An Overview Hillslope Hydrology
Mirò-BlueII
Riccardo Rigon
2nd International Summer School on
Water Research, Praia a Mare...
Goals
• Say what a hillslope is
• Talking about Richards equation
• Say what Hydrology on hillslope is concerned about
• S...
Hillslope Hydrology
Hillslope
Mirò-BlueII
Riccardo Rigon
2nd International Summer School on
Water Research, Praia a Mare, ...
4
What is a hillslope ?
MontgomeryandDietrich,WRR,1992
First you have to identify channels
R. Rigon
What we are talking ab...
5
Orlandinietal.,2011
R. Rigon
What we are talking about ?
Monday, July 8, 13
6
Hillslopes
R. Rigon
What we are talking about ?
Monday, July 8, 13
7
MontgomeryandDietrich,1989
Well ...
R. Rigon
What we are talking about ?
Monday, July 8, 13
daTarboton:www.cuahsi.org
8
R. Rigon
Hillslopes
Monday, July 8, 13
9
daTarboton:www.cuahsi.org
R. Rigon
Hillslopes
Monday, July 8, 13
10
daTarboton:www.cuahsi.org
R. Rigon
Hillslopes
Monday, July 8, 13
11
daTarboton:www.cuahsi.org
R. Rigon
Hillslopes
Monday, July 8, 13
12
daTarboton:www.cuahsi.org
R. Rigon
Hillslopes
Monday, July 8, 13
13
Dolomites- Duron Valley
R. Rigon
Hillslopes
Monday, July 8, 13
14
Soil depth
Soil
rocks
Where does water flow ?
R. Rigon
Soil
Monday, July 8, 13
15
Keep in mind the complexity
Courtesy of Enzo Farabegoli - Duron catchment
R. Rigon
The complexity of geology (and of ge...
Hillslope Hydrology
Hydrology
Mirò-BlueII
Monday, July 8, 13
17
How water moves in hillslopes ?
Turbulent flows - Laminar flows
Both are described by the Navier-Stokes equations
R. Ri...
18
2D - de Saint Venant equations
with some smart subgrid parameterization
(e.g. Casulli, 2009)
1D - Kinematic equation
So...
19
How water moves in hillslopes ?
Turbulent flows - Laminar flows
Darcy flows
R. Rigon
Fundamentals
Monday, July 8, 13
20
Darcy equations are OK
for saturated flow
They can be obtained from Navier-Stokes Equation
by*:
•introducing a resistan...
21
What about
unsaturated flow
R. Rigon
Fundamentals
Monday, July 8, 13
22
One idea is
that we can use Richards’ equation
So, on the earth what is
Richards’ equation ?
R. Rigon
Fundamentals
Mond...
23
Richards’ equation core
is that what it is true is this
Mass conservation (no nuclear reactions) !
but actually true if...
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)...
25
To obtain the last slide
One has to assume the validity of the Darcy-Buckingham law:
Darcy-Buckingham Law
Volumetric fl...
25
To obtain the last slide
One has to assume the validity of the Darcy-Buckingham law:
Darcy-Buckingham Law
Volumetric fl...
25
To obtain the last slide
One has to assume the validity of the Darcy-Buckingham law:
Darcy-Buckingham Law
Volumetric fl...
25
To obtain the last slide
One has to assume the validity of the Darcy-Buckingham law:
Darcy-Buckingham Law
Volumetric fl...
26
Ignore soil hysteresis
and think of the SWRC as a function that relates water content to matric
pressure
⇤ (⇥)
⇤t
=
⇤ (...
27
Assume a parametric form
of soil water retention curves
Se :=
w r
⇥s r
Parametric
van Genuchten
(1981)
C(⇥) :=
⇤ w()
⇤⇥...
28
A theory for getting hydraulic conductivity
from soil water retention curves
K( w) = Ks
⇧
Se
⇤
1 (1 Se)1/m
⇥m⌅2
Paramet...
