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

Abstract

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.

Finite Volume Methods for Well-Driven Flows in Anisotropic Porous Media

“…Corrections addressing this issue are available in the literature. We extended the schemes presented in [6] to an anisotropic medium. The rst scheme (WFC) reduced the ux error, but is not second-order accurate.…”

“…An approximation of the ux through well faces based on the analytical ow rate (6) in an isotropic medium ( = ) is proposed in [6] to be…”

“…The well face midpoint is denoted with x . The auxiliary hydraulic head values are eliminated in the same way as in [6]. From (2) we have…”

“…Non-linear two-point schemes [2,8,9,[13][14][15][16] were not originally formulated for the near-well discretization and thus loose their accuracy on coarse grids if a well is present. This type of scheme was adapted for well-driven ow through an isotropic medium in [6].…”

“…In this paper we extend the methods presented in [6] to the case of an anisotropic medium. The rst correction (see Section 3.1) is applied only to well faces, while the uxes through other faces are approximated using the unmodi ed non-linear two-point scheme [14].…”