Dynamics and stability of two-potential flows in the porous media.

The experimental and numerical results of the capillary-force-driven climb of wetting liquid in porous media, which is opposed by the gravity force, are analyzed with respect to the emergence of a multiphase flow front and flow stability of the climbing liquid. Two dynamic characteristics are used: (i) the multiphase flow front thickness as a function of time, and (ii) the capillary number as a function of Bond number, where both numbers are calculated from the harmonic average of pores radii. Throughout the climb, the influence of capillary, gravity, and viscous force variations on the flow behavior is investigated for different porous media. For a specific porous medium, a unique flow front power law function of time is observed for the capillary flow climbs with or without gravity force. Distinct dynamic flow front power law functions are found for different porous media. However, for capillary climb in different porous media, one is able to predict a unique behavior for the wetting height (the interface between wetted and dry regions of porous medium) using the capillary and Bond number. It is found that these two numbers correlate as a unique exponential function, even for porous media whose permeabilities vary for two orders of magnitude. For climbs without the gravity force (capillary spreads), the initial climb dynamics follows this exponential law, but for later flow times and when a significant flow front is developed, one observes a constant value of the capillary number. Using this approach to describe the capillary climb, only the capillary versus Bond number correlation is needed, which is completely measureable from the experiments.

Capillary climb dynamics in the limits of prevailing capillary and gravity force.

The dynamics of capillary climb of a wetting liquid into a porous medium that is opposed by gravity force is studied numerically. We use the capillary network model, in which an actual porous medium is represented as a network of pores and throats, each following a predefined size distribution function. The liquid potential in the pores along the liquid interface within the network is calculated as a result of capillary and gravity forces. The solution is general, and accounts for changes in the climbing height and climbing velocity. The numerical results for the capillary climb reveal that there are at least two distinct flow mechanisms. Initially, the flow is characterized by high climbing velocity, in which the capillary force is higher than the gravity force, and the flow is the viscous force dominated. For this single-phase flow, the Washburn equation can be used to predict the changes of climbing height over time. Later, for longer times and larger climbing height, the capillary and gravity forces become comparable, and one observes a slower increase in the climbing height as a function of time. Due to the two forces being comparable, the gas-liquid sharp interface transforms into flow front, where the multiphase flow develops. The numerical results from this study, expressed as the climbing height as a power law function of time, indicate that the two powers, which correspond to the two distinct mechanisms, differ significantly. The comparison of the powers with experimental data indicates good agreement. Furthermore, the power value from the Washburn solution is also analyzed, where it should be equal to 1/2 for purely viscous force driven flow. This is in contrast to the power value of ∼0.43 that is found experimentally. We show from the numerical solution that this discrepancy is due to the momentum dissipation on the liquid interface.

Infiltration time and imprint shape of a sessile droplet imbibing porous medium.

The infiltration of a sessile droplet into a homogeneous porous medium for a constant droplet base radius case is solved numerically, where the porous medium is represented as a capillary network consisting of pores and throats. For a homogeneous medium, the network is built of the spherical pores of constant radius, and the cylindrical throats of constant radius and height. Having such defined network, the droplet imbibes porous medium in a single-phase flow for which the free interface in porous medium is smooth, and the liquid phase permeability and the capillary pressure are constant. Using the numerical solution we carry out the parametric study in which: (i) liquid viscosity and surface tension, (ii) droplet volume and base radius, and (iii) porous medium porosity and permeability are varied. The droplet infiltration time, and the imprint shape that is given with two spheroid half-axes are calculated. The dimensionless analysis is utilized to correlate the droplet infiltration parameters from which master curves for the droplet infiltration time and the droplet imprint shape are obtained. Using the infiltration time correlation, both numerical and experimental results show a linear behavior.