Park, E., Elfeki, A. M. M., Dekking, F.M. (2003). Characterization of subsurface heterogeneity: Integration of soft and hard information using multi-dimensional Coupled Markov chain approach. Underground Injection Science and Technology Symposium, Lawrence Berkeley National Lab., October 22-25, 2003. p.49. Eds. Tsang, Chin.-Fu and Apps, John A.
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Characterization of Subsurface Heterogeneity: Integration of Soft and Hard Information using Multi-dimensional Coupled Markov Chain Approach
1. Characterization of Subsurface
Heterogeneity: Integration of Soft and Hard
Information using Multi-dimensional
Coupled Markov Chain Approach
Eungyu Park1, Amro Elfeki1, and Michel
Dekking2
1. Dept. of Hydrology and Ecology,
2. Dept. of Applied Probability,
Delft University of Technology
3. Subsurface Heterogeneity
One can easily experience the heterogeneity from
most fields by observing huge variation of its
properties from point to point
(Gelhar, 1993)
The heterogeneity of subsurface has been a long-
existing troublesome topic from the very beginning
of the subsurface hydrology
(Anderson, 1983)
4. Deterministic, Random, or
Stochastic?
Purely random?
No Regularity
Pure Random Process
Purely deterministic?
Deterministic Regularity
Pure Deterministic Process
Something in between?
Statistical Regularity
Stochastic Process
5. Stochastic Model
The word stochastic has its origin in the Greek
adjective στoχαστικoς which means skilful at
aiming or guessing
Mathematical models that employ stochastic
methods occupy an intermediate position in the
spectrum of dynamic models
Pure
Deterministic
Pure
Random
Stochastic
Model
6. Why do we need the Stochastic
Approach?
The erratic nature of the subsurface parameters
observed at field data
The uncertainty due to the lack of information
about the subsurface structure which is known
only at sparse sampled locations
7. Markov Chains
“…which may be regarded as a sequence or
chain of discrete state in time (or space) in
which the probability of the transition from
one state to a given state in the next step in the
chain depends on the previous state…”
(Harbaugh and Bonham-Carter, 1969 )
8. History of Markov Chains
Application in Subsurface
Characterization
Early Work (Traditional Markov Chain)
– Krumbein, 1-D (1967)
– Habaugh and Bonham-Carter, 1-D (1969)
Recent Work
– Elfeki, 2-D (1996): unconditional CMC
– Elfeki and Dekking, 2-D (2001): conditional CMC
– Carle and Fogg, 3-D Model (1996): MC + SIS +SA
9. CMC vs. Conventional Technique
CMC
• Based on conditional
probability (transition
matrix)
• Marginal Probability
• Converging Rate (2nd
largest eigen value)
• Asymmetry can be
described
• Model specification is
not necessary
Conventional
• Based on variogram or
autocovariance
• Sill
• Correlation Length
• Asymmetry is
impossible to be
described
• Model specification is
needed to be used in the
implementation
10. Advantages of Coupled Markov
Chain Model
Theory is simple and sound.
Implementation is easy.
Calculation procedure is efficient.
Geologic asymmetry can be modeled.
Conditioning is straightforward using
explicit formulae.
Model specification is unnecessary.
