3 geotop-summer-school2011

Riccardo Rigon, Stefano Endrizzi, Matteo Dall’Amico, Stephan Gruber, Cristiano Lanni

GEOtop: Richards

G. O

Kee

fe, S

ky

wit

h f

lat

wh

ite

clou

d, 1

96

2

Wednesday, June 29, 2011

“The medium is the message”Marshall MacLuham


3

Objectives

•Make a short discussion about Richards’ equation (full derivation is

left to textbooks)

•Describe a simple (simplified solution of the equation)

•Analyze a numerical simulation for a linear hillslope

•Drawing some (hopefully) non trivial conclusions

•Doing a brief discussion of what happens when the system becomes

saturated from saturated

Rigon et al.

Richards


4

What I mean with Richards ++

First, I would say, it means that it would be better to call it, for

instance: Richards-Mualem-vanGenuchten equation, since it is:

Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

C(ψ) :=∂θw()∂ψ

Rigon et al.

Richards


4




Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Water balance

C(ψ) :=∂θw()∂ψ

Rigon et al.

Richards


4




Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Water balance

ParametricMualem

C(ψ) :=∂θw()∂ψ

Rigon et al.

Richards


4




Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Water balance

ParametricMualem

Parametricvan Genuchten

C(ψ) :=∂θw()∂ψ

Rigon et al.

Richards


5

Parameters !

Se = [1 + (−αψ)m)]−n

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Se :=θw − θr

φs − θrC(ψ) :=

∂θw()∂ψ

Rigon et al.

Richards


6

dθw

dψ= −φs

α m n(α ψ)n−1

[1 + (α ψ)n]m+1(θr + φs)

The hydraulic capacity of soil is proportional to the pore-size distribution

SWRC

Derivative

Water content

Rigon et al.

Richards


7

Rigon et al.

Richards


8

Pedotransfer Functions

Nem

es (

20

06

)

Rigon et al.

Richards


9

X - 52 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES

Figure 2: Experimental set-up. (a) The infinite hillslope schematization. (b) The initial suction head profile.

Figure 3: The soil-pixel hillslope numeration system (the case of parallel shape is shown here). Moving from 0 to 900 (the total number of

soil-pixels), corresponds to moving from the crest to the toe of the hillslope

Table 1: Physical, hydrological and geotechnical parameters used to characterize the silty-sand soil

Parameter group Parameter name Symbol Unit Value

Physical Bulk density ρb (g/cm3) 2.0

% sand - - 60

% silt - - 40

Hydrological Saturated hydraulic conductivity Ksat (m/s) 10−4

Saturated water content Θsat (cm3/cm−3) 0.39Residual water content Θr (cm3/cm−3) 0.155

water retention curve parameter n [−] 1.881water retention curve parameter β (cm−1) 0.0688

Geotechnical Effective angle of shearing resistance φ� ◦ 38Effective cohesion c� kN/m2 0

D R A F T September 24, 2010, 9:13am D R A F T

Lanni and Rigon

Richards


C(ψ)∂ψ∂t = ∂

∂z

�Kz

�∂ψ)∂z − cosθ

��+ ∂

∂y

�Ky

∂ψ∂y

�+ ∂

∂x

�Kx

�∂ψ)∂x − sinθ

��

ψ ≈ (z − d cos θ)(q/Kz) + ψs

Bearing in mind the previous positions, the Richards equation, at hillslope

scale, can be separated into two components. One, boxed in red, relative

to vertical infiltration. The other, boxed in green, relative to lateral flows.

10

The Richards equation on a plane hillslope

Iver

son

, 20

00

; Cord

ano a

nd

Rig

on

, 20

08

Rigon et al.

Richards simplified


11

The Richards Equation!

Rigon et al.

Richards simplified


C(ψ)∂ψ

∂t=

∂

∂z

�Kz

�∂ψ

∂z− cos θ

��+ Sr

Vertical infiltration: acts in a

relatively fast time scale because

it propagates a signal over a

thickness of only a few metres

11


Rigon et al.

Richards simplified


Sr =∂

∂y

�Ky

∂ψ

∂y

�+

∂

∂x

�Kx

�∂ψ

∂x− sin θ

��

12


Rigon et al.

Richards simplified


Sr =∂

∂y

�Ky

∂ψ

∂y

�+

∂

∂x

�Kx

�∂ψ

∂x− sin θ

��

Properly treated, this is reduced to

groundwater lateral flow, specifically to the

Boussinesq equation, which, in turn, have

been integrated from SHALSTAB equations

12


Rigon et al.

Richards simplified


C(ψ)∂ψ

∂t=

∂

∂z

�Kz

�∂ψ

∂z− cos θ

��+ Sr

In literature related to the determination of slope stability this equation

assumes a very important role because fieldwork, as well as theory, teaches

that the most intense variations in pressure are caused by vertical infiltrations.

This subject has been studied by, among others, Iverson, 2000, and D’Odorico

et al., 2003, who linearised the equations.

13


Rigon et al.

Richards simplified


14

Decomposition of the Richards equation

In vertical infiltration plus lateral flow is possible under the assumption

that:

Time scale of infiltration

soil depth

constant diffusivity

time scale of lateral flow

hillslope length

Rigon et al.

Richards simplified


ψ ≈ (z − d cos θ)(q/Kz) + ψs

Iver

son

, 20

00

; D’O

dori

co e

t al

., 2

00

3,

Cord

ano a

nd

Rig

on

, 20

08

15

s

The Richards equation on a plane hillslope

Rigon et al.

Richards simplified


Assuming K ~ constant and neglecting the source terms

16

The Richards Equation 1-D

C(ψ)∂ψ

∂t= Kz 0

∂2ψ

∂z2

D0 :=Kz 0

C(ψ)

Rigon et al.

Richards super-simplified


Assuming K ~ constant and neglecting the source terms

∂ψ

∂t= D0 cos2 θ

∂2ψ

∂t2

16


C(ψ)∂ψ

∂t= Kz 0

∂2ψ

∂z2

D0 :=Kz 0

C(ψ)

Rigon et al.



∂ψ

∂t= D0 cos2 θ

∂2ψ

∂t2

17


Rigon et al.



The equation becomes LINEAR and, having found a solution

with an instantaneous unit impulse at the boundary, the

solution for a variable precipitation depends on the

convolution of this solution and the precipitation.

