Analytical Solution to Unsaturated Infiltration

Abstract

Rainfall infiltration in unsaturated soil slopes is a classic issue in geotechnical engineering (Conte and Troncone in Géotechnique 62:87–91, 2012; Morbidelli et al. in J Hydrol 557:878–886, 2018). Factors influencing the soil slope stability due to rainwater infiltration comprise the rainfall characteristics, the saturated permeability coefficient, the geometry of the slope, and the boundary and initial soil moisture conditions (Ali et al. in Comput Geotech 61:341–354, 2014; Wu et al. in Hydro-mechanical analysis of rainfall-induced landslides. Springer, 2020).

2.1 Introduction

Rainfall infiltration in unsaturated soil slopes is a classic issue in geotechnical engineering (Conte and Troncone 2012; Iverson 2000; Morbidelli et al. 2018). Factors influencing the soil slope stability due to rainwater infiltration comprise the rainfall characteristics, the saturated permeability coefficient, the geometry of the slope, and the boundary and initial soil moisture conditions (Ali et al. 2014; Wu et al. 2020).

The spatial and temporal evolution of unsaturated infiltration involves a governing partial differential equation that is expressed by the Richards’ equation (1931). The equation is highly nonlinear because the hydraulic conductivity and the pressure head depend on matric suction or moisture content. A number of exact and approximated analytical solutions to the infiltration equation were derived in past studies (e.g., Parlange et al. 1997; Basha 2011). Analytical solutions of the linearized infiltration equation were derived as an integral (Chen et al. 2003), as a Laplace transformation (Zhan et al. 2013), and as a Green’s function (Basha 1999). While numerical approaches can effectively simulate complex nonlinear infiltrations into an unsaturated soil (Tracy 2006; Wu et al. 2020), analytical solutions can verify these numerical procedures. The incorporation of physically based infiltration expressions better quantifies the infiltration models and causes a more reliable prediction of the water infiltration (Basha 2011). Compared with numerical solutions, the analytical methods are widely used because they can accurately check numerical methods, and concisely represent the pressure head variations with rainfall infiltration (Godt et al. 2012; Wu et al. 2020). Several analytical solutions to Richards’ equation were obtained using integral transform methods (Laplace and Fourier algorithms) and others (Qin et al. 2010). The interaction between soil infiltration and displacement was defined as hydro-mechanical coupling effect (Wu et al. 2020). The coupled equation is characterized by strong nonlinearity; thus, linearization is required when solving the equation (Li et al. 2013; Zhu et al. 2022). To derive the analytical solution to infiltration equation considering the coupled hydro-mechanical effect, the soil–water characteristic curve (SWCC) is represented using an exponential form, and then the analytical solution of pressure head with arbitrary initial conditions can be developed using integral transformation and other methods (Li and Wei. 2018; Wu et al. 2009, 2012, 2016, 2018, 2020).

Many natural slopes are covered with vegetation, which can hydraulically and mechanically affect the slope stability (Lynch 1995; Leung et al. 2017). Vegetation root water uptake changes the pore-water pressure or pressure head of slopes, which is defined as the hydraulic effect of vegetation on slopes (Leung et al. 2015; Ni et al. 2018; Wu et al. 2021). Water is transferred by roots from soil voids to upper stems and leaves. The root water uptake rate and hydraulic conductivities are influenced by a number of factors including the soil type, self-hydraulic resistance, and hydraulic resistance of surrounding soils (Nyambayo and Potts 2010). Based on exponential function SWCC, the analytical solutions for water unsaturated infiltration were developed using Green’s function (Ng et al. 2015). The effect of hydrological conditions on the stability of vegetated soil slopes was investigated (Liu et al. 2016; Wu et al. 2022).

This chapter is to derive an analytical solution incorporating both the root contribution and hydro-mechanical coupling. The governing equation considering vegetated root and coupled infiltration and deformation is developed. The analytical solution is obtained using Green’s function. Parametric analyses are carried out to investigate the effect of factors on the vegetated slope stability.

2.2 1D Analytical Solutions for Unsaturated Seepage

2.2.1 No Coupling

The one-dimensional infiltration equation can be expressed as (Richards 1931):

$$\frac{\partial }{\partial z}\left[ {K\left( h \right)\left( {\frac{\partial h}{{\partial z}} + 1} \right)} \right] = \frac{\partial \theta }{{\partial t}}$$

(2.1)

where h is the pressure head; z is the vertical direction; t is the rainfall time.

Based on Gardner model (1958) and variable definition ($h^{*} = {\text{e}}^{\alpha h} - \chi$ in Chap. 1, where α is the desaturation coefficient, $\chi$ = ${\text{e}}^{{\alpha h_{{\text{d}}} }}$), the linearized Richards’ equation can be obtained:

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

(2.2)

where $c = \alpha \left( {\theta_{{\text{s}}} - \theta_{{\text{r}}} } \right)/K_{{\text{s}}}$.

Tracy (2006) developed an analytical solution to transient infiltration in unsaturated soils:

$$h_{{\text{a}}} \left( {z,t} \right) = \frac{1}{\alpha }\ln \left( {h_{{\text{t}}}^{*} \left( {z,t} \right) + h_{{\text{s}}}^{*} \left( z \right) + {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)$$

(2.3)

in which,

$$h_{{\text{s}}}^{*} \left( z \right) = \left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right) \times \left( {1 - {\text{e}}^{ - \alpha z} } \right)/\left( {1 - {\text{e}}^{ - \alpha L} } \right)$$

(2.4)

$$h_{{\text{t}}}^{*} \left( {z,t} \right) = \frac{{2\left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)}}{Lc}{\text{e}}^{{\alpha \left( {L - z} \right)/2}} \sum\limits_{m = 1}^{\infty } {\left( { - \,1} \right)^{m} } \left( {\frac{{\lambda_{{\text{m}}} }}{{\mu_{{\text{m}}} }}} \right)\sin \left( {\lambda_{{\text{m}}} z} \right){\text{e}}^{{ - \mu_{{\text{m}}} t}}$$

(2.5)

where $\lambda_{{\text{m}}} = m\pi /L$ and $\mu_{{\text{m}}} = \left( {\alpha^{2} /4 + \lambda_{{\text{m}}}^{2} } \right)/c$.

