Invited lecture at the Stockholm International Meteorological Institute (IMI), Stockholm university, Sweden. This presentation is about the reduction of uncertainty in the geological structures on groundwater flow and contaminant transport.

1. Reducing Uncertainty of
Groundwater Contaminant Transport
Using
Markov Chains
Amro Elfeki
Dept. of Hydrology and Water Resources,
Faculty of Meteorology, Environment and Arid Land Agriculture,
KAU, Jeddah, KSA.
On leave of absence from:
Faculty of Civil Engineering,
Mansoura University, Egypt
Elfeki_amr@yahoo.co.uk

2. Outlines
• Definitions.
• 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 groundwater transport.
• 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.
Figure 1. Huesca outcrop, Spain, Courtesy Kees Geel
(from Dept. of Geology, Faculty of Applied Earth Sciences, TU Delft,
The Netherlands).
Geological Structure

5. Typical Problem of Groundwater
Contamination

6. 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
H o r iz o n t a l D is t a n c e b e t w e e n W e lls ( m )
- 5 0
0
Depth(m)
W e ll 1 W e ll 2
? K ( x ,y ,z ) ?
( x ,y ,z ) ?
C ( x ,y ,z ) ?
H = ?
H = ?
?
?
??
? ?
?
?
?
Classification of Uncertainty:
-Conceptual Model Uncertainty:
Darcy’s and Fick’s Laws.
-Geological Uncertainty:
Connectivity and dis-connectivity
of the layers, geological sequence,
boundaries between geological units.
-Parameter Uncertainty:
-K, porosity.
-Hydro-geological Uncertainty:
Constant head boundaries,
impermeable boundaries,
Plume boundaries, source area boundaries.
What is Uncertainty?

7. - The lack of information
about the subsurface
structure which is
known only at sparse
sampled locations.
- The erratic nature of
the subsurface
parameters observed
at field scale.
Why Addressing Uncertainty by
Stochastic Approach?
Courtesy lynn Gelhar

8. 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
t i m e = 1 6 0 0 d a y 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 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 4 0
- 2 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 4 0
- 2 0
0
G e o lo g y is C e r t a in a n d P a r a m e t e r s a r e U n c e r t a in
G e o lo g y is U n c e r t a in a n d P a r a m e t e r s a r e C e r t a in
0 0 . 0 1 0 . 1 1
C
C
actualC
σC
σC
Elfeki, Uffink and Barends, 1998
Geological Uncertainty:
Geological configuration.
Parameter Uncertainty:
Conductivity value of each unit.
Mg/l

9. Application of CMC at MADE Site
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
0 . 1
1
1 0
1 0 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
Elfeki, (2003 ) Journal of Hydraulic Reserac
Real field situation:
MAcro-Disperison
Experiment (MADE)
Columbus, Mississippi
Air Force Base Site in US
Data is in the form of
boreholes.
Geological prediction is
needed at unsampled
locations.
Boggs et al. (1990)

10. Application of CMC at MADE Site

11. ( )
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
(Elfeki and
Dekking, 2001)

12. 1,...n=l2,...n,=kpUp
k
q
lq
k
q
lq ,
1
1
1
∑∑ =
−
=
≤<
11 12 1
21
1
. .
. . . .
. . . .
. . . . .
. . .
n
lk
n nn
p p p
p
pl
p p
 
 
 
 =
 
 
  
1 2 ... ... n
1
2
.
n
p
11 11 12 1
1
21
1
1
1
. .
1
2 . . . .
. . . .
. . . . .
. . .
n
i
i
k
li
i
n
n ni
i
p p p p
p
p
l
n
p p
=
=
=
 
+ 
 
 
 
 =
 
 
 
 
 
  
∑
∑
∑
1 2 ... ... n
P
A B C D
One-dimensional Markov Chain
(Cont.)

13. D a r k G r e y ( B o u n d a r y C e lls )
L ig h t G r e y ( P r e v io u s ly G e n e r a t e d C e lls )
W h it e ( U n k n o w n C e lls )
i - 1 ,j i ,j
i ,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 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)

14. Concept of Unconditional
Realizations (CMC)

15. Concept of Unconditional
Realizations (CMC)-Cont.
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
1 2 3 4 5 6 7 8
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0

16. Concept of Conditional Realizations
(CMC)
1
2
3
4
5
6
7
8
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0

17. 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)

18. Schelluinen study area, The Netherlands
S t u d y A r e a
(Bierkens, 94).

19. 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 8 0 1 6 0 2 4 0
- 1 0
- 5
0
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0
- 1 0
- 5
0
1 2 3 4 5 6
Data from Bierkens, 1994

20. Parameter Estimation and Procedure
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
G e o l o g i c a l I m a g e
D o m a i n D i s c r e t i z a t i o n
G e n e r a t e d R e a l i z a t i o n
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
S u p e r p o s i t i o n o f t h e G r i d o v e r
t h e G e o l o g i c a l I m a g e a n d
E s t i m a t i o n o f T r a n s i t i o n P r o b a b i l i t y
B o r e h o l e s L o c a t i o n s
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
P a r a m e t e r s E s t i m a t i o n C o n d i t i o n a l S i m u l a t i o n
1
v
v lk
lk n
v
lq
q
T
p
T
=
=
∑