∂ψ

∂t= D0 cos2 θ

∂2ψ

∂t2

17


Rigon et al.



18


Rigon et al.



For a precipitation impulse of constant intensity, the solution can be

written:

ψ0 = (z − d) cos2 θ

D’O

dori

co e

t al

., 2

00

3

19

ψ = ψ0 + ψs

ψs =

qKz

[R(t/TD)] 0 ≤ t ≤ T

qKz

[R(t/TD)−R(t/TD − T/TD)] t > T


Rigon et al.



In this case the equation admits an analytical solution

D’O

dori

co e

t al

., 2

00

3

20

R(t/TD) :=�

t/(π TD)e−TD/t − erfc��

TD/t�

ψs =

qKz

[R(t/TD)] 0 ≤ t ≤ T

qKz

[R(t/TD)−R(t/TD − T/TD)] t > T

TD :=z2

D0


Rigon et al.



D’O

dori

co e

t al

., 2

00

3

21

TD

TD

TD

TD

Th

e R

ich

ard

s Eq

uat

ion

1

-D

Rigon et al.



The analytical solution methods for the advection-dispersion equation

(even non-linear), that results from the Richards equation, can be found

in literature relating to heat diffusion (the linearized equation is the

same), for example Carslaw and Jager, 1959, pg 357.

Usually, the solution strategies are 4 and they are based on:

- variable separation methods

- use of the Fourier transform

- use of the Laplace transform

- geometric methods based on the symmetry of the equation (e.g.

Kevorkian, 1993)

All methods aim to reduce the partial differential equation to a system

of ordinary differential equations

22

Th

e R

ich

ard

s Eq

uat

ion

1

-D

Rigon et al.

Richards 1D


23

Th

e R

ich

ard

s Eq

uat

ion

1

-D

Rigon et al.

Richards 1

D


Sim

on

i, 2

00

7

24

Th

e R

ich

ard

s Eq

uat

ion

1

-D

Rigon et al.

Richards 1D


25

Sim

on

i, 2

00

7

Th

e R

ich

ard

s Eq

uat

ion

1

-D

Rigon et al.

Richards 1D


26

A simple application ?








% sand - - 60

% silt - - 40






Rigon et al.

Richards 3D


27








% sand - - 60

% silt - - 40






Going back to the simple geometry case

Lanni and Rigon

Richards 3D for a hillslope


28

Conditions of simulation

Wet Initial Conditions

Dry Initial Conditions

Intense Rainfall

Low Rainfall

Moderate Rainfall

Lanni and Rigon



29


(a) DRY-Low (b) DRY-Med

(c) DRY-High (d) WET-Low

(e) WET-Med (f) WET-High

Figure 5: Values of pressure head developed at the soil-bedrock interface at each point of the subcritical parallel hillslope. The slope of

the pressure head lines represents the mean lateral gradient of pressure


Simulations result

Lanni and Rigon



30

Is the flow ever steady state ? X - 54 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES







Lanni and Rigon



31








Simulations result

Lanni and Rigon



32








Simulations result

Lanni and Rigon



33

LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES X - 55

(a) (b)

Figure 6: Temporal evolution of the vertical profile of hydraulic conductivity (a) and hydraulic conductivity at the soil-bedrock interface

(b) of a soil-pixel located in the mid-slope zone. Results are shown for the case representing DRY antecedent soil moisture conditions, Low

rainfall intensity and parallel hillslope shape of the subcritical (gentle) case


Three order of magnitude faster !

The key for understanding

Lanni and Rigon



34

When simulating is understanding

•Flow is never stationary

•For the first hours, the flow is purely slope normal with no lateral

movements

•After water gains the bedrock and a thin capillary fringe grows,

lateral flow starts

•This is due to the gap between the growth of suction with respect to

the increase of hydraulic conductivity

•The condition:

is not verified, since diffusivity in the slope normal direction is much lower

than in the lateral direction (after saturation is created)

Lanni and Rigon



35

Another issue

Extending Richards to treat the transition saturated to unsaturated zone.

Since :

At saturation: what does change in time ?

Rigon et al.

Saturation vs Vadose


36

Another issue

Extending Richards to treat the transition saturated to unsaturated zone. Which means:

Rigon et al.



37

If you do not have this extension you cannot deal properly with from

unsaturated volumes to saturated ones.

Or

Rigon et al.



GEOtop: Richards++

Law

ren

Har

ris,

Mou

nt

Rob

son

Riccardo Rigon, Stefano Endrizzi, Matteo Dall’Amico, Stephan Gruber


“I would like to have a smart phrase for any situation.

But I don’t . Actually, I think is not even necessary.

I learned that this save time to listening to what others

have say, and by be silent you learn ”

Riccardo Rigon


40

Objectives

•Make a short discussion about what happens when soil freezes

•Introduce some thermodynamics of the problem

•Discussing how Richards equation has to be modified to include soil

freezing.

•Treating some little concept behind the numerics

•Seeing a validation of the model

Rigon et al.

Richards ++


41


Extending Richards to treat the phase transition. Which means essentially to extend the soil water retention curves to become dependent on temperature.

Unsaturatedunfrozen

Freezingstarts

Freezingproceeds

UnsaturatedFrozen

Rigon et al.

Richards ++


42

Rigon et al.

The variable there !

Ice, soil, water and pores


43

Uc( ) := Uc(S, V, A, M)

Expression Symbol Name of the dependent variable∂SUc T temperature

- ∂V Uc p pressure∂AUc γ surface energy∂MUc µ chemical potential

dUc(S, V, A, M)dt

=∂Uc( )

∂S

∂S

∂t+

∂Uc( )∂V

∂V

∂t+

∂Uc( )∂A

∂A

∂t+

∂Uc( )∂M

∂M

∂t

Internal Energy

entropy

interfacial area

volume mass Independent variables

dUc(S, V, A, M) = T ( )dS − p( )dV + γ( ) dA + µ( ) dM

Rigon et al.



44

d[U( )+K( )+P ( )]dt = Φ( )

dS( )dt ≥ 0

dS(U, V, M) = 0

potential energy

kineticenergy

internalenergy

energy fluxes at the boundaries

the equilibrium relation becomes:

Assuming:

K( ) = 0 ; P ( ) = 0 ; Φ( ) = 0

Total Energy

Rigon et al.



45

At equilibrium. Gravity. One phase. No fluxes


Rigon et al.



