Viscous fingering in periodically heterogeneous porous media .2. Numerical simulations

We study nonlinear viscous fingering in heterogeneous media through direct numerical simulation. A pseudospectral method is developed and applied to our spatially periodic model introduced in Paper I [J. Chem. Phys. 107, 9609 (1997)]. The problem involves several parameters, including the Peclet number, Pe, the magnitude and wave numbers of the heterogeneity, sigma; n(x), n(y), respectively, and the log of the viscosity ratio R. Progress is made by fixing R at 3.0 and then working first with layered systems n(x)=0 and finally with ''checkerboard'' systems in which both wave numbers are nonzero. Strongly nonlinear finger dynamics are compared and contrasted with those occurring in the homogeneous case. For layered systems, it is found that very low levels of heterogeneity leads to an enhancement of the growth rate of the fingered zones, and that both harmonic and subharmonic resonances between the intrinsic scale of nonlinear fingering and those of the heterogeneity occur. We also fmd that the fingering regime of layered systems can be completely disrupted by modest levels of heterogeneity, leading to a ''channeling'' regime and dispersive behavior which is identified as a Taylor dispersion mechanism. The effective axial dispersion coefficient in this regime is found to be strongly dependent on the viscosity ratio. The situation becomes more complex for the checkerboard case. The channeling regime can in turn be disrupted by the axial dependence of the heterogeneity, which stimulates tip splitting and a return to complex nonlinear finger dynamics in regions of parameter space, including very large sigma, that would otherwise be strongly dispersive. The effectiveness of the axial variation in stimulating tip splitting is studied by a short parametric study in n(x), and is found to be maximized for certain axial frequencies in a manner similar to that found in Paper I. All our results are found to be in general qualitative agreement with available (but Limited) experimental visualizations. (C) 1997 American Institute of Physics.