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Reducing Concentration Uncertainty in Geological Structures
by Conditioning on Boreholes
Using The Coupled Markov Chain
Approach
Amro Elfeki Section Hydrology,
Dept. of Water Management, TU Delft,
The Netherlands.
Outlines• Motivation of this research.• Methodology: • Markov Chain in One-dimension.• Markov Chain in Multi-dimensions: Coupled Markov Chain (CMC). • Application of CMC at the Schelluinen study area (Bierkens,
94).• Comparison between:
CMC (Elfeki and Dekking, 2001) and
SIS (Sequential Indicator Simulation, Gomez-Hernandez and Srivastava, 1990) .
• Flow and Transport Models in a Monte-Carlo Framework.• Geostatistical Results.• Transport Results.
• Conclusions.
Motivation and Issues
Motivation of this research:
• Test the applicability of CMC model on field data at many sites.
• Compare CMC with SIS (well-known model in geostatistics).
• Incorporating CMC model in flow and transport models to study uncertainty in concentration fields.
• Deviate from the literature: - Non-Gaussian stochastic fields: (Markovian fields), - Statistically heterogeneous fields, and - Non-uniformity of the flow field (in the mean) due to boundary conditions.
Geological and Parameter Uncertainties
Unconditional CMC
1 2 3 4
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
tim e = 1 6 0 0 d ay s
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0-5 0
0
0 50 100 150 200 250 300
-40
-20
0
0 50 100 150 200 250 300
-40
-20
0
G eology is C erta in and Param eters are Uncerta in
G eology is U ncerta in and Param eters are C erta in
0 0.01 0.1 1
C
C
actualC
C
C
Elfeki, Uffink and Barends, 1998
Geological Uncertainty:
Geological configuration.
Parameter Uncertainty:
Conductivity value of each unit.
Application of CMC at MADE Site
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0
0.1
1
10
100
0 50 100 150 200 250
-10
-5
0
1
2
3
4
5
0 50 100 150 200 250
-10
-5
0
Elfeki, 2003 (in review)
Real field situation:Data is in the form of boreholes.Geological prediction is needed at unsampled locations.
( )
Markov property (One-Step transition probability)
Pr( )
Pr( ) : ,
Marginal Distribution
lim
Conditioning on the Fut
N
i i-1 i-2 i-3 0k l n pr
i i-1k l lk
Nklk
| , , S ,..., S S S SZ Z Z Z Z
| pS SZ Z
p w
( )
1 ( 1)
ure
Pr ( )N i
kq lki i Nk l q N i
lq
p p | , S S SZ Z Z
p
S So d
One-dimensional Markov Chain
Dark Grey (Boundary Cells)Light Grey (Previously Generated Cells)W hite (Unknown Cells)
i-1 ,j i,j
i,j-1
1 ,1
N x ,N y
N x ,1
1 ,N y
N x ,j
, , 1, , 1
, 1, , 1 ,,
Unconditioinal Coupled Markov Chains
: Pr( | , ) . 1,...
Conditioinal Coupled Markov Chains
: Pr( | , , )x
h vlk mk
lm k i j k i j l i j m h vlf mf
f
i j k i j l i j m N j qlm k q
hlk
. p pp Z S Z S Z S k n
. p p
p Z S Z S Z S Z S
.p
( )
( ) , 1,... .x
x
h N i vkq mk
h h N i vlf fq mf
f
. p pk n
. . p p p
Coupled Markov Chain “CMC” in 2D
(Elfeki and Dekking, 2001)
CMC vs. Conventional Methods
CMC Conventional Methods
Based on conditional probability (transition matrix).
Based on variogram or autocovariance.
Marginal Probability. Sill.
Asymmetry can be described.
Asymmetry is impossible to describe.
A model of spatial dependence is not necessary.
A model of spatial dependence is needed for implementation.
Compute only the one-step transition and the model takes care of the n-step transition probability.
Need to compute many lags for the variogram or auto-correlations. (unreliable at large lags)
Schelluinen study area, The Netherlands
Soil Coding
Soil description
1 Channel deposits (sand)
2 Natural levee deposits (fine sand, sandy clay, silty clay)
3 Crevasse splay deposits (fine sand, sandy clay, silty clay)
4 Flood basin deposits (clay, humic clay)
5 Organic deposits (peaty clay, peat)
6 Subsoil (sand)
0 80 160 240-10
-5
0
0 200 400 600 800 1000 1200 1400 1600-10
-5
0
1 2 3 4 5 6
Data from Bierkens, 1994
Parameter Estimation and Procedure
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
G eo log ica l Im age
D om ain D iscretiza tion
G en era ted R ea liza tion
0 50 100 150 200-10
-5
0
S u p erp o sition o f th e G rid ov er th e G eo log ica l Im age a n d E stim ation o f T ra n sition P rob ab ility
B oreh o les L oca tion s0 50 100 150 200
-10
-5
0
P a ra m eters E stim a tio n C o n d itio n a l S im u la tio n
1
vv lklk n
vlq
q
Tp
T
Horizontal transition probability matrix of 1650 m section calculated over sampling intervals of 25 m.
