U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow

TL;DR

U-FNO presents an enhanced surrogate for CO$_2$-water multiphase flow by integrating a U‑Net path into Fourier layers, yielding improved accuracy and data efficiency relative to both the original FNO and CNN benchmarks. The method demonstrates strong performance on 2D radial injection problems across wide permeability and porosity heterogeneity, significantly boosting plume-front accuracy for gas saturation and pressure buildup while enabling rapid inference far faster than conventional solvers. By leveraging an integral-kernel operator in Fourier space and a high-capacity local CNN path, U-FNO achieves mesh-flexible predictions and robust generalization, including unseen time steps. The work suggests substantial practical impact for probabilistic assessments, inversion, and site screening in geoscience applications, with publicly released code and web-accessible models for real-time predictions.

Abstract

Numerical simulation of multiphase flow in porous media is essential for many geoscience applications. Machine learning models trained with numerical simulation data can provide a faster alternative to traditional simulators. Here we present U-FNO, a novel neural network architecture for solving multiphase flow problems with superior accuracy, speed, and data efficiency. U-FNO is designed based on the newly proposed Fourier neural operator (FNO), which has shown excellent performance in single-phase flows. We extend the FNO-based architecture to a highly complex CO2-water multiphase problem with wide ranges of permeability and porosity heterogeneity, anisotropy, reservoir conditions, injection configurations, flow rates, and multiphase flow properties. The U-FNO architecture is more accurate in gas saturation and pressure buildup predictions than the original FNO and a state-of-the-art convolutional neural network (CNN) benchmark. Meanwhile, it has superior data utilization efficiency, requiring only a third of the training data to achieve the equivalent accuracy as CNN. U-FNO provides superior performance in highly heterogeneous geological formations and critically important applications such as gas saturation and pressure buildup "fronts" determination. The trained model can serve as a general-purpose alternative to routine numerical simulations of 2D-radial CO2 injection problems with significant speed-ups than traditional simulators.

Figures (10)

• Figure 1: Example of mapping between A. input to B. output gas saturation and C. pressure buildup. A. Field and scalar channels for each case. Note that the scalar variables are broadcasted into a channel at the same dimension as the field channels. B. Gas saturation evolution for 6 out of 24 time snapshots. C. Pressure buildup evolution for 6 out of 24 time snapshots.
• Figure 2: A. U-FNO model architecture. $a(x)$ is the input, $P$ and $Q$ are fully connected neural networks, and $z(x)$ is the output. B. Inside the Fourier layer, $\mathcal{F}$ denotes the Fourier transform, $R$ is the parameterization in Fourier space, $\mathcal{F}^{-1}$ is the inverse Fourier transform, $W$ is a linear bias term, and $\sigma$ is the activation function. C. Inside the U-FNO layer, $U$ denotes a two step U-Net, the other notations have identical meaning as in the Fourier layer.
• Figure 3: Training and validationtesting relative loss evolution vs. epoch for U-FNO, FNO, conv-FNO and CNN benchmark for A. gas saturation and B. pressure buildup.
• Figure 4: A. Gas saturation testing set plume mean absolute error ($MPE$) and plume $R^2$ scores ($R^2_{plume}$) using CNN, FNO, conv-FNO, and U-FNO. B. Pressure buildup field mean relative error ($MRE$) and $R^2$ scores using the same four models.
• Figure 5: Visualizations and scatter plots for example a to d. In each example, visualizations show the true gas saturation ($SG$), U-FNO predicted, U-FNO absolute error, CNN predicted, and CNN absolute error. The mean absolute error $\mu_{MAE}$ is labeled on the U-FNO and CNN absolute error plots. Scatter plots shows numerical simulation vs. predicted by U-FNO and CNN model on each grid. The legend for all of the scatter plots is shown in the bottom right.