prediction of contaminant plumes (shapes, spatial moments and macro-dispersion) in aquifers with...
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A.M.M. Elfeki Section Hydrology and Ecology, Dept. of Water
Management, Environmental and Sanitary Engineering, TU Delft, P.O. Box 5048, 2600 GA Delft,
The Netherlands.
Prediction of Contaminant Plumes (Shapes, Spatial Moments and Macro-
dispersion) in Aquifers with Insufficient Geological Information
Outline
• Markov Chain in One-dimension.• Markov Chain in Multi-dimension:
Coupled Markov Chain (CMC) Model. • Application of CMC to Characterize heterogeneity at
the MADE1 site:- Parameter Estimation.- Sensitivity Analysis.
• Hydrogeological conditions at MADE1.• Model Assumptions.• Simulation Results.• Conclusions.
One-dimensional Markov Chain
w p kN
lkN
)(lim
,:)Pr()Pr(
p S Z | S ZS Z ,..., S Z ,S Z ,S Z | S Z
lkl1-iki
p0r3-in2-il1-iki
)(Pr 1 qNliki S Z ,S Z | S Z
)1(
)(
1 )(Pr
iNlq
lkiN
kqqNliki p
pp S Z ,S Z | S Z
S So d
Coupled Markov Chain “CMC” in 2D
Dark Grey (Boundary Cells)Light Grey (Previously Generated Cells)W hite (Unknown Cells)
i-1 ,j i,ji,j-1
1 ,1
N x ,N y
N x ,1
1 ,N y
N x ,j
nkp . p
p . p SZSZSZ p
f
vmf
hlf
vmk
hlk
mjiljikjiklm ,...1.),|Pr(: 1,,1,,
.,...1,
),,|Pr(:
)(
)(
,1,,1,,
nkp . p . p
p .p . p
SZSZSZSZp
f
vmf
iNhfq
hlf
vmk
iNhkq
hlk
qjNmjiljikjiqklm
x
x
x
Procedure for Extracting a Final Geological Image
.0
1)(
otherwise
SZifZI
kij
ijk
MC
R
Rk
kR
kij ji
ji ZIMCMC
SZ
1
)()(
,
, 1}{#
}...,,max{ 21 nijijij
lij
Let the realizations be numbered 1,…, MC, and let Zij (R) be the
lithology of cell (i,j) in the Rth realization. The empirical relative frequency of lithology Sk at cell (i,j) is:
In the final image Z* the lithology at cell (i, j) will be the lithology which occurs most frequently in the MC realizations. So, if Sl is such that
Zij*= Sl.
MADE Site Data
0 50 100 150 200 250
-10
-5
0
0
1
2
3
4
5
Estimation of Vertical Transition Probability
0 50 100 150 200 250
-10
-5
0
0
1
2
3
4
5
S. 1 2 3 4 5 6 7
1 .879
.103
.009
.000
.009
.000
.000
2 .026
.911 .046
.009
.003
.000
.005
3 .003
.030
.897
.044
.010
.000
.016
4 .000
.006
.094
.869
.031
.000
.000
5 .000
.000
.003
.010
.961
.000
.026
6 .009
.014
.009
.005
.000
.963
.000
7 .000
.000
.000
.000
.000
.000
1.00
Vertical Sampling Interval=0.1 m
T
T = pvlq
n
q
vlkv
lk
1
Tlkv is the number of observed
transitions from Sl to Sk in the vertical direction.
