Miscible Displacements in Porous Media with Time-Dependent Injection Velocities

Miscible displacements in homogeneous porous media involving time-dependent injection velocities U(t) are analyzed. The displacements consist of periodic cycles that involve alternating stages of injection and production or of injection and soaking. Results of a linear stability analysis revealed that the growth rate of disturbances follows the overall trends of the velocity U(t) but with noticeable differences in periods of transition from production to injection. Furthermore, the growth rate of the time-dependent velocity was found to be smaller than that of a constant injection with a velocity equal to that corresponding to the minimum of U(t) while an overshoot was observed with respect to a displacement with a constant injection velocity equal to the maximum of U(t). Nonlinear simulations revealed that the dynamics of the flow can be drastically changed from those of the corresponding constant injection velocity and the changes depend on the period of the cycles, the amplitude of the velocity and on whether the displacement is initiated through an injection or a soaking stage. The enhancement or attenuation of the instability in comparison with the constant injection velocity increases with the cycles' period and the amplitude of the velocity in the injection stage, with the effects of the former being more prominent. It was also found that, beyond a certain critical cycle period, it is possible to observe instability and fingering in the case of time-dependent displacements that actually result in a net zero flow.