Sequential Physics-Constrained Neural Operator Forward Modeling for the textitNorne 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 46×112×22 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 W2 ≤ varepsilon(Ln-1)/(L-1), with exponential population-risk discrepancy for L>1; (3) physics-constrained spectral stability, showing PINO training with λR ≥ λ^*R reduces the learned Jacobian spectral radius to ρF + CλR-1/2, yielding uniform-in-time rollout error |δn| ≤ varepsilon/(1-ρ); and (4) K-step TBPTT gradient analysis, deriving geometric bias decay O(ρK), optimal window K^ = O(log(T/σ2)), and Adam convergence O(1/√t) + O(ρK^*). Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain R2>0.99 (oil), R2>0.90 (gas), R2≈ 0.80 (pressure), and monotonically improving R2 (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 ~104× wall-clock speedup over the OPM finite-volume simulator.