Second-order accurate finite volume method for well-driven flows

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 the well singularity. In the first method the flux through well faces is corrected using a logarithmic function, in a way related to the Peaceman model. Coupling this correction with a non-linear second-order accurate two-point scheme gives a greatly improved total well flux, but the resulting scheme is still inconsistent. 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.