11. A B C D
A 0 0 0 0
B 0 0 0 0
C 0 0 0 0
D 0 0 0 0
Transition Probability
Tally Matrix
to
Transition
Probability
Matrix
A B C D
A 0.6 0.1 0.2 0.1
B 0.375 0.625 0 0
C 0.1 0.1 0.8 0
D 0.5 0 0 0.5
p
x
A B C D
A 6 1 2 1
B 3 5 0 0
C 1 1 8 0
D 1 0 0 1
12. Transition Probability
Properties of Transition
Probability Matrix
A
B
lim
C
D
n
n
w
w
p
w
w
0.3659 0.1951 0.3659 0.0732w
1
1
n
lk
k
p
n n
p p
Row-sum
n-step transition
Marginal probability
equals to volume
proportion of each
lithology
n x
13. 1D Theory without Conditioning
By Markovian property the present is no longer
depend on the past history if the immediate past
is given
Pr
Pr
i i-1 i-2 i-3 0k l n pr
i i-1k l lk
| , , S , ,S S S SZ Z Z Z Z
| pS SZ Z
L
14. 1D Theory with Conditioning
If the future is given
1
1 1
1 1
Pr ( )
Pr ( | )Pr ( | )Pr ( )
Pr ( | )Pr( )
i i Nk l q
N i i i iq k k l l
N i iq l l
| ,S S SZ Z Z
S S S S SZ Z Z Z Z
S S SZ Z Z
( )
( 1)
N i
lk kq
lk q N i
lq
p p
p
p
If the future is given (Elfeki and Dekking, 2001)
→1 when N →∞
15. We assume that
kflmmilif1+ik1+i pSY,SX|SY,SX ,)Pr(
, 1, , 1
1 1
Pr ( )
Pr( )Pr( )
i j i j i jk l m
i k i l j k j m
,S S SZ Z Z
C X S X S Y S Y S
1
1
n
h v
lf mf
f
C .p p
where
Theory of Coupled Markov Chain in
2D x-chain
16. 2D CMC Theory Conditioned on
Future States
If the future in both directions x and y are given
(Elfeki and Dekking, 2001)
, 1, , 1 , ,
( ) ( )
( ) ( )
Pr( | , , , )
, 1,... .
y x
x x x y y y
x x x y y y
i j k i j l i j m i N p N j q
h h N i h h N j
lk kq mk kp
h h N i h h N j
lf fq mf fp
f
Z S Z S Z S Z S Z S
p p p p
k n
p p p p
x-chain
17. Calculation Sequence
Geologically plausible information transfer
– Information is sequentially transferred to
horizontal direction
– Information is sequentially transferred to
vertical direction
boundary
calculated
unknown
x
y
19. 3D Theory
Likewise
, , 1, , , 1, , , 1 , , , ,
, , 1, , , , , , , 1, , ,
, , , , 1
( )
( 1)
Pr , , , ,
Pr , Pr ,
Pr
x y
x y
x
x
i j k o i j k l i j k m i j k n N j k p i N k q
i j k o i j k l N j k p i j k o i j k m i N k q
i j k o i j k n
hx N ihx hy
lo op mo
hx N i
lp
Z S Z S Z S Z S Z S Z S
C Z S Z S Z S Z S Z S Z S
Z S Z S
p p p
C
p
( )
( 1)
y
y
hy N j
oq v
nohy N j
mq
p
p
p
where
1( )( )
( 1)( 1)
yx
yx
hy N jhx N ihx hy v
lr mr nr rp rq
hy N jhx N i
r lp mq
p p p p p
C
p p
21. General Algorithm
1. Discretizing domain
2. Saving conditioning data
3. Deriving transition probabilities
4. Applying 1-, 2-, and 3D equations to the domain
5. Generate equally probable single realizations
6. Repeat step 1 through 5, if multiple realization are
desired
7. Do Monte Carlo analysis if desired using equally
probable multiple realizations generated from step 6
22. Applications of Coupled Markov
Chain Model
MADE site (16 Boreholes)
Input Borehole Data
x=3 m
y=0.1 m
24. Improved Reproduction By New Scheme
•Original •Sparse
71% Reproduction of Original 78% Reproduction of Original
•Previous Scheme •New Scheme (Angle Tolerance)
25. Monte Carlo Simulation and Ensemble
Indicator Map
Multiple Realizations
1
2
3
4
5
6
7
lith
Ensemble Indicator Map
Ensemble Indicator Function
0 1
26. The theory of 3D
Coupled Markov
Chain Model (3D
CMC) is developed
and programmed
under MATLAB
environment
Development of 3D CMC
33. Future Improvement of 3D CMC
Developing error minimizing sequences
Enhancing data adaptability by incorporating data
from various sources (soft geologic data, i.e.
geophysical, GPR, CPT, seismic data)
Integrating flow simulator (FDM or FEM)
Integrating solute transport simulator (RWPT or
MOC)
Integrating with spatial database (GIS)
Need to verify the Model through various 3D
application using hard data as well as soft data and
cross-validation techniques
Optimum estimation of horizontal transition
probabilities
34. Concluding Remarks
3D Coupled Markov Chain (3D CMC) is
developed through this study
Efficient way of utilization of the horizontal
data is added to 2D CMC
MATLAB software CMC3D is developed
and flow and transport simulators are
attached