29
The last representation of mass conservation
is just matter of convenience
habits, and ignorance of some phenomena
•var...
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
R. Rigon
Monday, July 8, 13
Iverson,2000;CordanoeRigon,2008
32
The Richards equation made dimensionless
Richardsoniana
R. Rigon
Monday, July 8, 13
Iverson,2000;CordanoeRigon,2008
33
Richards eq. solution expressed in terms of
the asymptotic hydrostatic solution and a t...
34
Asymptotic solution
Transient solution
A lot of tricks here !
R. Rigon
Richardsoniana
Monday, July 8, 13
35
Depth from surface
Terrain Slope
Water table position
A lot of tricks here !
R. Rigon
Richardsoniana
Monday, July 8, 13
and one equation for
Iverson,2000;CordanoeRigon,2008
36
So Richards equation is
divided into one equation for
Richardsonia...
37
Interestingly
Water table was not present in the original Richards
equation
Hydrostatic hypothesis
R. Rigon
Richardsoni...
38
In detail:
initial condition
R. Rigon
Richardsoniana
Monday, July 8, 13
39
In detail:
transient situation
R. Rigon
Richardsoniana
Monday, July 8, 13
40
In detail:
final situation
R. Rigon
Richardsoniana
Monday, July 8, 13
41
In turn
“Short term
solution” Taylor’s
expansion
Water table
equation Taylor’s
expansion
Slope normal flow
time scale L...
42
Pay attention to this
Slope normal flow
time scale Lateral flow
time scale
Richardsoniana
R. Rigon
Hydraulic diffusivit...
43
Details
that can be found in Cordano and Rigon, 2008
in words
•Take the dimensionless Richards equation
•Substitute in ...
44
Neglecting those details
that can be found in Cordano and Rigon, 2008
Zeroth perturbation order
R. Rigon
Richardsoniana...
45
Neglecting those details
that can be found in Cordano and Rigon, 2008
Zeroth perturbation order
R. Rigon
1D-Richards eq...
46
Neglecting those details
that can be found in Cordano and Rigon, 2008
Water Table equation
R. Rigon
Richardsoniana - Iv...
47
Neglecting those details
that can be found in Cordano and Rigon, 2008
Zeroth perturbation order
First perturbation orde...
48
Integrating zeroth order solution in the column
Making a long story short
R. Rigon
Richardsoniana - Iversoniana
Monday,...
49
Integrating zeroth order solution in the column
R. Rigon
Richardsoniana - Iversoniana
Monday, July 8, 13
50
Integrating zeroth order solution in the column
Making a long story short
Topkapi* model
Liu and Todini, 2002
R. Rigon
...
51
Integrating first order solution slope-parallel
Making a long story short - II
Boussinesq equation
(e.g. Cordano and Ri...
52
Making a long story short - III
R. Rigon
Figure represents a map of a small catchment, river network and a hillslope (h...
53
Integrating Boussinesq
Making a long story short - III
HsB
Troch et al. 2003
R. Rigon
: is the so called width function...
54
Hillslopes Width function
R. Rigon
Richardsoniana - Iversoniana - and beyond
Monday, July 8, 13
55
Simplifying HsB assuming stationarity of fluxes
and neglecting diffusive terms
Making a long story short - IV and V
Top...
56
Simplifying HsB assuming stationarity of fluxes
and neglecting diffusive terms
Making a long story short - IV and V
ass...
57
Take home message
We can use Richards equation at various degree of simplification:
•1D (if we think that just slope-no...
58
Take home message
We can use various simplification of either 1D and 2D Beq together:
• 1D Complete + 2D asymptotic- st...
59
Take home message
We can also try a kinematic approximation of the Boussinesq equation, and
therefore:
• 1D Complete + ...
1D linear + 2D asymptotic
a.k.a D’Odorico et al., 2005
Mirò.The-nightingale-s-song-at-midnight-and-the-morning-rain
Monday...
