Full-scale temperature response function (G-function) for heat transfer by borehole ground heat exchangers (GHEs) from sub-hour to decades

Heat transfer by borehole ground heat exchangers involves diverse time-space scales and thus imposes a significant challenge to geothermal engineers. In order to overcome this challenge, this paper develops an analytical full-scale model from the idea of matched asymptotic expansion. The full-scale model is a composite expression consisting of a composite-medium line-source solution (inner solution), a finite line-source solution (outer solution), and an infinite line-source solution. The full-scale model is first verified by a frequency-decomposition method. Furthermore, the full-scale model is reformulated as a multi-stage model based on Duhamel's theorem to reduce the computational cost. The multi-stage model combines the three separate solutions in a sequential way, i.e., the inner solution for the short-time scale, the conventional infinite line-source solution for the intermediate time scale, and the outer solution for the long-time scale. Finally, we perform a parametric study on a ground heat exchanger with single U-shaped tube, by which the spacing between U-tube legs, the length-to-radius ratio of borehole, the ratios of thermal diffusivities and conductivities of the ground and backfilling material are analyzed.