Surface Permeability of Particulate Porous Media

Abstract

The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by super-fast non-linear diffusion partial differential equations. To enhance the predictive power of the mathematical model in practical applications, one requires the knowledge of the effective surface permeability of the particle-in-contact ensemble, which can be directly related with the macroscopic permeability of the particulate media. We have shown previously that permeability of a single particulate element can be accurately determined through the solution of the Laplace-Beltrami Dirichlet boundary-value problem. Here, we demonstrate how that methodology can be applied to study permeability of a randomly packed ensemble of interconnected particles. Using surface finite element techniques we examine numerical solutions to the Laplace-Beltrami problem set in the multiply-connected domains of interconnected particles. We are able to rigorously estimate tortuosity effects of the surface flows in a particle ensemble setting.

Capillary transport in paper porous materials at low saturation levels

“…After reaching a steady state, when the flux density is constant in the flow domain, the distribution of pressure as a function of saturation, Fig. 2, was found to be in very good agreement with that anticipated in the macroscopic model (3), which is in a non-dimensional form…”

“…The resultant non-linear diffusion equation ( 9) has a similar form as that studied in [1,4] in the case of particulate porous media. The main difference at this point is that the equation in the bulk has a constant coefficient of diffusion D 0 , which is defined by the connectivity of the porous network of fibres, while in particulate porous media, there is a weak logarithmic dependence on saturation, and the diffusivity is driven by the shape of the particles and their contact area, details can be found [1][2][3][4]. In a way, the situation is simpler in the case of fibrous materials than that in particulate media, since the connectivity parameter can be quite accurately found via a network model.…”

“…Liquid distributions and transport in particulate porous media, such as clay, loam and sand, at low saturation levels was found to have very distinctive features resulting in a special class of mathematical problems, superfast non-linear diffusion [1][2][3][4]. It has been established, both experimentally and theoretically, that any time any wetting liquid naturally (that is when there is no force wetting regime involved) spreads in a dry (or a nearly dry) porous particulate matrix, the moving front dynamics follows, after some time, the power evolution law dictated by the super-fast non-linear diffusion mechanisms.…”

Capillary transport in paper porous materials at low saturation levels: normal, fast or superfast?

“…The resultant nonlinear diffusion equation (2.9) has a similar form to that studied in [1,4] in the case of particulate porous media. The main difference at this point is that the equation in the bulk has a constant coefficient of diffusion D 0 , which is defined by the connectivity of the porous network of fibres, while in particulate porous media there is a weak logarithmic dependence on saturation and the diffusivity is driven by the shape of the particles and their contact area; details can be found in [1][2][3][4]. In a way, the situation is simpler in the case of fibrous materials than that in particulate media, since the connectivity parameter can be quite accurately found via a network model.…”

“…Apparently, liquid spreading in dry porous materials is not just characteristic of particulate porous media, such as sand, but also frequently occurs in other porous materials. Therefore, in this paper, the previously developed model [1][2][3][4] is generalized to another fairly common type of porous material consisting of fibre elements, such as papers and textiles, where a similar kind of nonlinear diffusion process is anticipated. We would like to understand how general and universal the superfast mechanism, first discovered in particulate porous media [1], actually is.…”