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

Abstract

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.

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

“…Theoretical justifications for the space-dependent transport parameters in almost all of the state-of-the-art, nonlocal transport models were articulated in Zhang et al 40 We will also test these hypotheses in Section 3. It is also noteworthy that eq 1a can be obtained by adding the subdiffusive term to the space fADE proposed by Zhang et al 7 A detailed derivation of this equation is also shown in Section S.1.4.…”

“…For example, eq 1a is analogous to the fractional-order advection−dispersion−reaction equation. 41 The nonlocal diffusive flux in eq 1d captures the contribution of diffusive flux from all upstream zones to the right (outlet) boundary due to nonlocal superdiffusive transport in the model domain [L, R] (see also Zhang et al 7 ), while the flux in eq 1c defines input of pollutant flux from outside the model domain, which is usually zero. In addition, the factor b and other parameters in eq 1a define the transport model: when b = 0, λ = 0, and κ = 0, model (1a) reduces to the standard spatiotemporal fractional-derivative model; 42 when b = 1, λ = 0, and α = 2, eq 1a reduces to the fractal mobile−immobile transport model (for the total phase) proposed by Schumer et al; 43 when b = 1, λ → ∞, and α = 2, eq 1a reduces to the classical advection−dispersion equation (ADE) with a retardation factor of 1 + β.…”

“…We combine the fractional-adjoint approach and the sensitivity analysis 7 to derive the backward model of (1). The mathematical background of "adjoint" is introduced in Section S.1.5 as well as the reason for combining the adjoint approach and the sensitivity analysis.…”

“…This study will convert the general, forward nonlocal transport model (here, "nonlocal" means that the current change of pollutant mass depends on the historical pollutant mass (which is called "time nonlocal") or spatial pollutant distribution (which is called "spatial nonlocal")) to its backward counterpart, which contains the abovementioned PSI models as end members. 7,29 Third, PSI in rivers and aquifers is constrained by the poor predictability of model parameters. On the one hand, backward tracking for PSI is not a physical process but a process involving probability propagation, since pollutants do not move backward in nature (except for local, reverse flow, or backward diffusion).…”

Numerical Simulation of Solute Transport in Saturated Porous Media with Bounded Domains

“…This means that the super‐diffusive jumps follow the main flow direction and the up‐location and upstream zones overlap each other (Zhang et al 2016). Therefore, the positive Riemann‐Liouville fractional derivative should be applied to describe the diffusive flux (Zhang et al 2016, 2018) because the positive fractional derivative in the s‐FADE model signifies that the diffusive flux of the contaminant source comes from the up‐location (Zhang et al 2016). According to the above debate, the one‐sided s‐FADE with a positive fractional derivative was used in this study.…”