基于格子Boltzmann法岩体裂隙粗糙特征对渗流的影响

王继刚; 方明松; 陈刚; 胡成

doi:10.19509/j.cnki.dzkq.tb20220190

基于格子Boltzmann法岩体裂隙粗糙特征对渗流的影响

doi: 10.19509/j.cnki.dzkq.tb20220190

王继刚^1,,
方明松²,
陈刚¹,
胡成^1, ,

1.
中国地质大学(武汉)环境学院, 武汉 430078
2.
惠州国储石油基地有限责任公司, 广东惠州 516086

基金项目:

国家自然科学基金项目 42022018

详细信息

作者简介:
王继刚(1995—)，男，现正攻读地质工程专业硕士学位，主要从事地下水数值模拟研究工作。E-mail: 1969560921@qq.com

通讯作者:
胡成(1976—)，男，副教授，主要从事地下水数值模拟研究工作。E-mail: hu_cheng@cug.edu.cn

中图分类号: P641
计量
- 文章访问数: 486
- PDF下载量: 55
- 被引次数: 0
出版历程
- 收稿日期: 2022-04-29
- 录用日期: 2022-08-31
- 修回日期: 2022-08-16

Influence of the fractures roughness of rock on fluid flow by the lattice Boltzmann method

1.
School of Environment Studies, China University of Geosciences(Wuhan), Wuhan 430078, China
2.
Huizhou State Reserve Petroleum Base Limited Liability Company, Huizhou Guangdong 516086, China

摘要

摘要:
裂隙在岩体中的形貌结构复杂, 岩体裂隙的粗糙特征对裂隙的渗透性存在较大影响。目前传统的数值模拟软件主要是以等效连续介质为基础的宏观评价, 无法模拟裂隙微小结构内的介观渗流特征; 虽然存在考虑裂隙粗糙特征的粗糙裂隙渗透评估模型, 但是将粗糙裂隙的剖面高度标准偏差值作为粗糙特征的定量表征缺乏物理意义并且存在局限性。首先运用W-M(Weierstrass-Mandelbrot)函数构建具有不同分形维数的二维粗糙单裂隙数字模型。其次基于格子Boltzmann法理论通过编程实现介观尺度的渗流模拟, 并结合以裂隙剖面高度标准偏差值作为粗糙特征定量表征的立方定律公式进行分析。结果表明: 以裂隙剖面高度标准偏差值作为粗糙特征定量表征的立方定律公式存在不足; 以分形维数作为粗糙特征定量表征的局部修正立方定律公式相对可行。研究对于地下水污染防治以及地下水资源评估有着重要的工程实际意义。
- 格子Boltzmann法 /
- 二维粗糙单裂隙 /
- 分形维数 /
- 立方定律
Abstract: Objective
The morphological structure of the fractures in the rock mass is complex, and the fissures rough characteristics of the rock have a great influence on the permeability of the fractures. The current traditional numerical simulation software is mainly based on the macroscopic evaluation of equivalent continuous media, which cannot simulate the mesoscopic fluid flow characteristics within the tiny structure of the fracture. Although there exist models for assessing the permeability of rough fractures considering fracture roughness characteristics, it lacks physical meaning and has limitations in taking standard deviation of the profile height of rough fractures as the quantitative representation of rough characteristics.
Methods
Firstly, the W-M (Weierstrass-Mandelbrot) function was applied to construct a numerical model of a two-dimensional rough single fracture with different fractal dimensions. Secondly, the simulation of fluid flow at the mesoscopic scale was realized by programming based on the lattice Boltzmann method theory and analysed by combining the cubic law formulation with the value of standard deviation of the fracture profile height as a quantitative characterization of roughness.
Results
The results show that the cubic law formula using standard deviation value of the fracture profile height as a quantitative characterization of the roughness feature is inadequate. It is feasible to use the fractal dimension as a local modified cubic law formulation for the quantization of rough features.
Conclusion
The study of rock fracture fluid flow has important engineering practical significance for groundwater pollution control and groundwater resource assessment.
- lattice Boltzmann method /
- two-dimensional rough single crack /
- fractal dimension /
- cubic law
注释:

