岩体粗糙裂隙几何特征对其Forchheimer型渗流特性的影响

周新; 盛建龙; 叶祖洋; 罗旺; 黄诗冰; 程爱平

doi:10.11779/CJGE202111014

岩体粗糙裂隙几何特征对其Forchheimer型渗流特性的影响 English Version

周新^{1, 2,},
盛建龙^{1, 2},
叶祖洋^{1, 2, ,},
罗旺^{1, 2},
黄诗冰^{1, 2},
程爱平^{1, 2}

1.
武汉科技大学资源与环境工程学院,湖北　武汉　430081
2.
冶金矿产资源高效利用与造块湖北省重点实验室,湖北　武汉430081

基金项目:

国家自然科学基金项目 42077243

国家自然科学基金项目 51709207

湖北省自然科学基金项目 2018CFB631

详细信息

作者简介:
周新(1996— ),男,硕士研究生,主要从事岩土渗流力学方面的研究。E-mail：zhouxin_wust@163.com

通讯作者:
叶祖洋, E-mail：yezuyang@wust.edu.cn

中图分类号: TU43
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出版历程
- 收稿日期: 2020-11-03
- 网络出版日期: 2022-12-01
- 刊出日期: 2021-10-31

Effects of geometrical feature on Forchheimer-flow behavior through rough-walled rock fractures

ZHOU Xin^{1, 2,},
SHENG Jian-long^{1, 2},
YE Zu-yang^{1, 2, ,},
LUO Wang^{1, 2},
HUANG Shi-bing^{1, 2},
CHENG Ai-ping^{1, 2}

1.
School of Resources and Environmental Engineering, Wuhan University of Science and Technology, Wuhan 430081, China
2.
Hubei Key Laboratory for Efficient Utilization and Agglomeration of Metallurgical Mineral Resources, Wuhan University of Science and Technology, Wuhan 430081, China

摘要

摘要: 为了研究岩体粗糙裂隙几何特征与其非线性渗流特性的相互关系,基于裂隙面分形特性提出了三维粗糙裂隙的几何结构表征模型,通过直接求解N-S（Navier-Stokes）方程,研究了不同开度均值、标准差和分形维数对岩体裂隙Forchheimer型渗流特性的影响规律,验证了Forchheimer方程描述流量与压力梯度非线性关系的有效性。研究结果表明：当流量较小时,随着裂隙开度均值减小、标准差增大,线性系数逐渐增大即水力开度逐渐减小,渗透能力下降,分形维数对其渗透能力的影响较小,并提出了水力开度与开度均值、标准差的经验关系式；当流量较大时,水流流态从线性流向非线性流转变,随着开度均值减小、标准差和分形维数的增大,非线性系数增大,临界雷诺数减小,测得Re_c范围为11.16～39.3。
- 岩体裂隙 /
- 几何特征 /
- 渗流特性 /
- 分形理论
Abstract: In order to study the relationship between the geometrical feature and the nonlinear flow properties of rough-walled rock fractures, a numerical model based on the fractal behavior is proposed to characterize the three-dimensional geometry of rough-walled fractures. By solving the N-S (Navier-Stokes) equation directly, the effects of mean aperture, standard deviation of aperture and different fractal dimensions on the Forchheimer flow characteristics of fractures are investigated. The Forchheimer equation is validated to describe the nonlinear relationship between the flow rate and the pressure gradient. The results show that with the lower flow rate, the linear coefficient increases and the hydraulic aperture decreases with the decreasing mean aperture and increasing standard deviation of the aperture, thus the empirical relation for the hydraulic aperture, the mean aperture and the standard deviation of aperture is put forward, while the effects of the fractal dimension almost can be ignored. On the contrary, with larger flow rate, for the flow pattern changing from linear to nonlinear flow, as the mean aperture decreases and the standard deviation of aperture and the fractal dimension increase, the nonlinear coefficient increases, and the critical Reynolds number decreases, with the range of Re_c being 11.16~39.3.
- rock fracture /
- geometrical feature /
- flow property /
- fractal theory

HTML全文

宋二祥老师对敝人《稳定安全系数计算公式中荷载与抗力错位影响探讨》^[1]（以下简称原文）提出了宝贵的指导及讨论意见,非常感谢！

宋文提出“边坡稳定安全系数的定义......是边坡土体所具有的抗剪强度与保持边坡刚好稳定所需要的强度之比,所以没有抗力与荷载错位的问题”。正如宋老师在其文献[4]^[2]所指出的,边坡稳定安全系数目前主要有两种定义方法：①为抗滑力矩与下滑力矩之比（可简化为抗力荷载比）,相应的稳定安全系数计算方法一般采用单一安全系数法；②即为上述的抗剪强度比,毕肖普、简布等将之定义为滑动面上的抗剪强度与实际产生的剪应力之比,相应的稳定安全系数计算方法一般采用强度（抗剪强度）折减法。宋文认为自然边坡等土工结构的稳定更适合采用第二种定义形式而不是第一种。原文没有讨论哪种安全系数定义更合理,讨论的是业界按第一种定义编写的安全系数计算公式有时并不完全符合第一种定义这种现象,现在讨论边坡稳定工程中按第二种定义会不会有抗力与荷载错位现象及强度折减法适用性问题。

