Asymptotic expansions for the ground heat transfer due to a borehole heat exchanger with a Neumann boundary condition

We model the heat transfer of a borehole heat exchanger considering a finite-line heat source inside a semi-infinite medium with constant thermal conductivity and geothermal gradient. As a novelty with respect to other models, periodic heat flux changes are allowed on the ground surface. From this model, we calculate analytic expressions for the average ground temperature by integrating the exact solution along the line-source. Also, asymptotic formulas for the intermediate and long timescales are derived.