Soil 1 2 3 4 5 6
1 0.979 0.004 0.001 0.006 0.009 0.001
2 0.020 0.965 0.001 0.008 0.006 0.000
3 0.003 0.002 0.966 0.013 0.016 0.000
4 0.000 0.001 0.009 0.983 0.007 0.000
5 0.001 0.001 0.006 0.007 0.984 0.001
6 0.000 0.000 0.001 0.000 0.002 0.997
Vertical transition probability matrix 1650 m section calculated over sampling intervals of 0.25 m.
Soil 1 2 3 4 5 61 0.945
0.000
0.009
0.000
0.009
0.037
2 0.071
0.796
0.021
0.041
0.071
0.000 3 0.000 0.000 0.797 0.086 0.089 0.028
4 0.003
0.013
0.041
0.714
0.222
0.007 5 0.004
0.012
0.047 0.119
0.768
0.050
6 0.000 0.000 0.000 0.000 0.000 1.000
Transition Probabilities (1650 x10 m)
Transition Probabilities (240 x10 m)
Horizontal transition probability matrix Vertical transition probability matrix
State 3 4 5 6 State 3 4 5 6
3 0.979 0.010 0.011 0.000 3 0.969 0.027 0.004 0.000
4 0.011 0.974 0.015 0.000 4 0.008 0.724 0.268 0.000
5 0.008 0.120 0.977 0.003 5 0.025 0.139 0.791 0.045
6 0.010 0.000 0.007 0.983 6 0.000 0.000 0.000 1.000
0 80 160 240-10
-5
0 3
4
5
6
Sampling intervalsDx = 2 m
Dy= 0.25 m
0.966 0.013 0.016 0.000
0.009 0.983 0.007 0.000
0.006 0.007 0.984 0.001
0.001 0.000 0.002 0.997
0.797 0.086 0.089 0.028
0.041 0.714 0.222 0.007
0.047 0.119 0.768 0.050
0.000 0.000 0.000 1.000
Horizontal Transition Probability from 1650x10
Vertical Transition Probability from 1650x10
Parameter Numerical ValueTime step 5 [day]Longitudinal dispersivity 0.1 [m]Transverse dispersivity 0.01 [m]Effective porosity 0.30 [-]Injected tracer mass 1000 [grams]Head difference at the site 1.0 [m]Monte-Carlo Runs 50 MCNumber of particles 10,000 [particles]
Physical and Simulation ParametersSoil Properties at the core scale from Bierkens, 1996 (Table 1).
Soil Coding
Soil type Wi
6 Fine & loamy sand 0.12 0.60 1.76 4.40 0.09
5 Peat 0.39 -2.00 1.7 0.30 2.99
3 Sand & silty clay 0.19 -4.97 3.49 0.1 5.86
4 Clay & humic clay 0.30 -7.00 2.49 0.01 10.1
2( )iLog K( )iLog K ( / )iK m day 2
iK
Convergence:~14000 IterationsAccuracy 0.00001
( , ) ( , ) 0
( , )
( , )
x
y
K x y K x yx x y y
K x yVx
K x yVy
Flow Model
Contam inant Source
P lum e at T im e, t
Im perm eable boundary
Im perm eable boundary
is the hydraulic head,
Vx and Vy are pore velocities, is the hydraulic conductivity, and is the effective porosity.
( , )K x y
Hydrodynamic Condition: Non-uniform Flow in the Mean due to Boundary Conditions.
Transport Model
Governing equation of solute transport :
C is concentration
Vx and Vy are pore velocities, and
Dxx , Dyy , Dxy , Dyx are pore-scale dispersion coefficients
x y xx xy yx yyC C C C C C CV V D D D Dt x y x x y y x y
* - i jmij ijL L T
VVD V D
V
*mD
ij
L
T
is effective molecular diffusion,
is delta function,
is longitudinal dispersivity, andis lateral dispersivity.
