BackgroundIs the dispersion coefficient of sorbing
solutes modified by retardation ? Inverse
modeling with HYDRUS-1D
V. Pot, P. Benoit, M. Le Menn

Solute transport models
based on the Convection Dispersion Equation (CDE)
TracerθSorbing solute∂c∂c∂ 2c= θ DLns 2 − q∂z
∂t
∂z

θ∂c
∂s∂c
∂ 2c
= θ DLs 2 − q − ρ∂z∂t∂t∂zs = kd c

UMR INRA AgroParisTech Environnement and Arable Crops

Equilibrium sorption
Retardation R=1+ρ kd/ θ∂s= α k (k d c − s ) Kinetic sorption∂t78850 Thiverval-Grignon France

O.M. Eklo, T. Sveistrup, J. Kværner
Norwegian Institute for Agricultural and Environment Research,
1432 Ås Norway
Classical hypothesis:
HYDRUS workshop – Prague - 28th march 2008

D Ls = D Lns
HYDRUS workshop – Prague - 28th march 2008

Background

Background

Taylor, 1953Aris, 1959

For viscous Poiseuille flow in cylindrical tubes DLns is:. effect of molecular diffusion in the liquid phase. effect of velocity variations with the radius of the tube.

Expression of Taylor dispersion for sorbing solutes
Retardation (equilibrium sorption) affects the value of DLs andIncreases it as compared to tracer for cylindrical pores.

Expression of dispersion coefficient that upscales the two effects
for asymptotic times: Taylor dispersion. It is a Fickian process.

D Ls = Dm + κ 1

DLns = Dm + κU 2a 2
DmU 2a2Dm⎛⎝

κ1 = κ ⎜11 −16 6 ⎞+⎟R R2 ⎠

κ = 1/48

HYDRUS workshop – Prague - 28th march 2008

This phenomenon is named: Fickian sorption-enhanced dispersion

HYDRUS workshop – Prague - 28th march 2008

1

From tube to natural fractures

From natural fractures to porousmedia?

Solute transport solved by
Lattice-Gas cellular automaton model
2,3 mm
1,2 mmIs the dispersion coefficient of sorbing solutes still modified by retardationin the case of the complicated geometry of porous media?

bmaxb(Harvey & Marsily, 1995)

velocity profiles are unknown in pores due to their highly varied nature
Pot & Genty, Transp Porous Media, 2005
Pot & Genty, Adv. in Water Resour., 2007
HYDRUS workshop – Prague - 28th march 2008

HYDRUS workshop – Prague - 28th march 2008

Experimental design

Transport models – HYDRUS-1DPhysical and chemical equilibrium

Sandy loam soil (Grue, Southern Norway)θ∂ 2c∂c∂s∂c= θ DLs 2 − q − μ l θ c+ρ∂z∂t∂t∂zs = kd c

CDE I model: (kd, μl)s
CDE II model: (D L ,kd, μl)

Physical equilibrium and chemical non-equilibrium

ht = -10 cm
hb = -10 cm

HYDRUS workshop – Prague - 28th march 2008

∂s
∂ 2c
∂c
∂c
θ + ρ f e e = θ DLs 2 − q − μ l θ c − ρα k [(1 − f e )k d c − s k ]∂t∂t∂z∂zs = se + s k
Two-site I model: (k , μ , f , α )∂s k= α k [(1 − f e )k d c − s k ]∂tdslekTwo-site II model: ( DL , kd, μl, fe, αk)
HYDRUS workshop – Prague - 28th march 2008

2Results: Bromide Transport

Results: Metribuzin transport

Physical equilibrium transport

HYDRUS workshop – Prague - 28th march 2008

Results: Metribuzin transportN∑ (P − O )E L2 = i =1 Ni

HYDRUS workshop – Prague - 28th march 2008

Results: Metribuzin transport

2i∑ (O )
i =12iPi are the model-simulated data
Oi are the measured data
N is the number of observations

HYDRUS workshop – Prague - 28th march 2008

Kd = 0.28 L/Kg
Batch kd = 0.36 L/Kg
DT50 = 4 days
Batch DT50 = 26 days

Kd = 0.13 L/Kg
Batch kd = 0.14 L/Kg

Kd = 0.15 L/Kg
Batch kd = 0.42 L/Kg

CDE II model

D Ls ≈ 3 D Lns

Two-site II modelDLs ≈ 2 D Lns

R≈2

HYDRUS workshop – Prague - 28th march 2008

3Results: Metribuzin transport
Two-site I modelTwo-site II modelfeαk (/h)

feαk (/h)

Grue A1

0.42 ± 0.01

0.10 ± 0.01

0.59 ± 0.07

0.08 ± 0.01

Grue B1

0.28 ± 0.02

0.12 ± 0.02

0.48 ± 0.05

0.07 ± 0.03

Grue C30.02 ± 0.01

0.73 ± 0.14

0.09 ± 0.06

1.06 ± 0.56

HYDRUS workshop – Prague - 28th march 2008

Conclusions• Physical equilibrium for the sandy loam soil
• Non-equilibrium two-site sorption combined with Fickian sorptionenhanced dispersion (Two-site II model): A and B horizons.
• Fickian sorption-enhanced dispersion (CDE II model): C horizon.
• Role of other non-equilibrium processes not included in the models
we used: - slow diffusion of metribuzin to the sorption places- intraparticle diffusion
higher effective dispersion• Need of more theoretical and modelling studies to elucidate thepotential dependency relation of a Fickian form between dispersion
and retardation in porous medium.
HYDRUS workshop – Prague - 28th march 20084