Efficient finite element formulation for geothermal heating systems. Part I: steady state

This paper presents the development of a computationally efficient finite element tool for the analysis of 3D steady state heat flow in geothermal heating systems. Emphasis is placed on the development of finite elements for vertical borehole heat exchangers and the surrounding soil layers. Three factors have contributed to the computational efficiency: the proposed mathematical model for the heat exchanger, the discretization of the spatial domain using the Petrov–Galerkin method and the sequential numerical algorithm for solving the resulting system of non-linear equations. These have contributed in reducing significantly the required number of finite elements necessary for describing the involved systems. Details of the mathematical derivations and some numerical examples are presented