Viscous fingering in miscible displacements: Unification of effects of viscosity contrast, anisotropic dispersion, and velocity dependence of dispersion on nonlinear finger propagation

The two‐dimensional (2‐D) isotropic simulations of Tan and Homsy [Phys. Fluids 31, 1330 (1988)] are extended to much broader and longer domains, and the 2‐D anisotropic simulations of Zimmerman and Homsy are extended to include a general velocity dependence. The mechanisms of nonlinear interaction of viscous fingers found for the first time in the anisotropic simulations recur in isotropic simulations, but at weaker levels of dispersion. An appropriate scaling to unify the average long time growth of the instability with both anisotropy in geometry and dispersion is provided. The long time growth of the instability from simulations agrees with acoustic measurements in 3‐D porous media, Bacri et al. [Phys. Rev. Lett. 67, 2005 (1991)], elucidating the effects of viscosity contrast, anisotropy, and velocity dependence of longitudinal dispersion. The combination of sufficiently high viscosity contrast, weak transverse dispersion, and strong dependence of longitudinal dispersion on velocity results in an augmentation to the long time growth of the instability. The associated critical parameter found by linear stability theory of Yortsos and Zeybek [Phys. Fluids 31, 3511 (1988)] predicts accurately this same long time growth increase.