The SLIP model, a physically based model for shallow landslide instability prediction
3.2 The SLIP model
SLIP was developed to model processes that have been confirmed by recent studies (Flury et. al 1994, Zhan et al. 2007, Krzeminska et al. 2013, Springman et al., 2013). The aim was to predict slope failure through a simplified formulation by modeling the main characteristics of the instability triggering without including too many elements of the complex mechanism. This choice was made because excessive parameters would limit the application of the model at the large scale. The model defines the safety factor Fs by applying the limit equilibrium method to an equivalent infinite slope
Chapter 3 The SLIP model, a physically based model for shallow landslide instability prediction
29
that is composed of two homogeneous soil portions: a partially saturated portion and a fully saturated portion representing the saturated zones around the macro channels. Homogenization is used to obtain, with respect to the original conditions, the loss of shear strength of an equivalent soil that is stable in the presence of both saturated and partially saturated zones; this is consistent with both the principles of soil mechanics and the application of the limit equilibrium method. For simplicity, the saturated zones are represented in the model by a saturated sub-layer of thickness mH (0 < m < 1) where H represents the potentially unstable top soil layer (first 1-1.5m). This sub-layer (mH) could be put in a random position inside the potentially unstable sub-layer (H) but, for simplicity and because this choice does not change the consistency of the approach, it is positioned at the base of the layer, growing upwards during rain infiltration from the underlying bedrock or denser soil. This choice avoids adding an insignificant parameter, i.e., the location of the saturated sub-layer. This assumption does not affect the solidity of the approach as the saturated sub-layer mH co-exists with an unsaturated sub-layer of thickness (1-m)H (Figure 3.1). The parameter mH is related to the total amount of rainwater, h, as follows:
(
Sr)
n H h
m ⋅ −
= ⋅
⋅ 1
β*
(3.1)
Where h is the time-varying rainfall depth, H is the thickness of the potentially unstable superficial soil, β* is the percentage of rain that infiltrates into the soil (being [1 - β*] the surface and subsurface runoff), n is soil porosity, and Sr is the degree of saturation.
Figure 3.1 Infinite model geometry, homogenized soil portions and forces involved in the calculation of Fs
The shear strength of the saturated sub-layer (mH) is described by the Mohr-Coulomb criterion:
' tan '
' σ ϕ
τ =c+ ⋅ (3.2)
Chapter 3 The SLIP model, a physically based model for shallow landslide instability prediction
30
The shear strength of the unsaturated soil is described by the simplified extended Mohr-Coulomb criterion for unsaturated soils (Fredlund & Rahardjo, 1993)
(
f)
f(
a w)
nff c σ u ϕ u u ϕ
τ = '+ − ⋅tan '+ − tan (3.3)
where σ’=(σf-u)f is the effective normal stress on the failure plane at failure, (ua - uw) is the matric suction, c’ is the effective cohesion, φ’ is the internal friction angle, and φn is the friction angle associated with the matric suction. The quantity (ua-uw)·tanφn is independent of σ’, and in Equation (3.3) it represents an “apparent cohesion”, which is named cψ (Fredlund & Rahardjo, 1993). Thus, the final shear strength criterion can be expressed as:
' tan '
' σ ϕ
τ =c+cψ + ⋅ (3.4)
The apparent cohesion cψ is a function of the matric suction; however, in SLIP, it is directly correlated to the degree of saturation Sr. This allows better control of the uncertainties in the evaluation of the soil state parameters because Sr varies over a smaller range than the matric suction. The link between cψ and the degree of saturation Sr is obtained from the results of experiments on different kinds of soils (Fredlund et al., 1996, Bogaard et al., 2014), such as medium and fine grained sands (Figure 3.2). By neglecting isteresis that occurs during the wetting/drying cycles, cψ* can be expressed as a function of Sr as:
( )
λψ A Sr Sr
c* = ⋅ ⋅ 1− (3.5)
Figure 3.2 Apparent Cohesion vs degree of Saturation for 3 soil samples (Fredlund et al., 1996, Bogaard et al., 2014)
where A and λ are model parameters that depends on the type of soil.
Chapter 3 The SLIP model, a physically based model for shallow landslide instability prediction
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The values of the parameters λ and A have been identified by experimental tests (Mari, 2000;
Montrasio and Valentino, 2003 and 2007) for the most common types of soil of the Italian territory and are shown in Table 2.1.
Types of soil λλλλ A(kPa) Clay OCR = 1 0.4 100 Clay 1 < OCR < 2 0.4 100 Clay OCR > 2 0.4 100 Silty Clay 0.4 80
Silt 0.4 80
Silty Sand 0.4 80 Loose sand 0.4 40
Sand 0.4 40
Packed Sand 0.4 40
Table. 3.1 Parameters λλλλ e A for different kinds of soil.
