Introduction

Rainfall-induced landslides are common in many regions under tropical or subtropical climates (Lumb 1962; Brand 1984; Fourie 1996; Sidle and Ochiai 2006; Schuster and Highland 2007; Glade et al. 2006; Zhang et al. 2016a; Chen et al. 2017; Salciarini et al. 2017; Tang et al. 2017). The slope failures may occur on natural terrain slopes in a variety of materials including residual and colluvial soils (Dai et al. 2003; Crosta and Frattini 2003; Mondal and Mandal 2017; Vahedifard et al. 2017). The natural terrain slopes are usually composed of mixed coarse and fine particles due to deposition or weathering processes. In-situ full scale and laboratory reduced scale model tests showed that under rainfall infiltration, fine particles were transported from the upper part of the slope and slope surface to the toe due to internal erosion under the seepage flow (Crosta and di Prisco 1999; Hu et al. 2005; Jian et al. 2005).

Internal erosion has been extensively investigated in the stability of dams, levees and streambank (Fell et al. 2003; Wilson et al. 2007; Fox and Wilson 2010; Zhang and Chen 2006; Chang and Zhang 2010). For the problem of dam stability, internal erosion refers to the loss of soil particles within a dam or its foundation by seepage forces, and can be initiated by concentrated leak erosion, backward erosion, contact erosion, or suffusion (Fell and Fry 2007; Xu and Zhang 2009; Peng and Zhang 2012; Zhang et al. 2016b). In this study, we focus on the internal erosion due to suffusion, which involves detachment and migration of fine particles within the matrix of coarse soil particles under seepage flow.

Geometric conditions (e.g., grain size distribution, pore size distribution, grain shape, and pore shape), hydraulic conditions (e.g., hydraulic gradient, seepage direction, pore fluid velocity, chemical property of the fluid), and mechanical conditions (e.g., compaction and stresses) affect the potential for internal erosion (Kenney et al. 1985; Wan and Fell 2008; Schuler 1995; Chang and Zhang 2011, 2013b). Tremendous research studies have been conducted to establish criteria of the potential of internal erosion based on laboratory tests, physical model tests and field observations (Istomina 1957; Kezdi 1969; Kenney and Lau 1985; Wan and Fell 2008; Fannin and Moffat 2006; Li and Fannin 2008; Indraratna et al. 2011; Sato and Kuwano 2015; Chang and Zhang 2011, 2013a). Substantial efforts have also been made to model particle transport and internal erosion problems such as piping (Fujisawa et al. 2010), streambank or riverbank stability (Wilson et al. 2007; Darby et al. 2007), ground subsidence (Sterpi 2003; Cividini et al. 2009) and sand production from oil wells (Papamichos and Vardoulakis 2005) through analytical or numerical methods.

The previous studies focus mostly on fully saturated porous media. Limited studies have been conducted to investigate the influences of internal erosion on unsaturated soil slopes, especially under the condition of rainfall infiltration along the slope surface. In the problem of slope stability under rainfall infiltration, the soils are unsaturated above the ground water level (Bass et al. 2017). The seepage in the vadose zone is important for soil shear strength and global slope stability (Ciervo et al. 2015; Giuseppe et al. 2016). When transport of fine particles occurs in a slope, the porosity and permeability may change due to variation of pore space and grain size distribution. This may further lead to variation of water flow or movement of a wetting front and affect the global stability during infiltration.

The objective of this study is to develop a model for coupled seepage and internal erosion in unsaturated soil slopes. The variation of porosity and grain size distribution due to particle transport is considered. A parametric study is conducted for a hypothetical slope to investigate the effect of internal erosion on water infiltration and slope stability.

