General spectral theory for the onset of instabilities in displacement processes in porous media

SummaryThe problem of penetration of a fluid into a porous medium containing a more viscous liquid is investigated. It is known that the displacement front may become unstable in this case as it may break up into «fingers». The problem of inception of fingers has been treated previously in the literature by describing the displacement front in terms of its Fourier transform. In the present paper, we generalize earlier procedures by making allowance for an arbitrary elemental growth law. Furthermore, we assume that the phenomenon of fingering is not solely governed by the prevailing flow potentials, but also by the spectrum of heterogeneities in the porous medium. This is achieved by introducing a constant characteristic of the frequency of the heterogeneities in the porous medium. It then turns out that the maximum rate of growth as a function of wave length is considerably shifted from that predicted in the literature. At the same time it is also shown that the difficulty encountered by other workers which consists of small wave lengths growing at an infinitely high rate, is being avoided.