Hillslope hydrologyandrichards
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Transcript of Hillslope hydrologyandrichards
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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
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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
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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
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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
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5
Orl
an
din
i et
al.,
20
11
R. Rigon
What we are talking about ?
Monday, July 8, 13
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Hillslopes
R. Rigon
What we are talking about ?
Monday, July 8, 13
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7
Mon
tgom
ery
and
Die
tric
h, 1
98
9
Well ...
R. Rigon
What we are talking about ?
Monday, July 8, 13
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da
Tar
boto
n: w
ww
.cu
ahsi
.org
8
R. Rigon
Hillslopes
Monday, July 8, 13
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9
da
Tar
boto
n: w
ww
.cu
ahsi
.org
R. Rigon
Hillslopes
Monday, July 8, 13
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10
da
Tar
boto
n: w
ww
.cu
ahsi
.org
R. Rigon
Hillslopes
Monday, July 8, 13
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da
Tar
boto
n: w
ww
.cu
ahsi
.org
R. Rigon
Hillslopes
Monday, July 8, 13
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da
Tar
boto
n: w
ww
.cu
ahsi
.org
R. Rigon
Hillslopes
Monday, July 8, 13
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Dolomites- Duron Valley
R. Rigon
Hillslopes
Monday, July 8, 13
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Soil depth
Soil
rocks
Where does water flow ?
R. Rigon
Soil
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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
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Hillslope HydrologyHydrology
Mir
ò -
Blu
e II
Monday, July 8, 13
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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
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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
R. RigonMonday, July 8, 13
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How water moves in hillslopes ?
Turbulent flows - Laminar flows
Darcy flows
R. Rigon
Fundamentals
Monday, July 8, 13
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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
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What aboutunsaturated flow
R. Rigon
Fundamentals
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One idea isthat we can use Richards’ equation
So, on the earth what is Richards’ equation ?
R. Rigon
Fundamentals
Monday, July 8, 13
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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
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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()⇤⇥
Process based models
R. RigonMonday, July 8, 13
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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
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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
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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
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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
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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
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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
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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
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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
Process based models
R. RigonR. RigonMonday, July 8, 13
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An example of top down derivationfrom Richards’ equation
Ch
imp
anzee
Con
go p
ain
tin
g
Monday, July 8, 13
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Iver
son
, 20
00
; Cord
ano e
Rig
on
, 20
08
31
The Richards equation on a plane hillslope
Richardsoniana
R. RigonMonday, July 8, 13
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Iver
son
, 20
00
; Cord
ano e
Rig
on
, 20
08
32
The Richards equation made dimensionless
Richardsoniana
R. RigonMonday, July 8, 13
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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
R. RigonMonday, July 8, 13
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Asymptotic solution
Transient solution
A lot of tricks here !
R. Rigon
Richardsoniana
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Depth from surface
Terrain Slope
Water table position
A lot of tricks here !
R. Rigon
Richardsoniana
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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
R. RigonMonday, July 8, 13
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Interestingly
Water table was not present in the original Richards equation
Hydrostatic hypothesis
R. Rigon
Richardsoniana
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In detail:initial condition
R. Rigon
Richardsoniana
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In detail:transient situation
R. Rigon
Richardsoniana
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In detail:final situation
R. Rigon
Richardsoniana
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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
R. RigonMonday, July 8, 13
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Pay attention to this
Slope normal flow
time scale Lateral flow
time scale
Richardsoniana
R. Rigon
Hydraulic diffusivityD(�) :=K(�)C(�)
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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
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Neglecting those detailsthat can be found in Cordano and Rigon, 2008
Zeroth perturbation order
R. Rigon
Richardsoniana - Iversoniana
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Neglecting those detailsthat can be found in Cordano and Rigon, 2008
Zeroth perturbation order
R. Rigon
1D-Richards equation A source term
(exchange with water table)
Richardsoniana - Iversoniana
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Neglecting those detailsthat can be found in Cordano and Rigon, 2008
Water Table equation
R. Rigon
Richardsoniana - Iversoniana
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Neglecting those detailsthat can be found in Cordano and Rigon, 2008
Zeroth perturbation order
First perturbation order
+ analogous for d*
R. Rigon
Richardsoniana - Iversoniana
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Integrating zeroth order solution in the column
Making a long story short
R. Rigon
Richardsoniana - Iversoniana
Monday, July 8, 13
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Integrating zeroth order solution in the column
R. Rigon
Richardsoniana - Iversoniana
Monday, July 8, 13
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Integrating zeroth order solution in the column
Making a long story short
Topkapi* model Liu and Todini, 2002
R. Rigon
*With some interpretation
Richardsoniana - Iversoniana
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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
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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
Richardsoniana - Iversoniana - and beyond
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Integrating Boussinesq
Making a long story short - III
HsBTroch et al. 2003
R. Rigon
: is the so called width function
Richardsoniana - Iversoniana - and beyond
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Hillslopes Width function
R. Rigon
Richardsoniana - Iversoniana - and beyond
Monday, July 8, 13
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Simplifying HsB assuming stationarity of fluxesand neglecting diffusive terms
Making a long story short - IV and V
TopogO’Loughlin, 1986
R. Rigon
Richardsoniana - Iversoniana - and beyond
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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
Richardsoniana - Iversoniana - and beyond
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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
Richardsoniana - Iversoniana - and beyond
Monday, July 8, 13
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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
* On this I will come back later
R. Rigon
Richardsoniana - Iversoniana - and beyond beyond
Monday, July 8, 13
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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
* On this I will come back later
R. Rigon
Richardsoniana - Iversoniana - and beyond beyond
Monday, July 8, 13
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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
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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 !
