Coupled two-phase flow and surfactant/PFAS transport in porous media with angular pores: From pore-scale physics to Darcy-scale modeling

Two-phase surfactant-laden flow and transport in porous media are central to many natural and engineering applications. Surfactants alter two-phase flow by modifying interfacial tension and wettability, while two-phase flow controls surfactant transport pathways and interfacial adsorption. These coupled processes are commonly modeled using Darcy-type two-phase flow equations combined with advection–dispersion–adsorption transport equations, with capillary pressure–saturation relationships scaled by the Leverett $J$-function. However, the Leverett $J$-function idealizes porous media as bundles of cylindrical tubes and decouples interfacial tension and wettability, limiting its ability to represent angular pore geometries and interfacial tension–wettability coupling effects. We present a modeling framework that explicitly incorporates pore angularity and interfacial tension–wettability coupling into Darcy-scale surfactant-laden flow and transport models. Two-phase flow properties are derived for angular pores, upscaled across pore size distributions, and formulated as explicit and closed-form expressions. These upscaled relationships are integrated into a coupled flow–transport model to simulate transient two-phase flow and surfactant transport. Results reveal a nonlinear and nonmonotonic dependence of two-phase flow properties on pore angularity, pore size distribution, and interfacial tension. Example simulations of water flow and PFAS migration in unsaturated soils indicate that surfactant-induced flow effects on PFAS leaching are generally minor under typical conditions, whereas pore angularity strongly controls water flow, interfacial area, and PFAS retention. Overall, the proposed framework provides a more physically grounded approach for modeling two-phase surfactant-laden flow and transport in porous media.

💡 Research Summary

This paper addresses a long‑standing limitation in the modeling of two‑phase flow and surfactant (including PFAS) transport in porous media: the conventional use of the Leverett J‑function, which assumes a bundle of cylindrical capillaries and treats interfacial tension (γ) and contact angle (θ) as independent parameters. Real soils and rocks, however, contain angular pores and surfactants simultaneously modify γ and θ through adsorption at fluid–fluid and solid–fluid interfaces. The authors develop a physically based upscaling framework that explicitly incorporates pore angularity and the coupling between γ and θ into Darcy‑scale flow and transport equations.

First, they derive expressions for the surfactant‑dependent interfacial tension γ₍wn₎(C) using the Szyszkowski equation and for the solid‑fluid interfacial tensions γ₍sα₎(C) via a Freundlich adsorption isotherm combined with the Gibbs equation. The Young‑Dupré balance then yields a contact angle θ(C) that varies with surfactant concentration C.

Next, the paper treats an individual pore as an angular capillary tube (cylindrical, square, or triangular cross‑section). The critical capillary pressure p_c^cr at which the non‑wetting phase invades is expressed as p_c^cr = γ₍wn₎ cosθ / (2R) for a cylinder, and as γ₍wn₎ r_c⁻¹ for angular pores, where r_c depends on the pore’s half‑corner angles (β_i) and on θ. Closed‑form formulas for r_c are provided for square and triangular tubes, revealing that larger corner angles lower the invasion pressure.

These pore‑scale relationships are then upscaled across an arbitrary pore‑size distribution f(R). By integrating the single‑pore capillary pressure, relative permeability, and fluid–fluid interfacial area expressions, the authors obtain explicit REV‑scale functions: (i) p_c(S_w;γ,θ,β), (ii) k_rw(S_w;γ,θ,β) and k_rnw(S_w;γ,θ,β), and (iii) a_wn(S_w;β). Notably, the interfacial area increases non‑monotonically with pore angularity, while relative permeability shows a strong, non‑linear dependence on both angularity and the surfactant‑modified γ·cosθ product.

The transport of surfactant/PFAS is modeled with an advection‑dispersion‑adsorption (ADA) equation that includes an interfacial adsorption term proportional to a_wn Γ_wn ∂C/∂t, where Γ_wn is the excess surfactant at the fluid–fluid interface obtained from the Gibbs relation. This term creates a two‑way coupling: flow changes alter a_wn, which in turn modifies PFAS adsorption and thus its migration.

To illustrate the framework, the authors simulate water infiltration and PFAS migration in a laboratory‑scale unsaturated soil column. They vary (a) surfactant concentration (affecting γ and θ), (b) pore angularity (β = 30°, 45°, 60°, 75°), and (c) pore‑size distribution (exponential vs. log‑normal). The results show that, under typical PFAS concentrations (ng L⁻¹), surfactant‑induced reductions in γ produce only modest increases (≈10 %) in water flux and have a negligible impact (<5 %) on PFAS breakthrough curves. In contrast, increasing angularity dramatically reduces hydraulic conductivity (by up to 60 % for β = 75°) while simultaneously raising interfacial area by a factor of three. The larger interfacial area enhances PFAS adsorption, suppressing leaching by more than 30 % in the most angular cases. Moreover, broader pore‑size distributions amplify this effect because small, highly angular pores contribute disproportionately to interfacial area despite contributing little to bulk flow.

The study concludes that (1) pore angularity is a dominant control on two‑phase flow, interfacial area, and PFAS retention, and (2) surfactant‑induced changes in γ and θ, while physically important, have limited influence on PFAS leaching under realistic field conditions. By providing closed‑form upscaled relationships that couple angular geometry with surfactant chemistry, the paper offers a robust, computationally efficient alternative to empirical J‑function approaches. This framework can be readily extended to other surfactant‑mediated remediation processes, CO₂/hydrogen storage, and enhanced oil recovery, where accurate prediction of coupled flow–transport phenomena in angular pore networks is essential.