Equilibrium distribution of the liquid phase in an unsaturated granular material

In an unsaturated wet granular material, the distribution of the liquid phase depends on the degree of saturation S, i.e. the ratio between the liquid volume and the pores volume. For low values of S, the liquid forms capillary bridges between the grains. As S increases, part of the pore space becomes saturated, and the spatial distribution of the liquid phase results from the competition between gravitational and capillary forces. We show that starting from an initial preparation in which water and grains are mixed homogeneously, the system converges to an equilibrium distribution that follows a Boltzmann law, where static disorder plays a role analogous to thermal agitation. We argue that this equilibrium distribution is determined by the transient flow occuring between the initial homogeneous preparation and the final stationary state. We propose a Langevin-like equation to describe this transient dynamics, and compare our predictions with conductivity measurements providing the local water content as a function of height in a granular medium. Experimental results obtained for samples with different polydispersities collapse onto a single master curve, in agreement with the proposed model.