A semi-analytical intermittent multilayer solution for borehole heat exchangers with Bayesian calibration under heat extraction and thermal recovery conditions

Ground source heat pump systems (GSHP) and borehole heat exchangers (BHEs) are commonly analyzed using analytical or semi-analytical models due to their rapid performance compared to full numerical simulations. Yet many existing solutions still treat either continuous operation, homogeneous ground conditions, or fixed parameter assumptions, which limit their applicability to layered and intermittently operated systems. This study develops a semi-analytical intermittent multilayer solution for vertical borehole heat exchangers by combining the analytical treatment of discontinuous heat extraction with a multilayer moving finite line-source formulation that accounts for groundwater-induced advection–dispersion and anisotropy in layered porous media. The intermittent load history is represented through step superpositions, while cross-layer thermal interactions are handled through a composite multilayer formulation. To improve agreement with numerical reference data, the direct transient response is corrected using layer-specific η parameters, introduced as effective time-scale corrections. Generalized here further to an operational condition-dependent form, with separate values for heat extraction and recovery periods. The unknown on and off η parameters are calibrated in a Bayesian framework using an affine-invariant ensemble sampler (AIES), applied to a reduced structural parameterization rather than a full physical-parameter inversion. The model is benchmarked against COMSOL Multiphysics results for four scenarios with different layer configurations and thermo-hydraulic contrasts. Bayesian calibration consistently reduces errors at the three target probes by 40–73.5%, while independent validation identifies limitations near layer interfaces and under strong advective-dispersive conditions. In the predominantly conductive scenario, the calibrated η values remain close to unity, indicating that the uncalibrated model already captures the main thermal response. The developed methodology provides a computationally efficient tool for analyzing and optimizing intermittently operated multilayer BHE systems.