Modeling Hydrologically Mediated Hot Moments of Transient Anomalous Diffusion in Aquifers Using an Impulsive Fractional-Derivative Equation

Hydrologically mediated hot moments (HM-HMs) of transient anomalous diffusion (TAD) denote abrupt shifts in hydraulic conditions that can profoundly influence the dynamics of anomalous diffusion for pollutants within heterogeneous aquifers. How to efficiently model these complex dynamics remains a significant challenge. To bridge this knowledge gap, we propose an innovative model termed "the impulsive, tempered fractional advection-dispersion equation" (IT-fADE) to simulate HM-HMs of TAD. The model is approximated using an L1-based finite difference solver with unconditional stability and an efficient convergence rate. Application results demonstrate that the IT-fADE model and its solver successfully capture TAD induced by hydrologically trigged hot phenomena (including hot moments and hot spots) across three distinct aquifers: (a) transient sub-diffusion arising from sudden shifts in hydraulic gradient within a regional-scale alluvial aquifer, (b) transient sub- or super-diffusion due to convergent or push-pull tracer experiments within a local-scale fractured aquifer, and (c) transient sub-diffusion likely attributed to multiple-conduit flow within an intermediate-scale karst aquifer. The impulsive terms and fractional differential operator integrated into the IT-fADE aptly capture the ephemeral nature and evolving memory of HM-HMs of TAD by incorporating multiple stress periods into the model. The sequential HM-HM model also characterizes breakthrough curves of pollutants as they encounter hydrologically mediated, parallel hot spots. Furthermore, we delve into discussions concerning model parameters, extensions, and comparisons, as well as impulse signals and the propagation of memory within the context of employing IT-fADE to capture hot phenomena of TAD in aquatic systems. Hydrologically mediated hot moments (HM-HMs) encompass sudden shifts in water flow conditions in aquifers, exerting substantial influence on the migration of pollutants. To enhance our comprehension and modeling of these occurrences, we propose a mathematical model termed the Impulsive, Tempered Fractional Advection-Dispersion Equation (IT-fADE). This model employs specialized mathematical operators to effectively capture the rapid changes caused by HM-HMs and to efficiently solve the associated equations. Our assessment of the IT-fADE model across three distinct aquifers demonstrates its aptitude in faithfully depicting the transient anomalous diffusion of pollutants resulting from HM-HMs. The IT-fADE model incorporates specific terms and operators that represent the dynamic nature of HM-HMs and their impact on pollutant mobility. In summary, the IT-fADE model furnishes a tool for comprehending and modeling the repercussions of hydrologically triggered hot events on pollutant migration in aquifers. This framework may help researchers and scientists better quantify how pollutants will behave in these environments whose conditions can change abruptly. An impulsive, tempered fractional model captured transient anomalous diffusion (TAD) due to hydrologically mediated hot moments (HM-HMs) The impulsive term and fractional differential operator in the new model describe the ephemeral and dynamic nature of HM-HMs of TAD The sequential HM-HM model fitted the pollutant breakthrough curves affected by parallel hydrologically mediated hot spots