Well Modelling By Means Of Coupled 1D-3D Flow Models

Summary In this work we present a new numerical method for solving coupled 1D-3D flow models, which can be used, for example, to model the flow in a well and the surrounding reservoir. Assuming 1D Poiseuille flow in the well, we use Stirling’s law of filtration to couple it to the 3D flow model used for the reservoir. The presence of a line source in the governing equations of the reservoir is known to cause a logarithmic type singularity in the solution around the well. For this reason, the solution is difficult to approximate numerically. We therefore introduce a decomposition technique where the solution is split into an explicitly known logarithmic term capturing the singularity and a well-behaved correction term. After a decoupling, the coupled 1D-3D model can then be reformulated as a fixed point iteration scheme iterating over the 1D pressure in the well and the 3D correction term for the reservoir. The iteration scheme can be implemented using both Galerkin and mixed finite element methods, the former of which leads to mass conservative solutions. The advantage of the decoupling and reformulation is twofold: Firstly, it recasts the model into a system for which the discretization schemes and solution methods are readily available. Secondly, it recovers optimal convergence rates without needing to perform a mesh-refinement around the well.