Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law...
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Transcript of Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law...
![Page 1: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/1.jpg)
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Theory of Forward MC: FMC
-1 -2 -3 0
-1
Pr( | , , ,..., )Pr( | ) : ,
i i i ik l n ar
i ik l lk
S S S S SZ Z Z Z Z pS SZ Z
SS S
i0 1 i+ 1i-1 N2
l k qdSSaN -1
S rSb
![Page 3: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/3.jpg)
Theory of Backward MC: BMC
1Pr( | ).i il kkl
S SZ Zp
lkp klp
SS S
i0 1 i+ 1i-1 N2
l k qdSSaN -1
S rSb
![Page 4: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/4.jpg)
Conditioning FMC (Elfeki and Dekking 2001)
( 1)
1 0 ( ): Pr ( ) ,N
ab bqNb a qab q N
aq
p pp | , S S SZ Z Z
p
SS S
i0 1 i+ 1i-1 N2
l k qdSSaN -1
S rSb
![Page 5: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/5.jpg)
Conditioning BMC
1 0Pr ( ).N Nr q aqr a | , SS SZ Z Zp
1 01 0
0
Pr( , )Pr ( ) .
Pr ( )N Nr q a
N Nr q aqr aN q a
, SS SZ Z Z | , S S SZ Z Zp , SSZ Z
1 0
1 0 1 0 0
0 0
Pr ( )
Pr ( ).Pr ( ).Pr( ).
Pr ( ).Pr( )
N Nr q aqr 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 qdSSaN -1
S rSb
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Conditioning BMC (cont.)
1 0
( 1)1 1 0
( )0
Pr ( )
Pr ( ).Pr ( ),
Pr ( )
N Nr q aqr a
NN N Nq r r rq ara
NN 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 qdSSaN -1
S rSb
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Coupling and Conditioning Two One-dimensional Markov Chains on
a Lattice System (CMC method)
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.,...1,
),,|Pr(:
)(
)(
,1,,1,,
nkp . p . p
p .p . p
SZSZSZSZp
f
vmf
iNhfq
hlf
vmk
iNhkq
hlk
qjNmjiljikjiqklm
x
x
x
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
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, 1, , 1 0,,
( )
( )
: Pr( | , , )
, 1,..., .
i j k i j d i j m j adm k a
h h i vkd ak mk
h h i vfd of mf
f
p Z S Z S Z S Z S
. . p p pk n
. . p p p
Coupled Markov Chain for Backward Conditioning on the Left Boundary.
i+ 1 ,ji,ji,j-1
1 ,1
N x ,N y
N x ,1
1 ,N y
N x ,j
Coupled M arkov Chain for Forward Conditioning on the Right Boundary.
i-1 ,j i,ji,j-1
1 ,1
N x ,N y
N x ,1
1 ,N y
N x ,j1 ,j
1 ,j
------------------>
------------------<
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Methods of Implementation FCMC, BCMC and FBCMC
? >? >
?< ?<
? > ?<
Forward M arkov Chain M odel (two forward steps)
Backw ard M arkov Chain M ode l (tw o backward steps)
Forward-Backward M arkov Chain M odel (one forward step and one backward step)
W ell (1) W ell (2) W ell (1) W ell (2)
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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.
![Page 12: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/12.jpg)
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.
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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)
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Case study no. 1 (Afsluitdijk-Lemmer), NL
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- 2 0
- 1 0
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
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
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
0
1 2 3 4 5 6
A
B
C
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Extracting a Final Image
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 5- 1 0
- 50
1 2 3 4 5 60
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
- 1 0
0
1
- 1 0
0
2
- 1 0
0
3
- 1 0
0
4
- 1 0
0
5
- 1 0
0
6
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Stability of Monte Carlo
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- 2 0
- 1 0
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- 2 0
- 1 0
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- 2 0
- 1 0
0
1 2 3 4 5 6
Results of 30 R ealiza tions
Resu lts of 100 R ealizations0 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
- 2 0
- 1 0
0
Results of 5 Realizations
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- 2 0
- 1 0
0
Results of 10 R ealizations0 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
- 2 0
- 1 0
0
Results of 1000 Realiza tions
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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
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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
E m pirica l Entropy States C oding
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=5m0 200 400 600 800 1000 1200 1400 1600
-10
-5
0
Application of the Concept of Walter’s Law
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Case study no. 2 (Casparde Roblesdijk part of
Waddenzeedijken,), NL
![Page 20: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/20.jpg)
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0- 1 5- 1 0
- 50
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 C odingEntropy M easure
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0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
1
2
3
4
5
6
7
8
9
10
11
dx =10 m
dx =20 m
dx =100 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0
0.25
0.5
0.75
1
1.25
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 1 5- 1 0
- 50
dx =40 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
dx = 5 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
Influence of Sampling Interval in Horizontal
![Page 22: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/22.jpg)
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
a
b
c
d
e
f
No. of Boreholes
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0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
1 2 3 4 5 6 7 8 9 10 11
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0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
1
2
3
4
5
6
7
8
9
10
11dx =10 m
dx =20 m
dx =100 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 1 5- 1 0
- 50
dx =40 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0
0.25
0.5
0.75
1
1.25
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
![Page 25: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/25.jpg)
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
1
2
3
4
5
6
7
8
9
10
11
dx =10 m
dx =20 m
dx =100 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 1 5- 1 0
- 50
dx =40 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0
