Well-posedness and finite element approximation for the steady-state closed-loop geothermal system

Abstract

In this article, we study a multi-physics model for the closed-loop geothermal system consisting of some pipelines and a reservoir. The mathematical model for simulating this system mainly focuses on heat transfer between porous media in the reservoir and fluid flow in the pipelines. Besides, to describe the fluid and heat transfer in two regions, the governing equations of the mathematical model are denoted by the steady Navier-Stokes/Darcy equations coupled with the convection-diffusion equations. Next, we design a perturbation system to prove the well-posedness of the governing equations. Moreover, in variational formulation, a Nitsche's penalty term is added to overcome the difficulty of instability caused by the interface condition of heat flux. Furthermore, a decoupled iterative algorithm is proposed to solve the considered equations based on the finite element method, and the stability and convergence of the presented algorithm are shown. Finally, some numerical tests are performed to verify the accuracy and efficiency of the proposed algorithm.