[Paper Review] Optimization of Closed-Loop Shallow Geothermal Systems Using Analytical Models

O. Heinzel, Smajil Halilovic+2 more|arXiv (Cornell University)|Mar 26, 2026

Geothermal Energy Systems and Applications0 citations

TL;DR

This paper presents a fast analytical framework combining a Finite Line Source soil model and a 1U borehole heat transfer model with a Modified Lloyd’s algorithm to optimally place BHEs in non-convex domains and then minimize uniform borehole length under fluid-temperature constraints.

ABSTRACT

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.

Motivation & Objective

• Develop a quick analytical model to predict 20-year BHE performance for 1U boreholes.
• Create a robust heuristic to autonomously locate optimal BHE coordinates in non-convex domains.
• Integrate spatial placement with a depth optimization to minimize total BHE length while respecting fluid temperature limits over the simulation horizon.

Proposed method

• Model soil temperature with a finite line source and Green’s function using the Method of Images.
• Model borehole-fluid coupling via a quasi-steady 1U transfer with boundary conditions and analytic outlet temperature (Tout).
• Compute soil-fluid interaction using self- and inter-borehole terms and a dual-grid convolution accelerated by FFT for O(Nt log Nt).
• Formulate BHE field design as a two-step optimization: (i) Maximize inter-BHE and boundary spacing via Modified Lloyd’s CVT in a non-convex domain, (ii) minimize uniform borehole length L subject to Tout within Tmin/Tmax over time.
• Optimize spatial distribution with a Non-convex CVT using a Modified Lloyd’s algorithm that projects centroids back into the domain.
• Solve the depth optimization with a Sequential Least Squares Programming (SLSQP) bound + nonlinear constrained problem to minimize L while keeping Tout within bounds.

Experimental results

Research questions

• RQ1 How can BHE fields be optimally placed in non-convex domains to maximize spacing and minimize interference?
• RQ2 What is the minimal uniform borehole length L that satisfies 20-year fluid-temperature constraints for a given heat demand profile?
• RQ3 How does the proposed CVT-based placement perform across different property geometries (grid, rectangular domains, L-shaped domains)?

Key findings

• The Modified Lloyd’s CVT method yields BHE placements in non-convex domains that closely resemble optimal grids, with average nearest-neighbor distance differences around 2 m in symmetric cases.
• Optimal borehole lengths vary with property geometry: Grid (5x5) 91.47 m; Rectangular Medium 91.44 m; Rectangular Small 101.86 m; Rectangular Large 84.07 m; L-shaped domain 80.72 m.
• A larger available area enables greater borehole spacing and lower L due to reduced thermal interference, while smaller areas require deeper bores to meet temperature constraints.
• The framework maintains Tout within Tmin/Tmax over a 20-year horizon, with Tmax becoming limiting in cooling-dominant profiles, illustrating the need for appropriate domain-aware depth design.