46

Ti = Tw

pi = pw

µi = µw

The equilibrium relation becomes:

At equilibrium. No gravity. No fluxes. Two phases

Rigon et al.



47


At equilibrium. Gravity. Two phases. No fluxes

Rigon et al.



48

At equilibrium. Gravity. Two phases. No fluxesSeen in the phases diagram.

Going deeper in the pool we move according to the arrows going from higher to lower positions

Rigon et al.



49

SdT ( )− V dp( ) + Mdµ( ) ≡ 0

dµw(T, p) = dµi(T, p)

Gibbs-Duhem identity

From the equilibrium condition

At equilibrium. Two phases. No gravity. No fluxes. And ... no interfaces.

The equilibrium equation between the phases allows to derive the equations for the curves separating the phases, i.e. to obtain the Clausius-Clapeyron equation:

Internal Energy

Rigon et al.



50

−hw( )T

dT + vw( )dp = −hi( )T

dT + vi( )dp

⇒ dp

dT=

sw( )− si( )vw( )− vi( )

=hw( )− hi( )

T [vw( )− vi( )]≡ Lf ( )

T [vw( )− vi( )]

At equilibrium. Gravity. Two phases. No gravity. No fluxes. And ... no interfaces.

Rigon et al.



51

At equilibrium. Two phases. No gravity. No fluxes. And interfaces.

U( ) := T ( )S − p( )V + γ( )A + µ( )M

If we assume existing a relation between the interfacial area A and the volume, the effect of the surface can be seen as a pressure

Rigon et al.



52

pw = pa − γwa∂Awa(r)∂Vw(r)

= pa − γwa∂Awa/∂r

∂Vw/∂r= pa − γwa

2r

:= pa − pwa(r)

That is, what is seen in the Young-Laplace equation

pa

pw

At equilibrium. Two phases. No gravity. No fluxes. And interfaces.

Rigon et al.



53

dS =�

1Tw

− 1Ti

�dUw +

�pw + γiw

∂Aiw∂Vw

Tw− pi

Ti

�dVw −

�µw

Tw− µi

Ti

�dMw = 0

Ti = Tw

pi = pw + γiw∂Aiw∂Vw

µi = µw

The equilibrium condition becomes:

and, finally:

Putting all together

Rigon et al.



54

pw0 = pa − γwa∂Awa(r0)

∂Vw= pa − pwa(r0) pi = pa − γia

∂Aia(r0)∂Vw

:= pa − pia(r0)

pw1 = pa − γia∂Aiar(0)

∂Vw− γiw

∂Aiw(r1)∂Vw

Two interfaces (air-ice and water-ice) should be considered!!!

A closer look

Rigon et al.



55

pi = pa − γia∂Aia(r0)

∂Vw≡ pa

pia(r) = −γia∂Aia(r0)

∂Vw← 0

pw1 = pw0 − γwa∂

∂Vw(Awa(r1)−Awa(r0)) = pw0 + pwa(r0)− pwa(r1)

pw1 = pw0 + ∆pfreez∆pfreez := −γwa∂ ∆Awa

∂Vw= pwa(r0)− pwa(r1)

Considering the assumption “freezing=drying” (Miller, 1963) the ice “behaves like air”:

Rigon et al.

freezing = drying


56

−hw( )T

dT + vw( )dpw = −hi( )T

dT + vi( )dpi

�dpw = dpfreez

dpi = 0

LfdT

T=

1ρw

dpfreez

pw1 ≈ pw0 + ρwLf

T0(T − T0)

From the equilibrium condition and the Gibbs-Duhem identity:

From the “freezing=drying” assumption:

freezing conditionunsaturated condition

Rigon et al.

freezing = drying


57

Unsaturatedunfrozen

Freezingstarts

Freezingprocedes

UnsaturatedFrozen

Rigon et al.

freezing = drying


58

Unsaturatedunfrozen

UnsaturatedFrozen

Freezingstarts

Freezingprocedes

Rigon et al.

freezing = drying


59

θw =θs

Aw |ψ|α + 1

θw = θr + (θs − θr) · {1 + [−α (ψ)]n}−m

θmaxw = θs ·

�Lf (T − Tm)

g T ψsat

�-1/b

C l a p p a n d H o r n b e r g e r (1978)

Unfrozen water content

soil water retention curve

thermodynamicequilibrium (Clausius Clapeyron)

+

Luo et al. (2009), Niu a n d Y a n g ( 2 0 0 6 ) , Zhang et al. (2007)

Gardner (1958) Shoop and Bigl (1997)

Van Genuchten (1980) Hansson et al (2004)

ψw =pw

ρw gpressure head:

θw(T ) = θw [ψw(T )]

Rigon et al.

freezing = drying


60

T ∗ := T0 +g T0

Lfψw0

Θ = θr + (θs − θr) · {1 + [−α · ψw0]n}−m

ice content: θi =ρw

ρi

�Θ− θw

�

θw = θr + (θs − θr) ·�

1 +�−αψw0 − α

Lf

g T0(T − T

∗) · H(T − T∗)

�n�−m

liquid water content:

Total water content:

depressed m e l t i n g point

A summary of the equations

Rigon et al.

freezing = drying


61

−0.05 −0.04 −0.03 −0.02 −0.01 0.00

0.1

0.2

0.3

0.4


temperature [C]

Thet

a_u

[−]

psi_m −5000

psi_m −1000

psi_m −100

psi_m 0

ice

air

water

...

Assume you have an initial condition of little more that 0.1 water content

In the graphics

Rigon et al.

freezing = drying


62

−0.05 −0.04 −0.03 −0.02 −0.01 0.00

0.1

0.2

0.3

0.4


temperature [C]

Thet

a_u

[−]

psi_m −5000

psi_m −1000

psi_m −100

psi_m 0

ice

air

water

...

There is a freezing point depression of less than 0.01 centigrades

In the graphics

Rigon et al.

freezing = drying


63

−0.05 −0.04 −0.03 −0.02 −0.01 0.00

0.1

0.2

0.3

0.4


temperature [C]

Thet

a_u

[−]

psi_m −5000

psi_m −1000

psi_m −100

psi_m 0

ice

air

water

...

