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 哈尔滨工业大学学报  2019, Vol. 51 Issue (8): 159-166  DOI: 10.11918/j.issn.0367-6234.201807209 0

### 引用本文

LI Mengtian, ZHANG Xiao, LI Shucai, ZHANG Qingsong, ZUO Jinxin, LAN Xiongdong. Grouting lifting numerical methods based on numerical simulation and model experiment[J]. Journal of Harbin Institute of Technology, 2019, 51(8): 159-166. DOI: 10.11918/j.issn.0367-6234.201807209.

### 文章历史

1. 山东大学 岩土与结构工程研究中心，济南 250061;
2. 中铁一院集团山东建筑设计院有限公司，山东 青岛 266031

Grouting lifting numerical methods based on numerical simulation and model experiment
LI Mengtian1, ZHANG Xiao1, LI Shucai1, ZHANG Qingsong1, ZUO Jinxin2, LAN Xiongdong1
1. Geotechnical and Structural Engineering Research Center, Shandong University, Jinan 250061, China;
2. Shandong Institute of Architectural Design Co., Ltd., of China Railway First Design Institute Group, Qingdao 266031, Shandong, China
Abstract: To solve the problem of surface lifting caused by grouting in urban shallow underground projects, grouting lift calculation method and simplified algorithm were established based on the theory of stochastic medium, and effects of compaction grouting and fracture grouting on surface uplift were studied. Through PFC simulation and laboratory model experiments, the suffering area of the surface uplift by ball compaction grouting and horizontal fracture grouting were studied, and the accuracy of the grouting lift calculation method of surface uplift displacement was verified. Effects of grouting depth, soil dry density, and soil water content on surface uplift during grouting were investigated by introducing the grouting uplift coefficient γ. Results show that the simplified method of theoretical prediction of surface uplift displacement was accurate when the depth of grouting was greater than 2 m, and it was not applicable when the depth of grouting was less than 2 m. The surface uplift caused by the two grouting methods was consistent with the results of the numerical simulation and the mathematical model calculation. Single hole grouting changed the displacement field of the overlying inverted cone soil. The maximum surface displacement caused by compaction grouting was large, so was the influence range of fracture grouting on the surface. Factors such as grouting depth, soil dry density, and soil water content all influenced the maximum surface displacement caused by grouting, among which grouting depth was the most important factor.
Keywords: grouting     surface uplift     mathematical model     numerical simulation     model experiment

1 注浆抬升计算方法

1.1 压密注浆造成地面抬升位移的计算方法

 图 1 压密注浆单元模型 Fig. 1 Compaction grouting unit model

 $W _ {\rm e } = \frac { 1 } { r ^ { 2 } ( z ) } \exp \left[ \frac { - \pi \left( x ^ { 2 } + y ^ { 2 } \right) } { r ^ { 2 } ( z ) } \right] \mathrm { d } \xi \mathrm { d } \zeta \mathrm { d } \eta.$ (1)

 $\begin{array} { c } { W ( x , y , 0 ) = \int _ { a _ { 1 } } ^ { b _ { 1 } } \int _ { c _ { 1 } } ^ { d _ { 1 } } \int _ { e _ { 1 } } ^ { f _ { 1 } } \frac { \tan ^ { 2 } \beta } { \eta ^ { 2 } } \exp \left\{ \frac { - \pi \tan ^ { 2 } \beta } { \eta ^ { 2 } } \right.} \cdot \\ { \left[ ( x - \xi ) ^ { 2 } + ( y - \zeta ) ^ { 2 } \right] \} \mathrm { d } \xi \mathrm { d } \zeta \mathrm { d } \eta - \int _ { a _ { 2 } } ^ { b _ { 2 } } \int _ { c _ { 2 } } ^ { d _ { 2 } } \int _ { e _ { 2 } } ^ { f _ { 2 } } \frac { \tan ^ { 2 } \beta } { \eta ^ { 2 } } } \cdot \\ { \exp \left\{ \frac { - \pi \tan ^ { 2 } \beta } { \eta ^ { 2 } } \left[ ( x - \xi ) ^ { 2 } + ( y - \zeta ) ^ { 2 } \right] \right\} \mathrm { d } \xi \mathrm { d } \zeta \mathrm { d } \eta } .\end{array}$ (2)

