Presentation given at the second I

Using complex models and conceptualizations
for modeling shallow landslides hydrology

R. Magritte - La grande marea, 1951

Riccardo Rigon e Cristiano Lanni

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

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 ?

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Richards

First, I would say, it means that it would be better to call it, for
instance: Richards-Mualem-vanGenuchten equation, since it is:

⇤⇥ ⇥
C(⇥) = ⇥ · K( w ) ⇥ (z + ⇥)
⇤t
n
Se = [1 + ( ⇥) )] m

⇧ ⇤ ⇥ m ⌅2
w) = Ks 1 (1 Se ) 1/m
K( Se

⇤ w () w r
C(⇥) := Se :=
⇤⇥ ⇥s r
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Richards


⇤⇥ ⇥
C(⇥) = ⇥ · K( w ) ⇥ (z + ⇥) Water balance
⇤t
n
Se = [1 + ( ⇥) )] m

⇧ ⇤ ⇥ m ⌅2
w) = Ks 1 (1 Se ) 1/m
K( Se

⇤ w () w r
C(⇥) := Se :=
⇤⇥ ⇥s r
4

Rigon & Lanni


Richards


⇤⇥ ⇥
⇤t
n
Se = [1 + ( ⇥) )] m

⇧ ⇤ ⇥ m ⌅2 Parametric
w) = Ks 1 (1 Se ) 1/m
K( Se Mualem

⇤ w () w r
C(⇥) := Se :=
⇤⇥ ⇥s r
4

Rigon & Lanni


Richards


⇤⇥ ⇥
⇤t
n
Se = [1 + ( ⇥) )] m Parametric
van Genuchten
⇧ ⇤ ⇥ m ⌅2 Parametric
w) = Ks 1 (1 Se ) 1/m
K( Se Mualem

⇤ w () w r
C(⇥) := Se :=
⇤⇥ ⇥s r
4

Rigon & Lanni

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

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

5

Rigon & Lanni Figure 12.1: Water ﬂuxes



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igure 2: Experimental set-up.The OpenBook schematization. (b) The initial suction head pr
(a) The inﬁnite hillslope hillslope

il-pixel hillslope numeration system (the case of parallel shape is shown here). Moving from 0 to 900
7

sponds to moving from the crest to the toe of the hillslope
Rigon & Lanni


Conditions of simulation

Homogeneous soil

Gentle slope Steep slope

Wet Initial Conditions Intense Rainfall

Moderate Rainfall
Dry Initial Conditions
Low Rainfall

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Initial Conditions

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- 54
Simulations result
LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES

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

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Rigon & Lanni

Richards 3D for a hillslope

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


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Lanni and Rigon
Rigon & Lanni


Simulations result

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

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Lanni and Rigon
Rigon & Lanni


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

(e) WET-Med (f) WET-High
F T September 24, 2010, 9:13am D 13 A
R
Values of pressure head developed at the soil-bedrock interface at each point of the subcritical parallel hillslope. The
Lanni and Rigon
Rigon & Lanni
e Monday, October 10, 11
head lines represents the mean lateral gradient of pressure


The key for understanding

Three order of magnitude faster !
(a) (b)
14

Lanni andTemporal evolution of the vertical proﬁle of hydraulic conductivity (a) and hydraulic conductivity at the soil-bedrock interface
Rigon & Lanni
Figure 6: Rigon


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

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The Richards equation on a plane hillslope

⇤ ⇥⌅ ⇤ ⌅ ⇤ ⇥⌅
⇥ ) ⇥ )
C(⇥) ⇥ = ⇥
Kz cos + ⇥
Ky ⇥ + ⇥
Kx sin
Iverson, 2000; Cordano and Rigon, 2008

⇥t ⇥z ⇥z ⇥y ⇥y ⇥x ⇥x

⇥ ⇥ (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.

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The Vertical Richards Equation

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⇤ ⇥⌅
⇤⇥ ⇤ ⇤⇥
C(⇥) = Kz cos + Sr
⇤t ⇤z ⇤z

Vertical infiltration: acts in a
relatively fast time scale because
it propagates a signal over a
thickness of only a few metres

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⇤ ⇥⌅
⇤⇥ ⇤ ⇤⇥
C(⇥) = Kz cos + Sr
⇤t ⇤z ⇤z

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.
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The Lateral Richards Equation

⇤ ⌅ ⇤ ⇥⌅
⇤ ⇤⇥ ⇤ ⇤⇥
Sr = Ky + Kx sin
⇤y ⇤y ⇤x ⇤x

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The Lateral Richards Equation

⇤ ⌅ ⇤ ⇥⌅
⇤ ⇤⇥ ⇤ ⇤⇥
Sr = Ky + Kx sin
⇤y ⇤y ⇤x ⇤x

Properly treated, this is reduced to
groundwater lateral flow, specifically to the
Boussinesq equation, which, in turn, have
been integrated from SHALSTAB equations

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The Decomposition of the Richards equation

In vertical infiltration plus lateral flow is possible under the assumption
that:
soil depth hillslope length

time scale of lateral flow

constant diffusivity
reference conductivity
Time scale of infiltration

reference hydraulic capacity
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•But Is the condition:

verified ?
courtesy of E. Cordano

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

saturation. However

The scale factor strongly varies with time 22

Rigon & Lanni



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

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

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So

What happens is that, at the beginning the conditions for considering
just the vertical flow are satisfied

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So

What happens is that, at the end the conditions for considering just the
vertical flow are NOT satisfied. Because D0b >> D0 top

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Therefore when a perched water table form

Instead

And lateral flow dominates (is as fast ) than infiltration

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IS THIS TRUE ?
We need to go back to the basics

⇥2
f (Se )
K(Se ) = v
K s Se
After Mualem, 1976

f (1)

Where v is an exponent expressing the
connectivity between pores, evaluated by Mualem
for various soil types.
Se
1
f (Se ) = dx
0 (x)

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IS THIS TRUE ?

