Calculating thermal response function of inclined geothermal boreholes with novel discretized formulation and acceleration techniques

This paper presents a novel and efficient approach to calculate the thermal response function ( g -function) of inclined geothermal boreholes. Traditional analytical methods based on the finite line source model require computationally expensive segment-to-segment numerical integration, which becomes particularly challenging for inclined boreholes due to the complex three-dimensional geometry. We propose a semi-analytical discretized formulation that eliminates the need for complicated double integrals by directly discretizing the governing spatio-temporal integral equations using numerical quadrature rules. This approach replaces segment-to-segment response factors with efficient point-to-point coupling factors that have closed-form analytical expressions. Additionally, we develop an intuitive distance aggregation technique that exploits the symmetrical properties of borehole layouts to reduce computational complexity. The proposed method unifies the treatment of vertical and inclined boreholes within a single computational framework without introducing additional complexity for inclined configurations. Numerical validation against the reference implementation of finite line source models demonstrates excellent agreement while achieving significant computational speedups up to three orders of magnitudes times faster than state-of-the-art finite line source methods.