A relationship between the fractal dimension and scaling groups of unstable miscible displacements

A detailed two-dimensional flow visualization study was performed to examine the dynamics of viscous fingering in miscible displacements. Detailed quantitative miscible displacement experiments using a microcomputer-based imaging workstation on a variety of oil recovery fluid systems were performed. The effect of two dimensionless scaling groups, namely gravity number and viscosity ratio, on the displacement behavior was investigated. Based on image analysis, the irregular fingering patterns of the flow visualization experiments were analyzed for fractal characteristics. Results indicate that the areal sweep efficiency of unstable miscible displacement follows a fractal scaling law with a fractal dimension and proportionality constant related to the gravity number and the viscosity ratio. The study shows that the fractal dimension decreases with decreasing gravity number and increasing viscosity ratio. This relationship was mapped by an artificial neural network model, which can be used to estimate the fractal dimension and the proportionality constant of miscible displacements as functions of the two scaling groups. These results have potential application in the mathematical modeling of unstable EOR displacements and in the scaling of laboratory displacements to field conditions. Received: 19 December 1999/Accepted: 26 January 2001