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

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 λ_R \geq λ^_R reduces the learned Jacobian spectral radius to ρ_F + Cλ_R^{-1/2}, yielding uniform-in-time rollout error |δ_n| \leq \varepsilon/(1-ρ); and (4) K-step TBPTT gradient analysis, deriving geometric bias decay O(ρ^K), optimal window K^ = O(\log(T/σ^2)), and Adam convergence O(1/\sqrt{t}) + O(ρ^{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.