hillslope hydrologyandrichards

An Overview Hillslope HydrologyM

irò -

Blu

e II

Riccardo Rigon

2nd International Summer School on Water Research, Praia a Mare, July 2013

Monday, July 8, 13

Goals

• Say what a hillslope is

• Talking about Richards equation

• Say what Hydrology on hillslope is concerned about

• Simplifying Richards’ equation

1

2

• Some reflections

• And Beyond ...

Welcome

R. RigonMonday, July 8, 13

Hillslope HydrologyHillslope

Mir

ò -

Blu

e II

Riccardo Rigon

2nd International Summer School on Water Research, Praia a Mare, July 2013

Monday, July 8, 13

4

What is a hillslope ?

Mon

tgom

ery

and

Die

tric

h, W

RR

, 19

92

First you have to identify channels

R. Rigon

What we are talking about ?

Monday, July 8, 13

5

Orl

an

din

i et

al.,

20

11

R. Rigon


Monday, July 8, 13

6

Hillslopes

R. Rigon


Monday, July 8, 13

7

Mon

tgom

ery

and

Die

tric

h, 1

98

9

Well ...

R. Rigon


Monday, July 8, 13

da

Tar

boto

n: w

ww

.cu

ahsi

.org

8

R. Rigon

Hillslopes

Monday, July 8, 13

http://www.cuahsi.org


9

da

Tar

boto

n: w

ww

.cu

ahsi

.org

R. Rigon

Hillslopes

Monday, July 8, 13



10

da

Tar

boto

n: w

ww

.cu

ahsi

.org

R. Rigon

Hillslopes

Monday, July 8, 13



11

da

Tar

boto

n: w

ww

.cu

ahsi

.org

R. Rigon

Hillslopes

Monday, July 8, 13



12

da

Tar

boto

n: w

ww

.cu

ahsi

.org

R. Rigon

Hillslopes

Monday, July 8, 13



13

Dolomites- Duron Valley

R. Rigon

Hillslopes

Monday, July 8, 13

14

Soil depth

Soil

rocks

Where does water flow ?

R. Rigon

Soil

Monday, July 8, 13

15

Keep in mind the complexity

Courtesy of Enzo Farabegoli - Duron catchment

R. Rigon

The complexity of geology (and of gelogists)

Monday, July 8, 13

Hillslope HydrologyHydrology

Mir

ò -

Blu

e II

Monday, July 8, 13

17

How water moves in hillslopes ?

Turbulent flows - Laminar flows

Both are described by the Navier-Stokes equations

R. Rigon

Fundamentals

Monday, July 8, 13

18

2D - de Saint Venant equations with some smart subgrid parameterization (e.g. Casulli, 2009)

1D - Kinematic equationSo many to cite here but ... Liu and Todini, 2002

R. Rigon

Less is more

Navier-Stokes equations are actually never used to do hillslope hydrology

For a synthesis see: abouthydrology.blogspot.com


http://www.abouthydrology

http://www.abouthydrology

19

How water moves in hillslopes ?

Turbulent flows - Laminar flows

Darcy flows

R. Rigon

Fundamentals

Monday, July 8, 13

20

Darcy equations are OKfor saturated flow

They can be obtained from Navier-Stokes Equation by*:

•introducing a resistance term

•assuming creep flow (neglecting kinetic terms)

•integrating over the Darcy scale

*Whitaker, 1966; Bear, 1988; Narsilio et al., 2009

R. Rigon

Fundamentals

Monday, July 8, 13

21

What aboutunsaturated flow

R. Rigon

Fundamentals

Monday, July 8, 13

22

One idea isthat we can use Richards’ equation

So, on the earth what is Richards’ equation ?

R. Rigon

Fundamentals

Monday, July 8, 13

23

Richards’ equation core

is that what it is true is this

Mass conservation (no nuclear reactions) !but actually true if the continuum (a.k.a. Darcy) hypothesis is valid

Process based models

R. RigonR. RigonMonday, July 8, 13

Not necessarily this:

24

Se = [1 + (��⇥)m)]�n

Se :=�w � �r

⇥s � �r

C(⇥)⇤⇥

⇤t= ⇥ ·

�K(�w) �⇥ (z + ⇥)

⇥

K(�w) = Ks

⇧Se

⇤�1� (1� Se)1/m

⇥m⌅2

SWRC + Darcy-Buckingham

(1907)

ParametricMualem (1976)

Parametricvan Genuchten

(1981)

C(⇥) :=⇤�w()⇤⇥



25

To obtain the last slide

One has to assume the validity of the Darcy-Buckingham law:

Darcy-Buckingham Law

Volumetric flow through the surface of the infinitesimal volume

Bu

ckin

gh

am, 1

90

7, R

ich

ard

s, 1

93

1

~Jv = K(✓w)~r h

Fundamentals

Monday, July 8, 13

25

To obtain the last slide

One has to assume the validity of the Darcy-Buckingham law:

Darcy-Buckingham Law

Volumetric flow through the surface of the infinitesimal volume

Hydraulic conductivity times gradient of the hydraulic headB

uck

ingh

am, 1

90

7, R

ich

ard

s, 1

93

1

~Jv = K(✓w)~r h

Fundamentals

Monday, July 8, 13

26

Ignore soil hysteresisand think of the SWRC as a function that relates water content to matric

pressure

⇤�(⇥)⇤t

=⇤�(⇥)⇤⇥

⇤⇥

⇤t� C(⇥)

⇤⇥

⇤t

Hydraulic capacity of the soil

R. Rigon

Fundamentals

Monday, July 8, 13

27

Assume a parametric form of soil water retention curves

Se :=�w � �r

⇥s � �r

Parametricvan Genuchten

(1981)

C(⇥) :=⇤�w()⇤⇥

Se = [1 + (��⇥)m)]�n

But other forms are possible ...

R. Rigon

Fundamentals

Monday, July 8, 13

28

A theory for getting hydraulic conductivity from soil water retention curves

K(�w) = Ks

⇧Se

⇤�1� (1� Se)1/m

⇥m⌅2Parametric

Mualem (1976)

But other forms are possible also here...

R. Rigon

Fundamentals

Monday, July 8, 13

29

The last representation of mass conservationis just matter of convenience

habits, and ignorance of some phenomena

•variable and changing temperature

•soil freezing

•transition to saturation

•preferential flow


R. RigonR. RigonMonday, July 8, 13

An example of top down derivationfrom Richards’ equation

Ch

imp

anzee

Con

go p

ain

tin

g

Monday, July 8, 13

Iver

son

, 20

00

; Cord

ano e

Rig

on

, 20

08

31

The Richards equation on a plane hillslope

Richardsoniana


Iver

son

, 20

00

; Cord

ano e

Rig

on

, 20

08

32

The Richards equation made dimensionless

Richardsoniana


Iver

son

, 20

00

; Cord

ano e

Rig

on

, 20

08

33

Richards eq. solution expressed in terms of

the asymptotic hydrostatic solution and a transient term:

See also. D’Odorico et al., 2003

Richardsoniana


34

Asymptotic solution

Transient solution

A lot of tricks here !

R. Rigon

Richardsoniana

Monday, July 8, 13

35

Depth from surface

Terrain Slope

Water table position

A lot of tricks here !

R. Rigon

Richardsoniana

Monday, July 8, 13

and one equation forIver

son

, 20

00

; Cord

ano e

Rig

on

, 20

08

36

So Richards equation is

divided into one equation for

Richardsoniana


37

Interestingly

Water table was not present in the original Richards equation

Hydrostatic hypothesis

R. Rigon

Richardsoniana

Monday, July 8, 13

38

In detail:initial condition

R. Rigon

Richardsoniana

Monday, July 8, 13

39

In detail:transient situation

R. Rigon

Richardsoniana

Monday, July 8, 13

40

In detail:final situation

R. Rigon

Richardsoniana

Monday, July 8, 13

41

In turn

“Short term

solution” Taylor’s

expansion

Water table

equation Taylor’s

expansion

Slope normal flow

time scale Lateral flow

time scaleSee also. D’Odorico et al., 2003

Richardsoniana


42

Pay attention to this

Slope normal flow

time scale Lateral flow

time scale

Richardsoniana

R. Rigon

Hydraulic diffusivityD(�) :=K(�)C(�)

Monday, July 8, 13

43

Detailsthat can be found in Cordano and Rigon, 2008

in words

•Take the dimensionless Richards equation

•Substitute in it the solution structure (asymptotic plus fast part)

•Here you obtain two coupled equations

•Further expand the solution structure in Taylor series

•Consider the terms which have the same expansion exponent in

•Solve each equation

R. Rigon

Richardsoniana - Iversoniana

Monday, July 8, 13

44

Neglecting those detailsthat can be found in Cordano and Rigon, 2008

Zeroth perturbation order

R. Rigon


Monday, July 8, 13

45



R. Rigon

1D-Richards equation A source term

(exchange with water table)


