Adjoint Subordination to Calculate Backward Travel Time Probability of Pollutants in Water with Various Velocity Resolutions

Backward probabilities, such as the backward travel time probability density function for pollutants in natural aquifers/rivers, have been used by hydrologists for decades in water quality applications. Calculating these backward probabilities, however, is challenging due to non-Fickian pollutant transport dynamics and velocity resolution variability at study sites. To address these 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 trace pollutant transport in diverse flow systems. The adjoint model subordinates to a reversed regional flow field, transforms forward-in-time boundaries into either absorbing or reflective boundaries, and reverses the tempered stable density to define backward mechanical dispersion. The corresponding Lagrangian solver efficiently projects backward super-diffusive mechanical dispersion along streamlines. Field applications demonstrate the adjoint subordination model's success with respect to recovering release history, groundwater age, and pollutant source locations for various flow systems. These include systems with upscaled constant velocity, nonuniform divergent flow fields, or fine-resolution velocities in a nonstationary, regional-scale aquifer, where non-Fickian transport significantly affects pollutant dynamics and backward probabilities. Caution is needed when identifying the phase-sensitive (aqueous vs. absorbed) pollutant source in natural media. The study also explores possible extensions of the adjoint subordination model for quantifying backward probabilities of pollutants in more complex media, such as discrete fracture networks.