Coupling of Darcy--Forchheimer and Compressible Navier--Stokes Equations with Heat Transfer

This paper is devoted to the coupling of a 2D reservoir model with a 1.5D vertical wellbore model, both written in axisymmetric form. The physical problems are respectively described by the Darcy-Forchheimer and the compressible Navier-Stokes equations, together with an exhaustive energy equation. Each model was previously studied and its finite element discretization was validated. The two weak problems are bound together by means of transmission conditions at the perforations, yielding a non standard mixed formulation. A technical analysis is then carried out and the well-posedness of the time-discretized coupled problem, in both the continuous and the discrete cases, is established. Numerical tests including physical cases are presented, validating the coupled code.