[S-RD24] Cutting-edge sensing technology applied to geology and resource exploration

[SRD24-04] Evaluation of the effect of boundary conditions in physics-informed neural network for geothermal modeling

★Invited Papers

*Takehiro Kobira¹, Kazuya Ishitsuka¹, Weiren Lin¹ (1.Graduate School of Engineering, Kyoto University)

Keywords:geothermal modeling, physics-informed neural network, boundary conditions

Numerical modeling of natural-state geothermal systems is important for geothermal resource evaluation and optimization of well locations. As a new method for the inverse modeling of temperature, pressure and permeability of geothermal systems, physics-informed neural networks (PINN), which considers physical laws and boundary conditions in the loss function, has been recently proposed. PINN have recently attracted attention as a method that can learn with following physical laws even when there are small amounts of data, and can learn with physical validity, enhancing the plausibility of its predictions. However, it is difficult to observe the temperature and pressure at the boundary of a region in geothermal system modeling at depth, and for practical use of PINN, the prediction accuracy must be maintained even if the number of boundaries to be considered is limited. Thus, in this study, the influence of boundary conditions on the prediction accuracy of the PINN was verified. Specifically, we evaluated the errors in the predicted quantities by the PINN with different boundary conditions using pseudo temperature, pressure, and permeability models. First, a permeability distribution was created by applying a disturbance to an empirical equation. The temperature and pressure distributions were calculated using the TOUGH2 hydrothermal simulation software by assigning the permeability to a 1 km (horizontal) × 1 km (vertical) two-dimensional region where other physical properties (porosity, density, thermal conductivity, and specific heat) were assumed to be uniform for simplicity. The boundary conditions were constant temperature and pressure at the top, constant heat flux at the bottom, and zero pressure gradient on both sides. After the pseudo-distribution was created, the temperature, pressure, and permeability obtained from a hypothetical well in the region were used as teacher data, and training was performed five times using a PINN, which takes into account mass and energy conservation laws. Eight types of boundary combinations were considered: all boundaries, top, bottom, sides, top and bottom, top and sides, sides and bottom, and no boundary conditions. By comparing the errors between the pseudo-distributions created and the temperature, pressure, and permeability distributions obtained in the study with the 8 PINNs, we consider the influence of the boundary conditions on the prediction errors (Analysis 1). For each condition, the study was conducted with 3, 5, and 8 wells in order to examine the influence of the number of wells. Finally, we also applied the training with different boundary condition to a pseudo-distribution with an area of 6 km (horizontal) × 3 km (vertical) calculated based on a geological model of the Lahendong geothermal area in Indonesia, to verify the applicability of this study to real data (Analysis 2). The average of the error obtained by averaging the difference between the observed and estimated temperature values in Analysis 1 over the entire grid was compared for each condition. The error in the three cases considering only the bottom, top and bottom, and sides and bottom, regardless of the number of wells, was smaller than that in the case considering all boundaries. The reason for the smaller errors in the three cases where the bottom was considered may be because the temperature gradient condition at the bottom worked as temperature-related teaching data in the extrapolated region where there were no wells. The reason why the error is smaller than when all boundaries are considered can be attributed to the fact that the loss function is simpler than when all boundaries are considered, which reduces the risk of falling into a local solution. For Analysis 2, there was almost no difference between the error when all boundaries were considered and the error when only the bottom or the top and bottom were considered. For pressure and permeability, there were a few kinds of PINNs where the average of the five training runs of the error was more than 1% larger than that of the case where all boundaries were considered. These results demonstrate that it is possible to perform accurate PINN modeling without specifying the boundary conditions of the entire domain.