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

•It is shown that the same closure problem arises for variance and PDF equations.•This link can be used to test new closure models with the simpler variance equation.•A new time depended closure model based on the IEM model is proposed.•The new model significantly improves the simulated concentration variances and PDFs.

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.