Gravity Fingering During Infiltration: Impact of Medium Heterogeneity

Infiltration of water into dry soil often leads to gravity fingering, a hydrodynamic instability that results in a columnar wetting front and focuses the infiltrating water through vertical macroscopic fingers. Despite the four-decade-long history of gravity fingering, and its widespread occurrence, a quantitative macroscopic modeling framework has proved elusive. Recently, we proposed a mathematical model that successfully captures and predicts the many facets of the phenomenon [1, 2]. The proposed framework is based on a phase field approach, built on the idea that the free energy of the system is nonlocal, which leads naturally to higher-order terms in the mass conservation equations. Our phase-field model contains a new term, a fourth-order spatial derivative of saturation, the magnitude of which is set by a proposed scaling. Our model reproduces the key features of unsaturated flow: a nonmonotonic saturation profile, and the formation and persistence of gravity fingers. It shows good quantitative agreement with experiments in terms of finger tip saturation, tip velocity and finger width. A rigorous linear stability analysis also confirms the experimental observation that saturation overshoot is a prerequisite for gravity fingering. Here, we address the question of how soil heterogeneity affects gravity fingering. We extend our phase-field model to the case of a spatially-varying permeability field by properly incorporating its effect on the energy functional. We perform high-resolution simulations and laboratory experiments on sand packs with heterogeneous patterns of grain size. We show that, even with contrasts in permeability of up to a factor of 50, heterogeneity has only a mild influence of the gravity fingering phenomenon, and that it is the hydrodynamic instability that overwhelmingly controls infiltration.