# A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers

## Abstract

Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. The mixing model, describing the transport of the PDF in concentration space, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modelling.

## Authors

L. Schüler
-Institute of Geosciences, Friedrich Schiller University Jena, Burgweg 11, 07749 Jena, Germany
-Department Computational Hydrosystems, Helmholtz Centre for Environmental Research – UFZ, Permoserstraße 15, 04318 Leipzig, Germany

N. Suciu
-Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germany
-Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

P. Knabner
-Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germany

S. Attinger
-Institute of Geosciences, Friedrich Schiller University Jena, Burgweg 11, 07749 Jena, Germany
-Department Computational Hydrosystems, Helmholtz Centre for Environmental Research – UFZ, Permoserstraße 15, 04318 Leipzig, Germany

## Keywords

PDF method; Variance; Solute transport; Heterogeneity; Mixing; Global random walk.

## Cite this paper as:

L. Schüler, N. Suciu, P. Knabner and S. Attinger, A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers, Adv. Water Resour., 96 (2016), 55-67

## PDF

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Elsevier

0169-3913

1573-1634