(所有作者声明不存在利益冲突)

HTML全文

图 1 D2Q9离散速度模型示意图

Figure 1. Schematic diagram of the D2Q9 discrete velocity model

下载: 全尺寸图片幻灯片

图 2 LBM数值流速与泊肃叶流解析流速误差分析图

Figure 2. Error analysis diagram of LBM numerical velocity and Poiseuille flow analytical velocity

下载: 全尺寸图片幻灯片

图 3 相同高度标准值偏差粗糙单裂隙数字模型

Figure 3. Digital model of a rough single crack with a standard deviation value of the same height

下载: 全尺寸图片幻灯片

图 4 不同位置处LBM数值模拟流速曲线

Figure 4. Velocity curves of LBM numerical simulation at different positions

下载: 全尺寸图片幻灯片

图 5 不同分形维数(D)粗糙单裂隙数字模型

Figure 5. Digital model of rough single crack with different fractal dimensions (D)

下载: 全尺寸图片幻灯片

图 6 6种分形维数(D)粗糙单裂隙不同位置处流速

Figure 6. Flow velocity at different positions of rough single crack with six fractal dimensions (D)

下载: 全尺寸图片幻灯片

图 7 不同分形维数下相应误差关系曲线

Figure 7. Corresponding error curves under different fractal dimensions

下载: 全尺寸图片幻灯片

图 8 不同分形维数(D)单裂隙流速分布图

Figure 8. Velocity distribution of a single crack with different fractal dimensions (D)

下载: 全尺寸图片幻灯片

图 9 粗糙因子f(D)与分形维数(D)关系曲线

Figure 9. Relation curve between rough factor f(D) and the fractal dimension (D)

下载: 全尺寸图片幻灯片

图 10 不同分形维数(D)，不同裂隙开度(b)粗糙单裂隙数字模型

Figure 10. Digital model of a rough single crack with different fractal dimensions (D) and crack openings (b)

下载: 全尺寸图片幻灯片

图 11 不同分形维数(D)，不同裂隙开度(b)粗糙单裂隙LBM模拟流速

Figure 11. Simulation flow rate of LBM in a single rough crack with different fractal dimensions (D) and crack openings (b)

下载: 全尺寸图片幻灯片

图 12 粗糙裂隙解析流速与数值流速对比图(D. 分形维数；b. 裂隙开度)

Figure 12. Comparison of analytic and numerical flow rates in rough cracks with different fractal dimensions and crack openings

下载: 全尺寸图片幻灯片

表 1 不同裂隙形态下LBM数值模拟数值流速对比数据

Table 1. LBM numerical simulation numerical flow rate comparison data under different crack morphologies

位置x/m	裂隙a平均流速/(10^-4m·s^-1)	裂隙b平均流速/(10^-4m·s^-1)	相对误差\|(v_a-v_b)/v_a\|/%
200	1.677 8	1.503 4	10.40
300	1.388 9	1.905 3	37.18
400	1.399 0	1.616 4	15.54
500	1.453 6	1.634 4	12.44
700	1.684 7	1.880 5	11.62
800	1.720 6	1.964 9	14.20

下载: 导出CSV

表 2 不同分形维数粗糙单裂隙数字模型参数

Table 2. Parameters of the digital model of a rough single crack with different fractal dimensions

分形维数D	1.01	1.1	1.2	1.3	1.4	1.5
裂隙开度b/m	0.002	0.002	0.002	0.002	0.002	0.002
裂隙长度l/m	0.2	0.2	0.2	0.2	0.2	0.2

下载: 导出CSV

表 3 不同分形维数、不同裂隙开度粗糙裂隙参数

Table 3. Parameters of the rough cracks with different fractal dimensions and different openings

