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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