Optimizing injection protocols in Hele-Shaw displacements: A trade-off between swept area and injection time

Viscous fingering instabilities reduce the efficiency of immiscible displacement processes in radial Hele-Shaw flows. Time-dependent injection protocols provide a practical means of mitigating these instabilities, and previous studies have shown that linearly increasing injection rates can strongly suppress finger growth when both the total injected volume and the total injection time are fixed. Here we reformulate this optimization problem by prescribing a fixed radial distance and terminating the injection when the most advanced finger reaches it, a condition inspired by breakthrough in displacement processes. We then introduce a linear-plateau injection protocol, in which the injection rate increases linearly up to a switching time t_{m} and subsequently remains constant at a maximum value Q_{max}, reflecting the finite capacity of pumping systems. Using linear stability analysis and fully nonlinear boundary-integral simulations, we show that this linear-plateau protocol stabilizes the interface and increases the swept area, even though the interface evolves for a longer time than in a reference protocol with constant injection Q_{max} throughout the process. Finally, we introduce a profitlike function that balances the gain in swept area against the cost associated with longer injection times. This formulation is motivated by potential applications of the proposed control strategy to displacement processes in porous media, such as enhanced oil recovery and CO_{2} injection for geological storage. Optimization of this function identifies the switching time that provides the best trade-off between displacement efficiency and injection duration. These results establish a Hele-Shaw framework for optimizing injection protocols under a swept-area-injection-time trade-off while accounting for a maximum allowable injection rate.