Presentation given at the second I

R. M

agri

tte

- L

a gra

nd

e m

area

, 19

51

Riccardo Rigon e Cristiano Lanni

Using complex models and conceptualizations for modeling shallow landslides hydrology

Monday, October 10, 11

“Tutto precipita”Gianni Letta

“Everything falls apart”Gianni Letta

Panta rei os potamòs Tutto scorre come un fiume

Everything flows as in a river

Eraclito (Sulla Natura)


IWL 2 Napoli, 28-30 Settembre 2011

Rigon & Lanni

3

Outline

•Hillslope Hydrology is tricky

•But, as well as landslide triggering, should be simple in simple settings

•About some consequences of the current parameterization of Richards equation

•So, from where all the complexity of real events comes from ?



Rigon & Lanni

4

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 + (��⇥)m)]�n

Se :=�w � �r

⇥s � �r

C(⇥)⇤⇥

⇤t= ⇥ ·

�K(�w) �⇥ (z + ⇥)

⇥

K(�w) = Ks

⇧Se

⇤�1� (1� Se)1/m

⇥m⌅2

C(⇥) :=⇤�w()⇤⇥



Rigon & Lanni

4

Richards



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

Water balance

C(⇥) :=⇤�w()⇤⇥



Rigon & Lanni

4

Richards



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

Water balance

ParametricMualem

C(⇥) :=⇤�w()⇤⇥



Rigon & Lanni

4

Richards



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

Water balance

ParametricMualem

Parametricvan Genuchten

C(⇥) :=⇤�w()⇤⇥



Rigon & Lanni

5

Does exist a free available and reliable solver of Richards equation ?

12. Surface Fluxes 12.2. Values of reference

Surface description z 0(cm) ReferenceMud flats, ice 0.001 Sutton (1953)Smooth tarmac 0.002 Bradley (1968)Large water surfaces 0.01 - 0.06 Numerous referencesGrass (lawn up to 1 cm) 0.1 Sutton (1953)Grass (artificial, 7.5 cm high) 1.0 Chamberlain (1966)Grass (thick up to 10 cm high) 2.3 Sutton (1953)Grass (thin up to 50 cm) 5 Sutton (1953)Trees (10-15 m high) 40-70 Fichtl and McVehil (1970)Large city 165 Yamamoto and Shimanuki (1964)

Table 12.9: Example of roughness parameters for various surfaces (Evaporation into the Atmosphere, Wilfried Brutsaert, 1984)

��

��

� ��

��

��

� ��

! �!��!� ��

��"��

�!��

��#$�%��"�� &��%��"��

� ��'�� !��!�

�(�%��"��

��! �� '

��! � ��

! �!��!� ��

� �� '

Figure 12.1: Water fluxes

page 54 of 92



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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



Rigon & Lanni

8

Conditions of simulation

Homogeneous soil

Gentle slope Steep slope

Wet Initial Conditions

Dry Initial Conditions

Intense Rainfall

Low Rainfall

Moderate Rainfall



Rigon & Lanni

9

Initial Conditions



Rigon & Lanni

10


(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



Rigon & Lanni

11

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







Lanni and Rigon

Richards 3D for a hillslope



Rigon & Lanni

12








Simulations result

Lanni and Rigon




Rigon & Lanni

13








Simulations result

Lanni and Rigon




Rigon & Lanni

14

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




Rigon & Lanni

15

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 & Lanni

C(⇥)⇥�⇥t = ⇥

⇥z

⇤Kz

�⇥�)⇥z � cos�

⇥⌅+ ⇥

⇥y

⇤Ky

⇥�⇥y

⌅+ ⇥

⇥x

⇤Kx

�⇥�)⇥x � sin�

⇥⌅

⇥ ⇥ (z � d cos �)(q/Kz) + ⇥s

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.

16

The Richards equation on a plane hillslope

Iver

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Rigon & Lanni

C(⇥)⇥�⇥t = ⇥

⇥z

⇤Kz

�⇥�)⇥z � cos�

⇥⌅+ ⇥

⇥y

⇤Ky

⇥�⇥y

⌅+ ⇥

⇥x

⇤Kx

�⇥�)⇥x � sin�

⇥⌅

⇥ ⇥ (z � d cos �)(q/Kz) + ⇥s




16


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Rigon & Lanni

C(⇥)⇥�⇥t = ⇥

⇥z

⇤Kz

�⇥�)⇥z � cos�

⇥⌅+ ⇥

⇥y

⇤Ky

⇥�⇥y

⌅+ ⇥

⇥x

⇤Kx

�⇥�)⇥x � sin�

⇥⌅

⇥ ⇥ (z � d cos �)(q/Kz) + ⇥s




16


Iver

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Rigon & Lanni

17

The Vertical Richards Equation

Iver

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Rigon & Lanni

C(⇥)⇤⇥

⇤t=

⇤

⇤z

⇤Kz

�⇤⇥

⇤z� cos �

⇥⌅+ Sr

Vertical infiltration: acts in a

relatively fast time scale because

it propagates a signal over a

thickness of only a few metres

17


Iver

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Rigon & Lanni

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.