C(⇥)
⇤⇥
⇤t
=
⇤
⇤z
⇤
Kz
⇤⇥
⇤z
cos
⇥⌅
+ Sr
In literature related to the determination of slope stability this equation
assum...
The analytical solution methods for the advection-dispersion equation
(even non-linear), that results from the Richards eq...
⇥ ⇥ (z d cos )(q/Kz) + ⇥s
Iverson,2000;D’Odoricoetal.,2003,
CordanoandRigon,2008
63
s
The Richards equation on a plane hil...
Assuming K ~ constant and neglecting the source terms
⇤⇥
⇤t
= D0 cos2 ⇤2
⇥
⇤t2
64
The Richards Equation 1-D
C( )
@
@t
= Kz...
The equation becomes LINEAR and, having found a solution
with an instantaneous unit impulse at the boundary, the
solution ...
66
The Richards Equation 1-D
D’Odoricoetal.,2003
R. Rigon
Linearize it !
Monday, July 8, 13
For a precipitation impulse of constant intensity, the solution can be
written:
⇥0 = (z d) cos2
D’Odoricoetal.,2003
67
= 0...
In this case the equation admits an analytical solution
D’Odoricoetal.,2003
68
R(t/TD) :=
⇤
t/( TD)e TD/t
erfc
⇤
TD/t
⇥
s ...
D’Odoricoetal.,2003
69
TD
TD
TD
TD
TheRichardsEquation1-D
R. Rigon
Linearize it !
Monday, July 8, 13
70
Second message
Why using other simplifying assumptions (like
Horton’s or Green-Ampt), if we have this ?
R. Rigon
Forget...
Just kidding!
Monday, July 8, 13
72
Did you care about hypotheses ?
Is it for any occasion realistic ? Look at the following sandy-loam:
Hypotheses counts
...
72
Did you care about hypotheses ?
Is it for any occasion realistic ? Look at the following sandy-loam:
Hypotheses counts
...
constant diffusivity
73
The Decomposition of the Richards equation
is possible under the assumption that:
Time scale of in...
Assuming hydrostatic conditions
74
Initial condition is then:
Consequently, at surface
Hypotheses counts
R. Rigon
Monday, ...
75
For the sandy-loam soil
assuming the water table at one meter depth
we have a vertical variation of hydraulic conductiv...
76
D which characterizes the time scales of flow is varying
with depth
Hypotheses counts
R. Rigon
So a D0 reference cannot...
77
Therefore
at surface
so, lateral flow at the water table level
has the same time scale vertical flow at
the surface (at...
78
igure 2: Experimental set-up. (a) The infinite hillslope schematization. (b) The initial suction head pr
il-pixel hillsl...
79
- 54 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(a) DRY-Low (b) DRY-Med
Simulations res...
80
At the beginning the pressure is constant
along the whole transect (except for
phenomena at the divide’s edge
Comparing...
81
After a certain amount of time (25h in this
simulation) pressures along the slope
differentiate. With a little of analy...
82
(a) (b)
Figure 6: Temporal evolution of the vertical profile of hydraulic conductivity (a) and hydraulic conductivity at...
83
56 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(a) (b)
(a)
(c)
Figure 7: Transient pore ...
84
When simulating is understanding
courtesyofE.Cordano
T’L can be very small indeed .....
Interpretations
R. Rigon
Monday...
85
Understanding from simulations
At the beginning of the infiltration process the situation in surface is
marked by the b...
86
When lateral flow start we are in the following situation
courtesyofE.Cordano
Understanding from simulations
R. Rigon
I...
87
At the beginning
The condition of the perturbative derivation are verified
courtesyofE.Cordano
R. Rigon
Interpretations...
88
At the end
courtesyofE.Cordano
Conditions for lateral flow are dominating. Actually the same
phenomenology deducted by ...
