Petrov–Galerkin finite element approximation for the three dimensional well model

Numerical simulation of fluid flow in a hydrocarbon reservoir has to account for the presence of wells. In petroleum extraction and exploitation, the well is usually treated as a point or line source, due to its radius is much smaller comparing with the scale of the whole reservoir. Especially, in 3-dimensional situation, the well is regarded as a line source. To capture the well singularity, a Petrov–Galerkin finite element scheme is developed. With this in hand, we then derive the error estimates and obtain the wellbore pressure directly without post-processing. Some numerical tests are presented to demonstrate the effectiveness of the proposed method.