### A Time Dependent Mixing Model for Concentration Fluctuations in Heterogeneous Aquifers

*Lennart Schüler*

^{1}, Nicolae Suciu

^{2}, Peter Knabner

^{2}, Sabine Attinger

^{1}

^{1}Universität Jena

^{2}Universität Erlangen-Nürnberg

P 2.1

*in*Aquifer systems in Europe and beyond

Geological formations are heterogeneous. Their properties are usually not measurable everywhere but only at some locations. This lack of knowledge implies uncertainty in aquifer parameters like hydraulic conductivity. As a consequence, transport of solutes through these formations is also uncertain and has to be described in a probabilistic sense. Mean and variance estimates of solute concentrations give some information on the probability distribution of concentrations.

We present analytical results for the concentration variance. Going beyond mean and variance estimates, we are able to state the transport equation for the whole probability density distribution (PDF) of the concentration by adopting an approach first introduced in turbulence theory by Colucci et al. (1998). However, we show that both, the variance and the PDF equations have the same closure problem. This link is used to propose and test an alternative closure strategy for the variance equation adopted to the special problems in groundwater transport. This new closure model is transfered to the PDF formulation.

One prominent result is the new time dependent dissipation model which takes the mixing length scales into account. This new model greatly improves the concentration variance prediction at early times. The transfer of the new model to the PDF description is also successful. Former attempts failed in reproducing the correct dissipation results which is now possible.

We also propose to make use of spatial filtering and filtered density functions (FDF). The FDF evolution equations are similar in form with those of probability density functions, but their solutions are still random and depend on the width of the spatial filter. The use of FDFs drastically reduces computational costs without the loss of accuracy.

COLUCCI, P. J. & JABERI, F. A. & GIVI, P. & POPE, S. B. (1998): Filtered density function for large eddy simulation of turbulent reacting flows – In: Phys. Fluids 10, 499-515