A deep Euler-moment model for gas–liquid two-phase pipe flow

Ultra-deep gas wellbores (length-to-diameter ratio > 60 000) lack a computationally efficient and physically consistent one-dimensional flow model that integrates both data-driven learning and conservation-law priors. This paper introduces a deep Euler-moment model leveraging a hybrid Physics-Informed Neural Network architecture that fuses physical conservation laws with sparse measurement data. The loss function is optimized via simulated annealing, and the network architecture is tuned against phase-space discretization levels. Validation against numerical solutions confirms predictive accuracy for two initial gas distributions. To replicate borehole measurement limitations, we evaluate spatiotemporal extrapolation under extreme data sparsity. Results show that the model maintains L2 relative errors within 1.14 × 10−3 for temporal extrapolation and 9.55 × 10−3 under anomalous inputs, with error growth well-bounded despite increased computational time. This work establishes a robust, data-physics hybrid approach that bypasses mesh constraints, demonstrating reliable extrapolation capability for deep-well and long-term predictions.