Elfeki, A. M. and Dekking F. M. (2005). Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy. Journal of Geotechnical and Geological Engineering, Vol. 23: pp.721-756.

2. Theory of Forward MC: FMC
-1 -2 -3 0
-1
Pr( | , , ,..., )
Pr( | ) : ,
i i i ik l n ar
i ik l lk
SS S S SZ Z Z Z Z
pS SZ Z
     
  
SS S
i0 1 i+1i-1 N2
l k qdSSa
N-1
SrSb

3. Theory of Backward MC: BMC
1Pr( | ).i il k
kl
S SZ Zp

  
lkp 
klp

SS S
i0 1 i+1i-1 N2
l k qdSSa
N-1
SrSb

4. Conditioning FMC (Elfeki and
Dekking 2001)
( 1)
1 0 ( )
: Pr ( ) ,
N
ab bq
Nb a qab q N
aq
p p
p | ,S S SZ Z Z
p

    
SS S
i0 1 i+1i-1 N2
l k qdSSa
N-1
SrSb

5. Conditioning BMC
1 0Pr ( ).N Nr q a
qr a
| , SS SZ Z Zp

   
1 0
1 0
0
Pr( , )
Pr ( ) .
Pr ( )
N Nr q a
N Nr q a
qr a
N q a
, SS SZ Z Z
| , SS SZ Z Zp , SSZ Z
 

  
    
 
1 0
1 0 1 0 0
0 0
Pr ( )
Pr ( ).Pr ( ).Pr( )
.
Pr ( ).Pr( )
N Nr q a
qr a
N N Nq r ra a a
N q a a
| , SS SZ Z Zp
| , S | S SS S SZ Z Z Z Z Z
| S SSZ Z Z


 
    
     
  
SS S
i0 1 i+1i-1 N2
l k qdSSa
N-1
SrSb

6. Conditioning BMC (cont.)
1 0
( 1)
1 1 0
( )
0
Pr ( )
Pr ( ).Pr ( )
,
Pr ( )
N Nr q a
qr a
N
N N Nq r r rq ara
N
N q a aq
| , SS SZ Z Zp
p p| | SS S SZ Z Z Z
| S pSZ Z



 
    
   

 
SS S
i0 1 i+1i-1 N2
l k qdSSa
N-1
SrSb

7. Coupling and Conditioning Two
One-dimensional Markov Chains on
a Lattice System (CMC method)

8. .,...1,
),,|Pr(:
)(
)(
,1,,1,,
nk
p.p.p
p.p.p
SZSZSZSZp
f
v
mf
iNh
fq
h
lf
v
mk
iNh
kq
h
lk
qjNmjiljikjiqklm
x
x
x


 


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

9. , 1, , 1 0,,
( )
( )
: Pr( | , , )
, 1,..., .
i j k i j d i j m j adm k a
h h i v
kd ak mk
h h i v
fd of mf
f
p Z S Z S Z S Z S
. .p p p
k n
. .p p p

      


Coupled Markov Chain for Backward Conditioning on the Left Boundary.
i+1,ji,j
i,j-1
1,1
Nx,Ny
Nx,1
1,Ny
Nx,j
Coupled Markov Chain for Forward Conditioning on the Right Boundary.
i-1,j i,j
i,j-1
1,1
Nx,Ny
Nx,1
1,Ny
Nx,j1,j
1,j
------------------>
------------------<

10. Methods of Implementation FCMC,
BCMC and FBCMC
? >? >
?< ?<
? > ?<
Forward Markov Chain Model (two forward steps)
Backward Markov Chain Model (two backward steps)
Forward-Backward Markov Chain Model (one forward step and one backward step)
Well (1) Well (2) Well (1) Well (2)

11. Procedure for Extracting a Final
Geological Image




 

.0
1
)(
otherwise
SZif
ZI
kij
ijk
 



MC
R
R
k
k
R
k
ij ji
ji
ZI
MCMC
SZ
1
)(
)(
,
, 1}{#

}...,,max{ 21 n
ijijij
l
ij  
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.