For unsaturated slopes (Fig. 2.1), the governing equations are modified as follows:

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

(2.6)

Fig. 2.1

Top and bottom flux boundaries for a soil profile with a finite thickness. The soil layer is between z = 0 and z = l, where z is the vertical coordinate: a impermeable boundary; and b groundwater level at the bottom

where $\beta$ is the slope angle. The analytical solution of the seepage equation along the z-axis can be expressed as:

$$h_{{\text{t}}}^{*} \left( {z,t} \right) = \frac{{2\left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)}}{Lc}{\text{e}}^{{\alpha \cos \beta \left( {L - z} \right)/2}} \sum\limits_{m = 1}^{\infty } {\left( { - \,1} \right)^{m} } \left( {\frac{{\lambda_{{\text{m}}} }}{{\mu_{{\text{m}}} }}} \right)\sin \left( {\lambda_{{\text{m}}} z} \right){\text{e}}^{{ - \mu_{{\text{m}}} t}}$$

(2.7)

$$h_{{\text{s}}}^{*} \left( z \right) = \left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)\frac{{1 - {\text{e}}^{ - \alpha \cos \beta z} }}{{1 - {\text{e}}^{ - \alpha \cos \beta L} }}$$

(2.8)

2.2.2 Hydro-mechanical Coupling

Based on Darcy’s law, mass and momentum conservation, the equation that governs 1D hydro-mechanical coupling in unsaturated soils, including the consideration of increases in the water table, can be given by (Lloret 1987; Wu et al. 2020):

$$\frac{\partial }{\partial z}\left[ {k\frac{\partial }{\partial z}\left( {h + z} \right)} \right] = \left( {n\beta_{{\text{w}}} S_{{\text{r}}} \gamma_{{\text{w}}} + n\frac{{\partial S_{{\text{r}}} }}{\partial h}} \right)\frac{\partial h}{{\partial t}} - S_{{\text{r}}} \alpha_{{\text{c}}} \frac{{\partial \varepsilon_{z} }}{\partial t}$$

(2.9)

$$\frac{\partial }{\partial z}\left[ {E\varepsilon_{\text{v}} - \frac{E}{F}\left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)} \right] + \left[ {nS_{{\text{r}}} \rho_{{\text{w}}} + \left( {1 - n} \right)\rho_{{\text{s}}} } \right]g = 0$$

(2.10)

where ε_v (ε_v = ε_z for one-dimensional problem) is the total volumetric strain of the soil mass, which is greater than zero during compression and less than zero during swelling; n is the percentage of voids; S_r is the saturation; (u_a − u_w) is the matric suction; α_c is the hydro-mechanical coupling coefficient (0 ≤ α_c ≤ 1), which is determined by the bulk modulus of the solid skeleton and the bulk modulus of the solid soil (Wu et al. 2020); E is the elastic modulus of the soil that takes account of alterations in the net normal stresses (Wu et al. 2020); F is the elastic modulus of the soil due to variations in matric suction, which is assumed to be a function of stress; β_w is the compressibility of the fluid; ρ_w is the density of water; ρ_s is the density of the soil phase; and g is the acceleration of gravity.

Equations (2.9) and (2.10) govern the coupling of 1D deformation and seepage in unsaturated soils and account for the groundwater table changes.

On the assumption that pore-air pressure in the soil mass is kept constant; the derivative of Eq. (2.10) with respect to t can be written as:

$$\frac{{\partial \varepsilon_{\text{v}} }}{\partial t} = - \frac{{\gamma_{{\text{w}}} }}{F}\frac{\partial h}{{\partial t}}$$

(2.11)

From Eqs. (2.9) and (2.11), the following equation can be derived:

$$\frac{\partial }{\partial z}\left[ {k\frac{\partial }{\partial z}\left( {h + z} \right)} \right] = \left( {\beta_{{\text{w}}} \frac{\partial \rho }{{\partial h}} + n\frac{{{\text{d}}S_{{\text{r}}} }}{{{\text{d}}h}}} \right)\frac{\partial h}{{\partial t}} + \frac{{\gamma_{{\text{w}}} S_{{\text{r}}} \alpha_{{\text{c}}} }}{F}\frac{\partial h}{{\partial t}}$$

(2.12)

Based on assumption that the coefficient of permeability and the moisture content vary exponentially with the pore-water pressure head, that is, k(h) = k_se^αh and θ(h) = θ_se^αh (h < 0, t > 0). Here, k_s is the saturated hydraulic conductivity, θ_s is the volumetric moisture at saturation. Because S_r = θ(h)/θ_s when the fluid compressibility is neglected (β_w = 0), Eq. (2.12) can be written as follows:

$$\frac{\partial }{\partial z}\left[ {k\frac{\partial }{\partial z}\left( {h + z} \right)} \right] = M{\text{e}}^{\alpha h} \frac{\partial h}{{\partial t}}\quad \left( {t > 0} \right)$$

(2.13)

where M = θ_sα + γ_wα_c/F.

With dimensionless variables Z = αz and T = α²k_st/M introduced into a function of W(Z, T) = e^αh · e^Z⁄2+T/4, Eq. (2.13) can be rewritten as:

$$\frac{\partial W}{{\partial T}} = \frac{{\partial^{2} W}}{{\partial Z^{2} }}$$

(2.14)

The boundaries comprise a base and top one in Fig. 2.1. In the literature of analytical solutions, the base boundary was usually assumed to coincide with a stationary groundwater table and the pressure head was set zero (Wu et al. 2020). However, in this book, a zero flux is considered at the base boundary. The hydraulic boundary condition is given by:

$$h\left| \begin{gathered} \hfill \\_{z = 0} \hfill \\ \end{gathered} \right. = 0$$

(2.15)

or

$$\, k\frac{\partial h}{{\partial z}} + k\left| \begin{gathered} \hfill \\_{z = 0} \hfill \\ \end{gathered} \right. = 0$$

(2.16)

The top boundary in Fig. 2.1 is controlled by pressure head or rainfall intensity (q) at the ground surface, and it is written as:

$$h\left| \begin{gathered} \hfill \\_{z = l} \hfill \\ \end{gathered} \right. = h_{0}$$

(2.17)

or

$$k\frac{\partial h}{{\partial z}} + k\left| \begin{gathered} \hfill \\_{z = l} \hfill \\ \end{gathered} \right. = q\left( t \right)$$

(2.18)

in which, l is the depth in the 1D unsaturated infiltration model.

