Geometric dispersion and unstable flow in porous media.

Unstable flow patterns are produced when a viscous fluid (o) is displaced by an inviscid one (s). These patterns, viscous fingers, become fractallike in media such as porous rock that promote extensive fluid-mobility fluctuations. The porous rock is also effective in dispersing the concentrations of the fluids. The rate of (geometric) dispersion competes with the growth rate of the fluctuations and establishes a lower limit to the length scale of the fractal pattern. We have solved the equations describing the coupled dynamics of unstable miscible flow and dispersion by a generalization of the previously reported ``probabilistic'' method. The inclusion of dispersion provides both a well-posed mathematical problem and a means for quantitative comparison with experiment. Numerical calculations show, for large viscosity ratio (M==${\ensuremath{\mu}}_{o}$/${\ensuremath{\mu}}_{s}$), that scaling of the pattern area with the amount of physical dispersion is the same as scaling with system size (in the absence of dispersion). In the case of small dispersion we confirm the validity of the discovery of a crossover from fractal scaling to stable flow based only on the finite viscosity ratio (${\ensuremath{\mu}}_{o}$g${\ensuremath{\mu}}_{s}$\ensuremath{\ne}0). We have designated this global-scale stabilization as viscous relaxation. In this limit of vanishing dispersion, we show that the individual finger width scales with the numerical grid size, but the areal density of the fingers increase with decreasing M. We have developed a simple analytic expression for the areal displacement density as a function of M. Only for M\ensuremath{\rightarrow}\ensuremath{\infty} does this density vanish (corresponding to fractal scaling). Comparisons are made with two-dimensional physical model flow experiments. Using independently determined dispersion coefficients we account quantitatively for flood displacement over a wide range of viscosity ratio M.