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

Backward 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.