Extraction of density-layered fluid from a porous medium

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear stratification case, there are certain values of flow rate for which a steady solution does not occur. A spectral method is then implemented to consider cases in which there is a stagnant region beneath a sharp interface between two layers of different, but constant, density. In this situation, flows also exist only for flow rates beneath a critical flux value, consistent with the results for the continuous density stratification