Elfeki, A. M. (2006). Reducing Concentration Uncertainty Using the Coupled Markov Chain Approach. Journal of Hydrology vol. 317, pp 1-6.

1. Reducing Concentration Uncertainty
Using
The Coupled Markov Chain
Approach
Amro Elfeki
Section Hydrology,
Dept. of Water Management,
TU Delft,
The Netherlands.

2. 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.

3. Motivation and Issues
Motivation 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.

4. Geological and Parameter Uncertainties
Unconditional CMC
1 2 3 4
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
time = 1600 days
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-40
-20
0
0 50 100 150 200 250 300
-40
-20
0
Geology is Certain and Parameters are Uncertain
Geology is Uncertain and Parameters are Certain
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.

5. ( )
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
N
klk
| , , S ,...,S S S SZ Z Z Z Z
| pS SZ Z
p w
     
  

( )
1 ( 1)
ure
Pr ( )
N i
kq lk
i i Nk l q N i
lq
p p
| ,S S SZ Z Z
p

  
   
S S
o d
One-dimensional Markov Chain

6. Dark Grey (Boundary Cells)
Light Grey (Previously Generated Cells)
White (Unknown Cells)
i-1,j i,j
i,j-1
1,1
Nx,Ny
Nx,1
1,Ny
Nx,j
, , 1, , 1
, 1, , 1 ,,
Unconditioinal Coupled Markov Chains
: Pr( | , ) . 1,...
Conditioinal Coupled Markov Chains
: Pr( | , , )x
h v
lk mk
lm k i j k i j l i j m h v
lf mf
f
i j k i j l i j m N j qlm k q
h
lk
.p p
p 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 v
kq mk
h h N i v
lf fq mf
f
.p p
k n
. .p p p




Coupled Markov Chain “CMC” in 2D
(Elfeki and Dekking, 2001)

7. 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)

8. 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

9. 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
Geological Image
Domain Discretization
Generated Realization
0 50 100 150 200
-10
-5
0
Superposition of the Grid over
the Geological Image and
Estimation of Transition Probability
Boreholes Locations
0 50 100 150 200
-10
-5
0
Parameters Estimation Conditional Simulation
1
v
v lk
lk n
v
lq
q
T
p
T




10. Horizontal transition probability matrix of 1650 m section
calculated over sampling intervals of 25 m.
Soil 1 2 3 4 5 6
1 0.979 0.004 0.001 0.006 0.009 0.001
2 0.020 0.965 0.001 0.008 0.006 0.000
3 0.003 0.002 0.966 0.013 0.016 0.000
4 0.000 0.001 0.009 0.983 0.007 0.000
5 0.001 0.001 0.006 0.007 0.984 0.001
6 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 6
1 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)

11. 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 intervals
Dx = 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

12. Parameter Numerical Value
Time 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 MC
Number of particles 10,000 [particles]
Physical and Simulation Parameters
Soil 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 Iterations
Accuracy 0.00001

13. ( , ) ( , ) 0
( , )
( , )
x
y
K x y K x y
x x y y
K x yV
x
K x yV
y
  
  
  
   
    
   






Flow Model













Contaminant Source
Plume at Time, t
Impermeable boundary
Impermeable 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.

14. 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 yy
C C C C C C CV V D D D D
t x y x x y y x y
   
   
   
   
   
             
        
* - i j
mij ijL L T
VV
D V D
V
   
       
     
*mD
ij
L

T

is effective molecular diffusion,
is delta function,
is longitudinal dispersivity, and
is lateral dispersivity.

15. 1 1
1 1
cos sin sin cos
. / . / . / . /
n n n n
p p x p p yL T L T
n n n n
p 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
    
 
         
         
6 4 4 4 44 7 4 4 4 4 486 7 8
dispersive termadvective term
    1 22 2xy yxx x
p p x L T
D VD V
X t t X t V t Z V t Z V t
x y V V
 
 
 
 
 

          
 
    1 22 2yx yy y x
p p y L T
D D V V
Y t t Y t V t Z V t Z V t
x 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

16. 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
Lithology Coding
6 5
T= 82 years
# drillings
2
3
5
9
25
31