Coupled model of unsaturated flow and internal erosion

Governing equations of flow and erosion

The soil is regarded as a mixture of solid and liquid which consists of soil skeleton phase, a water phase and a phase of liquefied particles, which are fine particles scoured by internal erosion from the soil skeleton and can move freely with water. The liquefied fine particles satisfy the mass conservation equation (Cividini and Gioda 2004):

$$ \frac{\partial {\rho}_{tr}}{\partial t}+\mathit{\operatorname{div}}\left({\rho}_{tr}{v}_i\right)={q}_{er}-{q}_{dp} $$
(1)

where ρtr is the density of liquefied fine particles; vi is fluid velocity in the i direction; qer is the volume flux of fine particles eroded from the soil skeleton; and qdp is the volume flux of fine particles deposited on the soil skeleton.

The density of fine soil particles in the soil skeleton, ρf, is the difference between the initial density of a fine particle ρf0 before the internal erosion starts and the density of liquefied fine particles ρtr:

$$ {\rho}_f={\rho}_{f0}-{\rho}_{tr} $$
(2)

Therefore, the governing equation for the fine particles in two dimensions can be expressed as:

$$ \frac{\partial {\rho}_f}{\partial t}+\frac{\partial }{\partial x}\left({\rho}_f{v}_x\right)+\frac{\partial }{\partial y}\left({\rho}_f{v}_y\right)=-{q}_{er}+{q}_{dp} $$
(3)

The mass conservation of the soil skeleton phase can be expressed in a similar form as follows (Papamichos and Vardoulakis 2005):

$$ \frac{\partial {\rho}_{sk}}{\partial t}+\mathit{\operatorname{div}}\left({\rho}_{sk}{v}_i^{sk}\right)={q}_{dp}-{q}_{er} $$
(4)

where ρsk is the density of the soil skeleton and visk is the velocity of the soil skeleton. The density of soil skeleton ρsk is equal to (1-n)ρs, where ρs is the density of soil particles and n is the porosity. The velocity of the soil skeleton phase can be assumed to be zero. Therefore, the governing equation of the soil skeleton can be expressed as follows:

$$ \frac{\partial n}{\partial t}=\frac{q_{er}-{q}_{dp}}{\rho_s} $$
(5)

Based on the mass conservation and Darcy’s law, the governing equation for two-dimensional water flow in an unsaturated soil can be written as (Fredlund and Rahardjo 1993):

$$ \frac{\partial }{\partial x}\left({k}_x\left(\psi \right)\frac{\partial H}{\partial x}\right)+\frac{\partial }{\partial y}\left({k}_y\left(\psi \right)\frac{\partial H}{\partial y}\right)=\frac{\partial {\theta}_w\left(\psi \right)}{\partial t} $$
(6)

where kx and ky are the coefficients of permeability in x and y directions, respectively; θw is the volumetric water content of the unsaturated soil. For an unsaturated soil, the volumetric water content θw and coefficient of permeability k are both related with matric suction ψ = ua - uw, where ua is the pore-air pressure and uw is the pore-water pressure. The relation between volumetric water content θw and matric suction ψ is named as a Soil Water Characteristic Curve (SWCC). The relation between the coefficient of permeability k and matric suction ψ is called a permeability function. In this study, the air phase in unsaturated soil is assumed to be continuous and connects with the atmosphere. Hence, the pore-air pressure ua is assumed to be zero. The SWCC and permeability function are, hence, expressed as functions of pore-water pressure uw. Equations (3), (5) and (6) govern the coupled infiltration and internal erosion problem. It should be noted that only the change of porosity of the soil skeleton is considered in this study. The effects of deformation and stress changes on internal erosion should be conducted in a further study.

Constitutive laws for internal erosion

The constitutive law of internal erosion describes the process in which the water phase transforms the fine particles into a liquefied particle phase by erosion. Various erosion laws have been proposed in the literature. Most of these laws are empirical relations which relate the erosion rate to the flow characteristics (i.e., velocity, hydraulic gradient), porosity, density of particles, and empirical erosion coefficients (Partheniades 1965; Vardoulakis et al. 1996; Papamichos and Vardoulakis 2005; Govindaraju et al. 1995; Hanson and Simon 2001; Sterpi 2003; Seghir et al. 2014; Chetti et al. 2016).