Horizontal Transition Probability MatricesS. 1 2 3 4 5 6 7
1 .500
.100
.100
.100
.100
.100
.000
2 .100
.500
.100
.100
.100
.000
.100
3 .100
.100
.500
.100
.100
.000
.100
4 .100
.100
.100
.500
.100
.000
.100
5 .100
.100
.100
.100
.500
.100
.000
6 .001
.001
.001
.001
.001
.994
.001
7 .001
.001
.001
.001
.001
.001
.994
S. 1 2 3 4 5 6 7
1 .879 .103 .009 .000 .009 .000 .000
2 .026 .911 .046 .009 .003 .000 .005
3 .003 .030 .897 .044 .010 .000 .016
4 .000 .006 .094 .869 .031 .000 .000
5 .000 .000 .003 .010 .961 .000 .026
6 .009 .014 .009 .005 .000 .963 .000
7 .001 .001 .001 .001 .001 .001 .994
S. 1 2 3 4 5 6 7
1 .922
.015
.015
.015
.015
.015
.003
2 .015
.922
.015
.015
.015
.015
.003
3 .015
.015
.922
.015
.015
.015
.003
4 .015
.015
.015
.922
.015
.015
.003
5 .015
.015
.015
.015
.922
.015
.003
6 .015
.015
.015
.015
.015
.922
.003
7 .001
.001
.001
.001
.001
.001
.994
Probability Maps under Different Transition Probabilities
12345a
- 1 0
- 5
0
c- 1 0
- 5
0
d
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
b
- 1 0
- 5
0
e
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
- 1 0
- 5
0
0.000.250.500.751.00
LITH O LO G Y 1
LITH O LO G Y 2
LITH O LO G Y 3
LITH O LO G Y 4
LITH O LO G Y 5
LITH O LO G Y
Probability
Comparison between Single Realizations and Final Images
0 50 100 150 200 250
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
1
2
3
4
5
Effect of Horizontal Transition Probability
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0 1
2
3
4
5
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
a
b
c
d
e
Effect of Number of Boreholes on Site Characterization
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
1
2
3
4
50 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
Comparison of the Proportion of Each Lithology
Conditioned on Number of Boreholes Lithology 16 Boreholes 9 Boreholes 6 Boreholes
1 0.046 0.038 0.0122 0.258 0.207 0.205
3 0.327 0.380 0.479
4 0.097 0.109 0.141
5 0.273 0.267 0.163
Hydraulic Conductivity Assigned to Facies (Lithofacies Definitions from Rehfeldt, et al. 1992 and Lithofacies
Values from Adams and Gelhar, 1992).
Facies Measured Conductivity (m/day) lower limit upper limit mid range
1. Open work gravel 86.4 864. 475.2
2. Fine gravel 8.64 86.4 47.52
3. Sand 0.864 8.64 4.752
4. Sandy gravel 0.0864 0.864 0.4752
5. Sandy clayey gravel 0.00864 0.0864 0.04752
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
b
1
2
3
4
5
Comparison of Statistics Computed from the Three Scenarios (with upper, lower and mid range
conductivities) and the values estimated from the MADE site (Rehfeldt et al., 1992)
ln( )K
Parameter
16 Boreholes 9 Boreholes 6 Boreholes MADE dataRehfeldt, et al., [1992].
-5.28 (Upper)-5.87 (Mid Range)-7.68 (Lower)
-5.44 (Upper)-6.03 (Mid Range) -7.43 (Lower)
-5.15 (Upper)-5.75 (Mid Range)-7.47 (Lower)
-5.2(-10.1 - 0.4)
8.19 (Upper)8.19 (Mid Range)7.31 (Lower)
7.43 (Upper)7.43 (Mid Range)7.43(Lower)
5.25 (Upper)5.25 (Mid Range)5.03 (Lower)
4.5(3.4 - 5.6)2
ln( )K
Variogram Analysis of Images
0 20 40 60 80Spatial Lag (m )
39.80
79.80
119.80
159.80
199.80
Vrai
ogra
m L
n(K)
in T
he V
ertic
al D
irect
ion
0 20 40 60 80Spatial Lag (m )
39.80
79.80
119.80
159.80
199.80
Vrai
ogra
m L
n(K)
in T
he H
oriz
onta
l Dire
ctio
n
16 boreholes (Table 4 and Table 1)9 boreholes (Table 4 and Table 1)6 boreholes (Table 4 and Table 1)16 boreholes (Table 3 and Table 1)9 boreholes (Table 3 and Table 1)6 boreholes (Table 3 and Table 1)16 boreholes (Table 2 and Table 1)9 boreholes (Table 2 and Table 1)6 boreholes (Table 2 and Table 1)
Comparison of various Models with CMC
0 50 100 150 200 250
-10
-5
0
0
1
2
3
4
5
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
MADE Site Data
0 50 100 150 200 250
-10
-5
0
0
1
2
3
4
5
Flow and Transport Models Assumptions
• 2D-Vertical Cross-Section.• Steady State Flow System:
(Seasonal Variability is Negligible).• Confined Aquifer Conditions.• Non-reactive Transport (Bromid Tracer).• Molecular Diffusion is Negligible.