Monday, July 8, 13

46


Water Table equation

R. Rigon


Monday, July 8, 13

47



First perturbation order

+ analogous for d*

R. Rigon


Monday, July 8, 13

48

Integrating zeroth order solution in the column

Making a long story short

R. Rigon


Monday, July 8, 13

49


R. Rigon


Monday, July 8, 13

50


Making a long story short

Topkapi* model Liu and Todini, 2002

R. Rigon

*With some interpretation


Monday, July 8, 13

51

Integrating first order solution slope-parallel

Making a long story short - II

Boussinesq equation(e.g. Cordano and Rigon, 2013)

R. Rigon

: dimensionless transmissivities

: drainable porosity

Richardsoniana - Iversoniana - and beyond

Monday, July 8, 13

52

Making a long story short - III

R. Rigon

Figure represents a map of a small catchment, river network and a hillslope (hollow type, in gray). The distance of any point (P in the figure) in the hillslope to the channel head (C in the figure) is evaluated along the path drawn following the steepest descent (the dashed line). The characteristic length of the hillslope L (the length of x axis in Figure) is the mean of hillslope to channel distance for any point in the hillslope. The x axis used in the paper is downward parallel to mean topographic gradient, the axis y is normal to x (parallel to contour lines in a planar hillslope) and the z axis orthogonal to the x and y axes downward.

Integrating again over the lateral dimensionfrom Boussinesq


Monday, July 8, 13

53

Integrating Boussinesq

Making a long story short - III

HsBTroch et al. 2003

R. Rigon

: is the so called width function


Monday, July 8, 13

54

Hillslopes Width function

R. Rigon


Monday, July 8, 13

55

Simplifying HsB assuming stationarity of fluxesand neglecting diffusive terms

Making a long story short - IV and V

TopogO’Loughlin, 1986

R. Rigon


Monday, July 8, 13

56

Simplifying HsB assuming stationarity of fluxesand neglecting diffusive terms

Making a long story short - IV and V

assuming an exponential decay of vertical hydraulic conductivity

TopmodelBeven and Kirkby, 1979

R. Rigon


Monday, July 8, 13

57

Take home message

We can use Richards equation at various degree of simplification:

•1D (if we think that just slope-normal infiltration counts

•1D + 2D Boussinesq (Beq) if we want to account for lateral flow

* On this I will come back later

R. Rigon


Monday, July 8, 13

58

Take home message

We can use various simplification of either 1D and 2D Beq together:

• 1D Complete + 2D asymptotic- stationary• 1D linearized + 2D asymptotic- stationary• 1D bulk* + 2D asymptotic- stationary

• 1D Complete + 2D full Beq• 1D linearized + 2D full Beq• 1D bulk* + 2D full Beq


R. Rigon

Richardsoniana - Iversoniana - and beyond beyond

Monday, July 8, 13

59

Take home message

We can also try a kinematic approximation of the Boussinesq equation, and

therefore:

• 1D Complete + 2D Kinematic• 1D linearized + 2D Kinematic• 1D bulk* + 2D Kinematic


R. Rigon

Richardsoniana - Iversoniana - and beyond beyond

Monday, July 8, 13

1D linear + 2D asymptotic a.k.a D’Odorico et al., 2005

Mir

ò. T

he-

nig

hti

ngal

e-s-

son

g-a

t-m

idn

igh

t-an

d-t

he-

morn

ing-r

ain

Monday, July 8, 13

C(⇥)⇤⇥

⇤t=

⇤

⇤z

⇤Kz

�⇤⇥

⇤z� cos �

⇥⌅+ 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.

61

The Richards Equation!

R. Rigon

Linearize it !

Monday, July 8, 13

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 linearised 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

62

Th

e R

ich

ard

s Eq

uat

ion

1

-D

R. Rigon

Linearize it !

Monday, July 8, 13

⇥ ⇥ (z � d cos �)(q/Kz) + ⇥s

Iver

son

, 20

00

; D’O

dori

co e

t al

., 2

00

3,

Cord

ano a

nd

Rig

on

, 20

08

63

s

The Richards equation on a plane hillslope

R. Rigon

Linearize it !

Monday, July 8, 13

Assuming K ~ constant and neglecting the source terms

⇤⇥

⇤t= D0 cos2 �

⇤2⇥

⇤t2

64

The Richards Equation 1-D

C( )@

@t= Kz 0

@2

@z2

D0 :=Kz 0

C( )

D’O

dori

co e

t al

., 2

00

3

R. Rigon

Linearize it !

Monday, July 8, 13

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.

65


D’O

dori

co e

t al

., 2

00

3

R. Rigon

Linearize it !

