

According to Mohr–Coulomb criterion and effective stress, the shear strength of saturated soils can be expressed as:

$$\tau_{{\text{f}}} = c^{\prime} + \sigma^{\prime}\tan \varphi^{\prime}$$
(1.30)

where \(\tau_{{\text{f}}}\) is the shear strength (kPa), c is the cohesion (kPa), \(\sigma_{{\text{n}}}\) is the normal stress acting on the failure surface (kPa), and \(\varphi\) is the angle of internal friction (°). Cohesion and cohesive shear strength are due to chemical bonding between soil particles and surface tension within the water films (Lu and Likos 2006). Frictional shear strength (\(\sigma_{{\text{n}}} \tan \varphi\)) is owing to internal friction between soil particles that depends on the normal stress acting on the failure surface.

Engineering practices indicate that the shear strength equation of saturated soils can meet the engineering requirements. The shear strength parameters are also influenced by matric suction. With an increase in matric suction, c and \(\varphi\) increase, which depends on soil texture and structure. Soil shear strength significantly increases with an increase in net normal stress, matric suction, and the parameters of shear strengths.

However, several phases of unsaturated soils make the shear strength equation of saturated soils difficult to apply. Therefore, some studies on the shear strength criteria of unsaturated soils have been carried out. There are main representative shear strength criteria here.

Bishop (1959) developed a shear strength criterion for unsaturated soils:

$$\tau_{{\text{f}}} = c^{\prime} + \left[ {\left( {\sigma - u_{{\text{a}}} } \right)_{{\text{f}}} + \chi \left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)_{{\text{f}}} } \right]\tan \varphi^{\prime}$$
(1.31)

where \(\tau_{{\text{f}}}\) is the shear strength of unsaturated soils; c′ and \(\varphi^{\prime}\) are the effective cohesion and friction angle, respectively; (ua − uw) is the matric suction; ua is the pore air pressure; uw is the pore-water pressure (h = uw/γw, γw = ρwg); and \(\chi\) is the function of the degree of saturation.

Based on two stress state variables, the following equation was developed to describe shear strength (Fredlund and Rahardjo 1993):

$$\tau_{{\text{f}}} = c^{\prime} + \left( {\sigma - u_{{\text{a}}} } \right)_{{\text{f}}} \tan \varphi^{\prime} + \left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)_{{\text{f}}} \tan \varphi^{b}$$
(1.32)

where \(\varphi^{b}\) is the internal friction angle due to the distribution of matric suction.

Lu and Likos (2004) proposed a unified form of shear strength equation:

$$\begin{aligned} \tau_{{\text{f}}} & = c^{\prime} + \chi_{{\text{f}}} \left( {\sigma - u_{{\text{a}}} } \right)_{{\text{f}}} \tan \varphi^{\prime} + \chi_{{\text{f}}} \left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)_{{\text{f}}} \tan \varphi^{\prime} \\ & = c^{\prime} + c^{\prime\prime} + \left( {\sigma - u_{{\text{a}}} } \right)_{{\text{f}}} \tan \varphi^{\prime} \\ \end{aligned}$$
(1.33)

in which

$$c^{\prime\prime} = \chi_{{\text{f}}} \left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)_{{\text{f}}} \tan \varphi^{\prime}$$
(1.34)

The first two terms in Eq. (1.33), c′ and c″, represent shear strength due to the so-called apparent cohesion in unsaturated soils. In an unsaturated soil, the third term represents frictional shearing resistance provided by the effective normal force at the grain contacts. The apparent cohesion captured by the first two terms includes the classical cohesion c′ representing shearing resistance arising from interparticle physicochemical forces, and the second term c″ describing shearing resistance arising from capillarity effects. The term c″ is defined as capillary cohesion hereafter. Physically, capillary cohesion describes the mobilization of suction stress \(\chi\)(ua − uw) in terms of shearing resistance. The relationship between capillary cohesion and the maximum suction stress at failure, \(\chi_{{\text{f}}} \left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)_{{\text{f}}}\) is defined as shear strength also affects the water movement of the soils (Eudoxie et al. 2012).

Slope failure in unsaturated soil regions induced by rainfall is due to shear strength of unsaturated soils (Fredlund and Rahardjo 1993; Lu and Likos 2004; Guzzetti et al. 2008; Muntohar and Liao 2009). Both rainfall characteristics (rainfall intensity and duration) and soil permeability may influence failure mechanism.

The soil slope stability was commonly followed by stability analysis according to the pressure head and/or the stress condition within the soil slope profile. Various techniques were employed to compute factor of safety (Fs), and the conventional limit equilibrium methods (Alonso et al. 2010). The limit equilibrium approach is mostly effective for slope failure with a small depth compared with their length and breadth. A slope sliding at a depth happens as the driving stress contributing to failure exceeds the anti-slip stress offered by the soil mass strength. Namely, sliding can occur at a particular depth as follows:

$$F_{{\text{s}}} \left( {z,t} \right) = \frac{{\tan \varphi^{\prime}}}{\tan \beta } + \frac{{c^{\prime} - h\left( {z,t} \right)\gamma_{{\text{w}}} \tan \varphi^{b} }}{W\sin \beta \cos \beta }$$
(1.35)

where \(F_{{\text{s}}} \left( {z,t} \right)\) is the safety factor over depth and time; W is the weight of the sliding mass; and γw represents the unit weights of water.