Based on a Fourier integral transformation (Ozisik 1989), the exact solution to Eq. (2.14) can be derived considering different boundaries (Wu et al. 2020).

2.3 2D Analytical Solutions of Rainfall Infiltration in Unsaturated Soils

The 2D Richards’ equation in mixed format is expressed as:

$$\frac{\partial }{\partial x}\left[ {K_{x} \left( h \right)\frac{\partial h}{{\partial x}}} \right] + \frac{\partial }{\partial z}\left[ {K_{z} \left( h \right)\left( {\frac{\partial h}{{\partial z}} + 1} \right)} \right] = \frac{\partial \theta }{{\partial t}}$$

(2.19)

in which, K_x and K_z are the hydraulic conductivities along x- and z-directions, respectively.

Similarly, the 2D linearized Richards’ equation can be written as:

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

(2.20)

The mathematical model is shown in Fig. 2.2. L and W represent the height and length, respectively. The normalized boundary conditions are as follows:

Fig. 2.2

Schematic diagram of 2D transient seepage model in unsaturated soils

$$h\left( {0,z,t} \right) = h_{{\text{d}}}$$

(2.21)

$$h\left( {W,z,t} \right) = h_{{\text{d}}}$$

(2.22)

$$h\left( {x,0,t} \right) = h_{{\text{d}}}$$

(2.23)

$$h\left( {x,L,t} \right) = \ln \left( {\left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)\sin \left( {\frac{\pi x}{W}} \right) + {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)/\alpha$$

(2.24)

The normalized analytical solutions $h_{{\text{t}}}^{*} \left( {x,z,t} \right)$ and $h_{{\text{s}}}^{*} \left( {x,z,t} \right)$ for this two-dimensional model and can be expressed as follows (Tracy 2006):

$$h_{{\text{t}}}^{*} \left( {x,z,t} \right) = \frac{{2\left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right)}}{Lc}\sin \left( {\frac{\pi x}{W}} \right){\text{e}}^{{\alpha \left( {L - z} \right)/2}} \sum\limits_{m = 1}^{\infty } {\left( { - \,1} \right)^{m} } \left( {\frac{{\lambda_{{\text{m}}} }}{{\mu_{{\text{m}}} }}} \right)\sin \left( {\lambda_{{\text{m}}} z} \right){\text{e}}^{{ - \mu_{{\text{m}}} t}}$$

(2.25)

$$h_{{\text{s}}}^{*} \left( {x,z,t} \right) = \left( {1 - {\text{e}}^{{\alpha h_{{\text{d}}} }} } \right){\text{e}}^{{\frac{{^{{\alpha \left( {L - z} \right)}} }}{2}}} \sin \left( {\frac{\pi x}{W}} \right)\frac{{\sinh \left( {\beta_{1} z} \right)}}{{\sinh \left( {\beta_{1} L} \right)}}$$

(2.26)

where $\beta_{1} = \sqrt {\alpha^{2} /4 + \left( {\pi /W} \right)^{2} }$.

2.4 Analytical Solution of Water Infiltration of Vegetated Slope Considering the Coupling Effects

Coupling between water infiltration and mechanical deformation in unsaturated soils is central to many natural and man-made systems in civil and environmental engineering. During water infiltration, the pore-water pressure or pressure head is redistributed, on one hand by the hydraulic properties of the unsaturated soils including retention characteristics and permeability, and, on the other hand, by the external loading due to climate conditions (rainfall intensity, duration, and evapotranspiration rate). Changes in the pore-water pressure or pressure head are generated by infiltration, which in turn modifies the hydraulic domain and induces deformations of the unsaturated soils. Alternatively, any variation in the mechanical loading can exert an effect on the infiltration process. It is indeed the hydro-mechanical coupled response of an unsaturated soil that is responsible for the most common instabilities associated with water infiltration: landslides and settlements, due to collapse or shear strength reduction (Thorel et al. 2011; Wu et al. 2020).

Recently, ecological protection technologies have become popular for slope environmental restoration and treatment (Tan et al. 2019; Broda et al. 2020). The stability analysis of vegetated slopes is a hot point in geotechnical engineering. Several studies have analyzed the stability of infinite vegetated slopes (Feng et al. 2020). However, few studies have been reported on the analytical solution of vegetated slope stability considering the hydro-mechanical coupling. Developing analytical solutions for water infiltration in unsaturated soil slopes is a significant issue in practical engineering. The main objective of this section is to derive an analytical solution considering both the root effect and hydro-mechanical behavior. The governing equation considering vegetated root and coupled infiltration-deformation is derived. The analytical solution is developed using Green’s function method, which is then compared with the numerical solution. Parametric analyses are performed to investigate the effect of factors on the infinite vegetated slope stability.

2.4.1 Governing Equations of Rooted Unsaturated Soils

To simplify the problem, the following assumptions are made:

• (1) The unsaturated slope is infinite, and the soil is isotropic and elastic. The groundwater level is assumed parallel to the slope surface, and the groundwater level is fixed.

• (2) The growth direction of vegetation roots is perpendicular to the slope surface, and the root water uptake is simulated by adding a sink term (Raats 1979) to Richards equation.

• (3) Soil skeleton is compressible, while water is incompressible.

