Sequential Physics-Constrained Neural Operator Forward Modeling for the 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 at most epsilon(L n-1)/(L-1), with exponential population-risk discrepancy for L>1; (3) physics-constrained spectral stability, showing PINO training with λ R at least λ* R reduces the learned Jacobian spectral radius to ρ F + Cλ R-1/2, yielding uniform-in-time rollout error δ n at most epsilon/(1-ρ); and (4) K-step TBPTT gradient analysis, deriving geometric bias decay O(ρ K), optimal window K = O((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 about 0.80 (pressure), and monotonically improving R squared (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 about 10 4 times wall-clock speedup over the OPM finite-volume simulator.