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

Clement Etienam; Juntao Yang; Oleg Ovcharenko; Nick Luiken; Tsubasa Onishi; Nefeli Moridis; Issam Said

cs.LGMay 27, 2026

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

Authors: Clement Etienam, Juntao Yang, Oleg Ovcharenko, Nick Luiken, Tsubasa Onishi, Nefeli Moridis, Issam Said

Organizations: NVIDIA Corporation

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 $λ_R \geq λ^*_R$ 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.

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