工程科学与技术   2020, Vol. 52 Issue (5): 125-135

1. 山东大学 岩土与结构工程研究中心，山东 济南 250061;
2. 纽卡斯尔大学 工程与建筑环境学院，澳大利亚 卡拉汉 NSW2308;
3. 山东建筑大学交通工程学院，山东 济南 250101;
4. 中国石油大学（华东） 储运与建筑工程学院，山东 青岛 266580;
5. 青岛地铁集团，山东 青岛 266000

Study on the Filtration Mechanism in Permeation Grouting Using the Particle Deposition Probability Model
ZHU Guangxuan1,2, ZHANG Qingsong1, FENG Xiao3, LIU Rentai1, ZHANG Lianzhen1,4, LIU Shiqun5, ZHANG Jianwei5
1. Geotechnical and Structural Eng. Research Center, Shandong Univ., Jinan 250061, China;
2. Faculty of Eng. and Built Environment, The Univ. of Newcastle, Callaghan NSW2308, Australia;
3. School of Transportation Eng., Shandong Jianzhu Univ., Jinan 250101, China;
4. College of Pipeline and Civil Eng., China Univ. of Petroleum, Qingdao 266580, China;
5. Qingdao Subway Group Co. Ltd., Qingdao 266000, China
Abstract: The filtration effect has an important effect on the diffusion of suspension permeation grouting. Filtration coefficient is the key parameter of permeation diffusion. Most of the existing studies assumed the coefficient as a constant, which had great limitations. Based on the mass conservation of each component, linear filtration law and homogeneous capillary model, the permeation diffusion equation of cement grout was established, using the particle deposition probability model to describe the filtered process of cement particles in porous media. Additionally, the determination method of model parameters was proposed. The 3D model test system of permeation grouting was developed. Using the model test system, the seepage grouting of sand with constant flow rate can be realized, and the multi-information in the grouting process can be collected. The model test of radial diffusion permeation grouting was carried out. The temporal and spatial variations of porosity and grout pressure were analyzed, combined with the theoretical model calculation results. The influences of grouting rate and water cement ratio on filtration mechanism were discussed, and the accuracy of theoretical calculation model was also verified. The results showed that the smaller the grouting rate, the bigger the porosity attenuation at the grouting hole, and the more severe the porosity attenuation along the slurry diffusion direction. The smaller the grout water-cement ratio, the faster the porosity attenuation and the more significant the filtration effect. Compared with the experimental results, the maximum calculation error of porosity and grouting pressure were less than 12% and 14.2%, respectively. The model can describe the radial diffusion process of cement slurry well.
Key words: permeation grouting    deposition probability model    filtration effect    radial diffusion    cement grout

1 渗透注浆扩散理论模型

1.1 基本假设

1）浆液、水均为不可压缩、均质各向同性的流体；

2）被注介质为均质和各向同性；

3）本文以工程中常用的水灰比大于1.0的水泥浆液为研究对象，该范围内的浆液可以视为宾汉流体[1]

4）注浆速率恒定，浆液流动为层流运动；

5）浆液在被注介质中扩散为均匀渗透扩散。

1.2 滤过效应理论描述

 图1 注浆区域各部分组成示意图 Fig. 1 Composition of grouted soil

 $n = {n_{\rm{c}}} + {n_{\rm{w}}}$ (1)

 $\frac{{\partial {\rho _{\rm{w}}}{n_{\rm{w}}}}}{{\partial t}} + {{{v}}_{\rm{w}}}\nabla ({\rho _{\rm{w}}}{n_{\rm{w}}}) = 0$ (2)

 $\frac{{\partial {\rho _{\rm{c}}}{n_{\rm{c}}}}}{{\partial t}} + {{{v}}_{\rm{c}}}\nabla ({\rho _{\rm{c}}}{n_{\rm{c}}}) = \eta$ (3)

 ${{{v}}_{\rm{w}}} = {{{v}}_{\rm{c}}} = {{v}}$ (4)

 $\frac{{\partial {\rho _{\rm{s}}}{n_{\rm{s}}}}}{{\partial t}} = - \eta$ (5)

 $\frac{{\partial {\rho _{\rm{c}}}{n_{\rm{s}}}}}{{\partial t}} = - \eta$ (6)

 $\frac{{\partial n}}{{\partial t}} = \frac{\eta }{{{\rho _{\rm c}}}}$ (7)

 $\frac{{\partial n}}{{\partial t}} + {{v}}\nabla (n) = \frac{\eta }{{{\rho _{\rm{c}}}}}$ (8)

 ${{v}}\nabla (n) = 0$ (9)

