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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
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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.
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
volatilisation
atmogenicinput
'DNAPL'
waterworks
punctual input
diffusionlow permeable regions
advectiondispersionretardation
contaminant releasedissolution - desorption
'LNAPL'
sorption
(after Schüth 94)
max. plume length ?safe location?
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
(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)
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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).
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
Outline• Problem description• Previous approaches• Derivation of the formula• Sensitivity analysis and applications• Conclusions
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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)
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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
[%
]
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
Outline• Problem description• Previous approaches• Derivation of the formula• Sensitivity analysis and applications• Conclusions
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
Sensitivity analysis I
-1 -0.5 0 0.5 1 1.5 2relative sensitivity coefficient (-)
aquifer thickness
verticaltransversedispersivity
acceptorconcentration
donorconcentration
stoichiometriccoefficient
longitudinaldispersivity
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
volatilisation
atmogenicinput
'DNAPL'
waterworks
punctual input
diffusionlow permeable regions
advectiondispersionretardation
contaminant releasedissolution - desorption
'LNAPL'
sorption
(after Schüth 94)
max. plume length ?
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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)
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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)
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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)
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
volatilisation
atmogenicinput
'DNAPL'
waterworks
punctual input
diffusionlow permeable regions
advectiondispersionretardation
contaminant releasedissolution - desorption
'LNAPL'
sorption
(after Schüth 94)
safe location?
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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)
Center of Applied Geoscience (ZAG)Hydrogeophysics - Geomathematics
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.