Hillslope hydrology
In this study, treatment of hillslope hydrology consists of two mathematical parts (equations), one is used to describe the seepage flow and the other is used to describe the overland water flow. The equation for seepage flow using in this study is the one presented by Rosso et al. (2006). The equation for overland water flow is derived based on the assumption of an uniform overland water flow.
The equation of seepage flow
The hillslope hydrology model presented by Rosso et al. (2006) established the equation that is used in this study to describe the seepage flow as a part of the model. Rosso’s seepage flow equation is derived by coupling the conservation of mass of soil water with the Darcy’s law. Here, the seepage flow equation is established using the assumptions that overland water flow is generated by saturation excess, subsurface flow is parallel to the slope surface, and the impermeable layer is shallow. Using topographic elements to divide the hillslope, as shown in Figure 1, the topographic elements are defined by the intersection of contour and flow tube boundaries orthogonal to the contours, and the equation of seepage flow is as follows.
As shown in Figure 1, p denotes the net rainfall, z denotes the thickness of the landslide, a is the upslope contributing area, b is the width of the topographic elements, and h is the height of the subsurface flow. By water balance and Darcy’s law, the following expression can be obtained
$$ ap-bhK \sin \theta =a\frac{e}{1+e}\left(1-{S}_r\right)\frac{dh}{dt} $$
(1)
where θ is the slope angle to the horizontal, s
r
is the average degree of saturation, e is the average void ratio above the groundwater table, K is the saturated conductivity of the soil, t is the rainfall duration time, and the other parameters are the same as in the previous.
Based on Equation (1) for the initial condition of stable piezometric condition at the depth of h (0) = h
0
, we obtain
$$ \begin{array}{l}h=\frac{apz}{Tb \sin \theta}\left[1- \exp \left(-\frac{1+e}{e-e{s}_r}\;\frac{Tb \sin \theta }{az}t\right)\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}+{h}_0 \exp \left(-\frac{1+e}{e-e{s}_r}\;\frac{Tb \sin \theta }{az}t,\right)\begin{array}{cc}\hfill \hfill & \hfill for\frac{ap}{Tb \sin \theta }>1\hfill \end{array}\end{array} $$
(2)
For the simple case of h
0
= 0, the height of the subsurface flow is expressed as follows:
$$ h=\left\{\begin{array}{c}\hfill \frac{apz}{Tb \sin \theta}\left[1- \exp \left(-\frac{1+e}{e-e{s}_r}\frac{Tb \sin \theta }{az}t\right)\right],t\le t*\hfill \\ {}\hfill z,\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill \end{array}\begin{array}{cc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill \end{array}t>t*\hfill \end{array}\right\} $$
(3)
and
$$ t*=-\frac{e-e{s}_r}{1+e}\frac{za}{Tb \sin \theta } \ln \left(1-\frac{Tb \sin \theta }{ap}\right) $$
(4)
where T is the hydraulic transmissivity, with T = Kz, here, and t* is the time of the ground water table rising up to the slope surface.
Equation of overland flow
In Rosso’s hillslope hydrology model, overland water flow is generated by saturation excess. That is to say, when the ground water rises up to the slope surface and rainfall continues, overland water flow is generated. Nevertheless, in Rosso’s hillslope hydrology model, no equation is given to describe overland water flow. Thus, an equation of overland flow should be derived to improve the hillslope hydrology model.
In order to obtain the equations for the relationship between overland water flow and rainfall, the following assumptions were made:
-
1)
No erosion occurs on the slope surface.
-
2)
The overland water flow is uniform flow, which means that streamlines are parallel with each other. That is to say, the overland flow is parallel to the slope surface.
Then, based on the assumptions and the principle of water balance, we obtain the following expression:
$$ ap-q-r=a\frac{dl}{dt} $$
(5)
where, q is the seepage flow discharge, by Darcy’s law q = bzKsinθ, r is the discharge of the overland flow, l is the depth of the overland flow, and the other parameters are as before.
Here, the overland water flow is assumed to be an uniform flow. So, based on the hydraulics, the overland water flow can be written as
where, v is the average velocity of overland flow, l is the depth of the overland water flow, and the other parameters as the same as in the previous.
