Effects of geometrical feature on Forchheimer-flow behavior through rough-walled rock fractures
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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Rec范围为11.16～39.3。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 Rec being 11.16~39.3.
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Keywords:
- rock fracture /
- geometrical feature /
- flow property /
- fractal theory
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0. 引言
西安地裂缝是20世纪50年代后期发现的,引起了人们的关注。1976年唐山地震后,西安地裂缝显著活动引起建筑物不均匀沉降而破坏。地质、工程界的研究,明确了西安地裂缝是地质构造运动而产生的认识[1-2]。20世纪80年代以来,进一步开展了西安地裂缝的原因、分布及活动规律的调查、监测研究,初步确定了其成因受南侧临潼-长安大断裂控制[2-5]。近年来,西安地裂缝活动加剧不仅受到周围地区地震作用影响,且与西安地区过量开采承压水产生相关。通过对西安地裂缝造成现有建(构)筑物破坏特征分析,地裂缝致灾机理是地裂缝上盘下沉,从而引起不均匀沉降、拉裂和错动位移,进而导致建筑物、地下洞室裂开和坍塌,路基、管道错动和断裂。同时,地表水沿地裂缝入渗和潜蚀,引起黄土湿陷不均匀沉降变形,对建(构)筑物造成二次破坏。西安地裂缝带来的危害性不仅体现在对于各类建(构)筑物生产建设的直接破坏,还会对工程场地土的性质与工程稳定性产生严重影响。从而严重地限制了建筑场地的使用,影响城市建设的合理布局[6-7, 9-10]。
对于这一特殊地质环境下的地铁隧道,特别是横穿地裂缝的隧道结构,衬砌结构沿纵向将承受比正常情况下大得多的附加应力和变形。附加作用表现为：①在地裂缝附近,上盘地层错动下沉可能脱空衬砌结构仰拱基底,或减小基底对衬砌结构的支撑作用,导致隧道衬砌发生剪切破坏；②地裂缝下盘对衬砌结构的约束作用,导致地裂缝附近衬砌结构承受拉应力；③上盘土体向下运动,引起该侧隧道结构变形,相当于弹性地基梁一端发生沉降的弯曲变形；④衬砌结构受地裂缝两侧土体的相运动,对隧道衬砌施加向下的作用力,引起隧道结构出现拉裂变形[6, 8]。地裂缝隧道设计了衬砌结构位移的预留空间[11]。由此可见,未采取措施的地裂缝隧道可能引起衬砌结构强度破坏和防水失效,不能保证正常运营。
为了揭示地裂缝隧道的力学特征和变形形态,需要分析地裂缝的产状和运动特征,以及地裂缝活动对衬砌结构的作用等。
1. 西安地裂缝分布与运动特征
西安地裂缝包括有显露地裂缝与隐伏地裂缝,是典型的黄土地区地质灾害现象。目前已发现14条地裂缝总延伸长度约103 km,分布在150 km2的黄土梁洼地貌范围内,单条地裂缝出露长度在2~12 km之间。其主要位于渭河断裂以南,临潼—长安断裂以北,向东西两侧(浐河至皂河)延伸。西安地裂缝由主裂缝、次生地裂缝和分枝地裂缝三部分组成,总体走向近似平行于临潼—长安断裂；总体倾向近似与临潼-长安断裂倾向相反,倾角约80°。西安地裂缝延伸长度可达数公里至十数公里,空间上呈不等间距平行排列,其分布范围内的地表由北向南呈逐渐升高的梁-洼地貌景观,如图1,2所示[5]。
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图 2 西安地裂缝南北走向地层剖面图Figure 2. Stratigraphic profile of north-south direction of ground fissures in Xi'an地裂缝大都发育在“黄土梁”地貌的南侧陡坡上这一特定地貌构造部位。其垂直位移单向累积,断距随深度的增大而增大。地裂缝发育剖面如图3所示。
2. 西安地铁隧道穿越地裂缝的设计
2.1 地裂缝最大竖向预估位移量
西安地裂缝的发展经历了从发生阶段发展至剧烈活动到成熟阶段,随后缓慢变形直至稳定这一过程。地裂缝的最大位移估算如表1所示。
表 1 西安地裂缝最大垂直预估位移量[8]Table 1. Maximum predicted vertical displacements of ground fissures(mm) 地裂缝编号 A(预估极限值) A×1.5(设计值) 地裂缝编号 A(预估极限值) A×1.5(设计值) f2 200 300 f8 100 150 f3 150 225 f10 150 225 f5 300 450 f11 300 450 f6/f6' 300(200) 450(300) f9/f9' 300(250) 450(375) f7 300 450 f12 100 150 2.2 地裂缝区间隧道结构设计
地铁隧道穿越地裂缝不可避免,应遵从以下原则：以结构措施适应变形为主前提下,在地裂缝处理段需须对结构进行分段预留必要的变形空间适应地裂缝的变形；加强断面结构抵抗变形对结构的破坏；变形缝处在结构发生变形时应当能够保持防水的效果。地裂缝活动主变形区范围根据地裂缝活动引起附近地层的活动变形范围确定为：上盘0~6 m,下盘0~4 m；微变形区上盘6~20 m,下盘4~15 m。上盘变形影响范围大于下盘。隧道衬砌结构为了适应地裂缝活动的变形应在地裂缝处应设置变形缝[6, 8]。如4, 5所示。
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图 5 地裂缝隧道纵坡调整示意图Figure 5. Schematic diagram of longitudinal slope adjustment of tunnel with ground fissures3. 地裂缝隧道的数值分析
3.1 地裂缝隧道衬砌结构及地层条件的模型
