Reducing Uncertainty of Groundwater Contaminant Transport

Using Markov Chains

Amro ElfekiDept. of Hydrology and Water Resources,

Faculty of Meteorology, Environment and Arid Land Agriculture, KAU, Jeddah, KSA.

On leave of absence from: Faculty of Civil Engineering, Mansoura University, Egypt

Elfeki_amr@yahoo.co.uk

Outlines• Definitions. • 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 IssuesMotivation of this research:

• Test the applicability of CMC model on field data at many sites.

• Incorporating CMC model in flow and transport models to study uncertainty in groundwater transport.

• 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.

 Figure 1. Huesca outcrop, Spain, Courtesy Kees Geel (from Dept. of Geology, Faculty of Applied Earth Sciences, TU Delft, The Netherlands). 

Geological Structure

Typical Problem of Groundwater Contamination

0 50 100 150 200 250

Horizontal Distance between W ells (m)

-50

0

Dep

th (m

)

W ell 1 W ell 2

? K(x,y,z)?

(x,y,z)?

C(x,y,z)?H=?

H=?

?

???

? ?

?

?

?

Classification of Uncertainty:

-Conceptual Model Uncertainty:Darcy’s and Fick’s Laws.

-Geological Uncertainty:Connectivity and dis-connectivity of the layers, geological sequence,boundaries between geological units.

-Parameter Uncertainty: -K, porosity.

-Hydro-geological Uncertainty:Constant head boundaries, impermeable boundaries, Plume boundaries, source area boundaries.

What is Uncertainty?

- The lack of information about the subsurface structure which is known only at sparse sampled locations.

- The erratic nature of the subsurface parameters observed at field scale.

Why Addressing Uncertainty by Stochastic Approach?

Courtesy lynn Gelhar

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

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0

G eology is Certa in and Param eters are U ncerta in

G eology is Uncerta 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.

Mg/l

Application of CMC at MADE Site

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Elfeki, (2003 ) Journal of Hydraulic Reserach

Real field situation:

MAcro-Disperison Experiment (MADE)Columbus, MississippiAir Force Base Site in US

Data is in the form of boreholes.

Geological prediction is needed at unsampled locations.

Boggs et al. (1990)

Application of CMC at MADE Site

( )

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

(Elfeki and Dekking, 2001)

1,...n= l 2,...n,=k p Upk

qlq

k

qlq ,

1

1

1

11 12 1

21

1

. .

. . . . . . . .

. . . . .. . .

n

lk

n nn

p p pp

pl

p p

1 2 ... ... n12

.n

p

11 11 12 11

21

1

11

. .12 . . . .

. . . .

. . . . .

. . .

n

ii

k

lii

n

n nii

p p p p

p

pl

np p

1 2 ... ... n

P

A B C D

One-dimensional Markov Chain (Cont.)

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

, , 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)

Concept of Unconditional Realizations (CMC)

Concept of Unconditional Realizations (CMC)-Cont.

0 50 100 150 200 250 300-80

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1 2 3 4 5 6 7 8

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Concept of Conditional Realizations (CMC)

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7

8

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0

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

Study Area

(Bierkens, 94).

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

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0

1 2 3 4 5 6

Data from Bierkens, 1994

Parameter Estimation and Procedure

  

0 50 100 15 0 20 0-10

-5

0

0 50 100 15 0 20 0-10

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0 50 100 150 200-10

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0

G eolo g ica l Im ag e

D om ain D iscretiza tion

G en erated R ealiza tio n

0 50 100 15 0 20 0-10

-5

0

S u p erp ositio n o f th e G rid o ver th e G eo log ica l Im ag e an d E stim a tio n o f T ran sition P rob a b ility

B o reh o les L o cation s0 50 10 0 15 0 200

-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

TpT

Horizontal transition probability matrix of 1650 m section calculated over sampling intervals of 25 m.

Soil 1 2 3 4 5 61 0.979 0.004 0.001 0.006 0.009 0.0012 0.020 0.965 0.001 0.008 0.006 0.0003 0.003 0.002 0.966 0.013 0.016 0.0004 0.000 0.001 0.009 0.983 0.007 0.0005 0.001 0.001 0.006 0.007 0.984 0.0016 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 0.0284 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 yV x

K x yV y

Groundwater 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 TVV

D V DV

*mDij

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 eologica l C ross Section

39 Boreholes for C ond ition ing C onditioned S ingle R ealiza tion

Effect of Conditioning on 240 x10m Sec.

0 50 100 150 200-10

-5

0

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-5

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0

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0

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-5

0

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-5

0

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0

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0

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0

1 2 3 4

Litho logy C oding

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-5

0

1

2

3

4

31 boreholes

25 boreholes

9 boreholes

2 boreholes

Effect of Conditioning on S. R. Plume

mg/lit

0 50 100 150 200-10

-5

0

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-5

0

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-5

0

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-5

0

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-5

0

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0

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0

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-5

0

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-5

0

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-5

0

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-5

0

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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

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0

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0

0 50 100 150 200-10

-5

0

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-5

0

0

0.1

1

10

3 4

Lithology 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

troid

(m)

O riginal SectionC ondition ing on 2 boreholesC ondition ing on 3 boreholesC ondition ing on 5 boreholesC ondition ing on 9 boreholesC ondition ing on 25 boreho les

0 10000 20000 30000 40000T im e (days)

-10

-8

-6

-4

-2

0

Y_C

oord

inat

e of

the

Cen

troid

(m)

O riginal SectionConditioning on 2 boreholesConditioning on 3 boreholesConditioning on 5 boreholesConditioning on 9 boreholesConditioning on 25 boreho les

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 SectionC onditioning on 2 boreholesC onditioning on 3 boreholesC onditioning on 5 boreholesC onditioning on 9 boreholesC onditioning on 25 boreholes

0 10000 20000 30000 40000T im e (days)

0

1000

2000

3000

4000

Var

ianc

e in

X_d

irect

ion

(m2 )

O rig inal SectionCondition ing on 2 boreho lesCondition ing on 3 boreho lesCondition ing on 5 boreho lesCondition ing on 9 boreho lesCondition ing on 25 boreholes

Trend is reached at 3 boreholes

Convergence at 5 and 25 boreholes

Convergence at 9 boreholes

Contam inant S ource

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

istri

butio

n

O rig inal SectionC onditioning on 2 boreholesC onditioning on 3 boreholesC onditioning on 5 boreholesC onditioning on 9 boreholesC onditioning 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

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-5

0

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0

0 0.1 1 10

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-5

0

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0

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0

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0

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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

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-5

0

0 50 100 150 200-10

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0

0 50 100 150 200-10

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0

0 50 100 150 200-10

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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 Y earst = 68.4 Y earst = 95.8 Y earst = 136.9 Y ears

max actualC C

max

1 for #boreholes 5 c

C

max

1 for #boreholes 5c

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. 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.

3. In non-Gaussian fields, single realization concentration fields and the ensemble concentration fields are non-Gaussian in space with peak skewed to the left.

4. 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).

Conclusions

5. 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.

6. Coefficient of variation of max concentration [CV(Cmax)] decreases significantly when conditioning is performed on more than 5

boreholes.

7. 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).