Temperature goes down to -0.015. Then, the water unfrozen remains 0.1

In the graphics

Rigon et al.

freezing = drying


64

Unsaturatedunfrozen

UnsaturatedFrozen

Freezingstarts

Freezingprocedes

An over and over again

Rigon et al.

freezing = drying


65

The overall relation between Soil water content, Temperature, and suction

Rigon et al.

freezing = drying


66

The overall relation between Soil water content, Temperature, and suction

Rigon et al.

freezing = drying


67

-3 -2 -1 0 1

0.0

0.2

0.4

0.6

0.8

1.0

n=1.5

temperature [C]

the

ta_

w/t

he

ta_

s [

-]

psi_w0=0 psi_w0=-1000

alpha=0.001 [1/mm]

alpha=0.01 [1/mm]

alpha=0.1 [1/mm]

alpha=0.4 [1/mm]

-10000 -8000 -6000 -4000 -2000 0

0.0

0.2

0.4

0.6

0.8

1.0

n=1.5

psi_w0 [mm]

the

ta_

w/t

he

ta_

s [

-]

T=2 T=-2

alpha=0.001 [1/mm]

alpha=0.01 [1/mm]

alpha=0.1 [1/mm]

alpha=0.4 [1/mm]

T > 0α [mm−1]

n 0.001 0.01 0.1 0.41.1 0.939 0.789 0.631 0.5491.5 0.794 0.313 0.099 0.0492.0 0.707 0.099 0.009 0.0022.5 0.659 0.032 0.001 1.2E-4

T = −2 ◦C

α [mm−1]n 0.001 0.01 0.1 0.41.1 0.576 0.457 0.363 0.3161.5 0.063 0.020 0.006 0.0032.0 4E-3 4E-4 4E-5 1E-52.5 2.5E-4 8E-6 2.5E-7 3.2E-8

24

θw/θs at ψw0=−1000 [mm]

Rigon et al.

freezing = drying


68

M. Dall’Amico et al.: Freezing unsaturated soil model 473


and consequently:

ψ =ψ(T ) =ψw0 +Lf

g T ∗(T −T

∗) (19)

The above equation is valid for T < T∗: in fact, when T ≥

T∗, the freezing process is not activated and the liquid water

pressure head is equal to the ψw0. Equations (17) and (19)

collapse in Eq. (15) for a saturated soil (i.e. ψw0 = 0). Thus

the formulation of the liquid water pressure head ψ(T ) under

freezing conditions, valid both for saturated and unsaturated

soils, becomes:

ψ(T ) =ψw0 + Lf

g T∗ (T −T∗) if T <T

∗

ψ(T ) =ψw0 if T ≥T∗

(20)

which can be summarized using the Heaviside function H( )as:

ψ(T ) =ψw0 +Lf

g T ∗(T −T

∗) ·H(T ∗−T ) (21)

If the soil water retention curve is modeled according to the

Van Genuchten (1980) model, the total water content be-

comes:

Θv = θr +(θs−θr) ·{1+[−α ψw0]n}−m

(22)

where θr (-) is the residual water content. The liquid water

content θw becomes:

θw = θr +(θs−θr) ·{1+[−α ψ(T )]n}−m(23)

Equation (23) gives the liquid water content at sub-zero

temperature and is usually called ”freezing-point depression

equation” (e.g. Zhang et al., 2007 and Zhao et al., 1997).

Differently from Zhao et al. (1997), it takes into account not

only the temperature under freezing conditions but also the

depressed melting temperature T∗, which depends on ψw0. It

comes as a consequence that the ice fraction is the difference

between Θv and θw:

θi = Θv(ψw0)−θw [ψ(T )] (24)

It results that, under freezing conditions (T < T∗), θw and

θi are function of ψw0, which dictates the saturation degree,

and T, that dictates the freezing degree. Equation (24), usu-

ally called “freezing-point depression equation”, relates the

maximum unfrozen water content allowed at a given temper-

ature in a soil. Figure 2 reports the freezing-point depression

equation for pure water and the different soil types according

to the Van Genuchten parameters given in Table 1.

Equations (21) and (17) represent the closure relations sought

for the differential equations of mass conservation (Eq. 6)

and energy conservation (Eq. 8).

Table 1. Porosity and Van Genuchten parameters for water and

different soil types as visualized in Fig. 2

θs θr α n source

(-) (-) (mm−1

) (-)

water 1.0 0.0 4E-1 2.50

sand 0.3 0.0 4.06E-3 2.03

silt 0.49 0.05 6.5E-4 1.67 (Schaap et al., 2001)

clay 0.46 0.1 1.49E-3 1.25 (Schaap et al., 2001)

−5 −4 −3 −2 −1 0 1

0.0

0.2

0.4

0.6

0.8

1.0

Temperature [ C]

wate

r con

tent

[−]

pure water

clay

silt

sand

Fig. 2. Freezing curve for pure water and various soil textures, ac-

cording to the Van Genuchten parameters given in Table 1.

5 The decoupled solution: splitting method

The final system of equations is given by the equations of

mass conservation (Eq. 6) and energy conservation (Eq. 8):

∂Θm(ψw0,T )∂t +∇•Jw(ψw0,T )+Sw = 0

∂U(ψw0,T )∂t +∇• [G(T )+J(ψw0)]+Sen = 0

(25)

The previous system is a function of T and ψw0 and can be

solved by the splitting method, as explained in Appendix B.

In the first half time step, the Richards’ equation is solved

and the internal energy is updated with only the advection

contribution. In the second half, no water flux is allowed,

which makes the volume a closed system, and the internal

energy is updated with the conduction flux in order to find

the new temperature and the new combination of water and

ice contents.

Fig. 2. Freezing curve for pure water and various soil textures, ac-cording to the Van Genuchten parameters given in Table 1.

The above equation is valid for T < T∗: in fact, when T ≥

T∗, the freezing process is not activated and the liquid water

pressure head is equal to the ψw0. Equations (17) and (19)collapse in Eq. (15) for a saturated soil (i.e. ψw0 = 0). Thusthe formulation of the liquid water pressure headψ(T ) underfreezing conditions, valid both for saturated and unsaturatedsoils, becomes:

ψ(T ) = ψw0+ Lfg T ∗ (T −T

∗) if T <T

∗

ψ(T ) = ψw0 if T ≥ T∗

(20)

which can be summarized using the Heaviside function H( )

as:

ψ(T ) = ψw0+Lf

g T ∗ (T −T∗) ·H(T

∗ −T ) (21)

If the soil water retention curve is modeled according to theVan Genuchten (1980) model, the total water content be-comes:

�v= θr+(θs−θr) ·�1+ [−α ψw0]n

�−m (22)

where θr (−) is the residual water content. The liquid watercontent θw becomes:

θw= θr+(θs−θr) ·�1+ [−α ψ(T )]n

�−m (23)

Equation (23) gives the liquid water content at sub-zerotemperature and is usually called “freezing-point depressionequation” (e.g. Zhang et al., 2007 and Zhao et al., 1997).Differently from Zhao et al. (1997), it takes into account notonly the temperature under freezing conditions but also the

Table 1. Porosity and Van Genuchten parameters for water anddifferent soil types as visualized in Fig. 2.

