Stability analysis of two-phase buoyancy-driven flow in the presence of a capillary transition zone

Natural convection that results from the dissolution of a diffusive species into a fluid saturated porous layer in the presence of a capillary transition zone is an important phenomenon in geological flows. The prediction of the onset of convection has remained elusive under the theory of gravitationally unstable multiphase flow in porous media. The present study offers a paradigm for the stability of two-phase buoyancy-driven flow in the presence of the capillary transition zone in a saturated porous medium, which allows for a quantitative description of the onset of natural convection. The analysis, which is based on the quasi-steady-state approximation, stresses the critical role of the capillary transition zone and the upward crossflow between the diffusive boundary layer and the capillary transition zone in the stability of the system, as well as in the transient growth of perturbations. We show that the instability problem can be characterized by capillary-dominant and buoyancy-dominant regimes with a transition in between. In the capillary-dominant regime, capillarity has a strong role in destabilizing the diffusive boundary layer. While in the buoyancy-dominant regime, the capillary transition zone is small enough that it can be ignored and the buoyancy force is the sole cause of the instability. Furthermore, our analysis shows that the capillary transition zone can potentially accelerate the evolution of the natural convection over 6 times faster than the buoyancy-dominant regime. Finally, the nonlinear dynamics of the system is studied using direct numerical simulations. The nonlinear simulations confirm the predictions from the linear stability analysis.