Shear stabilization of miscible displacement processes in porous media

Abstract

The interface region between two fluids of different densities and viscosities in a porous medium in which gravity is directed at various angles to the interface is analyzed. Under these conditions, base states exist that involve both tangential and normal velocity components. These base states support traveling waves. In the presence of a normal displacement velocity, the amplitude of these waves grows according to the viscous fingering instability. For the immiscible case, it can easily be shown that the growth rate is not affected by the tangential velocities, while surface tension results in the usual stabilization. For the case of two miscible fluids, the stability of the base states using the quasi-steady-state approximation is investigated. The resulting equations are solved analytically for time t=0 and a criterion for instability is formulated. The stability of the flow for times t≳0 is investigated numerically using a spectral collocation method. It is found that the interaction of pressure forces and viscous forces is modified by tangential shear as compared to the classical problem, resulting in a stabilizing effect of the tangential shear. The key to understanding the physical mechanism behind this stabilization lies in the vorticity equation. While the classical problem gives rise to a dipole structure of the vorticity field, tangential shear leads to a quadrupole structure of the perturbation vorticity field, which is less unstable. This quadrupole structure is due to the finite thickness of the tangential base state velocity profile, i.e., the finite thickness of the dispersively spreading front, and hence cannot emerge on the sharp front maintained in immiscible displacements.