Optimization of Closed-Loop Shallow Geothermal Systems Using Analytical Models

Closed-loop shallow geothermal systems are one of the key technologies for decarbonizing the residential heating and cooling sector. The primary type of these systems involves vertical borehole heat exchangers (BHEs). During the planning phase, it is essential to find the optimal design for these systems, including the depth and spatial arrangement of the BHEs. In this work, we have developed a novel approach to find the optimal design of BHE fields, taking into account constraints such as temperature limits of the heat carrier fluid. These limits correspond to the regulatory practices applied during the planning phase. The approach uses a finite line source model to simulate temperature changes in the ground in combination with an analytical model of heat transport within the boreholes. Our approach is demonstrated using realistic scenarios and is expected to improve current practice in the planning and design of BHE systems.

💡 Research Summary

This paper addresses the design optimization of closed‑loop shallow geothermal systems that rely on vertical borehole heat exchangers (BHEs), a technology pivotal for decarbonising residential heating and cooling. The authors recognize that conventional full‑scale numerical models (finite element or finite difference) provide high fidelity but are far too computationally intensive for iterative design optimization. To bridge this gap, they develop a compact analytical framework that couples a finite line source (FLS) model of ground heat transfer with an analytical 1U borehole fluid‑heat‑transfer model, and they embed this framework within a fast optimization loop.

The ground‑temperature model starts from the heat‑diffusion equation in a homogeneous porous medium, treating each borehole as a finite line source of length L. By applying the method of images, a Dirichlet condition at the ground surface is satisfied, leading to closed‑form vertical (Z) and horizontal (R) response kernels. The temperature field at any point and time is obtained by convolving the heat‑injection history q(t) with these kernels (Eq. 5‑7). Superposition allows the contribution of all N_b boreholes to be summed efficiently.

For the fluid side, the authors adopt the well‑known 1U configuration. They write two coupled first‑order ODEs for the down‑flow (T_i) and up‑flow (T_o) temperatures, incorporating pipe‑to‑borehole wall resistance (R_s) and inter‑leg resistance (R_inter). By defining dimensionless parameters β_s, β_inter and γ, they derive analytical expressions g₁, g₂, g₃ that capture the borehole’s geometric and thermal properties. The outlet temperature T_out(t) is expressed analytically as a weighted integral of the borehole wall temperature profile (Eq. 9‑10).

Coupling the two domains, the building energy demand E(t) is linked to a uniform heat‑extraction rate per unit length q(t)=−E(t)/(L N_b). The average outlet temperature ⟨T_out⟩ is then related to the energy balance (Eq. 11). Crucially, the total temperature response is split into a self‑interaction term Ψ_self (heat that a borehole feeds back to itself) and an interaction term Ψ_inter (heat exchanged between different boreholes). Ψ_self is evaluated at the borehole radius r_b and is essentially instantaneous, while Ψ_inter depends on the horizontal kernel R evaluated at inter‑borehole distances.

A major computational bottleneck would be the O(N_t N_b²) convolution required for a 20‑year hourly simulation of a field with ~25 boreholes. The authors overcome this by (1) separating Ψ_self and Ψ_inter in the frequency domain, (2) applying a dual‑grid strategy: Ψ_self is computed on the fine hourly grid to capture peak loads, whereas Ψ_inter is computed on a coarse monthly‑averaged grid because distant interactions act as a low‑pass filter, and (3) using the Fast Fourier Transform (FFT) together with the convolution theorem to reduce the complexity to O(N_t log N_t). This reduces the total operation count by several orders of magnitude, making the analytical model tractable within an optimization loop.

For spatial optimization, the paper moves beyond traditional rectangular or regular grid layouts, which struggle with irregular property boundaries, obstacles, or building footprints. It introduces a flexible layout algorithm based on Centroidal Voronoi Tessellations (CVT) generated via an adapted Lloyd’s algorithm. Each borehole is placed at the centroid of its Voronoi cell, maximizing inter‑borehole spacing and distance from domain boundaries while respecting a minimum spacing Δ_min. The algorithm iteratively updates cell centroids, simultaneously minimizing the uniform borehole length (i.e., total drilling depth) and enforcing fluid temperature constraints (T_min ≤ T_out ≤ T_max) throughout the simulation horizon.

The combined framework therefore delivers three key contributions: (i) a physically consistent, analytically solvable ground‑fluid heat‑transfer model; (ii) a computational acceleration scheme (FFT + dual‑time‑grid) that enables rapid evaluation of long‑term thermal behavior; and (iii) a geometry‑agnostic spatial optimization method that automatically generates near‑optimal borehole layouts for complex sites. The authors demonstrate the approach on realistic scenarios (e.g., 25–30 boreholes over a 20‑year horizon) and show that it can reduce drilling costs while guaranteeing compliance with temperature limits, outperforming existing design tools. Limitations include the assumption of homogeneous soil, uniform heat extraction across boreholes, and the focus on 1U configurations; these would need refinement for heterogeneous sites or alternative borehole designs. Overall, the paper presents a compelling, scalable methodology that bridges the gap between high‑fidelity thermal modeling and practical engineering optimization for shallow geothermal systems.