An unsaturated soil slops with vegetation is shown in Fig. 2.3, L^* is the thickness of a soil (m), $L_{1}^{*}$ is the depth of the rooted zone (m), $L_{2}^{*}$ is the thickness of the unrooted zone (m). The Richards’ equation in a reference coordinate system (oxz) can be expressed as:

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

(2.27)

Fig. 2.3

Graphical representation of rainfall-induced landslides

where h is the pressure head (m); k(h) is the hydraulic conductivity (m/s); $C(h) = {\text{d}}\theta /{\text{d}}h$ is the differential water capacity (m⁻¹); θ is the volumetric water content; t is the infiltration duration (s). Equation (2.27) needs to be modified to be suitable for rooted soil slopes. The transformation relationship between the slope coordinate system (ox*z*) and the reference coordinate system (oxz) is described as follows:

$$x^{ * } = x\cos \beta - z\sin \beta$$

(2.28a)

$$z^{ * } = x\sin \beta + z\cos \beta$$

(2.28b)

Substituting Eq. (2.28) into Eq. (2.27), according to Assumption (1), the improved Richards’ equation for rainfall infiltration into soil slopes can be given by:

$$\frac{\partial }{{\partial z^{ * } }}\left[ {k(h)\frac{\partial h}{{\partial z^{ * } }}} \right] + \frac{\partial k(h)}{{\partial z^{ * } }}\cos \beta = C(h)\frac{\partial h}{{\partial t}}$$

(2.29)

The sink term proposed by Raats (1979) is used to describe the water uptake of roots:

$$S\left( {z^{ * } } \right) = g\left( {z^{ * } } \right)T$$

(2.30)

where g(z^*) is the shape function of roots at depth z^* ([m/s]⁻¹); and T is the transpiration rate (m/s), which is affected by weather and leaf area index. Substituting Eq. (2.30) into Eq. (2.29), according to Assumption (2), the modified Richards equation considering the water uptake by vegetation roots can be obtained (Wu et al. 2022):

$$\frac{\partial }{{\partial z^{ * } }}\left[ {k(h)\frac{\partial \psi }{{\partial z^{ * } }}} \right] + \frac{\partial k(h)}{{\partial z^{ * } }}\cos \beta - S\left( {z^{ * } } \right)\left\langle {z^{ * } - L_{2}^{*} } \right\rangle = C(h)\frac{\partial h}{{\partial t}}$$

(2.31)

where $\left\langle {z^{ * } - L_{2}^{*} } \right\rangle = \left\{ {\begin{array}{*{20}l} {z^{ * } - L_{2}^{*} ,} & {z^{ * } \ge L_{2}^{*} } \\ {0,} & {z^{ * } < L_{2}^{*} } \\ \end{array} } \right.$ is the sign function.

Equation (2.31) ignores the hydraulic coupling in unsaturated soils. According to the law of conservation of mass, the strict expression of the term $C(h)\frac{\partial h}{{\partial t}}$ on the right side of Eq. (2.31) is $\frac{1}{\rho }\frac{\partial }{\partial t}(\rho nS_{{\text{r}}} )$ (Kim 2000; Wu et al. 2020, 2022), where $\rho$, $n$, and $S_{{\text{r}}}$ represent the density of water, soil porosity, and degree of saturation. S_r = (θ(h) − θ_r)/(θ_s − θ_r), θ_s is the saturated water content, and θ_r is the residual water content. Then one can obtain:

$$\frac{\partial }{\partial t}\left( {\rho nS_{{\text{w}}} } \right) = \theta \frac{\partial \rho }{{\partial t}} + \rho \frac{{\partial S_{{\text{r}}} }}{\partial t}n + \rho S_{{\text{r}}} \frac{\partial n}{{\partial t}}$$

(2.32)

the first term on the right side of Eq. (2.32) represents the water compression term, which is 0 according to Assumption (3); the second term is equivalent to $C(h)\frac{\partial h}{{\partial t}}$; the third term represents the soil skeleton term, and $\rho S_{{\text{r}}} \frac{\partial n}{{\partial t}} = \rho S_{{\text{r}}} \alpha_{{\text{c}}} \frac{{\partial \varepsilon_{\text{v}} }}{\partial t}$, where α_c is Biot’s hydro-mechanical coupling parameter (0 < α_c ≤ 1), ɛ_v is volumetric strain. Therefore, Eq. (2.31) can be rewritten as:

$$\frac{\partial }{{\partial z^{ * } }}\left[ {k(h)\frac{\partial h}{{\partial z^{ * } }}} \right] + \frac{\partial k(h)}{{\partial z^{ * } }}\cos \beta - S\left( {z^{ * } } \right)\left\langle {z^{ * } - L_{2}^{*} } \right\rangle = C(h)\frac{\partial h}{{\partial t}} - S_{{\text{r}}} \alpha_{{\text{c}}} \frac{{\partial \varepsilon_{\text{v}} }}{\partial t}$$

(2.33)

Substituting Eq. (2.11) into Eq. (2.33), the governing equation considering the hydro-mechanical coupling in unsaturated soil slopes can be obtained:

$$\frac{\partial }{{\partial z^{ * } }}\left[ {k(h)\frac{\partial h}{{\partial z^{ * } }}} \right] + \frac{\partial k(h)}{{\partial z^{ * } }}\cos \beta - S\left( {z^{ * } } \right)\left\langle {z^{ * } - L_{2}^{*} } \right\rangle = C(h)\frac{\partial h}{{\partial t}} + \frac{{\gamma_{{\text{w}}} (\theta - \theta_{{\text{r}}} )\alpha_{{\text{c}}} }}{{(\theta_{{\text{s}}} - \theta_{{\text{r}}} )F}}\frac{\partial h}{{\partial t}}$$

(2.34)

2.4.2 Analytical Solutions

According to Assumption (1), the bottom and surface boundary conditions of the slope are written as, respectively:

$$h\left( {0,t} \right) = 0\quad t > 0$$

(2.35)

$$\left[ {k(h)\frac{{\partial h\left( {L^{*} ,t} \right)}}{{\partial z^{ * } }} + k(h)\cos \beta } \right] = q\cos \beta \quad t > 0$$

(2.36)

where q is the rain intensity (m/s).

