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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Jean raynald de dreuzy g osciences rennes cnrs france

Jean-Raynald de Dreuzy

Géosciences Rennes, CNRS, FRANCE

Interpretation of data for field-scale modeling and predictions


Outline
Outline

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives


Contaminant containment
Contaminant containment

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


A decision based framework
A decision-basedframework

  • SM = C – L

    • SM: safety margin

    • C: capacity (SC)

    • L: load (SL)

  • Probability of failure


A decision based framework1
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
Accounting for uncertainty

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


Decision framework
Decisionframework



PhD. Etienne Bresciani (2008-2010)

Risk assessment for High Level Radioactive Waste storage


Outline1
Outline

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives


An elementary example
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
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
Equivalent permeability distribution for 10.000 realizations

Reality is a single realization


Consequences on transport
Consequences on transport

Reality is a single realization


Conditioning by permeability values
Conditioning by permeability values


Conditioning by head values
Conditioning BY head values



Outline2
Outline

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives


Data interpolation
Data interpolation

  • Accounting for correlation

  • Inverse of distance interpolation

  • Geostatistics

    • Kriging

    • Simulation

  • Field examples


What is correlation
Whatiscorrelation


Geostatistical simulation
Geostatistical simulation


Outline3
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
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 parameters1
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
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
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
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

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
Otherwise: Optimization of an objective

  • Objective function

    • Minimize head mismatch between model and data

[Carrera, 2005]


Inverse problem issues
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


Simulated annealing interlude on traveling salesman problem

GW Flow & Transport

Simulated annealing interlude on traveling salesman problem


Regularization

Regularization

plausibility

  • Which proportion between

  • goodness of fit

  • plausibility

  • l?


Illustration of regularization
Illustration of regularization

“True” medium

[Carrera, Cargèse, 2005]


Interpretation of regularization

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
Parameterization

  • Relevant parameterization depends

  • on data quantity

  • on geology

  • on optimization algorithm

[de Marsily, Cargèse, 2005]


Comparison of 7 inverse methods
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
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…


Test cases

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.


Results for test cases 1 and 3

GW Flow & Transport

Results for test cases 1 and 3


Outline4
Outline

A decision-based framework

An elementary example

Data interpolation

Inverse Problem

Conclusion: back to the objectives



Consequence of data scarcity and geological complexity uncertainty
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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