89
Take home message:
Never fully believe on the magic of simplifications
Detailed physics in models can help
R. Rigon
Mag...
90
Lateral Flow
•Can be fast, ... very fast, much faster than what happens in vadose
conditions
•In fact, to have the effe...
91
Inappropriate numerics (or gridding)
Can hide it!
R. Rigon
Interpretations
Monday, July 8, 13
Further investigations
MachaelLeong-Cuttingthetimewithaknife,2012
Monday, July 8, 13
93
CAPITOLO 5. IL BACINO DI PANOLA
Figura 5.2: Rappresentazione della profondit`a del suolo del pendio di Panola.
costante...
94
Terrain surface Bedrock surface Soil depth varies
Depression
Soil (sandy loam) Bedrock
Ksat = 10-4 m/s Ksat = 10-7 m/s
...
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
R. Rigon
Richards equati...
96
t=6h t=9ht=7h t=14h
Lannietal.,2011
With a rainfall of 6.5 mm/h and a duration of 9 hours
Tromp Van Meerveld et al., 20...
97
Q(m3/h)
t=9h
t=18h
t=22h
Lannietal.,2011
With a rainfall of 6.5 mm/h and a duration of 9 hours
R. Rigon
Richards equati...
98
1D
3D
 No role played by hillslope
gradient
First Slope Normal infiltration works
Then Lateral flow start
Infiltration...
99
Now we want a model that can run 100 times faster
In which we obviously use all the machinery of the
Richards’ equation...
100
Introducing the concept of concentration time
in subsurface flow
we have the distances from the channels
R. Rigon
Vari...
101
If we assume that water just move laterally in saturated
conditions, we can use Darcy law for getting the
velocities
p...
102
If we assume that water just move laterally in saturated
conditions, we can use Darcy law for getting the
velocities
A...
103
Then:
Time = Lengths/velocity
And, for any point:
is the max residence time*
R. Rigon
Variations
*The operator means t...
104
The largest time
is the concentration time
Up to concentration time
The area contributing to the discharge is not the
...
105
The area contributing to the discharge is not the
TOTAL upslope area
Lannietal.,2012a
R. Rigon
!(Steady state)
Monday,...
106
Actually there is a second issue
Water table cannot “exist” everywhere
Fig. 1. A flow chart depicting the coupled satur...
107
Ii.e. time to water table
development
Twt(x,y):= [Vwt(x,y)-V0(x,y)]/I
Initial conditions
(hydrostatic slope normal)
bo...
108
YES
t> Tmax
wt(x,y)
hydrologically
connected
A(x,y) >0
YES
NO
hydrologically
disconnected
A(x,y) =0
A heuristic model
...
109
YES
update soil
pressure
start lateral flow update soil
pressure
next
time
step
A heuristic model
Lannietal.,2012
R. R...
110
* Is not completely true.
I question also of personal attitude:
I understand (fluid) mechanics through
equations and I...
111
3968 C. Lanni et al.: Modelling shallow landslide susceptibility
1
2
3
Figure 7. Patterns of Return period TR (years) ...
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) deflus...
114
Macropore Flow
Initiation
Water supply to the
macropores
Interaction
Water transfer between
macropores and the
surroun...
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
Figur...
