Hamiltonian formulation towards minimization of viscous fluid fingering

TL;DR

This paper develops an exact Hamiltonian-based method to determine optimal fluid injection rates that minimize viscous fingering in Hele-Shaw cells and porous media, improving upon previous approximate approaches across all capillary numbers.

Contribution

It introduces a Hamiltonian formulation to derive exact solutions for injection rates that suppress fingering, extending previous high capillary number approximations to all regimes.

Findings

Exact injection rate expressions outperform approximate ones.

The method applies to both Hele-Shaw cells and porous media.

Significant reduction in fingering formation achieved.

Abstract

A variational approach has been recently employed to determine the ideal time-dependent injection rate Q(t) that minimizes fingering formation when a fluid is injected in a Hele-Shaw cell filled with another fluid of much greater viscosity. However, such a calculation is approximate in nature, since it has been performed by assuming a high capillary number regime. In this work, we go one step further, and utilize a Hamiltonian formulation to obtain an analytical exact solution for Q(t), now valid for arbitrary values of the capillary number. Moreover, this Hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three-dimensional porous media. An analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided.