Under Construction… still

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.  The diffusive exchange adds extraordinarily long residence times to the transport through such materials.  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.  Mathematical modeling of such exchange has a rich history starting apparently with Carslaw and Jaeger in transport and Villermaux in chromatography, leading to the general “Memory function” form for such exchange.  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).  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.

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) model is used here to simulate a groundwater injection/withdrawal experiment reported in Gouze et al. (2008).

Transport of solutes in flowing fluid involves only advection and diffusion. Because one cannot characterize all the variations in fluid velocity at small scales, when modeling transport at larger scales we manufacture an upscaled transport flux that is supposed to account for these variations, called dispersion, with classical models of dispersion adopted from Fick’s law of diffusion, with varying degrees of success in representing averaged concentrations. However this approach requires solutes to experience the full range of velocity variations before this (“asymptotic”) Fickian model can work. In many cases of environmental fluid mechanics this experience takes such a long time that so-called “preasymptotic” dispersion conditions prevail. Prior efforts to model preasymptotic dispersion involve making the dispersion coefficient a function of time-since-injection or of distance-transported, in bulk fluids and in groundwater. This approach is critiqued G. I. Taylor who created the concept of asymptotic dispersion because it requires multiply-defined dispersion coefficients for solutes injected at different start times but which overlap due to dispersive spreading.  We resolve this problem by using residence-time (“age”) as the independent variable for the dispersion coefficient.

The figure shows profiles at two times for 1D transport of two sequential injected solute pulses in a 16 m long flow at constant velocity of 12 cm/min and dispersion coefficient that relaxes exponentially to an asymptotic value of 12 cm^2/min.  The dispersive flux causes overlap of the two pulses in space, but this poses no problem when the dispersion coefficient is a function of the solute age.