Varado N., Ross P.J., Braud I., Haverkamp R., Kao C.
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Transcript of Varado N., Ross P.J., Braud I., Haverkamp R., Kao C.
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EVALUATION OF A FAST NUMERICAL SOLUTION OF THE 1D RICHARD’S
EQUATION AND INCLUSION OF VEGETATION PROCESSES
Varado N., Ross P.J., Braud I., Haverkamp R., Kao C.
Workshop DYNAS, December 6-8, 2004
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1. A fast non iterative solution of the 1D Richards’ equation (Ross, 2003)
2. How to evaluate the numerical solution ?– Use of analytical solutions:
• Moisture profile• Cumulative infiltration
– Use of a numerical h-iterative solution
3. A sink term to account for the water extraction by roots– Inclusion within the numerical solution– Test the accuracy of the vadose zone module
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1. Ross (2003) numerical solution (1)• 1D Richards equation
ee
e
b
ese
b
es
hhhh
hhhh
KKhh
hh
si 1 si 1
si si /32/1
eeses
e
h
hhhhKnhK
hhn
KhdhhK
si 1
si 1
• Brooks and Corey (1964) model to describe soil hydraulic properties:
• Kirchhoff potential or degree of saturation used as calculation variable:
1
zh)h(K
zt
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• Spatial discretisation :mass budget on layer n°i
iii qqQdtd
1
• Time discretisation:
0,1σ 1
ii qq
tQi
iiiiiii dScSbSa 11
• Tri-diagonal matrix:
• Taylor development at first order :
11
0i
i
ii
i
iii S
SqS
Sqqq
i-1
i
q i-1
q i
h i-1
h i
h i+1
xi
i+1
1. Ross (2003) numerical solution (2)
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• ADVANTAGES:
Non-iterative solution fastLayers thickness is allowed to be greater than in classical
modelsRobust
• Flux discretisation:Flux qi between layers i and i+1 is expressed from Darcy low written with Kirchhoff potential and hydraulic conductivity of each layer.
i
iiii
i
iiii Z
KKZ
Kq
11
12/1 1
• calculation : at each time step and for each node Hypothesis: if the pressure is hydrostatic, flux will be null
zKq
1. Ross (2003) numerical solution (3)
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1. A fast non iterative solution of the 1D Richards’ equation (Ross, 2003)
2. How to evaluate the numerical solution ?– Use of analytical solutions:
• Moisture profile• Cumulative infiltration
– Use of a numerical h-iterative solution
3. A sink term to account for the water extraction by roots– Inclusion within the numerical solution– Test the accuracy of the vadose zone module
![Page 7: Varado N., Ross P.J., Braud I., Haverkamp R., Kao C.](https://reader035.fdocuments.in/reader035/viewer/2022062316/568146b5550346895db3d64d/html5/thumbnails/7.jpg)
2.1. Analytical solutions
• With the Brooks and Corey model, no analytical solution describes the moisture profile.
– Moisture profile with simplified soil properties description: Basha (1999) : linear solution
– Cumulative infiltration with BC models: Parlange et al. (1985) Haverkamp et al. (1990)
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Basha (1999) analytical solution
• 8 soils with Gardner parameters (Mualem 1976 et Bresler 1978)
• Constant surface flux=15mm/h during 10h• Initially dry profile
hexpKhK s
hexprsr
Sols (m-1) Ks (m.sec-1) s Chino clay 0.0685 2.29E-07 0.532 Lamberg clay 32.7 3.34E-04 0.537 Peat 0.104 6.13E-07 0.47 Touched silt loam 1.56 4.86E-06 0.469 Oso Flasco fine
sand
7.2 2.00E-04 0.266 Crab Creek sand 46.6 1.27E-04 0.375 Rehovot sand 15.74 7.64E-05 0.44 Ida silt clay loam 6.7 4.17E-06 0.53
Gardner (1958) model: allows the analytical formulation of the Kirchhoff potential.
•Modification of the Ross (2003) numerical solution to deal with the same soils characteristics description•Huge simplification
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layer 1
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.10
0.20
layer 2
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.10
0.20
layer 3
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.10
layer 4
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.10
layer 5
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.06
0.12
layer 6
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.04
0.08
layer 7
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.02
0.05
layer 8
time (h)
wat
er c
onte
nt (
m3.
m-3
)
0 2 4 6 8 10
0.0
0.02
layer 9
time (h)
wat
er c
onte
nt (
m3.
