Abstract
We study the life-cycle of miscible fingering, from the early fingering initiation, through their growth and nonlinear interactions to their decay to a single finger at late times. Dimensionless analysis is used to relate the number of fingers, the nature of their nonlinear interactions (spreading, coalescence, tip splitting), and their eventual decay to the viscosity ratio, transverse Peclet number, and anisotropic dispersion. We show that the initial number of fingers that grow is approximately half that predicted by analytical solutions that neglect the impact of longitudinal diffusion smearing the interface between the injected solvent and the displaced fluid. The growth rates of these fingers are also approximately one quarter that predicted by these analyses. Nonetheless, we find that the dynamics of finger interactions over time can be scaled using the most dangerous wavenumber and associated growth rate determined from linear stability analysis. This subsequently allows us to provide a relationship that can be used to estimate when predict when the late time, single finger regime will occur.
Highlights
Viscous fingering, known as the Saffman-Taylor instability (Ref. 1), occurs when a less viscous fluid displaces a more viscous fluid in a porous medium
Before investigating different aspects of finger growth and decay, we provide a description of the overall finger dynamics using the different methods applied to analyze them
This is consistent with results obtained by many previous researchers (e.g., Refs. 28 and 29) who have shown that fractional flow based empirical models such as those of Koval25 and Todd and Longstaff26 can be used to estimate the average behavior of unstable viscous fingering in linear systems
Summary
Known as the Saffman-Taylor instability (Ref. 1), occurs when a less viscous fluid displaces a more viscous fluid in a porous medium. The average behavior of the flow in the different fingering regimes and the times at which the different regimes start are of particular interest to engineers as they typically use empirical models such as those of Koval or Todd and Longstaff to predict the impact of fingering on the trapping of carbon dioxide or oil recovery. These empirical models assume that the average concentration profile of the displacing fluid grows linearly with time. It is important to investigate simplified 2D systems before investigating more realistic 3D systems in order to ensure that we have a full description of the appropriate physics in these systems. 3D systems are both more complicated to analyze mathematically and more numerically intensive to simulate; previous studies (e.g., Ref. 33) have shown that, in the absence of gravitational effects, there is very little difference in the average behavior of 2D and 3D simulations
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