0.25
0.5
0.75
1
1.25
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
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0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
1
2
3
4
5
6
7
8
9
10
11dx =10 m
dx =20 m
dx =100 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 1 5- 1 0
- 50
dx =40 m
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
- 1 5- 1 0
- 50
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0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e
1 2 3 4 5 6 7
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
C ondition ing Boreholes
G enerated F inal Im ages
Boreholes for C ross-Validation
S im ula ted B oreholes from F ina l Im ages
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 0.25 0.5 0.75 1 1.25 1.5 1.75
![Page 28: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/28.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
C ondition ing Boreholes
G enerated F inal Im ages
Boreholes for C ross-Validation
S im ula ted B oreholes from F ina l Im ages
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 0.25 0.5 0.75 1 1.25 1.5 1.751 2 3 4 5 6 7
![Page 29: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/29.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e
1 2 3 4 5 6 7
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
Condition ing Boreho les
G enerated F inal Im ages
Boreholes for C ross-Validation
S im ulated Boreho les from F ina l Im ages
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
![Page 30: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/30.jpg)
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
1
2
3
4
5
6
7
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
e0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
dx = 400 feet
0
0.25
0.5
0.75
1
1.25
1.5
Forw ard-Backw ard M odel
Fully Backward M odel
Fully Forw ard M odel
![Page 31: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/31.jpg)
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
1234567
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
e0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 400 feetForw ard-Backw ard M odel
Fully Backw ard M odel
Fully Forw ard M odel
![Page 32: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/32.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 80 feet
0
0.25
0.5
0.75
1
1.25
1.5
1
2
3
4
5
6
7
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
![Page 33: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/33.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
1
2
3
4
5
6
7
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 280 feet
0
0.25
0.5
0.75
1
1.25
1.5
![Page 34: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/34.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
1
2
3
4
5
6
70 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 280 feet
0
0.25
0.5
0.75
1
1.25
1.5
Fully Forward M odel
Fully Backward M odel
Forward-Backw ard M odel
![Page 35: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/35.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
1234567
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 280 feet
Fully Forw ard M odel
Fully Backw ard M odel
Forward-Backw ard M odel
![Page 36: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/36.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
1
2
3
4
5
6
70 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 80 feet
0
0.25
0.5
0.75
1
1.25
1.5
Fully Forward M odel
Fully Backward M odel
Forw ard-Backward M odel
![Page 37: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/37.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
1
2
3
4
5
6
70 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
e0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 280 feet
0
0.25
0.5
0.75
1
1.25
1.5
Forw ard M odel
Backw ard M odel
Forw ard-Backw ard M odel
![Page 38: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/38.jpg)
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
a
b
c
d
e
f
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
pii = 0 .603
pii= 0.699
p ii = 0.801
pii = 0.908
0
0.25
0.5
0 .75
1
1.25
1.5
1 .75
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
pii = 0.986
1
2
3
4
5
6
7
![Page 39: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/39.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
a
b
c
d
e
f
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
pii = 0 .603
pii= 0.699
p ii = 0.801
pii = 0.908
0
0.25
0.5
0 .75
1
1.25
1.5
1 .75
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
pii = 0.986
1
2
3
4
5
6
7
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0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
1
2
3
4
5
6
7
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
a
b
c
e
d
e
f
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0
0.25
0.5
0.75
1
1.25
1.5
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
dx = 40 feet
dx = 80 fee t
dx = 280 feet
dx = 400 feet
![Page 41: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/41.jpg)
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
12345670 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
a
b
c
e
d
e
f
dx = 40 feet
dx = 80 feet
dx = 280 feet
dx = 400 feet
![Page 42: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/42.jpg)
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
a
b
e0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
c
d
e
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
dx = 100 feet
dx = 200 feet
dx = 400 feet
dx = 800 fee t
0
0.25
0.5
0.75
1
1.25
1.5
1.75
1
2
3
4
5
6
7
![Page 43: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/43.jpg)
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0- 3 0 0- 2 0 0- 1 0 0
0
a
b
e0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0 5 0 0 0 0
- 3 0 0- 2 0 0- 1 0 0
0
c
d
e
dx = 100 feet
dx = 200 feet
dx = 400 feet
dx = 800 feet
1
2
3
4
5
6
7
![Page 44: Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.](https://reader035.fdocuments.net/reader035/viewer/2022062905/5872466c1a28ab852f8b474f/html5/thumbnails/44.jpg)
0 4000 8000 12000 16000O verall H orizonta l F ie ld Scale (m eters)
0
0.01
0.02
0.03
0.04
dy/d
x
Afsluitd ijk- Lem m er (Netherlands)A fs luitd ijk C aspar de R oblesdijk (N etherlands)D elaware R ive r Aquife r (Long itud ina l-Section), USAD elaware R ive r Aquife r (C ross-Section),US AM AD E site a t Co lum bus, M ississ ipp i [E lfeki and R a jabian i, 2002]