Identification of Pollutant Source for Super‐Diffusion in Aquifers and Rivers with Bounded Domains

Abstract

AbstractBackward models for super‐diffusion in infinite domains have been developed to identify pollutant sources, while backward models for non‐Fickian diffusion in bounded domains remain unknown. To restrict possible source locations and improve the accuracy of backward probabilities, this technical note develops the backward model for super‐diffusion governed by the fractional‐divergence advection‐dispersion equation (FD‐ADE) in bounded domains. The resultant backward model is the fractional‐flux advection‐dispersion equation (FF‐ADE) with modified boundary conditions. In particular, the Dirichlet boundary condition in the forward FD‐ADE becomes a spatial‐nonlocal sink term in the backward FF‐ADE (to account for preferential flow), while the nonlocal, non‐zero‐value Neumann (or Robin) boundary condition in the forward FD‐ADE switches to the zero‐value Robin (or Neumann) boundary condition in the backward FF‐ADE (to eliminate pollutant source outside the domain). Field applications show that the backward location probability density function can approximate the point source location in a natural river or fluvial aquifer. The impact of reflective/absorbing boundaries and the upstream boundary location on the backward probability density function is also discussed.