Segmented Finite Line Source Analysis of Mid-Deep UBHE with Geothermal Gradient and Stratification

This study develops an analytical model for a mid-deep U-shaped borehole heat exchanger (UBHE) based on the segmented finite line source method, integrating geothermal gradient, five-layer geological stratification, and groundwater seepage within a unified framework. The injection, horizontal, and extraction sections are represented by independent local coordinates and coupled through the position- and time-dependent unit-length heat-transfer rate qlsn,t. Validation against the benchmark results of Bao et al. yields a mean absolute error of 0.87 °C and a mean relative error of 1.6%. Numerical-independence tests identify a 1 h time step and 100/50 m vertical/horizontal segment lengths as the adopted settings, and the iterative residual reaches 10−4 °C within eight iterations for the representative case. In a homogeneous, no-seepage limiting case, the model agrees with the classical finite line-source solution with an MAE of 0.012 °C and a maximum relative-error magnitude of 0.41%. The extraction-well fluid-temperature peak occurs at 600–800 m depth, whereas the local heat-transfer direction changes near 1600 m. For the investigated 2500–650–2500 m geometry, the highest cycle-averaged outlet temperature is obtained at an insulation length of 1600 m. Sensitivity analyses further quantify the effects of seepage velocity, equivalent thermal conductivity, geothermal gradient, and the validation insulation-length assumption.