Monday, July 8, 13

66


D’O

dori

co e

t al

., 2

00

3

R. Rigon

Linearize it !

Monday, July 8, 13

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

written:

⇥0 = (z � d) cos2 �

D’O

dori

co e

t al

., 2

00

3

67

= 0 + s

s =

8<

:

qKz

[R(t/TD)] 0 t T

qKz

[R(t/TD)�R(t/TD � T/TD)] t > T


R. Rigon

Linearize it !

Monday, July 8, 13

In this case the equation admits an analytical solution

D’O

dori

co e

t al

., 2

00

3

68

R(t/TD) :=⇤

t/(� TD)e�TD/t � erfc�⇤

TD/t⇥

s =

8<

:

qKz

[R(t/TD)] 0 t T

qKz

[R(t/TD)�R(t/TD � T/TD)] t > T

TD :=z2

D0


R. Rigon

Linearize it !

Monday, July 8, 13

D’O

dori

co e

t al

., 2

00

3

69

TD

TD

TD

TD

Th

e R

ich

ard

s Eq

uat

ion

1

-D

R. Rigon

Linearize it !

Monday, July 8, 13

70

Second message

Why using other simplifying assumptions (like Horton’s or Green-Ampt), if we have this ?

R. Rigon

Forget them!

Monday, July 8, 13

Just kidding!

Monday, July 8, 13

72

Did you care about hypotheses ?

Is it for any occasion realistic ? Look at the following sandy-loam:

Hypotheses counts


constant diffusivity

73

The Decomposition of the Richards equation

is possible under the assumption that:

Time scale of infiltration

soil depth

time scale of lateral flow

hillslope length

reference conductivity

reference hydraulic capacity

Iver

son

, 20

00

; C

ord

ano a

nd

Rig

on

, 2008

Hypotheses counts


Assuming hydrostatic conditions

74

Initial condition is then:

Consequently, at surface

Hypotheses counts


75

For the sandy-loam soilassuming the water table at one meter depth

we have a vertical variation of hydraulic conductivity of one order of magnitude !

Hypotheses counts


76

D which characterizes the time scales of flow is varying with depth

Hypotheses counts

R. Rigon

So a D0 reference cannot be significant

Monday, July 8, 13

77

Therefore

at surface

so, lateral flow at the water table level has the same time scale vertical flow at the surface (at least if we believe to Richards’ equation)

Hypotheses counts


78

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 ValuePhysical Bulk density ⇥b (g/cm3) 2.0

% sand - - 60% silt - - 40

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

Saturated water content �sat (cm3/cm�3) 0.39Residual water content �r (cm3/cm�3) 0.155

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

Geotechnical Effective angle of shearing resistance ⇤0 � 38Effective cohesion c0 kN/m2 0

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

The OpenBook hillslope in a 3D simulation

Comparing with 3D

R. Rigon

Lan

ni

and

Rig

on

, un

pu

bli

shed

Monday, July 8, 13

79


(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

Comparing with 3D

R. Rigon

Lan

ni

and

Rig

on

, un

pu

bli

shed

Monday, July 8, 13

80

At the beginning the pressure is constant

along the whole transect (except for

phenomena at the divide’s edge

Comparing with 3D


81

After a certain amount of time (25h in this

simulation) pressures along the slope

differentiate. With a little of analysis we

c a n d i s t i n g u i s h t w o r e g i o n s o f

differentiation. One controlled by the

boundary conditions at the bottom.

The second generated by lateral water

flow accumulation.

Comparing with 3D


82

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


Hidraulic conductivity is varying by three order of magnitude

at the bedrock interface.

The key to understand this phenomenology

Lan

ni

et a

l., 2012

Comparing with 3D


83


(a) (b)

(c) (d)

Figure 7: Transient pore pressure profiles related to points No. 300, 450 and 600 for the DRY -Low (a), DRY -High (b) and WET �High

(c) cases, and soil hydraulic conductivity function inferred using the Mualem model (d). The vertical lines at 0 pressure head indicate the

position of the water table at different timing



(a) (b)

(c) (d)

Figure 7: Transient pore pressure profiles related to points No. 300, 450 and 600 for the DRY -Low (a), DRY -High (b) and WET �High

(c) cases, and soil hydraulic conductivity function inferred using the Mualem model (d). The vertical lines at 0 pressure head indicate the

position of the water table at different timing


Another view

R. Rigon

Comparing with 3D

Monday, July 8, 13

84

When simulating is understanding

cou

rtes

y of

E. C

ord

ano

T’L can be very small indeed .....

