Coning during withdrawal from two fluids of different density in a porous medium

Abstract

Abstract. The steady response of the interface between two fluids with different density in a porous medium is considered during extraction through a line sink. Supercritical withdrawal, or coning as it is often called, in which both fluids are being withdrawn, is investigated using a coupled integral equation formulation. It is shown that for each entry angle of the interface into the sink there is a range of supercritical solutions that depend on the flow rate, and that as the flow rate decreases the cone narrows. As the magnitude of the entry angle increases this range of flow-rate values decreases to a narrow range as the entry becomes vertical. Only one branch of solutions (that with horizontal entry) has the property that the interface levels off at a finite height, and this is investigated as a separate branch of solution.

A Note on Axisymmetric Supercritical Coning in a Porous Medium

“…The results raise a clear point of difference between the withdrawal through a point sink and the earlier work for a line sink [12,17], in which there were multiple flow rate values valid for each entry angle of the interface. As the entry angle steepened the range of values of flow rate narrowed, seeming to approach a single value as α → π/2.…”

“…The case of supercritical flow, in which both layers are withdrawn simultaneously through a sink, was considered by Yu [16] and Henderson et al [7]. More recently, Hocking and Zhang [12] and Zhang et al [19] found that supercritical withdrawal into a line sink was possible over a range of different flow rates for a given entry angle of the interface. This result is contrary to the analogous surface water, supercritical case [4,9,10] in which solutions obtained with an integral equation technique showed that each flow rate into a line sink produced a unique angle of entry.…”

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.…”

Fluid injection through a line source into a wet porous medium

“…One such numerical method would be to use a finite-element method, a solid introduction to which can be found in the text by Wang and Anderson [9]. As an example of a different numerical approach, Hocking and Zhang [10] have applied boundary-integral techniques to find the shape of an up-coning More recently Forbes has used a spectral approach [11], which has advantages of allowing singularities and also of providing a straightforward relationship between solute concentration, stream function and velocity.…”