This research investigates the application of Multigrid Neural Operator
(MgNO), a neural operator architecture inspired by multigrid methods, in the
simulation for multiphase flow within porous media. The architecture is
adjusted to manage a variety of crucial factors, such as permeability and
porosity heterogeneity. The study extendes MgNO to time-dependent porous media
flow problems and validate its accuracy in predicting essential aspects of
multiphase flows. Furthermore, the research provides a detailed comparison
between MgNO and Fourier Neural Opeartor (FNO), which is one of the most
popular neural operator methods, on their performance regarding prediction
error accumulation over time. This aspect provides valuable insights into the
models' long-term predictive stability and reliability. The study demonstrates
MgNO's capability to effectively simulate multiphase flow problems, offering
considerable time savings compared to traditional simulation methods, marking
an advancement in integrating data-driven methodologies in geoscience
applications.

该研究探讨了多重网格神经算子（MgNO）在多孔介质内多相流模拟中的应用，该架构针对各种关键因素进行了调整，如渗透率和孔隙率的异质性。该研究将 MgNO 扩展到依赖时间的多孔介质流动问题，并验证了其在预测多相流的关键方面的准确性。此外，研究还详细比较了 MgNO 和傅里叶神经算子（FNO），这是其中一种最流行的神经算子方法，它们在随时间积累预测误差方面的性能表现。该研究展示了 MgNO 有效地模拟多相流问题的能力，相比传统模拟方法节省了大量时间，这标志着数据驱动方法在地球科学应用中的进步。

多孔介质中多相流的 MgNO 方法

A MgNO Method for Multiphase Flow in Porous Media

Pore-network models (PNMs) have become an important tool in the study of
fluid flow in porous media over the last few decades, and the accuracy of their
results highly depends on the extraction of pore networks. Traditional methods
of pore-network extraction are based on pixels and require images with high
quality. Here, a pixel-free method called the flashlight search medial axis
(FSMA) algorithm is proposed for pore-network extraction in a continuous space.
The search domain in a two-dimensional space is a line, whereas a surface
domain is searched in a three-dimensional scenario. Thus, the FSMA algorithm
follows the dimensionality reduction idea; the medial axis can be identified
using only a few points instead of calculating every point in the void space.
In this way, computational complexity of this method is greatly reduced
compared to that of traditional pixel-based extraction methods, thus enabling
large-scale pore-network extraction. Based on cases featuring two- and
three-dimensional porous media, the FSMA algorithm performs well regardless of
the topological structure of the pore network or the positions of the pore and
throat centers. This algorithm can also be used to examine both closed- and
open-boundary cases. Finally, the FSMA algorithm can search dead-end pores,
which is of great significance in the study of multiphase flow in porous media.

通过使用闪光灯搜索中轴（FSMA）算法，可以精确提取孔隙网络，从而实现对多孔介质中流体流动的研究。此算法不仅可以降低计算复杂性，而且可应用于不同维度的多孔介质，并能搜索闭合和开放边界情况下的孔隙，并在多相流研究中具有重要意义。

探照灯搜索介质轴：一种无像素孔隙网络提取算法

Flashlight Search Medial Axis: A Pixel-Free Pore-Network Extraction  Algorithm

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.

U-FNO 是一种基于傅里叶神经算子的新颖神经网络结构，可用于解决在多孔介质中的多相流问题，相较于传统卷积神经网络模型和 ML 模型，U-FNO 模型在 CO2 注入问题中具有更高的准确性和数据利用率，只需三分之一的训练数据即可实现与该项目等效的准确性。