17. Effect of Conditioning Single Realiz. Cmax
0 4 8 12 16 20 24 28 32
No. of Conditioning Boreholes
0
40
80
120
160
200
240
PeakConcentration(mg/lit)
Single Realization Cmax (t = 34.2 Years)
Single Realization Cmax (t = 68.4 Years)
Single Realization Cmax (t = 95.8 Years)
Single Realization Cmax (t = 136.9 Years)
Original Section (t = 34.2 Years)
Original Section (t = 68.4 Years)
Original Section (t = 95.8 Years)
Original Section (t = 136.9 Years)
Practical convergence
is reached after
about 21 boreholes
0 50 100 150 200
-10
-5
0

18. First Moment (Single Realization)
0 10000 20000 30000 40000
Time (days)
0
20
40
60
80
100
120
X_CoordinateoftheCentroid(m)
Original Section
Conditioning on 2 boreholes
Conditioning on 3 boreholes
Conditioning on 5 boreholes
Conditioning on 9 boreholes
Conditioning on 25 boreholes
0 10000 20000 30000 40000
Time (days)
-10
-8
-6
-4
-2
0
Y_CoordinateoftheCentroid(m)
Original Section
Conditioning on 2 boreholes
Conditioning on 3 boreholes
Conditioning on 5 boreholes
Conditioning on 9 boreholes
Conditioning on 25 boreholes
Trend is reached at
3 boreholes
Convergence at
9 boreholes













Contaminant Source
Plume at Time, t
Impermeable boundary
Impermeable boundary

19. Second Moment (Single Realization)
0 10000 20000 30000 40000
Time (days)
0
0.5
1
1.5
2
2.5
VarianceinY_direction(m2)
Original Section
Conditioning on 2 boreholes
Conditioning on 3 boreholes
Conditioning on 5 boreholes
Conditioning on 9 boreholes
Conditioning on 25 boreholes
0 10000 20000 30000 40000
Time (days)
0
1000
2000
3000
4000
VarianceinX_direction(m2)
Original Section
Conditioning on 2 boreholes
Conditioning on 3 boreholes
Conditioning on 5 boreholes
Conditioning on 9 boreholes
Conditioning on 25 boreholes
Trend is reached at
3 boreholes
Convergence at
5 and 25 boreholes
Convergence at
9 boreholes













Contaminant Source
Plume at Time, t
Impermeable boundary
Impermeable boundary

20. Breakthrough Curve (Single Realization)
0 10000 20000 30000 40000 50000
Time (days)
0
0.2
0.4
0.6
0.8
1
NormalizedMassDistribution
Original Section
Conditioning on 2 boreholes
Conditioning on 3 boreholes
Conditioning on 5 boreholes
Conditioning on 9 boreholes
Conditioning on 25 boreholes
0 50 100 150 200
-10
-5
0
Convergence at
25 boreholes

21. 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

22. 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 10
mg/lit
actualC C C

23. 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

24. 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

25. 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

26. Effect of Conditioning on Ensemble Cmax
0 4 8 12 16 20 24 28 32
No. of Conditioning Boreholes
0
10
20
30
40
50
60
70
80
90
100
110
EnsemblePeakConcentration(mg/lit)
Ensemble Cmax (t = 34.2 Years)
Ensemble Cmax (t = 68.4 Years)
Ensemble Cmax (t = 95.8 Years)
Ensemble Cmax (t = 136.9 Years)
Original Section (t = 34.2 Years)
Original Section (t = 68.4 Years)
Original Section (t = 95.8 Years)
Original Section (t = 136.9 Years)
0 4 8 12 16 20 24 28 32
No. of Conditioning Boreholes
0
1
2
3
4
5
6
CVofCmax
t = 34.2 Years
t = 68.4 Years
t = 95.8 Years
t = 136.9 Years
max actualC Cp
max
1 for #boreholes 5

 c
C
max
1 for #boreholes 5

c
C
p
max
time

 c
C

27. Conclusions
1. 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).

28. 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-
G a u s s i a n f i e l d s .
6. Coefficient of variation of max concentration [CV(Cmax)] decreases
significantly when conditioning is performed on more than 5
b o r e h o l e s .
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
r e q u i r e l e s s b o r e h o l e s ( 5 b o r e h o l e s ) .