 ${ c _ { 1 } = - \sqrt { ( A + \Delta A ) ^ { 2 } - ( H - \eta ) ^ { 2 } } } ,$
 ${ d _ { 1 } = \sqrt { ( A + \Delta A ) ^ { 2 } - ( H - \eta ) ^ { 2 } } } ,$
 ${ e _ { 1 } = - \sqrt { ( A + \Delta A ) ^ { 2 } - ( H - \eta ) ^ { 2 } - \zeta ^ { 2 } } },$
 $f_{1}=\sqrt{(A+\Delta A)^{2}-(H-\eta)^{2}-\zeta^{2}},$
 $a_{2}=H-A ; b_{2}=H+A, c_{2}=-\sqrt{A^{2}-(H-\eta)^{2}},$
 $d_{2}=\sqrt{A^{2}-(H-\eta)^{2}}, e_{2}=-\sqrt{A^{2}-(H-\eta)^{2}-\zeta^{2}},$
 $f_{2}=\sqrt{A^{2}-(H-\eta)^{2}-\zeta^{2}}.$

 $\begin{array}{l}{W(x, y, 0)=\left(\frac{4}{3} \pi(A+\Delta A)^{3}-\frac{4}{3} \pi A^{3}\right) \times} \\ {\frac{\tan ^{2} \beta}{H^{2}} \exp \left[\frac{-\pi \tan ^{2} \beta\left(x^{2}+y^{2}\right)}{H^{2}}\right]=} \\ {\quad \frac{4 \pi \tan ^{2} \beta \Delta A\left(3 A^{2}+3 A \Delta A+\Delta A^{2}\right)}{3 H^{2}} \times} \\ {\quad \exp \left[\frac{-\pi \tan ^{2} \beta\left(x^{2}+y^{2}\right)}{H^{2}}\right]}.\end{array}$ (3)

 $\begin{array}{c}{W_{\max }=W(0, 0, 0)=} \\ {\quad \frac{4 \pi \tan ^{2} \beta \Delta A\left(3 A^{2}+3 A \Delta A+\Delta A^{2}\right)}{3 H^{2}}}.\end{array}$ (4)
1.2 劈裂注浆造成地面抬升位移的计算方法

