Application of physics-informed neural networks for solution of the diffusivity equation and interpretation of well response

Abstract

The diffusivity equation is a widely accepted mathematical model for addressing the complex phenomenon of fluid flow through porous media. It has numerous applications in petroleum engineering, including reservoir simulation, reservoir characterization, and pressure and rate transient analysis. While analytical solutions to the diffusivity equation are available for simple boundary conditions, more complex problems typically require employment of numerical methods. Recently, there has been a growing focus on Physics- Informed Neural Networks (PINNs) due to their ability to integrate physical laws into the learning process, offering a more generalized approach compared to traditional numerical solvers. This study presents the effectiveness of Physics-Informed Neural Networks (PINNs) in solving flow through porous media problems and compares the results with available analytical and semi-analytical solutions. The proposed PINN framework demonstrates a mean percentage error of 0.04 % for simple one-dimensional linear flow scenarios and maintains errors below 0.5 % for various radial flow cases. Additionally, the effectiveness of transfer learning in solving the inverse problem of well test analysis is highlighted, with 0.1 % error in predicting the permeability and 11.96 % in skin factor prediction from noisy well test data. This research identifies the critical role of the physics loss weight factor (λP DE) in solution accuracy and its correlation with computational domain size. Furthermore, the study identifies the limitations and advantages of PINN models, providing a foundation for future advancements in the application of neural networks to complex fluid flow problems in porous media.