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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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5Freeze, R. A., et al. (1990), Hydrogeological Decision-Analysis .1. a Framework, Ground Water, 28(5), 738-766.
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):
6Freeze, R. A., et al. (1990), Hydrogeological Decision-Analysis .1. a Framework, Ground Water, 28(5), 738-766.
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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
A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives
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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
A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives
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Accounting for correlation Inverse of distance interpolation Geostatistics
Kriging Simulation
Field examples
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SGSIM seq_fill.mpeg
A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives
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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
18/05/2008GW Flow & Transport
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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18/05/2008GW Flow & Transport
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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
18/05/2008GW Flow & Transport
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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.
18/05/2008GW Flow & Transport
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A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives
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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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