18




Rigon & Lanni

Sr =⇤

⇤y

⇤Ky

⇤⇥

⇤y

⌅+

⇤

⇤x

⇤Kx

�⇤⇥

⇤x� sin �

⇥⌅

19

The Lateral Richards Equation



Rigon & Lanni

Sr =⇤

⇤y

⇤Ky

⇤⇥

⇤y

⌅+

⇤

⇤x

⇤Kx

�⇤⇥

⇤x� sin �

⇥⌅

Properly treated, this is reduced to

groundwater lateral flow, specifically to the

Boussinesq equation, which, in turn, have

been integrated from SHALSTAB equations

19

The Lateral Richards Equation



Rigon & Lanni

20

The 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

reference conductivity

reference hydraulic capacity

Iver

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Rigon & Lanni

21


•But Is the condition:

verified ?

cou

rtes

y of

E. C

ord

ano



Rigon & Lanni

22


The scale factor strongly varies with time

On the basis of the only MvG scheme, it is very difficult to say at

saturation. However

cou

rtes

y of

E. C

ord

ano



Rigon & Lanni

23


At the beginning, at the bedrock we are we are on the red line, at the

surface on the blue line

cou

rtes

y of

E. C

ord

ano



Rigon & Lanni

24


At the end, at the bedrock we are we are on the red line, at the surface

on the blue line

cou

rtes

y of

E. C

ord

ano



Rigon & Lanni

25

So

What happens is that, at the beginning the conditions for considering

just the vertical flow are satisfied

cou

rtes

y of

E. C

ord

ano



Rigon & Lanni

26

So

What happens is that, at the end the conditions for considering just the

vertical flow are NOT satisfied. Because D0b >> D0 top

cou

rtes

y of

E. C

ord

ano



Rigon & Lanni

27

Therefore when a perched water table form

Instead

And lateral flow dominates (is as fast ) than infiltration



Rigon & Lanni

K(Se) = KsSve

�f(Se)f(1)

⇥2

f(Se) =� Se

0

1�(x)

dx

Where v is an exponent expressing the connectivity between pores, evaluated by Mualem

for various soil types.

Aft

er M

ual

em, 1

97

6

IS THIS TRUE ?

28

We need to go back to the basics



Rigon & Lanni

K = Ks Kr

Having defined the relative hydraulic conductivity:

⇥ =1�

�S�1/m

e � 1⇥1/n

And expressed the suction in terms of van Genuchten’s expression::

The integral can be calculated:

Aft

er M

ual

em, 1

97

6

29

IS THIS TRUE ?



Rigon & Lanni

there results:

K(Se) = KsSve

⇤1�

�1� S1/m

e

⇥m⌅2

(m = 1� 1/n)

or, by expressing everything as a function of the suction potential:

K(⇥) =Ks

�1� (�⇥)mn [1 + (�⇥)n]�m

⇥2

[1 + (�⇥)n]mv (m = 1� 1/n)

30

PARAMETRIC FORMS OF THE HYDRAULIC CONDUCTIVITY



Rigon & Lanni

31

THEREFORE

•The results are strictly related to the validity of the MvG theory and

parameterization



Rigon & Lanni

32

Another issue

Extending Richards to treat the transition saturated to unsaturated zone.

Is it :

At saturation: what does change in time ?



Rigon & Lanni

33

Another issue

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

cou

rtes

y of

M. B

erti



Rigon & Lanni

34

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

unsaturated volumes to saturated ones.

Or

where we just saw most of the phenomena of interest happens

Obviously it can be done much better. Only in very special cases the specific

storage can be expressed in the way we showed (e.g. Green and Wang, 1990).



Rigon & Lanni

35

In any case

the question relies also in the reliability of the SWRC close to saturation (e.g. Vogel et al., 2000, Schaap and vanGenuchten, 2005; Romano, 2010)

cou

rtes

y of

M. B

erti



Rigon & Lanni

36

Stability onstage

The good old infinite slope



Rigon & Lanni

37

Infinite Slope with unsaturated conditionsThe equation

e.g. Lu and Godt, 2008



Rigon & Lanni

38

It is enough to say that a point is unstable to state that a landslide

occurs ?



