Chapter 3. Study Area & Methodology
3.11. Physically-based models
Infinite slope stability model
The infinite slope stability model (Figure 3.15) demands a mention at the onset since the TRIGRS models that have been applied in this study is particularly based on this stability concept. This model compares the destabilizing stresses and restorative strength of the material on an infinite plane parallel to the soil surface. No slope can be considered infinite in extent; as such, infinite slope is purely conceptual. However, when the depth of the slip surface is much less compared to the areal extent of the landslide and the field condition remains nearly identical, then such a mathematical model can be applied for estimating a close approximation of the stability condition (Bromhead, 1992). The other most
advantageous feature of this model is that, it is readily applicable to the grid-based GIS framework for determination of the stability condition on a spatially distributed setting. The Factor of Safety (FoS) can be calculated for each cell of the grid and the results presented in the form of a FoS–map.
Figure 3.15 Infinite slope model
Factor of Safety based on Limit-equilibrium concept is defined as ratio of the shear strength of the material to the destabilizing stresses along an assumed failure plane, and is expressed as in Equation 1.
s app
Shear Strength of concerned material FoS Applied Shear stress
(3.1)
Mohr–Coulomb failure criterion (Eq. 2) gives the shear strength of the material as a function of the soil cohesion and frictional resistance due to the effective normal stress on the failure plane
tans c n uw
(3.2)
where, τs is the shear strength of the soil, c' is the effective soil cohesion, φ' is the effective angle of internal friction, and, σn and uw are the total normal stress and pore water pressure on the failure plane.
The basic formulation of the FoS for an infinite slope can thus be expressed as:
tan' tan z h cos
sin gz
'
FoS c w
1 (3.3)
where ρ is the bulk density of soil, ρw is the bulk density of water, z is the vertical depth of failure plane from slope surface, β is the slope angle, h is the vertical depth of failure plane from ground water level, and g is the acceleration due to gravity. If the shear strength is greater than applied shear stress then FoS > 1, and the slope is considered stable from the viewpoint of limit equilibrium. If the applied shear stress exceeds the shear strength, or somehow the shear strength decreases due to increase in pore water pressure, then the existing applied shear stress, then the FoS falls below 1, and slope consequently is subjected to failure.
Selby (1993) proposed a modification of the infinite slope stability model for accommodating various in-situ factors
tan
tan z h cos
sin gz
c FoS c
s w s
s
r
1 (3.4)
where ρs is the saturated soil density, cr is the root cohesion, and cs is the soil cohesion.
Saturated soil density parameter is used due to its similarity to field conditions when a landslide is triggered. It must be noted that FoS value does not represent absolute stability or instability. The decrease of FoS occurs with an increase of the water level and corresponding reduction of effective stress. During a rainfall event, the water table elevation reduces FoS.
The probability of slope failure increases with increasing intensity and duration of the rainfall event. During dry season the water table recedes and the stability increases with an increase of FoS values (Selby, 1993).
Steady-state Sub-surface Hydrology
Hydrological factors have a direct influence on the slope stability–instability condition. Therefore, it is essential to apply a hydrological model for estimating the depth of ground water level. Both SHALSTAB and SINMAP is based on the hydrological concept as is described in TOPMODEL (Beven and Kirkby, 1979) and TOPOG (O’Loughlin, 1986) for modeling steady-state shallow subsurface ground water condition. The models map the spatial variation of the ground water table in the form of the Wetness Index derived from the upslope contributing catchment area, soil transmissivity and slope angle for a steady-state recharge. The final formulation of the Wetness Index is as that given in Equation 3.5.
z h sin
bT
W qa
(3.5)
T Kszcos (3.6) where, q is the steady state recharge, a is the upslope catchment area draining through contour length of b, T is the soil transmissivity and Ks is the saturated hydraulic conductivity, considered homogeneous along the soil depth. The area a represents the watershed elements obtained through the intersection of contours and drainage boundaries orthogonal to the contours. For the present study, the Multiple Flow Direction method, termed as D∞ method developed by Tarboton (1997), is applied for flow routing for all the three models. Based on the flow routing procedure, the upslope catchment area is computed using a recursive procedure proposed by Mark (1988). Wetness Index, W > 1 indicates saturation of the soil layer, and W is reset to 1 and the remaining water flows as surface run-off. The basic assumption of this model is that of constant soil transmissivity, which can ultimately be interpreted as constant soil thickness considering identical value of saturated permeability over the entire depth of soil layer.