The hydraulic conductivity and the volumetric water content of unsaturated soils can be expressed as (Gardner 1958):

$$k\left( h \right) = k_{{\text{s}}} {\text{e}}^{\alpha h}$$

(2.37)

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

(2.38)

where k_s represents the saturated hydraulic conductivity (m/s); α is the desaturation coefficient (kPa⁻¹). Substituting Eq. (2.38) into Eq. (2.34), one has:

$$\frac{\partial }{{\partial z^{ * } }}\left[ {k\frac{\partial h}{{\partial z^{ * } }}} \right] + \frac{\partial k}{{\partial z^{ * } }}\cos \beta - S\left( {z^{ * } } \right)\left\langle {z^{ * } - \, L_{2}^{*} } \right\rangle = M{\text{e}}^{\alpha h} \frac{\partial h}{{\partial t}}$$

(2.39)

where M = (θ_s − θ_r)α + γ_wα_c/F.

The soil–water uptake function (Eq. 2.30) can be simplified as (Lynch 1995):

$$S\left( {z^{ * } } \right) = T/L_{1}^{*}$$

(2.40)

Substituting Eqs. (2.37), (2.38), and (2.40) into Eq. (2.39) leads to:

$$\frac{{\partial^{2} k}}{{\partial z^{ * 2} }} + \frac{\partial k}{{\partial z^{ * } }}\alpha \cos \beta - T/L_{1}^{*} \left\langle {z^{ * } - \, L_{2}^{*} } \right\rangle = \frac{M}{{K_{{\text{s}}} }}\frac{\partial k}{{\partial t}}$$

(2.41)

Here, the variables are defined as:

$$\left\{ \begin{gathered} Z = z^{*} \cos \beta \hfill \\ H = L^{*} \cos \beta \hfill \\ H_{1} = L_{1}^{*} \cos \beta \hfill \\ H_{2} = L_{2}^{*} \cos \beta \hfill \\ K = k/k_{{\text{s}}} \hfill \\ Q_{{\text{a}}} = q_{{\text{a}}} /k_{{\text{s}}} \hfill \\ Q_{{\text{b}}} = q_{{\text{b}}} /k_{{\text{s}}} \hfill \\ P = k_{{\text{s}}} /M \hfill \\ \end{gathered} \right.$$

(2.42)

where q_a and q_b represent the previous rain intensity (m/s) and current rain intensity (m/s).

Substituting Eq. (2.42) into Eq. (2.41) leads to:

$$\frac{{\partial^{2} K}}{{\partial Z^{2} }} + \frac{\partial K}{{\partial Z}}\alpha - \frac{{S\left( {z^{ * } /\cos \beta } \right)\left\langle {z^{ * } /\cos \beta - L_{2}^{*} /\cos \beta } \right\rangle }}{{k_{{\text{s}}} \cos^{2} \beta }} = \frac{1}{{P\cos^{2} \beta }}\frac{\partial K}{{\partial t}}$$

(2.43)

By combining Eqs. (2.35), (2.36), and (2.43), the analytical solution of pressure head can be obtained using Green’s function as follows:

$$K_{{{\text{ste}}}} = \left\{ {\begin{array}{*{20}l} \begin{gathered} \exp \left( { - \alpha Z} \right) + Q_{{\text{a}}} \left[ {\exp \left( { - \alpha Z} \right) - 1} \right] \hfill \\ + \, \frac{T}{{K_{{\text{s}}} \cos \beta H_{1} }}\left[ {\exp \left( { - \alpha Z} \right) - 1} \right]\left( {H - H_{1} } \right), \hfill \\ \end{gathered} & {Z < H_{2} } \\ \begin{gathered} \exp \left( { - \alpha Z} \right) + Q_{{\text{a}}} \left[ {\exp ( - \alpha Z) - 1} \right] + \frac{T}{{K_{{\text{s}}} \cos \beta H_{1} }} \hfill \\ \left\{ \begin{gathered} \left[ {\exp ( - \alpha Z) - 1} \right](H - Z) + \exp ( - \alpha Z) \hfill \\ \left[ {Z - H_{1} - \alpha^{ - 1} \exp (\alpha Z) + \alpha^{ - 1} \exp (\alpha H_{1} )} \right] \hfill \\ \end{gathered} \right\}, \hfill \\ \end{gathered} & {Z \ge H_{2} } \\ \end{array} } \right.\quad \left( {\text{Steady-state}} \right)$$

(2.44)

$$\begin{aligned} K & = K_{{{\text{ste}}}} + 8\frac{\alpha P}{{k_{{\text{s}}} }}\cos^{2} \beta \exp \left[ {\frac{\alpha (H - Z)}{2}} \right] \\ & \quad \times \sum\limits_{i = 1}^{\infty } {\frac{{\left[ {\lambda_{i}^{2} + 0.25\alpha^{2} } \right]\sin \left( {\lambda_{i} H} \right)\sin \left( {\lambda_{i} Z} \right)}}{{2\alpha + \alpha^{2} H + 4H\lambda_{i}^{2} }}} G(t)\quad \quad \left( {{\text{Transient}}} \right) \\ \end{aligned}$$

(2.45)

where

$$G(t) = \int\limits_{0}^{t} {\left( {Q_{{\text{a}}} - Q_{{\text{b}}} } \right)k_{{\text{s}}} \exp \left[ { - P\cos^{2} \beta \left( {\lambda_{i}^{2} + 0.25\alpha^{2} } \right)\left( {t - \tau } \right)} \right]{\text{d}}\tau }$$

(2.46)

$\lambda_{i}$ is the ith root of the transcendental equation tan(λH) + (2λ/a) = 0. The transient solution changes with time. The “steady-state” infiltration evolves over formally infinite time, and thus the “steady-state” solution isn’t related to time. However, transient infiltration depends on time, and the analytical solution to transient infiltration is called transient solution. Rainfall infiltration into soil slopes is commonly a transient issue, thus transient solutions are greatly meaningful.

