Efficient solution of flow equations using a physics-informed Laplace convolutional neural network

Zhu Haifeng; Chen Zhiming; Wei Yu

doi:10.7623/syxb202609009

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Efficient solution of flow equations using a physics-informed Laplace convolutional neural network

更新时间：2026-10-17

- Efficient solution of flow equations using a physics-informed Laplace convolutional neural network
- ACTA PETROLEI SINICA Vol. 47, Issue 9, Pages: 1862-1873(2026)
- 作者机构：
  
  1. 中国石油大学（北京）石油工程学院,北京,102249
  2. 中国石油大学（北京）人工智能学院,北京,102249
- 作者简介：
- 基金信息：
- DOI：10.7623/syxb202609009
  CLC： TE319
- Online First：14 October 2026，
  
  Published：2026
- 稿件说明：
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朱海锋, 陈志明, Wei Yu, 张宝允. 基于物理信息约束的拉普拉斯卷积神经网络渗流方程高效求解[J]. 石油学报, 2026, 47(9): 1862-1873.

Zhu Haifeng, Chen Zhiming, Wei Yu. Efficient solution of flow equations using a physics-informed Laplace convolutional neural network[J]. ACTA PETROLEI SINICA, 2026, 47(9): 1862-1873.
朱海锋, 陈志明, Wei Yu, 张宝允. 基于物理信息约束的拉普拉斯卷积神经网络渗流方程高效求解[J]. 石油学报, 2026, 47(9): 1862-1873. DOI： 10.7623/syxb202609009.

Zhu Haifeng, Chen Zhiming, Wei Yu. Efficient solution of flow equations using a physics-informed Laplace convolutional neural network[J]. ACTA PETROLEI SINICA, 2026, 47(9): 1862-1873. DOI： 10.7623/syxb202609009.

摘要

人工智能在油气开发领域的应用逐步深入，在油藏数值模拟方面也展现出良好前景。渗流偏微分方程是油藏数值模拟的核心工具，但由于其机理复杂且计算耗时，其高效准确求解仍是油藏数值模拟智能化的关键。近年来随着油气人工智能技术的发展，以卷积神经网络（CNN）为代表的深度学习模型，因其对空间分布数据表征能力强，为渗流方程求解提供了新范式。基于此，提出了一种基于物理信息约束的拉普拉斯卷积神经网络（PILCNN）的渗流方程求解方法。该方法通过在常规CNN模块中嵌入拉普拉斯卷积算子，实现对渗流场局部特征的精细捕捉，并利用有限体积法构建损失函数，约束神经网络训练，实现渗流方程的求解。针对油气藏常见的均质、非均质、混合断层边界及变制度生产4个场景开展验证。结果表明：PILCNN在均质及非均质问题上的预测平均相对误差低至0.009.7%和0.005.9%，与参考解高度一致；对于混合边界及变制度生产问题，PILCNN仍表现出良好的稳定性。进一步基于不同参数开展了4组参数（渗透率、边界距离、井储系数及表皮系数）的敏感性分析。研究结果显示，PILCNN在不同参数下的求解稳定性良好。此外，对比了不同物理信息约束的卷积神经网络、残差神经网络及全连接神经网络的训练效果，PILCNN在全训练周期表现更优。

Abstract

The application of artificial intelligence in oil and gas development has been progressively expanding and demonstrates considerable potential in reservoir numerical simulation. Flow partial differential equations (PDEs) serve as the core tool for reservoir numerical simulation. However

due to the complex computational mechanisms and high computational cost of PDEs

obtaining efficient and accurate solutions to PDEs is the key to intelligent reservoir numerical simulation. In recent years

driven by the rapid development of artificial intelligence technologies in the oil and gas industry

deep learning models

particularly convolutional neural networks (CNNs)

have emerged as a novel paradigm for solving flow equations owing to their powerful capability in capturing and representing spatially distributed data. Consequently

a Physical-Informed Laplace Convolutional Neural Network (PILCNN) is proposed for solving the flow equation. By integrating a Laplace convolution operator into the conventional CNN architecture

the proposed method enables the accurate extraction of local features within flow fields. Meanwhile

the loss function is formulated via the finite volume method (FVM) to enforce physical constraints during neural network training

thereby facilitating the solution of flow equations. The proposed method is validated under four common scenarios in oil and gas reservoirs

including homogeneous reservoirs

heterogeneous reservoirs

reservoirs with mixed fault boundaries

and variable-rate production conditions. The results indicate that PILCNN achieves mean relative errors (MRE) as low as 0.009.7% and 0.005.9% in predicting homogeneous and heterogeneous cases

respectively

showing excellent agreement with the reference solutions. In addition

PILCNN exhibits robust stability when applied to mixed boundary and variable-rate production problems. Furthermore

four groups of parameter sensitivity analyses are conducted by varying permeability

boundary distance

wellbore storage coefficient

and skin factor. The results show that PILCNN maintains reliable solution stability under different parameter configurations. Comparative training experiments are performed among the Physics-Informed Convolutional Neural Network (PICNN)

Physics-Informed Residual Neural Network (PIResNet) and Physics-Informed Neural Network (PIetNN). PILCNN demonstrates preferable performance throughout the entire training process.

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references

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