Crossover from fractal to compact flow from simulations of two-phase flow with finite viscosity ratio in two-dimensional porous media

Abstract

The effect of the viscosity ratio (M\ensuremath{\equiv}${\mathrm{\ensuremath{\mu}}}_{\mathit{D}}$/${\mathrm{\ensuremath{\mu}}}_{\mathit{I}}$) in changing the nature of viscous fingering was studied using a simple, physical model of miscible, dispersionless, two-phase, linear flow in model two-dimensional porous media. For all viscosity ratios, the initial flows had an unstable, fractal character which crossed over to stable, compact flow on a time (or length) scale which increased with the viscosity ratio. An empirical scaling of the data enables an asymptotic characterization of both this time scale \ensuremath{\tau}\ensuremath{\approxeq}${\mathit{M}}_{\mathit{t}}^{\mathrm{\ensuremath{\varphi}}}$, where ${\mathrm{\ensuremath{\varphi}}}_{\mathit{t}}$=0.17\ifmmode\pm\else\textpm\fi{}0.03, as well as the front velocity v\ensuremath{\approxeq}${\mathit{M}}_{\mathit{t}}^{\mathrm{\ensuremath{\epsilon}}\mathrm{\ensuremath{\varphi}}}$ where \ensuremath{\epsilon}${\mathrm{\ensuremath{\varphi}}}_{\mathit{t}}$=0.07\ifmmode\pm\else\textpm\fi{}0.02. A comparison with identical simulations of radial flow indicates that the same characteristic length scale applies in both linear and radial geometries l\ensuremath{\approxeq}${\mathit{M}}_{\mathit{l}}^{\mathrm{\ensuremath{\varphi}}}$ where ${\mathrm{\ensuremath{\varphi}}}_{\mathit{l}}$=0.24\ifmmode\pm\else\textpm\fi{}0.06, while the time scales differ because of the different relations between time and size in the two geometries.