Fluid flow in heterogeneous porous media is unique in that the physical structure of the host medium completely dictates the attendant flow properties, including fluid mixing. Whilst many models of solute transport and reaction assume well-mixed conditions within a solute plume, in reality these plumes typically exhibit incomplete mixing due to the strong fluid stretching dynamics within such heterogeneous flows. These heterogeneities have been observed to generate several fluid stretching regimes (sublinear, linear, superlinear) which directly impact the mixing and dispersion dynamics. Such incomplete mixing has a very significant impact on reactive transport, where the observed effective reaction rate is often much slower than that corresponding to well-mixed conditions. Hence models of incomplete mixing are required which capture the impacts of the heterogeneous medium.
We present an overview of recent developments on this topic, and show how fluid mixing can be predicted directly from the medium conductivity structure (e.g. conductivity variance, correlation structure) via the use of a continuous time random walk (CTRW) approach to solving the governing stochastic models. This approach provides ab initio predictions [1,2] of the different stretching regimes observed in Darcy flow from the conductivity structure  and provides significant insights into the mechanisms which control fluid stretching and mixing. In turn, these stretching rates provide the inputs to lamellar models of fluid mixing [4,5] which provide predictions of the evolving concentration PDF within a evolving plume.
These advances not only provide the ability to predict fluid mixing and dilution directly from the Darcy-scale medium properties, but generate deep insights into the physical mechanisms which govern fluid mixing and dilution in these systems. By using a first-principles approach to quantitatively link conductivity structure and fluid mixing it is possible to directly identify the medium properties which govern fluid mixing and dilution, and also develop uncertainty analysis for cases where these properties are not well-quantified.