Thank you for your attention
116
G.Ulrici-
R. Rigon
Slides on http://abouthydrology.blogspot.com
Monday, July 8, 13
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Hillslope hydrologyandrichards

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Hillslope hydrologyandrichards

  1. 1. An Overview Hillslope Hydrology Mirò-BlueII Riccardo Rigon 2nd International Summer School on Water Research, Praia a Mare, July 2013 Monday, July 8, 13
  2. 2. 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
  3. 3. Hillslope Hydrology Hillslope Mirò-BlueII Riccardo Rigon 2nd International Summer School on Water Research, Praia a Mare, July 2013 Monday, July 8, 13
  4. 4. 4 What is a hillslope ? MontgomeryandDietrich,WRR,1992 First you have to identify channels R. Rigon What we are talking about ? Monday, July 8, 13
  5. 5. 5 Orlandinietal.,2011 R. Rigon What we are talking about ? Monday, July 8, 13
  6. 6. 6 Hillslopes R. Rigon What we are talking about ? Monday, July 8, 13
  7. 7. 7 MontgomeryandDietrich,1989 Well ... R. Rigon What we are talking about ? Monday, July 8, 13
  8. 8. daTarboton:www.cuahsi.org 8 R. Rigon Hillslopes Monday, July 8, 13
  9. 9. 9 daTarboton:www.cuahsi.org R. Rigon Hillslopes Monday, July 8, 13
  10. 10. 10 daTarboton:www.cuahsi.org R. Rigon Hillslopes Monday, July 8, 13
  11. 11. 11 daTarboton:www.cuahsi.org R. Rigon Hillslopes Monday, July 8, 13
  12. 12. 12 daTarboton:www.cuahsi.org R. Rigon Hillslopes Monday, July 8, 13
  13. 13. 13 Dolomites- Duron Valley R. Rigon Hillslopes Monday, July 8, 13
  14. 14. 14 Soil depth Soil rocks Where does water flow ? R. Rigon Soil Monday, July 8, 13
  15. 15. 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
  16. 16. Hillslope Hydrology Hydrology Mirò-BlueII Monday, July 8, 13
  17. 17. 17 How water moves in hillslopes ? Turbulent flows - Laminar flows Both are described by the Navier-Stokes equations R. Rigon Fundamentals Monday, July 8, 13
  18. 18. 18 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 R. Rigon Less is more Navier-Stokes equations are actually never used to do hillslope hydrology For a synthesis see: abouthydrology.blogspot.com R. Rigon Monday, July 8, 13
  19. 19. 19 How water moves in hillslopes ? Turbulent flows - Laminar flows Darcy flows R. Rigon Fundamentals Monday, July 8, 13
  20. 20. 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 R. Rigon Fundamentals Monday, July 8, 13
  21. 21. 21 What about unsaturated flow R. Rigon Fundamentals Monday, July 8, 13
  22. 22. 22 One idea is that we can use Richards’ equation So, on the earth what is Richards’ equation ? R. Rigon Fundamentals Monday, July 8, 13
  23. 23. 23 Richards’ equation core is that what it is true is this Mass conservation (no nuclear reactions) ! but actually true if the continuum (a.k.a. Darcy) hypothesis is valid Process based models R. RigonR. Rigon Monday, July 8, 13
  24. 24. 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() ⇤⇥ Process based models R. Rigon Monday, July 8, 13
  25. 25. 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 Monday, July 8, 13
  26. 26. 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 Monday, July 8, 13
  27. 27. 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 Monday, July 8, 13
  28. 28. 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 Monday, July 8, 13
  29. 29. 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 R. Rigon Fundamentals Monday, July 8, 13
  30. 30. 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 ... R. Rigon Fundamentals Monday, July 8, 13
  31. 31. 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... R. Rigon Fundamentals Monday, July 8, 13
  32. 32. 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 Process based models R. RigonR. Rigon Monday, July 8, 13
  33. 33. An example of top down derivation from Richards’ equation ChimpanzeeCongopainting Monday, July 8, 13
  34. 34. Iverson,2000;CordanoeRigon,2008 31 The Richards equation on a plane hillslope Richardsoniana R. Rigon Monday, July 8, 13
  35. 35. Iverson,2000;CordanoeRigon,2008 32 The Richards equation made dimensionless Richardsoniana R. Rigon Monday, July 8, 13