θs θr α n source(−) (−) (mm−1) (−)

water 1.0 0.0 4×10−1 2.50sand 0.3 0.0 4.06×10−3 2.03silt 0.49 0.05 6.5×10−4 1.67 (Schaap et al., 2001)clay 0.46 0.1 1.49×10−3 1.25 (Schaap et al., 2001)

depressed melting temperature T∗, which depends onψw0. It

comes as a consequence that the ice fraction is the differencebetween �v and θw:

θi= �v(ψw0)−θw[ψ(T )] (24)

It results that, under freezing conditions (T < T∗), θw and

θi are function of ψw0, which dictates the saturation degree,and T , that dictates the freezing degree. Equation (24), usu-ally called “freezing-point depression equation”, relates themaximum unfrozen water content allowed at a given temper-ature in a soil. Figure 2 reports the freezing-point depressionequation for pure water and the different soil types accordingto the Van Genuchten parameters given in Table 1.Equations (21) and (17) represent the closure relations

sought for the differential equations of mass conservation(Eq. 6) and energy conservation (Eq. 8).

5 The decoupled solution: splitting method

The final system of equations is given by the equations ofmass conservation (Eq. 6) and energy conservation (Eq. 8):

∂�m(ψw0,T )

∂t+∇•Jw(ψw0,T )+Sw= 0

∂U(ψw0,T )

∂t+∇• [G(T )+J (ψw0)]+Sen= 0

(25)

The previous system is a function of T and ψw0 and can besolved by the splitting method, as explained in Appendix B.In the first half time step, the Richards’ equation is solvedand the internal energy is updated with only the advectioncontribution. In the second half, no water flux is allowed,which makes the volume a closed system, and the internalenergy is updated with the conduction flux in order to findthe new temperature and the new combination of water andice contents.

5.1 Step 1: water and advection flux

Let us indicate with the superscript “n” the quantities at thetime step n, with “n+1” the quantities at the time step n+1:then t

n+1 = tn +�t (�t being the integration interval), and

with “n+1/2” the quantities at the end of the first step (tem-porary quantities). In the first step of the splitting method,

www.the-cryosphere.net/5/469/2011/ The Cryosphere, 5, 469–484, 2011

Dependence on texture

Rigon et al.

freezing = drying


69

∂

∂t

�θfl

w(ψw1)

�− �∇ •

�KH

�∇ ψw1 + KH�∇ zf

�+ Sw = 0

Liquid water may derive from

ice me l t ing : ∆θph

water flux: ∆θfl

Volume conservation:

0 ≤ θr ≤ Θ ≤ θs ≤ 1

θr −�θw0 + θi0 +

�1− ρi

ρw

�∆θph

i

�≤ ∆θfl

w ≤ θs −�θw0 + θi0 +

�1− ρi

ρw

�∆θph

i

�

Mass conservation (Richards, 1931) equation:

The Equations: the mass budget

Rigon et al.

freezing = drying


70

U = Cg(1− θs) T + ρwcwθw T + ρiciθi T + ρwLfθw

∂U

∂t+ �∇ • (�G + �J) + Sen = 0

�G = −λT (ψw0, T ) · �∇T

�J = ρw · �Jw(ψw0, T ) · [Lf + cw T ]

0 assuming freezing=drying

U = hgMg + hwMw + hiMi − (pwVw + piVi) + µwMphw + µiM

phi

no expansion: ρw=ρi

assuming:0 no flux during phase change

Eventually:

0 assuming equilibrium thermodynamics: µw=µi and Mw

ph = -Miph

conduction

advection

The Equations: the energy budget

Rigon et al.

freezing = drying consequences


71

dU

dt= CT

dT

dt+ ρw

�(cw − ci) · T + Lf

�∂θw

∂t

∂θw [ψw1(T )]∂t

=∂θw

∂ψw1· ∂ψw1

∂T· ∂T

∂t= CH(ψw1) · ∂ψfreez

∂T· dT

dt

dU

dt=

�CT + ρw

�Lf + (cw − ci) · T

�· CH(T ) · ∂ψfreez(T )

∂T

�· dT

dt

-3 -2 -1 0 1

020

40

60

80

100

140

alpha= 0.01 [1/mm] n= 1.5 theta_s= 0.4

Temp. [ C]

U [M

J/m

3]

psi_w0=0

psi_w0=-100

psi_w0=-1000

psi_w0=-10000

-3 -2 -1 0 1

alpha= 0.01 [1/mm] n= 1.5 C_g= 2300000 [J/m3 K]

Temp. [ C]

C_

a [

MJ/m

3 K

]

1e+01

1e+02

1e+03

psi_w0=0 psi_w0=-1000

theta_s= 0.02

theta_s= 0.4

theta_s= 0.8 {Capp

The Equations: the energy budget

Rigon et al.

freezing = drying consequences


72

∂U(ψw0,T )∂t

− ∂

∂z

�λT (ψw0, T ) · ∂T

∂z− J(ψw0, T )

�+ Sen = 0

∂Θ(ψw0)∂t

− ∂

∂z

�KH(ψw0, T ) · ∂ψw1(ψw0,T )

∂z−KH cos β

�+ Sw = 0

What we do in reality (GEOtop) is 1D

Rigon et al.

Equations and Numerics


73

turbulent fluxesInput

boundaryconditions

energybudget

snow/glaciers

precipitation water budget

Outputnew time step

GEOtopworkflow

Rigon et al.



74

• Finite difference discretization, semi-implicit Crank-Nicholson method;

• Conservative linearization of the conserved quantity (Celia et al, 1990);

• Linearization of the system through Newton-Raphson method;

• when passing from positive to negative temperature, Newton-Raphson method is subject to big oscillations (Hansson et al, 2004)

Numerics

Rigon et al.