Rigon & Lanni

39

LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES X - 59

Table 3: A matrix of the times needed to achieve specific percentages of destabilized hillslope area for a continuous rainfall simulation for

a 5-day period.

A.C. RAIN SHAPE TF5% TF10% TF15% TF30% TF50%

DR

Y

Low

Divergent 41h � � � �Parallel 41h � � � �

Convergent 41h 60h � � �

Med

Divergent 14-15h 15-16h 17-18h � �Parallel 14-15h 15-16h 16-17h 18h �

Convergent 14-15h 14-15h 14-15h 15h �H

igh Divergent 7-8h 8-9h 9-10h 10-11h 12h

Parallel 7-8h 8h 8-9h 8-9h 8-9h

Convergent 7-8h 7-8h 7-8h 7-8h 8-9h

WE

T

Low

Divergent 3-4h � � � �Parallel 3-4h � � � �

Convergent 3-4h 4-5h � � �

Med

Divergent 2-3h 3-4h 4-5h � �Parallel 2-3h 3h 3-4h 4-5h �

Convergent 2-3h 2-3h 2-3h 2-3h �

Hig

h Divergent 1-2h 1-2h 1-2h 3h 5h

Parallel 1-2h 1-2h 1-2h 2-3h 2-3h

Convergent 1-2h 1-2h 1-2h 1-2h 1-2h

�60h - - 20 h - - - 10 h - - - 5 h - 0h not achieved


Lan

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Rigon & Lanni

40

Total volume of waterin hillslope

T o t a l v o l u m e o f r a i n f a l l w a t e r i n hillslope

Total volume of water in hillslope before the event remained inside the hillslope



Rigon & Lanni

41


Table 4: A matrix of the rain volumes RFi and total water volume VFi (Rain volume + Pre-rain soil-water volume) needed to achieve

specific percentages of hillslope area for a continuous rainfall simulation for a 5-day period.

RAIN SHAPEF5% F10% F15% F30% F50%

DRY WET DRY WET DRY WET DRY WET DRY WET

RF

i(m

3)

Low

Divergent � � � � � � � �Parallel � � � � � � � �

Convergent � � � � � �

Med

Divergent � � � �Parallel � �

Convergent � �H

igh Divergent

Parallel

Convergent

VF

i(m

3)

Low

Divergent � � � � � � � �Parallel � � � � � � � �

Convergent � � � � � �

Med

Divergent � � � �Parallel � �

Convergent � �

Hig

h Divergent

Parallel

Convergent

�15m3 - - 125m3 - - 230m3 - - 350m3 - - > 520m3 not achieved

Table 5: A matrix of the times needed to achieve specific percentages of destabilized hillslope area for a continuous rainfall simulation for

a 5-day period. The case of steep hillslopes

A.C. RAIN SHAPE FT( 5%) FT(10%) FT(15%) FT(30%) FT(50%)

DRYLow Parallel 32h 35.5h 38h 39h �High Parallel 7h 7h 7h 7h 7h

WETLow Parallel 0.25h 0.25h 0.25h 0.25h 0.25h

High Parallel 0.25h 0.25h 0.25h 0.25h 0.25h

�60h - - - - - - - - - - - > 0h not achieved


Lan

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Rigon & Lanni

42

So simple, too simple ?

• (The evident and little informative statement) We found that wet volumes causes faster obtaining of instability

•However, the it seems that in simple settings the total volume

of water required to destabilized a certain percentage of area is

not very much variable (variation is included in 10%)



Rigon & Lanni

43

Ground surfaceBedrock surface

Soil-depth variability

Bedrock depression

Panola and the soil depth question



Rigon & Lanni

44

α = 13° α = 20° α = 30°

Soil (sandy-silt) Ksat = 10-4 m/s

Bedrock Ksat = 10-7 m/s

Rain Intensity = 6.5 mm/h Duration = 9 hours

Slope

Soil properties



Rigon & Lanni

45

Q (m

3 /h)

t=9h

t=18h

t=22h

Hillslope water dischargeo 2 peaks α = 13°

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

Lan

ni

et a

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01

1



Rigon & Lanni

46

time

t=6h

t=9h

t=7h

Saturated area at the soil-bedrock interface increases very rapidly…..

