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Some Difficulties in Modeling Water and Solute Transport in Soils Ph. ACKERER IMFS STRASBOURG [email protected]. With the help of B. Belfort, H. Beydoun, F. Lehmann and A. Younès. Hillslope hydrology. 0.36 km 2 , 1000-750 m. - PowerPoint PPT Presentation
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DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Some Difficulties in Modeling Water and Solute Transport in Soils
Ph. ACKERERIMFS [email protected]
With the help of B. Belfort, H. Beydoun, F. Lehmann and A. Younès.
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
0.36 km2, 1000-750 m
Contact: Bruno AMBROISE Contact: Bruno AMBROISE (IMFS)(IMFS)
Hillslope hydrology
The Ringelbach catchment
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Saturated area
Discharge
(from B. Ambroise, IMFS)
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Hillslope hydrology
Mathematical models– Darcy – Richards eq. – Soil hydraulic properties
Parameter measurements– Direct methods – Indirect methods
Numerical methods – Highly non linear PDEs– Very strong parameters contrasts– Long term simulation– ‘Flat’ geometry
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
(from UMR LISAH, Montpellier)
Usual concepts and mathematical models__________________________________________________________________________________
Model concept
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
101
10-1
10-3
10-5
10-7
Sc
ale
(m
)
Model scale
Q
1
Continuum Mec.(Stokes, Hagen-Poiseuille, …)
KT
KL
REVDarcy, Richards,Water retention curves , ….
Usual concepts and mathematical models__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Usual mathematical models – conservation laws__________________________________________________________________________________
( )+ .( v)= f
t
Mass conservation
(p )v gz
k
Generalized Darcy’s law
1 V
h hC( h ). K( h ).( ) S with C( h )
t z z h
V
hD( ). K( ) S with D( ) K( )
t z z
1 V
hK( h ).( ) S
t z z
Richards’ equation
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
re
s r
S
21
1 1
0 0
( ) /eS
lr eh = S h dS h dSK
Mualem, 1976
11 1/
[1 ]e n m
S = m n | h|
1/ 2( ) [1 (1 )]mLr e e e = SS SK
Van Genuchten, 1981
Pore-size distribution models
Usual mathematical models – Soil hydraulic properties__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Particle-size distribution (Arya & Paris, 1981)
1/ 2
(1 )4 / 6p b
i i ib
r = R n
Pore radius Ri: average particle radius for fraction ib : soil densityp : particle densityn : number of particle : 1.35 – 1.40
p bivi
p b
WV =
Water content
W: fraction of particle distribution
2 cos( )i
w i
h = g r
Water pressure
: surface tension : contact angle
Usual mathematical models – Soil hydraulic properties__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Robbez-Masson, UMR LISAH, Montpellier
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Macropores in un-colonised and colonised soil (from Pierret et al., 2002)
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Hierarchy of flow/transport models for variably-saturated structured media (after Altman et al., 1996)
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
From Tuller & Or, 2001
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
New mathematical models
Richard’s equation with alternative h() and K()
Network models
Alternative models
Some recent concepts__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Pore-size distribution models
( )
( ) ( )( )
R h
sm
m
KK h = K h
K h
Modified Van Genuchten, Vogel et al. (1998, 2001)
Soil Hydraulic Properties, h() and K()__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
ln( / )me
h hS Q
0.5 2( ) (2 ) exp( / 2)x
Q x u du
20.5ln( / ) ln( / )( ) m m
rh h h hh = Q QK
Kosugi, 1996
Soil Hydraulic Properties, h() and K()__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Soil Hydraulic Properties, h() and K()__________________________________________________________________________________
Pore-scale models (Tuller & Or, 2002)
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Soil Hydraulic Properties, h() and K()__________________________________________________________________________________
(a) Fitted liquid saturation for silt loam soil with biological macropores. (b) Predicted relative hydraulic conductivity. (Note that 1 J kg-1 = 10-2 bar.)(from Tuller & Or, 2002)
Pore-scale models (Tuller & Or, 2002)
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Pedotransfer functions (Wösten, 2001)
Soil Hydraulic Properties, h() and K()__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Soil Hydraulic Properties, h() and K()__________________________________________________________________________________
Smooth functions
Prunty & Casey, 2002
0.52
0 11
n
e i i ii
S a a h b h h d
21
1 1
0 0
( ) /eS
lr eh = S h dS h dSK
Mualem, 1976
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Kinematic–dispersive wave model (Di Pietro et al., 2003)
Network models __________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
From Pan et al., 2004
Alternative models__________________________________________________________________________________
Two-phase flow using Lattice Boltzmann approach
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Alternative models__________________________________________________________________________________
Water retention curve from Pan et al., 2004.
