Capillary transport in particulate porous media at low levels of saturation

Abstract

We have established previously, that the spreading of liquids in granular porous media at low levels of saturation, typically less than 10% of the available void space, has very distinctive features in comparison to that at higher saturation levels. In particular, we showed that the spreading is controlled by a special type of diusional process, that its physics can be captured by an equation of the super-fast diusion class, and these ndings were supported by rst-of-a-kind experiments. In this paper, we take these ndings to the next level including deeper examination and exposition of the theory, an expanded set of experiments to address scaling properties, and systematic evaluations of the predictive performance against these experimental data, keeping in mind also potential practical applications.

Surface Permeability of Particulate Porous Media

“…Our main concern here is the wetting cycle, when the liquid spreads over a dry porous matrix or over a matrix with a very low background saturation level up to s r ≈ 2%. These conditions are similar to those in the case studied previously experimentally and theoretically in 6 . The main driving force of the dispersion process, as is often the case during the wetting cycle, is capillary pressure developed at the moving front in the process of wetting of dry porous matrix, while the liquid bridges play a role of variable liquid reservoirs of uniform surface curvature.…”

“…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 details of derivation of (1) can be found in 5,6 , here we note that, the resultant governing non-linear equation (1) directly follows from the conservation of mass principle…”

“…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 .…”

“…Note, in that respect, that in the domain of spreading liquid bridges are supposed to never vanish, so that the condition s > s 0 is always fulfilled in the model, and there is no actual singularity of the mathematical description 5,6 . Specifically, in the macroscopic approximation, that is after averaging over some volume element containing many particles of the porous medium, the diffusion process in the slow creeping flow conditions can be described by the following non-linear diffusion equation…”