c. lanni e. cordano , r. rigon , a. tarantino -...
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C. Lanni(1), E. Cordano(1), R. Rigon(1), A. Tarantino(2)
(1) Dipartimento di Ingegneria Civile ed Ambientale (2) Dipartimento di Ingegneria Meccanica e Strutturale
University of Trento (Italy)
Landslide Processes: from geomorphologic mapping to dynamic modelling (6-7 February, 2009 - Strasbourg, France) A tribute to Prof. Dr. Theo van Asch
GEOtop model (Rigon et al., 2006) - www.geotop.org -
To assess the water-pressure field within soil thickness Ψ(x,t)
GEOtop solves both Energy and Water Balance.- 1D solution for the Energy Balance equation- 3D solution for the Mass Balance equation
Every soil pixel is composed of a number of layer chosen by the user and the field equations are
solved using the Finite Difference Method (FDM)
• Two plane slopes converging to a central channel
On the sides AB, BC, CD, DENo flux through soil-bedrock interface
On the sides AF and EF
q = Irain ⋅ cosα ψ =ψtop
ψtop = surface water head
or
q = 0
• Boundary Conditions
How Does GEOtop solve Richards’ Equation?In GEOtop vertical and lateral subsurface water flow are decoupling
So, Richards’ Equation is written in 1D form :
C(ψ)∂θ∂t
=∂∂z
−Kz(ψ) ∂ψ∂z
− cosα⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎤
⎦ ⎥ + S Sink TermIt contains the effect of
lateral flow and energy flux on mass balance
Lateral Flows are computed in explicit manner, using Darcy law:
S =
K( ˜ ψ 1)ψ i+1, j −ψ i, j
Δx+ K( ˜ ψ 2)
ψi, j −ψ i−1, j
Δx⎡
⎣ ⎢
⎤
⎦ ⎥ Δy ⋅ Δz( )+
K( ˜ ψ 3)ψ i, j +1 −ψ i, j
Δy+ K( ˜ ψ 4 )
ψi, j −ψ i, j−1
Δy⎡
⎣ ⎢
⎤
⎦ ⎥ Δx ⋅ Δz( )
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
⎫
⎬ ⎪ ⎪
⎭ ⎪ ⎪
/ Δx ⋅ Δy ⋅ Δz( )
with z = normal slope direction
How Do we define the C(ψ) and K(ψ) functions?
Using van Genuchten-Mualem Model
ψ =1β
θ −θr
θsat −θr
⎛
⎝ ⎜
⎞
⎠ ⎟
1 m
−1⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1n
VAN GENUCHTEN (1980) for the Hydraulic Capacity function
θ Actual volumetric water content
Saturates volumetric water content
Residual volumetric water content
β L−1[ ],m,n Shape parameters of the model, with m=1-1\n
K(ψ) = Ksatθ −θr
θsat −θr
⎛
⎝ ⎜
⎞
⎠ ⎟
0.5
1− 1−θ −θr
θsat −θr
⎛
⎝ ⎜
⎞
⎠ ⎟
1 m⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
m⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
2
Ksat LT−1[ ]Saturated Hydraulic conductivity
MUALEM (1976) for the Hydraulic Conductivity function
−[ ]θsat −[ ]θr −[ ]
χ valued according to Khalili & Kabbaz empirical relation (1998)
χ =ua − uw
ua − uw( )b
⎡
⎣ ⎢
⎤
⎦ ⎥
−0.55
if ua − uw( )b < ua − uw( )b
χ =1 if ua − uw( )b ≥ ua − uw( )b
ua − uw( )b FL−2[ ]where:
is the air-entry value matrix suction
Mohr-Coulomb failure criterion extended to unsaturated conditions
τ = c'+ σ − ua( )+ χ ψ( ) ua − uw( )[ ]⋅ tanφ'
FL−2[ ]
shear strenghτ
effective cohesionc'effective angle of shear strenghφ'
total stressσ
σ − ua( )ua − uw( )
net stress
matrix suction
χ Parameter ranging between 0 and 1, depending on the degree of saturation
FL−2[ ]FL−2[ ]
FL−2[ ]FL−2[ ]
the Indefinite Slope Stability Model
FS =tanφ 'tanα
+γwψ( )0.45 γwψb( )0.55
γ ⋅ h ⋅ sinα ⋅ cosαtanφ'
FS =tanφ 'tanα
+γwψ( )
γ ⋅ h ⋅ sinα ⋅ cosαtanφ'
if ψ <ψb
if ψ ≥ψb
ψb L[ ]=ua − uw( )b
γw Air-entry value suction head
assuming cohesionless soil (c’=0) and ua=uatm
H << L
Angle of the slopeSoil TypeAntecedent Soil Moisture ConditionsRainfall Intensity and Duration
The goal of the study is to investigate the role of some factors on the processes of pore-water pressure redistribution and, hence, on safety factor of the slope.
