Upscaling and effective properties in saturated zone transport

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Upscaling and effective properties in saturated
zone transport

• Wolfgang Kinzelbach
• IHW, ETH Zürich

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Contents

• Why do we need upscaling
• Methods
• Examples where we have been successful
• When does upscaling not work
• Conclusions

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Dilemma in Hydrology

• Point-like process information available
• Regional statement required
• Point-like information is highly variable and
  stochastic
• Solutions to inverse problem are non-unique
• Predictions based on non-unique model are doubtful

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Multiscale processes

• Turbulence
• Catchment hydrology
• Flow and transport in porous media

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Possibilities for going from one scale to another

• Same law different parameter
• Diffusion-Dispersion
• Average transmissivity
• Different law
• Molecular dynamics-Gas law
• Fractal geometries
• Radioactive decay of mixture of radionuclides
• No general law for larger scale
• Singular features, non-linear processes
• Small cause - big effect situations

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Common Problem

• Few coefficients for summing up complex subscale
  processes
• No clear separation of scales
• Way out scale dependent coefficients

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Effective parameters in transport

• Ensemble mixing versus real mixing

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homogen
heterogen
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Grossskalige Heterogenität
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Heterogeneity and effective parameters
Cutoff
Small scale Details unknown Stochastic Repetiti
ve Modelled implicitly by parametrization
Large scale Explicitly known Deterministic Sing
ular features Modelled explicitly by flowfield
Differential advection Only after a long
distance (asymptotic regime) Equivalent to a
diffusive process called dispersion After a
shorter distance (preasymptotic) equivalent to a
dual-porous medium
mobile
immobile
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Hint for practical work After design on the
assumption of homogeneity, test your design with
a set of randomly generated media
An ideally designed dipole may possibly look like
that
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A robust design would be the one which survives a
large majority of a class of realistic random
samples
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Ways out

• New sources of conditioning information for some
  processes airborne geophysics, remote sensing
  from satellite of airplane platforms,
  environmental tracers
• Simulation of small scale and Monte Carlo
• Back to much simpler conceptual models
• Computations only with error estimate

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Model Concepts
How to cope with parameter uncertainty ?
Stochastic Modelling
Large Scale Modelling
Main interest on large observational scales
Quantification of the impact of Uncertainty
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Stochastic Modelling Approach
Different realizations of a catchment zone
Stauffer et al., WRR, 2002
Risk Assessment (question 3)
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Large Scale Modelling

• Large Scale Predictions
• Model on a regional scale 50 small
  scale lengths
• Resolution
  small scale length/5
• Number of unknowns
•  
• 
  
  

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Homogenization

• Large Scale Flow Models with effective
  conductivity

fine grid model
large grid model
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Homogenization
Asymptotic theory (scale separation between
observation scale and heterogeneity scale)

• Homogenization Theory
• Volume averaging
• Ensemble Averaging (if system ergodic)

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Homogenization
Large Scale Transport with effective
(advection-enhanced) dispersion
fine grid model
large grid model
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Limitations of Homogenization

• Problems where the scale of heterogeneities
• is not well separated from
• Observation scale
• Process scale velocity gradients, concentration
  gradients, mixing length scale

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Limitations of Homogenization
Natural Media Multiscale Media with Scale
Interactions, (no scale separation)
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Limitations of Homogenization
Question How to model scale interactions
(continuum of scales) ?
pre-asymptotic system with scale dependent
parameters
After Schulze-Makuch et al., GW, 1999
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Limitations of Homogenization
process scale 1. by flow geometry 2. by mixing
length scale (transient) 3. by concentration
fronts
Question How to avoid artificial averaging
effects?
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Multiscale Modelling

• Improved Approach accounts for pre-asymptotic
  effects
• Coarse Graining
  (Filter) Methods

fine grid model
coarse grid model
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Multiscale Modelling Theory

• Idea
• Spatial filter over all length scales smaller
  than cut off length scale

Equivalent in Fourier Space to
Attinger, J. Comp. GeoSciences,2003
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Multiscale Modelling Flow

• Fine scale flow model
• Filtered flow model

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Multiscale Modelling Flow

• Scale dependent mean conductivity
• (subscale effects)
• D2
•  
•  
• D3
•  

Attinger, J. Comp. GeoSciences,2003
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Multiscale Modelling Flow

• Statistical properties of the filtered
  conductivity fields

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Multiscale Modelling Transport
Fine scale transport model
   
Filtered transport model
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Multiscale Modelling Transport
Scale dependent macro dispersivities real
dispersivities plus artificial mixing
(centre-of-mass fluctuations)
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Multiscale Modelling Transport
Scale dependent real dispersivities
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Multiscale Modelling Transport
Transport Codes
Model with artificial dilution
Model with real dilution
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Reactive Fronts

• Travelling fronts
• Introduction of generalised spatial moment
  analysis
• (Attinger et al., MMS, 2003)

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Reactive Fronts
Travel time differences lead to artificial mixing
by Large Scale Filtering
Fine scale model
Large scale Model
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Reactive Fronts
Local Mixing Real Mixing
Fine scale model
Large scale Model
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Reactive Fronts
Travel time differences nonlocal macrodispersive
flux
Real mixing local real dispersive flux
Attinger et al., MMS, 2003
Dimitrova et al., AWR, 2003