 图 2 劈裂注劈裂浆单元模型 Fig. 2 Fracture grouting unit model

 $b=\sqrt[4]{\frac{24 G q \mu}{\pi} \ln \left(\frac{r_{\max }}{r}\right)}\left(r_{0} 式中：G为软如地层介质参数；q为单位时间的注浆量；μ为水泥浆的黏度；rmax为浆液扩散的最大半径，其值与浆液压力有关，即 $ r_{\mathrm{max}}=r_{0} \exp \left[\frac{\pi G^{3}\left(p_{\mathrm{s}}-\sigma_{3}\right)^{4}}{H^{2}}\right]. $(6) 式中：ps为浆液压力；σ3为最小主应力. 水平辐射圆饼浆脉所引起的地面抬升量为土体劈裂所导致的地面抬升量和浆液水平扩散之前半径为r0浆球所造成的地面抬升量之差，表达式为 $ W(x, y, 0)=\int_{a_{3}}^{b_{3}} \int_{c_{3}}^{d_{3}} \int_{e_{3}}^{f_{3}} \frac{\tan ^{2} \beta}{\eta^{2}} \exp \left\{\frac{-\pi \tan ^{2} \beta}{\eta^{2}}\right. \cdot \\ \left[(x-\xi)^{2}+(y-\zeta)^{2}\right] \}_{\mathrm{d} \xi \mathrm{d} \zeta \mathrm{d} \eta}-\\\int_{a_{4}}^{b_{4}} \int_{c_{4}}^{d_{4}} \int_{e_{4}}^{f_{4}} \frac{\tan ^{2} \beta}{\eta^{2}} \exp \left\{\frac{-\pi \tan ^{2} \beta}{\eta^{2}}\right.\cdot \\\left[(x-\xi)^{2}+(y-\zeta)^{2}\right] \}_{\mathrm{d} \xi \mathrm{d} \zeta \mathrm{d} \eta}. $(7) 取值范围如下:a3=H-bmax, b3=H+bmax, $ c_{3}=-\sqrt{r_{\max }^{2}\left[1-\frac{(\eta-H)^{2}}{b_{\max }^{2}}\right]}, $$ d_{3}=\sqrt{r_{\max }^{2}\left[1-\frac{(\eta-H)^{2}}{b_{\max }^{2}}\right]}, $$ e_{3}=-\sqrt{r_{\max }^{2}\left[1-\frac{(\eta-H)^{2}}{b_{\max }^{2}}\right]-\zeta^{2}}, $$ f_{3}=\sqrt{r_{\max }^{2}\left[1-\frac{(\eta-H)^{2}}{b_{\max }^{2}}\right]-\zeta^{2}}, $$ a_{4}=H-r_{0}, b_{4}=H+r_{0}, c_{4}=-\sqrt{r_{0}^{2}-(H-\eta)^{2}}, $$ d_{4}=\sqrt{r_{0}^{2}-(H-\eta)^{2}}, $$ e_{4}=-\sqrt{r_{0}^{2}-(H-\eta)^{2}-\zeta^{2}}, $$ f_{4}=\sqrt{r_{0}^{2}-(H-\eta)^{2}-\zeta^{2}}. $为了省略式(7)中复杂的积分计算，参考压密注浆抬升位移量计算方法的简化，将加固区视作一个微元，把劈裂注浆造成的地表抬升位移等效为加固区的整体体积和微元体造成的抬升位移量的乘积.圆饼可设为长轴为rmax、短轴为bmax的椭圆绕短轴旋转一周形成的椭球体，简化后的地表抬升公式为 $ \begin{array}{c}{W(x, y, 0)=\left(\frac{4}{3} \pi r_{\max }^{2} b_{\max }-\frac{4}{3} \pi r_{0}^{3}\right) ) \times \frac{\tan ^{2} \beta}{H^{2}}} \cdot \\ {\exp \left[\frac{-\pi \tan ^{2} \beta\left(x^{2}+y^{2}\right)}{H^{2}}\right]=\frac{4 \pi \tan ^{2} \beta\left(r_{\max }^{2} b_{\max }-r_{0}^{3}\right)}{3 H^{2}}} \cdot \end{array}\\ \exp \left[\frac{-\pi \tan ^{2} \beta\left(x^{2}+y^{2}\right)}{H^{2}}\right]. $(8) 由式(8)可知，在地表距浆脉圆饼地表投影的中心点距离越近，地面抬升位移量就越大，且当D相等时$\left(D = \sqrt { \left(x ^ { 2 } + y ^ { 2 } \right) } \right)$为圆饼浆脉在地表投影的任一点距注浆点中心的距离)，地面抬升位移量也相同.地面抬升位移量最大值在D=0位置，最大抬升位移量的表达式为 $ W_{\max }=W(0, 0, 0)=\frac{4 \pi \tan ^{2} \beta\left(r_{\max }^{2} b_{\max }-r_{0}^{3}\right)}{3 H^{2}}. $(9) 2 模型实验 2.1 模型实验系统 为了验证压密注浆和劈裂注浆引起地面抬升位移量的计算公式及其简化公式，设计了室内球形压密注浆和劈裂注浆模型实验.实验被注介质为土，浆液为两种水灰比的水泥单液浆，采集注浆过程中的地表位移量和土压力，研究注浆参数对注浆抬升位移量的影响及注浆抬升过程，室内注浆模型实验系统如图 3所示.  图 3 注浆模型实验系统 Fig. 3 Experiment system for the grouting model 实验模型箱由亚克力板箱和钢结构外框构成，注浆设备为MY-JD3242型注浆泵及其控制系统.数据监测系统包括KTR2-50型位移传感器、土压传感器、渗压传感器、CCD相机等.9个位移传感器按照3×3方阵布置，每隔100 mm一个，位移、土压、渗压数据由XL2101G型静态电阻应变仪接收.辅助装置包括水平尺、偏光激光仪，用来保证土体表面的初始水平. 2.2 实验材料参数标定 注浆加固抬升实验用土取自某施工现场，对所取土样的密度、无侧限抗压强度、抗剪强度、液塑限进行实验测定，得到表 1中实验用土的参数.本实验中所用水泥为济南某水泥厂生产的P.O.32.5普通硅酸盐水泥，其性能指标见表 2. 表 1 原状土基本物理力学参数 Tab. 1 Basic physical and mechanical parameters of undisturbed soil 表 2 试验用注浆材料主要性能指标 Tab. 2 Performance of the grouting material used in experiment 2.3 注浆模型实验 2.3.1 压密注浆模型实验 在压密注浆模型试验中，为了实现球形压密注浆，注浆出口处绑定一个球形气囊，在注浆过程中浆液填满气囊使其呈球形膨胀.选取实验用土的干密度、注浆口的埋深及实验用土的含水率为压密注浆模型实验的考察因素.实验用土干密度ρs采用1.07，1.27及1.47 g/cm33种，注浆口的埋深H分别设置150，200及250 mm 3种，实验用土的含水率w采用12%，15%及18% 3种，考虑3种影响因素共实验27组. 2.3.2 劈裂注浆模型实验 土体填充为水平方向逐层压实填充，使水平应力为最大主应力，从而得到水平劈裂浆脉.劈裂注浆模型实验同样选取实验用土的干密度、注浆口的埋深及实验用土的含水率为劈裂注浆模型实验的考察因素，条件取值与压密实验方案一致，考虑3种影响因素共实验27组. 2.4 实验结果 2.4.1 注浆实验抬升现象 根据注浆实验抬升现象的记录可知，在注浆量不断增大的过程中，土体表面发生逐渐抬升现象，注浆管口在土体表面的投影处抬升位移量最大，距离投影点的位置越远抬升位移量越小.图 4(a)中线条标出了压密注浆影响的土层变形范围，图 4(b)中线条标出了劈裂注浆的土层变形范围，可以发现，压密注浆导致的地表最大抬升量要比劈裂注浆大，劈裂注浆地表土层抬升范围要比压密注浆大.  图 4 注浆引起的地表抬升 Fig. 4 Surface uplift caused by grouting 2.4.2 注浆抬升影响因素分析 在本文实验中，影响地表抬升最大位移Wmax的因素主要有Hρsw，为了研究各因素与Wmax的相关性，引入抬升系数γ1.在压密注浆中令Wmax=γ1RmaxRmax为球型浆泡的最大半径，其中0＜γ1＜1，表 3为压密注浆过程统计参数. 表 3 压密注浆实验参数及抬升系数γ1 Tab. 3 Compaction grouting experimental parameters and uplift coefficient γ 采用Minitab 16统计软件对γ1Hρsw，进行回归拟合，可得其关系为 $ r_{1}=0.52-0.00713 \rho_{\rm s}+0.147 H-0.00956 w. $(10) 经过计算R2=97.2%，拟合公式的标准化残差分布服从正态分布，上述公式拟合较好，有较好的统计意义. 对于劈裂注浆地表抬升实验，令γ2为注浆抬升土层引起的地表抬升最大位移Wmax与水平劈裂圆饼浆脉最大厚度bmax的比值，即Wmax=γ2bmaxγ2的取之范围为0＜γ2＜1，表 4为劈裂注浆过程统计参数.可得拟合公式为 $ \gamma_{2}=0.748-0.00839 \rho_{\rm s}+0.165 H-0.0123 w. \$ (11)

