Hillslope hydrologyandrichards

An Overview Hillslope Hydrology
Mirò-BlueII
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. Rigon
Monday, July 8, 13

Hillslope Hydrology
Hillslope
Mirò-BlueII
Riccardo Rigon
2nd International Summer School on
Water Research, Praia a Mare, July 2013
Monday, July 8, 13

4
What is a hillslope ?
MontgomeryandDietrich,WRR,1992
First you have to identify channels
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What we are talking about ?
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Orlandinietal.,2011
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Hillslopes
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MontgomeryandDietrich,1989
Well ...
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daTarboton:www.cuahsi.org
8
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Hillslopes
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Hillslopes
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Hillslopes
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Hillslopes
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Hillslopes
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Dolomites- Duron Valley
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Hillslopes
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Soil depth
Soil
rocks
Where does water flow ?
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Soil
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15
Keep in mind the complexity
Courtesy of Enzo Farabegoli - Duron catchment
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The complexity of geology (and of gelogists)
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Hillslope Hydrology
Hydrology
Mirò-BlueII
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How water moves in hillslopes ?
Turbulent flows - Laminar flows
Both are described by the Navier-Stokes equations
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Fundamentals
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2D - de Saint Venant equations
with some smart subgrid parameterization
(e.g. Casulli, 2009)
1D - Kinematic equation
So many to cite here but ... Liu and
Todini, 2002
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Less is more
Navier-Stokes equations are actually never used to do
hillslope hydrology
For a synthesis see: abouthydrology.blogspot.com
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How water moves in hillslopes ?
Turbulent flows - Laminar flows
Darcy flows
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Fundamentals
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20
Darcy equations are OK
for 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
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Fundamentals
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What about
unsaturated flow
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Fundamentals
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22
One idea is
that we can use Richards’ equation
So, on the earth what is
Richards’ equation ?
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Fundamentals
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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
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Monday, 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)
Parametric
Mualem (1976)
Parametric
van Genuchten
(1981)
C(⇥) :=
⇤ w()
⇤⇥
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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
Buckingham,1907,Richards,1931
~Jv = K(✓w)~r h
Fundamentals
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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 head
Buckingham,1907,Richards,1931
~Jv = K(✓w)~r h
Fundamentals
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26
Ignore soil hysteresis
and think of the SWRC as a function that relates water content to matric
pressure
⇤ (⇥)
⇤t
=
⇤ (⇥)
⇤⇥
⇤⇥
⇤t
C(⇥)
⇤⇥
⇤t
Hydraulic capacity of
the soil
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Fundamentals
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27
Assume a parametric form
of soil water retention curves
Se :=
w r
⇥s r
Parametric
van Genuchten
(1981)
C(⇥) :=
⇤ w()
⇤⇥
Se = [1 + ( ⇥)m
)]
n
But other forms are possible ...
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Fundamentals
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28
A theory for getting hydraulic conductivity
from soil water retention curves
K( w) = Ks
⇧
Se
⇤
1 (1 Se)1/m
⇥m⌅2
Parametric
Mualem (1976)
But other forms are possible also here...
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Fundamentals
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29
The last representation of mass conservation
is just matter of convenience
habits, and ignorance of some phenomena
•variable and changing temperature
•soil freezing
•transition to saturation
•preferential flow
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Monday, July 8, 13

An example of top down derivation
from Richards’ equation
ChimpanzeeCongopainting
Monday, July 8, 13

Iverson,2000;CordanoeRigon,2008
31
The Richards equation on a plane hillslope
Richardsoniana
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The Richards equation made dimensionless
Richardsoniana
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Richards eq. solution expressed in terms of
the asymptotic hydrostatic solution and a transient
term:
See also. D’Odorico et al., 2003
Richardsoniana
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34
Asymptotic solution
Transient solution
A lot of tricks here !
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Richardsoniana
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35
Depth from surface
Terrain Slope
Water table position
A lot of tricks here !
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Richardsoniana
Monday, July 8, 13

