  We present a framework for systems in which diffusion-advection transport of
a tracer substance in a mobile zone is interrupted by trapping in an immobile
zone. Our model unifies different model approaches based on distributed-order
diffusion equations, exciton diffusion rate models, and random walk models for
multi-rate mobile-immobile mass transport. We study various forms for the
trapping time dynamics and their effects on the tracer mass in the mobile zone.
Moreover we find the associated breakthrough curves, the tracer density at a
fixed point in space as function of time, as well as the mobile and immobile
concentration profiles and the respective moments of the transport.
Specifically we derive explicit forms for the anomalous transport dynamics and
an asymptotic power-law decay of the mobile mass for a Mittag-Leffler trapping
time distribution. In our analysis we point out that even for exponential
trapping time densities transient anomalous transport is observed. Our results
have direct applications in geophysical contexts but also in biological, soft
matter, and solid state systems.
