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Filtered density function uncertainty assessments for reactivetransport in groundwater

Fachliche Zuordnung Hydrogeologie, Hydrologie, Limnologie, Siedlungswasserwirtschaft, Wasserchemie, Integrierte Wasserressourcen-Bewirtschaftung
Förderung Förderung von 2012 bis 2018
Projektkennung Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 221406899
 
Erstellungsjahr 2016

Zusammenfassung der Projektergebnisse

Monitoring groundwater contamination and remediation scenarios requires mathematical models and numerical solutions for reactive transport in groundwater systems. In most relevant cases, the spatial heterogeneity of such systems at the field scale cannot be completely described and parameterized to allow deterministic descriptions. Stochastic approaches are therefore intensively used. One way to consider randomness in modeling transport in hydro-geological systems is through stochastic parameterizations of incompletely known hydraulic conductivity fields. This parameterization induces the randomness of the Darcy velocity field which in turn implies the randomness of the observables of the transport process. Beyond mean and variance, traditionally used in stochastic approaches, the full one-point onetime concentration probability density (PDF) is needed to estimate exceedance probabilities in assessments of groundwater contamination. The PDF approach is mainly useful in case of reactive transport: because reaction terms are in a closed form there is no need to upscale them, as in case of modeling the mean behavior of species concentrations. Solving such evolution equations also avoids the cumbersome Monte Carlo simulations often used to assess concentration PDFs. We have shown that evolution equations for concentration PDFs weighted by a conserved scalar (e.g. the density or the sum of the reactant species concentrations) take the form of a Fokker-Planck equation. Consequently, the associated systems of Itô equations can be used to estimate the weighted concentration PDF. We used a global random walk (GRW) algorithm consisting of a superposition of large numbers of solutions for the system of Itô equations on regular lattices embedded in Cartesian products of physical and concentration spaces. This procedure results in a huge reduction of computational costs. Using a spatial filtering operation to infer the statistics of the transport process, we further developed a filtered density function (FDF) approach. The FDF equations are formally identical to the PDF equations and can be solved by the same GRW algorithm. The PDF/FDF equations contain unclosed terms, depending on multi-point PDFs, which have to be modeled. We used upscaling methods to model the terms describing the transport in the physical space. To close the terms which govern the transport in the concentration space, we proposed a mixing model with a variable time scale, appropriate for groundwater systems. The modeled PDF/FDF equations were validated by comparisons with mean values and PDFs estimated from Monte Carlo simulations. The convergence with increasing filter width of the FDF solution to the PDF solution demonstrates the consistency of the FDF approach.

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