and one equation for
36
So Richards equation is
divided into one equation for
Richardsoniana
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37
Interestingly
Water table was not present in the original Richards
equation
Hydrostatic hypothesis
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Richardsoniana
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In detail:
initial condition
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Richardsoniana
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In detail:
transient situation
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Richardsoniana
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In detail:
final situation
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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
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42
Pay attention to this
Slope normal flow
time scale Lateral flow
time scale
Richardsoniana
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Hydraulic diffusivityD( ) :=
K( )
C( )
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43
Details
that 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
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Richardsoniana - Iversoniana
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Neglecting those details
Zeroth perturbation order
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45
R. Rigon
1D-Richards equation A source term
(exchange with water table)
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Water Table equation
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47
First perturbation order
+ analogous for d*
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Integrating zeroth order solution in the column
Making a long story short
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50
Making a long story short
Topkapi* model
Liu and Todini, 2002
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*With some interpretation
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51
Integrating first order solution slope-parallel
Making a long story short - II
Boussinesq equation
(e.g. Cordano and Rigon, 2013)
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: dimensionless transmissivities
: drainable porosity
Richardsoniana - Iversoniana - and beyond
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52
Making a long story short - III
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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 dimension
from Boussinesq
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53
Integrating Boussinesq
Making a long story short - III
HsB
Troch et al. 2003
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: is the so called width function
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Hillslopes Width function
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Simplifying HsB assuming stationarity of fluxes
and neglecting diffusive terms
Making a long story short - IV and V
Topog
O’Loughlin, 1986
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Simplifying HsB assuming stationarity of fluxes
and neglecting diffusive terms
Making a long story short - IV and V
assuming an exponential decay of vertical hydraulic
conductivity
Topmodel
Beven and Kirkby, 1979
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Monday, July 8, 13

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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
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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
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Richardsoniana - Iversoniana - and beyond beyond
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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
R. Rigon
Richardsoniana - Iversoniana - and beyond beyond
Monday, July 8, 13

1D linear + 2D asymptotic
a.k.a D’Odorico et al., 2005
Mirò.The-nightingale-s-song-at-midnight-and-the-morning-rain
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
TheRichardsEquation1-D
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Linearize it !
Monday, July 8, 13

⇥ ⇥ (z d cos )(q/Kz) + ⇥s
Iverson,2000;D’Odoricoetal.,2003,
CordanoandRigon,2008
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’Odoricoetal.,2003
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
R. Rigon
Linearize it !
Monday, July 8, 13

66
R. Rigon
Linearize it !
Monday, July 8, 13

For a precipitation impulse of constant intensity, the solution can be
written:
⇥0 = (z d) cos2
67
= 0 + s
s =
8
<
:
q
Kz
[R(t/TD)] 0  t  T
q
Kz
[R(t/TD) R(t/TD T/TD)] t > T
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Linearize it !
Monday, July 8, 13

In this case the equation admits an analytical solution
68
R(t/TD) :=
⇤
t/( TD)e TD/t
erfc
⇤
TD/t
⇥
s =
8
<
:
q
Kz
[R(t/TD)] 0  t  T
q
Kz
[R(t/TD) R(t/TD T/TD)] t > T
TD :=
z2
D0
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Linearize it !
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69
TD
TD
TD
TD
TheRichardsEquation1-D
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Linearize it !
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70
Second message
Why using other simplifying assumptions (like
Horton’s or Green-Ampt), if we have this ?
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Forget them!
Monday, July 8, 13

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
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Monday, July 8, 13

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
Iverson,2000;CordanoandRigon,2008
Hypotheses counts
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Monday, July 8, 13

Assuming hydrostatic conditions
74
Initial condition is then:
Consequently, at surface
Hypotheses counts
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75
For the sandy-loam soil
assuming the water table at one meter depth
we have a vertical variation of hydraulic conductivity of one order of magnitude !
Hypotheses counts
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Monday, July 8, 13

76
D which characterizes the time scales of flow is varying
with depth
Hypotheses counts
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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
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Monday, July 8, 13