α = 13°



Rigon & Lanni

47

1D

3D

No role played by hillslope gradient

1° STEP:

Vertical rain-infiltration

2° STEP: Lateral-flow

Infiltration-front propagation

Downslope drainagelimited by bedrock topography

Same as in the ideal planar case

Lan

ni

et a

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01

1



Rigon & Lanni

48Downslope Drainage efficiency

Pressure growing

α = 13° α = 20° α = 30°

Lan

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1



Rigon & Lanni

49

time

t=6h

t=9h

t=7h

α = 13°

…..and then the average value of positive pore-water pressure continues to grow

Pressure growing

Lanni et al., 2011



Rigon & Lanni

50

At the time of the simulations

We were not looking at this but, please observe that, increasing slope

decreases instability but drainage is more efficient.

Therefore there should be a specific slope angle which is, given the

condition of the simulation the more unstable.



Rigon & Lanni

51

(FS=1)

(1<FS<1.05)

t=10h

α = 30°

c’ = 0 kPaφ’ = 30°

If you tilt you slide



Rigon & Lanni

52

In complex topographyof the bedrock

•Topography commands the patterns of instability and convergence of

fluxes can increase instability (so obvious again!)

•The temporal dynamics of instabilities is also affected due to the

filling and spilling effect, and different parts of the hillslope can

become unstable at different times

•However, there is an interplay between slope and bumpiness of the bedrock

which is not trivial at all.

•The mechanism where infiltration comes first and lateral flow later continues

to be valid



Rigon & Lanni

53

Lessons Learned

• Simple stability analysis can be successful. Probably not for the right

reasons

• Simple settings give simple results (the total weight of water commands

the creation of large instabilities)

•This is due in the model to the compound of the vanGenuchten and

Mualem theory (which could not be real)

•Soil depths counts

•On small scales instabilities could be controlled by constraints of local

topography

•Boundary conditions matter (trivial kinematic approaches could not work)



Rigon & Lanni

54

Another case and its complexity: Duron



Rigon & Lanni

55

Duron stratigraphyFa

rab

egoli

et

al.,

20

11



Rigon & Lanni

56

Duron soil depthFa

rab

egoli

et

al.,

20

11



Rigon & Lanni

57

Duron geomorphologyFa

rab

egoli

et

al.,

20

11



Rigon & Lanni

58

Duron soil coverFa

rab

egoli

et

al.,

20

11



Rigon & Lanni

59

Duron land useFa

rab

egoli

et

al.,

20

11



Rigon & Lanni

60

And a tentative association of those maps withhydrological characters

Wit

h D

all’A

mic

o ,F

arab

egoli

et

al.,

20

11



Rigon & Lanni

61

Forecasting of temperaturein a point

In time

Wit

h D

all’A

mic

o ,F

arab

egoli

et

al.,

20

11



Rigon & Lanni

62

Soil water content at different depthin a point

Wit

h D

all’A

mic

o ,F

arab

egoli

et

al.,

20

11



Rigon & Lanni

63

Pro

bab

ilit

y of

lan

dsl

idin

gSi

mon

i et

al, 2

00

8



Rigon & Lanni

64

Duron

Pro

bab

ilit

y of

lan

dsl

idin

gSi

mon

i et

al, 2

00

8



Rigon & Lanni

65

Duron

Pro

bab

ilit

y of

lan

dsl

idin

gSi

mon

i et

al, 2

00

8



Rigon & Lanni

66

Duron

Pro

bab

ilit

y of

lan

dsl

idin

gSi

mon

i et

al, 2

00

8



Rigon & Lanni

67

Duron

Pro

bab

ilit

y of

lan

dsl

idin

gSi

mon

i et

al, 2

00

8



Rigon & Lanni

68

And the snow again !

Duron



Rigon & Lanni

69

Duron

Temperature of snow !



Rigon & Lanni

70

Lessons Learned

• Cows count ;-)

•Landslide forecasting is complex for dynamical reasons

•But also because it is a local phenomena where a lot of “accidents” (i.e.

land-use-landcover) modify the local hydrology and the “cohesion of soils”

•There is a missing link between all of those characteristics and

hydrological, and geotechnical parameters

•Cohesion exists but its estimation is kind of elusive when we are talking

about turfs and root strength



Rigon & Lanni

71

Credits

We are indebted to Emanuele Cordano for the participation to some early

stage of this research, and providing at late request, some plots of

hydraulic diffusivity.

We thank Enzo Farabegoli, Giuseppe Onorevoli and Martina Morandi for

allowing to use the maps of Duron catchment which resulted after three

years of detailed surveys.



Rigon & Lanni

Thank you for your attention.

G.U

lric

i -

20

00

?

72


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