A WEAKLY NONLINEAR INSTABILITY OF MISCIBLE DISPLACEMENTS IN A RECTILINEAR POROUS MEDIA FLOW

Abstract

The linear stability theory of Tan & Homsy [1986] is extended to include the effects of weak nonlinear coupling between mass flux and viscous effects when the viscous fingers grow from a slowly diffusing, nearly flat displacement front. A regular perturbation scheme combined with a similarity-separation of variables technique leads to a Landau equation for the amplitude of the disturbance. The Landau constant has a simple pole for a given wavenumber within the linear theory cutoff wavenumber for growth. An argument is given that this pole leads to pairing of fingers while the instability remains small. Comparison of the length scale of the pole of the Landau constant with experimental measurements of finger scale shows good agreement where plausibly finite-amplitude effects might come into play, but with the linear theory otherwise.