Jean raynald de dreuzy g osciences rennes cnrs france
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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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Interpretation of data for field-scale modeling and predictions

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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

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)

    • L: load (SL)

  • 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


Optimal and Acceptable risks


PhD. Etienne Bresciani (2008-2010)

Risk assessment for High Level Radioactive Waste storage


Outline

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

L=1km, Porosity=20%, head gradient=0.01

Localization of Ka and Kb?


Some flow and transport values

  • Extremal values

    • Kmin=Kb

    • Kmax=Ka

  • A random case

    • K~2.6 m/hr

    • Advection times


Equivalent permeability distribution for 10.000 realizations

Reality is a single realization


Consequences on transport

Reality is a single realization


Conditioning by permeability values


Conditioning BY head values


Solution

Ka


Outline

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives


Data interpolation

  • Accounting for correlation

  • Inverse of distance interpolation

  • Geostatistics

    • Kriging

    • Simulation

  • Field examples


Whatiscorrelation


Geostatistical simulation


Outline

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives


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.


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


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


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)


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


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


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]


Otherwise: Optimization of an objective

  • Objective function

    • Minimize head mismatch between model and data

[Carrera, 2005]


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


GW Flow & Transport

Simulated annealing interlude on traveling salesman problem


  • Addition of a permeability term

Regularization

plausibility

  • Which proportion between

  • goodness of fit

  • plausibility

  • l?


Illustration of regularization

“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]


Parameterization

  • Relevant parameterization depends

  • on data quantity

  • on geology

  • on optimization algorithm

[de Marsily, Cargèse, 2005]


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


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…


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.


GW Flow & Transport

Results for test cases 1 and 3


Outline

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives


Gary Larson, The far side gallery


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.


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