Dispersive Mixing in Unstable Displacement

The stable displacement of miscible fluids through a porous medium that has many small-scale permeability variations exhibits dispersive mixing between the fluids. The unstable displacement exhibits two flow regimes: one in which the displacement is dominated by viscous fingers and one in which the displacement is dominated by dispersive mixing due to the permeability variations. In the viscous finger-dominated regime the mixing zone expands linearly in time; in the dispersive-mixing-dominated regime the mixing zone expands as the square root of time. We have estimated the condition of transition between the two flow regimes. This condition has been tested by monitoring the development of the mixing zone in detailed numerical simulations of our own and by evaluating simulations reported by Araktingi and Orr, by Crump and by Moissis, et al. In addition, we found that when the unstable displacement is dominated by dispersive mixing, the expansion of the dispersive-mixing zone can be calculated according to the analytical model proposed by Kempers (1989). This model should be considered as a better alternative to the conventional Koval or Todd and Longstaff models for displacements with moderately mobility ratios.