Singularities in weakly compressible flow through a porous medium

Abstract

The flow of weakly compressible fluid in an annular domain filled with a porous medium admits a wide class of solutions, whose axial symmetry is broken by azimuthally varying permeability. The structural parameters governing the underpinning dynamical system are the angles of sectors containing media of distinct permeability and values of the permeabilities themselves. Albeit from the vantage point of physics there is no apparent reason that certain configurations be singled out, mathematical analysis reveals an intriguingly complex pattern of singularities. The singular manifolds are loci, in whose vicinity saddle points of the underlying pressure field shift abruptly or appear/disappear. Stunningly the shift angle is independent of all configuration parameters, equalling . When the system represents an injection or extraction well, the distance of the saddle points therefrom assumes the physical interpretation of the well’s reach, also known as the radius of influence. Therefore two infinitesimally close configurations might have disparate flow fields and in particular zones of influence. This analysis casts in a new light the baffling difficulties encountered in the construction of aquifer sparging wells and landfill or natural gas collection wells, whose reach is known to be poorly predictable, and the failure or success of the injection/suction to induce the motion of adequate amounts of fluid proved moot. The traditional concept of the radius of influence is shown to be ill-posed and its unpredictability in practice intimately connected to the structural singularities of the anisotropic porous medium.