A free energy based model for water transfer in amphiphilic soils

Abstract

A 3D macroscopic gradient-dynamics model is developed and applied to sandy soil in presence of exopolysaccharides (EPS), to mimic a soil influenced by root exuded mucilages (rhizospheric soil). Depending on water content, amphiphilic soil has a hydrophilic or hydrophobic behavior which impacts water transfer and retention. To model this saturation-dependent wettability, we propose a nonequilibrium thermodynamic approach based on the definition of the free energy of the system. The free energy functional contains gravity energy, the free surface energy of the water and the effective interaction (attractive and repellent) between water and the porous matrix with the amphiphilic matter. The latter defines the wettability of the porous medium. Water flow dynamics is derived from Onsager’s variational principle leading to a non-linear fourth order PDE on the saturation generalizing the Richards equation. The new formulation reproduces a range of water flow regimes encountered in soil with EPS: (i) the stoppage of imbibition front in a homogeneous soil leading to equilibrium where moist regions coexist with a dry region, (ii) a decrease in capillary height in comparison with a sand without amphiphilic matter and (iii) the existence of a threshold of amphiphilic concentration for which the capillary rise is stopped at the dry layer containing the amphiphilic matter. After calibrating the parameters of our model, numerical simulation is in qualitative and quantitative agreement with experiments from the literature.