When $Q_{{\text{b}}}$ is a constant, Eq. (2.45) can be simplified as:

$$\begin{aligned} K & = K_{{{\text{ste}}1}} - 8\left( {Q_{{\text{a}}} - Q_{{\text{b}}} } \right)\alpha \cos^{2} \beta \exp \left[ {\frac{\alpha (H - Z)}{2}} \right] \\ & \quad \times \sum\limits_{i = 1}^{\infty } {\frac{{\sin (\lambda_{i} H)\sin (\lambda_{i} Z)\exp \left[ { - P\cos^{2} \beta \left( {\lambda_{i}^{2} + 0.25\alpha^{2} } \right)t} \right]}}{{2\alpha + \alpha^{2} H + 4H\lambda_{i}^{2} }}} \\ \end{aligned}$$

(2.47)

where

$$K_{{{\text{ste}}1}} = \left\{ {\begin{array}{*{20}l} \begin{gathered} \exp \left( { - \alpha Z} \right) + Q_{{\text{b}}} \left[ {\exp ( - \alpha Z) - 1} \right] \hfill \\ + \frac{T}{{k_{{\text{s}}} \cos \beta H_{1} }}\left[ {\exp ( - \alpha Z) - 1} \right]\left( {H - H_{1} } \right), \hfill \\ \end{gathered} & {Z < H_{2} } \\ \begin{gathered} \exp \left( { - \alpha Z} \right) + Q_{{\text{b}}} \left[ {\exp ( - \alpha Z) - 1} \right] + \frac{T}{{k_{{\text{s}}} \cos \beta H_{1} }} \, \cdot \hfill \\ \left\{ \begin{gathered} \left[ {\exp ( - \alpha Z) - 1} \right](H - Z) + \exp ( - \alpha Z) \hfill \\ \left[ {Z - H_{1} - \alpha^{ - 1} \exp (\alpha Z) + \alpha^{ - 1} \exp (\alpha H_{1} )} \right] \hfill \\ \end{gathered} \right\}, \hfill \\ \end{gathered} & {Z \ge H_{2} } \\ \end{array} } \right.$$

(2.48)

The pore-water pressure head is calculated as follows:

$$h = \frac{\ln \left( K \right)}{\alpha }$$

(2.49)

2.4.3 Comparison of Analytical and Numerical Solutions

The finite element method is employed to obtain the numerical solutions to Eqs. (2.34)–(2.36). In this book, COMSOL software is employed to implement numerical solutions, which has been used in geotechnical, hydraulic, and civil engineering.

The parameters adopted here are listed in Table 2.1, which includes the hydraulic parameters (k_s, α, θ_s, and θ_r) (Liu et al. 2016). The parameters describe the hydraulic properties of loess, and the parameters of vegetation roots are determined according to the statistics of shrubs (Feng et al. 2020).

Table 2.1 Input parameters (Liu et al. 2016; Wu et al. 2022)

The analytical and numerical solutions of pressure head at t = 0 h, 20 h, and 30 h are shown in Fig. 2.4. The analytical solutions at t = 0 h are obtained by Eq. (2.44), and those at t = 20 h and 30 h are obtained by Eq. (2.45). According to the root mean square error RMSE and coefficient of determination (R²) shown in Fig. 2.4, it can be seen that RMSE between the numerical and analytical solutions is less than 0.005 m, and R² is very close to 1. That is, the error between the numerical and analytical solutions is very small. Compared with the numerical solution, the analytical solution concisely describes the pressure head variations with rainfall infiltration.

Fig. 2.4

Comparison of the analytical and numerical solutions

2.4.4 Parametric Analyses

Factor of safety (F_s) is an important indicator used to assess the stability of slopes, and slopes with F_s greater than 1 are generally considered to be stable, and vice versa. F_s is calculated based on the limit equilibrium method and strength theory of rooted unsaturated soils (Fredlund and Rahardjo 1993; Ku et al. 2018), as follows:

$$F_{{\text{s}}} = \frac{{c + c_{{\text{r}}} }}{{\left( {L^{*} - z^{*} } \right)\gamma \cos \beta \sin \beta }} + \frac{\tan \varphi }{{\tan \beta }} - \frac{{\gamma_{{\text{w}}} h\left[ {(\theta - \theta_{{\text{r}}} )/(\theta_{{\text{s}}} - \theta_{{\text{r}}} )} \right]\tan \varphi }}{{\left( {L^{*} - z^{*} } \right)\gamma \cos \beta \sin \beta }}$$

(2.50)

where $c_{{\text{r}}}$ is the root cohesion (kPa), obtained by Wu–Waldron criterion (Wu 1976), i.e., $c_{{\text{r}}} = \varsigma T_{{\text{s}}}$, $T_{{\text{s}}}$ is the root tensile strength (kPa), $\varsigma$ is the ratio of root cross-sectional area, which is 0.00025 in this book (Leung 2014). c and φ are the soil effective cohesion and effective frictional angle, while the rooted soil strength contributes to root cohesion. In this section, the effects of slope angle, rainfall intensity, transpiration rate, and suction-based modulus of elasticity on F_s and F_s ratio will be investigated. The F_s ratio, defined as the ratio of F_s during rainfall to the initial value (t = 0), shows the variations in F_s during rainfall and is also an important parameter for analyzing rainfall-include landslides.

2.4.4.1 Effect of Slope Angle

Here, the effects of slope angle on the pressure head, F_s and F_s ratio (defined as the ratio of F_s during rainfall to the initial value (t = 0)) are investigated. The parameters are as follows: Rain intensity is 0.7k_s, the tensile strength of the roots is 10 MPa (this parameter is determined by the statistics of shrubs (Leung 2014)), the unit weight of the soil (γ) is 20 kN/m³, the soil effective cohesion and effective frictional angle are 18 kPa and 28°, and other parameters are listed in Table 2.1.

Figure 2.5 represents the pressure head in slopes with slope angles of 10, 30, and 50° during rainfall. The conclusions can be drawn as follows (Fig. 2.5): (i) the pressure head of slopes decreases during rainfall; (ii) the pressure head in the shallow slope is more sensitive than that in the deep slope during rainfall; (iii) the smaller the slope angle, the larger the pressure head at the same position, which is mainly caused by the initial pressure head profile (the initial pressure head of the slope is negatively correlated with slope angle); (iv) the smaller the slope angle, the greater the variation in the pressure head at the same time, this is because the small slope angle has a positive effect on rainfall infiltration.