Then, substituting Equation (6) into Equation (5) yields
$$ ap-bzK \sin \theta -vbl=a\frac{dl}{dt} $$
(7)
The Chezy formula provides the average velocity in uniform flow. Thus, we have
$$ v=C\sqrt{RJ}=C\sqrt{l \sin \theta } $$
(8)
where,the Chezy coefficient C can be given by the Manning formula,
$$ C=\frac{1}{n}{l}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.} $$
(9)
Where n is the roughness coefficient.
Then, substituting Equations (8) and (9) into Equation (7) yields
$$ ap-bzK \sin \theta -\frac{1}{n}{bl}^{5/3}{\left( \sin \theta \right)}^{1/2}=a\frac{dl}{dt} $$
(10)
The finite difference method can be used to solve Equation (10), and Equation (10) can be discretized as follows.
$$ \mathrm{ap}\Delta \mathrm{t}-bzK \sin \theta \Delta t-\frac{1}{n}bl{(t)}^{5/3}{\left( \sin \theta \right)}^{1/2}\Delta t=a\left[l\left(t+1\right)-l(t)\right] $$
(11)
By solving Equation (11), the depth of overland water flow with different rainfall and time can be obtained.
Thus, Equations (3) and (10) compose the hillslope hydrology model. By the hillslope hydrology model of this study, groundwater height or the depth of overland water flow can be obtained at any time, and the processes of groundwater rising and overland water flow being generated can be calculated. Here, we take a simple case to illustrate this process. We choose the thickness of a landslide as 1 m and use the other parameters shown in Figures 2 and 3.
Figure 2 shows the relations of h + l against rainfall intensity. For a given slope, h + l increase with increasing duration of rainfall. The larger rainfall intensity, the faster the h + l increase rate, and the earlier the time for overland water flow generation, resulting in higher overland water flow depth.
In order to analyse the relationship of overland water flow depth with rainfall intensity, the overland water flow in Figure 3 is scaled up in Figure 2. Here, it should be noted that time begin with the groundwater table rising to the slope surface, that is overland water flow is beginning to be generated. In Figure 3, the time for overland water flow to begin to generate t* is given for different rainfall intensities. The figure shows that the larger the rainfall intensity, the higher the overland flow depth. The depth of overland water flow increases with increase rainfall duration time and then becomes nearly constant. Here, the depth of overland water flow is quite small, several centimeters or millimeters, and the result is in accordance with the actual case.
Slope stability
In traditional analysis of slope stability, overland water flow is not taken into account. As is well known, the depth of overland water flow is only several centimeters or millimeters, which may have little effect on a stable slope with a large safety factor; however, for a slope in the limiting equilibrium condition, the effect of overland water flow should not be ignored. This is because the overland water flow with only several centimeters or millimeters may cause the so-called stable slope to become an unstable slope. Especially, in landslide hazard assessment, not considering the effect of overland water flow may result in an incorrect assessment and lead to serious damage in the so-called safety area, which is obtained by landslide hazard assessment without consideration of the effect of overland water flow on the stability. Thus, in order to avoid the aforementioned accidents, the effect of overland water flow should be taken into account in slope stability analysis.
As we all know, if the thickness of a sliding mass on a slope is much smaller than the slope’s length, the slope can be called an infinite slope. For an infinite slope, edge effects can be neglected in stability analysis. That is to say, the safety factor of the slope can be determined by analysis of a rigid wedge or rigid slice of material of unit width and unit thickness. Here, in an area scale, the thickness of the landslide is much smaller than the length. Thus, the infinite slope assumption can be used in this study.
In Figure 4, a rigid slice with unit width and unit thickness is chosen to evaluate the safety factor for a slope. The depth of overland water flow is shown by l, the depth of the slope is z, the height of the subsurface flow is h, and τ
w
is the shear stress of the overland water flow acting on the slope. Overland water flow may add to the pore pressure and the flow motion causes a shear stress acting on the slope. These are both disadvantages for slope stability.
On the basis of the Mohr-Coulomb theory, the shear stress is expressed as
$$ {\tau}_f=c+\left(\sigma -u\right) \tan \phi $$
(12)
where, τ
f
is the shear strength of the soil, c is the cohesion of the soil, σ is the normal total stress, u is the pore water pressure,and φ is the internal friction angle of the soil.
