U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow (2109.03697v3)

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.

An Analysis of U-FNO: Advancements in Fourier Neural Operator-Based Modeling for Multiphase Flow

The paper presents U-FNO, a novel deep learning architecture designed to enhance the Fourier Neural Operator (FNO) framework for simulating multiphase flow phenomena within complex geological formations. The authors address the significant computational challenges associated with traditional numerical models for multiphase flow by leveraging machine learning methods to achieve superior performance in terms of accuracy, speed, and efficiency. This analysis focuses on evaluating the paper’s claims, examining the U-FNO model's development, and discussing the broader implications for the field of computational geosciences.

The motivation for this research stems from the complexity and computational expense of simulating multiphase flow in porous media, a process crucial for applications such as carbon capture and storage (CCS), hydrogen storage, and oil extraction. Traditional simulation methods, which rely on fine spatio-temporal discretizations, struggle with computational feasibility, particularly under constraints of non-linearity and heterogeneity in subsurface systems. To address these challenges, the authors promote a learning-based alternative that could replace repetitive simulations with a general-purpose, data-driven model.

U-FNO Model Architecture

The U-FNO model builds upon the original FNO by introducing a U-Fourier layer that integrates a U-Net with convolutional layers into its architecture. This hybrid approach retains the beneficial characteristics of both the FNO — its ability to efficiently handle functional mappings in a data-driven manner — and CNNs, known for their capacity to decode high-frequency spatial features. Consequently, U-FNO enhances generalization, benefiting from the mesh-free resolve of FNO and the localized convolutional operations of CNNs. The architecture's balance between Fourier transforms and U-Net convolution provides a significant boost in representational power, particularly for high-frequency information.

Performance Evaluation and Results

The U-FNO achieves marked improvements over both the original FNO and leading CNN benchmarks in predicting multiphase flow outcomes such as gas saturation and pressure buildup. Notably, it demonstrates a striking reduction in mean absolute error (MPE) for gas saturation and significant improvement in mean relative error (MRE) for pressure buildup, outperforming traditional simulation models. Furthermore, U-FNO requires only a third of the training data that CNN models necessitate to reach equivalent accuracy levels, illustrating its data efficiency.

Beyond accuracy metrics, the U-FNO excelled in computational efficiency. The U-FNO model evaluation is 6x to 10x faster than traditional simulators, proving invaluable in environments where rapid modeling is required for scenarios like history matching and probabilistic assessment. This advantage is particularly pronounced given the high dimensionality of input spaces in subsurface modeling, which typically pose risks of overfitting and impractical data handling.

Implications and Future Directions

The development of U-FNO represents a significant step forward in the application of neural operators to multiphase fluid modeling. Its application to a complex CO2-water system in geological storage contexts demonstrates potential for the approach to serve as a robust simulator substitute. These advancements suggest promising opportunities for U-FNO in broader contexts beyond subsurface flow, potentially offering insights into any system governed by similarly complex differential equations.

In terms of future advancements, extending U-FNO’s applicability to three-dimensional and highly heterogeneous formations could further enhance its utility. Additionally, refining the resolution invariance of the U-FNO, allowing it to accurately predict across varying spatio-temporal scales without additional training, could open new frontiers in predictive modeling. By providing a foundation for accessible simulations of complex physical processes, U-FNO unveils new possibilities for computational models across multiple engineering and scientific domains. This aligns with the growing shift towards machine learning-augmented modeling practices that emphasize efficiency and adaptability in high-dimensional simulation scenarios.