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Parameter estimation
Spatial variability and scales__________________________________________________________________________________
Direct measurements and interpolation
Indirect estimation by inverse approach
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
U n it 2
U n it 9
U n it 4
U n it 1 0
U n it 7
U n it 6
U n it 1
U n it 3
B 1 B 2
B 3
0 2 4 x [m ]
4
2
0
z [m
]
6 8 1 0 1 2 1 4
8
6
4
2
0
z [m
]
0 .0 7
0 .0 8
0 .0 9
0 .1 0
0 .11
0 .1 2
U n it 2U n it 4
v [m /n s]
B 1 B 2 B 3
S o il
sand
y an
d pa
rtly
sil
ty g
rave
ls
unsa
tura
ted
satu
rate
d
Grain size. In-well Pumping
K (m/s) K(m/s) K (m/s) Nb. of meas. 318 207 20 Minimum 1.5 10-5 3.4 10-5 8.7 10-4 Maximum 0.21 1.9 10-2 3.9 10-3 Average (geo.) 1.5 10-3 1.9 10-3 2.5 10-3 Variance (LnK) 2.56 1.43 0.264
(Ptak, Teutsch, 1994)
Spatial variability and scales__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
.
...
Measurement locations
Probability distribution of indicator 1
1 0
0
0 1
1 0
Conditioning
1 0.21 0 0.13 0.03
0.52 0.48 0.32 0.15 0
0 0.66 1 0.65 0.03
0.42 1 0.84 0.53 0
Interpolation
0 1
1
0 0
0 0
Conditioning
0 0.55 1 0.82 0.75
0.02 0.32 0.28 0.66 1
0 0.12 0 0.52 0.41
0.01 0 0.22 0.31 0
Interpolation
Spatial variability and scales__________________________________________________________________________________
Probability distribution of indicator 2
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
0 0.55 1 0.82 0.75
0.02 0.32 0.28 0.66 1
0 0.12 0 0.52 0.41
0.01 0 0.22 0.31 0
.
.
.
1 0.21 0 0.13 0.03
0.52 0.48 0.32 0.15 0
0 0.66 1 0.65 0.03
0.42 1 0.84 0.53 0
Probability normalization
Pk = Pk / (Pi)
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,10,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
Faciès 5Faciès 4Faciès 3Faciès 2Faciès 1
Pro
babi
lité
Integrated density function
Spatial variability and scales__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Spatial variability and scales__________________________________________________________________________________
Experimental site in Alsace
Ksat
init (30 cm)
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Spatial variability and scales__________________________________________________________________________________
Fluxes after 8 weeks
Water Nitrate
Water Nitrate
Fluxes after 16 weeks
Water
Fluxes after 20 weeks
Nitrate
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Inverse methods__________________________________________________________________________________
Parameter identification by inverse approaches
Generalized least-square approach
j j
2 2nth nh ntθ nθ
n+1 n+1 n+1 n+1 n+1j k j j k j k j
n=0 j=1 n=0 j=1h θ
1 1ˆ ˆJ p = h p -h + θ p ,h p -θσ σ
n+1j k
V
1h p = h(z,t)dV
V n+1 n+1j k j k
V
1θ p ,h p = θ(z,t) dV
V
bn 1
yi ia y
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Inverse methods__________________________________________________________________________________
d = 3.5cm S10 C10 S9 C9
S8 C8
L = 100 cm S7 C7
S6 C6 S5 C5 S4 C4
S3 C3
S2 C2
S1 C1
e = 0.7cm C0
M (g, t)
Sable
Δh imposée
Balance
10cm
Déversoir
Capteur de pression
Sonde capacitive
Plaque poreuse
Eau
Vanne d’ouverture
Alimentation en eau
Experimental set-up
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Computed and measured variables
Inverse methods__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Parameter Init. Est. Min Max
θr (cm3 /cm3) 0.045 0.1272 0.053 0.201
θs (cm3/cm3) 0.43 0.418 0.286 0.55
α (cm-1) 0.145 0.054 0.051 0.057
n 2.68 7.85 7.50 8.20
K1s(cm/h) 29.67 16.68 12.82 20.54
KPs(cm/h) 0.004 0.0049 0.0039 0.0059
Inverse methods__________________________________________________________________________________
Parameter estimation and validation
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
First order confidence interval
1 1 1 1
1 2
( )
1 2
.. ..