仍以瑞典条分法为例,当滑弧中心点O位于边坡上方时,如图1所示,土条1～（m-1）的重力产生下滑力 $\sum_{i = 1}^{m - 1} G ti$ ,土条m～n的重力产生抗滑力 $\sum_{i = m}^{n} G_{ti}$ ,两者作用方向相反,稳定安全系数K计算公式按第一种定义、当抗力与荷载发生原文所示第1类错位现象时可写为

图 1 瑞典条分法边坡稳定分析简图

Figure 1. Sketch about slope stability analysis by Swedish Slicing Method

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$\sum_{i = 1}^{m - 1} G_{t i} - \sum_{i = m}^{n} G_{t i} = \frac{\sum_{i = 1}^{n} (G_{n i} \tan φ_{i} + c_{i} l_{i})}{K}$ 。

(1)

按笔者建议的抗力与荷载归位时可写为

$\sum_{i = 1}^{m - 1} G_{t i} = \sum_{i = 1}^{n} \frac{G_{n i} \tan φ_{i} + c_{i} l_{i}}{K} + \frac{\sum_{i = m}^{n} G_{t i}}{K}$ ,

(2)

按强度折减法可写为

$\sum_{i = 1}^{m - 1} G_{t i} = \sum_{i = 1}^{n} (G_{n i} \frac{\tan φ_{i}}{K} + \frac{c_{i}}{K} l_{i}) + \frac{\sum_{i = m}^{n} G_{t i}}{1}$ 。

(3)

将式（1）～（3）进行比较可知：

（1）式（3）与式（1）相同。式（3）在形式上用强度指标除以K,直观地表达了强度折减法,因为K只涉及到抗剪强度而没有涉及其它抗力或荷载,即 $\sum_{i = m}^{n} G_{ti}$ 等作为荷载还是抗力都不会影响到K计算结果,也就不存在错位与否,故从第二种定义角度来看,式（1）所表达的第1类错位问题不存在。

（2）原文中总结了抗力与荷载错位现象的3类基本形式及2类组合形式（原文中式（5）～（9））,其中第2类形式以图1为例可写为

$\sum_{i = 1}^{m - 1} G_{t i} = \sum_{i = m}^{n} G_{t i} = \sum_{i = 1}^{n} (G_{n i} \frac{\tan φ_{i}}{K} + \frac{c_{i}}{K} l_{i})$ 。

(4)

第2类错位现象把部分抗力 $\sum_{i = m}^{n} G_{t i}$ 作为荷载直接与荷载 $\sum_{i = 1}^{m - 1} G_{t i}$ 相加,可认为是对抗力与荷载概念的理解有误或故意为之（目的是在目标安全系数K维持不变时提高设计抗力、以使工程变得更安全）,这与采用第一种或第二种定义无关,故采用第二种定义亦不能解决；第3类错位现象对部分荷载也除以了安全系数,采用第二种定义时因为安全系数不涉及荷载,故也不存在这类错位现象；第4类错位现象是第1类与第3类的组合,从第二种定义角度来看也没有错位问题；第5种错位现象是第2类与第3类的组合,存在着与第2类同样的问题,具体不再赘述。总之,在对抗力与荷载的概念理解及应用无误时,采用第二种定义的安全系数计算公式,如宋文所言,确实不存在采用第一种定义时的抗力与荷载错位问题。

（3）和式（3）及式（1）相比,式（2）对 $\sum_{i = m}^{n} G_{t i}$ 项等所有抗力均除以了同一安全系数K,符合单一安全系数法规定的安全系数为抗力与荷载之比这个定义,故为单一安全系数法。从概率设计的角度,可认为式（3）中 $\sum_{i = m}^{n} G_{t i}$ 项的抗力分项系数为1,而抗剪强度的抗力分项系数为K,因两者不等且通常K>1,故式（3）表达的强度折减法具有了概率含义,并非传统意义上的单一安全系数法。

采用强度折减法时：

（1）因岩土重力的变异性小于抗剪强度的,式（3）中 $\sum_{i = m}^{n} G_{t i}$ 项对应的抗力分项系数取1,小于K看起来也合理,但仅取1,没有一点安全裕度是否合适?式（3）比式（2）计算得到的K值更高,对于工程而言更偏于不安全。