分形维数	D=1.1	D=1.15	D=1.2	D=1.25
剖面高度标准偏差/m	0.003 438	0.003 592	0.003 66	0.003 738
裂隙开度b/m	0.005	0.003	0.005	0.003
裂隙水平长度/m	0.2	0.2	0.2	0.2

下载: 导出CSV

表 4 各前人立方定律解析流速与LBM数值模拟平均数值流速相对误差

Table 4. Relative errors between the analytical velocity of each previous cubic law and the average numerical velocity of the LBM numerical simulation

分形维数D	裂隙开度b/m	修正立方定律相对误差/%	Renshaw立方定律相对误差/%	Lomize立方定律相对误差/%
1.10	0.005	7.91	152.15	57.85
1.15	0.003	37.01	51.40	75.30
1.20	0.005	9.16	143.68	60.18
1.25	0.003	56.65	69.25	71.99

下载: 导出CSV

参考文献(32)

[1]	Lomize G M. Flow in fractured rocks[M]. Moscow: Gesenergoizdat, 1951: 127-129.
[2]	葛洪魁, 申颍浩, 宋岩, 等. 页岩纳米孔隙气体流动的滑脱效应[J]. 天然气工业, 2014, 34(7): 46-54. https://www.cnki.com.cn/Article/CJFDTOTAL-TRQG201407012.htm Ge H K, Shen Y H, Song Y, et al. Slippage effect of gas flow in shale nano-pores[J]. Natural Gas Industry, 2014, 34(7): 46-54(in Chinese with English abstract). https://www.cnki.com.cn/Article/CJFDTOTAL-TRQG201407012.htm
[3]	郭保华, 田采霞. 岩石单裂隙水力特性试验研究进展[J]. 河南理工大学学报: 自然科学版, 2009, 28(1): 39-44. doi: 10.3969/j.issn.1673-9787.2009.01.008 Guo B H, Tian C X. Experimental research progress on hydraulic characteristics of rock single fracture[J]. Journal of Henan Polytechnic University: Natural Science Edition, 2009, 28(1): 39-44(in Chinese with English abstract). doi: 10.3969/j.issn.1673-9787.2009.01.008
[4]	耿克勤, 陈凤翔, 刘光廷, 等. 岩体裂隙渗流水力特性的实验研究[J]. 清华大学学报: 自然科学版, 1996, 36(1): 102-106. doi: 10.3321/j.issn:1000-0054.1996.01.008 Geng K Q, Chen F X, Liu G T, et al. Experimental study on seepage hydraulic characteristics of rock mass fracture[J]. Journal of Tsinghua University: Science and Technology Edition, 1996, 36(1): 102-106(in Chinese with English abstract). doi: 10.3321/j.issn:1000-0054.1996.01.008
[5]	宋羿. 分叉-交叉裂隙中优势流及示踪试验与模拟研究[D]. 合肥: 合肥工业大学, 2019. Song Y. Experimental and simulation study on dominant flow and tracer in bifurcation-cross fracture[D]. Hefei: Hefei University of Technology, 2019(in Chinese with English abstract).
[6]	许光祥, 张永兴, 哈秋舲. 粗糙裂隙渗流的超立方和次立方定律及其试验研究[J]. 水利学报, 2003, 34(3): 74-79. doi: 10.3321/j.issn:0559-9350.2003.03.014 Xu G X, Zhang Y X, Ha Q L. Super-cubic and sub-cubic law of rough fracture seepage and its experimental study[J]. Journal of Hydraulic Engineering, 2003, 34(3): 74-79(in Chinese with English abstract). doi: 10.3321/j.issn:0559-9350.2003.03.014
[7]	Dou Z, Chen Z, Zhou Z, et al. Influence of eddies on conservative solute transport through a 2D single self-affine fracture[J]. International Journal of Heat and Mass Transfer, 2018, 121: 597-606. doi: 10.1016/j.ijheatmasstransfer.2018.01.037
[8]	Lee S H, Yeo I W, Lee K K, et al. Tail shortening with developing eddies in a rough-walled rock fracture[J]. Geophysical Research Letters, 2015, 42(15): 6340-6347. doi: 10.1002/2015GL065116
[9]	Lee S H, Yeo I W, Lee K K, et al. The role of eddies in solute transport and recovery in rock fractures: Implication for groundwater remediation[J]. Hydrological Processes, 2017, 31(20): 3580-3587. doi: 10.1002/hyp.11283
