Novel discretization method to calculate g-functions of vertical geothermal boreholes with improved accuracy and efficiency

The calculation of g-functions is essential for the design and simulation of geothermal boreholes. However, existing methods, such as the stacked finite line source (SFLS) model, face challenges regarding computational efficiency and accuracy, particularly with fine-grained discretization. This paper introduces a novel discretization method to address these limitations. We reformulate the g-function calculation as the solution to spatio-temporal integral equations and employ the Gauss–Legendre quadrature to approximate the spatial integrals, avoiding the assumption of a stepwise structure of heat extraction rate, thereby improving accuracy. This formulation transforms the time-consuming segment-to-segment integral calculations in SFLS model into simpler and analytical point-to-point response factors, which tackles the long-standing issue that higher discretization orders generally reduce computational error at the cost of speed. Furthermore, we identify that the governing integral equations are of the Fredholm first kind, leading to ill-conditioned linear systems that can cause g-function to diverge at high discretization orders. To address this, a regularization technique is implemented to ensure stable and convergent solutions. Numerical tests demonstrate that the proposed method is significantly more efficient, achieving comparable or improved accuracy at speeds 20 to 200 times faster than the SFLS model with optimized nonuniform discretization schemes.