Jean-Raynald de Dreuzy Géosciences Rennes, CNRS, FRANCE

Jean-Raynald de DreuzyGéosciences Rennes, CNRS, FRANCE

A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives

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3Freeze, R. A., et al. (1990), Hydrogeological Decision-Analysis .1. a Framework, Ground Water, 28(5), 738-766.

SM = C – L SM: safety margin C: capacity (SC) L: load (SL)

Probability of failure

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Objective function of alternative j: j Benefits of alternative j: Bj Costs of alternative j: Cj Risk of alternative j: Rj Probability of failure: Pf Cost associated with failure: Cf Utility function (risk aversion):


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PhD. Etienne Bresciani (2008-2010)

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Ris

k ass

ess

ment

for

Hig

h L

evel R

ad

ioact

ive W

ast

e s

tora

ge

ran g e o f scén a rio s (~ 1 0 .0 0 0 )

P erm eab ility

Im p erv io u s m ed ia P erm eab le m ed iam /cen tu ry m /y ea r m /d ay

L eakag e riskN a tu ra l m ed ium (un kn ow n )

C lay

G ran ite

P erm eab ility

Im p erv io u s m ed ia P erm eab le m ed iam /cen tu ry m /y ea r m /d ay

L eakag e riskN a tu ra l m ed ium (un kn ow n )

G ran ite


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Binary distribution of permeabilities Ka=10m/hr, Kb=2.4 m/dayL=1km, Porosity=20%, head gradient=0.01

Localization of Ka and Kb?

Extremal values Kmin=Kb Kmax=Ka

A random case K~2.6 m/hr Advection times

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Valeur de t Probabilité p(t)7.106 s=83 jours

=0.23 ans

97 %

11.6 ans 2%7.108 s=23 ans 1%

anst 88.0 anst 8.2

13ba KKK *Reality

is a sin

gle realiza

tion

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Reality is

a single re

alizatio

n

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Conditioning by <<t>R>NR <t>R]NR

Absence 5,2 4,3KA 5,7 4,2KB 2,4 4,3KD 6,2 4,3KA, KB 4,9 4,1KA, KC 4,8 4,1KB, KC 4,5 4KB, KD 5,7 4,3KA, KB, KD 4,3 3,9KA, KB, KC, KD 4,5 3,8

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Conditioning by NR <<t>R>NR <t>R]NR

Absence 2 105 5,2 4,3[hA] 2.5 104 4,3 4,3[hB] 2 104 2,8 3,4[hD] 1602 4,4 4,4[hA], [hB] 700 2,7 3,3[hA], [hC] 1100 2,9 3,8[hB], [hC] 200 1,4 1,7[hB], [hD] 172 1,7 1,9[hA], [hB], [hC] 400 1,6 1,8[hA], [hB], [hD] 36 1,3 0,2[hA], [hB], [hC], [hD] 17 1,4 0,1

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grid flow

Ka


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Accounting for correlation Inverse of distance interpolation Geostatistics

Kriging Simulation

Field examples

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SGSIM seq_fill.mpeg


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18/05/2008GW Flow & Transport

23Carrera, J., A. Alcolea, A. Medina, J. Hidalgo, and L. J. Slooten (2005), Inverse problem in hydrogeology, Hydrogeology Journal, 13, 206-222.

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conditionsinitial

conditionsboudarybc

Qt

hS

y

hT

yx

hT

x

:

T: transmissivityS: storage coefficient

Q: source termsbc: boudary conditions

h: head

direct problem

inverse problem

Trial and error approach: manually change T, S, Q in order to reach a good fit with h

Inverse problem: automatic algorithm

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conditionsboudarybc

y

hT

yx

hT

x

:

0

T: transmissivitybc: boudary conditions

h(xi)T(xi)

i:1…nbc?

direct problem

inverse problem

Ill-posed problemUnder-constrainted (more unknowns than data)

Model uncertainty: structure of the medium (geology, geophysics) not known accurately (soft data)

Heterogeneity: T varies over orders of magnitude Low sensitivity: data (h) may contain little

information on parameters (T) Scale dependence: parameters measured in the

field are often taken at a scale different from the mesh scale

Time dependence: data (h) depend on time Different parameters (unrelated): beyond T,

porosity, storativity, dispersivity Different data: simultaneous integration of

hydraulic, geophysical, geochemical (hard data)