1 1
1 1
cos sin sin cos
. / . / . / . /
n n n np p x p p yL T L T
n n n np p x x y p p y y xL T L T
X X V t Z Z Y Y V t Z Z
X X V t Z V V Z V V Y Y V t Z V V Z V V
dispersive termadvective term
1 22 2xy yxx xp p x L T
D VD VX t t X t V t Z V t Z V tx y V V
1 22 2yx yy y xp p y L T
D D V VY t t Y t V t Z V t Z V tx y V V
The displacement is a normally distributed random variable, whose mean is the advective movement and whose deviation from the mean is the dispersive movement.
instantaneous injection + uniform flow
Random Walk Method
Application of SIS at the Site
Geological Section
Deterministic and
Stochastic Zones
InSIS Model
Bierkens, 1996
Comparison between CMC and SIS (1)
0 200 400 600 800 1000 1200 1400 1600-10
-5
0
Conditioning on half of the drillings
SIS Model Simulation
CMC Model Simulation
Geological Section
Bierkens, 1996
Comparison between CMC and SIS (2)
0 200 400 600 800 1000 1200 1400 1600-10
-5
0
Conditioning on all drillings
SIS Model Simulation
CMC Model Simulation
Geological Section
Bierkens, 1996
Monte-Carlo on CMC
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
1 2 3 4 5 6
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
0
0.25
0.5
0.75
1
0.00 500.00 1000.00 1500.00-10.00
-5.00
0.00
State # 6
S tate # 1 S tate # 2
S tate # 3
S tate # 4
S tate # 5
C oding of the S tates
Schem atic Im age of the G eolog ica l C ross Section
39 Boreholes for C ondition ing C onditioned S ing le R ealization
Effect of Conditioning on 240 x10m Sec.
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
1 2 3 4
Litho logy C oding
0 80 160 240-10
-5
0
1
2
3
4
31 boreholes
25 boreholes
9 boreholes
2 boreholes
Ensemble Indicator Function
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
2 Boreholes
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0.00 50.00 100.00 150.00 200.00-10.00
-5.00
0.00
0 0.25 0.5 0.75 1
3 Boreholes
5 Boreholes
9 Boreholes
21 Boreholes
25 Boreholes
31 Boreholes
0 50 100 150 200-10
-5
0
1 2 3 4
0 50 100 150 200 250Lag (m )
0
0.5
1
1.5
2
2.5
Gam
ma
(Ln
K)
O rig inal Section
Condition ing on 2 boreholes
Condition ing on 3 boreholes
Condition ing on 5 boreholes
Condition ing on 9 boreholes
Condition ing on 25 boreholes
0 2 4 6 8 10Lag (m )
0
4
8
12
16
Gam
ma
(Ln
K)
O rig inal Section
Condition ing on 2 boreholes
Condition ing on 3 boreholes
Condition ing on 5 boreholes
Condition ing on 9 boreholes
Condition ing on 25 boreholes
Effect of Conditioning on Variogram
( )
2
1
1( ) ( ) ( )
2 ( )
n
i ii
= Y Yn
s
s x x ss
Measure of Variability
Effect of Conditioning on S. R. Plume
mg/lit
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0
0.1
1
10
3 4
Litho logy C oding
6 5
T= 82 years
# drillings
2
3
5
9
25
31
Effect of Conditioning Single Realiz. Cmax
0 4 8 12 16 20 24 28 32No. of Conditioning Boreholes
0
40
80
120
160
200
240
Pea
k C
once
ntra
tion
(mg/
lit)
Single Realization C m ax (t = 34.2 Years)
S ingle Realization C m ax (t = 68.4 Years)
S ingle Realization C m ax (t = 95.8 Years)
S ingle Realization C m ax (t = 136.9 Years)
O rig inal Section (t = 34.2 Years)
O rig inal Section (t = 68.4 Years)
O rig inal Section (t = 95.8 Years)
O rig inal Section (t = 136.9 Years)
Practical convergence is reached after
about 21 boreholes
0 50 100 150 200-10
-5
0
First Moment (Single Realization)
0 10000 20000 30000 40000T im e (days)
0
20
40
60
80
100
120
X_C
oord
inat
e of
the
Cen
troi
d (m
)
O riginal Section
C ondition ing on 2 boreholes
C ondition ing on 3 boreholes
C ondition ing on 5 boreholes
C ondition ing on 9 boreholes
C ondition ing on 25 boreholes
0 10000 20000 30000 40000T im e (days)
-10
-8
-6
-4
-2
0
Y_
Coo
rdin
ate
of t
he
Ce
ntr
oid
(m
)
O riginal Section
C onditioning on 2 boreholes
C onditioning on 3 boreholes
C onditioning on 5 boreholes
C onditioning on 9 boreholes
C onditioning on 25 boreholes
Trend is reached at 3 boreholes
Convergence at 9 boreholes
Contam inant Source
P lum e at T im e, t
Im perm eable boundary
Im perm eable boundary
Second Moment (Single Realization)
0 10000 20000 30000 40000T im e (days)
0
0.5
1
1.5
2
2.5
Var
ianc
e in
Y_d
irect
ion
(m2)
O riginal Section
C ondition ing on 2 boreholes
C ondition ing on 3 boreholes
C ondition ing on 5 boreholes
C ondition ing on 9 boreholes
C ondition ing on 25 boreholes
0 10000 20000 30000 40000T im e (days)
0
1000
2000
3000
4000
Va
rian
ce in
X_
dir
ect
ion
(m
2)
O rig inal Section
C ondition ing on 2 boreholes
C ondition ing on 3 boreholes
C ondition ing on 5 boreholes
C ondition ing on 9 boreholes
C ondition ing on 25 boreholes
Trend is reached at 3 boreholes