In this study, the erosion law modified from Cividini and Gioda (2004) is used:

$$ {q}_{er}={\beta}_{er}\left|v\right|\left({\rho}_f-{\rho}_{f\infty}\right) $$
(7)

where βer is the coefficient of the erosion rate; |v| is the magnitude of flow velocity; ρf is the density of the fine particles in the soil; and ρf∞ is the long-term density of fine particles which can be expressed as:

$$ {\rho}_{f\infty }=\left\{\begin{array}{ll}{\rho}_{f0}-\left({\rho}_{f0}-{\rho}_f^{\ast}\right)i/{i}^{\ast }& 0\le i\le {i}^{\ast}\\ {}{\rho}_f^{\ast }-{\alpha}_{er}\log \left(i/{i}^{\ast}\right)& i>{i}^{\ast}\end{array}\right. $$
(8)

where i* is the hydraulic gradient when erosion starts; ρ*f is the density of fine particles corresponding to i*; and αer is a parameter of the erosion law. This erosion law implies that if sufficient time is allowed, the erosion process will remove the fine particles until the long-term density ρf∞ is reached (ρf0 → ρf∞).

In the original model proposed by Cividini and Gioda (2004), the flow velocity is used in quantifying ρf∞. Based on previous studies, the hydraulic gradient is often used to describe the critical hydraulic conditions which govern the onset of internal erosion. Therefore, a critical hydraulic gradient i* instead of a critical velocity v* is used to describe the initiation of erosion in this study. Figure 1 shows the long-term density of fine particles ρf∞ varies with the hydraulic gradient according to Eq. (8). The curve is based on fitting the laboratory test results of silty sands (Sterpi 2003; Cividini and Gioda 2004). In the experimental work by Sterpi (2003), the soil from the urban area of Milano (Italy) was used. The in situ soil consisted of well graded sand and gravel. For the laboratory tests, only the material passing through an ASTM 10 sieve (D = 2 mm) was considered and the soil particles passing through the standard ASTM 200 sieve (D = 0.074 mm), were referred to as fine particles. A standard ASTM 200 sieve was placed in the outflow and used to separate the fine particles from coarser grains possibly eroded from the soil primary fabric. The amount of coarser grains collected in the sieve was found to be negligible in all the series of experiments, even for the highest applied gradients. The initial fine particle content is 20% of ρt, where ρt is the total density of the soil. ρ*f is assumed to be 96.5% of ρf0, i.e., 19.3% of ρt. The critical hydraulic gradient i* is 0.18 and αer is 4.76. As shown in the graph, ρf∞ is reduced drastically if the hydraulic gradient is greater than i*.

Fig. 1
figure 1

Relation between the long-term density of fine particles ρf∞ and hydraulic gradient i

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Hydraulic functions of unsaturated soil

In this study, the van Genuchten-Mualem (van Genuchten 1980; Mualem 1976) model, is adopted for the SWCC and the permeability function of unsaturated soil. The van Genuchten-Mualem SWCC model is expressed as:

$$ {\theta}_w=\left\{\begin{array}{ll}{\theta}_r+\frac{\theta_s-{\theta}_r}{{\left[1+{\left[{\alpha}_w\left(-{u}_w\right)\right]}^{n_w}\right]}^{m_w}}& {u}_w<0\\ {}{\theta}_s& {u}_w\ge 0\end{array}\right. $$
(9)

where θs is the saturated volumetric water content; θr is the residual volumetric water content; αw is a curve fitting parameter inversely related to the air-entry value; nw is a curve-fitting parameter related to the pore size distribution; and mw is a parameter with mw = 1–1/nw (Mualem 1976).

