Semi-Analytical Solutions for PFAS Transport in an Aquifer Coupled with Vadose Zone Diffusion

We present a suite of semi-analytical solutions for the transport and fate of per- and polyfluoroalkyl substances (PFAS) within a coupled vadose zone-aquifer system, addressing a scenario where an initially uncontaminated vadose zone is subjected to upward diffusion from a source area located in the underlying saturated zone. While hydrodynamic dispersion governs saturated-zone transport, solid-water and air-water interfacial adsorption in the vadose zone creates a critical storage reservoir that controls PFAS mass exchange across the water table. To facilitate the application of these analytical solutions, transport in the vadose zone is modeled as vertical diffusion, thereby allowing saturation-dependent storage and migration processes to be evaluated. To determine the saturation profile, the transport model incorporates distinct soil-water characteristics. An exact analytical solution is derived for the linear capillary pressure function, while an approximate solution is developed for the van Genuchten model. A comprehensive parametric analysis demonstrates that while vadose zone retention significantly governs the diffusive mass transfer rate, solid-water interface adsorption generally exerts a more pronounced impact than air-water interfacial adsorption. However, the influence of air-water interfacial adsorption becomes exceptionally prominent under the linear capillary model compared to the van Genuchten model. This elevated air-water interfacial impact is highly sensitive to scale: it is strictly constrained to configurations with a thin vadose zone, as a thicker vadose zone diminishes the relative influence of the linear capillary model. These solutions provide a straightforward tool for long-term environmental assessments, offering critical mechanistic insights into the prolonged plume tailing driven by localized back-diffusion. Because they assume stable saturation profiles, these solutions apply only to sites where vadose zone flow is negligible.