Simulation Parameters
Parameter Numerical ValueTime step 0.5 [day]Longitudinal dispersivity 0.1 [m]Transverse dispersivity 0.01 [m]Effective porosity 0.35 [-]Injected tracer mass 2500 [grams]Head difference at the site 0.7 [m]Gradient 0.0025 [-] Number of particles 1000,000 [particles]K (Open work gravel ) 864. [m/day]K (Fine gravel) 86.4 [m/day]K (Sand) 8.64 [m/day]K(Sandy gravel ) 0.864 [m/day]K(Sandy clayey gravel ) 0.0864 [m/day]
Simulation of MADE1Experement under Different Values of Conductivities
0 0.1 1 10 100
-10
-5
0
1 2 3 4 5
-10
-5
0
-10
-5
0
49 days
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
594 days
-10
-5
0
-10
-5
0
49 days
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
594 days
-10
-5
0
-10
-5
0
49 days
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
594 days
Lithology Coding
Concentration Scale (m g/L)
Effect of Number of Boreholes on the Simulated Plume Upper conductivity
49 days
0 50 100 150 200 250
279 days
0 50 100 150 200 250594 days
0 50 100 150 200 250
49 days49 days
279 days279 days
594 days594 days
1 2
3
45
6
7 8 9 10
11
12
13
14
15
161
3
5 7 9
11 13 15
16161 5 7
11 13
-10
-5
0
-10
-5
0
0 0.1 1 10 100
-10
-5
0
-10
-5
0
49 days
0 50 100 150 200 250
-10
-5
0
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
594 days
-10
-5
0
-10
-5
0
-10
-5
0
0 50 100 150 200 250
-10
-5
0
-10
-5
0
49 days49 days
279 days279 days
594 days594 days
Effect of Number of Boreholes on the Simulated PlumeLower Conductivity
-10
-5
0
-10
-5
0
0
0.1
1
10
100
-10
-5
0
-10
-5
0
-10
-5
0
49 days
0 50 100 150 200 250
-10
-5
0
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
594 days
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
0 50 100 150 200 250
-10
-5
0
-10
-5
0
49 days49 days
279 days279 days
594 days594 days
Effect of Number of Boreholes on the Simulated Plume Mid Range Counductivity
-10
-5
0
-10
-5
0
0
0.1
1
10
100
-10
-5
0
-10
-5
0
-10
-5
0
49 days
0 50 100 150 200 250
-10
-5
0
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
594 days
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
-10
-5
0
0 50 100 150 200 250
-10
-5
0
-10
-5
0
49 days49 days
279 days279 days
594 days594 days
Comparison between Measured and Simulated Mean Displacement
Comparison between Measured and Simulated Longitudinal Variance
Effect of Number of Boreholes on Lateral Variance
Effect of Number of Boreholes on The Angle of Rotation of The Plume
Effect of Number of Boreholes on Longitudinal Marco-Dispersion
Effect of Number of Boreholes on Lateral Marco-Dispersion
Effect of Number of Boreholes on The Breakthrough Curves
Observed, Predicted and Simulated Macro-dispersivities (with upper and mid range conductivities)
Dispersivity(m)
Observed Adams& Gelhars[1992]
This study(16 boreholes)
This study(9 boreholes)
This study(6 boreholes)
A11 5-10 1.48 -1.5 13.25 (Upper)4.3 (Mid Range)
8.75 (Upper)4.7 (Mid Range)
16.18 (Upper4.5 (Mid Range)
A33 not computed < 0.005 0.005 (Upper)~ 0 (Mid Range)
0.003 (Upper) ~ 0 (Mid Range)
0.0016 (Upper)0.0015 (Mid Range)
Conclusions
1. The CMC model has shown successful results in delineating the complex geological configuration of the aquifer at the MADE1 site.
2. Piih = 0.922 produces the main heterogeneous features in the
site when conditioned on 16 boreholes.
3. Flow and transport simulations capture the salient features of the flow field and the large scale plume behaviour at the site, although some assumptions are made.
4. Reducing the number of Conditioned boreholes from 16, 9 to 6 still shows reasonably the same plume behaviour in terms of average longitudinal and vertical extensions specially in the far-field. This gives more reliability on the use of CMC model for predicting plume configuration at field sites.
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