[10]	Louis C. A study of groundwater flow in jointed rock and its influence on the stability of rock masses[M]. London: Imperial College of Science and Technology, 1969.
[11]	Mourzenko V V, Thovert J F, Adler P M. Conductivity and transmissivity of a single fracture[J]. Transport in Porous Media, 2018, 123(2): 235-256. doi: 10.1007/s11242-018-1037-y
[12]	Chen Y F, Zhou J Q, Hu S H, et al. Evaluation of Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures[J]. Journal of Hydrology, 2015, 529: 993-1006. doi: 10.1016/j.jhydrol.2015.09.021
[13]	Poon C Y, Sayles R S, Jones T A. Surface measurement and fractal characterization of naturally fractured rocks[J]. Journal of Physics D Appli-ed Physics, 2000, 25(8): 1269-1275.
[14]	Brown S R, Scholz C H. Broad bandwidth study of the topography of natural rock surfaces[J]. Journal of Geophysical Research: Solid Earth, 1985, 90(B14): 12575-12582. doi: 10.1029/JB090iB14p12575
[15]	Brown S R. Fluid flow through rock joints: The effect of surfaceroughness[J]. Journal of Geophysical Research: Solid Earth, 1987, 92(B2): 1337-1347. doi: 10.1029/JB092iB02p01337
[16]	Odling N E. Natural fracture profiles, fractal dimension and joint roughness coefficients[J]. Rock Mechanics and Rock Engineering, 1994, 27(3): 135-153. doi: 10.1007/BF01020307
[17]	Keller K, Bonner B P. Automatic, digital system for profiling roughsurfaces[J]. Review of Scientific Instruments, 1985, 56(2): 330-331. doi: 10.1063/1.1138299
[18]	Brown S R, Scholz C H. Broad bandwidth study of the topography of natural rock surfaces[J]. Journal of Geophysical Research, 1985, 90(B14): 12575. doi: 10.1029/JB090iB14p12575
[19]	Schmittbuhl J, Steyer A, Jouniaux L, et al. Fracture morphology and viscous transport[J]. International Journal of Rock Mechanics and Mining Sciences, 2008, 45(3): 422-430. doi: 10.1016/j.ijrmms.2007.07.007
[20]	Chen Y D, Liang W G, Lian H J, et al. Experimental study on the effect of fracture geometric characteristics on the permeability in deformable rough-walled fractures[J]. International Journal of Rock Mechanics and Mining Sciences, 2017, 98: 121-140. doi: 10.1016/j.ijrmms.2017.07.003
[21]	Jin Y, Dong J B, Zhang X Y, et al. Scale and size effects on fluid flow through self-affine rough fractures[J]. International Journal of Heat and Mass Transfer, 2017, 105: 443-451. doi: 10.1016/j.ijheatmasstransfer.2016.10.010
[22]	Wang L, Cardenas M B. Development of an empirical model relating permeability and specific stiffness for rough fractures from numerical deformation experiments[J]. Journal of Geophysical Research-Solid Earth, 2016, 121(7): 4977-4989. doi: 10.1002/2016JB013004
[23]	Wang M, Chen Y F, Ma G W, et al. Influence of surface roughness on nonlinear flow behaviors in 3D self-affine rough fractures: Lattice Boltzmannsimulations[J]. Advances in Water Resources, 2016, 96: 373-388. doi: 10.1016/j.advwatres.2016.08.006
