Stochastic Simulation and Viscous Fingering

The extreme case of fluid displacements at infinitely adverse viscosity ratio is discussed in the light of results from the equivalent diffusion limited aggregation model. In this limit the pattern of displacing fluid is dominated by fingers on all scales and numerical simulation is greatly facilitated by the availability of simple random walker analogues. The possibility of extending these to the finite viscosity ratio case will also be discussed, along with its major difficulty. The origin of the fingering instability when a viscous fluid is displaced from a pDrous medium by a less viscous one is very simple, namely that flow occurs preferentially in regions where the less viscous fluid has already penetrated furthest. This mechanism is governed by a single dimensionless parameter, the viscosity ratio, and of itself contains no preferred lengthscale. Thus for example, fingers of thickness w develop in the tine required for the mean position of the displacement front to advance a distance of order w, and the finest scale fingers develop fastest.