The coefficient of permeability function is expressed as (van Genuchten 1980; Mualem 1976):

$$ k=\left\{\begin{array}{ll}{k}_s\frac{{\left\{1-{\left(-{\alpha}_w{u}_w\right)}^{n_w-1}{\left[1+{\left(-{\alpha}_w{u}_w\right)}^{n_w}\right]}^{-{m}_w}\right\}}^2}{{\left[1+{\left(-{\alpha}_w{u}_w\right)}^{n_w}\right]}_{m_w/2}}& {u}_w<0\\ {}{k}_s& {u}_w\ge 0\end{array}\right. $$
(10)

where ks is the saturated coefficient of permeability.

To consider the effect of internal erosion on the hydraulic properties, the saturated coefficient of permeability is defined as a function of porosity based on the Kozeny-Carman equation (Kozeny 1927; Carman 1937, 1956):

$$ {k}_s=\frac{n^3}{{\left(1-n\right)}^2}\times \frac{{\left(1-{n}_0\right)}^2}{{n_0}^3}{k}_{s0} $$
(11)

where n0 is the initial porosity before internal erosion starts; ks0 is the initial saturated coefficient of permeability corresponding to n0.

Finite element model and soil parameters

Figure 2 shows the finite element model of a hypothetical slope with a slope angle β of 35°. The slope is composed of residual soils. The thickness of the soil layer is 6 m. The finite element model is composed of 1100 quadrilateral elements.

Fig. 2
figure 2

Finite element model of the illustrative example

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For the governing equation of water flow in Eq. (6), the initial ground water table is along BC. Under the rainfall infiltration, the water levels along the boundaries AB, BC and CD are fixed as the initial ground water table. The infiltration flux along the slope surface AD is defined as (Chui and Freyberg 2009):

$$ flux={m}_N\left({u}_w\right)q+{m}_b\left({u}_w\right){R}_b\left(-{u}_w/{\gamma}_w\right) $$
(12)

where flux = infiltration flux; mN, mb = complementary smoothing functions (Fig. 3); Rb = external resistance; and q = rain intensity. This infiltration flux boundary condition implies that when the pore water pressure at the slope surface AD is negative, the infiltration flux is equal to rainfall intensity q. When the pore water pressure at the slope surface AD is positive, the infiltration flux is negative to guarantee no ponding on the slope surface.

Fig. 3
figure 3

Function mN and mb

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The governing equation of ρf, Eq. (3) is a first-order partial differential equation. The mass transport is due to convection. The initial density of fine particles is set to a given value ρf0. The velocity of a liquefied particle is zero along the boundaries AB, BC and AD. The density of fine particles at the lower vertical boundary CD is set to be 50% ρf0 in order to prevent accumulation of particles at the lower boundary. The governing equation of porosity n, Eq. (5), is an ordinary differential equation. The variation of porosity only depends on the internal erosion. The initial condition is a homogenous porosity field and no boundary conditions are necessary.

The unsaturated hydraulic parameters are assumed to represent silty sands (Santoso et al. 2011). The SWCC and permeability function of the soil are shown in Fig. 4. The saturated permeability of the soil is assumed to be 2 × 10−4 m/s. The erosion parameters are assumed based on the laboratory test results of silty sands in Sterpi (2003). The initial fine particle content is 20% of ρt, where ρt is total density of the soil. ρ*f is assumed to be 96.5% of ρf0, i.e., 19.3% of ρt. The other parameters are obtained by fitting laboratory test curves (Cividini and Gioda 2004). The critical hydraulic gradient i* is 0.18. The αer and βer are 4.76 and 1.95 × 10−3, respectively. Table 1 summarizes the input parameters in the numerical model.