If we use the τ
s
to denote the shear stress, the safety factor can be written as follows.
$$ {F}_s=\frac{\tau_f}{\tau_s} $$
(13)
Here, the expressions for total stress σ, u, and τ
s
are
$$ \sigma =\left\{\begin{array}{c}\hfill \left[\left(z-h\right)\gamma +h{r}_{sat}\right]{ \cos}^2\theta, \begin{array}{cc}\hfill \hfill & \hfill h\le z\hfill \end{array}\hfill \\ {}\hfill z{r}_{sat}{ \cos}^2\theta +{\gamma}_wl{ \cos}^2\theta, \begin{array}{cc}\hfill \hfill & \hfill l\ge 0\hfill \end{array}\hfill \end{array}\right. $$
(14)
$$ u=\left\{\begin{array}{c}\hfill h{r}_w{ \cos}^2\theta, \begin{array}{cc}\hfill \hfill & \hfill h\le z\hfill \end{array}\hfill \\ {}\hfill \left(z+l\right){r}_w{ \cos}^2\theta, \begin{array}{cc}\hfill \hfill & \hfill l\ge 0\hfill \end{array}\hfill \end{array}\right. $$
(15)
$$ {\tau}_s=\left\{\begin{array}{c}\hfill \left[\left(z-h\right)\gamma +h{\gamma}_{sat}\right] \cos \theta \sin \theta, \begin{array}{cc}\hfill \hfill & \hfill h\le z\hfill \end{array}\hfill \\ {}\hfill z{\gamma}_{sat} \cos \theta \sin \theta +{\gamma}_wl \cos \theta \sin \theta +{\tau}_w,\begin{array}{cc}\hfill l\ge 0\hfill & \hfill \hfill \end{array}\hfill \end{array}\right. $$
(16)
where γ is the average unit weight of the soil, γ
sat
is the saturated unit weight of the soil, and γ
w
is the unit weight of water.
In Equation (16), the shear stress of the overland water flow acting on the slope τ
w
can be derived by using hydraulics, as follows,
$$ {\tau}_w={\gamma}_wRJ={\gamma}_wl \sin \theta $$
(17)
Then, with the assumption that the slice is rigid and by substituting Equations (14)-(16) into Equations (12) and (13), the expression for the safety factor of a slope can be obtained as the following.
$$ \begin{array}{l}{F}_s=\\ {}\frac{c+\left[\left(z-h\right)\gamma +h\gamma \hbox{'}\right]{ \cos}^2\theta \tan \phi }{\left[\left(z-h\right)\gamma +h{\gamma}_{sat}\right] \sin \theta \cos \theta +{\gamma}_wl \cos \theta \sin \theta +{\gamma}_wl \sin \theta}\end{array} $$
(18)
Figure 5 shows the variability of the safety factor F
s
at different rainfall intensities. For a given slope, F
s
decreases with increasing of rainfall duration time, and the rate of decrease of F
s
becomes faster for larger rainfall intensity. Additionally, in Figure 5, the F
s
for h = z and Fs = 1 is shown. Fs = 1 is the limiting equilibrium condition of slope stability (Montgomery and Dietrich 1994; Rosso et al. 2006; Tsai and Yang 2006; Chang and Chiang 2009). Thus, the figure shows that when the groundwater table rises to the slope surface, the safety factor is larger than 1 for each rainfall intensity. That is the slope is stable. However, with continuing rainfall, overland water flow is generated and the safety factor decreases continuously until the depth of overland water flow remains constant. The figure shows that the safety factor decreases to values smaller than 1 for each rainfall intensity under overland water flow, i.e. the slope is now unstable.
In the existing research, if the ground water rises up to the slope surface, and the slope safety factor F
s
≥ 1, thus the slope is unconditionally stable for shallow landslide hazard assessment (Rosso et al. 2006; Chang and Chiang 2009). However, from Figure 5, we see that overland water flow causes the stable slope to become unstable. Hence, the “unconditionally stable” slope is not unconditionally stable. Considering the influence of overland water flow is quite important for shallow landslide hazard assessment.
