13 Jul 2023
 | 13 Jul 2023
Status: this preprint is currently under review for the journal HESS.

Adjoint subordination to calculate backward travel time probability of pollutants in water with various velocity resolutions

Yong Zhang, Graham E. Fogg, Hongguang Sun, Donald M. Reeves, Roseanna M. Neupauer, and Wei Wei

Abstract. Backward probabilities such as backward travel time probability density function for pollutants in natural aquifers/rivers had been used by hydrologists for decades in water-quality related applications. Reliable calculation of backward probabilities, however, has been challenged by non-Fickian pollutant transport dynamics and variability in the resolution of velocity at study sites. To address these two issues, we built an adjoint model by deriving a backward-in-time fractional-derivative transport equation subordinated to regional flow, developed a Lagrangian solver, and applied the model/solver to backtrack pollutant transport in various flow systems. The adjoint model applies subordination to a reversed regional flow field, converts forward-in-time boundaries to either absorbing or reflective boundaries, and reverses the tempered stable density to define backward mechanical dispersion. The corresponding Lagrangian solver is computationally efficient in projecting backward super-diffusive mechanical dispersion along streamlines. Field applications demonstrate that the adjoint subordination model can successfully recover release history, dated groundwater age, and spatial location(s) of pollutant source(s) for flow systems with either upscaled constant velocity, non-uniform divergent flow field, or fine-resolution velocities in a non-stationary, regional-scale aquifer, where non-Fickian transport significantly affects pollutant dynamics and backward probability characteristics. Caution is needed when identifying the phase-sensitive (aqueous versus absorbed) pollutant source in natural media. Possible extensions of the adjoint subordination model are also discussed and tested for quantifying backward probabilities of pollutants in more complex media, such as discrete fracture networks.

Yong Zhang et al.

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on hess-2023-131', Anonymous Referee #1, 15 Sep 2023
    • AC1: 'Reply on RC1', Yong Zhang, 22 Sep 2023
  • RC2: 'Comment on hess-2023-131', Anonymous Referee #2, 18 Sep 2023
    • AC2: 'Reply on RC2', Yong Zhang, 22 Sep 2023

Yong Zhang et al.

Yong Zhang et al.


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Short summary
Pollutant release history and source identification are helpful for managing water resources, but it remains a challenge to reliably identify such information for real-world, complex transport processes in rivers and aquifers. In this study, we filled this knowledge gap by deriving a general backward governing equation and developing the efficient solver. Field applications showed that this model and solver are applicable for a broad range of flow systems, dimension, and spatiotemporal scales.