The Transport of Heat in a Hydraulic Fracture Caused by a Variable Flowrate: A Reduced-Order Lauwerier/Gringarten Extended Model

Abstract

A reduced-order model for thermal transport through a single hydraulic fracture in hot dry rock is presented, allowing for fully variable flow rates, including flow reversals. This model extends the classical Lauwerier/Gringarten framework, which has served for decades as a reliable, conservative tool for estimating heat harvesting efficiency under the assumption of constant flow. However, modern geothermal strategies – such as Huff-n-Puff methods, thermal storage or variable charging and discharging cycles designed to deliver flexible generation profiles for power-grid demand – require models that accommodate arbitrary, time-dependent flowrates and fracture apertures. The developed model simulates these complex operations by reducing the coupled heat-transport equations into a one-dimensional advection-memory equation. This approach enables the fast design and analysis of injection-backflow tests and evaluation of long-term thermal depletion without the computational burden of full-physics simulators. Crucially, the model accounts for time-dependent fracture apertures, enabling it to simulate the physical ‘breathing’ of fractures caused by pressure fluctuations during charging and discharging. Finally, the applicability of the developed model is validated by first reproducing the test case presented in Juliusson and Horne (2012) and secondly by considering a test case that models Huff-n-Puff operations. The findings highlight that the reduced-order model is efficient, reliable and robust so that it may be used for stochastic or long-term simulations.