75

if ||�Γ(η)m+1|| > ||�Γ(η)m|| ⇒ �ηm+1 � �ηm − �∆η · δ

globally convergent Newton-Raphson

reduction factor δ with 0 ≤ δ ≤ 1. If δ = 1 the scheme is the normal Newton-Raphson scheme

∆η

Numerics

Rigon et al.



76

• Moving boundary condition between the two phases, where heat is liberated or absorbed

• Thermal properties of the two phases may be different

v1 = v2 = Tref (t > 0, z = Z(t))

v2 → Ti (t > 0, z →∞)

v1 = Ts (t > 0, z = 0)

λ1∂v1∂z − λ2

∂v2∂z = Lf ρw θs

dZ(t)dt (t > 0, z = Z(t))

∂v1∂t = k1

∂2v1∂z2 (t > 0, z < Z(t))

∂v2∂t = k2

∂2v2∂z2 (t > 0, z > Z(t))

v1 = v2 = Ti (t = 0, z)( Carlslaw and Jaeger, 1959, Nakano and Brown, 1971 )

The Stefan problem

Rigon et al.



77

M. Dall’Amico et al.: Freezing unsaturated soil model 477M. Dall’Amico et al.: Freezing unsaturated soil model 9

!"#$%&'()*+

,%-./

0

0 10

23 4"#

!"#$%&'()*+

,%-./

0

0 10

23 4"#! "

Fig. 4. Comparison between the analytical solution (dotted line) and the simulated numerical (solid line) at various depths (m). The

numerical model in panel (A) uses Newton C-max the one in panel (B) uses Newton Global. Both have a grid spacing of 10 mm and 500

cells. Oscillations are present in panel (B) but not in panel (A) where no convergence is reached.

Fig. 5. Comparison between the simulated numerical and the an-

alytical solution. Soil profile temperature at different days. Grid

size=10 mm, N=500 cells

semi-infinite region given by Neumann. The features of this

problem are the existence of a moving interface between

Fig. 6. Cumulative error associated with the the globally convergent

Newton method. Solid line: cumulative error (J), dotted line: cumu-

lative error (%) as the ratio between the error and the total energy

of the soil in the time step. � was set to 1E-8.

the two phases, in correspondence of which heat is liberated

Fig. 4. Comparison between the analytical solution (dotted line) and the simulated numerical (solid line) at various depths (m). Thenumerical model in panel (A) uses Newton C-max the one in panel (B) uses Newton Global. Both have a grid spacing of 10mm and 500cells. Oscillations are present in panel (B) but not in panel (A) where no convergence is reached.

conductive heat flow in both the frozen and thawed regions,(3) change of volume negligible, i.e. ρw= ρi and (4) isother-mal phase change at T = Tm, i.e. no unfrozen water existsat temperatures less then the melting temperature Tm. Theisothermal phase change and uniform thermal characteristicsin the frozen and unfrozen state, may be assumed by im-posing a discontinuity on the freezing front line z = Z(t):θw(z) = 0, θi(z) = 1 λ(z) = λi, CT(z) = ρici for (t > 0, z <

Z(t)) and θw(z) = 1, θi(z) = 0 λ(z) = λw, CT(z) = ρwcwfor (t > 0, z ≥ Z(t)), respectively. The initial conditionsare: Ti(t = 0,z > 0) = +2 ◦C and a substance completely un-frozen: θw(t = 0,z) = 1 and θi(t = 0,z) = 0. The boundaryconditions of Dirichlet type: Ts(t > 0,z = 0) = −5 ◦C andTbot(t > 0,z → ∞) = +2 ◦C for the top and bottom bound-ary, respectively. As Nakano and Brown (1971) did for thecase of an initially frozen soil, in Appendix C we reportedthe complete derivation of the solution both for freezing andthawing processes.As the analytical solution considers the freezing of pure

water, in the numerical scheme we have considered a soilwith porosity θs = 1 characterized by a very steep soil waterretention curve with no residual water content, approachinga step function (Table 1).The domain is composed of 500 cells characterized by a

uniform depth �z = 10mm; the integration time �t = 10 s.We have already shown in Fig. 4a the results of the test with-out the globally convergent method. Figure 5 shows the com-parison between the numerical and the analytical solutionsof the soil temperature profile using the globally convergent

Newton’s method. The analytical solution is represented bythe dotted line and the simulation according the numericalmodel by the solid line. The results are much improved withthe globally convergent method, as the simulated tempera-ture follows the analytical solution very well. The temper-ature evolution shows a change in the slope that coincideswith the separation point between the upper frozen and thelower thawed part. Figure 4b reports the comparison on thetime axis (days) at different depths (m). The numerical sim-ulation result shows oscillations, which begin at the time ofphase change and then dampen with time. In the numericalsolution the temperature starts decreasing only when all thewater in the grid cell has been frozen. Furthermore, Tl isinfluenced by the phase change of Tl+1 by the release of la-tent heat and thus the temperature oscillation continues alsoin the frozen state. Therefore, oscillation amplitude is bothlinked to the grid size and to the time: increasing the gridsize, the oscillation amplitude increases, as the mass of wa-ter to freeze increases before the temperature may decrease.The oscillation amplitude dampens with time as the freezingfront moves away from zl ; it may be reduced but not elimi-nated, as it is embedded with the fixed-grid Eulerian method,where the freezing front may move in a discrete way and notin a continuum as in the reality. In order to test the energyconservation capabilities of the algorithm, the error, definedas the difference between the analytical and the simulated so-lution, was calculated at each time step as the p-norm (p = 1)of all the components and was cumulatively summed for theduration of the simulation. The tolerance � on the energy


Rigon et al.



78

478 M. Dall’Amico et al.: Freezing unsaturated soil model





−5 −4 −3 −2 −1 0 1 2

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

Temp [ C]

soil

dept

h [m

]

03

15

40

75

SimAn











Fig. 5. Comparison between the simulated numerical and the an-alytical solution. Soil profile temperature at different days. Gridsize= 10mm, N = 500 cells.

balance was set to 1×10−8. Figure 6 shows the cumulatederror in J (solid line) and in percentage as the ratio betweenthe error and the total energy of the soil in the time step. With� set to 1×10−8, after 75 days of simulation, the error in per-centage remains very low (< 1×10−10), suggesting a goodenergy conservation capability of the algorithm.