  36. 36. Iverson,2000;CordanoeRigon,2008 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. Rigon Monday, July 8, 13
  37. 37. 34 Asymptotic solution Transient solution A lot of tricks here ! R. Rigon Richardsoniana Monday, July 8, 13
  38. 38. 35 Depth from surface Terrain Slope Water table position A lot of tricks here ! R. Rigon Richardsoniana Monday, July 8, 13
  39. 39. and one equation for Iverson,2000;CordanoeRigon,2008 36 So Richards equation is divided into one equation for Richardsoniana R. Rigon Monday, July 8, 13
  40. 40. 37 Interestingly Water table was not present in the original Richards equation Hydrostatic hypothesis R. Rigon Richardsoniana Monday, July 8, 13
  41. 41. 38 In detail: initial condition R. Rigon Richardsoniana Monday, July 8, 13
  42. 42. 39 In detail: transient situation R. Rigon Richardsoniana Monday, July 8, 13
  43. 43. 40 In detail: final situation R. Rigon Richardsoniana Monday, July 8, 13
  44. 44. 41 In turn “Short term solution” Taylor’s expansion Water table equation Taylor’s expansion Slope normal flow time scale Lateral flow time scaleSee also. D’Odorico et al., 2003 Richardsoniana R. Rigon Monday, July 8, 13
  45. 45. 42 Pay attention to this Slope normal flow time scale Lateral flow time scale Richardsoniana R. Rigon Hydraulic diffusivityD( ) := K( ) C( ) Monday, July 8, 13
  46. 46. 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 R. Rigon Richardsoniana - Iversoniana Monday, July 8, 13
  47. 47. 44 Neglecting those details that can be found in Cordano and Rigon, 2008 Zeroth perturbation order R. Rigon Richardsoniana - Iversoniana Monday, July 8, 13
  48. 48. 45 Neglecting those details that 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 Monday, July 8, 13
  49. 49. 46 Neglecting those details that can be found in Cordano and Rigon, 2008 Water Table equation R. Rigon Richardsoniana - Iversoniana Monday, July 8, 13
  50. 50. 47 Neglecting those details that can be found in Cordano and Rigon, 2008 Zeroth perturbation order First perturbation order + analogous for d* R. Rigon Richardsoniana - Iversoniana Monday, July 8, 13
  51. 51. 48 Integrating zeroth order solution in the column Making a long story short R. Rigon Richardsoniana - Iversoniana Monday, July 8, 13
  52. 52. 49 Integrating zeroth order solution in the column R. Rigon Richardsoniana - Iversoniana Monday, July 8, 13
  53. 53. 50 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 Monday, July 8, 13
  54. 54. 51 Integrating first order solution slope-parallel Making a long story short - II Boussinesq equation (e.g. Cordano and Rigon, 2013) R. Rigon : dimensionless transmissivities : drainable porosity Richardsoniana - Iversoniana - and beyond Monday, July 8, 13
  55. 55. 52 Making a long story short - III R. Rigon Figure represents a map of a small catchment, river network and a hillslope (hollow type, in gray). The distance of any point (P in the figure) in the hillslope to the channel head (C in the figure) is evaluated along the path drawn following the steepest descent (the dashed line). The characteristic length of the hillslope L (the length of x axis in Figure) is the mean of hillslope to channel distance for any point in the hillslope. The x axis used in the paper is downward parallel to mean topographic gradient, the axis y is normal to x (parallel to contour lines in a planar hillslope) and the z axis orthogonal to the x and y axes downward. Integrating again over the lateral dimension from Boussinesq Richardsoniana - Iversoniana - and beyond Monday, July 8, 13
  56. 56. 53 Integrating Boussinesq Making a long story short - III HsB Troch et al. 2003 R. Rigon : is the so called width function Richardsoniana - Iversoniana - and beyond Monday, July 8, 13
  57. 57. 54 Hillslopes Width function R. Rigon Richardsoniana - Iversoniana - and beyond Monday, July 8, 13
  58. 58. 55 Simplifying HsB assuming stationarity of fluxes and neglecting diffusive terms Making a long story short - IV and V Topog O’Loughlin, 1986 R. Rigon Richardsoniana - Iversoniana - and beyond Monday, July 8, 13
  59. 59. 56 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 R. Rigon Richardsoniana - Iversoniana - and beyond Monday, July 8, 13
