Autoregressive Physics-Informed Neural Operator Surrogates and Joint 4-D Seismic-Production Bayesian History Matching

Abstract

We present a surrogate-accelerated Bayesian history-matching framework, demonstrated on a realistic synthetic model ($N=100,188$ cells). The forward map $\mathcal{G}$ is replaced by a Fourier neural operator ${\mathcal{G}}_{\theta }$ trained in a closed loop, ${\hat{x}}_{n+1}={\mathcal{G}}_{\theta }(u,{\hat{x}}_n)$, with the discrete black-oil residual $R(x_{n+1},x_n)$ as a physics-informed neural operator (PINO) penalty, truncated backpropagation and noise injection. We assimilate production data and 4-D seismic using adaptive regularised ensemble Kalman inversion (aREKI), and synthesise $\Delta V_p$ inside the rollout via a closed-form Gassmann transform of the surrogate, under a two-stage schedule (aREKI on production, then an ensemble smoother with multiple data assimilation, ES-MDA, on both). To temper the ill-posedness, we invert $(\log K,\varphi )$ not in ${\mathbb{R}}^{2N}$, where $N_e\ll 2N$ leaves a rank-deficient covariance and a null space the data never inform, but on a chart $u=\Psi (\tilde{m})$, $\tilde{m}\in {\mathbb{R}}^d$, $d\ll 2N$, given by a Parseval-exact truncated cosine transform or a variational auto-encoder decoder. The ensemble then spans its own search space; the directions the data cannot constrain are cut to those the chart can express, and every member decodes to a geologically plausible field. For the rollout we prove an exposure bound $W_1\le {\sum }_j{L}^{n-1-j}\mathbb{E}{e}_j$, a physics-penalty contraction $\Vert D_x{\mathcal{G}}_{\theta }\Vert \le \alpha <1$ under a small residual defect, and an $\mathcal{O}({\rho }^K)$ truncation bias. A Loewner-order inequality ${\mathcal{I}}_p⪯{\mathcal{I}}_p+{\mathcal{I}}_s$ explains why joint assimilation is no less informative. On held-out data, ${\mathcal{G}}_{\theta }$ attains $R^2>0.999$ on pressure and saturation and rolls out in $0.579$ s, a $\sim 2500\times$ speed-up, bringing an $N_e=190$-member, $N_a=10$-iteration joint inversion within one GPU-hour.

arXiv comment40 pages, 15 figures, 6 tables, 4 algorithms