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 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

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 ModellingLarge Scale Modelling

Stochastic Modelling Approach Stauffer et al., WRR, 2002 Different realizations of a catchment zone Risk Assessment (question 3)

Large Scale Modelling Large Scale Predictions Model on a „regional“ scale : 50 „small“ scale lengths Resolution : „small“ scale length/5 Number of unknowns

Homogenization Large Scale Flow Models with effective conductivity fine grid model large grid model

Homogenization  Homogenization Theory  Volume averaging  Ensemble Averaging (if system ergodic) = Asymptotic theory (scale separation between observation scale and heterogeneity scale)

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

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

Limitations of Homogenization Natural Media = Multiscale Media with Scale Interactions, (no scale separation)

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

Multiscale Modelling Improved Approach: accounts for pre-asymptotic effects Coarse Graining (Filter) Methods fine grid model coarse grid model

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

Multiscale Modelling: Flow Fine scale flow model Filtered flow model

Multiscale Modelling: Flow Scale dependent mean conductivity (subscale effects) D=2: D=3: Attinger, J. Comp. GeoSciences,2003

Multiscale Modelling: Flow Statistical properties of the filtered conductivity fields

Multiscale Modelling: Transport Fine scale transport model Filtered transport model

Multiscale Modelling: Transport Scale dependent macro dispersivities: real dispersivities plus artificial mixing (centre-of-mass fluctuations)

Multiscale Modelling: Transport Scale dependent real dispersivities

Model with real dilution Model with artificial dilution Transport Codes Multiscale Modelling: Transport

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

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

Reactive Fronts Fine scale model Large scale Model 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