基于PI-DeepONet的储层应力场预测与泛化训练策略研究

doi:10.3969/j.issn.2096-1693.2026.03.015

石油科学通报 ›› 2026, Vol. 11 ›› Issue (3): 894-909. doi: 10.3969/j.issn.2096-1693.2026.03.015

基于PI-DeepONet的储层应力场预测与泛化训练策略研究

季源¹(), 陈掌星²^,³, 李俊⁴, 彭岩⁵^,^*(), 吴克柳⁵, 王笑涵⁵

¹ 中国石油大学(北京)人工智能学院，北京 102249
² 中国石油大学(北京)油气资源与工程全国重点实验室，北京 102249
³ 宁波东方理工大学工学部，宁波 315200
⁴ 浙江化工工程地质勘察院有限公司，杭州 310000
⁵ 中国石油大学(北京)石油工程学院，北京 102249

收稿日期:2026-04-23 修回日期:2026-05-14 出版日期:2026-06-15 发布日期:2026-06-30
通讯作者: *彭岩(1987年—)，教授，博士生导师，主要从事石油工程岩石力学等研究，yan.peng@cup.edu.cn。
作者简介:季源(1996年—)，博士研究生，主要从事石油工程岩石力学及人工智能研究，yuan_ji_cupb@163.com。
基金资助:
新疆维吾尔自治区重点研发项目(2024B01013-1);天山英才培养计划(T2024TSYCCX0070)

Reservoir stress-field prediction and generalization-oriented training strategies based on PI-DeepONet

JI Yuan¹(), CHEN Zhangxing²^,³, LI Jun⁴, PENG Yan⁵^,^*(), WU Keliu⁵, WANG Xiaohan⁵

¹ College of Artificial Intelligence, China University of Petroleum, Beijing 102249, China
² National Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum, Beijing 102249, China
³ College of Engineering, Eastern Institute of Technology, Ningbo 315100, China
⁴ Zhejiang Chemical Engineering Geological Prospecting Institute, Hangzhou 310000, China
⁵ China College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

Received:2026-04-23 Revised:2026-05-14 Online:2026-06-15 Published:2026-06-30
Contact: *yan.peng@cup.edu.cn

摘要/Abstract

摘要：

在储层应力场模拟中，物理信息神经网络(PINN)虽能实现高精度无监督求解，但其模型将计算域与物理参数深度耦合，导致训练完成的模型仅适用于一组固定的材料属性，泛化能力不足。为提高PINN的泛化能力，本文基于物理信息深度算子网络(PI-DeepONet)构建了一种智能计算方法，通过引入分支-主干网络结构与Hadamard乘积实现参数与坐标的特征融合，建立从储层物性参数到应力-位移场的端到端映射关系；同时融合硬约束机制与分阶段渐进训练策略，形成具备强泛化能力的应力场求解算子模型。结果表明，该方法克服了传统PINN无法泛化的局限，在稀疏物理空间离散密度下，分阶段训练效率提升约30.6%；在稠密物理空间离散密度下，采用硬约束机制使模型精度提升约62.4%。本研究为油气藏高效开发与CO₂地质封存评估提供了可靠的智能计算方法。

关键词: 物理信息深度算子网络, 物理信息神经网络, 储层应力场, 硬约束, 模型泛化性

Abstract:

In reservoir stress field simulation, Physics-Informed Neural Networks (PINN) can achieve high-precision unsupervised solutions; however, their model structure tightly couples the computational domain with physical parameters, resulting in applicability only to a fixed set of material properties. This leads to limited generalization under varying working conditions. To enhance the generalization capability of PINN, this study develops an intelligent computational approach based on the Physics-Informed Deep Operator Network (PI-DeepONet). By introducing a branch-trunk architecture and employing the Hadamard product to fuse parameter and coordinate features, an end-to-end mapping from reservoir physical parameters to the stress-displacement field is established. Furthermore, a hard-constraint mechanism and a staged progressive training strategy are integrated to construct a stress field operator model with strong generalization capacity. The results demonstrate that this method overcomes the non-generalizability of conventional PINN, achieving approximately 30.6% improvement in training efficiency under sparse physical-space discretization, and around 62.4% enhancement in prediction accuracy under dense discretization by applying hard constraints. This research provides a reliable intelligent computational framework for efficient hydrocarbon reservoir development and CO₂ geological storage assessment.