  60. 60. 57 Take home message We can use Richards equation at various degree of simplification: •1D (if we think that just slope-normal infiltration counts •1D + 2D Boussinesq (Beq) if we want to account for lateral flow * On this I will come back later R. Rigon Richardsoniana - Iversoniana - and beyond Monday, July 8, 13
  61. 61. 58 Take home message We can use various simplification of either 1D and 2D Beq together: • 1D Complete + 2D asymptotic- stationary • 1D linearized + 2D asymptotic- stationary • 1D bulk* + 2D asymptotic- stationary • 1D Complete + 2D full Beq • 1D linearized + 2D full Beq • 1D bulk* + 2D full Beq * On this I will come back later R. Rigon Richardsoniana - Iversoniana - and beyond beyond Monday, July 8, 13
  62. 62. 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
  63. 63. 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
  64. 64. 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
  65. 65. 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 R. Rigon Linearize it ! Monday, July 8, 13
  66. 66. ⇥ ⇥ (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
  67. 67. 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
  68. 68. 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’Odoricoetal.,2003 R. Rigon Linearize it ! Monday, July 8, 13
  69. 69. 66 The Richards Equation 1-D D’Odoricoetal.,2003 R. Rigon Linearize it ! Monday, July 8, 13
  70. 70. For a precipitation impulse of constant intensity, the solution can be written: ⇥0 = (z d) cos2 D’Odoricoetal.,2003 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 The Richards Equation 1-D R. Rigon Linearize it ! Monday, July 8, 13
  71. 71. In this case the equation admits an analytical solution D’Odoricoetal.,2003 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 The Richards Equation 1-D R. Rigon Linearize it ! Monday, July 8, 13
  72. 72. D’Odoricoetal.,2003 69 TD TD TD TD TheRichardsEquation1-D R. Rigon Linearize it ! Monday, July 8, 13
  73. 73. 70 Second message Why using other simplifying assumptions (like Horton’s or Green-Ampt), if we have this ? R. Rigon Forget them! Monday, July 8, 13
  74. 74. Just kidding! Monday, July 8, 13
  75. 75. 72 Did you care about hypotheses ? Is it for any occasion realistic ? Look at the following sandy-loam: Hypotheses counts R. Rigon Monday, July 8, 13
  76. 76. 72 Did you care about hypotheses ? Is it for any occasion realistic ? Look at the following sandy-loam: Hypotheses counts R. Rigon Monday, July 8, 13
  77. 77. 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 R. Rigon Monday, July 8, 13
  78. 78. Assuming hydrostatic conditions 74 Initial condition is then: Consequently, at surface Hypotheses counts R. Rigon Monday, July 8, 13
  79. 79. 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 R. Rigon Monday, July 8, 13
  80. 80. 76 D which characterizes the time scales of flow is varying with depth Hypotheses counts R. Rigon So a D0 reference cannot be significant Monday, July 8, 13
  81. 81. 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. Rigon Monday, July 8, 13
  82. 82. 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 R. Rigon LanniandRigon,unpublished Monday, July 8, 13
  83. 83. 79 - 54 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES (a) DRY-Low (b) DRY-Med Simulations result Comparing with 3D R. Rigon LanniandRigon,unpublished Monday, July 8, 13
  84. 84. 80 At the beginning the pressure is constant along the whole transect (except for phenomena at the divide’s edge Comparing with 3D R. Rigon Monday, July 8, 13
  85. 85. 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. Rigon Monday, July 8, 13
  86. 86. 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 R. Rigon Monday, July 8, 13
  87. 87. 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 R. Rigon Comparing with 3D Monday, July 8, 13
  88. 88. 84 When simulating is understanding courtesyofE.Cordano T’L can be very small indeed ..... Interpretations R. Rigon Monday, July 8, 13
  89. 89. 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 R. Rigon Interpretations Monday, July 8, 13
  90. 90. 86 When lateral flow start we are in the following situation courtesyofE.Cordano Understanding from simulations R. Rigon Interpretations Monday, July 8, 13