12. Sensitivity Analysis
• Various sampling intervals.
• Various horizontal transition probability matrices.
• Various degrees of diagonal dominancy of the horizontal transition
matrix.
• Use of Walther’s law to account for horizontal variability.
• Effect of conditioning on the model performance.
• Sensitivity of the Monte Carlo realizations.
• Various implementation strategies: forward, backward and
forward-backward methods.
• Use of cross validation to evaluate the model performance.

13. Case Studies
• Case study no. 1 (Afsluitdijk-Lemmer), NL
• Case study no. 2 (Casparde Roblesdijk part
of Waddenzeedijken,), NL
• Case study no. 3 (The Delaware river and
its underlying aquifer system in the vicinity
of the Camden metroplitan area, New
Jersey)

14. Case study no. 1 (Afsluitdijk-
Lemmer), NL
0 200 400 600 800 1000 1200 1400 1600
-20
-10
0
0 200 400 600 800 1000 1200 1400 1600
-10
0
0 200 400 600 800 1000 1200 1400 1600
-10
0
0 200 400 600 800 1000 1200 1400 1600
-10
0
1 2 3 4 5 6
A
B
C

15. Extracting a Final Image
0 200 400 600 800 1000 1200 1400 1600
-15
-10
-5
0
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
-10
0
1
-10
0
2
-10
0
3
-10
0
4
-10
0
5
-10
0
6

16. Stability of Monte Carlo
0 200 400 600 800 1000 1200 1400 1600
-20
-10
0
0 200 400 600 800 1000 1200 1400 1600
-20
-10
0
0 200 400 600 800 1000 1200 1400 1600
-20
-10
0
1 2 3 4 5 6
Results of 30 Realizations
Results of 100 Realizations
0 200 400 600 800 1000 1200 1400 1600
-20
-10
0
Results of 5 Realizations
0 200 400 600 800 1000 1200 1400 1600
-20
-10
0
Results of 10 Realizations
0 200 400 600 800 1000 1200 1400 1600
-20
-10
0
Results of 1000 Realizations

17. 0 200 400 600 800 1000 1200 1400 1600
-15
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-15
-10
-5
0
1
2
3
4
5
6
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=10m
dx=40m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=20m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=5m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0
0.25
0.5
0.75
1
dx=2.5m
Influence of Sampling Interval in
Horizontal

18. 0 200 400 600 800 1000 1200 1400 1600
-15
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-15
-10
-5
0
1 2 3 4 5 6
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=10m
dx=40m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=20m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=5m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 0.25 0.5 0.75 1
dx=2.5m
Empirical Entropy States Coding
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=2.5m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
dx=5m
0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
Application of the Concept of
Walter’s Law

19. Case study no. 2 (Casparde
Roblesdijk part of
Waddenzeedijken,), NL

20. 0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 0.25 0.5 0.75 1 1.25 1.5 1 2 3 4 5 6 7 8 9 10 11 12 13
Lithology CodingEntropy Measure

21. 0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
1
2
3
4
5
6
7
8
9
10
11
dx =10 m
dx =20 m
dx =100 m
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0
0.25
0.5
0.75
1
1.25
0 500 1000 1500 2000
-15
-10
-5
0
0 500 1000 1500 2000
-15
-10
-5
0
dx =40 m
0 500 1000 1500 2000 2500
-15
-10
-5
0
dx = 5 m
0 500 1000 1500 2000 2500
-15
-10
-5
0
Influence of Sampling Interval in
Horizontal

22. 0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
a
b
c
d
e
f
No. of Boreholes

23. 0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
1 2 3 4 5 6 7 8 9 10 11

24. 0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
1
2
3
4
5
6
7
8
9
10
11
dx =10 m
dx =20 m
dx =100 m
0 500 1000 1500 2000
-15
-10
-5
0
dx =40 m
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0
0.25
0.5
0.75
1
1.25
0 500 1000 1500 2000
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0

25. 0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
1
2
3
4
5
6
7
8
9
10
11
dx =10 m
dx =20 m
dx =100 m
0 500 1000 1500 2000
-15
-10
-5
0
dx =40 m
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0
0.25
0.5
0.75
1
1.25
0 500 1000 1500 2000
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0

