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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 predictionsOutline

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

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

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

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

- Reduce parameter number drastically

Simulated annealing interlude on traveling salesman problem

Regularization

plausibility

- Which proportion between
- goodness of fit
- plausibility
- l?

p2

p2

p1

p1

Interpretation of regularizationp2

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…

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

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.