m-3
)0 2 4 6 8 10
0.0
0.01
0
Ross (2003)
Basha (1999)
Touched Silt Loam α=1.56x10-2 cm-1
Ks=4.86x10-4 cm.s-1
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• I(t), I(q)
• 3 characteristics soils (sand, clay, loam)• θ(z=0)=θs
• Initially dry profile, hsurf=0
123
4
5
6
7
8
9
10
x=20cm
x=40cm
x=10cm exp * 11* * ln
1I
t I
Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990
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clay
time (h)
cum
ulat
ive
infil
tratio
n (m
m)
0 2 4 6 8 10
020
4060
8010
012
014
0
Ross (2003)analytical solution
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• I(t), I(q)
• 3 characteristics soils (sand, clay, loam)• θ(z=0)=θs
• Initially dry profile, hsurf=0
123
4
5
6
7
8
9
10
x=20cm
x=40cm
x=10cm
Results on infiltration are sensitive to the discretization, especially on clayey soils:
A finer discretization is needed close to the soil surface
exp * 11* * ln1
It I
Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990
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clay 15 layers
time (h)
cum
ulat
ive
infil
tratio
n (m
m)
0 5 10 15 20
050
100
150
200
Ross (2003)analytical solution
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Haverkamp (personal communication): moisture profile with the Brooks and Corey model.
• z(q, θ )
• Initially dry profile, θ(z=0)=θs, hsurf=0• 3 characteristics soils (sand, clay, loam)
42 ** * *
* * *
1ln 111 1 1
1 2 4
z
z z
cc cz z
z
qc czq q c q
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1 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.28
Profile 10 layers
HaverkampRoss (2003)
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2 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.44
Profile 10 layers
HaverkampRoss (2003)
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3 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.60
Profile 10 layers
HaverkampRoss (2003)
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Haverkamp (personal communication): moisture profile with the Brooks and Corey model.
• z(q, θ )
• Initially dry profile, θ(z=0)=θs, hsurf=0• 3 characteristics soils (sand, clay, loam)
• The soil column needs to be homogeneously discretized from the surface to the bottom.
42 ** * *
* * *
1ln 111 1 1
1 2 4
z
z z
cc cz z
z
qc czq q c q
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1 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.96
HaverkampRoss (2003)
Profile 100 layers
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2 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.96
Profile 100 layers
HaverkampRoss (2003)
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3 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.97
Profile 100 layers
HaverkampRoss (2003)
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4 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.97
Profile 100 layers
HaverkampRoss (2003)
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5 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.98
Profile 100 layers
HaverkampRoss (2003)
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6 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.98
Profile 100 layers
HaverkampRoss (2003)
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7 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.97
Profile 100 layers
HaverkampRoss (2003)
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8 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.97
Profile 100 layers
HaverkampRoss (2003)
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9 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.98
Profile 100 layers
HaverkampRoss (2003)
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10 h
teneur en eau (m^3/m^3)
prof
onde
ur (m
)
0.0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.5
-1.0
-0.5
0.0
E=0.98
Profile 100 layers
HaverkampRoss (2003)
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• Comparison with a SVAT model: SiSPAT (Braud et al., 1995), which provides a reference h-iterative solution (Celia et al. 1990)– Coupled resolution of heat and water transfers – Fine discretization (around 1 cm)– Numerous validations under distinct pedo-climatic
conditions.
• Raining and evaporation periods• Systematic tests on 3 characteristic soil types,
various climate forcing and initial conditions
• Systematic underestimation of the evaporation flux (-2%) and overestimation of water content in the first layer (8%)
2.2. Another reference numerical solution
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1. A fast non iterative solution of the 1D Richards’ equation (Ross, 2003)
2. How to evaluate the numerical solution ?– Use of analytical solutions:
• Moisture profile• Cumulative infiltration
– Use of a numerical h-iterative solution
3. A sink term to account for the water extraction by roots– Inclusion within the numerical solution– Test the accuracy of the vadose zone module
![Page 31: Varado N., Ross P.J., Braud I., Haverkamp R., Kao C.](https://reader035.fdocuments.in/reader035/viewer/2022062316/568146b5550346895db3d64d/html5/thumbnails/31.jpg)
• Inclusion of a sink term within the Richards’ equation (Feddes et al. 1978).
• Does not affect the resolution of the tridiagonal matrix
• Ex(z,t) from literature: Li et al. (2001) account for water stress and provides a compensation by the deeper layers still humid.
• Linear function of a PET
• Interception like a reservoir• No resolution of the energy budget; use of a partition law:
( ) 1 ,hK h Ex z tt z z
3. Account for vegetation processes (1)
1 2, , ,Ex z t z z g z TP
(1 exp( ))exp( )
bl
bl
TP ETP a LAIEP ETP a LAI
iiiiiii dScSbSa 11
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• Test of the accuracy of the vadose zone module with the SiSPAT model
• Test on a soybean dataset
– Underestimation of soil evaporation greater than on bare soil
– Overestimation of water content in the first layer– Low relative error on transpiration– Different partition of the energy between the use of a
PET or the resolution of the energy budget.
3. Account for vegetation processes (2)
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Conclusion• Fast, accurate and robust numerical solution• Validation against analytical solutions and a
numerical solution.• Inclusion of a sink term to account for vegetation
processes
– Another formulation of the evaporation flux?– Problem of partition of the energy
• Vadose zone module.• Inclusion within a large scale hydrological model