Sequential Physics-Constrained Neural Operator Forward Modeling for the $\textit{Norne}$ Reservoir System

Researchers developed a mathematical framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using Fourier Neural Operators (FNO) and physics-informed neural operators (PINO), applied to the Norne benchmark reservoir on a 113,344-cell grid over 3,298 days. The autoregressive PINO surrogate achieved R² above 0.99 for oil, 0.90 for gas, and approximately 0.80 for pressure across the full production horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1,000-member ensemble ran in under one minute on a single B200 GPU, delivering roughly a 10,000-fold wall-clock speedup over the OPM finite-volume simulator.

arXiv:2605.28909v1 Announce Type: new Abstract: We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators FNO and their physics-informed variant PINO . The application focus is the Norne benchmark reservoir, defined on a heterogeneous $46\times112\times22$ grid $N=113,344$ cells , with a production history spanning $T=30$ timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: 1 functional-analytic formulation in a product-Sobolev-space setting, including well-posedness of the implicit timestep map and sharp local Lipschitz estimates; 2 covariate shift quantification, proving that the Wasserstein-2 distance grows as $W 2 \leq \varepsilon L^n-1 / L-1 $, with exponential population-risk discrepancy for $L 1$; 3 physics-constrained spectral stability, showing PINO training with $\lambda R \geq \lambda^ R$ reduces the learned Jacobian spectral radius to $\rho F + C\lambda R^{-1/2}$, yielding uniform-in-time rollout error $|\delta n| \leq \varepsilon/ 1-\rho $; and 4 $K$-step TBPTT gradient analysis, deriving geometric bias decay $O \rho^K $, optimal window $K^ = O \log T/\sigma^2 $, and Adam convergence $O 1/\sqrt{t} + O \rho^{K^ } $. Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain $R^2 0.99$ oil , $R^2 0.90$ gas , $R^2\approx 0.80$ pressure , and monotonically improving $R^2$ water across the full 3298-day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000-member ensemble runs in under one minute on a single B200 GPU, giving a ${\sim}10^4\times$ wall-clock speedup over the OPM finite-volume simulator.