Study of the relationship between fractal dimension and viscosity ratio for viscous fingering with a modified DLA model

The morphological evolution characters of viscous fingering, which occurred in two-dimensional Hele-Shaw cells with various viscosity ratio of the displaced-to-driving fluids from 1 to 10 000, were simulated by a modified DLA model on off-lattices. The model can reproduce experimental patterns conveniently and generate various finger growth aggregations from skeletal patterns to fleshy patterns. Both the shapes and the fractal dimension of fractal clusters obtained by controlling the parameters in the present model are in good agreement with experiments. The present work indicates that the effective fractal dimension of viscous fingering decreases and the corresponding morphologies of finger growth vary from fleshy patterns to skeletal patterns when viscosity ratio VR is increased in laboratory scale. Low fractal dimension corresponds to very ramified viscous fingering fractal structure and short breakthrough time. In small scale, the effective fractal dimension can be reasonably regarded as a useful parameter for the fluid displacement process. In the scale of laboratory experiments, the unfavorable viscous fingering can be controlled by reducing the viscosity ratio, thus, increasing the effective fractal dimension of viscous fingering.