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Interpolation of heads Determination of flow tubes

Each tube contains a known permeability value Determination of head everywhere by:

Drawbacks Instable (small h0 errors induce large T0 errors) Strong unrealistic transmissivity gaps between

flow tubes Independence between transmissivity obtained

between flow tubes

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h

hdh

h

h

T

T0

2

2

0

ln

Principle: express permeability as a linear function of known permeability and head values

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n

i

m

jjjiii YhxY

1 10

ˆ

The Co-kriging equation uses the measured values of Φ =h-H, of Y, and the strcutures (covaraince, variogram, cross-variogram of Y, Φ and Y- Φ) which are known (Y) or calculated analytically from the stochastic PDE.

The inverse problem is thus solved without having to run the direct problem and to define an objective function.

Sometimes the covariance of Y is assumed known with an unknown coefficient which is optimized by cross-validation at points of known Y


29[Kitanidis,1997]

Advantages No direct problem Almost analytical Additional knowledge on uncertainties

Drawbacks Limited to low heterogeneitiesRequires lots of data

Objective function Minimize head mismatch between model

and data

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2

1

h

obsn

i i

im

iobs

objhh

pF

[Carrera, 2005]

Unstable parameters from data Restricts instability of the objective funtion Solution: regularization

More parameters than data (under-constrained) Reduce parameter number drastically

Reduce parameter space Acceptable number of parameters

gradient algorithms requiring convex functions: <5-7 parameters

Monté-Carlo algorithms: <15-20 parameters Solution: parameterization

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Addition of a permeability term

2

1

2

1

K

obs

h

obs n

i i

im

iobs

n

i i

im

iobs

objKKhh

pF

plausibility

Which proportion between goodness of fit plausibility?

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B-1B-2

B-3

O-1

O-2

O-3O-4O-5

O-6

O-7

O-8O-9

O-10

0.5 -0.46 -1.4 -2.4 -3.3 -4.3 -5.3 -6.2 -7.2 -8.1 -9.1

“True” medium

[Carrera, Cargèse, 2005]

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p2

p1 p2

p1

t* 1p pF *p- p V p- p

t 1 *h hF *h- h V h- h

p2

p1

phF F F

Reduces uncertainty

Smooths long narrow valleys

Facilitates convergence

Reduces instability and non-uniqueness

Long narrow valleys

Hard convergence and instability

[Carrera, Cargese, 2005]

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Relevant parameterization depends on data quantity on geology on optimization algorithm

[de Marsily, Cargèse, 2005]

Zimmerman, Marsily, Carrera et al, 1998 for stochastic simulations, 4 test problems 3 based on co-kriging Carrera-Neuman, Bayesian, zoning Lavenue-Marsily, pilot points Gomez-Hernandez, Sequantial non

Gaussian Fractal ad-hoc method

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If test problem is a geostatistical field, and variance of Y not too large, (variance of Log10T less than 1.5 to 2) all methods perform well Importance of good selection of variogram Co-kriging methods that fit the variogram by

cross-validation on both Y and h’ data perform better

For non-stationary “complex” fields The linearized techniques start to break down Improvement is possible, e.g. through zoning Non-linear methods, and with a careful fitting of

the variogram, perform better The experience and skill of the modeller makes

a big difference…38


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K: 6 teintes de gris couvrant chacune un ordre de grandeur entre 10-7 et 10-2. cas 1: champ gaussien de log transmissivité (log10(T)) moyenne et variance de -5.5 et 1.5 respectivement et de longueur de corrélation 2800 m. cas 2 moyenne et variance plus importantes de -1.26 et 2.39. cas 3 comprend un milieu hétérogéne de log transmissivité et variance -5.5 et 0.8 et des chenaux de log transmissivité -2.5. cas 4 comprend de larges chenaux de forte transmissivité avec une distribution de log transmissivité de moyenne et variance -5.3 et 1.9.


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Gary Larson, The far side gallery 42

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Example of protection zone delineation

Pochon, A., et al. (2008), Groundwater protection in fractured media: a vulnerability-based approach for delineating protection zones in Switzerland, Hydrogeology Journal, 16(7), 1267-1281.

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Jean-Raynald de Dreuzy Géosciences Rennes, CNRS, FRANCE