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Jean-Raynald de Dreuzy Géosciences Rennes, CNRS, FRANCE

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A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives 2

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

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SM = C – L SM: safety margin C: capacity (SC) L: load (SL) Probability of failure 4

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5 Freeze, 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):

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6 Freeze, 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) 9 Risk assessment for High Level Radioactive Waste storage

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A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives 10

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11 Binary distribution of permeabilities Ka=10m/hr, Kb=2.4 m/day L=1km, Porosity=20%, head gradient=0.01 Localization of Ka and Kb?

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Extremal values Kmin=Kb Kmax=Ka A random case K~2.6 m/hr Advection times 12 Valeur de tProbabilité p(t) 7.10 6 s=83 jours =0.23 ans 97 % 11.6 ans2% 7.10 8 s=23 ans1%

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13 Reality is a single realization

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14 Reality is a single realization

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15 Conditioning by R > NR R ] NR Absence5,24,3 KAKA 5,74,2 KBKB 2,44,3 KDKD 6,24,3 K A, K B 4,94,1 K A, K C 4,84,1 K B, K C 4,54 K B, K D 5,74,3 K A, K B, K D 4,33,9 K A, K B, K C, K D 4,53,8

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16 Conditioning byNR R > NR R ] NR Absence2 10 5 5,24,3 [hA][hA]2.5 10 4 4,3 [hB][hB]2 10 4 2,83,4 [hD][hD]16024,4 [h A ], [h B ]7002,73,3 [h A ], [h C ]11002,93,8 [h B ], [h C ]2001,41,7 [h B ], [h D ]1721,71,9 [h A ], [h B ], [h C ]4001,61,8 [h A ], [h B ], [h D ]361,30,2 [h A ], [h B ], [h C ], [h D ]171,40,1

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17 gridflow Ka

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A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives 18

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Accounting for correlation Inverse of distance interpolation Geostatistics Kriging Simulation Field examples 19

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A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives 22

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18/05/2008GW Flow & Transport 23 Carrera, 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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24 T: transmissivity S: storage coefficient Q: source terms bc: 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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25 T: transmissivity bc: boudary conditions h(x i ) T(x i ) i:1…n bc? direct problem inverse problem Ill-posed problem Under-constrainted (more unknowns than data) km

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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) 26

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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 h 0 errors induce large T 0 errors) Strong unrealistic transmissivity gaps between flow tubes Independence between transmissivity obtained between flow tubes 27

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Principle: express permeability as a linear function of known permeability and head values 28 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

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18/05/2008GW Flow & Transport 29 [Kitanidis,1997] Advantages No direct problem Almost analytical Additional knowledge on uncertainties Drawbacks Limited to low heterogeneities Requires lots of data

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Objective function Minimize head mismatch between model and data 30 [Carrera, 2005]

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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 31

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18/05/2008GW Flow & Transport 32

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33 Addition of a permeability term plausibility Which proportion between goodness of fit plausibility ?

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34 True medium [Carrera, Cargèse, 2005]

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35 p2 p2 p1p1 p 2 p1p1 p1p1 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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36 Relevant parameterization depends on data quantity on geology on optimization algorithm [de Marsily, Cargèse, 2005]

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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 37

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If test problem is a geostatistical field, and variance of Y not too large, (variance of Log 10 T 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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18/05/2008GW Flow & Transport 39 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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18/05/2008GW Flow & Transport 40

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A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives 41

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Gary Larson, The far side gallery 42

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43 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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