Equation-Informed Machine Learning for Reliable Temperature Forecasting in Enhanced Geothermal Systems

Reliable temperature forecasting in Enhanced Geothermal Systems (EGS) is critical for design and economic assessment, yet existing decline-curve and machine-learning (ML) approaches often neglect geothermal-specific physics. This study proposes a unified, physics-consistent framework that advances both decline-curve analysis (DCA) and surrogate modeling for geothermal temperature forecasts. First, the four Arps decline families (exponential, harmonic, hyperbolic, and stretched exponential) are extended by introducing an equilibrium-temperature term derived from Newton-type cooling, enforcing finite long-term temperature limits appropriate for geothermal reservoirs while recovering the classical petroleum forms as a special case. Second, these extended decline equations are embedded directly inside an equation-informed neural network (EINN) so that gradients propagate through the physical decline laws, contrasting with prior work in which physics appears only through external loss regularization or fixed analytical parametrizations. Third, this equation-embedded network is compared with a Gaussian Process Regression surrogate that delivers rapid, uncertainty-aware multi-horizon forecasts. All models are trained and evaluated on a fully coupled thermo–hydro–mechanical (THM) dataset that maps hydraulic fracture count, fracture spacing, well spacing, thermal conductivity, and circulation rate to full 0–60 month temperature profiles. The results show that the modified DCA models achieved near-perfect fits (median R² = 0.999, median RMSE = 0.071°C). The stretched-exponential form was preferred in 60% of cases, followed by the hyperbolic model in 37%. The EINN yielded test MAE = 2.50°C and RMSE = 3.71°C. The GPR surrogate delivered robust generalization on hold-out data (RMSE = 3.39°C, MAE = 2.34°C) across prediction horizons.