金毅, 郭安琪, 董佳斌,等.规则自相似多孔介质中分形行为对渗透率的影响[J].河南理工大学学报（自然科学版）,2026,45(3):133-142.

JIN Y, GUO A Q, DONG J B,et al.Effects of fractal behavior on the permeability in regular self-similar porous media[J].Journal of Henan Polytechnic University(Natural Science) ,2026,45(3):133-142.

规则自相似多孔介质中分形行为对渗透率的影响

金毅1,2,3, 郭安琪1, 董佳斌1,2, 刘丹丹1, 赵静妍1, 郑军领1, 吴影1

1.河南理工大学 资源环境学院，河南 焦作 454003；2.河南省煤系非常规资源成藏与开发重点实验室，河南 焦作  454003；3.煤炭安全生产与清洁高效利用省部共建协同创新中心，河南 焦作  454003

摘要:目的 为探究渗流行为与孔隙结构分形拓扑结构的关系，开展规则自相似多孔介质中分形行为对渗透率的影响研究。  方法 依据分形拓扑理论和谢尔宾斯基地毯模型构建思想，结合多孔介质间的串并联关系构建规则自相似多孔介质及其串并联几何模型，融合圆柱绕流中颗粒直径-渗透率关系和渗透率尺度缩放规律，推导规则自相似多孔介质及串并联介质的渗透率计算模型。  结果 结果表明：多孔介质串联情况下，总体渗透率k与两子片段参数（k₁，l₁）和（k₂，l₂）满足k=1/(1/k₁×l₁/L+1/k₂×l₂/L），其中，总长度L=l₁+l₂，渗透率并联模型为k=1/(k₁×l₁+k₂×l₂)/L。此外，缩放间隙度P与渗透率呈指数为2的乘幂关系，相对直径为D/L的圆柱绕流渗透率满足k_l=0.14l²(1-D/L)^{2.784 4}。计算得到二级与三级分形迭代渗透率与多孔介质模拟值误差分别为0.051 9，0.067 3，V4S2L2Rd，V5S3L2Rd，V6S4L2Rd，V6S2L2Rd模型的误差分别为-0.040 1，-0.079 1，-0.001 2，-0.098 0，说明迭代结构越复杂，误差越大。  结论 依据以上关系和孔隙结构原始尺度与目标尺度之间的缩放比例可以计算规则分形多孔介质渗透率，这为复杂多孔介质非均质渗流规律研究提供了新的思路与方法。

关键词:分形拓扑理论;多孔介质;渗透率;流阻;串并联关系

doi:10.16186/j.cnki.1673-9787.2023120057

基金项目:国家自然科学基金资助项目（41972175， 42502167）；河南省高校科技创新团队项目（21IRTSTHN007）；河南省高校基本科研业务费专项项目（NSFRF220204，NSFRF220427）

收稿日期:2023/12/09

修回日期:2024/05/10

出版日期:2026/04/28

Effects of fractal behavior on the permeability in regular self-similar porous media

Jin Yi1,2,3, Guo Anqi1, Dong Jiabin1,2, Liu Dandan1, Zhao Jingyan1, Zheng Junling1, Wu Ying1

1.School of Resources and Environment， Henan Polytechnic University， Jiaozuo  454003， Henan， China；2.Henan Key Laboratory of Coal Measure Unconventional Resources Accumulation and Exploitation， Jiaozuo  454003， Henan， China；3.Collaborative Innovation Center of Coal Work Safety and Clean High Efficiency Utilization， Jiaozuo  454003， Henan， China

Abstract: Objectives To investigate the relationship between seepage behavior and the fractal topological structure of the pore spaces， the influence of fractal behavior on permeability was studied.  Methods Based on the fractal topological theory and the construction concept of the Sierpinski carpet， regular self-similar porous media and their series-parallel geometric models were constructed by combining the series-parallel relationships between multiple porous media. A permeability calculation model for regular self-similar porous media and series-parallel media was derived by integrating the particle diameter-permeability relationship in cylinder flow and the permeability scaling law.  Results For porous media in series， the overall permeability k satisfies k=1/(1/k₁×l₁/L+1/k₂×l₂/L） where the total length L=l₁+l₂， and the parameters of the two sub-segments are （k₁，l₁） and （k₂，l₂）； for the parallel case， the permeability model is k=1/（k₁×l₁+k₂×l₂）/L. In addition，there is a power relationship with an exponent of 2 between the scaling lacunarity P and the permeability， and the permeability of cylinder flow with a relative diameter of D/L satisfies k_l=0.14l²(1-D/L)^{2.784 4}.The calculated permeability errors between the second-order and third-order fractal iterations and the simulated values of porous media are 0.051 9 and 0.067 3， respectively； the errors for the V4S2L2Rd， V5S3L2Rd， V6S4L2Rd， and V6S2L2Rd models are -0.040 1， -0.079 1， -0.001 2， and -0.098 0， respectively， indicating that a more complex iterative structure leads to a larger error. Conclusions The permeability of regular fractal porous media can be calculated based on the above relationships and the scaling ratio between the original scale of the pore structure and the target scale.  Conclusions This provides a new idea and method for studying the heterogeneous seepage behavior of complex porous media.

Key words:fractal topological theory;porous media;permeability;flow resistance;series-parallel relationship