Morteza Sadeghi and Scott B. Jones

Dept. Plants, Soils and Climate, Utah State University

Scaling Solute Transport during the Soil-Water Redistribution Process

Large amounts of chemicals are applied in agriculture, industry, and transportation for use in the topsoil.

Chemicals are transported to greater depths, leading to contamination of soils and groundwater.

*Fertilizer and Pesticide

Application

Fertilizers & Pesticides

*Accidental Chemical Spills

Accidental chemical spills

Leakage from corroded tanks

Hanford Site

Road Deicing

Nitrate risk in shallow groundwater

Fertilizers are a main source of Nitrate contamination

Arsenic concentration in groundwater

USGS: high arsenic concentration in groundwater associated with landfills and

arsenical pesticides is common.

To Manage,

Quantifying Solute Transport in soil is of

paramount importance for a wide range of

environmental and agricultural issues.

Solute Transport is one of the most complex phenomena in vadose zone!!!

cj D qc

z

mnrs

r h ])(1[

)(

2/

21

])(1[

}])(1[)(1{mn

mnn

s h

hhKK

L w w

qD D D

7/3

2ws

1R

c: solute concentration (mass-per-solvent volume)

hq K K

z

q

t z

cR j

t z

Solute flux

Ret

arda

tion

Wat

er fl

ow

VG models

Diffusivity

Tort

ousi

ty

The System is highly highly nonlinear

There are analytical solutions only for simplified

cases (simple hydraulic models, neglecting the

dispersion/diffusion process or solute reaction).

Tedious numerical calculations have to be

repeated for any soils and any initial/boundary

conditions separately.

TO overcome this complexity,

We introduce a method for scaling different

soils into a unique non-dimensional

medium so that one numerical solution of any soil

can be used for many other soils.

One scenario of interest concentration

dept

h

Solution to this case is important to manage solutes movement to avoid moving beyond the root zone for use only by plant roots.

Solutes are incorporated in irrigation water

an initial wetted zone is created Irrigation water is redistributed

carrying solutes to deeper depths.

θfiθi θ

q=0

zfi

θ = θi

z

qfi

cfici c

j=0

zfi

c = ci

z

Initial and Boundary Conditions:

Water Flow Solute Transport

* i

fi i

fiz

zz *

*

fi

qq

q

* fi

fi i fi

qt t

z

/fi

i

h

fi fi fihq Kdh z K

* i

fi i

c cc

c c

*

fi fi

J JJ

J q

We propose scaling variables as follows:

where:

Scaled water content: Scaled concentration:

Scaled depth: Scaled time:

Scaled water flux: Scaled solute flux:

* * *

* *

c J

t z

* *

* *

q

t z

* *(0, ) 0q t

* *( , ) 0t *

* **

1, 0< < 1( ,0)

0, > 1z

zz

* *( , ) 0c t

* *(0, ) 0J t

** *

*

1, 0< < 1( ,0)

0, > 1z

c zz

A scale-invariant system is obtained:

The only remaining soil-dependent variable is φ (normalized retardation/exclusion) :

i

fi i

ε: Solute reaction coefficient

θi: Initial water content at dry zone

θfi: Initial water content at wet zone

0.0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

80

90

100

loam

clay

clay loam

sandy clay

θz

(cm

)

Water content profile(t = 5 day)

HYDRUS-1D results:

Concentration profile(t = 5 day)

0.0 2.0 4.0 6.0 8.0 10.00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

c (mmol)

z (c

m)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

c*

z*

Scaled results ( φ is the same for all cases)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

*θ

z*

Scaled water content profile(t *= 5) Scaled concentration profile(t* = 5)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

c*z*

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

c*

smallest φ

(t* = 5) (t* = 10)

Effect of φ (normalized retardation/exclusion) on scaled data: i

fi i

largest φ

smallest φ

largest φ

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

40

* or c*θ

t = 65.95 d

t = 6.59 d

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

40

* or c*θ

t = 22.56 d

t = 2.25 d

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

40

* or c*θ

z (c

m)

t = 48.59 d

t = 4.85 d

(φ = 0.54)

(φ = 0.13)

(φ = 2.34)

Effect of φ:when φ > 0.5, Solute movement is slower than wetting front when φ < 0.5, Solute movement is faster than wetting front

dotted lines: water content profilessolid lines: solute concentration profiles

0.0 0.2 0.4 0.6 0.8 1.010

20

30

40

c*

z (c

m)

1370 (loam)t = 2.16 day

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

40

c*

Pima clay loamt = 7.57 day

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

60

70

80

c*

Beit Netofa clayt = 504.92 day

Approximate solutions using the proposed scaling method:

dashed lines: HYDRUS simulations solid lines: approximate solutions

0.0 20.0 40.0 60.0 80.0 100.0 120.01.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

t*

zf*

Solute front (φ = 0.1)a = 0.542b = 0.255

Wetting fronta = 0.413b = 0.299

Solute front (φ = 10)a = 0.013b = 0.407

Scaled wetting and solute front depths vs scaled time.

* *1b

fz at

An Empirical Solution for solute penetration depth:By fitting a curve to the scaled results of one case

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

wetting front

solute front ( = 0.1)φ

solute front ( = 10)φ

zf by Hydrus (cm)

zf b

y em

piri

cal e

quat

ion

/

1

fi

i

bh

fi fih

f fi

fi i fi

Kdh z Kz z a t

z

* *1b

fz at De-scaling

A new method is proposed for scaling coupled water flow and solute transport during soil water redistribution.

The scaled solutions are invariant for a wide range of soils and initial conditions when the scaled exclusion/retardation term, φ, is identical for all the cases.

The invariance of the scaled solutions provides an insight to the factors influencing solute transport.

The new method provides opportunities to easily obtain approximate solutions of the highly non-linear governing equations.

Summary & Conclusions

Future Studies

The new scaling method considers a single irrigation

event. It is worthwhile to apply such a method to

frequent applications of irrigation water to track the

solute front in long run.

So far, we have not been able to do so.

For more Details read:

Sadeghi, M., and S.B. Jones. 2012. Scaled

Solutions to Coupled Soil-Water Flow and

Solute Transport during the Redistribution

Process. Vadose Zone Journal, 11(4): -.