Fast Mixing in Heterogeneous Media Characterized by Fractional Derivative Model

Abstract

This study aims at investigating non-Fickian temporal scaling of fast mixing processes using fractional advection dispersion equation (FADE) model in Indiana carbonate, multi-lognormal hydraulic conductivity field, self-affine fractures and cemented porous media, in which the fundamental solution of the FADE model is a standard symmetric Lévy stable distribution. The temporal scaling of the scalar dissipation rate (SDR) induced by the FADE model is a function of fractional derivative order $\alpha$, $\chi (t) \sim t^{{ - \frac{\alpha + 1}{\alpha }}}$ ($1 \le \alpha \le 2$), and it reduces to the Fickian scaling $t^{{ - \frac{3}{2}}}$ when $\alpha = 2$. Smaller values of $\alpha$ reflect more efficient and a better mixing state at early time. The fitted results show that the FADE model is much more accurate than the traditional model, which can also well interpret the fast mixing scaling from clearer physical mechanism than the empirical power law fitting line. The fitted values of $\alpha$ capture the complexity of the heterogeneous media, which are consistent with the existing empirical results. Thus, the SDR of the FADE model is feasible to describe the temporal scaling of the fast mixing for the tracer transport in heterogeneous media.