This study presents a data-driven spatial interpolation algorithm based on physics-informed graph neural networks used to develop national temperature-at-depth maps for the conterminous United States. The model was trained to approximately satisfy the three-dimensional heat conduction law by simultaneously predicting subsurface temperature, surface heat flow, and rock thermal conductivity. In addition to bottomhole temperature measurements, we incorporated other physical quantities as model inputs, such as depth, geographic coordinates, elevation, sediment thickness, magnetic anomaly, gravity anomaly, gamma-ray flux of radioactive elements, seismicity, and electric conductivity. We constructed surface heat flow, and temperature and thermal conductivity predictions for depths of 0-7 km at an interval of 1 km with spatial resolution of 18 km$^2$ per grid cell. Our model showed superior temperature, surface heat flow and thermal conductivity mean absolute errors of 4.8{\deg} C, 5.817 mW/m$^2$ and 0.022 W/(C-m)$, respectively. The predictions were visualized in two-dimensional spatial maps across the modeled depths. This thorough modeling of the Earth's thermal processes is crucial to understanding subsurface phenomena and exploiting natural underground resources.

The goal of this paper is to assess the utility of Reduced-Order Models (ROMs) developed from 3D physics-based models for predicting transient thermal power output for an enhanced geothermal reservoir while explicitly accounting for uncertainties in the subsurface system and site-specific details. Numerical simulations are performed based on Latin Hypercube Sampling (LHS) of model inputs drawn from uniform probability distributions. Key sensitive parameters are identified from these simulations, which are fracture zone permeability, well/skin factor, bottom hole pressure, and injection flow rate. The inputs for ROMs are based on these key sensitive parameters. The ROMs are then used to evaluate the influence of subsurface attributes on thermal power production curves. The resulting ROMs are compared with field-data and the detailed physics-based numerical simulations. We propose three different ROMs with different levels of model parsimony, each describing key and essential features of the power production curves. ROM-1 is able to accurately reproduce the power output of numerical simulations for low values of permeabilities and certain features of the field-scale data, and is relatively parsimonious. ROM-2 is a more complex model than ROM-1 but it accurately describes the field-data. At higher permeabilities, ROM-2 reproduces numerical results better than ROM-1, however, there is a considerable deviation at low fracture zone permeabilities. ROM-3 is developed by taking the best aspects of ROM-1 and ROM-2 and provides a middle ground for model parsimony. It is able to describe various features of numerical simulations and field-data. From the proposed workflow, we demonstrate that the proposed simple ROMs are able to capture various complex features of the power production curves of Fenton Hill HDR system. For typical EGS applications, ROM-2 and ROM-3 outperform ROM-1.