21. 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)

22. 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 8 0 1 6 0 2 4 0
- 1 0
- 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

23. 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 Kσ( )iLog K ( / )iK m day 2
iKσ
Convergence:
~14000 Iterations
Accuracy 0.00001

24. ( , ) ( , ) 0
( , )
( , )
x
y
K x y K x y
x x y y
K x yV
x
K x yV
y
  
 ÷ ÷
 ÷ ÷
   
∂ ∂Φ ∂ ∂Φ+ =
∂ ∂ ∂ ∂
∂Φ=−
∂
∂Φ=−
∂
ε
ε
Groundwater Flow Model













C o n t a m in a n t S o u r c e
P lu m e a t T im e , t
I m p e r m e a b le b o u n d a r y
I m p e r m e a b le b o u n d a r y
is the hydraulic head,
Vx and Vy are pore velocities,
is the hydraulic conductivity, and
is the effective porosity.
Φ
( , )K x y
ε
Hydrodynamic Condition:
Non-uniform Flow in the Mean
due to Boundary Conditions.

25. 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
   
 ÷  ÷ ÷   
= + +α δ α α
*mD
ijδ
L
α
T
α
is effective molecular diffusion,
is delta function,
is longitudinal dispersivity, and
is lateral dispersivity.

26. 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
ϕ ϕ ϕ ϕ+ +
+ +
= + ∆ + − = + ∆ + +
= + ∆ + − = + ∆ + +
6444447444448678
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
α α
 
 ÷
 ÷
 
∂∂
+ ∆ = + + + ∆ + ∆ − ∆
∂ ∂
( ) ( ) 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
α α
 
 ÷
 ÷
 
∂ ∂
+ ∆ = + + + ∆ + ∆ + ∆
∂ ∂
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

27. Application of SIS at the Site
Geological
Section
Deterministic
and
Stochastic
Zones
In
SIS Model
Bierkens, 1996

28. Comparison between CMC and SIS (1)
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0
- 1 0
- 5
0
Conditioning on half of the drillings
SIS Model
Simulation
CMC Model
Simulation
Geological
Section
Bierkens, 1996

29. Comparison between CMC and SIS (2)
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0
- 1 0
- 5
0
Conditioning on all drillings
SIS Model
Simulation
CMC Model
Simulation
Geological
Section
Bierkens, 1996

30. Monte-Carlo on CMC
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
1 2 3 4 5 6
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
0
0 . 2 5
0 . 5
0 . 7 5
1
0 . 0 0 5 0 0 . 0 0 1 0 0 0 . 0 0 1 5 0 0 . 0 0
- 1 0 . 0 0
- 5 . 0 0
0 . 0 0
S t a t e # 6
S t a t e # 1 S t a t e # 2
S t a t e # 3
S t a t e # 4
S t a t e # 5
C o d i n g o f t h e S t a t e s
S c h e m a t i c I m a g e o f t h e G e o l o g i c a l
C r o s s S e c t i o n
3 9 B o r e h o l e s f o r C o n d i t i o n i n g C o n d i t i o n e d S i n g l e R e a l i z a t i o n

31. Effect of Conditioning on 240 x10m Sec.
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
1 2 3 4
L it h o lo g y C o d in g
0 8 0 1 6 0 2 4 0
- 1 0
- 5
0
1
2
3
4
31 boreholes
25 boreholes
9 boreholes
2 boreholes

32. Effect of Conditioning on S. R. Plume
mg/lit
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0
0 . 1
1
1 0
3 4
L it h o lo g y C o d in g
6 5
T= 82 years
# drillings
2
3
5
9
25
31

33. Effect of Conditioning Single Realiz. Cmax
0 4 8 1 2 1 6 2 0 2 4 2 8 3 2
N o . o f C o n d it io n in g B o r e h o le s
0
4 0
8 0
1 2 0
1 6 0
2 0 0
2 4 0
PeakConcentration(mg/lit)
S in g le R e a liz a t io n C m a x ( t = 3 4 .2 Y e a r s )
S in g le R e a liz a t io n C m a x ( t = 6 8 .4 Y e a r s )
S in g le R e a liz a t io n C m a x ( t = 9 5 .8 Y e a r s )
S in g le R e a liz a t io n C m a x ( t = 1 3 6 .9 Y e a r s )
O r ig in a l S e c t io n ( t = 3 4 . 2 Y e a r s )
O r ig in a l S e c t io n ( t = 6 8 . 4 Y e a r s )
O r ig in a l S e c t io n ( t = 9 5 . 8 Y e a r s )
O r ig in a l S e c t io n ( t = 1 3 6 . 9 Y e a r s )
Practical convergence
is reached after
about 21 boreholes
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0