26. 0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
0 500 1000 1500 2000 2500
-15
-10
-5
0
1
2
3
4
5
6
7
8
9
10
11
dx =10 m
dx =20 m
dx =100 m
0 500 1000 1500 2000
-15
-10
-5
0
dx =40 m
0 500 1000 1500 2000 2500
-15
-10
-5
0

27. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
1 2 3 4 5 6 7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
Conditioning Boreholes
Generated Final Images
Boreholes for Cross-Validation
Simulated Boreholes from Final Images
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 0.25 0.5 0.75 1 1.25 1.5 1.75

28. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
Conditioning Boreholes
Generated Final Images
Boreholes for Cross-Validation
Simulated Boreholes from Final Images
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 0.25 0.5 0.75 1 1.25 1.5 1.75
1 2 3 4 5 6 7

29. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
1 2 3 4 5 6 7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
Conditioning Boreholes
Generated Final Images
Boreholes for Cross-Validation
Simulated Boreholes from Final Images
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0

30. 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
1
2
3
4
5
6
7
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
e
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
dx = 400 feet
0
0.25
0.5
0.75
1
1.25
1.5
Forward-Backward Model
Fully Backward Model
Fully Forward Model

31. 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
1
2
3
4
5
6
7
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
e
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
dx = 400 feet
Forward-Backward Model
Fully Backward Model
Fully Forward Model

32. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
dx = 80 feet
0
0.25
0.5
0.75
1
1.25
1.5
1
2
3
4
5
6
7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0

33. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
1
2
3
4
5
6
7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
dx = 280 feet
0
0.25
0.5
0.75
1
1.25
1.5

34. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
1
2
3
4
5
6
7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
dx = 280 feet
0
0.25
0.5
0.75
1
1.25
1.5
Fully Forward Model
Fully Backward Model
Forward-Backward Model

35. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
1
2
3
4
5
6
7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
dx = 280 feet
Fully Forward Model
Fully Backward Model
Forward-Backward Model

36. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
1
2
3
4
5
6
7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
dx = 80 feet
0
0.25
0.5
0.75
1
1.25
1.5
Fully Forward Model
Fully Backward Model
Forward-Backward Model

37. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
1
2
3
4
5
6
7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
e
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
dx = 280 feet
0
0.25
0.5
0.75
1
1.25
1.5
Forward Model
Backward Model
Forward-Backward Model

38. 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
a
b
c
d
e
f
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
pii = 0.603
pii= 0.699
pii = 0.801
pii = 0.908
0
0.25
0.5
0.75
1
1.25
1.5
1.75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
pii = 0.986
1
2
3
4
5
6
7

39. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
a
b
c
d
e
f
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
pii = 0.603
pii= 0.699
pii = 0.801
pii = 0.908
0
0.25
0.5
0.75
1
1.25
1.5
1.75
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
pii = 0.986
1
2
3
4
5
6
7

40. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
1
2
3
4
5
6
7
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
a
b
c
e
d
e
f
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0
0.25
0.5
0.75
1
1.25
1.5
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
dx = 40 feet
dx = 80 feet
dx = 280 feet
dx = 400 feet

41. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
1
2
3
4
5
6
70 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
-300
-200
-100
0
a
b
c
e
d
e
f
dx = 40 feet
dx = 80 feet
dx = 280 feet
dx = 400 feet

42. 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
a
b
e
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
c
d
e
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
dx = 100 feet
dx = 200 feet
dx = 400 feet
dx = 800 feet
0
0.25
0.5
0.75
1
1.25
1.5
1.75
1
2
3
4
5
6
7

43. 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
a
b
e
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
-300
-200
-100
0
c
d
e
dx = 100 feet
dx = 200 feet
dx = 400 feet
dx = 800 feet
1
2
3
4
5
6
7

44. 0 4000 8000 12000 16000
Overall Horizontal Field Scale (meters)
0
0.01
0.02
0.03
0.04
dy/dx
Afsluitdijk- Lemmer (Netherlands)
Afsluitdijk Caspar de Roblesdijk (Netherlands)
Delaware River Aquifer (Longitudinal-Section), USA
Delaware River Aquifer (Cross-Section),USA
MADE site at Columbus, Mississippi [Elfeki and Rajabiani, 2002]