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# Interpretation of data for field-scale modeling and predictions - PowerPoint PPT Presentation

Jean- Raynald de Dreuzy Géosciences Rennes, CNRS, FRANCE. Interpretation of data for field-scale modeling and predictions. Outline. A decision-based framework An elementary example Data interpolation Inverse Problem Conclusion: back to the objectives. Contaminant containment.

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Jean-Raynald de Dreuzy

Géosciences Rennes, CNRS, FRANCE

### Interpretation of data for field-scale modeling and predictions

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives

Contaminant containment

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

A decision-basedframework

• SM = C – L

• SM: safety margin

• C: capacity (SC)

• Probability of failure

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

Accounting for uncertainty

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

Decisionframework

Risk assessment for High Level Radioactive Waste storage

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives

An elementaryexample

Binary distribution of permeabilities Ka=10m/hr, Kb=2.4 m/day

Localization of Ka and Kb?

Some flow and transport values

• Extremal values

• Kmin=Kb

• Kmax=Ka

• A random case

• K~2.6 m/hr

Equivalent permeability distribution for 10.000 realizations

Reality is a single realization

Consequences on transport

Reality is a single realization

Conditioning by permeability values

Ka

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives

• Accounting for correlation

• Inverse of distance interpolation

• Geostatistics

• Kriging

• Simulation

• Field examples

Whatiscorrelation

Geostatistical simulation

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives

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

inverse problem (Identification of parameters)

direct problem

T: transmissivity

S: storage coefficient

Q: source terms

bc: boudary conditions

inverse problem

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

Inverse problem: automatic algorithm

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

• 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)

• 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

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

Example of CokrigING

• No direct problem

• Almost analytical

• Drawbacks

• Limited to low heterogeneities

• Requires lots of data

[Kitanidis,1997]

• Objective function

• Minimize head mismatch between model and data

[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

Simulated annealing interlude on traveling salesman problem

Regularization

plausibility

• Which proportion between

• goodness of fit

• plausibility

• l?

“True” medium

[Carrera, Cargèse, 2005]

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]

• 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

• 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…

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.

Results for test cases 1 and 3

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives

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.