（2）如图1所示,按式（3）计算第二种定义的安全系数, $\sum_{i = 1}^{m - 1} G_{t i}$ = $\sum_{i = m}^{n} G_{t i}$ 时K值无穷大, $\sum_{i = 1}^{m - 1} G_{t i}$ < $\sum_{i = m}^{n} G_{t i}$ 时K值为负,但显然安全系数不能无穷大甚至为负。这种情况而非完全虚拟,例如土石坝拦挡淤泥工程, $\sum_{i = m}^{n} G_{t i}$ 项抗力由土石坝产生,土石坝足够稳定时就有可能发生类似计算结果；再如原文所示的锚固结构,锚杆提供的抗力足够大时也可能发生类似计算结果,也就是说当岩土体抗剪强度以外的因素产生了抗滑力且较大时,式（3）所示的强度折减法也可能会产生安全系数计算结果不合理问题,如同发生抗力与荷载错位现象一样；但如果按原文所建议的抗力与荷载归位后的计算公式则没有类似问题。故强度折减法可能更适用于仅由岩土体抗剪强度提供抗滑力时的自然边坡稳定计算,这也许就是现有技术标准中不太采用第二种定义形式对有支挡结构的人工边坡进行稳定验算的主要原因。

总结：①原文边坡稳定安全系数计算公式中的抗力与荷载错位现象是按第一种定义总结的,部分错位现象是对抗力与荷载概念的理解及应用不当造成的,采用第二种定义并不能解决因此而导致的安全系数不准确问题；②在对抗力与荷载概念的理解及应用无误时,采用第二种定义则不存在第一种定义时抗力与荷载错位问题,宋文观点是正确的；③有岩土体抗剪强度以外的因素提供抗滑力且较大时,强度折减法计算结果与发生了错位现象的抗力荷载比法一样,都存在着稳定安全系数计算结果不合理现象,对有支挡的人工边坡有时可能不太适用。

以上观点不妥之处,敬请宋老师及读者们继续批评指正。

图 1 二维SRAM示意图

Figure 1. Illustration of two-dimensional SRAM

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图 2 u=0.8 mm,δ=0.21 mm时裂隙模型形貌对比（红色代表大开度,蓝色代表小开度）

Figure 2. Comparison of morphologies of fracture models for u=0.8 mm, $δ$ =0.21 mm (red represents large aperture, blue represents small aperture)

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图 3 三维粗糙裂隙模型构建流程图

Figure 3. Modeling process for three-dimensional rough fractures

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图 4 流量与压力梯度关系

Figure 4. Relationship between flow rate and pressure gradient

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图 5 速度局部分布图（x=0~20 mm,y=20 mm）

Figure 5. Local distribution of velocity (x=0~20 mm, y=20 mm)

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图 6 流线局部分布图（y=20 mm, x=25~32 mm）

Figure 6. Local distribution of streamline (y=20 mm, x=25~32 mm)

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图 7 等效水力开度与几何特征的关系

Figure 7. Relationship between hydraulic aperture and geometric characteristics

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图 8 非达西惯性系数与几何特征的关系

Figure 8. Relationship between non-Darcy inertia coefficient and geometric characteristics

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图 9 临界雷诺数与几何特征的关系

Figure 9. Relationship between critical Reynolds number and geometric characteristics

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表 1 网格无关性分析

Table 1 Analysis of grid independence

单元尺寸/mm	网格数量/10⁴	求解时间	物理内存/GB	结果/Pa
0.250	66.58	8分59秒	4.68	110.92
0.230	80.94	12分28秒	4.86	103.77
0.200	113.52	19分22秒	5.82	95.112
0.180	143.68	34分23秒	6.09	90.053
0.175	152.47	59分10秒	6.34	89.12
0.170	164.69	90分46秒	6.30	87.509
0.160	189.04	165分22秒	6.56	86.55

