A new method is presented to evaluate thermal interactions between vertical geothermal boreholes. The finite line source (FLS) solution is extended to consider thermal interactions between groups of boreholes. Groups of boreholes that share similar temperatures and heat extraction rates are identified using hierarchical agglomerative clustering, and each group is represented in the model as a single equivalent borehole. Each equivalent borehole is split into segments, and temporal and spatial superposition of the FLS solution are employed to calculate the total temperature change along the length of the equivalent boreholes. The new method is shown to provide an accurate calculation of the g-function, with a mean absolute percentage error below 0.612% on the g-functions of regular borefields of up to 144 boreholes using only 3 to 5 equivalent boreholes. Calculation times are significantly reduced: the g-function of a borefield of 1024 randomly positioned boreholes is calculated in 3.65 s.

This study presents a physics-informed neural network (PINN) framework for predicting open channel flow velocity, integrating traditional hydraulic principles with modern deep learning techniques when limited data are available. The framework combines a multi-branch neural architecture with a composite loss function that enforces physical consistency through established hydrological principles, including the continuity condition, location-dependent Manning's equation, and geometric constraints. Unlike conventional PINN implementations that require dense spatiotemporal discretization, this approach is specifically designed for sparse data conditions (or local data), utilizing a surrogate model that assimilates limited sensor-based measurements while embedding key physical–empirical laws. The model achieves superior predictive performance compared to traditional machine learning baselines, demonstrating a 40% reduction in mean absolute percentage error relative to standard approaches. Moreover, the framework maintains physical consistency, predicting location-dependent Manning coefficients that align with established hydraulic engineering literature and achieving accurate hydraulic radius predictions across diverse channel configurations. These results suggest that integrating physical constraints effectively compensates for data sparsity, offering a promising solution for applications such as hydrokinetic power assessment where both accuracy and physical plausibility are essential.

The study of heat transfer in ground heat exchangers (GHEs) considering the fluid advection inside the pipes; the heat transfer between the fluid and the ground through the grout material; and the thermal interaction between GHEs is a challenging task. The present paper presents a new semi-analytical method that takes into account the aforementioned effects to consider both the short- to long-term effects of GHEs. The heat transfer between the fluid and grout was studied by a transient multipole expansion considering time-dependent fluid temperatures and an advection model for the pipes obtained from an energy balance on the heat carrier fluid. Thermal interactions were analyzed using an equivalent borehole method while penalizing the transient multipole expansion to include thermal interaction effects. Validation of the short-term predictions was performed by comparing the proposed model to experimental data found in the literature and to an FEA model. The proposed model was then compared with a FEA model in long-term simulations of a geothermal field comprised of 24 GHEs for multi-annual simulation. The method resulted in substantial reduction in computational time while preserving good accuracy when compared with the FEA model.