Finally, the apparent cohesion of the homogenized equivalent slope, cψ, depends on both cψ and the thickness m⋅H of the saturated sub-layer. The expression for cψ was obtained by performing a series of experiments on stratified soils in a flume (Montrasio & Valentino, 2007; Silva 2000) and is given by:
( )
αψ
ψ c m
c = * 1− (3.6)
The safety factor is time-dependant through water variations in the top soil. Saturation ratio increases in time due to rain infiltration and decreases by evapo-transpiration, down flow and percolation. The decrease of water content is modeled in a simplified way represented by the decrease of m with time, which is obtained using a negative exponential function of time:
( ) m
(
k t)
mt = 0⋅exp − t⋅ (3.7)
where kt represents the global drainage capability of the slope, which takes into account evapo-transpiration, down flow and percolation. The value of kt is derived from the back-analysis of shallow landslides that occur on different kinds of soils. Finally, the formulation of the safety factor (Fs), according to the limit equilibrium method is the following:
W w
S m n
C n
F m
⋅ + Γ
Ω
⋅ +
−
⋅ + Γ
⋅
= cotβ⋅tanϕ'[ ( 1)] '
(3.8)
Where:
r
S (1 n) n S
Γ =G ⋅ − + ⋅ (3.9)
Chapter 3 The SLIP model, a physically based model for shallow landslide instability prediction
32 )
S 1 ( n
nW = ⋅ − r (3.10)
γW
H β 2 sin Ω 2
⋅
= ⋅ (3.11)
] ) 1 ( ) 1 ( '
[ ] ' [
' c cψ L c A S S λ m α
C = + ⋅ = + ⋅ r⋅ − r ⋅ − (3.12)
( ) ∑ [ ( ) ]
=
−
− −
⋅
= ⋅ β
ω1
*
1 exp )
(
i
i t r
t t S k
H t n
m
(3.13)⋅ β is the slope angle of the potentially unstable slope;
⋅ φ' is the shear resistance angle of the soil;
⋅ γw is the weight per unit volume of water;
⋅ H is the thickness of the potentially unstable layer;
⋅ L is the length of the soil layer;
⋅ n is the porosity of the soil;
⋅ GS is the ratio between the specific unit weight of the soil and water unit weight;
⋅ Sr is the degree of saturation of the soil;
⋅ c' is the effective cohesion of the soil;
⋅ cψ is the apparent cohesion given by the partial saturation of the soil;
⋅ kt is the global permeability of the soil;
⋅ A, λ are shear strength parameters under unsaturated conditions;
⋅ α is a model parameter that is related to the homogenization.
⋅ β* is the amount of rain water that infiltrates into soil
⋅ ξ is the amount of rain water that runs off (1- β*)
Slope instability is reached when Fs < 1
Fs is a function of slope geometry (β, H), soil state (Gs, n, Sr), the shear strength parameters of the soil under saturated conditions (φ’, c’), the shear strength under unsaturated conditions (A, λ), a model parameter that is related to the homogenization modeling (α), the drainage capability of the slope (kt), the water’s unit weight (γw) and the rainfall depth (h).
The model has been previously applied at several scales, including the field scale (single slope, territorial, regional, national) and the laboratory scale (flume tests). At the field scale, the approach used to evaluate the model parameters and to calculate Fs differs depending on the size of the studied area. At the scale of a single slope, the model parameters are obtained by a geotechnical
Chapter 3 The SLIP model, a physically based model for shallow landslide instability prediction
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characterization of the soil (lab and in situ tests), and Fs is a deterministic value that unequivocally defines the moment of failure when its value reaches 1. At the larger scale, the parameters must be linked to geological, lithological, land use, vegetation cover and geometry (DEM - Digital Elevation Model) information, and the safety factor is defined in areas of fixed dimensions, called cells through a nxn grid (the same as the DEM). At the larger scale, the meaning of the approach is different and is linked to the susceptibility of a certain area to the occurrence of hydro-geological instability. A complete forecast of instability requires additional methods. In this case, the output of the model can be represented both as time varying safety factor maps, which give a value of Fs for each elementary cell with the same resolution as the DEM (non-aggregated results), and “instability index” maps (aggregated results). Thus, for operational purposes, the safety factors obtained on the elementary cells can be aggregated by considering larger reference areas. The “instability index” is defined as the ratio between the number of elementary cells in which Fs < 1 (instability condition) and the total number of cells in each reference area. Laboratory scale investigations such as flume tests can be used to determine the influence of the model parameters and to calibrate them. As described above, the model assumes double porosity in the soil, which strongly influences the infiltration; in lab tests, if some arrangements aren’t made, the soil (even if it has the same matrix as the in situ soil) doesn’t reproduce the macro-porosity which implies a process that is different from that in the field.
In chapter 7 the results of some laboratory flume tests on a silty sand, with and without these arrangements are presented. The arrangements consisted of inserting in the homogeneous soil matric some preferential infiltration channels made by gravel inclusions. The double porosity of the soil can be well seen in these tests.