Abstract
Linear stability analysis and nonlinear simulations have been carried out to analyze the Rayleigh–Taylor instability in homogeneous porous media under time-dependent flow displacements. The flow processes consist of a sinusoidal time-dependent velocity characterized by its period (T) and amplitude (Γ) and ensure that the same amount of fluid is injected over a full flow period. A new, more efficient approach to determine instability characteristics has been developed for the stability analysis of these time-dependent injection flows and showed a growth rate that varies in time like the displacement velocity. The effects of the period T and amplitude Γ as well as the fluids’ viscosity (R) and density differences (ΔG) have been analyzed. Consistent with constant injection displacements, a larger ΔG leads to stronger instabilities. Furthermore, it is found that a larger R tends to attenuate the instability during extraction and soaking periods and to enhance it during injection. This study also revealed that for a given total injection time, the time-dependent flow can be less or more unstable than its constant injection counterpart. In particular, for Γ < −1, larger periods lead to stronger instabilities with longer more developed fingers. For Γ > 1, on the other hand, it is found that larger periods tend to attenuate the instability resulting in a smaller number of fingers and a more diffused front. Flows with unit amplitude (Γ = 1) exhibit the same qualitative trends as but are overall more unstable than their counterparts with Γ > 1.
Highlights
Flow instabilities develop at the interface between fluids during displacement processes in porous media when a less viscous fluid displaces a more viscous one or as a result of density mismatch
We examine the dynamics of buoyancy-driven instabilities in a two-dimensional porous medium under timedependent flow displacements
It was found that timeaveraged growth rates over the maximum of all considered periods offer the best characterization of the instability and result in trends that are in better qualitative agreement with those observed in the non-linear simulations, σave(R, ΔG, T, Γ)
Summary
Flow instabilities develop at the interface between fluids during displacement processes in porous media when a less viscous fluid displaces a more viscous one or as a result of density mismatch. Inspired by the effectiveness of the time-dependent control strategies employed in immiscible displacements by Li et al. and Dias and Miranda, Chen et al. investigated a similar control strategy but for a miscible displacement in which the injection rate varies as Q(t) ∼ t−1/3 Their non-linear simulation results confirmed the stabilizing effects of such control strategies, Q(t) ∼ t−1/3, and it was reported that the development of intricate fingering patterns, such as fingers’ splitting and merging, was all suppressed, compared to those of the constant injection scenario. Yuan and Azaiez were the first to employ a time-dependent injection strategy based on a sinusoidal velocity model in miscible horizontal displacements They observed that the sweep efficiency of their sinusoidal model was less than that of the constant injection counterpart. The effectiveness of the time-dependent injection scheme in attenuating or enhancing the instability relative to the constant injection scheme will be analyzed
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