A semi-analytical model for detailed 3D heat flow in shallow geothermal systems

A semi-analytical model for simulating transient conductive-convective heat flow in a three-dimensional shallow geothermal system consisting of multiple borehole heat exchangers (BHE) embedded in a multilayer soil mass is introduced. The model is formulated in three steps, starting from an axial symmetric system and ending in a 3D multilayer, multiple BHE system. In step 1, the model is formulated as a single BHE embedded in an axial symmetric homogeneous soil layer, and the governing heat equations are solved analytically using the fast Fourier transform, the eigenfunction expansion and the modified Bessel function. In step 2, the model is extended to incorporate multiple layers using the spectral element method. And in step 3, the model is extended to incorporate multiple borehole heat exchangers using a superposition technique suitable for Dirichlet boundary conditions. The ensuing computational model solves detailed three-dimensional heat flow using minimal CPU time and capacity. The number of the required spectral elements is equal to the number of soil layers embedded in which any number of borehole heat exchangers with any layout configuration. A verification example illustrating the model accuracy and numerical examples illustrating its computational capabilities are given. Despite the apparent rigor of the proposed model, its high accuracy and computational efficiency make it suitable for engineering practice.