Fig. 2.5

Effect of slope angle on the pressure head

F_s (related to depth) with slope angles of 10, 30, and 50° during rainfall are described in Fig. 2.6a. Figure 2.6a states the following points: (a) F_s of the shallow slope is greater than that of the deep slope; (b) F_s decreases during rainfall, which can be explained in Fig. 2.5; (c) increase at z = 4.5 m due to the reinforcement effect of vegetation roots. F_s ratio F_s ratio with slope angles of 10, 30, and 50° during rainfall are represented in Fig. 2.6b. The larger the slope angle, the larger F_s ratio in Fig. 2.6b. This demonstrates that the larger the slope angle, the smaller F_s. In Fig. 2.6a, F_s suddenly the stability of a gentle slope is more sensitive to rainfall than that of the deep slope.

Fig. 2.6

Effect of slope angle on a F_s and b F_s ratio

2.4.4.2 Effect of Rain Intensity

The variations in the dimensionless rainfall intensity over the duration of the rainfall event for both coupled and uncoupled analyses are shown in Fig. 2.7. The parameters are k_s = 10⁻⁵ m/s, θ_s = 0.4, |F| = 5 × 10³ kPa, and α = 0.01 cm⁻¹ (Van Genuchten 1980). The model height was 400 cm (Fig. 2.7). The value of F can be positive or negative. A negative F means an expansive soil where a suction decrease leads to soil volume increase, while a positive F denotes a collapsible soil where a soil suction decrease leads to soil volume decrease (Wu et al. 2020). Compared with the bottom boundary of a stationary groundwater table, the impermeable bottom boundary leads to more pronounced coupling effects in the lower part of the soil layer. It is also noted that the groundwater ponding occurs at the bottom boundary (t = 15 h, Fig. 2.7), particularly for an expansive soil (F < 0). Waterfall intensity plays a significant role in the advancement of the wetting front, and the pressure head profile moves more quickly as the rainfall intensity increases. The coupling effect is also closely linked with the rainfall intensity (Wu et al. 2020).

Fig. 2.7

Changes in the pressure head profile over time under coupled and uncoupled states

Here, the effect of rain intensity on the pressure head, F_s and F_s ratio are investigated. Figure 2.8 describes the pressure head of the slope with rainfall intensities of 0.5k_s, 0.7k_s and 0.9k_s. The remaining parameters are listed in Table 2.1. In Fig. 2.8, the smaller the rainfall intensity, the larger the pressure head at the same position.

Fig. 2.8

Effect of rainfall intensity on the pressure head

Figure 2.9a, b describe F_s, and F_s ratio of the slope with rain intensities of 0.5k_s, 0.7k_s, and 0.9k_s, respectively. The variation in F_s with time and depth is similar to that in Fig. 2.6a. The larger the rain intensity, the smaller F_s or F_s ratio of the slope. The stability of the shallow slope is more sensitive to that the deep slope under conductivity is an important parameter of soil seepage capacity, different rainfall intensities (Fig. 2.6). Saturated hydraulic conductivity is an important parameter of soil seepage capacity. Figure 2.10 represents the influence of saturated hydraulic conductivity on F_s and F_s ratio of rooted soil slopes. With increasing saturated hydraulic conductivity, F_s and F_s ratio at the same depth decrease.

Fig. 2.9

Effect of rainfall intensity on the a F_s and b F_s ratio

Fig. 2.10

Effect of saturated hydraulic conductivity on a F_s and b F_s ratio

2.4.4.3 Effect of Transpiration Rate

The transpiration rate is an important parameter. The effect of transpiration rate on the pressure head, F_s and F_s ratio, is investigated here. The parameters are as follows: Rain intensity is 0.7k_s, and other parameters are the same as those in Sect. 2.4.4.2.

Figure 2.11 represents the pressure head of unsaturated soil slopes with transpiration rates of 3, 4.5, and 6 mm/d. Transpiration rates depend on the vegetation and environmental conditions, and the transpiration rates are determined according to the statistics of shrubs (Leung and Ng 2013). The pressure head varies over time, and depth is the same as that in Sect. 2.4.4.1. The smaller the transpiration rate, the smaller the pressure head at the same position. The transpiration of vegetation reduces water content in slopes, causing an increase in soil suction. Figure 2.12a depicts F_s of slopes with transpiration rates of 3, 4.5, and 6 mm/d, respectively. The larger the transpiration rate, the larger safety factor of soil slopes. This is because the vegetation root uptake water reduces the water content of soil slopes. Figure 2.12b represents F_s ratio of slopes with transpiration rates of 3, 4.5, and 6 mm/d, respectively. The larger the transpiration rate, the smaller F_s ratio at the same position.

Fig. 2.11

Effect of transpiration rate on the pressure head

Fig. 2.12

Effect of transpiration intensity on a F_s and b F_s ratio

2.4.4.4 Effect of the Suction-Based Modulus of Elasticity

The suction-based modulus of elasticity (F) is key for hydro-mechanical coupling. The effect of suction-based modulus of elasticity on the pressure head, F_s, and F_s ratio, was investigated here. Three cases (F = 10³ kPa, − 10² kPa, and no hydro-mechanical coupling) are considered here. The governing equation without the hydro-mechanical coupling is Eq. (2.31), and the boundary conditions are the same as those considering the coupling effect. The transpiration rate in this section is 4.5 mm/d, and the other parameters are kept unchanged as Sect. 2.4.4.3.

Figure 2.13 represents the pressure head in soils with different suction-based moduli of elasticity. The variations of pressure head with suction-based modulus of elasticity are summarized as follows: (i) When F is positive, the hydro-mechanical coupling causes the increase of pore-water pressure head in the slope. Water flow in expansive soils is faster than that in collapsible soils. (ii) The effect of hydro-mechanical coupling on the pressure head in the slope becomes obvious with the decrease of the absolute value F.

