High-order finite line source solution for the simulation of straight geothermal boreholes
The finite line source solution is routinely used for thermal simulations of geothermal boreholes, for instance in the evaluation of g -functions. This solution models thermal response factors that give the ground temperature variation due to heat extraction at a borehole and is the fundamental building block for borehole simulations based on semi-analytical methods. In typical approaches, piecewise-uniform heat transfer rates are assumed along boreholes and the resulting temperature variations are averaged along borehole segments. Recently, a simulation method that assumes piecewise-polynomial heat transfer rates along the borehole was introduced. Temperature variations are evaluated at nodes along the boreholes, instead of averaged along segments, facilitating the formulation of the finite line source solution. In this paper, a specialized high-order finite line source solution is developed for straight boreholes, including vertical and inclined boreholes. A recurrence formula is presented to evaluate higher order solutions from the zeroth and first order solutions, with only the zeroth order solution necessitating the numerical evaluation of an integral. Compared to earlier methods, the new solution decreases the number of integral evaluations to one evaluation per ground temperature from a number equal to the number of state variables (i.e. number of discrete values of heat extraction rates) per ground temperature. While the specialized solution correctly evaluates high-order finite line source solutions, it is shown that numerical errors are amplified when using the recurrence formula.