Different Values of these features are chosen as described below
Angle of the slope
Two cases analyzed: steep and gentle slope
i. STEEP SLOPE, when the angle of the slope is bigger than the frictional angle of the soil
ii. GENTLE SLOPE, when the angle of the slope is smaller (or is the same) than the frictional angle of the soil
tanφ'tanα1
= 0.7
tanφ'tanα2
=1.0
STEEP SLOPE
GENTLE SLOPE
Physical, Mechanical and Hydraulic features
Two cases analyzed: SANDY SOIL and SANDY-SILT SOIL
Through physical properties and soil texture it is possible toget the shape parameters of the van Genuchten model usingVereecken PTF (1989)
SANDY SOILS1
φ'= 35o
% sand = 80% silt = 20
Ksat =10−4 m /s
SANDY-SILT SOILS2
φ'= 30o
% sand = 40% silt = 60
Ksat =10−6 m /s
COARSE-GRAINED SOIL
FINE-GRAINED SOIL
CI1 – Wet Antecedent Condition
CI2 – Moderately Wet Antecedent Condition
CI3 – Dry Antecedent Condition
Tree different initial condition considered in the analysis:
Water Content Profile
Initial Soil Moisture Conditions are implemented by linear pore pressure profile
0.7 if steep slope case1.0 if gentle slope case
0.35 (steep) 0.05 (gentle) for CI10.40 (steep) 0.10 (gentle) for CI20.50 (steep) 0.20 (gentle) for CI3
ψ(z) =ψbottom + γw ⋅ H − z( )
Chosen so as to obtain the following values of initial safety factor of the slope:
FS=1.05 for CI1 initial conditionFS=1.10 for CI2 initial condition
FS=1.20 for CI3 initial condition
FS =tanφ 'tanα
+γwψ( )0.45 γwψb( )0.55
γ ⋅ h ⋅ sinα ⋅ cosαtanφ'
Initial Water-porepressure profile
Rainfall DATA by Paneveggio Station (in Alpine region)
elevation: 1760 m a.s.lcoordinate (Gauss-Boaga): Est 1711557
North 5132115
tp1 High Intensity Short Durationtp2 Medium Intensity Medium Durationtp3 Low Intensity Long Duration
Rainfall Intensity (mm/h) Duration (h)
tp1 24 2tp2 10 6tp3 7 12
Province of Trento (Italy)
Return Time = 100 years
1. NEGLIGIBLE EFFECTS OF LATERAL-FLOW ON THETRIGGERING CONDITIONS
negligible effects of the lateral water flow on negligible effects of the lateral water flow on the reaching of the failure conditionsthe reaching of the failure conditions
C(ψ) ∂θ∂t
=∂∂z
−Kz(ψ) ∂∂z
ψ − cosα( )⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎤
⎦ ⎥ + S
ψ(z,t) ≅ψ(x,z,t)
2. HIGH INCREASE OF THE PRESSUREHEAD ON THE FIRST LAYER
CI1 – WET ANTECEDENT CONDITION and tp1 – SHORT RAINFALL DURATION
Until failure timeUntil failure time
Amount of water needed to reach the failure
VCI1tp1 = I tp1 ⋅ t failure
tp1 = 24 *1.7 = 41 mm
ΨΨfailurefailure
1. Negligible effect of LATERAL-FLOW on the triggering conditions
Amount of water needed to reach the failure
VCI1tp 3 = I tp 3 ⋅ t failure
tp 3 = 7 * 2.9 = 20 mm
2. Increase of pressure head at the first layer lower than the tp1 case
< VCI1tp1
CI1 – WET ANTECEDENT CONDITION and tp3 – LONG RAINFALL DURATION
qinf = I tpi = −Kz (ψ) ∂ψ∂z
− cosα⎛ ⎝ ⎜
⎞ ⎠ ⎟
∂ψ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟
tp1
>∂ψ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟
tp 3
VCI1tp3<VCI1
tp1
Same initial value of k(Ψ), but Itp1>Itp3. So:
• Why these differences ?