Fig. 4
figure 4

The SWCC and permeability function of the residual soil

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Table 1 List of parameters in the numerical model
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Numerical simulation and slope stability analysis

A finite element model based on the coupled governing equations [Eqs. (3), (5), and (6)] is developed in the multiphysics modeling finite element program, COMSOL (COMSOL 2010). First, the partial differential equations (PDEs) [Eqs. (3), (5), and (6)] are formulated in the PDE coefficient form module using the graphical user interface (GUI). The boundary conditions and initial conditions are also formulated in PDE coefficient forms. Then, the Galerkin method is used to discretize the partial differential equations. A nonlinear differential algebraic equation solver IDA that was created by the Lawrence Livermore National Laboratory (Hindmarsh et al. 2005) is used as a time integrator for the differential algebraic system, in which the backward differentiation formulas (BDF) are used to discretize the time derivative terms. The solver name IDA stands for Implicit Differential-Algebraic. As the time stepping scheme is implicit, the damped Newton method (Deuflhard 1974) is then used to solve the resulting nonlinear equations. Within each step of the Newton iteration, the most recently updated nodal values of the dependent variables are used to compute nonlinear coefficients. The nonlinear iteration at each time step is continued until the numerical solutions satisfy prescribed convergence criteria for pressure head and displacement. In this study, the absolute tolerance is set to be 0.001, and the relative tolerance is set to be 0.01.

Since slope failures induced by rainfall infiltration are usually shallow, infinite slope stability analysis may be appropriate for evaluating the stability of soil slopes under rainfall. The safety factor for the slip surface at depth h (Figure 2) is (Iverson 2000; Cho and Lee 2002):

$$ {F}_s=\frac{c^{\prime }}{\gamma_th\sin \beta \cos \beta }+\frac{\tan {\phi}^{\prime }}{\tan \beta }-\frac{h_w{\gamma}_w\tan {\phi}^b}{\gamma_th\tan \beta } $$
(13)

where β is the slope angle; γt is the total unit weight of the soil; c’ is the effective cohesion; ϕ′ is the effective friction angle; h is the depth of the slip surface (Fig. 2); hw is the pore water pressure head; γw is the unit weight of water; and ϕb is the angle indicating the rate of increase in shear strength related to matrix suction. In the example shown in Fig. 2, c′ is assumed to be 2 kPa. The value of ϕb depends on the matric suction and is generally around 1/3 to 2/3 of ϕ′; here, ϕ′ and ϕb are assumed to be 34° and 25°, respectively. The distribution of the pore water pressure head from the numerical simulation is used in calculating the safety factor.

Analysis results and discussion

Effects of internal erosion on infiltration and slope stability

The slope is subject to a rainfall with a constant flux of q = 2×10−4 m/s. To analyze the influence of internal erosion on the slope stability, the results of the coupled analysis of flow and internal erosion are compared with those without considering internal erosion.

As shown in Fig. 5, the depth of the wetting front is deeper when considering internal erosion. After 68 min of rainfall, the unsaturated zone is fully saturated if internal erosion is considered. However, there is still a 2.0 m thick soil that is unsaturated if the internal erosion is neglected. The differences between the coupled and uncoupled analysis become more significant as the infiltration progresses.

Fig. 5
figure 5

Variation of pore-water pressure profile along cross-section X–X′ in Fig. 2 (ks0 = 2 × 10−4 m/s, βer = 1.95 × 10−3)

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Figure 6 shows the variation of fine particle content and porosity along the cross-section X–X′ in the middle part of the slope. It can be seen that internal erosion occurs mainly at the shallow depths of the slope. The variation of the fine content and porosity in the shallow soil develop almost simultaneously with the advance of the wetting front. The fine particles around the groundwater table are also eroded slightly. This is mainly because, when the soil is almost saturated, the flow velocity is large enough to transport the fine particles based on the erosion law in Eq. (7).

Fig. 6
figure 6

Variation of (a) fine particle content ρf/ρf0 and (b) porosity with time along cross-section X–X′ (ks0 = 2 × 10−4 m/s, βer = 1.95 × 10−3)

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Figure 7 presents the profiles of the hydraulic gradient at different times of rainfall. The hydraulic gradients are all greater than 0.57 in the unsaturated zone. Extremely large hydraulic gradients of more than 8 appear near the wetting fronts. As the unsaturated zone is fully saturated (t = 68 min), the gradient is reduced significantly but is still greater than 1.0.

