Theoretical model for the wetting of a rough surface

Many applications would benefit from an understanding of the physical mechanism behind fluid movement on rough surfaces, including the movement of water or contaminants within an unsaturated rock fracture. Presented is a theoretical investigation of the effect of surface roughness on fluid spreading. It is known that surface roughness enhances the effects of hydrophobic or hydrophilic behavior, as well as allowing for faster spreading of a hydrophilic fluid. A model is presented based on the classification of the regimes of spreading that occur when fluid encounters a rough surface: microscopic precursor film, mesoscopic invasion of roughness and macroscopic reaction to external forces. A theoretical relationship is developed for the physical mechanisms that drive mesoscopic invasion, which is used to guide a discussion of the implications of the theory on spreading conditions. Development of the analytical equation is based on a balance between capillary forces and frictional resistive forces. Chemical heterogeneity is ignored. The effect of various methods for estimating viscous dissipation is compared to available data from fluid rise on roughness experiments. Methods that account more accurately for roughness shape better explain the data as they account for more surface friction; the best fit was found for a hydraulic diameter approximation. The analytical solution implies the existence of a critical contact angle that is a function of roughness geometry, below which fluid will spread and above which fluid will resist spreading. The resulting equation predicts movement of a liquid invasion front with a square root of time dependence, mathematically resembling a diffusive process.