Interpretations


85

Understanding from simulations

At the beginning of the infiltration process the situation in surface is

marked by the blue line, the situation at the bedrock is marked by the

red line

cou

rtes

y of

E. C

ord

ano

R. Rigon

Interpretations

Monday, July 8, 13

86

When lateral flow start we are in the following situation

cou

rtes

y of

E. C

ord

ano

Understanding from simulations

R. Rigon

Interpretations

Monday, July 8, 13

87

At the beginning

The condition of the perturbative derivation are verified

cou

rtes

y of

E. C

ord

ano

R. Rigon

Interpretations

Monday, July 8, 13

88

At the end

cou

rtes

y of

E. C

ord

ano

Conditions for lateral flow are dominating. Actually the same

phenomenology deducted by the perturbation theory! But obtained for a

different reason.

R. Rigon

Interpretations

Monday, July 8, 13

89

Take home message:

Never fully believe on the magic of simplifications

Detailed physics in models can help

R. Rigon

Magic ad Mermeids do not exist (Sponge Bob)

Monday, July 8, 13

90

Lateral Flow

•Can be fast, ... very fast, much faster than what happens in vadose

conditions

•In fact, to have the effects just described, we have to believe to the form

that Soil Water retention Curves have.

•Other soils behave differently

•If macropores or cracks are present, vertical infiltration can still remain

faster

R. Rigon

Interpretations

Monday, July 8, 13

91

Inappropriate numerics (or gridding)

Can hide it!

R. Rigon

Interpretations

Monday, July 8, 13

Further investigationsM

ach

ael

Leon

g -

Cu

ttin

g t

he

tim

e w

ith

a k

nif

e, 2

01

2

Monday, July 8, 13

93

CAPITOLO 5. IL BACINO DI PANOLA

Figura 5.2: Rappresentazione della profondita del suolo del pendio di Panola.

costante su un campione prelevato a 10 cm di profondita, risulta pari a 64 [cm/h]; per cio che concerneil valore della conducibilita idraulica a saturazione del bedrock, non esistono misure dirette e↵ettuatesu campioni prelevati in sito; tuttavia si stima che il suo valore sia 2-3 ordini di grandezza inferiorerispetto a quella del terreno soprastante. Entrambi i valori di conducibilita idraulica satura (del bedrocke del terreno) saranno comunque oggetto di calibrazione numerica all’atto delle simulazioni svolte conGEOtop, utilizzando come valori di partenza quelli qui citati.Va infine evidenziata la presenza di cinque macropori (si veda la fig. 5.2 per la loro posizione), didiametro compreso tra i 10 ed i 60 mm, che a�orano sulla trincea e che contribuiscono attivamentealla generazione del deflusso.

5.2 Analisi delle precipitazioni e del deflusso subsuperficiale

Il versante di Panola e tipicamente ben drenato e rimane insaturo per la maggior parte dell’anno.Durante il pediodo di studio (gennaio - fine maggio 2002), il contenuto d’acqua medio del terrenomostra un andamento fortemente stagionale, passando da valori relativi prossimi all’80% del periodoinvernale, a valori di circa il 40% ad inizio estate, calando poi ulteriormente nei mesi piu caldi: in fig.(5.3) si riporta l’andamento appena descritto.Il deflusso sub-superficiale misurato presso la trincea e intermittente e si verifica solo in risposta aprecipitazioni intense; uno studio approfondito della relazione tra entita delle precipitazioni e deflussosub-superficiale e stato condotto da Tromp-van Meerveld e McDonnell, (2006a) [24], e nel seguito se neriportano i risultati, necessari a comprendere la dinamica idrologica del pendio.Nello studio condotto da Tromp-van Meerveld e McDonnell sono stati esaminati 2 anni di dati dideflusso sub-superficiale, dal 19 febbraio 1996 al 10 maggio 1998, periodo nel quale si sono verificati147 temporali. L’analisi correlata delle precipitazioni e del deflusso indica che il 22% degli eventi

79

Panola’s hillslope

R. Rigon

Richards equation is still valid here ?

