An Effective Continuum Model for the Liquid-To-Gas Phase Change in a Porous Medium Driven by Solute Diffusion: I. Constant Pressure Decline Hates

Abstract

Abstract We derive an effective continuum model to describe the nucleation and subsequent growth of a gas phase from a supersaturated, slightly compressible binary liquid in a porous medium, driven by solute diffusion. The evolution of the gas results either from the reduction of the system pressure at a constant rate. The model addresses two stages before the onset of bulk gas flow, nucleation and gas phase growth. We assume negligible gradients due to gravity or viscous forces, thus the critical gas saturation, which signals the onset of bulk gas flow, is only a function of the nucleation fraction. We show that the important quantities characterizing the process, such as the fraction of pores that host activated sites, the deviation from thermodynamic equilibrium, the maximum supersaturation in the system and the critical gas saturation depend crucially on the nucleation characteristics of the medium. We use heterogeneous nucleation models primarily in the form of pre-existing gas, trapped in hydrophobic cavities, but also in terms of a rate-dependent nucleation, to investigate in detail the nucleation behavior. Using scaling analysis and a simpler analytical model we show that the relevant quantities during nucleation can be expressed in terms of a simple combination of dimensionless parameters, which include rate effects, for either type of nucleation model. The theory predicts that the maximum supersaturation in the system is a weakly increasing function of rate, which in the region of typical experimental parameters, can be approximated as a power law with a small exponent. This function depends sensitively on the probability density function of the nucleation cavity sizes. It also predicts that the final nucleation fraction, thus the critical gas saturation, is a power law of the decline rate. The theoretical exponents are shown to be in good agreement with experimental data. The subsequent evolution of the gas phase and the approach to the critical gas saturation is also described.