78
igure 2: Experimental set-up. (a) The inﬁnite hillslope schematization. (b) The initial suction head pr
il-pixel hillslope numeration system (the case of parallel shape is shown here). Moving from 0 to 900
sponds to moving from the crest to the toe of the hillslope
The OpenBook hillslope in a 3D
simulation
Comparing with 3D
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LanniandRigon,unpublished
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79
- 54 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(a) DRY-Low (b) DRY-Med
Simulations result
Comparing with 3D
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LanniandRigon,unpublished
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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
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Monday, July 8, 13

82
(a) (b)
Figure 6: Temporal evolution of the vertical proﬁle of hydraulic conductivity (a) and hydraulic conductivity at the soil-bedrock interface
Hidraulic conductivity is varying by three order of magnitude
at the bedrock interface.
The key to understand this phenomenology
Lannietal.,2012
Comparing with 3D
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Monday, July 8, 13

83
56 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(a) (b)
(a)
(c)
Figure 7: Transient pore pressure proﬁles related to points No. 300, 450
(c) cases, and soil hydraulic conductivity function inferred using the Mu
position of the water table at different timing
D R A F T September 24, 2
Another view
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Comparing with 3D
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84
When simulating is understanding
courtesyofE.Cordano
T’L can be very small indeed .....
Interpretations
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Monday, July 8, 13

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
courtesyofE.Cordano
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Interpretations
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86
When lateral flow start we are in the following situation
courtesyofE.Cordano
Understanding from simulations
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Interpretations
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87
At the beginning
The condition of the perturbative derivation are verified
courtesyofE.Cordano
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Interpretations
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88
At the end
courtesyofE.Cordano
Conditions for lateral flow are dominating. Actually the same
phenomenology deducted by the perturbation theory! But obtained for a
different reason.
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Interpretations
Monday, July 8, 13

89
Take home message:
Never fully believe on the magic of simplifications
Detailed physics in models can help
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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
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Interpretations
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91
Inappropriate numerics (or gridding)
Can hide it!
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Interpretations
Monday, July 8, 13

Further investigations
MachaelLeong-Cuttingthetimewithaknife,2012
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93
CAPITOLO 5. IL BACINO DI PANOLA
Figura 5.2: Rappresentazione della profondità del suolo del pendio di Panola.
costante su un campione prelevato a 10 cm di profondità, risulta pari a 64 [cm/h]; per ciò che concerne
il valore della conducibilità idraulica a saturazione del bedrock, non esistono misure dirette e↵ettuate
su campioni prelevati in sito; tuttavia si stima che il suo valore sia 2-3 ordini di grandezza inferiore
rispetto a quella del terreno soprastante. Entrambi i valori di conducibilità idraulica satura (del bedrock
e del terreno) saranno comunque oggetto di calibrazione numerica all’atto delle simulazioni svolte con
GEOtop, utilizzando come valori di partenza quelli qui citati.
Panola’s hillslope
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Richards equation is still valid here ?
Monday, July 8, 13

94
Terrain surface Bedrock surface Soil depth varies
Depression
Soil (sandy loam) Bedrock
Ksat = 10-4 m/s Ksat = 10-7 m/s
Panola’s hillslope
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Monday, July 8, 13

95
Q(m3/h)
t=9h
t=18h
t=22h
With a rainfall of 6.5 mm/h and a duration of 9 hours
Lannietal.,2011
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Monday, July 8, 13

96
t=6h t=9ht=7h t=14h
Lannietal.,2011
Tromp Van Meerveld et al., 2006 call it filling and spilling
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Monday, July 8, 13

97
Q(m3/h)
t=9h
t=18h
t=22h
Lannietal.,2011
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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 case
Lannietal.,2011
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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
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Monday, July 8, 13

100
Introducing the concept of concentration time
in subsurface flow
we have the distances from the channels
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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:
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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
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Variations
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103
Then:
Time = Lengths/velocity
And, for any point:
is the max residence time*
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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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104
The largest time
is the concentration time
Up to concentration time
The area contributing to the discharge is not the
TOTAL upslope area
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Variations
Monday, July 8, 13

