Gradient governed growth : the effect of viscosity ratio on stochastic simulations of the Saffman-Taylor instability

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Abstract
We report numerical simulations related to two-fluid flow in porous media. We assume that Darcy's law holds for the flow of each of the two fluids, one of which we are injecting into the porous medium, the other being that displaced. We therefore initially solve Laplace's equation for the pressure. At each time step we then advance the interface between the two fluids by a discrete step at a single point chosen with probability proportional to the pressure gradient. We examine the full range of viscosity ratios, covering both stable and unstable cases, and measure the rate of growth of the length of the interface between the two fluids. When the viscosity of the injected fluid is low compared with that of the fluid which is being displaced, the predicted fingering instability is similar to results obtained via the Witten and Sander model of diffusion limited aggregation. As the viscosity of the injected fluid increases, the fingers become thicker and they grow more slowly in length