Fig. 7
figure 7

Hydraulic gradients along cross-section X–X′ (ks0 = 2 × 10−4 m/s, βer = 1.95 × 10−3)

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Figure 8 shows the safety factor of the slope along various depths of the slip surface. The soil slope reaches an unstable state more rapidly when considering the internal erosion. After 68 min of infiltration, the critical slip surface is around 3.0 m, and the factor of safety of the slope is less than 1.0. However, the factor of safety of the slope without internal erosion is still greater than 1.0. This result implies that internal erosion may lead a more abrupt landslide.

Fig. 8
figure 8

Profile of safety factor at different times (ks0 = 2 × 10−4 m/s, βer = 1.95 × 10−3)

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Effects of erosion parameters

As shown in Eqs. (7) and (8), the four parameters, βer, i*, ρ*f, and αer can influence the development of erosion. The ρ*f represents the fine particle density corresponding to the initiation of erosion and is generally very close to the initial value of ρf. Hence, in this example, ρ*f is assumed to be constant.

Terzaghi (1939) defined a critical hydraulic gradient for the upward flow condition in which the soil is unstable when the effective stress becomes zero. The critical hydraulic gradient is typically about 1.0. Skempton and Brogan (1994) showed that the critical hydraulic gradient for internally unstable sandy gravels is only about one fifth to one third (i.e., 0.2 to 0.34) of the theoretical value given by Terzaghi (1939). Wan and Fell (2008) investigated silt-sand-gravel or clay-silt-sand-gravel soils and found loose higher porosity soils began to erode at gradients less than 0.3. The critical hydraulic gradient in a horizontal flow condition for very unstable gravel soils could be even smaller (i.e., 0.16 as reported in Adel et al. 1988, samples are of large dimensions and in a loose state of packing). The explanation for the small critical gradients is that a major part of the overburden load is carried by the skeleton of gravel, leaving the fine particles under relatively small pressures (Skempton and Brogan 1994). These previous studies confirm that the measured value of 0.18 for i* in Sterpi (2003) is reasonable for unstable soils.

Very limited studies have laboratory measurements of the amount of removed particles lost with time (Chang and Zhang 2013a). Therefore, the possible range of the fitting erosion parameters βer and αer can hardly be obtained. The hole erosion test (HET) and the slot erosion test (SET) have been developed to study rates of erosion and the critical hydraulic shear stress to initiate piping erosion. An approximately linear relation between the rate of erosion and the applied hydraulic shear stress has been found based on the laboratory test results (Hanson and Simon 2001; Briaud et al. 2001; Wan and Fell 2004; Bonelli and Brivois 2008; Chang et al. 2011). The empirical relation is expressed as:

$$ \dot{\varepsilon}={C}_e\left(\tau -{\tau}_c\right) $$
(14)

where \( \dot{\varepsilon} \) is the erosion rate of the soil per unit surface area of the slot/hole (kg/s/m2); Ce is the coefficient of soil erosion; τc is the critical shear stress of soil at initiation of erosion (N/m2); and τ is the hydraulic shear stress along the slot/hole (N/m2) which can be obtained based on the hydraulic gradient across the soil sample. For example, in the HET, the hydraulic shear stress along the pre-formed hole is:

$$ {\tau}_{HET}={\gamma}_w{i}_{HET}\frac{\phi_{HET}}{4} $$
(15)

where τHET is the hydraulic shear stress on the surface of the pre-formed hole; iHET is the hydraulic gradient across the soil sample; and ϕHET is the diameter of the hole, assumed to remain circular.

According to Wan and Fell (2004), the coefficient of soil erosion Ce obtained for the various HETs or SETs are on the order of 10−1 to 10−6. Equation (7) is similar to Eq. (14) as both equations are approximately linear. Hence, coefficient βer is similar to the coefficient of soil erosion Ce and can vary within a large range depending on the basic soil properties such as grain size distribution, relative density, etc. In this example, the effects of the erosion parameters βer and αer on erosion and infiltration are investigated by parametric studies.

