  We consider a finite volume method for a well-driven fluid flow in a porous
medium. Due to the singularity of the well, modeling in the near-well region
with standard numerical schemes results in a completely wrong total well flux
and an inaccurate hydraulic head. Local grid refinement can help, but it comes
at computational cost. In this article we propose two methods to address well
singularity. In the first method the flux through well faces is corrected using
a logarithmic function, in a way related to the Peaceman correction. Coupling
this correction with a second-order accurate two-point scheme gives a greatly
improved total well flux, but the resulting scheme is still not even first
order accurate on coarse grids. In the second method fluxes in the near-well
region are corrected by representing the hydraulic head as a sum of a
logarithmic and a linear function. This scheme is second-order accurate.
