Reducing Uncertainty of Groundwater Contaminant Transport Using Markov Chains
Reducing Concentration Uncertainty Using the Coupled Markov Chain Approach
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Reducing Concentration Uncertainty Using The Coupled Markov Chain Approach Amro Elfeki Section Hydrology, Dept. of Water Management, TU Delft, The Netherlands.
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Transcript of Reducing Concentration Uncertainty Using the Coupled Markov Chain Approach
Reducing Concentration UncertaintyUsing
The Coupled Markov Chain Approach
Amro Elfeki Section Hydrology,
Dept. of Water Management, TU Delft,
The Netherlands.
Outlines• Motivation of this research.• Methodology: • Markov Chain in One-dimension.• Markov Chain in Multi-dimensions: Coupled Markov Chain (CMC). • Application of CMC at the Schelluinen study area (Bierkens,
94).• Comparison between:
CMC (Elfeki and Dekking, 2001) and SIS (Sequential Indicator Simulation, Gomez-Hernandez and Srivastava, 1990) .
• Flow and Transport Models in a Monte-Carlo Framework.• Geostatistical Results.• Transport Results.
• Conclusions.
Motivation and IssuesMotivation of this research:
• Test the applicability of CMC model on field data at many sites.
• Incorporating CMC model in flow and transport models to study uncertainty in concentration fields.
• Deviate from the literature: - Non-Gaussian stochastic fields: (Markovian fields), - Statistically heterogeneous fields, and - Non-uniformity of the flow field (in the mean) due to boundary conditions.
Geological and Parameter Uncertainties
Unconditional CMC
1 2 3 4
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
tim e = 1 6 0 0 d ay s
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 50 100 150 200 250 300
-40
-20
0
0 50 100 150 200 250 300
-40
-20
0
G eology is Certa in and Param eters are U ncerta in
G eology is Uncerta in and Param eters are Certa in
0 0.01 0.1 1
C
C
actualC
C
C
Elfeki, Uffink and Barends, 1998
Geological Uncertainty: Geological configuration.
Parameter Uncertainty: Conductivity value of each unit.
( )
Markov property (One-Step transition probability)Pr( )Pr( ) : ,
Marginal Distribution
lim
Conditioning on the Fut
N
i i-1 i-2 i-3 0k l n pr
i i-1k l lk
Nklk
| , , S ,..., S S S SZ Z Z Z Z | pS SZ Z
p w
( )
1 ( 1)
ure
Pr ( )N i
kq lki i Nk l q N i
lq
p p | , S S SZ Z Z
p
S So d
One-dimensional Markov Chain
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
, , 1, , 1
, 1, , 1 ,,
Unconditioinal Coupled Markov Chains
: Pr( | , ) . 1,...
Conditioinal Coupled Markov Chains: Pr( | , , )
x
h vlk mk
lm k i j k i j l i j m h vlf mf
f
i j k i j l i j m N j qlm k q
hlk
. p pp Z S Z S Z S k n
. p p
p Z S Z S Z S Z S
.p
( )
( ) , 1,... .x
x
h N i vkq mk
h h N i vlf fq mf
f
. p pk n
. . p p p
Coupled Markov Chain “CMC” in 2D
(Elfeki and Dekking, 2001)
CMC vs. Conventional Methods
CMC Conventional Methods
Based on conditional probability (transition matrix).
Based on variogram or autocovariance.
Marginal Probability. Sill.
Asymmetry can be described.
Asymmetry is impossible to describe.
A model of spatial dependence is not necessary.
A model of spatial dependence is needed for implementation.
Compute only the one-step transition and the model takes care of the n-step transition probability.
Need to compute many lags for the variogram or auto-correlations. (unreliable at large lags)
Schelluinen study area, The Netherlands
Soil Coding
Soil description
1 Channel deposits (sand)
2 Natural levee deposits (fine sand, sandy clay, silty clay)
3 Crevasse splay deposits (fine sand, sandy clay, silty clay)
4 Flood basin deposits (clay, humic clay)
5 Organic deposits (peaty clay, peat)
6 Subsoil (sand)
0 80 160 240-10
-5
0
0 200 400 600 800 1000 1200 1400 1600-10
-5
0
1 2 3 4 5 6
Data from Bierkens, 1994
Parameter Estimation and Procedure
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
G eo lo g ica l Im a g e
D o m ain D iscretiza tio n
G en era ted R ea liza tion
0 50 100 150 200-10
-5
0
S u p erp o sitio n o f th e G rid o ver th e G eo lo g ica l Im a ge a n d E stim ation o f T ran sition P rob a b ility
B o reh o les L ocatio n s0 50 100 150 200
-10
-5
0
P a ra m eters E stim a tio n C o n d itio n a l S im u la tio n
1
vv lklk n
vlq
q
TpT
Horizontal transition probability matrix of 1650 m section calculated over sampling intervals of 25 m.