Key words: PI-DeepONet, PINN, reservoir stress field, hard constraint mechanism, model generalization

中图分类号:

TE311
TP18

季源, 陈掌星, 李俊, 彭岩, 吴克柳, 王笑涵. 基于PI-DeepONet的储层应力场预测与泛化训练策略研究[J]. 石油科学通报, 2026, 11(3): 894-909.

JI Yuan, CHEN Zhangxing, LI Jun, PENG Yan, WU Keliu, WANG Xiaohan. Reservoir stress-field prediction and generalization-oriented training strategies based on PI-DeepONet[J]. Petroleum Science Bulletin, 2026, 11(3): 894-909.

https://sykxtb.cup.edu.cn/CN/Y2026/V11/I3/894

图/表 21

图1 PINN网络结构示意图

Fig. 1 Schematic diagram of the PINN architecture

图2 PI-DeepONet网络结构示意图

Fig. 2 Schematic diagram of the PI-DeepONet architecture

图3 案例参数设置示意图

Fig. 3 Schematic diagram of the case parameter settings

图4 PINN预测结果

Fig. 4 Prediction results of the PINN model

图5 储层岩石参数(λ=1.0，μ=0.5)下PINN计算结果与有限元的差值百分比

Fig. 5 Percentage difference between the PINN results and the FEM solution under reservoir rock parameters (λ=1.0, μ=0.5)

图6 储层岩石参数(λ=0.8，μ=0.3)下PINN计算结果与有限元的差值百分比

Fig. 6 Percentage difference between the PINN results and the FEM solution under reservoir rock parameters (λ=0.8, μ=0.3)

图7 储层岩石参数(λ=1.0，μ=0.5)下PI-DeepONet计算结果与有限元的差值百分比

Fig. 7 Percentage difference between the PI-DeepONet results and the FEM solution under reservoir rock parameters (λ=1.0, μ=0.5)

图8 储层岩石参数(λ=0.8，μ=0.3)下PI-DeepONet计算结果与有限元的差值百分比

Fig. 8 Percentage difference between the PI-DeepONet results and the FEM solution under reservoir rock parameters (λ=0.8, μ=0.3)

表1 PINN对不同区块预测误差

Table 1 Prediction errors of the PINN model for different reservoir blocks

误差绝对平均值/%	σ_xx	σ_yy	τ_xy	ε_xx	ε_yy	γ_xy	u	v	平均值
λ=1.0，μ=0.5	8.25	0.6	0.56	2.36	0.46	0.59	1.29	0.45	1.82
λ=0.8，μ=0.3	17.77	5.21	6.25	75.08	60.55	76.07	80.24	64.20	48.17

表2 PI-DeepONet对不同区块预测误差

Table 2 Prediction errors of the PI-DeepONet model for different reservoir blocks

误差绝对平均值/%	σ_xx	σ_yy	τ_xy	ε_xx	ε_yy	γ_xy	u	v	平均值
λ=1.0，μ=0.5	9.80	1.06	0.65	4.12	0.40	0.67	1.23	1.15	1.135
λ=0.8，μ=0.3	4.81	0.64	0.62	0.78	0.88	0.59	0.69	0.34	1.168

图9 软约束、硬约束与有限元结果对比

Fig. 9 Comparison between soft constraint, hard constraint, and FEM results

图10 不同离散空间下随机训练策略对应力σ_xx的预测误差

Fig. 10 Prediction error of random training strategy for stress σ_xx under different discrete spaces