  91. 91. 87 At the beginning The condition of the perturbative derivation are verified courtesyofE.Cordano R. Rigon Interpretations Monday, July 8, 13
  92. 92. 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. R. Rigon Interpretations Monday, July 8, 13
  93. 93. 89 Take home message: Never fully believe on the magic of simplifications Detailed physics in models can help R. Rigon Magic ad Mermeids do not exist (Sponge Bob) Monday, July 8, 13
  94. 94. 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
  95. 95. 91 Inappropriate numerics (or gridding) Can hide it! R. Rigon Interpretations Monday, July 8, 13
  96. 96. Further investigations MachaelLeong-Cuttingthetimewithaknife,2012 Monday, July 8, 13
  97. 97. 93 CAPITOLO 5. IL BACINO DI PANOLA Figura 5.2: Rappresentazione della profondit`a del suolo del pendio di Panola. costante su un campione prelevato a 10 cm di profondit`a, risulta pari a 64 [cm/h]; per ci`o che concerne il valore della conducibilit`a 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 conducibilit`a 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 R. Rigon Richards equation is still valid here ? Monday, July 8, 13
  98. 98. 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 R. Rigon Richards equation is still valid here ? Monday, July 8, 13
  99. 99. 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 R. Rigon Richards equation is still valid here ? Monday, July 8, 13
  100. 100. 96 t=6h t=9ht=7h t=14h Lannietal.,2011 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
  101. 101. 97 Q(m3/h) t=9h t=18h t=22h Lannietal.,2011 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
  102. 102. 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 R. Rigon Richards equation is still valid here ? Monday, July 8, 13
  103. 103. 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
  104. 104. 100 Introducing the concept of concentration time in subsurface flow we have the distances from the channels R. Rigon Variations Monday, July 8, 13
  105. 105. 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
  106. 106. 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
  107. 107. 103 Then: Time = Lengths/velocity And, for any point: is the max residence time* R. Rigon Variations *The operator means that we are looking for the maximum of T choosing it from all the possible path that we can define upstream of the point i Monday, July 8, 13
  108. 108. 104 The largest time is the concentration time Up to concentration time The area contributing to the discharge is not the TOTAL upslope area R. Rigon Variations Monday, July 8, 13
  109. 109. 105 The area contributing to the discharge is not the TOTAL upslope area Lannietal.,2012a R. Rigon !(Steady state) Monday, July 8, 13
  110. 110. 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 R. Rigon !(Steady state) Monday, July 8, 13
  111. 111. 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 R. Rigon Monday, July 8, 13
  112. 112. 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 R. Rigon Faster is better Monday, July 8, 13
  113. 113. 109 YES update soil pressure start lateral flow update soil pressure next time step A heuristic model Lannietal.,2012 R. Rigon Faster is better Monday, July 8, 13
  114. 114. 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
  115. 115. 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
  116. 116. Further investigations -II MachaelLeong-Cuttingthetimewithaknife,2012 Monday, July 8, 13
  117. 117. 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 `e 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
  118. 118. 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
  119. 119. 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. pu`o 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, `e che nella creazione della geometria di calcolo 3D utilizzata da GEOtop non `e DaPrà,2013 Certainly the volumes of water cannot be simulated with the only Richards equation No way! R. Rigon Macropores Monday, July 8, 13
  120. 120. Thank you for your attention 116 G.Ulrici- R. Rigon Slides on http://abouthydrology.blogspot.com Monday, July 8, 13

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