Advancing Petroleum Engineering Solutions: Integrating Physics-Informed Neural Networks for Enhanced Buckley-Leverett Model Analysis

Physics-Informed Neural Networks (PINNs) integrate physical principles-typically mathematical models expressed as differential equations-into the machine learning (ML) processes to guarantee the physical validity of ML model solutions. This approach has gained traction in science and engineering for modeling a wide range of physical phenomena. Nonetheless, the effectiveness of PINNs in solving nonlinear hyperbolic partial differential equations (PDEs), is found challenging due to discontinuities inherent in such PDE solutions. While previous research has focused on advancing training algorithms, our study highlights that encoding precise physical laws into PINN framework suffices to address the challenge. By coupling well-constructed governing equations into the most basic, simply structured PINNs, this research tackles both data-independent solution and data-driven discovery of the Buckley-Leverett equation, a typical hyperbolic PDE central to modeling multi-phase fluid flow in porous media. Our results reveal that vanilla PINNs are adequate to solve the Buckley-Leverett equation with superior precision and even handling more complex scenarios including variations in fluid mobility ratios, the addition of a gravity term to the original governing equation, and the presence of multiple discontinuities in the solution. This capability of PINNs enables accurate, efficient modeling and prediction of practical engineering problems, such as water flooding, polymer flooding, inclined flooding, and carbon dioxide injection into saline aquifers. Furthermore, the forward PINN framework with slight modifications can be adapted for inverse problems, allowing the estimation of PDE parameters in the Buckley-Leverett equation from observed data. Sensitivity analysis demonstrate that PINNs remain effective under conditions of slight data scarcity and up to a 5% data impurity.