Fig. 2.13

Effect of suction-based elastic moduli on the pressure head

Figure 2.14a represents F_s of soil slopes with different suction-based moduli of elasticity. Variation in safety factor with F is as follows: (i) When F is positive (negative), the hydro-mechanical coupling effect results in increases (decreases) of F_s. (ii) The effect of hydro-mechanical coupling on F_s becomes marked with decreasing absolute value of F.

Fig. 2.14

Effect of suction-based elastic moduli on a F_s and b F_s ratio

Figure 2.14b describes F_s ratios of soil slopes with different suction-based moduli of elasticity. Figure 2.14b clearly demonstrates that small absolute value of F will significantly reduce the factor of safety of slopes.

2.5 Discussions and Conclusions

2.5.1 Discussions

F_s ratios caused by the hydro-mechanical coupling effect of bare slopes (transpiration rate is 0) and rooted slopes (transpiration rate is 4.5 mm/d) are compared. The evaporation of bare soils caused by temperature is ignored. The F_s ratios of the bare and rooted slopes at t = 20 h and 40 h are described in Fig. 2.15. The parameters are the same as those in Sect. 2.4.4.4. In Fig. 2.15a, F_s ratios of the rooted slope are smaller than those of the bare slope. The stability changes caused by the hydro-mechanical coupling in rooted soil slopes are greater than those of bare slopes. A similar conclusion can be drawn from Fig. 2.15b.

Fig. 2.15

F_s ratios of the bare slope and vegetated slope: a t = 20 h, b t = 40 h

It should be emphasized that the proposed analytical solution has two main limitations: (i) This solution is only suitable for rooted slopes with uniform root architecture. Existing results (e.g., Ng et al. 2015; Liang et al. 2017) have indicated that vegetation roots have complex shape, including exponential root, triangular root, and uniform root. The book tries to obtain analytical solutions considering different root architecture functions in future work. (ii) The boundary condition of soil slopes is transformed from flow boundary to pressure head boundary due to saturated hydraulic conductivity less than rain intensity, which may affect the results of analytical solutions. Although there still occur some limitations, the proposed analytical solution incorporates the root effect and hydro-mechanical coupling for the first time. In actual engineering, the proposed analytical solution can be easily applied to examine rainfall-induced landslides in rooted soil regions.

Figure 2.16 represents the variations in the dimensionless rainfall intensity over the duration of the rainfall event for both the coupled and uncoupled conditions. The parameters used are k_s = 10⁻⁵ m/s, θ_s = 0.4, |F| = 5 × 10³ kPa, and α = 0.01 cm⁻¹ (Van Genuchten 1980; Wu et al. 2020). The value of F can be positive or negative; a negative F means an expansive soil where a suction decrease leads to soil volume increase, while a positive F denotes a collapsible soil where a suction decrease leads to soil volume decrease (Wu et al. 2020). When t = 1 h, the wetting front moves to a depth of 120 cm in both the uncoupled analysis and coupled analysis with F > 0. However, with F < 0 the wetting front reaches 180 cm in the coupled analysis. When t = 1 h, the difference in the pressure head for the coupled (F < 0) and uncoupled conditions is 27.5 cm at the upper boundary, and the maximum difference in the pressure head within the unsaturated zone is 75.1 cm at a soil depth of 340 cm. When t = 15 h, the difference in the pressure head at the top boundary between the coupled (F > 0) and uncoupled conditions is 13.7 cm, and the maximum difference in the pressure head within the soil is 61.8 cm at a depth of 160 cm. The coupling effect becomes more apparent with increasing time, particularly in the bottom part of the soil layer. Compared with the case where the base boundary coincides with the stationary groundwater table (Wu et al. 2020), the case with zero flux at the base boundary leads to more pronounced coupling effects in the lower part of the soil column. It is also noted that groundwater ponding occurs at the base boundary (t = 15 h, Fig. 2.16).

Fig. 2.16

Pore-water pressure profile for different boundaries

As the infiltration time increases and the wetting front moves downward, the pressure head increases rapidly in the shallow depths of the unsaturated soils. Rainfall intensity plays an important role in the advancement of the wetting front, and the pressure head profile moves more quickly as the rainfall intensity increases. The coupling effect is also closely linked with the rainfall intensity. Other studies reported that the coupling effect is more noticeable in a shallow layer of an unsaturated soil (Wu et al. 2020). However, in this book, the coupling effect becomes more noticeable in the base layer, particularly with increasing time. As the rainfall accumulates at the base boundary, the coupled effect becomes more apparent there. Figure 2.16 describes the effect of boundary on the pressure profile considering the coupling effect. The boundary effect is more marked than the coupling. The zero-flux base boundary also leads to groundwater ponding in the lower part of the soils. The pressure head profile moves more quickly as the thickness of the soil layer decreases.

2.5.2 Conclusions

The governing equation considering hydro-mechanical coupling of unsaturated rooted slopes is derived. The analytical solution is obtained using Green’s function method. This solution is compared with the finite element method. The parametric studies were carried out to investigate the effect of slope angle, rain intensity, suction-based elastic modulus, and transpiration rate on the pore-water pressure head and slope safety factor. The following conclusions can be obtained:

• (1) We have developed an analytical solution that described transient groundwater flow in unsaturated soils using a Fourier integral transform, in which the groundwater level is allowed to advance, due to the zero flux boundary. The results indicate that the lower boundary condition plays a considerable role in the pressure head profile and that the coupling effect is more pronounced for a zero-flux boundary than a zero-pressure-head boundary. The coupling effect becomes more noticeable when ponding occurs.

• (2) The proposed analytical solution is explicit and provides a basis for the numerical solution for rainfall infiltration into rooted soil slopes. The proposed analytical solution is simple in form and has few input parameters, which is a simple and effective method for analyzing rainfall infiltration in vegetated slopes.

• (3) The larger the slope angle or rain intensity, the lower the pressure head, resulting in a smaller factor of safety for slopes. The larger the transpiration rate, the larger the pressure head, which causes an increase of safety factor of rooted soil slopes. The lower the absolute value of the suction-based elastic modulus, the more obvious the effect of hydro-mechanical coupling on the safety factor of rooted soil slopes.