Viscous fingering with partially miscible fluids

Abstract:

When a less viscous fluid displaces a more viscous fluid, the contrast in viscosity destabilizes the interface between the two fluids, leading to the formation of fingers. Experimental and numerical studies of viscous fingering have focused on fluids that are either fully miscible (e.g. water and glycerol) or perfectly immiscible (e.g. water and oil). In practice, however, the miscibility of two fluids can change appreciably with temperature and pressure, and often falls into the case of partial miscibility, where two fluids have limited solubility in each other (e.g. CO2 and water).

Following our recent work for miscible systems (Jha et al., PRL 2011, 2013) and immiscible systems (Cueto-Felgueroso and Juanes, PRL 2012, JFM 2014), here we propose a phase-field model for fluid-fluid displacements in a porous medium, when the two fluids have limited (but nonzero) solubility in one another. In our model, partial miscibility is characterized through the design of the thermodynamic free energy of the two-fluid system. We express the model in dimensionless form and elucidate the key dimensionless groups that control the behavior of the system. We present high-resolution numerical simulations of the model applied to the viscous fingering problem. On one hand, we demonstrate the effect of partial miscibility on the hydrodynamic instability. On the other, we elucidate the role of the degree of fingering on the rate of mutual fluid dissolution.

Figure caption: final snapshots in simulations of viscous fingering with a two-fluid system mimicking that of CO2 and water. The colormap corresponds to the concentration of CO2. A band of less viscous gas phase rich in CO2 (red) displaces through the more viscous liquid phase that is undersaturated with CO2 (blue). At the fluid interface, an exchange of CO2 occurs as a result of local chemical potentials that drives the system towards thermodynamic equilibrium. This results in a shrinkage of gas phase as well as a local increase in CO2 concentration within the liquid phase (the white diffusive regions). The viscosity contrast R, defined here as the natural log of the viscosity ratio, increases from left to right.