Having defined the relative hydraulic conductivity:
After Mualem, 1976

K = Ks Kr

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

1 ⇥1/n
⇥= Se 1/m
1

The integral can be calculated:

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PARAMETRIC FORMS OF THE
HYDRAULIC CONDUCTIVITY
there results:

⇤ ⇥m ⌅2
K(Se ) = v
K s Se 1 1 1/m
Se (m = 1 1/n)

or, by expressing everything as a function of the suction
potential:
⇥2
mn n m
Ks 1 ( ⇥) [1 + ( ⇥) ]
K(⇥) = n mv (m = 1 1/n)
[1 + ( ⇥) ]

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THEREFORE

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

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Another issue
Extending Richards to treat the transition saturated to unsaturated zone.
Is it :

At saturation: what does change in time ?

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Another issue
Extending Richards to treat the transition saturated to unsaturated zone.
Which means:
courtesy of M. Berti

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Or

If you do not have this extension you cannot deal properly with from
unsaturated volumes to saturated ones.

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).

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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)
courtesy of M. Berti

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Stability onstage

The good old infinite slope 36

Rigon & Lanni


Infinite Slope with unsaturated conditions
The equation

e.g. Lu and Godt, 2008

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

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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 SHAP E TF 5% TF 10% TF 15% TF 30% TF 50%
Divergent 41h

Low
P arallel 41h

Convergent 41h 60h

DRY Divergent 14-15h 15-16h 17-18h

M ed
P arallel 14-15h 15-16h 16-17h 18h

Convergent 14-15h 14-15h 14-15h 15h
High Divergent 7-8h 8-9h 9-10h 10-11h 12h
P arallel 7-8h 8h 8-9h 8-9h 8-9h

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

Divergent 3-4h
Low
Lanni and Rigon, 2011

P arallel 3-4h

Convergent 3-4h 4-5h
Divergent 2-3h 3-4h 4-5h
W ET

M ed

P arallel 2-3h 3h 3-4h 4-5h

Convergent 2-3h 2-3h 2-3h 2-3h
Divergent 1-2h 1-2h 1-2h 3h 5h
High

P arallel 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 - - - 5h - 0h not achieved

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

Total volume of water Total volume of
in hillslope rainfall water in
hillslope

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Table 4: A matrix of the rain volumes RF i and total water volume VF i (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.

F 5% F 10% F 15% F 30% F 50%
RAIN SHAPE
DRY WET DRY WET DRY WET DRY WET DRY WET
Divergent

Low
P arallel

RF i (m3 ) Convergent
Divergent
M ed P arallel
Convergent
Divergent
High

P arallel
Convergent
Lanni and Rigon, 2011

Divergent
Low

P arallel
Convergent
VF i (m3 )

Divergent
M ed

P arallel
Convergent
Divergent
High

P arallel
Convergent

41
15m3 - - 125m3 - - 230m3 - - 350m3 - - > 520m3 not achieved

Rigon & Lanni


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%)

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Panola and the soil depth question

Soil-depth variability

Ground surface
Bedrock surface
Bedrock depression

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Soil properties

Soil (sandy-silt) Ksat = 10-4 m/s
Bedrock Ksat = 10-7 m/s
Rain Intensity = 6.5 mm/h
Duration = 9 hours

Slope α = 13° α = 20° α = 30°

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 Hillslope water discharge
o 2 peaks
α = 13°

t=9h

t=6h t=7h t=9h t=14h
t=22h
Q (m3/h)

t=18h
Lanni et al., 2011

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α = 13°

tim
e

t=6h

t=7h

t=9h
 Saturated area at the soil-bedrock interface increases very
rapidly…..
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Same as in the ideal planar case
1° STEP:
1D
Vertical rain-infiltration

Infiltration-front propagation
 No role played by hillslope
gradient
2° STEP:
3D Lateral-flow
Lanni et al., 2011

Downslope drainage
limited by bedrock topography 47

Rigon & Lanni


Pressure growing
Lanni et al., 2011

α = 13° α = 20° α = 30°
Downslope Drainage efficiency 48

Rigon & Lanni


Pressure growing

α = 13°

tim
e

t=6h

t=7h

Lanni et al., 2011
t=9h
 …..and then the average value of positive pore-water pressure
continues to grow
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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.

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If you tilt you slide
α = 30°
(FS=1)
(1<FS<1.05)

c’ = 0 kPa
φ’ = 30°

t=10h
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In complex topography
of 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

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

•However, there is an interplay between slope and bumpiness of the bedrock
which is not trivial at all.

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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)

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Another case and its complexity: Duron

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Farabegoli et al., 2011
Duron stratigraphy

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Farabegoli et al., 2011 Duron soil depth

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Duron geomorphology

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Duron soil cover

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Duron land use

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And a tentative association of those maps with
hydrological characters
With Dall’Amico ,Farabegoli et al., 2011

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Forecasting of temperature
in a point

In time
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Soil water content at different depth
in a point

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Probability of landsliding
Simoni et al, 2008

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Duron

Simoni et al, 2008

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Duron

Simoni et al, 2008

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Rigon & Lanni

Duron

Simoni et al, 2008

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Rigon & Lanni

Duron

Simoni et al, 2008

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Duron

And the snow again !

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Duron

Temperature of snow !

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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

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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.

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Thank you for your attention.

G.Ulrici - 2000 ?

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