Some Difficulties in Modeling Water and Solute Transport in Soils Ph. ACKERER IMFS STRASBOURG

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.


0.36 km2, 1000-750 m

Contact: Bruno AMBROISE Contact: Bruno AMBROISE (IMFS)(IMFS)

Hillslope hydrology

The Ringelbach catchment


Saturated area

Discharge

(from B. Ambroise, IMFS)


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


(from UMR LISAH, Montpellier)

Usual concepts and mathematical models__________________________________________________________________________________

Model concept


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__________________________________________________________________________________


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


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__________________________________________________________________________________


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__________________________________________________________________________________


Robbez-Masson, UMR LISAH, Montpellier


Macropores in un-colonised and colonised soil (from Pierret et al., 2002)


Hierarchy of flow/transport models for variably-saturated structured media (after Altman et al., 1996)


From Tuller & Or, 2001


New mathematical models

Richard’s equation with alternative h() and K()

Network models

Alternative models

Some recent concepts__________________________________________________________________________________


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()__________________________________________________________________________________


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




Pore-scale models (Tuller & Or, 2002)



(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)


Pedotransfer functions (Wösten, 2001)




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


Kinematic–dispersive wave model (Di Pietro et al., 2003)

Network models __________________________________________________________________________________


From Pan et al., 2004

Alternative models__________________________________________________________________________________

Two-phase flow using Lattice Boltzmann approach


Alternative models__________________________________________________________________________________

Water retention curve from Pan et al., 2004.


Parameter estimation

Spatial variability and scales__________________________________________________________________________________

Direct measurements and interpolation

Indirect estimation by inverse approach


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)



.

...

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


Probability distribution of indicator 2


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




Experimental site in Alsace

Ksat

init (30 cm)



Fluxes after 8 weeks

Water Nitrate

Water Nitrate


Water


Nitrate


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


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


Computed and measured variables

Inverse methods__________________________________________________________________________________


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


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__________________________________________________________________________________


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


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


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) +


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__________________________________________________________________________________


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


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

Documents

Some Difficulties in Modeling Water and Solute Transport in Soils Ph. ACKERER IMFS STRASBOURG