Sequential Physics-Constrained Neural Operator Forward Modeling for the $\textit{Norne}$ 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 $46\times112\times22$ 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.