A Simple Formula to Estimate the Maximum Contaminant Plume Length · 2007. 4. 9. · Center of Applied Geoscience (ZAG) Hydrogeophysics - Geomathematics A Simple Formula to Estimate

Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics

A Simple Formula to Estimate theMaximum Contaminant Plume

Length

P. Dietrich (1), A. J. Valocchi (2), R. Liedl (1) , P. Grathwohl (1)

(1) Center for Applied Geoscience, University of Tuebingen (2) Dept. of Civil and Environmental Eng., Univ. of Illinois, Urbana-Champaign


Outline• Problem description• Previous approaches• Derivation of the formula• Sensitivity analysis and applications• Conclusions

More on this topic will be presented by David Lerner, Olaf Cirpka and Uli Maierin their talks tomorrow.


volatilisation

atmogenicinput

'DNAPL'

waterworks

punctual input

diffusionlow permeable regions

advectiondispersionretardation

contaminant releasedissolution - desorption

'LNAPL'

sorption

(after Schüth 94)

max. plume length ?safe location?


(after Teutsch and Rügner, 1999)

Temporal development of plumes

plume shrinks(due to decreasing emission from source)

t1

t3

t4

sourceflow direction

t2

contaminant plume develops

starting point: contaminant source

plume growth slows down

plume approaches steady state(release from source = contaminant degradation)


Previous approaches• Numerical simulations based on first-order degradation

or Monod kinetics• Analytical solutions are mostly based on “Domenico-

like” approaches (involving exponential and error functions).

• Modelled contaminant concentrations never are exactly equal to zero (“infinite plume”).

• Some threshold concentration has to be defined in order to “make modelled plumes finite”.

Problems


Previous approaches• Numerical simulations based on first-order degradation

or Monod kinetics• Analytical solutions are mostly based on “Domenico-

like” approaches (involving exponential and error functions).

• Ham et al. (2004) were the first to employ a sharp reaction front (2D horizontal, infinite domain).




Model assumptions

z = M

z

xx = L0

source ofelectron

donor(concentr. cD

0)

source of electron acceptor (concentr. cA0)

steady-statereaction

front

impervious layer

uniform flow field(dispersivities αL, αT)

γ = stoichiometric ratio [-] (= number of moles of acceptor needed to annihilate 1 mole of donor)


Why are model assumptionsappropriate to estimate

the maximum plume length?

• 3D instead of 2D-> decrease of plume length

• consideration of biodegradation in the plume -> decrease of plume length

• source zone does not extend over entire aquifer thickness-> decrease of plume length


Mathematical solution

( ) ( )[ ]00

0

1

211211

4121

222

AD

A

n

Mn

n

ccce

nL

TL

+=

−

−∑∞

=

−−+−−

γπ

ααα

π

Implicit representation of plume length L:

Liedl et al., 2005submitted to WRR

Explicit formula (keeping only first term in infinite series):

+

−+=

0

00

22

14ln

11

2A

AD

TL

L

ccc

M

L γπαα

π

α

L


Sufficiency of the term L1

1x10-5 1x10-4 1x10-3 1x10-2

αLαT/M²

0.01

0.1

1

10

devi

atio

n of

L1 f

rom

L25

[%]

0.5

0.6

0.7

0.8

0.9

cA0/(γcD0+cA0) =0.95


Elimination of longitudinal dispersivity

Explicit formula:

+

=∗0

00

2

2

14ln4

A

AD

T cccML γ

παπ

Neglecting longitudinal dispersion:

Liedl et al., 2005submitted to WRR

+

++=

+

−+

= 0

00

22

2

2

0

00

22

14ln1124ln

11

2A

ADTL

TA

AD

TL

L

ccc

MM

ccc

M

L γπ

ααπ

απγ

πααπ

α

≈ 2


Impact of longitudinal dispersivity

1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 1x10-2

αLαT/M²

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

1x100

(L1 -

L1* )/L 1

[%

]




Sensitivity analysis I

-1 -0.5 0 0.5 1 1.5 2relative sensitivity coefficient (-)

aquifer thickness

verticaltransversedispersivity

acceptorconcentration

donorconcentration

stoichiometriccoefficient

longitudinaldispersivity


volatilisation

atmogenicinput

'DNAPL'

waterworks

punctual input




'LNAPL'

sorption

(after Schüth 94)

max. plume length ?


Application of formula I

1x10-5 1x10-4 1x10-3 1x10-2 1x10-1

αT /M2 (1/m)

1x100

1x101

1x102

1x103

1x104

L 1* (

m)

0.50.1

0.01 0.001

cA0/(γcD0+cA0) =

0.30.7

0.0001

Estimation of max. plume length

+= 0

00

2

2 4ln4A

AD

T cccML

γ

παπ

+= 0

00

2

2 4ln4A

AD

T cccML

γ

παπ

0.13 (e.g. Maier, 2004: cNH4

+ = 15 mg/L, cO2

= 8 mg/L, γ = 3.5)


Sensitivity analysis II

1x10-5 1x10-4 1x10-3 1x10-2 1x10-1

αT /M2 (1/m)

1x100

1x101

1x102

1x103

1x104

L 1* (

m)

0.50.1

0.01 0.001

cA0/(γcD0+cA0) =

0.30.7

0.0001

Thickness of aquifer

0.13 (e.g. Maier, 2004: cNH4+ = 15 mg/L,

cO2 = 8 mg/L, γ = 3.5)


Sensitivity analysis III

1x10-5 1x10-4 1x10-3 1x10-2 1x10-1

αT /M2 (1/m)

1x100

1x101

1x102

1x103

1x104

L 1* (

m)

0.50.1

0.01 0.001

cA0/(γcD0+cA0) =

0.30.7

0.0001

Transverse dispersion

0.13 (e.g. Maier, 2004: cNH4+ = 15 mg/L,

cO2 = 8 mg/L, γ = 3.5)


volatilisation

atmogenicinput

'DNAPL'

waterworks

punctual input




'LNAPL'

sorption

(after Schüth 94)

safe location?


1x10-5 1x10-4 1x10-3 1x10-2 1x10-1

αT /M2 (1/m)

1x100

1x101

1x102

1x103

1x104

L 1* (

m)

0.50.1

0.01 0.001

cA0/(γcD0+cA0) =

0.30.7

0.0001

Application of formula IIRisk assessment

+= 0

00

2

2 4ln4A

AD

T cccML

γ

παπ

+= 0

00

2

2 4ln4A

AD

T cccML

γ

παπ

0.13 (e.g. Maier, 2004: cNH4+ = 15 mg/L,

cO2 = 8 mg/L, γ = 3.5)


Conclusions• Steady-state plumes are finite if there is a binary reaction

between electron acceptor and donor.• As opposed to other analytical concepts, our modelling

approach involves a sharp reaction front in a vertical model domain.

• The maximum plume length can be estimated based on an easy-to-use formula.

• The actual plume length could be shorter due to a smaller extent of the source, the three-dimensional plume geometry or biodegradation occurring inside the plume.

• The plume length is actually most sensitive to aquifer thickness, but values for transverse dispersivities are much more uncertain.

A Simple Formula to Estimate the Maximum Contaminant Plume Length · 2007. 4. 9. · Center of Applied Geoscience (ZAG) Hydrogeophysics - Geomathematics A Simple Formula to Estimate

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