ΔV
Ψ(z) at the failure time
Ψfailure
ΔΨ ΔV
Itp1>Itp3
tp3tp1
1. NOT NEGLIGIBLE effect of LATERAL-FLOW on the triggering conditions
at the failure time
ψcenter ≠ψ toe
It needs to solve 3D RichardsIt needs to solve 3D Richards’’ EquationEquation
C(ψ)∂θ∂t
=∂∂z
−Kz(ψ) ∂∂z
ψ − cosα( )⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎤
⎦ ⎥ + S
ψ(z,t) ≠ψ(x,z,t)
∂ψ∂x
≠ 0 ; ∂ψ∂z
≠ 0
PIEZOMETRIC LINEPIEZOMETRIC LINE
CI3 – DRY ANTECEDENT CONDITION and tp3 – LONG RAINFALL DURATION
Suction profile Suction profile ΨΨ(z)(z)at the failure timeat the failure time
CenterCenterΨΨ=875 mm=875 mm
ToeToeΨΨ=844 mm=844 mm
ToeToeΨΨ=844 mm=844 mm
CenterCenterΨΨ=855 mm=855 mm
CI1(wet) vs CI3(dry)tp1 – short rainfall duration
Dry Antecedent Soil Moisture Conditionsamplify the role of lateral flow on instability conditions ΨΨcentercenter−− ΨΨbottombottom = 31 mm= 31 mm
ΨΨcentercenter−− ΨΨbottombottom = 11 mm= 11 mm
CenterCenterΨΨ=930 mm=930 mm
ToeToeΨΨ=844 mm=844 mm
ToeToeΨΨ=844 mm=844 mm
CenterCenterΨΨ=870 mm=870 mm
Suction profile Suction profile ΨΨ(z)(z)at the failure timeat the failure time
CI1(wet) vs CI3(dry)tp3 – long rainfall duration
Long Rainfall duration amplify the role of lateral flow on the instability conditions ΨΨcentercenter−− ΨΨbottombottom = 86 mm= 86 mm
ΨΨcentercenter−− ΨΨbottom bottom = 26 mm= 26 mm
1. Negligible effects of LATERAL-FLOW
C(ψ)∂θ∂t
=∂∂z
−Kz(ψ) ∂∂z
ψ − cosα( )⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎤
⎦ ⎥ + S
negligible effects of lateral water flownegligible effects of lateral water flow ψ(z,t) ≅ψ(x,z,t)
∂ψ∂x
= 0 ; ∂ψ∂z
≠ 0
2. Very High Increase of pressure head at the first layer
At any time during the simulation
ψcenter ≠ψ toe
ΨΨfailurefailure
A. GENERALLY THE COLLAPSE DEPENDS ON VERTICAL PORE-WATER PRESSURE REDISTRIBUTION
B. THESE FEATURES INCREASE THE POSSIBILITY OF SLOPE FAILURE IN THE UPPER LAYERS OF THE SOIL TICKNESS
1. WET ANTECEDENT SOIL MOISTURE CONDITION (CI3)2. SHORT RAINFALL DURATION (AND HIGH INTENSITY) (tp1)3. FINE-GRAINED SOIL TYPE (S2)
IN CASE OF:
OTHERWISE:
1. DRY ANTECEDENT SOIL MOISTURE CONDITION (CI1)2. LONG RAINFALL DURATION (AND HIGH INTENSITY) (tp1)3. COARSE-GRAINED SOIL TYPE (S1)
A. LATERAL FLOW PLAYS AN IMPORTANT ROLE ON THE SLOPE STABILITY CONDITIONS
B. GENERALLY SLOPE FAILURE OCCOURS NEAR THE SOIL BEDROCK INTERFACE
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