Physics-Informed Neural Networks for Porosity Prediction and Uncertainty Quantification

Effective porosity is a key petrophysical parameter, yet a well without a measured sonic (DT) curve cannot support sonic porosity calculation or seismic-to-well ties. This study presents a physics-informed neural network (PINN) that jointly synthesizes a DT log and predicts porosity for three Niger Delta wells, none of which carries a measured sonic curve. A shared-trunk, dual-head architecture produces both outputs from one representation. Gardner’s equation, the Wyllie time-average equation, and a density-porosity cross-check act as physics terms in the loss function. Aleatoric uncertainty comes from a heteroscedastic likelihood and epistemic uncertainty from a five-member deep ensemble with Monte Carlo dropout, which together give a predictive uncertainty band for every porosity value. These bands are Gaussian-approximation predictive intervals, not confidence intervals, and they have not been recalibrated. Under leave-one-well-out cross-validation against a no-physics ablation and a Random Forest baseline, the PINN outperforms the no-physics ablation on every held-out well (R² = 0.938, 0.682, and 0.806 on Jay_01, Jay_02, and Jay_03, versus 0.881, 0.592, and 0.691) and is competitive with the Random Forest baseline, trailing narrowly on Jay_01 (0.938 vs 0.943), leading narrowly on Jay_02 (0.682 vs 0.678), and leading clearly on Jay_03 (0.806 vs 0.753). No measured sonic log exists for validation, so the accuracy of the synthetic DT curves remains unverified.