geohydrology ii (3)

Lecture (3)Lecture (3)

Uncertainty in Groundwater Flow and Transport Models.

0 50 100 150 200 250Horizontal 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=?

?

???

? ?

?

?

?

Layout of the LectureLayout of the Lecture

• What is uncertainty?What is uncertainty?• Why addressing the uncertainty by the Why addressing the uncertainty by the

stochastic approach?stochastic approach?• How to model uncertainty?How to model uncertainty?

Monte-Carlo sampling.Monte-Carlo sampling.• How to quantify uncertainty? How to quantify uncertainty?

Stochastic Differential Eqs. & Monte Carlo Stochastic Differential Eqs. & Monte Carlo method.method.

• How to reduce uncertainty?How to reduce uncertainty?• Some Applications.Some Applications.

What is Uncertainty?What is Uncertainty?

0 50 100 150 200 250Horizontal 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 bounaries.

Why Addressing the Uncertainty by the Why Addressing the Uncertainty by the Stochastic Approach?Stochastic Approach?

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

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

Monte-Carlo SamplingMonte-Carlo Sampling

Uniform random number generator:

Multiplicative Congruence Method developed by Lehmer [1951].

1( )i i-

i i

= MODULO A. B , MN N = /MU N

Ni is a pseudo-random integer, i is subscript of successive pseudo-random integers produced, i-1 is the immediately preceding integer, M is a large integer used as the modulus, A and B are integer constants used to govern the relationship in company with M, Ui is a pseudo-random number in the range {0,1}, and " MODULO" notation indicates that Ni is the remainder of the division of (A.Ni-1) by M.

Uniform Random Number ExampleUniform Random Number Example

1.0,5.0,3.0,9.0.......5,3,9,1,5,3,9:

)3(710737319*8

)9(0109911*8

)1(410414115*8

)5(210252513*8

)3(710737319*8

9)()10()18(

0

1

sequence

remainder

remainder

remainder

remainder

remainder

seedNMODULO N = N i-i

Generation of a Random Variable from any Generation of a Random Variable from any DistributionDistribution

Inverse of Distribution Function.

Transformation Method.

Acceptance-Rejection Method.

Inverse of Distribution FunctionInverse of Distribution Function

)()(

')'()(

UF = F = U

αdαf= αF

1-

α

-

Transformation MethodTransformation Method

Random number generator for normal distribution

(from central limit theory):" Observations which are the sum of many independently operating processes tend to be normally distributed as the number of effects becomes large"

12

21

m/

- m/Uε =

m

i=i

with mean (μ=0) and unit standard deviation (σ=1), Ui is the i-th element of a sequence of random numbers from a uniform distribution in the range {0,1}, and m is the number of Ui to be used.

612

1

- Uε = i

i

If m is 12, a normal distribution with tails truncated at six times standard deviation is produced

σ + εμα = αα

Example of IDF in Discrete 1D Markov ChainExample of IDF in Discrete 1D Markov Chain

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

qlq

k

qlq ,

1

1

1

A B C D

A B C D

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

Stochastic Differential Equations (Stochastic Differential Equations (SDEsSDEs))

Stochastic differential equation (SDE) = Differential equations for random functions (stochastic processes)

= Classical differential equation (DE) +

Random functions, coefficients, parameters and boundary or initial values,

e.g.

( , ) ( , ) 0

where ( , ) ( , ) are random space functions. and or stationary processes.

xx yy

xx yy

x y x y ΩK Kx x y y

x y x yKK

Solving SDEs (Stochastic Forward Problem)Solving SDEs (Stochastic Forward Problem)

Analytical Approaches G reen 's F u n c tion A p p roach

P ertu rb a tion M eth odS p ec tra l M eth od

Num erical ApproachesM on teC arlo M eth od

S o lvin g S D E s

Monte-Carlo MethodMonte-Carlo Method1. Generate a realization of a random field of the parameter under study.2. A classical numerical flow or/and transport model is run on the random field

and a set of results is obtained. 3. Another random field is made and the model is run again, and so on. 4. It's necessary to have a very large number of runs, and the output model

results corresponding to each input is obtained. 5. Statistical analysis of the ensemble of the output can be made to get the

mean, the variance, the covariance or the probability density function for each node with a location in the grid.

