Rise velocity of an air bubble in porous media: Theoretical studies

Abstract

The rise velocity of injected air phase from the injection point toward the vadose zone is a critical factor in in‐situ air sparging operations. It has been reported in the literature that air injected into saturated gravel rises as discrete air bubbles in bubbly flow of air phase. The objective of this study is to develop a quantitative technique to estimate the rise velocity of an air bubble in coarse porous media. The model is based on the macroscopic balance equation for forces acting on a bubble rising in a porous medium. The governing equation incorporates inertial force, added mass force, buoyant force, surface tension and drag force that results from the momentum transfer between the phases. The momentum transfer terms take into account the viscous as well as the kinetic energy losses at high velocities. Analytical solutions are obtained for steady, quasi‐steady, and accelerated bubble rise velocities. Results show that air bubbles moving up through a porous medium equilibrate after a short travel time and very short distances of rise. It is determined that the terminal rise velocity of a single air bubble in an otherwise water saturated porous medium cannot exceed 18.5 cm/s. The theoretical model results compared favorably with the experimental data reported in the literature. A dimensional analysis conducted to study the effect of individual forces indicates that the buoyant force is largely balanced by the drag force for bubbles with an equivalent radius of 0.2–0.5 cm. With increasing bubble radius, the dimensionless number representing the effect of the surface tension force decreases rapidly. Since the total inertial force is quite small, the accelerated bubble rise velocity can be approximated by the terminal velocity.