105
The area contributing to the discharge is not the
TOTAL upslope area
Lannietal.,2012a
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!(Steady state)
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106
Actually there is a second issue
Water table cannot “exist” everywhere
Fig. 1. A ﬂow chart depicting the coupled saturated/unsaturated hydrological model developed in this study.
1
2
3
4
Figure 2. The concept of hydrological connectivity. Lateral subsurface flow occurs at point5
(x,y) when this becomes hydrologically connected with its own upslope contributing area6
A(x,y).7
8
Fig. 2. The concept of hydrological connectivity. Lateral subsurface
ﬂow 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 water
table (i.e. zero-pressure head) at the soil–bedrock interface
(Fig. 3); and I [LT 1] is the rainfall intensity assumed to be
uniform 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
1
2
3
4
Figure 3. i(z) and i(z) are, respectively, the in5
head vertical profiles. wt(z) and wt(z) represe6
head vertical profiles associated with zero-suction7
8
Fig. 3. ✓i(z) and i(z) are, respecti
and the initial suction head vertical p
resents the linear water content and
associated with zero-suction head at
the relation between [L] and
equilibrium:
= (z = 0) + z = b + z,
Lannietal.,2012b
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!(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
Lannietal.,2012
Slope Normal
unsaturated flow
A heuristic model
for each
time
step
Faster is better
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Monday, July 8, 13

108
YES
t> Tmax
wt(x,y)
hydrologically
connected
A(x,y) >0
YES
NO
hydrologically
disconnected
A(x,y) =0
A heuristic model
Lannietal.,2012
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Faster is better
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109
YES
update soil
pressure
start lateral flow update soil
pressure
next
time
step
A heuristic model
Lannietal.,2012
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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.
Doodley,Muttley,andtheirflyingmachines
R. Rigon
Attitudes
Monday, July 8, 13

111
3968 C. Lanni et al.: Modelling shallow landslide susceptibility
1
2
3
Figure 7. Patterns of Return period TR (years) of the critical rainfalls for shallow landslide4
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 of
landslide 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 Ca Lb Ca Lb Ca Lb
Years Category % % % % % %
Uncond Unstable 9.9 60.2 7.7 77.7 8.5 56.8
0–10 Very high 20.3 26.9 16.1 18.5 13.5 39.2
10–30 High 7.8 0.0 5.6 1.5 5.8 4.0
Lannietal.,2012
However, it works
R. Rigon
Faster is better if it works (Klemes fogive me!)
Monday, July 8, 13

Further investigations -II
MachaelLeong-Cuttingthetimewithaknife,2012
Monday, July 8, 13

113
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 è stata suddivisa la trincea e (b) numero di eventi meteorici che
producono deflussi misurabili.
5.2.1 Il ruolo dei macropori
TrompVanMeerveldetal.,2006
And finally macropores
R. Rigon
Macropores
Monday, July 8, 13

114
Macropore Flow
Initiation
Water supply to the
macropores
Interaction
Water transfer between
macropores and the
surrounding soil matrix
M.Weiler,fromMochaproject
Macropores!
R. Rigon
Macropores
Monday, July 8, 13

115
0.00
date (dd/mm) 2002
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
Figura 5.16: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trincea
alla base del pendio.
0.000.020.040.060.080.10
Simulazione 0 - evento 6 febbraio
date (dd/mm) 2002
portate[l/s]
05/02 06/02 07/02 08/02 09/02 10/02 11/02 12/02
Flussi misurati
Simulazione 0
0.000.020.040.060.080.10
Simulazione 0 - evento 30 marzo
date (dd/mm) 2002
portate[l/s]
29/03 30/03 31/03 01/04 02/04 03/04 04/04 05/04 06/04 07/04
Flussi misurati
Simulazione 0
Figura 5.17: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trincea
alla base del pendio: a sinistra si riporta l’evento del 6 febbraio 2002, a destra quello del 31 marzo.
può essere causata da diversi fattori, quali un’errata assegnazione delle caratteristiche del suolo o del
bedrock, 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 presentati
successivamente, è che nella creazione della geometria di calcolo 3D utilizzata da GEOtop non è
DaPrà,2013
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.Ulrici-
R. Rigon
Slides on http://abouthydrology.blogspot.com
Monday, July 8, 13

Hillslope hydrologyandrichards

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