Interpretation of data for field-scale modeling and predictions

1. Jean-Raynald de Dreuzy Géosciences Rennes, CNRS, FRANCE Interpretation of data for field-scale modeling and predictions

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

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

4. A decision-basedframework • SM = C – L • SM: safety margin • C: capacity (SC) • L: load (SL) • Probability of failure

5. A decision-basedframework Freeze, R. A., et al. (1990), Hydrogeological Decision-Analysis .1. a Framework, Ground Water, 28(5), 738-766. • Objective function of alternative j: Fj • Benefits of alternative j: Bj • Costs of alternative j: Cj • Risk of alternative j: Rj • Probability of failure: Pf • Costassociatedwithfailure: Cf • Utility function (risk aversion): g

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

7. Decisionframework

8. Optimal and Acceptable risks

9. PhD. Etienne Bresciani (2008-2010) Risk assessment for High Level Radioactive Waste storage

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

11. An elementaryexample 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?

12. Some flow and transport values • Extremal values • Kmin=Kb • Kmax=Ka • A random case • K~2.6 m/hr • Advection times

13. Equivalent permeability distribution for 10.000 realizations Reality is a single realization

14. Consequences on transport Reality is a single realization

15. Conditioning by permeability values

16. Conditioning BY head values

17. Solution Ka

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

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

20. Whatiscorrelation

21. Geostatistical simulation

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

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

24. inverse problem (Identification of parameters) direct problem T: transmissivity S: storage coefficient Q: source terms bc: boudary conditions h: head inverse problem Trial and error approach: manually change T, S, Q in order to reach a good fit with h Inverse problem: automatic algorithm

25. inverse problem (Identification of parameters) h(xi) T(xi) i:1…n bc? direct problem T: transmissivity bc: boudary conditions inverse problem Ill-posed problem Under-constrainted (more unknowns than data) km

26. Specificities of inverse problem in hydrogeology • 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)

27. First approach: Cauchy problem • 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

28. Use of geostatistics and cokrigeage • 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 • Principle: express permeability as a linear function of known permeability and head values

29. GW Flow & Transport Example of CokrigING • Advantages • No direct problem • Almost analytical • Additional knowledge on uncertainties • Drawbacks • Limited to low heterogeneities • Requires lots of data [Kitanidis,1997]

30. Otherwise: Optimization of an objective • Objective function • Minimize head mismatch between model and data [Carrera, 2005]

31. Inverse problem issues • 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

32. GW Flow & Transport Simulated annealing interlude on traveling salesman problem

33. Addition of a permeability term Regularization plausibility • Which proportion between • goodness of fit • plausibility • l?

34. Illustration of regularization “True” medium [Carrera, Cargèse, 2005]

35. p2 p2 p1 p1 Interpretation of regularization p2 Long narrow valleys Hard convergence and instability p1 Reduces uncertainty Smooths long narrow valleys Facilitates convergence Reduces instability and non-uniqueness [Carrera, Cargese, 2005]

36. Parameterization • Relevant parameterization depends • on data quantity • on geology • on optimization algorithm [de Marsily, Cargèse, 2005]

37. Comparison of 7 inverse methods • 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

38. Results • 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…

39. GW Flow & Transport Test cases 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.

40. GW Flow & Transport Results for test cases 1 and 3

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

42. Gary Larson, The far side gallery

43. Consequence of data scarcity and geological complexity: UNCERTAINTY 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.