图11 不同离散空间下分阶段训练策略对应力σ_xx的预测误差

Fig. 11 Prediction error of the stage-wise training strategy for stress σ_xx under different discrete spaces

表3 不同离散空间下随机训练策略的预测误差与训练时间

Table 3 Prediction error and training time of random training strategies under different discrete spaces

误差绝对平均值/%	σ_xx	σ_yy	τ_xy	ε_xx	ε_yy	γ_xy	u	v	平均值	时间/min
30×30	4.92	1.06	0.46	0.46	0.85	0.85	0.66	0.12	1.1725	963
40×40	4.81	0.64	0.62	0.78	0.88	0.59	0.69	0.34	1.168	1100
50×50	2.33	0.95	0.63	6.89	0.28	0.33	0.42	0.08	1.488	1094
60×60	1.65	0.92	0.64	0.71	1.10	0.43	1.43	0.08	0.87	1165
70×70	1.49	0.82	0.57	0.64	0.99	0.39	1.29	0.07	0.78	1356
80×80	0.94	1.29	0.43	0.96	0.07	0.93	0.34	0.05	0.597	1482

表4 不同离散空间下分阶段训练策略的预测误差与训练时间

Table 4 Prediction error and training time of stage-wise training strategies under different discrete spaces

误差绝对平均值/%	σ_xx	σ_yy	τ_xy	ε_xx	ε_yy	γ_xy	u	v	平均值	时间/min
30×30	0.62	0.26	0.44	0.39	0.21	0.47	0.52	6.03	1.1175	632
40×40	1.52	0.46	0.23	1.51	0.59	0.54	2.77	4.02	1.455	744
50×50	1.76	0.30	0.45	0.68	0.32	1.49	0.49	0.14	0.703	1296
60×60	1.38	0.19	0.88	1.20	0.25	1.66	1.00	0.53	0.886	1853
70×70	1.24	0.17	0.79	1.08	0.22	1.49	0.90	0.47	0.795	2675
80×80	2.45	0.67	0.49	1.89	0.91	0.88	2.38	0.81	1.31	3650

图12 随机训练策略的预测误差百分比(40×40)

Fig. 12 Prediction error percentage of the random training strategy (40×40)

图13 分阶段渐进训练策略的预测误差百分比(40×40)

Fig. 13 Prediction error percentage of the staged progressive training strategy (40×40)

表5 40×40密度下，不同训练策略的预测误差与训练时间

Table 5 Prediction errors and training time of different training strategies (40×40 density)

误差绝对平均值/%	σ_xx	σ_yy	τ_xy	ε_xx	ε_yy	γ_xy	u	v	平均值	时间/min
随机训练策略	1.51	0.37	0.25	0.19	0.20	0.11	0.06	0.07	0.345	1813
分阶段渐进训练策略	0.21	0.15	0.15	0.49	0.30	0.37	0.71	0.51	0.361	1258

图14 随机训练策略的预测误差百分比(80×80)

Fig. 14 Prediction error percentage of the random training strategy (80×80)

图15 分阶段渐进训练策略的预测误差百分比(80×80)

Fig. 15 Prediction error percentage of the staged progressive training strategy (80×80)

表6 80×80密度下，不同训练策略的预测误差与训练时间

Table 6 Prediction errors and training time of different training strategies (80×80 density)

误差绝对平均值/%	σ_xx	σ_yy	τ_xy	ε_xx	ε_yy	γ_xy	u	v	平均值	时间/min
随机训练策略	0.23	0.07	0.15	0.09	0.05	0.19	0.04	0.03	0.106	2737
分阶段渐进训练策略	0.20	0.19	0.12	0.16	0.31	0.34	0.43	0.51	0.282	6440

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基于PI-DeepONet的储层应力场预测与泛化训练策略研究

Reservoir stress-field prediction and generalization-oriented training strategies based on PI-DeepONet

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