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

TL;DR

The paper tackles the computational burden of 4D multiphase flow simulations in geological carbon sequestration by introducing Fourier-MIONet, a hybrid of MIONet and U-FNO that treats time continuously as a trunk input. By using separate branches for field and scalar inputs and a time-focused decoder based on 2D FFT and U-FNO, Fourier-MIONet achieves comparable accuracy to U-FNO with far fewer parameters and substantially lower memory and training time. It also demonstrates strong generalization to unseen times and benefits from nonuniform time sampling, enabling reliable long-term predictions using only a small number of time snapshots. This approach offers a practical path toward real-time optimization and robust uncertainty handling in safety-critical GCS deployments.

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.

Figures (10)

• Figure 1: An example of inputs and outputs. (A) Example of field inputs. (B) Example of scalar inputs. (C) Gas saturation evolution at 6 out of 24 time snapshots. (D) Pressure buildup evolution at 6 out of 24 time snapshots.
• Figure 2: Architecture of MIONet. All the branch nets and the trunk net have the same number of outputs, which are merged together via the element-wise product.
• Figure 3: Architecture of U-FNO. (A) $v(x)$ is the input function, $P$ and $Q$ are fully connected neural networks, and $u(x)$ is the output function. (B) Inside each Fourier layer, $\mathcal{F}$ denotes the Fourier transform, $\mathcal{R}$ is a weight matrix, $\mathcal{F}^{-1}$ is the inverse Fourier transform, $W$ is an another weight matrix, and $\sigma$ is the activation function. (C) Inside each U-Fourier layer, $U$ denotes a U-Net block. Figure is adapted from wen2022u.
• Figure 4: Fourier-MIONet architecture. (A) $\mathbf{v}_1$ is field input and $\mathbf{v}_2$ is scalar input. (B) The input $t$ is time. (C) Fourier layer. (D) U-Fourier layer.
• Figure 5: Effect of $batch_{time}$ for Fourier-MIONet. (A) $R^2$. (B) MAE. (C) GPU memory usage. (D) Training time per epoch. (E) Minimum number of training epochs needed. (F) Total training time. The solid blue curves and shaded regions represent the mean and one standard deviation of 3 runs of Fourier-MIONet. The precise values in this figure can be found in Table \ref{['table2:detail main results']}. U-FNO does not have time batch size.