Monday, July 8, 13

94

Terrain surface Bedrock surface Soil depth varies

Dep

ress

ion

Soil (sandy loam) Bedrock

Ksat = 10-4 m/s Ksat = 10-7 m/s

Panola’s hillslope

R. Rigon


Monday, July 8, 13

95

Q (m

3 /h)

t=9h

t=18h

t=22h

With a rainfall of 6.5 mm/h and a duration of 9 hoursLa

nn

i et

al., 2

01

1

R. Rigon


Monday, July 8, 13

96

t=6h t=9ht=7h t=14h

Lan

ni

et a

l., 2

01

1

With a rainfall of 6.5 mm/h and a duration of 9 hours

Tromp Van Meerveld et al., 2006 call it filling and spilling

R. Rigon


Monday, July 8, 13

97

Q (m

3 /h)

t=9h

t=18h

t=22h

Lan

ni

et a

l., 2

01

1

With a rainfall of 6.5 mm/h and a duration of 9 hours

R. Rigon


Monday, July 8, 13

98

1D

3D

No role played by hillslope gradient

First Slope Normal infiltration works

Then Lateral flow start

Infiltration front propagate

Drainage is controlled by the bedrock form

As in the open book caseLa

nn

i et

al., 2

01

1

R. Rigon


Monday, July 8, 13

99

Now we want a model that can run 100 times faster

In which we obviously use all the machinery of the Richards’ equation, i.e. hydraulic conductivity and soil

water retention curves

R. Rigon


Monday, July 8, 13

100

Introducing the concept of concentration timein subsurface flow

we have the distances from the channels

R. Rigon

Variations

Monday, July 8, 13

101

If we assume that water just move laterally in saturated conditions, we can use Darcy law for getting the

velocities

possibly in its more traditional form:

R. Rigon

Variations

Monday, July 8, 13

102

If we assume that water just move laterally in saturated conditions, we can use Darcy law for getting the

velocities

And assuming Dupuit approximation, i.e. hydrostatic distribution of pressures

R. Rigon

Variations

Monday, July 8, 13

103

Then:

Time = Lengths/velocity

And, for any point:

is the max residence time*

R. Rigon

Variations

*The operator means that we are looking for the maximum of T choosing it from all the possible path that we can define upstream of the point i

Monday, July 8, 13

104

The largest time

is the concentration time

Up to concentration time

The area contributing to the discharge is not the TOTAL upslope area

R. Rigon

Variations

Monday, July 8, 13

105

The area contributing to the discharge is not the TOTAL upslope area

Lan

ni

et a

l., 2

01

2a

R. Rigon

!(Steady state)

Monday, July 8, 13

106

Actually there is a second issue

Water table cannot “exist” everywhere

C. Lanni et al.: Modelling shallow landslide susceptibility 3961

Fig. 1. A flow chart depicting the coupled saturated/unsaturated hydrological model developed in this study.

35

1

2

3 4

Figure 2. The concept of hydrological connectivity. Lateral subsurface flow occurs at point 5

(x,y) when this becomes hydrologically connected with its own upslope contributing area 6

A(x,y). 7

8

Fig. 2. The concept of hydrological connectivity. Lateral subsurfaceflow occurs at point (x,y) when this becomes hydrologically con-nected with its own upslope contributing area A(x,y).

storage of soil moisture needed to produce a perched watertable (i.e. zero-pressure head) at the soil–bedrock interface(Fig. 3); and I [LT�1] is the rainfall intensity assumed to beuniform in space and time. Computation of V0 and Vwt re-quire the use of a relationship between soil moisture content✓ [�] and suction head [L], and a relationship between and the vertical coordinate (positive upward) z [L] (Fig. 3).By using the assumption that the suction head profile (z)

changes from one steady-state situation to another over time,

36

1

2

3 4

Figure 3. i(z) and i(z) are, respectively, the initial water content and the initial suction 5

head vertical profiles. wt(z) and wt(z) represents the linear water content and suction 6

head vertical profiles associated with zero-suction head at the soil-bedrock interface. 7

8

Fig. 3. ✓i(z) and i(z) are, respectively, the initial water contentand the initial suction head vertical profiles. ✓wt(z) and wt(z) rep-resents the linear water content and suction head vertical profilesassociated with zero-suction head at the soil–bedrock interface.

the relation between [L] and z [L] is that of hydraulicequilibrium:

= (z = 0) + z = b + z, (2)

where b = (z = 0) is the suction head at the soil–bedrockinterface. Bierkens (1998) argued that this assumption isvalid for a shallow system where redistribution of soil-water

www.hydrol-earth-syst-sci.net/16/3959/2012/ Hydrol. Earth Syst. Sci., 16, 3959–3971, 2012

Lan

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01

2b

R. Rigon

!(Steady state)

Monday, July 8, 13

107

Ii.e. time to water table development

Twt(x,y):= [Vwt(x,y)-V0(x,y)]/I

Initial conditions(hydrostatic slope normal)

boundary conditions(including rainfall, I)

t> Twt(x,y)

