Two-phase flow in Hele-Shaw cells: numberical studies of sweep efficiency in a five-spot pattern

Abstract

AbstractThe unsteady Hele-Shaw problem is a model nonlinear system that, for a certain parameter ranger, exhibits the phenomenon known as viscous fingering. While not directly applicable to multiphase porous-media flow, it does prove to be an adequate mathematical model for unstable dieplacement in laboratory parallel-plate devices. We seek here to determine, by use of an accurate boundary-integral frount-tracking scheme, the extent to which the simplified system captures the canonical nonlinear behavior of displacement flows and, in particular, to ascertain the role of noise in such systems. We choose to study a particular pattern of injection and production “wells.” The pattern chosen is the isolated “five-spot,” that is a single source surrounded by four symmetrically-placed sinks in an infinite two-dimensional “reservoir.” In cases where the “pusher” fluid has negligible viscosity, sweep efficiency is calculated for a range of values of the single dimensionless parameter τ, an inverse capillary number. As this parameter is reduced, corresponding to increased flow rate or reduced interfacial tension, this efficiency decreases continuously. For small values of τ, these stable displacements change abruptly to a regime characterized by unstable competing fingers and a significant reduction in sweep efficiency. A simple stability argument appears to correctly predict the noise level required to transit from the stable to the competing-finger regimes. Published compilations of experimental results for sweep efficiency as a function of viscosity ratio showed an unexplained divergence when the pusher fluid is less viscous. Our simulations produce a similar divergence when, for a given viscosity ratio, the parameter τ is varied.