Numerical modeling of fluid displacement in Hele-Shaw cells: a gap-averaged approach for power-law and Newtonian fluids

This work presents a physics-based two-dimensional model for simulating displacement flows of power-law fluids in Hele-Shaw cells. The model is derived by approximating fully developed velocity profiles across the gap-wise direction and averaging the mass and momentum conservation equations, resulting in a two-dimensional formulation that efficiently captures complex fluid dynamics. Implemented in OpenFOAM, this approach achieves computational speeds over 200 times faster than comparable 3D simulations, while preserving the accuracy of displacement dynamics. Validated against 3D DNS results and experimental data, this 2D model accurately replicates observed flow phenomena. Simulations of over 70 cases examined the effect of the ratio of friction pressure gradients (RFG) between fluid pairs on interface stability. Results show that RFGs below unity maintain a flat interface, while higher values induce viscous fingering. In cases with RFG closer to unity, a longer duct or extended displacement time is required for significant finger growth.