Evidence for a universal saturation profile for radial viscous fingers

Area of Science:

• Fluid Dynamics
• Complex Systems
• Non-Newtonian Fluid Mechanics

Background:

• Viscous fingering occurs when a low viscosity fluid displaces a high viscosity fluid in porous media.
• This phenomenon leads to complex, fractal-like patterns.
• Understanding these patterns is crucial for various applications, including oil recovery and microfluidics.

Purpose of the Study:

• To investigate the fractal dimension and saturation profiles of viscous fingering patterns.
• To explore the universality of these patterns under different injection conditions.
• To assess the effectiveness of control strategies based on linear stability theory.

Main Methods:

• Combined experimental approaches with constant injection rate, constant source pressure, and linearly increasing injection rate.
• Employed numerical simulations based on a diffusion-limited aggregation (DLA) model.
• Analyzed the azimuthally-averaged fraction of area occupied by fingers, S(r,t).

Main Results:

• Identified a fractal dimension of approximately 1.7 for the finger patterns.
• Characterized the saturation profile into three distinct regions: fully saturated, frozen finger, and growing finger.
• Developed a relationship between finger radius, injected volume, and inner region radius: r_f = 0.4r_b (V/r_b^2)^(1/1.7).

Conclusions:

• Viscous fingering instabilities exhibit remarkable universality and order, despite their complex fractal nature.
• Control strategies based on linear stability theory are less effective for fully nonlinear fingering patterns.
• The findings provide a deeper understanding of pattern formation in multiphase flow systems.