Prediction of gas production for shale matrix–fracture systems using deep operator network-ensembled physics-informed neural networks

Shale gas production forecasting remains challenging because flow is governed by strong nonlinearity, matrix–fracture interaction, and regime-dependent transport behaviour, while available production histories are often limited and erratic. Purely data-driven machine learning models can approximate complex input–output mappings but may generalize poorly when physical consistency is not enforced. In this study, we develop a physics-informed deep operator learning framework for single-phase gas flow that couples a spatiotemporal sampling strategy with a forward mapping from key matrix and fracture properties to cumulative production response. Mass conservation and momentum closure are embedded as soft constraints via automatic differentiation, penalizing the residuals of the governing equations in the training objective. We further quantify the impact of non-Darcy fracture flow by incorporating a Forchheimer momentum closure and comparing it against a Darcy-based formulation, showing improved short-term predictive performance when inertial effects are non-negligible. A different field-scale simulation benchmark based on the Marcellus shale model dataset has been used to validate the model. Across multiple forecasting horizons, the proposed approach achieves high agreement with reference cumulative gas production, with R 2 values above 0.97 and normalized error metrics ( NRMSE , NMAE , and WAPE ) on the order of ∼4%, outperforming comparable standalone deep-learning baselines. Overall, the results indicate that embedding single-phase physics constraints improves model robustness and predictive accuracy, particularly when production data is limited.