[24]	Guo Z L, Shi B C, Wang N C. Lattice BGK model for incompressible Navier-Stokes equation[J]. Journal of Computational Physics, 2000, 165(1): 288-306. doi: 10.1006/jcph.2000.6616
[25]	Witherspoon P A, Wang J S Y, Iwai K, et al. Validity of Cubic Law for fluid flow in a deformable rockfracture[J]. Water Resources Research, 1980, 16(6): 1016-1024. doi: 10.1029/WR016i006p01016
[26]	Renshaw C E. On the relationship between mechanical and hydraulic apertures in rough-walled fractures[J]. Journal of Geophysical Research: Solid Earth, 1995, 100(B12): 24629-24636. doi: 10.1029/95JB02159
[27]	谢和平, 于广明, 杨伦, 等. 采动岩体分形裂隙网络研究[J]. 岩石力学与工程学报, 1999, 18(2): 29-33. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX902.006.htm Xie H P, Yu G M, Yang L, et al. Study on fractal fracture network of mining-induced rock mass[J]. Chinese Journal of Rock Mechanics and Engineering, 1999, 18(2): 29-33(in Chinese with English abstract). https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX902.006.htm
[28]	张永波, 靳钟铭, 刘秀英. 采动岩体裂隙分形相关规律的实验研究[J]. 岩石力学与工程学报, 2004, 23(20): 3426-3429. doi: 10.3321/j.issn:1000-6915.2004.20.006 Zhang Y B, Jin Z M, Liu X Y. Experimental study on fractal correlation of mining-induced rock fracture[J]. Chinese Journal of Rock Mechanics and Engineering, 2004, 23(20): 3426-3429(in Chinese with English abstract). doi: 10.3321/j.issn:1000-6915.2004.20.006
[29]	杨庆, 娄峰, 栾茂田. 岩体断裂面分维数测量精度的改进方法[J]. 岩石力学与工程学报, 2003, 22(2): 246-249. doi: 10.3321/j.issn:1000-6915.2003.02.016 Yang Q, Lou F, Luan M T. An improved method for measuring accuracy of tractal dimension of rock fracture surface[J]. Chinese Journal of Rock Mechanics and Engineering, 2003, 22(2): 246-249(in Chinese with English abstract). doi: 10.3321/j.issn:1000-6915.2003.02.016
[30]	冯增朝, 赵阳升, 文再明. 岩体裂缝面数量三维分形分布规律研究[J]. 岩石力学与工程学报, 2005, 24(4): 601-609. doi: 10.3321/j.issn:1000-6915.2005.04.010 Feng Z C, Zhao Y S, Wen Z M. Study on three-dimensional fractal distribution of fracture surface number in rock mass[J]. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(4): 601-609(in Chinese with English abstract). doi: 10.3321/j.issn:1000-6915.2005.04.010
[31]	朱忠谦, 张啸枫, 张承泽, 等. 裂缝-孔隙模型中气水两相分形渗流实验和数学模型建立[J]. 地质科技情报, 2014, 33(4): 86-90. https://www.cnki.com.cn/Article/CJFDTOTAL-DZKQ201404014.htm Zhu Z Q, Zhang X F, Zhang C Z, et al. Experimental study and mathematical model establishment of gas-water two-phase fractal seepage in fracture-pore model[J]. Geological Science and Technology Information, 2014, 33(4): 86-90(in Chinese with English abstract). https://www.cnki.com.cn/Article/CJFDTOTAL-DZKQ201404014.htm
[32]	钟嘉高, 梁杏, 任志刚. 岩体裂隙等效水力隙宽的统计确定方法[J]. 地质科技情报, 2007, 26(4): 103-106. https://www.cnki.com.cn/Article/CJFDTOTAL-DZKQ200704020.htm Zhong J G, Liang X, Ren Z G. Statistical determination method of equivalent hydraulic gap width of rock mass fracture[J]. Geological Science and Technology Information, 2007, 26(4): 103-106(in Chinese with English abstract). https://www.cnki.com.cn/Article/CJFDTOTAL-DZKQ200704020.htm