 $\frac{{\partial n}}{{\partial t}} = \frac{\eta }{{{\rho _{\rm{c}}}}} = - \lambda \delta$ (10)

 $\delta = \frac{{{n_{\rm{c}}}}}{n}$ (11)

 $n\frac{{\partial \delta }}{{\partial t}} + {{v}}n\nabla (\delta ) = \lambda \delta (\delta - 1)$ (12)

 $v(r,t)n(r,t) = {v_0}\frac{{{r_0}}}{r}$ (13)

 ${\;\;\;\;\;\;\;\;\;\;\;\;n}\frac{{\partial \delta }}{{\partial t}} + {v_0} \cdot \frac{{{r_0}}}{r}\frac{{\partial \delta }}{{\partial r}} = - \lambda \delta (\delta - 1)$ (14)

 $\delta ({r_0},t) = {\delta _0}$ (15)

 $n(r \ge {r_0}) = {n_0}$ (16)
 $\delta (r \ge {r_0}) = 0$ (17)

 $\theta = {\theta _0} \cdot \exp \bigg( - \frac{v}{{{v_{{\rm{cr}}}}}}\bigg)$ (18)

Reddi等[29]针对土滤问题，考虑多孔介质颗粒级配分布，给出了滤过系数 $\lambda$ 与颗粒沉积概率 $\theta$ 之间的关系如下：

 \begin{aligned}[b] \lambda =& \frac{v}{{{a^*}{{\rm{e}}^{2({b^2} + m)}}}} \cdot \\ &\left[ {4{{\left( {a\theta } \right)}^2} - 4{{\left( {a\theta } \right)}^3}{{\rm{e}}^{({b^2} - 2m)/2}} + {{\left( {a\theta } \right)}^4}{{\rm{e}}^{2({b^2} - 2m)}}} \right] \end{aligned} (19)

1.3 浆液扩散运动方程

 ${\;\;\;\;\;\;\;\;\;\;\;\;v} = \frac{{{b^2}}}{{8\mu \left( \delta \right)}}\left( { - \frac{{{\rm{d}}p}}{{{\rm{d}}r}} - \frac{{8{\tau _0}\left( \delta \right)}}{{3{b_0}}}} \right)$ (20)

 $k = \frac{{nb_{\rm{0}}^{\rm{2}}}}{8}$ (21)

 $q = 2\text{π}r{l_0}vn$ (22)

 ${\;\;\;\;\;\;\;\;\;\;\;\frac{{{\rm{d}}p}}{{{\rm{d}}r}}} = - \frac{{\mu \left( \delta \right){v_0}{r_0}}}{{krn}} - \frac{{2{\tau _0}\left( \delta \right)}}{3}\sqrt {\frac{{2n}}{k}} ,r \le {r_t}$ (23)

 $p{|_{r = {r_t}}} = {p_{\rm{w}}}$ (24)

 $k = \frac{{{k_0}}}{{1 - \beta \left( {n - {n_0}} \right)}}$ (25)

2 模型参数试验确定方法

2.1 1维渗透注浆试验 2.1.1 试验材料

 $\delta = \frac{1}{{1 + \dfrac{{{\rho _{\rm{c}}}}}{{{\rho _{\rm{w}}}}} \cdot {R_{{\rm{WC}}}}}}$ (26)

 ${\tau _0} = 4.57 \times {10^{ - 6}} \times {{\rm{e}}^{47.84\delta }}$ (27)
 ${\;\;\;\;\;\;\;\;\;\;\;\mu} = {\mu _{\rm{w}}} + 0.127\;9\delta - 0.263\;1{\delta ^2}$ (28)

2.1.2 试验装置及方案

1维渗透注浆试验装置有两节钢套筒组成，钢套筒内径10 cm，分别在距离浆液入口30 cm和60 cm处设置浆液出流口及开关。浆液由恒流量注浆泵从左侧进浆口泵入，右侧出流口流出。1维试验装置示意图如图2所示。

 图2 1维渗透注浆试验装置示意图 Fig. 2 Schematic diagram of one-dimensional permeation grouting test device

2.1.3 试验结果

 图3 浆液黏度随时间变化曲线 Fig. 3 Change of grout viscosity with time

 图4 孔隙率随时间变化曲线 Fig. 4 Change of porosity with time

 图5 渗透率随时间变化曲线 Fig. 5 Change of permeability with time

2.2 参数 ${a{{ \theta} _{\bf 0}}}$ ${{ v}_{\bf{cr}}}$ $\beta$ 的确定方法 2.2.1 参数 $a{\theta _0}$ 、临界速度 ${v_{\rm cr}}$ 求解