Figure 9 shows the variation of fine content and pore-water pressure profiles when βer is increased to 0.005 and 0.01. Comparing Fig. 9 to Fig. 5, the dissipation of matric suction is much faster, and the difference between the coupled and uncoupled analysis is greater when βer is increased from 1.95 × 10−3 to 0.01. The effect of erosion on the wetting front movement is more significant with a larger value of βer. The main reason is that when βer is increased, the erosion rate increases (Eq. 7). Consequently, the porosity is increased more significantly and the hydraulic conductivity in the unsaturated zone is significantly increased.

Fig. 9
figure 9

Variation of pore-water pressure profiles along cross-section X–X′ (a) βer = 0.005 and (b) βer = 0.01

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Figure 10 shows the pore water pressure profiles in the unsaturated zone with different values of i*. When the critical gradient of initiation is increased to 1.0, the effect of the internal erosion on the advance of the wetting front is reduced. The result of i* of 0.5 is not much different from that of 1.0.

Fig. 10
figure 10

Comparison of pore-water pressure profiles with different values of i*

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As shown in Eq. (8), the erosion parameter αer influences the long term density of fines ρf∞. Figure 11 shows that after the initiation of erosion, the slope of the curve for ρf∞ is gentler if αer is smaller. Hence, the long-term density of fine particles, which can be maintained in the slope, is greater even if the flow field does not change. Figure 12 compares the pore water pressure profiles for different values of αer. As shown in the graph, when αer is 0.5, the effect of erosion on pore water pressure distributions is reduced.

Fig. 11
figure 11

Effect of the parameter αer on the erosion function ρf∞

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Fig. 12
figure 12

Comparison of pore-water pressure profiles for different values of αer

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Comparing Figs. 9, 10, and 12, βer is the main factor that affects the internal erosion in a slope. The effect of erosion on the wetting front movement is more significant with large values of βer. The effects of i* and αer are less significant when compared with βer.

Effect of soil hydraulic parameters

To investigate the effects of soil hydraulic properties on the coupled analysis, the saturated coefficient of permeability and the air-entry value parameter of the SWCC model, which have been identified as the two most important hydraulic parameters (Zhang et al. 2016a), are selected for a parametric study.

Three different values of the initial saturated coefficient of permeability ks0, 5 × 10−5, 2 × 10−4 and 8 × 10−4 m/s are adopted in the parametric study. The other parameters are the same as those in Table 1. Figure 13 shows the pore water pressure profiles with different values of ks0. For a soil slope with ks0 equal to 5 × 10−5 m/s (Fig. 13a), the internal erosion has little effect on the pore water pressure profile before 80 min of rainfall. After 80 min, the advance of the wetting front becomes faster and the effect of internal erosion is more obvious. However, the effect of internal erosion is still less significant than the case with ks0 equal to 2 × 10−4 m/s. When ks0 is increased to 8 × 10−4 m/s, the rainfall intensity is smaller than the saturated permeability, and; hence, the matric suction can be maintained even after a long duration of rainfall (Zhang et al. 2004). The internal erosion increases the depth of the wetting front only slightly for this case. Comparing the three graphs in Fig. 13, the influence of internal erosion on water infiltration is significant only when the rainfall flux is equal or greater than the saturated coefficient of permeability (q/ks0 ≥1).

Fig. 13
figure 13

Variation of the pore-water pressure profile along cross-section X–X′ with different values of ks0: a 5 × 10−5 m/s, b 2 × 10−4 m/s, and c 8 × 10−4 m/s (βer = 1.95 × 10−3)

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Figure 14 presents the comparison of the pore water pressure profiles with different values of ks0 for a more instable soil (βer = 0.01). It shows that the internal erosion is more significant for the two cases with q/ks0 ≥1. There are some effects of erosion for the case with ks0 equal to 8 × 10−4 m/s (q/ks0 < 1). However, this minor effect of internal erosion can be considered as negligible even for the more instable soil.

