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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 Repetitive Modelled implicitly by parametrization Large scale Explicitly known Deterministic Singular 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 Quantification of the impact of Uncertainty Main interest on large observational scales How to cope with parameter uncertainty ? Stochastic ModellingLarge Scale Modelling

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Stochastic Modelling Approach Stauffer et al., WRR, 2002 Different realizations of a catchment zone 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 Homogenization Theory Volume averaging Ensemble Averaging (if system ergodic) = Asymptotic theory (scale separation between observation scale and heterogeneity scale)

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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 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

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

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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) D=2: D=3: 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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Model with real dilution Model with artificial dilution Transport Codes Multiscale Modelling: Transport

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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 Fine scale model Large scale Model Travel time differences lead to artificial mixing by Large Scale Filtering

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Reactive Fronts Fine scale model Large scale Model 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

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