When a mobile fluid is in contact with a material that allows only diffusive transport, there occurs diffusive exchange between the mobile phase and the diffusive “immobile” phase.\u00a0 The diffusive exchange adds extraordinarily long residence times to the transport through such materials.\u00a0 Examples include transport in: fractured and porous material, groundwater in a porous medium with low-conductivity inclusions or even microscopically-permeable solid phases, river corridors with exchange between river and sediment bed domains, pipes lined with biofilms, gas phases over the ocean or lakes, as well as biological, industrial, and chemical engineering applications.\u00a0 Mathematical modeling of such exchange has a rich history, leading to the general “Memory function” form for such exchange.\u00a0 Memory function models encapsulate a wide range of models used not only for mobile-immobile exchange but also so-called anomalous dispersion or non-Fickian transport (multirate mass transfer, time-fractional advective-dispersive transport, and many continuous-time random walk models).\u00a0 However, the memory function operator induces anomalous delays (and not anomalous transport) which when coupled with conventional transport model operations can give rise to arbitrarily specified (e.g., power-law) tailing.<\/p>\n\n\n\n
All memory function models can be cast as first-order exchange models where the rate of immobile-mobile zone exchange depends on time spent in the immobile zone. Such a “phase exposure-dependent exchange” (PhEDExWRR2017<\/a>) model is used here to simulate a groundwater injection\/withdrawal experiment reported in Gouze et al. (2008)<\/a>.<\/p>\n\n\n\n