Superfast Nonlinear Diffusion: Capillary Transport in Particulate Porous Media

Abstract

The migration of liquids in porous media, such as sand, has been commonly considered at high saturation levels with liquid pathways at pore dimensions. In this Letter, we reveal a low saturation regime observed in our experiments with droplets of extremely low volatility liquids deposited on sand. In this regime, the liquid is mostly found within the grain surface roughness and in the capillary bridges formed at the contacts between the grains. The bridges act as variable-volume reservoirs and the flow is driven by the capillary pressure arising at the wetting front according to the roughness length scales. We propose that this migration (spreading) is the result of interplay between the bridge volume adjustment to this pressure distribution and viscous losses of a creeping flow within the roughness. The net macroscopic result is a special case of nonlinear diffusion described by a superfast diffusion equation for saturation with distinctive mathematical character. We obtain solutions to a moving boundary problem defined by superfast diffusion equation that robustly convey a time power law of spreading as seen in our experiments.

Surface Permeability of Particulate Porous Media

“…Theoretically, the superfast non-linear diffusion equation belongs to a special class of mathematical models. Unlike in the standard porous medium equation 7 , in this special case, the non-linear coefficient of diffusion D(s) demonstrates divergent behaviour as a function of saturation s, D(s) ∝ (s − s 0 ) −3/2 , where s 0 is some minimal saturation level (s 0 ≈ 0.5%), which could be only achieved in a state when the liquid bridges cease to exist completely [3][4][5][6] .…”

“…The analysis of this regime of wetting, which is crucial for studies of biological processes and spreading of non-volatile liquids in arid natural environments and industrial installations, has shown that the liquid dispersion has many distinctive features and can be accurately described by the so-called superfast non-linear diffusion equation 5,6 .…”

Data assimilation for moving mesh methods with an application to ice sheet modelling

“…The major advantage of our moving mesh method is that only a small number of mesh steps are needed to accurately determine the positions of the boundaries, unlike fixed or adaptive mesh methods (Berger and Oliger, 1984;Li et al, 2014;Cornford et al, 2013Cornford et al, , 2016Gladstone et al, 2010). Our moving mesh method has been successfully applied to a number of moving boundary problems, including one-and two-dimensional models of ice sheet flow, tumour growth and chemical spreading (Partridge, 2013;Bonan et al, 2016;Lukyanov et al, 2012;Lee et al, 2013).…”

Role of contact-angle hysteresis for fluid transport in wet granular matter

“…Our model for equilibration dynamics is based on the assumption that the fluid transport is entirely driven by differences in the local capillary pressure. For wetting liquids with a low vapor pressure, this transport proceeds mainly through thin wetting films in the roughness of the beads [3,6]. A diffusive transport through the vapor phase has to be considered for volatile liquids that exhibit a sufficiently high vapor pressure [15,16].…”

“…The structure of the fluid interfaces in a wet granular bed essentially determines the strength of capillary cohesion and thus plays a key role in the prediction and mitigation of devastating events such as landslides or the clogging of industrial pipes [1,2]. For a thorough understanding, a realistic model of fluid movement is indispensable: The capillary pressure-driven fluid transport in wet granulates [3,4] changes the fluid distribution, which in turn suggests that the mechanical properties are affected by fluid displacement.…”