Chapter 5. Local Slope Stability Analysis
5.2. Rainfall Induced Slope Instability Analysis
5.2.1. Transient Seepage Analysis
SEEP/W is a code to perform numerical analysis of 2D steady state or transient seepage analyses within porous materials through the application of finite element method (FEM) (GEO-SLOPE International Ltd., 2010a). SEEP/W is capable of analysing isotropic or anisotropic seepage through a complex geometry, made up of either homogeneous or inhomogeneous material. In this section, the theory, meshing, necessary material properties, initial conditions, boundary conditions, and interpretation of results, associated with SEEP/W are briefly described in relation to transient seepage analyses.
Flow through soil
The formulation describing the flow of water through saturated and unsaturated soils in the SEEP/W code is based on Darcy’s Equation of flow through saturated soil and can be represented by the following equation (GEO-SLOPE International Ltd., 2010).
q k i
(5.1)where,
q Specific discharge matrix
k Hydraulic conductivity matrix
i h;l
Unit hydraulic head gradient matrix
Richards (1931) modified the Darcy’s Equation to be applied to the flow of water through unsaturated soil. Under the conditions of unsaturated flow, the hydraulic conductivity is expressed as a function of either the volumetric water content or the pore-water pressure (Childs and Collins-George, 1950). The general governing differential equation for two- dimensional seepage can thus, be expressed as:
x y
h h
k k q
x x y y t
(5.2)where,
h Hydraulic head
kx Hydraulic conductivity in the x-direction ky Hydraulic conductivity in the y-direction q Specific discharge or applied boundary flux
Volumetric water content t Time
Equation 2 essentially states that the sum of the rate of flow changes in x and y directions and the external applied flux, is equal to the rate of change of the volumetric water content with respect to time.
The change in volumetric water content is then related to the change in pore-water pressure by the following expression
w w
m u
(5.3)where,
w w
u hy ; Pore-water pressure
mw Slope of the volumetric water storage curve
w Unit weight of water y Elevation head Substituting the above in Equation 2;
x y w w
h y
h h
k k q m
x x y y t
(5.4)At an element, located at a particular position the elevation head will always remain constant and therefore the governing differential equation used in SEEP/W finite element formulation, is:
x y w w
h h h
k k q m
x x y y t
(5.5)However, changes in the stress state and the density or porosity of the soil affect the changes in volumetric water content. Fredlund and Morgenstern (1976 and 1977) stated regarding both condition of saturated and unsaturated, the stress state can be presented by two state variables as follow: (σ – ua) and (ua – uw), where σ is the total stress, ua is the pore-air pressure, and uw is the pore-water pressure, (ua – uw) is the matric suction.
Thus, while formulating the governing differential equation describing the flow of water through unsaturated soil, the following assumptions are made in the SEEP/W code.
a. (σ – ua) remains constant and has no effect on changes in volumetric water content.
b. The hydraulic conductivity is a function of the matric suction; changes in the hydraulic conductivity and the volumetric water content are consequently dependent only on changes in the matric suction.
c. With the pore-air pressure, (ua) remaining constant, total stress is treated as constant and the change in volumetric water content is a function only of pore-water pressure changes.
SEEP/W uses the Backward Difference Method for the time integration, and thus in order to solve for the new head at the end of the time increment, it is necessary to know the head at the start of the increment. In other words, the initial conditions must be known or specified in order to perform a transient analysis. The simulations result is significantly affected by the initial conditions.
Soil Water Retention Characteristics
The main soil parameters required in the SEEP/W are the saturated hydraulic conductivity, the Soil Water Characteristic Curve (SWCC) and the Unsaturated Hydraulic
Conductivity Curve (UHCC). The SEEP/W has two method based on form equations by Fredlund and Xing (1994) and Van Genuchten (1980) respectively to define the SWCC.
Van Genuchten, (1980) SWCC model
1
s r
w r n m
a
(5.6)
where,
Negative pore-water pressure
w Volumetric water content corresponding to ψ
s Volumetric water content at saturated soil condition
r Residual volumetric water content , ,
a n m Curve fitting parameters Fredlund and Xing, (1994) SWCC model
ln
s
w m
n
C
e a
(5.7)
where,
C Correction function
a i; eThe natural number (2.71828) 3.67 ln s
i
m ,
1.31 1
3.72
m
i s
n s
m
i The suction pressure corresponding to the water content occurring at the inflection point of the
s The slope of the line tangent to the function that passes through the inflection point
Apart from the above-mentioned empirical SWCC models, a predictive method using grain size data and Liquid limit of the soil as developed by Aubertin et al., (2003), modified from the method proposed by Kovacs (1981), is implemented in the SEEP/W code. Data points obtained through experimental procedure can be input into SEEP/W and interpolated to form the SWCC. Along with the above-mentioned methods, SEEP/W also has several
“typical” sample water content functions for different types of soils in its database.
Unsaturated Hydraulic Conductivity
In a saturated soil, all the pore spaces between the solid particles are filled with water.
Once the air-entry value is exceeded, air enters the largest pores and the air-filled pores than act as obstruction to the flow and increase the tortuosity of the flow path. As a result, the ability of the soil to transport water (i.e., the hydraulic conductivity) decreases. As pore-water pressures become increasingly more negative, more pores become air-filled and the hydraulic conductivity decreases further. The ability of water to flow through a soil profile depends on how much water is present in the soil, which is represented by the volumetric water content function. Actual measurement of the unsaturated hydraulic conductivity function is a very complex, time-consuming and expensive procedure. The unsaturated hydraulic conductivity function is therefore developed using one of several predictive methods that utilize either a grain-size distribution curve or a measured volumetric water content function and the saturated hydraulic conductivity. SEEP/W has built-in predictive methods that can be used to estimate the hydraulic conductivity function once the volumetric water content function and the saturated hydraulic conductivity value have been specified.
Van Genuchten (1980), UHCC model
Van Genuchten (1980) proposed the following closed form equation to describe the hydraulic conductivity of a soil as a function of matric suction; the hydraulic conductivity function of a soil can be estimated once the saturated conductivity and the two curve fitting parameters, a and n = 1/(1-m), are known.
2 1
2
1 1
1
n n m
w s m
n
a a
k k
a
(5.8)kw The calculated conductivity for a specified water content or negative pore-water pressure (m/s),
ks The measured saturated conductivity (m/s), Fredlund et al., (1994) UHCC model
The method consists of developing the unsaturated hydraulic conductivity function by integrating along the entire curve of the volumetric water content function. The governing equation is:
i i
i i
N y
y y
i j
w s N y
s y
y i j
e e
k k e
e e
e
(5.9) The first derivative of the Fredlund and Xing, (1994) SWCC model Apart from the above-mentioned UHCC model, SEEP/W also has the Green and Corey, (1971) UHCC model.