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表 2 线性系数A和非线性系数B

Table 2 Values of linear coefficient A and nonlinear coefficient B

开度均值/mm	标准差/mm	D=2.1		D=2.2		D=2.3		D=2.4		D=2.5
开度均值/mm	标准差/mm	A	B	A	B	A	B	A	B	A	B
0.70	0.09	7.9881	0.8631	8.0081	0.9063	7.9850	0.9245	8.0201	1.0886	7.9810	1.1948
	0.12	8.1592	1.0319	8.1831	1.0395	8.3549	1.1386	8.1942	1.2533	8.3403	1.2722
	0.15	8.3122	1.0427	8.5025	1.0754	8.4001	1.1766	8.4931	1.3884	8.6016	1.5342
	0.18	8.7742	1.1996	8.7086	1.2634	8.7903	1.5410	8.8561	1.5730	9.0268	1.8872
	0.21	9.3183	1.3650	9.0407	1.9623	8.9406	2.3172	9.0750	2.5956	9.6689	2.6839
0.75	0.09	6.5077	0.6359	6.5082	0.6476	6.5181	0.6737	6.5301	0.6708	6.5595	0.7189
	0.12	6.6114	0.7013	6.5928	0.8232	6.6005	0.8222	6.6883	0.8853	6.6451	0.9675
	0.15	6.7716	0.7821	6.7630	0.8663	6.8099	0.9138	6.8346	1.0207	6.9403	1.0424
	0.18	7.0165	0.9409	7.0303	0.9758	7.0799	1.1197	7.1480	1.1841	7.2179	1.3674
	0.21	7.3457	1.0103	7.3609	1.1502	7.3260	1.3037	7.4201	1.4182	7.6003	1.6886
0.80	0.09	5.4034	0.4757	5.4458	0.4823	5.3659	0.5485	5.3438	0.5740	5.3992	0.5922
	0.12	5.4444	0.5018	5.4855	0.5276	5.4405	0.5953	5.4694	0.6206	5.5305	0.6438
	0.15	5.5328	0.6351	5.5351	0.6505	5.5571	0.6903	5.6169	0.7380	5.6962	0.7999
	0.18	5.7546	0.6519	5.6861	0.7334	5.7750	0.8330	5.7980	0.8662	5.9162	0.9422
	0.21	5.9371	0.7400	5.9073	0.7968	5.9457	0.9684	6.0859	1.0399	6.1693	1.2410
0.85	0.09	4.4743	0.3796	4.4783	0.3949	4.4466	0.4224	4.5273	0.4204	4.4770	0.4775
	0.12	4.5770	0.3917	4.6725	0.4285	4.6931	0.4406	4.6128	0.4738	4.5644	0.4814
	0.15	4.6240	0.4168	4.6540	0.4926	4.6321	0.5152	4.7123	0.5528	4.7514	0.5866
	0.18	4.7719	0.4638	4.7216	0.5441	4.7519	0.6109	4.8300	0.6132	4.9185	0.6987
	0.21	4.8836	0.5452	4.8808	0.5772	4.9173	0.6806	4.9958	0.7364	5.1056	0.9202
0.90	0.09	3.7816	0.2981	3.7857	0.2999	3.7941	0.2996	3.8015	0.3186	3.7873	0.3434
	0.12	3.8605	0.3122	3.8300	0.3352	3.8388	0.3376	3.8633	0.3639	3.9077	0.3656
	0.15	3.9244	0.3393	3.9207	0.3507	3.9756	0.3782	3.9765	0.4225	4.0178	0.4993
	0.18	3.9775	0.3738	3.9765	0.3965	4.0512	0.4002	4.0533	0.4836	4.1303	0.5552
	0.21	4.0892	0.4038	4.0805	0.4779	4.1200	0.5300	4.2182	0.5797	4.3172	0.6987
注：A=10⁸·kg/s¹/m⁵；B=10¹⁴·kg/m⁸。

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表 3 不同开度均值拟合参数对比

Table 3 Comparison of fitting parameters with different apertures

开度均值/mm	$η$	m	n	c	R²
0.70	1251	-2294	-78.76	197.5	0.8792
0.75	846.2	-1563	-60.45	164.2	0.9763
0.80	603.1	-1084	-34.12	102.4	0.9580
0.85	536.7	-1015	-32.87	100.9	0.9372
0.90	534.5	-1003	-29.43	97.9	0.9432

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表 4 岩体裂隙渗流研究中关于临界雷诺数的文献总结

Table 4 Summary of critical Reynolds number of flow in rock fractures in existing literatures

文献	材料	开度/ $μm$	Re_c	研究方法
Konzuk等^[24]	石灰岩张拉裂隙	均值=381	2.8~14.3	室内试验
Javadi 等^[20]	花岗岩张拉裂隙	—	0.001~25	室内试验
Zhang等^[3]	砂岩张拉裂隙	6.14~18.95	3.5~24.8	室内试验
Zimmerman等^[2]	三维裂隙	均值=149	1~10,20	数值模拟
Chen等^[25]	沉积岩和侵入岩	—	25~66	现场试验
Qian等^[26]	人工粗糙平行板模型	1000~2500	245~759	室内试验
Rong等^[27]	花岗岩张拉裂隙	—	1.5~13	室内试验
本研究	三维粗糙裂隙	700~900	11.16~39.3	数值模拟

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参考文献(27)

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