Integrodifferential formulations of the continuous-time random walk for solute transport subject to bimolecular A+B→0 reactions: From micro- to mesoscopic

We develop continuous-time random walk (CTRW) equations governing the transport of two species that annihilate when in proximity to one another. In comparison with catalytic or spontaneous transformation reactions that have been previously considered in concert with CTRW, both species have spatially variant concentrations that require consideration. We develop two distinct formulations. The first treats transport and reaction microscopically, potentially capturing behavior at sharp fronts, but at the cost of being strongly nonlinear. The second, mesoscopic, formulation relies on a separation-of-scales technique we develop to separate microscopic-scale reaction and upscaled transport. This simplifies the governing equations and allows treatment of more general reaction dynamics, but requires stronger smoothness assumptions of the solution. The mesoscopic formulation is easily tractable using an existing solution from the literature (we also provide an alternative derivation), and the generalized master equation (GME) for particles undergoing A+B→0 reactions is presented. We show that this GME simplifies, under appropriate circumstances, to both the GME for the unreactive CTRW and to the advection-dispersion-reaction equation. An additional major contribution of this work is on the numerical side: to corroborate our development, we develop an indirect particle-tracking-partial-integro-differential-equation (PIDE) hybrid verification technique which could be applicable widely in reactive anomalous transport. Numerical simulations support the mesoscopic analysis.