3.11.1. SHALSTAB
Numerical models of instability Expressing Equation 4 in terms of h/z for FoS = 1 results in the saturation amount necessary for the landslide to occur:
tan cos
gz c tan
tan z
h
w w s
1 2
(3.7)
Thus, replacing the term h/z from Equation 5 into Equation 7 and rearranging, the infinite slope and steady-state hydrological models are coupled and the final formulation of SHALSTAB is presented in the form of the steady-state recharge required to initiate slope instability (Montgomery and Dietrich, 1994):
tan cos
gz c tan
tan a
sin q bT
w w s
1 2 (3.8)
Therefore, SHALSTAB classifies the region based on the amount of recharge required to initiate instability. The output from this model is expressed in [L/T] of steady- state recharge, lower values indicate a greater propensity for instability and otherwise.
However, the problem lies in the proper categorization of the range of recharge for dividing the region into zones of higher and lower instability. As is noted by Montgomery and Dietrich (1994), the steady-state recharge cannot be considered as a long-term average of
inclination, surface vegetation cover, infiltration characteristics of the soil and soil and bed rock transmissivity. For correct interpretation of the SHALSTAB results, correlation of these various factors needs to be considered (Pack et al., 1998).
3.11.2. SINMAP
The Stability Index Mapping (SINMAP), like SHALSTAB, applies a steady-state hydrological model coupled to an infinite slope model within a probabilistic framework (Pack et al., 1998). The output of the SINMAP model is in the form of Stability Index (SI), which represents the probability that the FoS of a slope will be greater than 1.0, assuming uniform probability distribution of the parameters over the uncertainty range. However, unlike SHALSTAB, the maximum and minimum value of the ratio Tb/qa over the uncertainty range is to be input into the SINMAP model. The steady-state hydrological model as used in SHALSTAB, (Eq. 5) is applied to estimate the soil saturation, assuming that the maximum value is equal to 1. The FoS is thus expressed in the form as in Equation 9.
sin
tan r sin ,
bT Min qa cos
c FoS
a
1 1
(3.9)
where, ca is the non-dimensional cohesion, and r is the ratio of water and wet soil density, given by
gzsin
c c c
s s r a
(3.10)
w
r s
(3.11)
The input parameters Tb/qa, ca and φ are described in terms of maximum and minimum thresholds, except for ρs, for which its mean value is used. The uncertainty ranges of the input parameters generate a probability distribution of slope stability (in terms of evaluated FoS) between the threshold boundaries. If the lowest value of FoS obtained for any combination of the input parameters within the range of uncertainty, is greater than 1, the output is denoted symbolically by FoSmin [FoSmin = (lowest calculated value of FoS) > 1]. At any locations where the minimum FoS < 1, there is a probability of failure, and the Stability Index (SI) value is then expressed as in Equation 12.
Prob 1
SI FoS (3.12)
Pack et al. (1998) proposes six classification ranges for zonation of the region into stability classes, based on SI and FoSmin values. For SI < 0.5, the location is classified unstable and stabilizing efforts are needed; while for SI values ranging from 0.5–1.0 and FoSmin values ranging from 1.0–1.25, the locations are classified as quasi-stable to moderately stable and slope destabilization factors are required for landsliding. For FoSmin
values ranging from 1.25–1.5 and greater than 1.5, the area is classified as moderately stable and unconditionally stable respectively.
3.11.3. TRIGRS
TRIGRS (Transient Rainfall Infiltration and Grid-based Regional Slope-stability) (Baum et al., 2002; Savage et al., 2004; Baum et al., 2008) is a FORTRAN code developed for the purpose of evaluating the transient pore pressure response to rainfall infiltration, and thereby, simulate the temporal and spatial distribution of shallow, rainfall-induced landslides expressed as decrease in the Factor of Safety values. TRIGRS is capable of computing transient pore-pressure changes and changes in the factor of safety due to rainfall infiltration, combined with a simple runoff routing scheme. Complex rainfall histories can be implemented as time step function with varying intensity and duration. A detailed description of the TRIGRS model is presented in Chapter 6.