[Paper Review] Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration

Zhongyi Jiang, Min Zhu+1 more|arXiv (Cornell University)|Mar 8, 2023

CO2 Sequestration and Geologic InteractionsEnvironmental Science3 citations

TL;DR

This paper proposes Fourier-MIONet, a Fourier-enhanced multiple-input neural operator that efficiently learns the solution operator of multiphase flow in porous media for geological carbon sequestration. By integrating U-Fourier neural operator (U-FNO) into the MIONet framework, it achieves 90% fewer parameters, 3.5× faster training, and up to 85% lower CPU and 64% lower GPU memory usage than U-FNO, while predicting 30-year simulations with only six time snapshots and maintaining high accuracy and generalizability due to physical continuity of PDE solutions.

ABSTRACT

Geologic carbon sequestration (GCS) is a safety-critical technology that aims to reduce the amount of carbon dioxide in the atmosphere, which also places high demands on reliability. Multiphase flow in porous media is essential to understand CO$_2$ migration and pressure fields in the subsurface associated with GCS. However, numerical simulation for such problems in 4D is computationally challenging and expensive, due to the multiphysics and multiscale nature of the highly nonlinear governing partial differential equations (PDEs). It prevents us from considering multiple subsurface scenarios and conducting real-time optimization. Here, we develop a Fourier-enhanced multiple-input neural operator (Fourier-MIONet) to learn the solution operator of the problem of multiphase flow in porous media. Fourier-MIONet utilizes the recently developed framework of the multiple-input deep neural operators (MIONet) and incorporates the Fourier neural operator (FNO) in the network architecture. Once Fourier-MIONet is trained, it can predict the evolution of saturation and pressure of the multiphase flow under various reservoir conditions, such as permeability and porosity heterogeneity, anisotropy, injection configurations, and multiphase flow properties. Compared to the enhanced FNO (U-FNO), the proposed Fourier-MIONet has 90% fewer unknown parameters, and it can be trained in significantly less time (about 3.5 times faster) with much lower CPU memory ($<$ 15%) and GPU memory ($<$ 35%) requirements, to achieve similar prediction accuracy. In addition to the lower computational cost, Fourier-MIONet can be trained with only 6 snapshots of time to predict the PDE solutions for 30 years. The excellent generalizability of Fourier-MIONet is enabled by its adherence to the physical principle that the solution to a PDE is continuous over time.

Motivation & Objective

• To address the high computational cost and data inefficiency of conventional reservoir simulators and existing deep learning models for simulating multiphase flow in porous media.
• To develop a surrogate model that enables real-time prediction of CO2 migration and pressure fields in heterogeneous, anisotropic reservoirs under varying injection and rock properties.
• To reduce training data requirements and computational resources while maintaining high accuracy for long-time-scale simulations (e.g., 30 years).
• To improve generalization to unseen time points by leveraging the physical continuity of PDE solutions over time.
• To enable scalable, low-memory inference for 4D (3D space + time) multiphase flow problems in large-scale geological formations.

Proposed method

• Integrates the multiple-input neural operator (MIONet) framework with the U-Fourier neural operator (U-FNO) to handle multiple input functions (e.g., permeability, porosity, injection rates) on different domains.
• Employs a trunk network to encode input functions and a Fourier-based merge network with 3D or 2D FFTs to learn the solution operator in spectral space.
• Uses non-uniform time sampling with denser snapshots at early times and sparser ones later to improve generalization to unseen time points.
• Leverages the physical principle that PDE solutions are continuous over time to enhance generalization without requiring additional data.
• Trains the model using only six time snapshots per simulation case, significantly reducing data and computational demands.
• Optimizes memory usage by using 2D FFT instead of 4D FFT when z-coordinate and time are jointly encoded in the trunk net.

Experimental results

Research questions

• RQ1 Can a deep neural operator be designed to efficiently and accurately simulate long-term multiphase flow in heterogeneous porous media for geological carbon sequestration?
• RQ2 How can computational cost and data requirements be reduced without sacrificing prediction accuracy in 4D PDE simulations?
• RQ3 Can non-uniform time sampling improve generalization to unseen time points in PDE solution learning?
• RQ4 To what extent can the physical continuity of PDE solutions be exploited to enhance model generalization with minimal training data?
• RQ5 Can the proposed architecture scale to large-scale 3D+time problems with significantly reduced memory and parameter usage compared to state-of-the-art models?

Key findings

• Fourier-MIONet reduces the number of trainable parameters by 90% compared to U-FNO, achieving 4.68 million parameters versus 46.67 million.
• The model requires 85% less CPU memory and 64% less GPU memory than U-FNO, with GPU memory usage dropping to under 30% of the baseline.
• Training is 3.5 times faster than U-FNO, with a 24 GiB GPU capable of training the model efficiently.
• The model achieves an R² of 0.986 for pressure buildup prediction and maintains high accuracy even with only six time snapshots for training.
• Non-uniform time sampling with dense early snapshots and sparse later ones enables accurate prediction at both seen and unseen time points.
• The 2D FFT variant of Fourier-MIONet reduces parameters to 3.3% and GPU memory to 14.7% of the original FNO, showing strong potential for 4D problems.