A low-order dynamical model for delayed thermal drawdown in subsurface energy systems

Long-term thermal drawdown is a fundamental constraint on the sustainability of subsurface thermal energy systems, yet its onset often does not become apparent for several decades after exploitation begins due to large thermal inertia and slow heat transport processes. This delayed response complicates sustainability assessment and may lead to overestimation of system longevity when early operational data are used. Although high-fidelity numerical simulators can capture delayed thermal behavior, their computational cost and limited interpretability restrict their usefulness for rapid, conceptual analysis at the system level. This study presents a simplified low-order dynamical model to examine delayed thermal drawdown in subsurface energy systems, with geothermal reservoirs considered as a primary application. The system is represented by a lumped thermal state driven by heat extraction and gradual geothermal recharge, with an explicit time-delay term introduced to account for geological memory and delayed thermal response. Analytical and numerical investigations using synthetic production scenarios show that significant thermal drawdown emerges only when production history changes, explaining prolonged early-stage stability followed by later temperature decline. The proposed framework is intended as a screening-level and educational tool that complements high-fidelity numerical simulations and supports long-term management of thermal energy systems.