3 数值模拟 3.1 数值模拟参数设置

3.2 数值模拟结果 3.2.1 压密注浆抬升模拟结果

 图 5 压密注浆土体位移场数值模拟 Fig. 5 Numerical simulation of soil displacement field of compaction grouting
 图 6 注浆深度在250 mm时地表位移随时间的变化 Fig. 6 Surface displacement over time when the grouting depth is 250 mm
3.2.2 劈裂注浆抬升模拟结果

 图 7 劈裂注浆土体位移场数值模拟 Fig. 7 Numerical simulation of soil displacement field of fracture grouting
3.2.3 结果分析

 图 8 注浆抬升距中心点的对阵分布 Fig. 8 Contrast distribution of grouting lift from center point
3.3 简化计算方法可行性验证

 图 9 数值计算与简化计算结果对比 Fig. 9 Comparison between numerical calculation and simplified calculation

4 结论

1) 分析了注浆导致地表抬升的力学机理，并得到了压密注浆、劈裂注浆导致地表抬升位移量的计算方法及其简化算法.

2) 两种注浆方式引起的地表抬升与数学模型计算结果、数值模拟结果吻合性较好；通过对注浆抬升系数γ的回归分析，得到抬升系数的表达式，注浆深度、土体干密度及含水率等因素均影响注浆引起的地表最大抬升量，其中注浆深度影响最大.

3) 数值模拟验证了压密注浆、劈裂注浆导致的地表抬升位移量的计算方法及其简化算法的准确性，单孔注浆使上覆倒锥形土体位移场发生变化，两种注浆方式对比来看，压密注浆地表最大抬升较大，劈裂注浆影响范围较大.

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