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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
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⇥ ⇥ (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
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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
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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
The Richards Equation 1-D
D’O
dori
co e
t al
., 2
00
3
R. Rigon
Linearize it !
Monday, July 8, 13
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66
The Richards Equation 1-D
D’O
dori
co e
t al
., 2
00
3
R. Rigon
Linearize it !
Monday, July 8, 13
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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
The Richards Equation 1-D
R. Rigon
Linearize it !
Monday, July 8, 13
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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
The Richards Equation 1-D
R. Rigon
Linearize it !
Monday, July 8, 13
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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
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70
Second message
Why using other simplifying assumptions (like Horton’s or Green-Ampt), if we have this ?
R. Rigon
Forget them!
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Just kidding!
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72
Did you care about hypotheses ?
Is it for any occasion realistic ? Look at the following sandy-loam:
Hypotheses counts
R. RigonMonday, July 8, 13
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72
Did you care about hypotheses ?
Is it for any occasion realistic ? Look at the following sandy-loam:
Hypotheses counts
R. RigonMonday, July 8, 13
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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
R. RigonMonday, July 8, 13
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Assuming hydrostatic conditions
74
Initial condition is then:
Consequently, at surface
Hypotheses counts
R. RigonMonday, July 8, 13
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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
R. RigonMonday, July 8, 13
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76
D which characterizes the time scales of flow is varying with depth
Hypotheses counts
R. Rigon
So a D0 reference cannot be significant
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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
R. RigonMonday, July 8, 13
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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
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79
X - 54 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(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
D R A F T September 24, 2010, 9:13am D R A F T
Simulations result
Comparing with 3D
R. Rigon
Lan
ni
and
Rig
on
, un
pu
bli
shed
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80
At the beginning the pressure is constant
along the whole transect (except for
phenomena at the divide’s edge
Comparing with 3D
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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
R. RigonMonday, July 8, 13
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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
D R A F T September 24, 2010, 9:13am D R A F T
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
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83
X - 56 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(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
D R A F T September 24, 2010, 9:13am D R A F T
X - 56 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(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
D R A F T September 24, 2010, 9:13am D R A F T
Another view
R. Rigon
Comparing with 3D
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84
When simulating is understanding
cou
rtes
y of
E. C
ord
ano
T’L can be very small indeed .....
Interpretations
R. RigonMonday, July 8, 13
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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
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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
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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
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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
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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)
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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
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91
Inappropriate numerics (or gridding)
Can hide it!
R. Rigon
Interpretations
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Further investigationsM
ach
ael
Leon
g -
Cu
ttin
g t
he
tim
e w
ith
a k
nif
e, 2
01
2
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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 ?
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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
Richards equation is still valid here ?
Monday, July 8, 13
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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
Richards equation is still valid here ?
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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
Richards equation is still valid here ?
Monday, July 8, 13
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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
Richards equation is still valid here ?
Monday, July 8, 13
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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
Richards equation is still valid here ?
Monday, July 8, 13
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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
Richards equation is still valid here ?
Monday, July 8, 13
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100
Introducing the concept of concentration timein subsurface flow
we have the distances from the channels
R. Rigon
Variations
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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
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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
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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
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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
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The area contributing to the discharge is not the TOTAL upslope area
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2a
R. Rigon
!(Steady state)
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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.
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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,
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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
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!(Steady state)
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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
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Slope Normalunsaturated flow
A heuristic model
for each timestep
Faster is better
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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
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YES
update soil pressure
start lateral flow update soil pressure
next timestep
A heuristic modelLa
nn
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2
R. Rigon
Faster is better
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* 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
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3968 C. Lanni et al.: Modelling shallow landslide susceptibility
40
1
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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/
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However, it works
R. Rigon
Faster is better if it works (Klemes fogive me!)
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Further investigations -IIM
ach
ael
Leon
g -
Cu
ttin
g t
he
tim
e w
ith
a k
nif
e, 2
01
2
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CAPITOLO 5. IL BACINO DI PANOLA
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
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Mee
rvel
d e
t al
., 2
00
6
And finally macropores
R. Rigon
Macropores
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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
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CAPITOLO 5. IL BACINO DI PANOLA
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
Flussi misuratiSimulazione 0
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
Flussi misuratiSimulazione 0
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
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3
Certainly the volumes of water cannot be simulated with the only Richards equation
No way!
R. Rigon
Macropores
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Thank you for your attention
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G. U
lric
i -
R. Rigon
Slides on http://abouthydrology.blogspot.com
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