Gravitational fingering often occurs for water flow in the vadose zone and accurate modeling of this important flow process remains a significant scientific challenge. This paper presents the latest theoretical developments of the optimality-based active region model (ARM), a macroscopic framework developed for describing gravitational fingering flow in the vadose zone. ARM divides the soil into active (fingering) and inactive regions, introducing a relationship between water flux and hydraulic gradient derived from the principle of optimality that the system self-organizes to maximize water flow conductivity. Unlike traditional models, ARM’s hydraulic conductivity depends on both capillary pressure or water saturation and water flux, reflecting the unstable nature of fingering flow. The paper provides an updated mathematical derivation of ARM relationships using calculus of variations and extends ARM to account for small water flux in the non-fingering zone, resulting in a dual-flow field model. These new developments should make ARM more rigorous and realistic for field-scale applications.
Liu et al. (Sun,) studied this question.