 图6 滤过系数拟合曲线 Fig. 6 Fitting of inversion data

 \begin{aligned}[b] &\lambda = \frac{v}{{{a^*}{{\rm{e}}^{2({b^2} + m)}}}}{\left( {a{\theta _0}} \right)^2}{\rm{\cdot}} \\ &\;\;\;\;\;\;\left[ {4{{\rm{e}}^{\big( - \frac{{2v}}{{{v_{{\rm{cr}}}}}}\big)}} - 4\left( {a{\theta _0}} \right){{\rm{e}}^{\big( - \frac{{3v}}{{{v_{{\rm{cr}}}}}}\big) + ({b^2} - 2m)/2}} + {{\left( {a{\theta _0}} \right)}^2}{{\rm{e}}^{\big( - \frac{{4v}}{{{v_{{\rm{cr}}}}}}\big) + 2({b^2} - 2m)}}} \right] \\ \end{aligned} (29)

2.2.2 参数 $\;\beta$ 求解

 图7 渗透率–孔隙率变化关系 Fig. 7 Relationship of permeability and porosity

3 模型试验系统研发

3.1 模型试验系统构成

 1.伺服控制系统；2.油压管路；3.钢质活塞；4.液压站；5.储浆桶；6.液压千斤顶；7.高强支架；8.输浆管路；9.注浆孔；　　10.引线孔；11.渗压传感器；12.被注砂样；13.试验腔；14.出浆口；15.废液收集容器；16.电阻式应变箱；17.电脑。 图8 3维渗透注浆试验系统示意图 Fig. 8 Schematic diagram of 3D permeation grouting test system

3.2 供浆单元构成

 图9 压浆装置 Fig. 9 Grouting device

3.3 3维注浆试验台构成

3维渗透注浆试验台由多层筒体组成，每层筒体内径150 cm、外径160 cm、高度30 cm，各层之间采用高强螺栓连接。试验台研发过程见文献[31]。其实物图及结构示意图见图1011

 图10 3维注浆试验台[31] Fig. 10 3D permeation grouting test bench[31]

 1.引线孔；2.连接螺栓；3.出浆口；4.试验腔；5.试验腔底板；　 6.钢质注浆花管；7.承载板；8.圆形底座。 图11 3维渗透注浆试验台结构设计 Fig. 11 Structure design of 3D permeation grouting test bench

 1.PVC注浆管；2.橡胶阻隔环；3.浆液出流孔；4.封闭端。 图12 PVC注浆内管结构示意图 Fig. 12 Structure diagram of PVC grouting inner pipe

3.4 多元信息采集

 图13 监测点平面布置图 Fig. 13 Layout plan of monitoring elements

4 3维渗透注浆模型试验

4.1 试验材料

4.2 试验方案

4.3 操作步骤

1）填砂及渗压传感器埋设

2）压水试验

3）渗透注浆

4）取样及孔隙率测试

4.4 模型试验值与理论计算值对比分析

4.4.1 浆液扩散形态分析

 图14 浆液扩散边界测量值及计算值 Fig. 14 Measurement and calculation values of grout diffusion boundary

4.4.2 浆液压力及孔隙率时空演化规律

 图15 孔隙率时空变化 Fig. 15 Temporal and spatial variations in porosity

 图16 浆液压力时空分布 Fig. 16 Temporal and spatial variations in grout pressure

4.4.3 注浆速率影响

 图17 不同注浆速率孔隙率空间分布 Fig. 17 Spatial distribution of porosity with different grouting rates

4.4.4 水灰比影响

 图18 孔隙率空间分布（t=120 s，q=15 L/min） Fig. 18 Spatial distribution of porosity（t=120 s，q=15 L/min）

5 结　论

1）考虑渗流域内各组分质量守恒，引入线性滤过定律，采用颗粒沉积概率模型描述水泥颗粒在多孔介质内沉积吸附行为，结合均匀毛细管模型，建立了水泥浆液渗透注浆柱形扩散模型。并且，提出了模型参数反分析确定方法。

2）开展了3维渗透注浆扩散模型试验，获得了砂层孔隙率及浆液压力时空变化规律，试验结果与理论模型计算结果对比，所建立模型的孔隙率最大计算误差小于13%，注浆压力最大计算误差小于14.2%，所建立模型可较好地描述水泥浆液多孔介质柱形扩散过程。

3）注浆速率越小，注浆口处孔隙率衰减量越大，同时孔隙率沿浆液扩散方向衰减越剧烈；在确保注浆压力小于地层启劈压力前提下，提高注浆速率，可有效减小滤过效应，有利于浆液充分扩散。浆液水灰比越小，孔隙率衰减越快，滤过效应越显著。

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