Monte-Carlo Method (Flow)Monte-Carlo Method (Flow)

1

22

1

1( ) ( ),

1( ) ( ) ( )

MC

kk

MC

kk

= MC

= MC

x x

x x x

Ensemble Flow Statistics

is the hydraulic head at a location x in the kth realization.

The ensemble average( )k x

( ) x

Coefficient of variation

2 ( ) x

2 2( ) 2 ( ) ( ) ( ) 2 ( ) x x x x x

( )( )( )

CV

xx

x

represents the uncertainty in the predictions.

Monte-Carlo Method (Transport)Monte-Carlo Method (Transport)

1

22

1

1( ) ( ),

1( , ) ( ) ( )

MC

kk

MC

C kk

C ,t = C ,tMC

t = C ,t C ,tMC

x x

x x x

Ensemble Transport Statistics

( )kC ,tx

( )C ,t x

is the concentration time t and a location x in the kth realization.

The ensemble average,

represents the uncertainty in the predictions. 2 ( , )C t x

2 2( ) 2 ( , ) ( ) ( ) 2 ( , )C CC ,t t C ,t C ,t t x x x x x

( , )( , )( , )

CC

tCV tC t

xx

x

Example of Monte Carlo Method for FLOWExample of Monte Carlo Method for FLOW

0 5 10 15 20 25-25

-20

-15

-10

-5

0

- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

2 . 5

3

3 . 5

4

4 . 5

5

Sing le Realiza tion ln (K )

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0 5 10 15 20 25-25

-20

-15

-10

-5

0

10 20 KBAR SDK

6 NO. OF CLASSES

25 25 LX LY

3 3 1000 lx ly Mc

1 1 dx dy

0.001 10000 eps maxit

5 4 upstream downstream

1 12 seed knorm

1 porosity

Random Field CorrelationRandom Field Correlation

0 5 10 15 20 25Separation Lag (m)

-0 .4

0

0.4

0.8

1.2

Aut

o_C

orre

latio

n {l

og(K

)}

Single R ealizationTheoretical C urveEnsem ble

Flow and Transport Domain Flow and Transport Domain

Lx = dx (N x-1)

Ly =

dy

(Ny-

1)

X

Y

(0,0)

Yo

XoDo

W o

H up Hdn

Single Realization Head FieldSingle Realization Head Field

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0 5 10 15 20 25-25

-20

-15

-10

-5

0

3.95 4.05 4.15 4.25 4.35 4.45 4.55 4.65 4.75 4.85 4.95

0 5 10 15 20 25-25

-20

-15

-10

-5

0

-0.14

-0.1

-0.06

-0.02

0.02

0.06

0.1

0.14

0.18

0.22

S ingle Realiza tion ln (K )

S ing le Realization (Head) Theoretica l Ensem ble Head Head Perturbation

Lx = dx (N x-1)

Ly =

dy

(Ny-

1)

X

Y

(0,0)

Yo

XoDo

W o

Hup Hdn

Single Realization of Darcy’s Fluxes Single Realization of Darcy’s Fluxes

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0 5 10 15 20 25-25

-20

-15

-10

-5

0

Darcy’s fluxes superimposed over the heterogeneity

Darcy’s fluxes superimposed over the hydraulic heads

Single Realization Head Gradient Profile Single Realization Head Gradient Profile

0 4 8 12 16 20Distance in the mean Flow direction (m )

-4

-2

0

2

4

Hea

d (m

)

M ean H eadS ingle R ealizationlog(K )- R ealization

0 4 8 12 16 20Distance in the mean Flow direction (m )

4

4.2

4.4

4.6

4.8

5

Hea

d (m

)

M ean HeadS ingle R ealiza tion

Ensemble Head FieldEnsemble Head Field

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0 5 10 15 20 25-25

-20

-15

-10

-5

0

3.95 4 .05 4.15 4.25 4.35 4.45 4.55 4.65 4.75 4.85 4 .95

0

0.005

0.01

0.015

0.02

0.025

0 5 10 15 20 25-25

-20

-15

-10

-5

0

H ead VarianceEnsem ble H ead

( , ) ( )up up downx

xH x y H H HL

1

22

1

1( ) ( ),

1( ) ( ) ( )