Equation (1.35) can be re-arranged as:

$$h\left( {z,t} \right) = \frac{{c^{\prime}}}{{\gamma_{{\text{w}}} \tan \varphi^{b} }} - \frac{{\gamma_{{{\text{sat}}}} z\sin \beta \cos \beta }}{{\gamma_{{\text{w}}} \tan \varphi^{b} }}\left( {F_{{\text{s}}} - \frac{{\tan \varphi^{\prime}}}{\tan \beta }} \right)$$
(1.36)

in which, γsat represents the unit weight of the saturated soil. When Fs approaches 1, the infinite soil slope reaches a limit state. Based on Eq. (1.36), the limit-state pore-water pressure head can be obtained.

Rainfall-induced landslides may occur in unsaturated soils above the groundwater table, usually with shallow sliding surfaces parallel to the slope surface (Lu and Godt 2008), which involves 2D and 3D problems. However, an infinite slope model is usually used as a simplified model of the 2D or 3D issues with simple geometry and ignores the stress concentration, the practice sometime demonstrates its effectiveness for assessing shallow slope stability (Michalowski 2018).

Slope instabilities are often hydrologically initiated by the advancement of the wetting front alone (Muntohar and Liao 2010), a rise in groundwater level (Asch et al.1999; Montgomery et al. 2009), and positive pore-water pressure on the soil–bedrock boundaries (Baum et al. 2010). The most common mechanism for rainfall-induced landslides occurs when the soil slides on a low-conductivity layer. Rainfall infiltration leads to a rise in the pressure head, resulting in positive pore-water pressures (Iverson 2000; Muntohar and Liao 2010).

Generally, unsaturated soil slope failures happen most frequently during or after rain periods (Wu et al. 2020). The characteristics of water flow, change of pore-water pressure, and shear strength of soils are the major parameters related to slope stability analysis involving unsaturated soils that are directly affected by the boundary conditions (i.e., infiltration and evaporation) at the soil–atmosphere interface. The relative importance of soil properties, rainfall intensity, initial water table location, and slope geometry in inducing instability of soil slopes under different rainfall was investigated through a series of studies. Soil properties and rainfall intensity were found to be the primary factors controlling the slope instability due to rainfall, while the initial water table location and slope geometry only played a secondary role (Rahardjo et al. 2007).

The Green–Ampt model is a typical approximate infiltration model. Due to the simplicity and few parameters, the approximate infiltration model has become popular (Grimaldi et al. 2013). The classic GA model is only suitable for infiltration in horizontal soils. Therefore, modified GA models have been developed to describe the water infiltration in layered soils and slopes (e.g., Mein and Larson 1973; Chen and Young 2006; Kale and Sahoo 2011). Some modified infiltration models that account for rainwater redistribution have also been proposed (e.g., Corradini et al. 1997; Dou et al. 2014). These infiltration models have been extended to regional rainfall-runoff models for the hydrological prediction of catchments (Yuan et al. 2019). However, the actual infiltration process is very complicated and affected by many factors such as soil heterogeneity and rainfall conditions, and becomes difficult to be described accurately based on theoretical formulations (Srivastava et al. 2020). These theoretical equations generally tend to overestimate the factor of safety of soil slopes, resulting in slides and geological hazards (Kim et al. 2012). Some intelligent methods have been developed to predict the water infiltration into soils using machine learning techniques (Sihag et al. 2018).

Hydrological responses and slope factor of safety due to rainfall are concerned from a perspective of hydro-mechanical coupling. Coupled and uncoupled hydro-mechanical behaviors in unsaturated soils have been carried out to characterize the physical responses of unsaturated infiltration (i.e., variation of soil moisture, matric suction, effective stress, shear strength, and slope stability) (Casini 2013). The coupled issues are strongly linked in unsaturated soil slopes due to water infiltration, and the coupled poromechanical model actually examines the behavior and stability of rooted soils subjected to rainfall (Kim et al. 2012). Pressure heads generated in the uncoupled analysis are employed to examine deformation or soil slope stability (Cai and Ugai 2004; Yoo and Jung 2006). The accuracy and computational efficiency of the uncoupled analysis highly depend on the selected time increments (Huang and Lo 2013). The soil hydraulic and mechanical responses are calculated simultaneously in the coupled analysis. The coupled analysis produced a reasonably well defined wetting front and a lower critical Fs for unsaturated soil slopes. The coupled investigation could produce more accurate assessment of soil slope stability due to water infiltration and demonstrate a better physical representation of water infiltration and stress variation within unsaturated soil slopes.

More and more methods highlight the role of vegetation because of their interception role of the canopy and the root characteristics. Meanwhile, recent studies indicate that vegetation cannot control the rainfall-induced shallow landslide distribution (Emadi-Tafti et al. 2021). Some researches focus on the effect of roots on root–soil composite strength, or saturated hydraulic conductivity (Alessio 2019). The more complex the root architecture is, the stronger the root-composite strength becomes, while the faster the rainfall infiltrates. It has generally been concluded that vegetation roots mechanically and hydrologically affect slope stability. The plant roots seem act as a positive function in root-composite strength, while a negative role in water infiltration. Plant roots have various architectures in different land ecosystems and climatic conditions (Ma et al. 2018). Increasing studies related to soil–root complex focus on the root architectures (Burylo et al. 2011; Li et al. 2016a, b). One major controversy exists, e.g., the plant roots play positive role and enhance slope strength (Arnone et al. 2016). The roots could advance rainfall infiltration, thus contributing an adverse effect on slope stability (Ghestem et al. 2011; Garg et al. 2015). The root–soil composite strength and the hydraulic conductivity are of utmost importance for the rooted soil slope stability.