Simulations of Viscous Fingering in a Random Network

We present a computer study of viscous fingering in an idealized porous medium. The porous medium is represented by an isotropic network of up to 80,000 nodes connected by thin tubes, which is modelled in two dimensions by the Delaunay triangulation of points placed at random in a circular region, and in three dimensions by a Voronoi tessellation in a sphere. We then simulate two-fluid displacements in this network and are able to demonstrate the effects of viscous and capillary forces. In the absence of capillary forces we show that the boundary between the two fluids is fractal. Furthermore we use the local average flow rates and pressures to calculate effective saturation dependent fractional flows. Using a radial Buckley-Leverett theory, the mean saturation profile can be inferred from the solution of the fractional flow equation, which is consistent with the measured saturation.