西安地铁二号线地裂缝区段隧道一般采用马蹄形隧道断面,以正交穿越地裂缝的地铁2号线为分析对象,地裂缝区间隧道采用CRD开挖方法,最大断面净空宽为8.3 m,高为8.45 m；初衬为C25喷射混凝土,厚30 cm；二衬为C30模注钢筋混凝土,厚50 cm。衬砌结构沿纵向每10 m或15 m预设10 cm宽的变形缝,充填密封防渗材料封闭变形缝。以便衬砌结构适应地裂缝上下盘土体相对运动,避免衬砌结构附加拉应力,防止基底出现脱空现象。并且在衬砌结构端部局部加厚以便适应可能出现的应力集中现象。
西安地铁地裂缝隧道通过FLAC-3D有限差分软件建立了计算模型。计算模型隧道埋深为10 m,横断面内水平向宽度为80 m,竖向高度为60 m,轴向长度为200 m。模拟地层埋深0~7.5 m为晚更新世风积黄土,埋深7.5~25.5 m为晚更新世粉质黏土及古土壤层,埋深25.5~30.5 m为中更新世黄土及古土壤层,以及埋深30.5 m以下为中更新世粉质黏土。地裂缝采用库仑摩擦接触面模拟；应力应变关系采用莫尔-库仑屈服条件的弹塑性模型描述；地层及二次衬砌结构采用实体单元模拟；初期衬砌结构采用壳单元模拟。
在地裂缝活动导致自由场地地面不均匀沉降如图6所示条件下,衬砌结构错动位移如图7,8和图9所示。衬砌结构变形缝最大挤压变形为3.6 cm,位于地裂缝处变形缝两侧衬砌拱底；最大张拉变形为5.7 cm,位于上盘内邻近地裂缝衬砌结构变形缝的拱底处。衬砌结构变形缝预留10 cm,满足最大挤压变形的要求。衬砌结构大、小主应力如图10,11所示。大主应力受拉的集中区域主要分布在衬砌结构内侧腰部,最大值为2.02 MPa；小主应力受拉的区域也分布于此。地裂缝两侧衬砌结构内侧腰部受拉,应进行加强处理。隧道衬砌结构采用C30混凝土,其抗压强度为30 MPa,抗拉强度为2.01 MPa,添加钢纤维可满足受拉的强度要求。
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图 6 地裂缝自由场地地表沉降Figure 6. Surface settlements of free site under acting ground fissures![]()
图 7 隧道衬砌拱顶纵轴线沉降分布曲线Figure 7. Distribution of settlement at arch of lining structure and on surface![]()
图 8 隧道衬砌结构顶、底水平位移分布曲线Figure 8. Distribution of horizontal displacement at vault and arch bottom of lining structure along axial direction of tunnel![]()
图 9 地裂缝附近隧道衬砌结构变位示意图Figure 9. Differential settlements of segmented lining near ground fissures4. 地裂缝隧道结构与轨道工程措施
在地裂缝区间段隧道运行100 a后,地裂缝会导致隧道下沉500 mm。为保证隧道具有列车运行的空间,在隧道截面扩构段,二衬结构加大截面厚度及增加配筋,提高纵向分布筋直径及间距的方法抵抗扭转、剪切对结构的影响。地裂缝隧道段的初期支护和内衬之间增设沥青混凝土复合衬砌,在初期支护和二次衬砌之间形成夹层,利用沥青混凝土特有的延展性、流变性,密封衬砌结构变形缝。随着地裂缝活动,沥青混凝土在围岩压力作用下沿侧向产生挤出变形,从围岩压力大的部位向围岩压力小的部位流动,使得围岩压力趋于均匀化。当地裂缝活动导致衬砌结构错动变形时,沥青混凝土易产生流变剪切变形,适应衬砌结构变形缝的变化,可发挥其防渗能力。沥青混凝土变形缝构造与变形模型试验结果如图12,13所示。
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图 12 加筋沥青混凝土防渗变形缝结构Figure 12. Scheme of reinforced asphalt concrete and anti-seepage deformation joint structure5. 结论
(1)西安临潼—长安断裂带是地裂缝产生的构造活动,过量开采承压水产生不均匀沉降是地裂缝发展的附加作用。准确预测地裂缝的位移量,是地裂缝隧道结构设计的重要依据。
(2)西安地铁二号线地裂缝影响段65 m设变形缝,在地裂缝影响范围内,主变形段通常占地裂缝80%~90%的总垂直位移量,是主要的设防区,按10 m进行隧道分段。微变形段垂直位移量仅占10%~20%的总位移量,按10~15 m进行隧道分段。
(3)地裂缝隧道结构应采取衬砌结构适应地裂缝变形的原则。在地裂缝处理段需对结构进行分段,预留必要的变形空间作为变形缝以适应地裂缝的变形。加强变形缝断面的结构,以便满足抵抗变形对结构破坏的要求。
(4)衬砌结构内侧拱腰分布拉应力集中区,需提高衬砌结构混凝土的抗拉强度。采取加筋沥青混凝土复合衬砌及沥青玛蹄脂填充变形缝处理,可改善衬砌结构受力条件,密封衬砌结构变形缝,保持防水的效果。
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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,
=0.21 mm (red represents large aperture, blue represents small aperture) ![]()
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
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 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 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
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