Effects of the Three-Dimensional Tensor Character of Dispersion on the Viscous Fingering Instability in Miscible Displacement.

The effect of the tensor character of dispersion on the viscous fingering instability in miscible displacement is examined. All experiments and field applications of miscible displacement exhibit anisotropy in the dispersion characteristics of the medium and three-dimensionality in the evolution of the instability. We investigate these important features by large scale numerical simulation by spectral methods based on Hartley transforms. Since this is a complex problem, we used several models in our simulations, each comprising a different contribution to dispersion. The models were 2-D isotropic dispersion with weaker levels of dispersion than simulated before, 2-D anisotropic with velocity dependence appropriate to the Hele-Shaw cell, 2-D anisotropic with general velocity dependence, and 3-D isotropic. All cases required homogeneous porous media. These simulations were executed with a variety of initial conditions and over a wide region of parameter space. The nature of nonlinear viscous fingering was determined. Several nonlinear finger interactions were observed. Shielding, spreading, tip-splitting, and pairing of viscous fingers were observed. Multiple coalescence and fading were seen here for the first time. Transversely and longitudinally averaged one-dimensional concentration histories demonstrate the rate at which the mixing zone broadens and the increase in lateral scale as the fingers evolve when no tip-splitting occurs. These properties are insensitive to both the dispersion anisotropy and the Peclet number at high Peclet number and long times. This suggests the dominance of finger interaction mechanisms that are essentially independent of details of the concentration fields and governed fundamentally by pressure fields.