MC

kk

MC

kk

= MC

= MC

x x

x x x

0 2 4 6 8 10 12 14 16 18 20-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0

0.005

0.01

0.015

0.02

0.025

0 5 10 15 20 25-25

-20

-15

-10

-5

0

Ensem ble H ead

Head Variance ProfileHead Variance Profile

0 5 10 15 20 25-25

-20

-15

-10

-5

0

0

0.005

0.01

0.015

0.02

0.025

H ead Variance

2 2 2 2_

2 2 2 2_

2 2_ _

0.21 ln 0.2 sin bounded domain

0.46 unbounded domain

at 40

xh bounded x Y Y

Y x

h unbounded x Y Y

h bounded h unbounded x Y

L xJL

J

L

Lx = dx (N x-1)

Ly =

dy

(Ny-

1)

X

Y

(0,0)

Yo

XoD o

W o

Hup Hdn

0 2 4 6 8 10 12 1 4 16 18 20-2 0

-1 8

-1 6

-1 4

-1 2

-1 0

-8

-6

-4

-2

0

0

0.005

0.01

0.015

0.02

0.025

0 5 10 15 20 25-25

-20

-15

-10

-5

0

Ensem ble Head

0 4 8 12 16 20Distance in the mean Flow direction (m)

0

0.005

0.01

0.015

0.02

0.025

Var (

h)

X-direction

Solute Transport EquationSolute Transport Equation

Dispersion DiffusionAdvection

ij ii j i

C C CVDt x x x

where C is the concentration field at time t, Dij is the hydrodynamic dispersion tensor, i, j are counters, Vi is the component of the Eulerian interstitial velocity in xi direction defined as follows,

j

iji x

K

- = V

where Kij is the hydraulic conductivity tensor, and is the porosity of the medium.

Set-up of the Monte Carlo Transport Set-up of the Monte Carlo Transport Experiment Experiment

.Xc (t)

tx x

y y t(Xo,Yo)

(Xo,Yo) Initial Source Location.

Xc(t) is Plume centroid in X-direction.

2xx(t) is Plume longitudinal variance.

2yy(t) is Plume transverse variance.

Heterogeneous FieldHeterogeneous Field

2 7 12 17 22 27 32 37 42 47

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

K-field

Flow field

K (m/day)

Single Realization SimulationSingle Realization Simulation

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 .00 0 .50 5 .00 50 .00

tim e = 10 0 d ays

tim e = 40 0 d ays

tim e = 10 00 da ys

tim e = 13 00 da ys

C on cen tra tio n in m g /l

tim e = 60 0 days

Monte-Carlo Method ResultsMonte-Carlo Method Results

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 4 0 60 80 10 0 12 0 14 0 1 60 1 80 20 0-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 4 0 6 0 80 1 00 1 20 14 0 1 60 1 80 20 0-30

-20

-10

0

0 .00 0 .50 5 .0 0 50 .00

tim e = 100 days

tim e = 400 days

tim e = 100 0 days

tim e = 130 0 days

C oncentra tion in m g/l

tim e = 6 00 d ays

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 2 0 40 60 8 0 10 0 1 20 14 0 16 0 18 0 2 00-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

2 7 12 17 22 27 32 37 42 47

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200-30

-20

-10

0

0 2 0 4 0 6 0 8 0 10 0 12 0 1 40 16 0 18 0 20 0-30

-20

-10

0

C <C> C C

<C>____

Quantification of Uncertainties using Quantification of Uncertainties using CMCCMC

0 50 100 150 200 250 300-50

0

Single realization of the geological structure used in the experimentsFigure 1.

Classifications of Uncertainties:

Geological Uncertainty: Geological configuration.

Parameter Uncertainty: Conductivity value of each unit.

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 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.