Soil 1 2 3 4 5 61 0.979 0.004 0.001 0.006 0.009 0.0012 0.020 0.965 0.001 0.008 0.006 0.0003 0.003 0.002 0.966 0.013 0.016 0.0004 0.000 0.001 0.009 0.983 0.007 0.0005 0.001 0.001 0.006 0.007 0.984 0.0016 0.000 0.000 0.001 0.000 0.002 0.997
Vertical transition probability matrix 1650 m section calculated over sampling intervals of 0.25 m.
Soil 1 2 3 4 5 61 0.945
0.000
0.009
0.000
0.009
0.037
2 0.071
0.796
0.021
0.041
0.071
0.000 3 0.000 0.000 0.797 0.086 0.089 0.028
4 0.003
0.013
0.041
0.714
0.222
0.007 5 0.004
0.012
0.047 0.119
0.768
0.050
6 0.000 0.000 0.000 0.000 0.000 1.000
Transition Probabilities (1650 x10 m)
Transition Probabilities (240 x10 m)
Horizontal transition probability matrix Vertical transition probability matrix
State 3 4 5 6 State 3 4 5 6
3 0.979 0.010 0.011 0.000 3 0.969 0.027 0.004 0.000
4 0.011 0.974 0.015 0.000 4 0.008 0.724 0.268 0.000
5 0.008 0.120 0.977 0.003 5 0.025 0.139 0.791 0.045
6 0.010 0.000 0.007 0.983 6 0.000 0.000 0.000 1.000
0 80 160 240-10
-5
0 3
4
5
6
Sampling intervalsDx = 2 m
Dy= 0.25 m
0.966 0.013 0.016 0.000
0.009 0.983 0.007 0.000
0.006 0.007 0.984 0.001
0.001 0.000 0.002 0.997
0.797 0.086 0.089 0.028
0.041 0.714 0.222 0.007
0.047 0.119 0.768 0.050
0.000 0.000 0.000 1.000
Horizontal Transition Probability from 1650x10
Vertical Transition Probability from 1650x10
Parameter Numerical ValueTime step 5 [day]Longitudinal dispersivity 0.1 [m]Transverse dispersivity 0.01 [m]Effective porosity 0.30 [-]Injected tracer mass 1000 [grams]Head difference at the site 1.0 [m]Monte-Carlo Runs 50 MCNumber of particles 10,000 [particles]
Physical and Simulation ParametersSoil Properties at the core scale from Bierkens, 1996 (Table 1).
Soil Coding
Soil type Wi
6 Fine & loamy sand 0.12 0.60 1.76 4.40 0.09
5 Peat 0.39 -2.00 1.7 0.30 2.99
3 Sand & silty clay 0.19 -4.97 3.49 0.1 5.86
4 Clay & humic clay 0.30 -7.00 2.49 0.01 10.1
2( )iLog K( )iLog K ( / )iK m day 2
iK
Convergence:~14000 IterationsAccuracy 0.00001
( , ) ( , ) 0
( , )
( , )
x
y
K x y K x yx x y y
K x yV x
K x yV y
Flow Model
Contam inant Source
P lum e at T im e, t
Im perm eable boundary
Im perm eable boundary
is the hydraulic head, Vx and Vy are pore velocities, is the hydraulic conductivity, and is the effective porosity.
( , )K x y
Hydrodynamic Condition: Non-uniform Flow in the Mean due to Boundary Conditions.
Transport Model
Governing equation of solute transport :
C is concentration Vx and Vy are pore velocities, and Dxx , Dyy , Dxy , Dyx are pore-scale dispersion coefficients
x y xx xy yx yyC C C C C C CV V D D D Dt x y x x y y x y
* - i jmij ijL L TVV
D V DV
*mDij
L
T
is effective molecular diffusion,
is delta function,is longitudinal dispersivity, andis lateral dispersivity.
1 1
1 1
cos sin sin cos
. / . / . / . /
n n n np p x p p yL T L T
n n n np p x x y p p y y xL T L T
X X V t Z Z Y Y V t Z Z
X X V t Z V V Z V V Y Y V t Z V V Z V V
dispersive termadvective term
1 22 2xy yxx xp p x L T
D VD VX t t X t V t Z V t Z V tx y V V
1 22 2yx yy y xp p y L T
D D V VY t t Y t V t Z V t Z V tx y V V
The displacement is a normally distributed random variable, whose mean is the advective movement and whose deviation from the mean is the dispersive movement.
