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

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 $46\times 112\times 22$ 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 $W_2\le \epsilon (L^n-1)/(L-1)$, with exponential population-risk discrepancy for $L>1$; (3) physics-constrained spectral stability, showing PINO training with ${\lambda }_R\ge {\lambda }_R^{\ast }$ reduces the learned Jacobian spectral radius to ${\rho }_F+C{\lambda }_R^{-1/2}$, yielding uniform-in-time rollout error $|{\delta }_n|\le \epsilon /(1-\rho )$; 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 $O(1/\sqrt{t})+O({\rho }^{K^{\ast }})$. 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.