Abstract
We investigate the dynamics, interactions, and decay of immiscible viscous fingers in two and three dimensions over time in a high-aspect ratio (up to 100:1) system. The behavior is related to the viscosity ratio and a macroscopic capillary number. The same four fingering regimes are observed as in miscible displacements (spreading of the interface between wetting and non-wetting fluid but no fingers, the growth of many fingers that can be described by perturbation analysis, non-linear interactions between fingers and decay to a single finger) for low viscosity ratio and high capillary to viscous ratios. At higher viscosity ratios and lower capillary to viscous ratios, periodic tip-splitting and decay results in a fluctuation between one and two fingers at late time. This has not been seen in miscible displacements. We provide a stability plot that can be used to identify when this will occur. Similar behaviors were seen in both two and three dimensions, suggesting that learnings from two-dimensional (2D) linear displacements can be applied to similar three-dimensional (3D) flows. In particular, the square root of the number of fingers seen in the 3D simulations and their decay with time was almost identical to 2D.
Highlights
Viscous fingering, the Saffman–Taylor instability, describes the instabilities that occur when a less viscous fluid displaces a more viscous fluid in a porous medium.[1,2] It is distinct from the Rayleigh–Taylor instability where the fingering is driven by density differences[2] and capillary fingering which is fractal in nature and occurs at the pore scale when the capillary number is very low.[3]
The typical life cycle of immiscible viscous fingering from the start of fingering to very late times is shown in Fig. 5, for a large aspect ratio simulation with a viscosity ratio of 300 and a capillary to viscous number of Nca 1⁄4 3:6 Â 10À5
We have investigated the full life span of immiscible viscous fingering on the continuum scale, in 2- and 3D, focusing on rectilinear displacements
Summary
The Saffman–Taylor instability, describes the instabilities that occur when a less viscous fluid displaces a more viscous fluid in a porous medium.[1,2] It is distinct from the Rayleigh–Taylor instability where the fingering is driven by density differences[2] and capillary fingering which is fractal in nature and occurs at the pore scale when the capillary number is very low.[3]. Viscous fingering instabilities are challenging to predict as they grow and develop in a non-linear fashion, depending on the mobility ratio between the displacing and displaced fluids and mixing via diffusion (miscible fluids) or capillary pressure (immiscible fluids). Analytical investigations[7,8,9] all show that initially, there is a particular finger wavelength (the most “dangerous” wavelength) that grows most quickly and this wavelength depends upon the mobilities of the fluids, the speed of the displacement, and the capillary pressure This is similar to the case for miscible displacements, where the most dangerous wavenumber depends upon the viscosity ratio of the fluids and the diffusive Peclet number.[19]. We identify the non-linear interaction mechanisms between fingers at intermediate times and provide a stability plot that can be used to estimate when the fingers will merge into a single finger at late time
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