Convergence at 5 and 25 boreholes
Convergence at 9 boreholes
Contam inant Source
P lum e at T im e, t
Im perm eable boundary
Im perm eable boundary
Breakthrough Curve (Single Realization)
0 10000 20000 30000 40000 50000T im e (days)
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
Mas
s D
istr
ibut
ion
O rig inal Section
Condition ing on 2 boreholes
Condition ing on 3 boreholes
Condition ing on 5 boreholes
Condition ing on 9 boreholes
Condition ing on 25 boreholes
0 50 100 150 200-10
-5
0
Convergence at 25 boreholes
Conditioning on 2 boreholes (Ensemble )
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 0.1 1 10
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
CactualC C
mg/lit
T = 4.1 years
T = 82.2 years
T = 136.9 years
Conditioning on 5 boreholes (Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 0.1 1 10mg/lit
actualC C C
Conditioning on 9 boreholes (Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
actualC C C
Conditioning on 21 boreholes(Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
actualC C C
Conditioning on 31 boreholes(Ensemble)
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
0 50 100 150 200-10
-5
0
actualC C C
Effect of Conditioning on Ensemble Cmax
0 4 8 12 16 20 24 28 32No. of Conditioning Boreholes
0
10
20
30
40
50
60
70
80
90
100
110
Ens
embl
e P
eak
Con
cent
ratio
n (m
g/lit
)
Ensem ble C m ax (t = 34.2 Years)
Ensem ble C m ax (t = 68.4 Years)
Ensem ble C m ax (t = 95.8 Years)
Ensem ble C m ax (t = 136.9 Years)
O rig inal Section (t = 34.2 Years)
O rig inal Section (t = 68.4 Years)
O rig inal Section (t = 95.8 Years)
O rig inal Section (t = 136.9 Years)
0 4 8 12 16 20 24 28 32No. of Conditioning Boreholes
0
1
2
3
4
5
6
CV
of C
max
t = 34.2 Years
t = 68.4 Years
t = 95.8 Years
t = 136.9 Years
max actualC C
max
1 for #boreholes 5
c
C
max
1 for #boreholes 5
c
C
max
time
c
C
Conclusions1. CMC model proved to be a valuable tool in predicting heterogeneous
geological structures which lead to reducing uncertainty in concentration distributions of contaminant plumes.
2. Comparison between SIS and CMC have shown more or less similar results in terms of the geological configuration of the confining layers. However, CMC has more merits over the SIS:
-some parts of the confining layers are treated determistically in SIS method which is not the case in CMC method.
-non-stationarity in the confining layers is treated straightforwardly by CMC, it is inherited in the method, while in SIS model subdivision into three sub-layers has to be performed.
-three variogram models with different parameters have been used in SIS, while direct transition probabilities were used in CMC.
3. Convergence to actual concentration is of oscillatory type, due to the fact that some layers are connected in one scenario and disconnected in another scenario.
Conclusions
4. In non-Gaussian fields, single realization concentration fields and the ensemble concentration fields are non-Gaussian in space with peak skewed to the left.
5. Reproduction of peak concentration, plume spatial moments and breakthrough curves in a single realization requires many conditioning boreholes (20-31 boreholes). However, reproduction of plume shapes require less boreholes (5 boreholes).
6. Ensemble concentration and ensemble variance have the same pattern. Ensemble variance is peaked at the location of the peak ensemble concentration and decreases when one goes far from the peak concentration. This supports early work by Rubin (1991). However, in Rubin’s case the maximum concentration was in the center of the plume which is attributed to Gaussian fields. The non-centered peak concentration, in this study, is attributed to the non-Gaussian fields.
Conclusions
7. Coefficient of variation of max concentration [CV(Cmax)] decreases significantly when conditioning on more than 5 boreholes.
5. Reproduction of peak concentration, plume spatial moments and breakthrough curves in a single realization requires many conditioning boreholes (20-31 boreholes). However, reproduction of plume shapes require less boreholes (5 boreholes).
6. Ensemble concentration and ensemble variance have the same pattern. Ensemble variance is peaked at the location of the peak ensemble concentration and decreases when one goes far from the peak concentration. This supports early work by Rubin (1991). However, in Rubin’s case the maximum concentration was in the center of the plume which is attributed to Gaussian fields. The non-centered peak concentration, in this study, is attributed to the non-Gaussian fields.