Geological and Parameter UncertaintiesGeological and Parameter Uncertainties

Monte-Carlo Results for Geological Monte-Carlo Results for Geological UncertaintyUncertainty

0 50 10 0 150 200 250 30 0-50

0

0 50 10 0 150 200 250 30 0-50

0

0 50 10 0 150 200 250 30 0-50

0

0 50 10 0 150 200 250 30 0-50

0

0 50 100 150 200 250 300-50

0

0.00 0.01 0 .10 1.00

tim e = 200 da ys

tim e = 1000 days

tim e = 2000 days

tim e = 3000 days

C oncen tra tion in m g /l

tim e = 1600 days

0 50 100 1 50 200 25 0 30 0-50

0

0 50 100 1 50 200 25 0 30 0-50

0

0 50 100 1 50 200 25 0 30 0-50

0

0 50 100 1 50 200 25 0 30 0-50

0

0 50 10 0 150 200 250 300-50

0

0 .00 0.01 0 .10 1 .00

tim e = 20 0 d ays

tim e = 1000 days

tim e = 2000 days

tim e = 3000 days

C oncen tra tion in m g /l

tim e = 1600 days

0 50 100 150 200 250 300-50

0

0 50 100 150 200 250 300-50

0

0 50 100 150 200 250 300-50

0

0 50 100 150 200 250 300-50

0

0 50 100 150 200 250 300-50

0

0 .00 0 .01 0.10 1.00

tim e = 200 days

tim e = 1000 days

tim e = 2000 days

tim e = 3000 days

C oncen tra tion in m g/l

tim e = 1600 days

Conditional Simulations Conditional Simulations

From a practical point of view, it is desirable that the random fields

not only

reproduce the spatial structure of the field

but also

honour the measured data and their locations.

This requires an implementation of some kind of conditioning, so that the generated realizations are constrained to the available field measurements.

Representation of a Conditional SimulationRepresentation of a Conditional Simulation

Methods of ConditioningMethods of Conditioning

D ire c t M e th o ds"M e trica l M e th o d s"

In d ire c t M e th o ds"K rig ing M eth o d"

M e tho d s o f C o nd it ion ing

Conditioning in One-dimensional Markov Conditioning in One-dimensional Markov ChainChain

i0 1 i+ 1i-1 N2

S kS l S q

)1(

)(

)1(

)(

1

11

111

1

11

1

111

1

11

1

)(Pr

)Pr().|(Pr)(Pr).|(Pr).|(Pr

)(Pr

),(Pr),(Pr).|(Pr

)(Pr

),(Pr),(Pr),|(Pr

)(Pr

),(Pr),,Pr(

)(Pr

)(Pr

iNlq

iNkqlk

qlk

iNlq

lkiN

kqqNliki

liliqN

lilikikiqNqNliki

qNli

kilikiqNqNliki

qNli

kilikiliqNqNliki

qNli

qNkiliqNliki

qNliki

ppp

p

ppp

S Z ,S Z | S Z

SZS Z S Z S Z S Z S Z S Z S Z

S Z ,S Z | S Z

S Z S Z S Z S Z S Z S Z

S Z ,S Z | S Z

S Z S Z

S Z S Z S Z S ZS Z S Z ,S Z | S Z

S Z S Z S ZS ZS Z S Z ,S Z | S Z

S Z ,S Z | S Z

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

Reducing Geological Uncertainty by Reducing Geological Uncertainty by Conditioning (5 boreholes)Conditioning (5 boreholes)

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

Reducing Geological Uncertainty by Reducing Geological Uncertainty by Conditioning (9 boreholes)Conditioning (9 boreholes)

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

Reducing Geological Uncertainty by Reducing Geological Uncertainty by Conditioning (21 boreholes)Conditioning (21 boreholes)

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

Reducing Geological Uncertainty by Reducing Geological Uncertainty by Conditioning (31 boreholes)Conditioning (31 boreholes)

Simulation of the MADE1 Experiment

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

Plume Simulation in a Heterogeneous Plume Simulation in a Heterogeneous Aquifer (the MADE site)Aquifer (the MADE site)

0 50 100 150 200 250

-10

-5

0

Conclusions Conclusions

- Uncertainty is always present in any modelling step and in the collected data.

- The Stochastic approach is capable of modelling uncertainty regarding data, heterogeneity etc..

- The Monte Carlo method is a tool to quantify uncertainty.

- Reduction of uncertainty can be archived via conditioning on all the available data.