  In reservoir simulations, the radius of a well is inevitably going to be
small compared to the horizontal length scale of the reservoir. For this
reason, wells are typically modelled as lower-dimensional sources. In this
work, we consider a coupled 1D-3D flow model, in which the well is modelled as
a line source in the reservoir domain and endowed with its own 1D flow
equation. The flow between well and reservoir can then be modelled in a fully
coupled manner by applying a linear filtration law.
  The line source induces a logarithmic type singularity in the reservoir
pressure that is difficult to resolve numerically. We present here a
singularity removal method for the model equations, resulting in a reformulated
coupled 1D-3D flow model in which all variables are smooth. The singularity
removal is based on a solution splitting of the reservoir pressure, where it is
decomposed into two terms: an explicitly given, lower regularity term capturing
the solution singularity and some smooth background pressure. The singularities
can then be removed from the system by subtracting them from the governing
equations. Finally, the coupled 1D-3D flow equations can be reformulated so
they are given in terms of the well pressure and the background reservoir
pressure. As these variables are both smooth (i.e. non-singular), the
reformulated model has the advantage that it can be approximated using any
standard numerical method. The reformulation itself resembles a Peaceman well
correction performed at the continuous level.
