Phase exposure‐dependent exchange

Abstract

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.

Modeling Bimolecular Reactive Transport With Mixing‐Limitation: Theory and Application to Column Experiments

“…Porta et al () also note that the tailing simulated in their disordered (randomly placed solid circles, not always in contact with each other), two‐dimensional simulations is not apparent in the experimental data from the well‐ordered (glass bead pack), three‐dimensional porous medium of Gramling et al () (Raje and Kapoor () also used glass beads), and suggest that the tailing is due to “poorly connected and almost immobile” or “essentially stagnant” (Porta et al, , p. 10) zones in their two‐dimensional model porous media. This seems a reasonable conjecture that would call for separate incorporation of mobile‐immobile mass transfer (e.g., Ginn et al, ), in addition to the mobile‐mobile mass transfer exploited here.…”

“…This transport model is applied to all points in space, assuming transport begins at time t = 0, so the time spent in ballisticules is the same as absolute time. It is possible to formulate all the following using an additional dimension of exposure time for time spent in ballisticules (in which case the following is a modification of the PhEDEX model of Ginn et al, ) but for the conditions here of uniform steady one‐dimensional flow this added complexity is not necessary, as pointed out by an anonymous reviewer.…”

Statistical Formulation of Generalized Tracer Retention in Fractured Rock

“…The significance of the forward and reverse rates for modeling more complex mass transfer kinetics has been emphasized by Ginn () and recently Ginn et al (). In the work of Ginn et al (), the partition function g is expressed in an alternative manner from the usual multirate formulation, by a constant forward and a time‐dependent reverse rate, (using the notation in Ginn et al ()); their expression equation defines g given the function of the reverse rate. Thus, they shift the experimental inference to the reverse rate and its time dependence.…”

“…Thus, they shift the experimental inference to the reverse rate and its time dependence. Ginn et al (, Table ) summarizes standard models and g forms, expressed as time‐dependent reverse rates. Our new expressions of g functions can be easily cast into a time‐dependent reverse rate using their equation ; for instance, the multirate Pareto g yields a simple yet general form of (see Ginn et al, , Table ): …”

A hierarchy of models for simulating experimental results from a 3D heterogeneous porous medium

“…Early investigations of bimodal systems accounted for the influence of immobile regions on transport phenomena by representing the immobile region with a stagnant volume fraction which is coupled to the mobile region with a constant mass transfer coefficient α [15,14,8,50]; this idea has been extended to more general multiple-region models [27,5], and models that include convection and dispersion in both regions [53,1,27,25,28,23,22]. Reviews of much of the literature on this topic have been presented by Cherblanc et al [7] and Fernàndez-Garcia and Sanchez-Vila [18].…”