Upscaling and effective properties in saturated zone transport Wolfgang Kinzelbach IHW, ETH Zürich.
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Transcript of Upscaling and effective properties in saturated zone transport Wolfgang Kinzelbach IHW, ETH Zürich.
![Page 1: Upscaling and effective properties in saturated zone transport Wolfgang Kinzelbach IHW, ETH Zürich.](https://reader030.fdocuments.us/reader030/viewer/2022032517/56649cb65503460f9497b8c9/html5/thumbnails/1.jpg)
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
Repetitive
Modelled implicitlyby parametrization
Large scale
Explicitly known
Deterministic
Singular features
Modelledexplicitly byflowfield
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
Quantification of the impact of Uncertainty
Main interest on large observational scales
How to cope with parameter uncertainty ?
Stochastic Modelling Large Scale Modelling
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Stochastic Modelling Approach
500 1000 1500
x_1 [m ]
200
700
1200
x_2
[m]
Stauffer et al., WRR, 2002
Different realizations of a catchment zone
Risk Assessment (question 3)
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Large Scale ModellingLarge Scale Predictions
• Model on a „regional“ scale : 50 „small“ scale lengths• Resolution : „small“ scale length/5
• Number of unknowns
00062515550 3 .. d
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HomogenizationLarge Scale Flow Models with effective conductivity
fine
grid
mod
el large grid m
odel
0 )()(~ xxKK 000 )(xK
K x( )
0K
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Homogenization
Homogenization Theory Volume averaging Ensemble Averaging (if system ergodic)
= Asymptotic theory (scale separation between observation scale and heterogeneity scale)
l L0
lL
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HomogenizationLarge Scale Transport with effective (advection-enhanced) dispersion
00 0 0 0 0tc x t u c x t D c x t( , ) ( , ) ( , )
fine
grid
mod
el large grid m
odel
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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
0l
Lobs
l L Lpr obs
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Limitations of HomogenizationNatural Media = Multiscale Media with Scale Interactions,
(no scale separation)
1l 2l 3l1
12
0ll
22
3
0ll
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Limitations of Homogenization
After Schulze-Makuch et al., GW, 1999
Question: How to model scale interactions
(continuum of scales) ?
pre-asymptotic
system
with scale dependent parameters
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Limitations of Homogenization
Question: How to avoid artificial averaging effects?
process scale
1. by flow geometry
2. by mixing length scale (transient)
3. by concentration fronts
l LLpr obs
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Multiscale ModellingImproved Approach: accounts for pre-asymptotic effects
Coarse Graining (Filter) Methods
fine
grid
mod
el coarse grid m
odel
0 )()(~ xxKK 0
)()(~)(eff xxKK
l L
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Multiscale Modelling: Theory
Idea:
Spatial filter over all length scales smaller than cut –off length scale
λ
)()( yxfydxf d
d
2
1
Attinger, J. Comp. GeoSciences,20031kf k( )
Equivalent in Fourier Space to
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Multiscale Modelling: Flow
Fine scale flow model
Filtered flow model
0 )()(~ xxKK
0 )()(~)(eff xxKK
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Multiscale Modelling: Flow
Scale dependent mean conductivity
(subscale effects)
D=2:
D=3:
)exp()(eff22
22
2
1
l
lKK fg
)exp()(/
eff
23
22
222
3
1
6
1
l
lKK ffg
Attinger, J. Comp. GeoSciences,2003
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Multiscale Modelling: Flow
Statistical properties of the filtered conductivity fields
2
2222
1
1/
/
d
ff l
21221/
/ lll
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Multiscale Modelling: Transport
0t c x t u x c x t D t c x teff( , ) ( ) ( , ) ( , ) ( , )
Fine scale transport model
Filtered transport model
0tc x t u x c x t D c x t( , ) ( ) ( , ) ( , )
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Multiscale Modelling: Transport
211 11 1 22
2
11
1Hf d
T
t l
l
macro( )/( , )
Scale dependent macro dispersivities:
real dispersivities plus artificial mixing (centre-of-mass
fluctuations)
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Multiscale Modelling: Transport
11 11 11
211 11 1 2
2
211 1 2 1 2
2
2 22
114
1
1 114
11
f dT
f d dT
t t t
t lD tl
t lD t
ll
eff eff eff
eff( )/
eff( )/ ( )/
( , ) ,
( )
,
Scale dependent real dispersivities
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Model with real dilution
Model with artificial dilution
Transport Codes
1/2
2V
T
1/2
2H
T
H2f
011
eff11
l
t4D1
l
t4D1
11lσα(t)α
VH
Multiscale Modelling: Transport
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Reactive Fronts
Travelling fronts• Introduction of generalised spatial moment analysis
(Attinger et al., MMS, 2003)
cDcxuckc ptdt 1
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Reactive Fronts
Fin
e s
cale
mo
del L
arge
scale
Mo
del
'''' xcxxDxdcuckc dptdt 01
Travel time differences lead to artificial mixing by Large Scale Filtering
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Reactive Fronts
Fin
e s
cale
mo
del L
arge
scale
Mo
del
cDcuckc ptdt eff01
Local Mixing = Real Mixing
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Reactive Fronts
Travel time differences
=nonlocal macrodispersive flux
Real mixing
= local real dispersive flux
Attinger et al., MMS, 2003Dimitrova et al., AWR, 2003