Characterization of Subsurface Heterogeneity: Integration of Soft and Hard Information using Multi-dimensional Coupled Markov Chain Approach

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

Contents
1. Introduction (Heterogeneity/
Stochastic/Markov Chain)
2. Development of Coupled Markov Chain
Theory
3. 2D Implementation
4. 3D Implementation
5. Ensemble Indicator
6. Concluding Remarks

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)

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

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

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

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 )

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

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

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.

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
 
 
 
 
 
 

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 0.0732w
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

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

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 →∞

 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

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

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

3D Theory
y
x
z
x-chain
z-chain

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

  
 
  
 
 


Active Conditioning
Use the tolerance angle
along scanning and
perpendicular to scanning
direction
x
y

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

Applications of Coupled Markov
Chain Model
 MADE site (16 Boreholes)
Input Borehole Data
x=3 m
y=0.1 m

Geological
Map
(Adams and
Gelhar, 1992)
Simulated
Image using
CMC

Improved Reproduction By New Scheme
•Original •Sparse
71% Reproduction of Original 78% Reproduction of Original
•Previous Scheme •New Scheme (Angle Tolerance)

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

 The theory of 3D
Coupled Markov
Chain Model (3D
CMC) is developed
and programmed
under MATLAB
environment
Development of 3D CMC

3D Single Realization using 3D CMC

3D Ensemble Indicator Function
of Lithology 3

Statistical Mapping of 3D Ensemble
Indicator Function of Lithology 3
 10%+ certainty range  90%+ certainty range

10 %
90 %
50 %
Statistical Mapping of 3D Ensemble
Indicator Function of Lithology 2
ABC

Sample Flow Simulation
 MODFLOW by USGS

Sample Transport Simulation
 Random Walk Particle Tracking Method
x
y

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

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

Characterization of Subsurface Heterogeneity: Integration of Soft and Hard Information using Multi-dimensional Coupled Markov Chain Approach

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Characterization of Subsurface Heterogeneity: Integration of Soft and Hard Information using Multi-dimensional Coupled Markov Chain Approach