Super-Fast Non-Linear Diffusion and Surface Permeability

Another essential element of the model, which has to be determined in practical applications of the model, is the coefficient of permeability K = K(s), which is

expected to be some function of saturation.

Previously, the super-fast diffusion model (1.29) has been validated against ex- periments using a simplifying assumption that K = const (Lukyanov et al. 2012). The study has demonstrated sufficiently good comparison, though it was under- stood that any enhancement of the model predictive power can be only achieved if the permeability coefficient K as a function of the liquid content (saturation s) is determined. This is the main topic of this study. Obviously, the curved shape of the particles should affect the ability of those surfaces to conduct the flow. In what follows, we will first generalise the macroscopic description to the case of fi- brous porous materials, where the permeability coefficient can be recovered using a relatively simple network model. Then, we will analyse mathematical problems related with the definition of the coefficient permeability K in particulate porous media.

1.4

Thesis Overview

In this thesis, we will consider several aspects of mathematical modelling liquid dispersion processes at low saturation levels. After an introduction in Chapter 1, in Chapter 2, we deal with the macroscopic description of the low saturation regime of spreading in fibrous porous materials using a mesoscopic network model. We demonstrate how the macroscopic properties of fibrous materials (common paper would be a good example) and the super-fast non-linear diffusion model, initially developed to describe liquid dispersion in particulate porous media, can be recovered on the basis of a mesoscopic network description.

In Chapter 3 and 4, we return to the mathematical description of liquid dis- persion in particulate porous media and examine surface transport over a single constituent element of the porous matrix to estimate effective coefficient of per- meability, which in turn appears in the macroscopic super-fast diffusion model. We demonstrate that the permeability coefficient of an element can be accurately determined on the basis of the analysis of the Laplace-Beltrami boundary value problem set on the curved surface of the element. We analyse weak formulation of the problem and its approximation via surface finite element techniques including error analysis using appropriate norms.

In Chapter 5, we examine the surface transport and the Laplace-Beltrami problem over a set of coupled elements.

Finally, in Chapter 6, we apply the developed technique and its finite element realisation to consider a representative ensemble of randomly packed intercon- nected particles. We summarise the analysis and demonstrate how the results can be directly used in practical estimations of the permeability coefficients of particulate porous media at low saturation levels. In Chapter 7, we briefly discuss how the results can be generalised and transferred to different settings.

The main results presented in Chapter 4 have been published in (Sirimark et al. 2018a,b). The results obtained in a randomly packed particle ensemble, Chapter 6, have been submitted to a journal (Sirimark et al. 2019).

Chapter 2

Capillary Transport in Fibrous

Porous Materials at Low Levels

of Saturation

The problem of capillary transport in particulate porous materials, such as sand, at low levels of saturation can be simulated using a macroscopic math- ematical model, the superfast non-linear diffusion (Lukyanov et al., 2012). The mathematical model is based on several characteristic features of the phenomenon. First of all, in particulate porous media at low saturation levels, the liquid is con- centrated in the pendular rings formed at the point of contact of the constituent particles. The pendular rings are the reservoirs bounded by a constant-curvature interface, which has a characteristic pressure-volume relationship. At the same time, the flow predominantly occurs over the surface elements of the particles characterised by a constant saturation level s0. Apparently, such separation of

the functional elements can occur in other types of porous media. In this chapter, we will study the process of diffusion in one of those system, the fibrous porous materials, like textile or paper.

The structure of fibrous porous materials is quite different from that of par- ticulate porous media (Alava and Niskanen 1994; Eichhorn and Sampson 2005; Herminghaus 2005; Niskanen and Alava 1994a; Rasi 2013; Sampson 2003). Yet, all the main elements of the super-fast diffusion model are present. At low satura- tion levels, the liquid is only located on the rough surfaces of the fibres (including intrafibre pores) and in the liquid bridges formed at the intersections of the fi- bres (Sauret et al. 2014, 2015; Soleimani et al. 2015). The microscopic surface roughness of the fibres generates the capillary pressure to drive the liquid flow through the network, where the liquid bridges, as in the case of particulate me- dia, play the role of variable volume reservoirs. We further assume that the liquid at least partially wets the fibres, so that the contact angle on rough surfaces of the fibres would be small (close to zero) or zero.

2.1

Morphology of Liquid Distributions in Fi-

brous Materials.

The morphology of the liquid bridges formed between the crossing fibres in the wetting case is found to be in general more complex than that observed between the particles (Alava and Niskanen 1994; Eichhorn and Sampson 2005; Herming- haus 2005; Niskanen and Alava 1994a; Rasi 2013; Sampson 2003; Scheel et al. 2008a,b). The liquid volume at the crossing of two rigid fibres can take under

the action of surface tension forces several distinct morphologies depending on the amount of the liquid VB, the separation distance and the angle between the

fibres θf: a long liquid column, a mixed morphology state that consists of a drop

on one side together with a small amount of liquid on the other side and a drop or a compact hemispherical drop or a pendular ring (Sauret et al. 2014, 2015; Soleimani et al. 2015). In general, the elongated liquid columns are only formed at small angles θf ≤ 20◦ between the crossing fibres (Sauret et al. 2015).