A Note on Axisymmetric Supercritical Coning in a Porous Medium

Abstract

The steady response of a fluid with two layers of different density in a porous medium is considered during extraction through a point sink. Supercritical withdrawal in which both layers are being withdrawn is investigated using a spectral method. We show that for each withdrawal rate, there is a single entry angle of the interface into the point sink.As the flow rate decreases the angle of entry steepens until it becomes almost vertical, at which point the method fails. This limit is shown to correspond to the upper bound on sub-critical (single-layer) flow.2010 Mathematics subject classification: 0103.

Critical Surface Coning Due to a Line Sink in A vertical Drain Containing a Porous Medium

“…Zhang et al [11,25] considered this problem in a vertically confined two-dimensional aquifer and used hodograph and numerical methods to find limiting solutions as the flow rate was increased where the interface attached to the horizontal, impermeable boundary at some distance from the sink. Flows in which the limiting steady flow rates were exceeded were computed by Yu [22] and also by Zhang et al [12,26] and in three dimensions by Hocking and Zhang [13], and in these papers there were two flowing layers of different density entering the sink. These solutions approached the limiting single layer steady state as the flow rate decreased, although there was some "noise" in the computations that did not allow an accurate determination of the critical values.…”

The Effect of Surface Tension on Free-Surface Flow Induced by a Point Sink

“…Recent work by Hocking and Zhang [13] in the analogous flows into a point sink in porous media have provided solutions in the steady, supercritical flow regime and shown the relationship between single-layer and two-layer flows, but such solutions have not been obtained in the surface-water case.…”