Asymptotic solutions of miscible displacements in geometries of large aspect ratio

Abstract

Asymptotic solutions are developed for miscible displacements at Stokes flow conditions between parallel plates or in a cylindrical capillary, at large values of the geometric aspect ratio. The single integro-differential equation obtained is solved numerically for different values of the Péclet number and the viscosity ratio. At large values of the latter, the solution consists of a symmetric finger propagating in the middle of the gap or the capillary. Constraints on conventional convection-dispersion-equation approach for studying miscible instabilities in planar Hele–Shaw cells are obtained. The asymptotic formalism is next used to derive—in the limit of zero diffusion— a hyperbolic equation for the cross-sectionally averaged concentration, the solution of which is obtained by analytical means. This solution is valid as long as sharp shock fronts do not form. The results are compared with recent numerical simulations of the full problem and experiments of miscible displacement in a narrow capillary.