YES

NO

Lan

ni

et a

l., 2

01

2

Slope Normalunsaturated flow

A heuristic model

for each timestep

Faster is better


108

YES

t> Tmaxwt(x,y)

hydrologicallyconnectedA(x,y) >0

YES

NO hydrologicallydisconnected

A(x,y) =0

A heuristic modelLa

nn

i et

al., 2

01

2

R. Rigon

Faster is better

Monday, July 8, 13

109

YES

update soil pressure

start lateral flow update soil pressure

next timestep

A heuristic modelLa

nn

i et

al., 2

01

2

R. Rigon

Faster is better

Monday, July 8, 13

110

* Is not completely true.

I question also of personal attitude:

I understand (fluid) mechanics through

equations and I try to interpret observations

through equations.

Someone else (i.e. many of my students)

simply did not have the training for that and

prefer to rebuilt the physics of the problem by

small pieces.

This has a certain appealing to many (especially

to natural scientists and geologists), and can

indeed be useful to see thing from different

perspectives.

Dood

ley,

Mu

ttle

y, a

nd

th

eir

flyi

ng m

ach

ines

R. Rigon

Attitudes

Monday, July 8, 13

111

3968 C. Lanni et al.: Modelling shallow landslide susceptibility

40

1

2

3

Figure 7. Patterns of Return period TR (years) of the critical rainfalls for shallow landslide 4

triggering (i.e., FS≤1) and associated levels of landslide susceptibility obtained by means 5

of QDSLaM. 6

7

Fig. 7. Patterns of return period TR (years) of the critical rainfalls for shallow landslide triggering (i.e. FS 1) and associated levels oflandslide susceptibility obtained by means of QDSLaM.

Table 3. Percentages of catchment area (C) and observed landslide area (L) in each range of critical rainfall frequency (i.e. return period TR)for QDSLaM.

Susceptibility Pizzano Fraviano Cortina

TR level C

aL

bC

aL

bC

aL

b

Years Category % % % % % %

Uncond Unstable 9.9 60.2 7.7 77.7 8.5 56.80–10 Very high 20.3 26.9 16.1 18.5 13.5 39.210–30 High 7.8 0.0 5.6 1.5 5.8 4.030–100 Medium 6.0 9.7 5.9 2.3 6.7 0.0> 100 Low 42.9 3.2 53.5 0.0 54.7 0.0Uncond. stable Very low 13.1 0.0 11.2 0.0 10.8 0.0

aC = catchment area; b L= landslide area

high slope failure hazard) of CI-SLAMwith that of the quasi-dynamic model QDSLaM (Borga et al., 2002b).QDSLaM is based on coupling a hydrological model to

a limit-equilibrium slope stability model to calculate thecritical rainfall necessary to trigger slope instability at anypoint in the landscape. The hydrological model assumesthat flow infiltrates to a lower conductivity layer and fol-lows topographically-determined flow paths to map the spa-tial pattern of soil saturation based on analysis of a “quasi-dynamic” wetness index. With respect to the model pro-posed in this paper, QDSLaM does not consider the follow-ing aspects: (i) vertical rainwater infiltration into unsaturatedsoil; (ii) analysis of the connectivity to compute the quasi-dynamic wetness index; and (iii) soil depth variability. All re-maining aspects of the modelling framework are described ina consistent way by the two models. Both models have beenapplied to the three catchments by using the same parameters

set. A map of shallow landsliding susceptibility obtained byusing QDSLaM is reported in Fig. 7, whereas Table 3 reportsthe corresponding percentages of slope-stability categories interms of catchment area and observed landslide area in eachrange of critical rainfall frequency (i.e. return period TR). Ex-amination of Fig. 7 and of Table 3 shows the considerableimpact of the areas considered unconditionally unstable inQDSLaM. The percentage of topographic elements consid-ered unconditionally unstable ranges from 7.7% (Fraviano)to 9.9% (Pizzano). This overrepresentation is reported bothin the lower and in the upper portions of the catchments,where high local slope values are present. CI-SLAM doesnot predict unconditionally unstable locations. In fact, thecontribution of negative pressure head (Eq. 15a) ensures thestability of steeper topographic elements (i.e. locations withtan � � tan '

0 for cohesionless soils) that would be otherwiseclassified as unconditionally unstable by QDSLaM (as well

Hydrol. Earth Syst. Sci., 16, 3959–3971, 2012 www.hydrol-earth-syst-sci.net/16/3959/2012/

Lan

ni

et a

l., 2

01

2

However, it works

R. Rigon

Faster is better if it works (Klemes fogive me!)

