Control of viscous fingering through variable injection rates and time-dependent viscosity fluids: Beyond the linear regime

The critical mixed transport process in remediation agent radial injection into contaminated aquifer plumes

Accurately depicting the subsurface mixing of radially injected remediation agents with contaminated plumes remains paramount yet challenging for understanding and simulating reactive transport. To address this, the present research employed the mixing dynamics of a potassium permanganate plume injected into a pre-existing contaminated plume. Through combining colour deconvolution and thresholding, we effectively isolated local mixing values within the Gaussian annular narrow mixing zone from the noise of mixed double-plume images. Key findings revealed increasing injection rate promotes plume mixing while adding xanthan gum to increase fluid viscosity moderates interface mixing, reducing mixing zone width by 25.3% and 37.4% for 100 mg/L and 400 mg/L xanthan gum, respectively. Grain size is pivotal, with a 30% increase in mixing areas observed in coarse-grained sands over medium-grained sands. Balancing sufficient mixing and preventing contaminated plume growth is essential for effective remediation. Injection rates below 5 mL/min may suppress contaminated plume expansion, albeit at the possible cost of protracted remediation durations. For the attainment of optimal remediation, it's imperative to harmonize robust mixing processes with the mitigation of contaminated plume expansion - a balance that adding xanthan gum during the initial injection phase seems poised to achieve (xanthan gum optimized the average mixing index (AMI)). These findings provide valuable insights into groundwater plume mixing, supporting effective remediation strategies.

Fingering stabilization and adhesion force in the lifting flow with a fluid annulus

The lifting Hele-Shaw cell flow commonly involves the stretching of a viscous oil droplet surrounded by air, in the confined space between two parallel plates. As the upper plate is lifted, viscous fingering instabilities emerge at the air-oil interface. Such an interfacial instability phenomenon is widely observed in numerous technological and industrial applications, being quite difficult to control. Motivated by the recent interest in controlling and stabilizing the Saffman-Taylor instability in lifting Hele-Shaw flows, we propose an alternative way to restrain the development of interfacial disturbances in this gap-variable system. Our method modifies the traditional plate-lifting flow arrangement by introducing a finite fluid annulus layer encircling the central oil droplet, and separating it from the air. A second-order, perturbative mode-coupling approach is employed to analyze morphological and stability behaviors in this three-fluid, two-interface, doubly connected system. Our findings indicate that the intermediate fluid ring can significantly stabilize the interface of the central oil droplet. We show that the effectiveness of this stabilization protocol relies on the appropriate choice of the ring's viscosity and thickness. Furthermore, we calculate the adhesion force required to detach the plates, and find that it does not change significantly with the addition of the fluid envelope as long as it is sufficiently thin. Finally, we detect no distinction in the adhesion force computed for stable or unstable annular interfaces, indicating that the presence of fingering at the ring's boundaries has a negligible effect on the adhesion force.

Control of interfacial instabilities through variable injection rate in a radial Hele-Shaw cell: A nonlinear approach for late-time analysis

In this paper, the nonlinear behavior of immiscible viscous fingering in a circular Hele-Shaw cell under the action of different time-dependent injection flow rate schemes is assessed numerically. Unlike previous studies which addressed the infinite viscosity ratio (inviscid-viscous flow), the problem is tackled by paying special attention to flows with finite viscosity ratio (viscous flow) in which the viscosity of the displacing and the displaced fluids can have any arbitrary value. Systematic numerical simulations based on a complex-variable formulation of Cauchy-Green barycentric coordinates are performed at different mobility ratios and capillary numbers with a focus on the late-time fully nonlinear regime. Additionally, numerical optimization is used to obtain the optimal flow rate schedule through a second-order weakly nonlinear stability analysis in contrast to previous studies in which the optimal flow rate was obtained entirely based on linear stability analysis. It is demonstrated that, irrespective of the values of the mobility ratio and/or the capillary number, for patterns whose constant injection counterpart exhibits linear flow regime, the curvature-driven relaxation time is comparable with the operational time of the time-dependent injection flow rate controlling schemes, and most of the controlling schemes work very well and suppress the fingering phenomenon remarkably with the maximum recovery improvement of 15%. As the nonlinearity of the system increases, the schemes may still perform well, but their effectiveness is more pronounced in patterns with less nonlinearity in their constant injection counterpart than those with higher nonlinearity. As the nonlinearity increases, the curvature-driven relaxation time becomes longer than the operational time of the schemes, leading to a reduction in their effectiveness. Additionally, it is shown that employment of the second-order weakly nonlinear stability analysis to formulate the objective function does not result in any remarkable variation in the obtained optimal flow rate schedule.

Stabilization of periodically forced Hele-Shaw flows by means of a nonmonotonic viscosity profile

The onset of viscous fingering in the presence of a viscosity profile is investigated theoretically for two immiscible fluids undergoing a time-dependent injection. Here, we show that the presence of a positive viscosity gradient at the interface between both fluids stabilizes the interface facilitating the spread of the perturbation. This effect is much more pronounced in the case of sinusoidal injection flows. The influence of the viscosity gradient on the dispersion relation is analyzed. Numerical simulations of the Navier-Stokes equation confirm the linear stability analysis.