General Backward Model to Identify the Source for Contaminants Undergoing Non-Fickian Diffusion in Water

Pollutant source identification (PSI) has been conducted for four decades for tracking Fickian diffusive pollutants, while PSI for non-Fickian diffusion, well-documented in aquifers and rivers, requires novel, predictive models. To enable PSI for non-Fickian diffusive pollutants, this study derived a general backward model using the fractional-adjoint approach in sensitivity analysis for dissolved contaminants with transport governed by the spatiotemporal fractional advection-dispersion equation (fADE). The backward fADE contains a self-adjoint time-fractional term for subdiffusion and direction-dependent, non-self-adjoint space-fractional terms for superdiffusion. Field applications showed that the resultant backward location probability density function identified the point source location in all three test cases, one alluvial aquifer and two rivers. The backward model and boundary conditions derived in this study made it possible to reliably and efficiently backtrack pollutants (and may include other constituents, such as bedload) undergoing mixed sub- and superdiffusion in natural aquatic systems. The classical PSI model, however, underestimated the source location since it did not account for solute retention and preferential flow. In addition, the measured tracer snapshots (if available before PSI) can enhance the parameter predictability and improve the applicability of backward fADE PSI. Most importantly, a spatially variable dispersion coefficient is needed in the backward fADE since PSI is most likely scale dependent in natural hydrologic systems.