7.2 Coupled water and energy flux: experimental data

In order to test the splitting time method for solving the cou-pled water and energy conservation, as done by Daanen et al.(2007), the model was tested against the experiment of Hans-son et al. (2004). The soil considered represents a Kana-gawa sandy loam, with the following parameters: θs= 0.535,θr= 0.05, α = 1.11×10−3 mm−1, n = 1.48, Cgs= 2.3×106,Jm−3 K−1, λgs = 2.5Wm−1 K−1 and saturated hydraulicconductivity KH(sat)= 0.0032mm s−1. The column wasconsidered initially unfrozen, with a uniform total water con-tent �v= 0.33 which, given the parameters of the soil reten-tion curve, corresponds to ψw0 = −2466.75mm, and initialtemperature T = 6.7 ◦C uniform. The boundary conditionsare of Neumann type: for the energy balance, at the top aflux F = −28 ·(T1+6) was considered, where T1 is the tem-perature of the first layer, and a zero flux condition at thebottom. For the mass balance equation, a zero flux at bothtop and bottom boundaries were used.Figure 7 shows the comparison of the profile of the total

water content �v. Starting from a thawed condition and auniform water content �v = 0.33, the liquid water content










0.00

00.

005

0.01

00.

015

0.02

0

time (days)

Cum

ulat

ive e

rror (

J)

0 15 30 45 60 75

Error (%)Error (J)

5e−1

35e−1

11e−1

0C

umul

ative

erro

r (%

)






Fig. 6. Cumulative error associated with the the globally convergentNewton method. Solid line: cumulative error (J), dotted line: cumu-lative error (%) as the ratio between the error and the total energyof the soil in the time step. � was set to 1×10−8.

decreases from above due to the increase of ice content. It isvisible that the freezing of the soil sucks water from below.The increase in total water content reveals the position ofthe freezing front: after 12 h it is located about 40mm fromthe soil surface, after 24 h at 80mm and finally after 50 h at140mm. Similar to Hansson et al. (2004), the results wereimproved by multiplying the hydraulic conductivity by animpedance factor, as described is Sect. 3.1. It was found thatthe value of ω that best resembles the results is 7.Figure 8 shows the cumulative number of iterations re-

quired by the Newton C-max and the Newton global schemesto converge. It is clear that the number of iterations of thenew method is much lower than the previous, indicating thatthis method provides improved performance on the total sim-ulation time.

7.3 Infiltration into frozen soil

The coupled mass and energy conservation algorithm wasfinally tested with simulated rain (infiltration) during thethawing of a frozen soil. The soil geometry is a 20 cmdepth column discretized in 800 layers of 0.25mm depth;the top boundary condition for the energy equation is ofNeumann type, with 10Wm−2 constant incoming flux (nodaily cycle) and the bottom boundary condition is a zeroenergy flux. The initial conditions on the temperature are:Ti(t = 0,z > 0) = −10 ◦C; the soil is considered initially un-saturated, with water pressure head ψw0 given by a hydro-static profile based on a water table at 5m depth. This profile

The Cryosphere, 5, 469–484, 2011 www.the-cryosphere.net/5/469/2011/

Rigon et al.



79

M. Dall’Amico et al.: Freezing unsaturated soil model 479M. Dall’Amico et al.: Freezing unsaturated soil model 11

water content [−]

soil

dept

h [m

m]

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

0.25 0.30 0.35 0.40 0.45 0.50 0.55

after 12 hours!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

Sim! Meas

water content [−]

soil

dept

h [m

m]

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

0.25 0.30 0.35 0.40 0.45 0.50 0.55

after 24 hours!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

Sim! Meas

water content [−]

soil

dept

h [m

m]

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

0.25 0.30 0.35 0.40 0.45 0.50 0.55

after 50 hours!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

Sim! Meas

Fig. 7. Comparison between the numerical (solid line) and the experimental results (points) obtained by Hansson et al. (2004) of the totalwater (liquid plus ice) after 12 (left), 24 (center) and 50 hours (right).

Fig. 8. Cumulative number of iterations of the Newton C-max andthe Newton global methods on the simulation test based on the ex-perimental results obtained by Hansson et al. (2004)


The coupled mass and energy conservation algorithm was fi-nally tested with simulated rain (infiltration) during the thaw-ing of a frozen soil. The soil geometry is a 20 cm depthcolumn discretized in 800 layers of 0.25 mm depth; the topboundary condition for the energy equation is of Neumanntype, with 10 W m−2 constant incoming flux (no daily cycle)and the bottom boundary condition is a zero energy flux. Theinitial conditions on the temperature are: Ti(t = 0,z > 0) =−10◦C; the soil is considered initially unsaturated, with wa-ter pressure head ψw0 given by a hydrostatic profile based ona water table at 5 m depth. This profile corresponds, accord-ing to Eq. (23) and (24), to the water and ice contents at eachlevel. The soil texture and thermal parameters are as in theexperiment described in Section 7.2, whereas the saturatedhydraulic conductivity is 0.3 mm s−1. As far as the bound-ary condition on the mass is concerned, the bottom bound-ary is characterized by a no flux condition, whereas the topboundary condition varies along two simulations: zero flux(without rain) and constant 10 mm h−1 flux, resembling aconstant precipitation (with rain). As can be seen in Fig. 9,in the first 6-7 days the behaviors with rain (solid line) andwithout precipitation (dotted line) are almost equal, becausehydraulic conductivity is so low due to the fast ice-saturationof the first layer during cold conditions neat the beginning ofthe experiment. Then, when some ice is melted, hydraulic

Fig. 7. Comparison between the numerical (solid line) and the experimental results (points) obtained by Hansson et al. (2004) of the totalwater (liquid plus ice) after 12 (left), 24 (center) and 50 h (right).


Fig. 7. Comparison between the numerical (solid line) and the experimental results (points) obtained by Hansson et al. (2004) of the totalwater (liquid plus ice) after 12 (left), 24 (center) and 50 hours (right).