.. .. .. .. .. ..
.. . .. . .. .
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. ..
k np
acn np
n n n n
k np
h h h h
p p p p
J
h h h h
p p p p
Sensitivity matrix
11TC J W J
Covariance matrix
( )k kkp J p C
Parameter uncertainty
Inverse methods__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Inverse methods__________________________________________________________________________________
Paramètres Ks Ks(P) r s n
Ks 1 -0,449 0,779 0,103 0,336 0,148
Ks(P) 1 -0,325 0,473 -0,591 0,212
r 1 -0,082 -0,018 0,543
s 1 0,162 -0,131
1 -0,707
n 1
Correlation matrix
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Inverse methods__________________________________________________________________________________
Parameters and computed variablepc,1 Yc,1
Min(J(p))
Virtual data set P, y(p)
Measurements: Ym,1 = y(p) + 1
Measurements: Ym,n = y(p) + n
Parameters and computed variablepc,n Yc,n
Min(J(p))
Measurements: Ym,i = y(p) + i
Parameters and computed variablepc,i Yc,i
Min(J(p))
Exp. Covariance matrix
First Monte Carlo approach
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Virtual data set P, y(p)
Inverse methods__________________________________________________________________________________
Measurements: Ym,1 = Yo + 1
Parameters and computed variablepc,1 Yc,1
Min(J(p))
Measurements: Ym,n = Yo + n
Parameters and computed variablepc,n Yc,n
Min(J(p))
Exp. Covariance matrix
Parameters and computed variablepc,i Yc,i
Min(J(p))
Measurements: Ym = Yo + i
Second Monte Carlo approach
ObservationsYo = y(p) +
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Ksat(cm/j)3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
r
0.084
0.086
0.088
0.090
0.092
0.094
0.096
0.098
0.100
0.102
0.104
1ère Méthode de Monte-Carlo
Méthode de Linéarisation2ième Méthode de Monte-Carlo
Comparison between 1er order and Monte Carlo Approaches
Inverse methods__________________________________________________________________________________
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
Conclusions__________________________________________________________________________________
Many challenges remain:
Understanding of processes and their mathematical modelling
Parameter scaling: from measurements to element size
Soil heterogeneity description
Accurate of numerical codes will be of great help
DYNAS 04 Workshop Ph. Ackerer, IMFS - STRASBOURG
References
Frontis Workshop on Unsaturated-Zone Modeling: Progress, Challenges and Applications, Wageningen, The Netherlands 3-5 October 2004. http://library.wur.nl/frontis/unsaturated/
Arya & Paris, Soil Sci. Soc. Am. J.,1981Binayak P. Mohanty, Water Res. Res, 1999Di Pietro et al., J. of Hydrology ,2003Pan et al., Water Res. Res., 2004Pierret et al., Géoderma, 2002Prunty & Casey, Vadose Zone J, 2002Tulle & Or, Vadose Zone J, 2002Vogel et al., Adv. Water Res., 2001Vogel & Roth, J of Hydrology, 2003Wösten, J. of Hydrology., 2001