Phase exposure-dependent exchange

Solutes and suspended material often experience delays during exchange between phases one of which may be moving. Consequently transport often exhibits combined effects of advection/dispersion, and delays associated with exchange between phases. Such processes are ubiquitous and include transport in porous/fractured media, watersheds, rivers, forest canopies, urban infrastructure systems, and networks. Upscaling approaches often treat the transport and delay mechanisms together, yielding macroscopic anomalous transport models. When interaction with the immobile phase is responsible for the delays, it is not the transport that is anomalous, but the lack of it, due to delays. We model such exchanges with a simple generalization of first-order kinetics completely independent of transport. Specifically, we introduce a remobilization rate coefficient that depends on the time in immobile phase. Memory-function formulations of exchange (with or without transport) can be cast in this framework, and can represent practically all time-nonlocal mass balance models including multirate mass transfer and its equivalent counterparts in the continuous time random walk and time-fractional advection dispersion formalisms, as well as equilibrium exchange. Our model can address delayed single-/multievent remobilizations as in delay-differential equations and periodic remobilizations that may be useful in sediment transport modeling. It is also possible to link delay mechanisms with transport if so desired, or to superpose an additional source of nonlocality through the transport operator. This approach allows for mechanistic characterization of the mass transfer process with measurable parameters, and the full set of processes representable by these generalized kinetics is a new open question.