Monday, July 8, 13

Further investigations -IIM

ach

ael

Leon

g -

Cu

ttin

g t

he

tim

e w

ith

a k

nif

e, 2

01

2

Monday, July 8, 13

113


Figura 5.4: Immagine tratta da Tromp-van Meerveld e McDonnell, (2006a) [24]; (a) deflusso sub-superficiale totale per i segmenti in cui e stata suddivisa la trincea e (b) numero di eventi meteorici cheproducono deflussi misurabili.

5.2.1 Il ruolo dei macropori

Secondo l’interpretazione di Tromp-van Meerveld e McDonnell, il flusso attraverso i cinque macroporicontribuisce in maniera significativa alla generazione del deflusso totale: rispetto al flusso totalemisurato durante i 147 eventi meteorici, il 42% deriva proprio da essi; in particolare il macroporodenominato M14 (si veda la fig. 5.2 per la sua localizzazione) e responsabile del 25% del deflussototale durante il periodo di analisi. Si e dimostrato come sussista una relazione lineare molto robusta(r2=0.96) tra il deflusso sub-superficiale totale ed il deflusso totale a carico dei macropori (fig. 5.5),il che suggerisce che vi sia un meccanismo comune che innesca entrambi i processi. Va evidenziatocomunque come vi sia una forte stagionalita nella caratterizzazione del deflusso ad opera dei macropori,che sarebbero responsabili del 50% del deflusso totale durante l’autunno e del 41% durante l’inverno,mentre in primavera ed estate contribuiscono solo per lo 0-2 %.

5.2.2 Soglia di risposta del pendio

Le analisi condotte da Tromp-van Meerveld e McDonnell suggeriscono come vi sia una soglia abbastanzanetta per innescare significativi (> 1mm) deflussi sub-superficiali nel versante di Panola: deflussisignificativi avvengono solo a seguito di eventi meteorici il cui apporto sia maggiore di 55 mm di

81

Tro

mp

Van

Mee

rvel

d e

t al

., 2

00

6

And finally macropores

R. Rigon

Macropores

Monday, July 8, 13

114

Macropore Flow InitiationWater supply to the macropores

InteractionWater transfer between macropores and the surrounding soil matrix

M. W

eile

r, f

rom

Moch

a p

roje

ct

Macropores!

R. Rigon

Macropores

Monday, July 8, 13

115


0.00

0.02

0.04

0.06

0.08

0.10

Flussi alla base del pendio - Simulazione 0

date (dd/mm) 2002po

rtate

[l/s

]

01/01 11/01 21/01 31/01 10/02 20/02 02/03 12/03 22/03 01/04 11/04 21/04 01/05 11/05 21/05

Flussi misuratiSimulazione 0

Figura 5.16: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trinceaalla base del pendio.

0.00

0.02

0.04

0.06

0.08

0.10

Simulazione 0 - evento 6 febbraio

date (dd/mm) 2002

porta

te [l

/s]

05/02 06/02 07/02 08/02 09/02 10/02 11/02 12/02


0.00

0.02

0.04

0.06

0.08

0.10

Simulazione 0 - evento 30 marzo

date (dd/mm) 2002

porta

te [l

/s]

29/03 30/03 31/03 01/04 02/04 03/04 04/04 05/04 06/04 07/04


Figura 5.17: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trinceaalla base del pendio: a sinistra si riporta l’evento del 6 febbraio 2002, a destra quello del 31 marzo.

puo essere causata da diversi fattori, quali un’errata assegnazione delle caratteristiche del suolo o delbedrock, oppure un errore nello stabilire la condizione iniziale circa la quota della falda.Un aspetto decisamente importante da considerare, tanto in questi risultati quanto in quelli presentatisuccessivamente, e che nella creazione della geometria di calcolo 3D utilizzata da GEOtop non estato possibile inserire la presenza dei macropori che si a↵acciano sulla trincea alla base del pendio.Tale mancanza puo avere un peso notevole all’atto di calcolare i volumi defluiti, in quanto, comeevidenziato alla sezione (5.2.1), nel periodo invernale essi contribuiscono a generare circa il 40% del

95

Da

Prà

, 2

01

3

Certainly the volumes of water cannot be simulated with the only Richards equation

No way!

R. Rigon

Macropores

Monday, July 8, 13

Thank you for your attention

116

G. U

lric

i -

R. Rigon

Slides on http://abouthydrology.blogspot.com

Monday, July 8, 13

http://abouthydrology.blogspot.com

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