Physics-Informed Neural Network for Robust Petrophysical Interpretation with Learnable Physical Parameters

Abstract Accurate estimation of porosity (ϕ), water saturation (SW), and shale volume (VSH) is crucial for quantitative reservoir characterization. Traditional deterministic models, such as Archie's equation and shaly-sand formulations (e.g., Simandoux, Waxman-Smits, and Dual Water models), rely on fixed parameters that are often uncertain, while purely data-driven machine learning models can act as black boxes and lack physical consistency. To address these issues, we introduce a physics-informed neural network (PINN) framework that combines petrophysical constraints with data-driven learning. The model uses a deep feedforward neural network as its core, with physics-based losses derived from Archie's resistivity law and gamma-ray shale volume relationships incorporated into the training process. Formation water resistivity (RW) and the cementation exponent (m) are treated as learnable parameters, allowing for automatic calibration to reservoir-specific conditions. The model is trained on 46,034 standardized well log data points from nine wells, utilizing seven input logs: compressional slowness (DTC), shear slowness (DTS), bulk density (DEN), gamma ray (GR), neutron porosity (NEU), photoelectric factor (PEF), and deep resistivity (RDEP). An independent test set of 11,275 depth samples from four additional wells is used for evaluation. To assess the contribution of the physics-informed formulation, a baseline multilayer perceptron (MLP) with the same architecture but without physics-based constraints was trained for comparison. The baseline MLP achieved R2 values of 0.9702 for ϕ, 0.9863 for Sw, and 0.9802 for VSH. In contrast, the proposed PINN achieved higher R2 values of 0.9825, 0.9911, and 0.9885, respectively, along with consistently lower mean squared errors (MSE) and mean absolute errors (MAE). These results demonstrate that incorporating physics-based constraints improves both predictive accuracy and physical consistency. The model also learns formation-specific parameters, converging to RW = 0.0541 Ω · m and m = 1.2851, enhancing interpretability and adaptability to field-specific conditions. Overall, the findings highlight that physics-informed machine learning effectively bridges the gap between deterministic petrophysical models and black-box neural networks, providing a robust and interpretable framework for reducing uncertainty in reservoir evaluation and supporting more reliable subsurface decision-making.