instantaneous injection + uniform flow
Random Walk Method
Effect of Conditioning on S. R. Plume
mg/lit
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0
0.1
1
10
3 4
Litho logy C oding
6 5
T= 82 years
# drillings
2
3
5
9
25
31
Effect of Conditioning Single Realiz. Cmax
0 4 8 12 16 20 24 28 32No. of Conditioning Boreholes
0
40
80
120
160
200
240
Pea
k C
once
ntra
tion
(mg/
lit)
Single Realization C m ax (t = 34.2 Years)S ingle Realization C m ax (t = 68.4 Years)S ingle Realization C m ax (t = 95.8 Years)S ingle Realization C m ax (t = 136.9 Years)O rig inal Section (t = 34.2 Years)O rig inal Section (t = 68.4 Years)O rig inal Section (t = 95.8 Years)O rig inal Section (t = 136.9 Years)
Practical convergence is reached after
about 21 boreholes
0 50 100 150 200-10
-5
0
First Moment (Single Realization)
0 10000 20000 30000 40000T im e (days)
0
20
40
60
80
100
120
X_C
oord
inat
e of
the
Cen
troid
(m)
O riginal SectionC ondition ing on 2 boreholesC ondition ing on 3 boreholesC ondition ing on 5 boreholesC ondition ing on 9 boreholesC ondition ing on 25 boreho les
0 10000 20000 30000 40000T im e (days)
-10
-8
-6
-4
-2
0
Y_C
oord
inat
e of
the
Cen
troid
(m)
O riginal SectionConditioning on 2 boreholesConditioning on 3 boreholesConditioning on 5 boreholesConditioning on 9 boreholesConditioning on 25 boreholes
Trend is reached at 3 boreholes
Convergence at 9 boreholes
Contam inant Source
P lum e at T im e, t
Im perm eable boundary
Im perm eable boundary
Second Moment (Single Realization)
0 10000 20000 30000 40000T im e (days)
0
0.5
1
1.5
2
2.5
Var
ianc
e in
Y_d
irect
ion
(m2 )
O riginal SectionC onditioning on 2 boreholesC onditioning on 3 boreholesC onditioning on 5 boreholesC onditioning on 9 boreholesC onditioning on 25 boreholes
0 10000 20000 30000 40000T im e (days)
0
1000
2000
3000
4000
Var
ianc
e in
X_d
irect
ion
(m2 )
O rig inal S ectionC ondition ing on 2 boreho lesC ondition ing on 3 boreho lesC ondition ing on 5 boreho lesC ondition ing on 9 boreho lesC ondition ing on 25 boreholes
Trend is reached at 3 boreholes
Convergence at 5 and 25 boreholes
Convergence at 9 boreholes
Contam inant S ource
P lum e at T im e, t
Im perm eable boundary
Im perm eab le boundary
Breakthrough Curve (Single Realization)
0 10000 20000 30000 40000 50000T im e (days)
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
Mas
s D
istri
butio
n
O rig inal SectionC onditioning on 2 boreholesC onditioning on 3 boreholesC onditioning on 5 boreholesC onditioning on 9 boreholesC onditioning on 25 boreholes
0 50 100 150 200-10
-5
0
Convergence at 25 boreholes
Conditioning on 2 boreholes (Ensemble )
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 0.1 1 10
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
CactualC C
mg/lit
T = 4.1 years
T = 82.2 years
T = 136.9 years
Conditioning on 5 boreholes (Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 0.1 1 10mg/lit
actualC C C
Conditioning on 9 boreholes (Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
actualC C C
Conditioning on 21 boreholes(Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
actualC C C
Conditioning on 31 boreholes(Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
actualC C C
Effect of Conditioning on Ensemble Cmax
0 4 8 12 16 20 24 28 32No. of Conditioning Boreholes
0
10
20
30
40
50
60
70
80
90
100
110
Ens
embl
e P
eak
Con
cent
ratio
n (m
g/lit
)
Ensem ble C m ax (t = 34.2 Years)Ensem ble C m ax (t = 68.4 Years)Ensem ble C m ax (t = 95.8 Years)Ensem ble C m ax (t = 136.9 Years)O rig inal Section (t = 34.2 Years)O rig inal Section (t = 68.4 Years)O rig inal Section (t = 95.8 Years)O rig inal Section (t = 136.9 Years)
0 4 8 12 16 20 24 28 32No. of Conditioning Boreholes
0
1
2
3
4
5
6
CV
of C
max
t = 34.2 Yearst = 68.4 Yearst = 95.8 Yearst = 136.9 Years
max actualC C
max
1 for #boreholes 5 c
C
max
1 for #boreholes 5c
C
max
time c
C
Conclusions1. CMC model proved to be a valuable tool in predicting heterogeneous
geological structures which lead to reducing uncertainty in concentration distributions of contaminant plumes.
2. Convergence to actual concentration is of oscillatory type, due to the fact that some layers are connected in one scenario and disconnected in another scenario.
3. In non-Gaussian fields, single realization concentration fields and the ensemble concentration fields are non-Gaussian in space with peak skewed to the left.
4. Reproduction of peak concentration, plume spatial moments and breakthrough curves in a single realization requires many conditioning boreholes (20-31 boreholes). However, reproduction of plume shapes require less boreholes (5 boreholes).
Conclusions
5. Ensemble concentration and ensemble variance have the same pattern. Ensemble variance is peaked at the location of the peak ensemble concentration and decreases when one goes far from the peak concentration. This supports early work by Rubin (1991). However, in Rubin’s case the maximum concentration was in the center of the plume which is attributed to Gaussian fields. The non-centered peak concentration, in this study, is attributed to the non-Gaussian fields.
6. Coefficient of variation of max concentration [CV(Cmax)] decreases significantly when conditioning is performed on more than 5 boreholes.
7. Reproduction of peak concentration, plume spatial moments and breakthrough curves in a single realization requires many conditioning boreholes (20-31 boreholes). However, reproduction of plume shapes require less boreholes (5 boreholes).
