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Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.

A
Amro Elfeki

The document discusses various methods for modeling Markov chains, including forward and backward conditioning approaches, as well as sensitivity analyses. It also presents case studies demonstrating the application of these models in geological settings, highlighting the influence of sampling intervals and transition probabilities on model performance. The methodology aims to extract a final geological image through Monte Carlo simulations and empirical frequency evaluations.

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Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.
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
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
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
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
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
Coupling and Conditioning Two One-dimensional Markov Chains on a Lattice System (CMC method)
.,...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
, 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 ------------------> ------------------<
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)
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.
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.
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)
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
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
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
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
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
Case study no. 2 (Casparde Roblesdijk part of Waddenzeedijken,), NL
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
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
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
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
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
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
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 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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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]

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Modeling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy.

  • 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]