This paper presents a comprehensive formulation of a finite element for the modelling of borehole heat exchangers. This work focuses on the accurate modelling of the grouting and the field of temperature near a single borehole. Therefore the grouting of the BHE is explicitly modelled. The purpose of this work is to provide tools necessary to the further modelling of thermo-mechanical couplings. The finite element discretises the classical governing equation of advection-diffusion of heat within a 1D pipe connected to ground nodes. Petrov-Galerkin weighting functions are used to avoid numerical disturbances. The formulation is able to capture highly transient and steady-state phenomena. The proposed finite element is validated with respect to analytical solutions. An example consisting of a 100 m depth U-pipe is finally simulated. A first continuous heating simulation highlights the non symmetric distribution of temperature inside and near the borehole. An estimation of the error on the results as a function of the resolution parameters is also carried out. Finally simulations of cyclic thermal loading exhibit the need to take into account all daily variations if the grouting behaviour must be modelled. This is true especially in case of freeze-thaw damaging risk. (C) 2016 Elsevier Ltd. All rights reserved.