Model of hydrophobic porous media applied to stratified media: Water trapping, intermittent flow and fingering instability

A macroscopic model of water infiltration is developed and applied to a stratified porous medium containing hydrophilic and hydrophobic layers. The governing equation is a gradient dynamics as in the Richards model but in which a wettability term replaces the suction pressure, taking into account the effective action of the hydrophobic matter. The resulting governing equation is a fourth-order PDE similar to the lubrication model or Cahn-Hilliard equations known to reproduce complex dynamics such as intermittency and fingering instability. The model makes it possible to simulate the finite spreading of water in horizontal directions when the porous medium is hydrophobic. Applied to a stratified medium alternating hydrophilic and hydrophobic layers, we highlight, depending on the mean water content, phenomena of water trapping in hydrophilic layers, intermittent flow and fingering instability. These regimes are studied using time integration bifurcation and numerical continuation tools.