Fig. 14
figure 14

Variation of the pore-water pressure profile along cross-section X–X′ with different values of ks0: a 5 × 10−5 m/s, b 2 × 10−4 m/s, and c 8 × 10−4 m/s (βer = 0.01)

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The air-entry value (AEV) of a soil is the matric suction where air starts to enter the largest pores in the soil. In the van Genuchten-Mualem SWCC model, the curve-fitting parameter αw is inversely related to the air-entry value. In this study, the responses of soil slopes with different values of αw, 0.04, 0.08 and 0.12, which correspond to AEV values of 25,12.5 and 8.33 kPa, respectively, are compared. The soil water characteristic curves and unsaturated permeability functions of the unsaturated soils with different values of αw are presented in Fig. 15. Figure 16 shows the pore water pressure profiles in the soil slopes with different values of αw. To demonstrate the effect of the parameter αw more clearly, βer is taken as 0.01. As shown in the graphs, with a smaller αw (a larger AEV), the wetting process is faster in the unsaturated zone. When αw is equal to 0.04, the effect of internal erosion on infiltration is initially very limited and becomes observable only when the unsaturated zone is almost fully saturated. When αw is 0.12, the influence of internal erosion on infiltration is observable after the depth of wetting front is less than 2 m. As a coarser soil is usually related with a smaller AEV and, hence, a larger αw, these results imply that the internal erosion may be more influential in a coarse soil slope.

Fig. 15
figure 15

Soil water characteristic curves and unsaturated permeability functions of the unsaturated soils with various values of αw

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Fig. 16
figure 16

Pore-water pressure profiles with different values of αw: a 0.04, b 0.08 c 0.12 (βer = 0.01)

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Conclusions

In this study, a coupled model of seepage and internal erosion in unsaturated soil is established to investigate the effects of internal erosion on water infiltration and slope stability. The major conclusions are as follows:

  1. 1.

    The soil porosity increases with the advance of the water front and the procession of internal erosion. The advance of the wetting front in the slope is more rapid and the slope reaches an unstable state more quickly due to internal erosion.

  2. 2.

    The parameter βer of the erosion law is the major factor that affects the internal erosion in a slope. The effect of erosion on the wetting front movement is more significant with large values of βer. The effects of parameters i* and αer are less significant when compared with βer.

  3. 3.

    When the rainfall flux is equal or greater than the saturated coefficient of permeability (q/ks0 ≥1), the influence of internal erosion on water infiltration and slope stability is more significant. The effect of internal erosion can be neglected as long as the rainfall flux is less than the saturated coefficient of permeability.

  4. 4.

    When the soil is coarser and the air-entry value of the soil is smaller, the influence of internal erosion on infiltration becomes more significant.

In this study, the measurement or estimation of the erosion and hydraulic parameters, especially βer, ks0 and the air entry value parameter αw, are essential when analyzing a slope problem in practice using the proposed model. It is suggested the saturated permeability and the erosion parameters should be measured through laboratory column tests. The unsaturated hydraulic functions can be estimated through the grain size distribution or soil basic properties (Fredlund et al. 2002; Johari et al. 2006; Vereecken et al. 2010; Zhang and Chen 2005; Li et al. 2014).

The significance of the proposed method is that the method provides a practical and quantitative tool to evaluate the effects of internal erosion on rainfall-induced landslides. It should be noted the proposed coupled model of flow and internal erosion is based on the basic assumptions of unsaturated transient flow, e.g., Darcy’s law, and the mass conservation of soil skeleton. The erosion law adopted is an empirical relation based on experimental results of a specific soil. The particle transport under the condition of non-Darcy’s flow (Beven and Germann 1982; Chen et al. 2015) is not considered. The effect of particle deposition (Chen and Zhang 2015) and the deformation of the soil skeleton (Cho and Lee 2001; Alonso et al. 2003; Ye et al. 2005; Chen et al. 2009; Wu and Zhang 2009; Borja and White 2010; Cascini et al. 2010; Xiong et al. 2014) are not considered.