Upscaling and effective properties in saturated zone

transport

Wolfgang Kinzelbach

IHW, ETH Zürich

Contents

• Why do we need upscaling

• Methods

• Examples where we have been successful

• When does upscaling not work

• Conclusions

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

Multiscale processes

• Turbulence

• Catchment hydrology

• Flow and transport in porous media

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

Common Problem

• Few coefficients for summing up complex subscale processes

• No clear separation of scales

• Way out: scale dependent coefficients

Effective parameters in transport

• Ensemble mixing versus real mixing

homogen heterogen

Grossskalige Heterogenität

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

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:

A robust design would be the one which survives a large majority of a class of realistic random samples

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

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

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)

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

HomogenizationLarge Scale Flow Models with effective conductivity

fine

grid

mod

el large grid m

odel

0 )()(~ xxKK 000 )(xK

K x( )

0K

Homogenization

Homogenization Theory Volume averaging Ensemble Averaging (if system ergodic)

= Asymptotic theory (scale separation between observation scale and heterogeneity scale)

l L0

lL

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

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

Limitations of HomogenizationNatural Media = Multiscale Media with Scale Interactions,

(no scale separation)

1l 2l 3l1

12

0ll

22

3

0ll

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

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

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

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

Multiscale Modelling: Flow

Fine scale flow model

Filtered flow model

0 )()(~ xxKK

0 )()(~)(eff xxKK

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

Multiscale Modelling: Flow

Statistical properties of the filtered conductivity fields

2

2222

1

1/

/

d

ff l

21221/

/ lll

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( , ) ( ) ( , ) ( , )

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)

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

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

Reactive Fronts

Travelling fronts• Introduction of generalised spatial moment analysis

(Attinger et al., MMS, 2003)

cDcxuckc ptdt 1

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

Reactive Fronts

Fin

e s

cale

mo

del L

arge

scale

Mo

del

cDcuckc ptdt eff01

Local Mixing = Real Mixing

Reactive Fronts

Travel time differences

=nonlocal macrodispersive flux

Real mixing

= local real dispersive flux

Attinger et al., MMS, 2003Dimitrova et al., AWR, 2003