34. First Moment (Single Realization)
0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0
T im e ( d a y s )
0
2 0
4 0
6 0
8 0
1 0 0
1 2 0
X_CoordinateoftheCentroid(m)
O r ig in a l S e c t io n
C o n d it io n in g o n 2 b o r e h o le s
C o n d it io n in g o n 3 b o r e h o le s
C o n d it io n in g o n 5 b o r e h o le s
C o n d it io n in g o n 9 b o r e h o le s
C o n d it io n in g o n 2 5 b o r e h o le s
0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0
T im e ( d a y s )
- 1 0
- 8
- 6
- 4
- 2
0
Y_CoordinateoftheCentroid(m)
O r ig in a l S e c tio n
C o n d itio n in g o n 2 b o r e h o le s
C o n d itio n in g o n 3 b o r e h o le s
C o n d itio n in g o n 5 b o r e h o le s
C o n d itio n in g o n 9 b o r e h o le s
C o n d itio n in g o n 2 5 b o r e h o le s
Trend is reached at
3 boreholes
Convergence at
9 boreholes













C o n t a m in a n t S o u r c e
P lu m e a t T im e , t
I m p e r m e a b le b o u n d a r y
I m p e r m e a b le b o u n d a r y

35. Second Moment (Single Realization)
0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0
T im e ( d a y s )
0
0 . 5
1
1 . 5
2
2 . 5
VarianceinY_direction(m2)
O r ig in a l S e c t io n
C o n d it io n in g o n 2 b o r e h o le s
C o n d it io n in g o n 3 b o r e h o le s
C o n d it io n in g o n 5 b o r e h o le s
C o n d it io n in g o n 9 b o r e h o le s
C o n d it io n in g o n 2 5 b o r e h o le s
0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0
T im e ( d a y s )
0
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
VarianceinX_direction(m2
)
O r ig in a l S e c t io n
C o n d it io n in g o n 2 b o r e h o le s
C o n d it io n in g o n 3 b o r e h o le s
C o n d it io n in g o n 5 b o r e h o le s
C o n d it io n in g o n 9 b o r e h o le s
C o n d it io n in g o n 2 5 b o r e h o le s
Trend is reached at
3 boreholes
Convergence at
5 and 25 boreholes
Convergence at
9 boreholes













C o n t a m in a n t S o u r c e
P lu m e a t T im e , t
I m p e r m e a b le b o u n d a r y
I m p e r m e a b le b o u n d a r y

36. Breakthrough Curve (Single Realization)
0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0
T im e ( d a y s )
0
0 . 2
0 . 4
0 . 6
0 . 8
1
NormalizedMassDistribution
O r ig in a l S e c tio n
C o n d it io n in g o n 2 b o r e h o le s
C o n d it io n in g o n 3 b o r e h o le s
C o n d it io n in g o n 5 b o r e h o le s
C o n d it io n in g o n 9 b o r e h o le s
C o n d it io n in g o n 2 5 b o r e h o le s
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
Convergence at
25 boreholes

37. Conditioning on 2 boreholes (Ensemble )
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 0 . 1 1 1 0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
CactualC σC
mg/lit
T = 4.1 years
T = 82.2 years
T = 136.9 years

38. Conditioning on 5 boreholes (Ensemble)
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 0 . 1 1 1 0
mg/lit
actualC C σC

39. Conditioning on 9 boreholes (Ensemble)
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
actualC C σC

40. Conditioning on 21 boreholes(Ensemble)
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
actualC C σC

41. Conditioning on 31 boreholes(Ensemble)
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
actualC C σC

42. Effect of Conditioning on Ensemble Cmax
0 4 8 1 2 1 6 2 0 2 4 2 8 3 2
N o . o f C o n d it io n in g B o r e h o le s
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
1 0 0
1 1 0
EnsemblePeakConcentration(mg/lit)
E n s e m b le C m a x ( t = 3 4 . 2 Y e a r s )
E n s e m b le C m a x ( t = 6 8 . 4 Y e a r s )
E n s e m b le C m a x ( t = 9 5 . 8 Y e a r s )
E n s e m b le C m a x ( t = 1 3 6 . 9 Y e a r s )
O r ig in a l S e c tio n ( t = 3 4 . 2 Y e a r s )
O r ig in a l S e c tio n ( t = 6 8 . 4 Y e a r s )
O r ig in a l S e c tio n ( t = 9 5 . 8 Y e a r s )
O r ig in a l S e c tio n ( t = 1 3 6 . 9 Y e a r s )
0 4 8 1 2 1 6 2 0 2 4 2 8 3 2
N o . o f C o n d it io n in g B o r e h o le s
0
1
2
3
4
5
6
CVofCmax
t = 3 4 .2 Y e a r s
t = 6 8 .4 Y e a r s
t = 9 5 .8 Y e a r s
t = 1 3 6 .9 Y e a r s
max actualC Cp
max
1 for #boreholes 5
σ
≤ ≥c
C
max
1 for #boreholes 5
σ
≥c
C
p
max
time
σ
↑ ↑c
C

43. 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).

44. 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).