010

000

2000

030

000

4000

0

time (days)

cum

ulat

ive n

umbe

r of i

tera

tion

0 1 2 3 4 5

Newton C=Cmax

Newton global

Fig. 8. Cumulative number of iterations of the Newton C-max andthe Newton global methods on the simulation test based on the ex-perimental results obtained by Hansson et al. (2004)


The coupled mass and energy conservation algorithm was fi-nally tested with simulated rain (infiltration) during the thaw-ing of a frozen soil. The soil geometry is a 20 cm depthcolumn discretized in 800 layers of 0.25 mm depth; the topboundary condition for the energy equation is of Neumanntype, with 10 W m−2 constant incoming flux (no daily cycle)and the bottom boundary condition is a zero energy flux. Theinitial conditions on the temperature are: Ti(t = 0,z > 0) =−10◦C; the soil is considered initially unsaturated, with wa-ter pressure head ψw0 given by a hydrostatic profile based ona water table at 5 m depth. This profile corresponds, accord-ing to Eq. (23) and (24), to the water and ice contents at eachlevel. The soil texture and thermal parameters are as in theexperiment described in Section 7.2, whereas the saturatedhydraulic conductivity is 0.3 mm s−1. As far as the bound-ary condition on the mass is concerned, the bottom bound-ary is characterized by a no flux condition, whereas the topboundary condition varies along two simulations: zero flux(without rain) and constant 10 mm h−1 flux, resembling aconstant precipitation (with rain). As can be seen in Fig. 9,in the first 6-7 days the behaviors with rain (solid line) andwithout precipitation (dotted line) are almost equal, becausehydraulic conductivity is so low due to the fast ice-saturationof the first layer during cold conditions neat the beginning ofthe experiment. Then, when some ice is melted, hydraulic

Fig. 8. Cumulative number of iterations of the Newton C-max andthe Newton global methods on the simulation test based on the ex-perimental results obtained by Hansson et al. (2004).

corresponds, according to Eqs. (23) and (24), to the waterand ice contents at each level. The soil texture and thermalparameters are as in the experiment described in Sect. 7.2,whereas the saturated hydraulic conductivity is 0.3mm s−1.As far as the boundary condition on the mass is concerned,the bottom boundary is characterized by a no flux condition,whereas the top boundary condition varies along two simula-tions: zero flux (without rain) and constant 10mmh−1 flux,resembling a constant precipitation (with rain). As can beseen in Fig. 9, in the first 6–7 days the behaviors with rain(solid line) and without precipitation (dotted line) are almostequal, because hydraulic conductivity is so low due to the fastice-saturation of the first layer during cold conditions neatthe beginning of the experiment. Then, when some ice ismelted, hydraulic conductivity increases so that some watercan infiltrate. At this point some incoming water may freezebecause the soil is still cold, the ice content is increased (notshown here) and the zero-curtain effect is prolonged. As aresult, infiltrating water provides energy (latent), so that tem-perature rises above 0 ◦C earlier than in the case of withoutrain. This earlier complete thaw is more evident at greaterdepths: in the case without rain, one has complete thawing at≈ −0.1 ◦C (change of slope of dashed curve) because the soilremains unsaturated and is characterized by T ∗ < Tf, while inthe case with rain complete thawing occurs at 0 ◦C becausethere is saturation. When soil is thawed, in the with rain casesoil temperature rises more slowly due to the lower thermaldiffusivity of the soil. It is interesting to notice that wateronly partially infiltrates into the frozen soil (see that dotted


Rigon et al.



80

Endrizzi, Quinton, Marsh, Dall’Amico

Siksik creek (NWT, Canada) photo Endrizzi

Siksik

Applications


81

What are the key factors controlling the ground thaw in organic-covered permafrost terrain?

• Peat thickness? thermal and hydraulic properties of the ground (Quinton et al, 2008)

snow evolution, soil moisture, evapotranspiration...

position of the water- and frost-table

• Water movement?

•Topographic complexity (slope, aspect)?


Siksik

Applications


82

0 10 20 30 40 50 60

01

02

03

04

0

thaw depth [cm]

rela

tive

fre

qu

en

cy [

%]

!

!

!

! !

!

! week1

week2

week3

week4

week5

week6

week7

week8

week9

Thaw depth relative frequencies: weekly average


Siksik

Applications


83

• lateral water flow• variable peat thickness

high spatial variability of thaw depth


Siksik thaw depth

Applications


84

week 1

___ model- - - meas

week 3

___ model- - - meas


Siksik thaw depth

Applications


85

week 7

___ model- - - meas

week 9

___ model- - - meas

Siksik thaw depth

Applications


86

Stuff learned

•Soil freezing affects infiltration dramatically

•For describing soil freezing you need to deep in thermodynamics

and make several assumptions about the reference state. In this case

the freezing equal drying assumption has been made, and the

Clausius-Clapeyron equation has been generalized (this can be useful

for ET too).

•Phase transitions introduce discontinuities that must be treated

properly from a numerical point of view

Rigon et al.

Applications


Riccardo Rigon, Cristiano Lanni, Silvia Simoni

GEOtop: stability of slopes

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“Often meteo hydrologic forecasting looks

perturbed by imponderable pangs*, as they were

influenced by horoscopes and astrology”

Valentino Zeichen

* Correction by the authors. The aphorism in its triviality, can in fact be adopted for many sciences, or even for scientists. It is certainly true that “scientific” forecasting are not better than hydrology, above all if they are applied blindly to real system, and the modeler does not visit everyday her problem with its difficulties and irreducible issues. Forecasting is difficult and full of uncertainties. However, o not believe to the contrary: reading horoscopes does not helps you in understanding hydrology. Landslide occurence either.


89

•Introducing the problem of physically modelling ladslides with a

physically based model

•Discussing some conceptual result

•Discussing some effects of soil depth variability

•Seeing some application

Objectives

Rigon et al.

Landsliding


90

When simulating is understanding

•Flow is never stationary

•For the first hours, the flow is purely slope normal with no lateral

movements

•After water gains the bedrock and a thin capillary fringe grows,

lateral flow starts

•This is due to the gap between the growth of suction with respect to

the increase of hydraulic conductivity

Rigon et al.

Landsliding


91

The three dimensional structure of the problem

Landsliding


R

A stability criterion

92

Landsliding

Lanni and Rigon


93

Symbol Name nickname UnitFoS Factor of Safety fos [/]c� cohesion chsn [M L2 T−2]φc columbian friction angle cfa [/]ηw position of the water table surface pwts [L]z depth of soil ds [L]γs soil/terrain density std [M L−1 T−2 ]γw density of liquid water dlw [M L−1 T−2 ]ζs slope of terrain surface sts [/]

FoS =c�

γs z cos ζs sin ζs+

tanφc

tan ζs− γwηw tanφc

γs z tan ζs

Rigon

Infinite SlopeThe equation

Landsliding
