黏时变渗透分段注浆压力的上、下限解

王军辉; 韩煊

doi:10.11779/CJGE20220582

黏时变渗透分段注浆压力的上、下限解 English Version

王军辉,
韩煊

北京市勘察设计研究院有限公司, 北京 100038

详细信息

作者简介:
王军辉(1973—)，男，博士，教授级高级工程师，主要从事水文地质与岩土工程方面的科学研究与工程咨询工作。E-mail: wjh1223@sina.com

中图分类号: TU432
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出版历程
- 收稿日期: 2022-05-07
- 网络出版日期: 2023-02-23
- 刊出日期: 2023-08-31

Upper- and lower-bound solutions for sectional penetration grouting pressure under time-dependent viscosity of slurry

BGI Engineering Consultants Ltd., Beijing 100038, China

摘要

摘要: 在浆液的黏时变效应下，渗透注浆压力（p）和注浆速率（q）都是随时间t变化的，给注浆工程的设计计算带来困难。首先，考虑到设计的便利，利用定积分原理，提出了p-q双时间变量下对应的分段注浆压力（P）和注浆速率（Q）的工程定义。其次，根据黏时变理论和Darcy定律，建立了p-q双时间变量下的黏时变渗透注浆解析模型（包括物理方程、几何方程和边界条件），揭示了黏时变渗透注浆的复杂数学物理过程，讨论了模型解的存在性与唯一性。再次，利用P-Q的工程定义式和渗透注浆解析模型进行P的理论解研究，充分利用积分不等式的性质，分别得到了P的下限与上限通解，讨论解的科学性与普适性。最后，结合当前实际工程需要，进一步讨论了指数黏时变函数P的上、下限特解（包括球面和柱面两种扩散模式）。
- 注浆 /
- 渗透 /
- 注浆压力 /
- 注浆速率 /
- 注浆持续时间 /
- 黏滞系数 /
- 黏时变 /
- 积分不等式
Abstract: Under the time-dependent viscosity of slurry, the grouting pressure (p) and the grouting rate (q) are both time-dependent, which makes design and calculation more difficult in grouting project. Firstly, the engineering definitions of grouting pressure (P) and grouting rate (Q) for a sectional grouting under the double-time-variable of p-q are introduced by using the defined integral. Secondly, according to the theory of time-dependent viscosity and the Darcy's law, an analytical model for the penetration grouting under time-dependent viscosity of slurry considering the double-time-variable of p-q is established (including physical equation, geometrical equation and boundary condition), with which the complex process of penetration grouting under time-dependent viscosity of slurry is discovered, and its solutions are discussed systematically. Thirdly, the theoretical solutions for the sectional grouting pressure P are deduced based on its engineering definition and the above analytical model, and with the property of integral inequality and the systematic theoretical derivation, the general limit solutions (including upper- and lower-bound solutions) of P are obtained, and their scientificity and universality are discussed. Finally, considering the actual engineering needs, the special upper- and lower-bound solutions of P for the two modes of spherical and cylindrical diffusions under exponential time-dependent viscosity function are further discussed.
- grouting /
- penetration /
- grouting pressure /
- grouting rate /
- grouting duration /
- viscosity coefficient /
- time-dependent viscosity /
- integral inequality

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图 1 单孔注浆浆液流向截面示意图

Figure 1. Section of flow direction of slurry under single hole-grouting

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表 1 球面扩散模式下P的上、下限特解

Table 1 Special limit solution of P under spherial diffusion mode

解的类型	解的表达式	符号说明
下限解	$\begin{gathered} H_{{\text{g1}}}^{\text{s}} = \frac{{{\mu _{{\text{g0}}}}aT\exp (aT)}}{{{\mu _{\text{w}}}[\exp (aT) - 1]K}} \cdot \hfill \\ \left\{ {\frac{Q}{{4{\text{π }}}}\left[ {\frac{1}{{{r_g}}} - \frac{1}{{{R^s}(T)}}} \right]} \right. - \hfill \\ \left. {\frac{{{n_{\text{e}}}{{[{R^{\text{s}}}(T) - {r_{\text{g}}}]}^2}[{R^s}(T) + 2{r_{\text{g}}}]}}{{6{R^{\text{s}}}(T)T}}} \right\} + {h_0} \hfill \\ \end{gathered}$	$\begin{aligned} &R^{\mathrm{s}}(T)=\\ &\sqrt[3]{\frac{Q T}{4 \pi n_{\mathrm{e}}}+r_{\mathrm{g}}^3} \end{aligned}$
上限解	$H_{{\text{g2}}}^{\text{s}} = \frac{{{\mu _{{\text{g0}}}}Q}}{{4{\text{π }}{\mu _{\text{W}}}TK}} \cdot$ $\left\{ {\frac{{[\exp (aT) - 1]}}{{a{r_{\text{g}}}}} - \int_0^T {\frac{{{\text{exp(}}a{\text{t)}}}}{{{{\bar R}^{\text{s}}}{\text{(}}t{\text{)}}}}{\text{d}}t} } \right\}$	$\begin{aligned} &\bar{R}^{\mathrm{s}}(t)=\\ &\sqrt[3]{\frac{Q t}{4 \pi n_{\mathrm{e}}}+r_{\mathrm{g}}^3} \end{aligned}$

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表 2 柱状扩散模式下P的上、下限特解

Table 2 Special limit solution of P under cylindrical diffusion mode

解的类型	解的表达式	符号说明
下限解	$\begin{gathered} H_{{\text{g1}}}^{\text{c}} = \frac{{{\mu _{{\text{g0}}}}aT\exp (aT)}}{{2{\mu _{\text{w}}}K[\exp (aT) - 1]}}\left\{ {\left( {\frac{Q}{{{\text{π }}L}} + \frac{{{n_{\text{e}}}}}{T}r_{\text{g}}^{\text{2}}} \right) \cdot } \right. \hfill \\ \ln \frac{{{R^{\text{c}}}(T)}}{{{r_{\text{g}}}}} - \left. {\frac{{{n_{\text{e}}}}}{{2T}}[{R^{\text{c}}}{{(T)}^2} - r_{\text{g}}^{\text{2}}]} \right\} \hfill \\ \end{gathered}$	$\begin{gathered} {R^{\text{c}}}(T) = \hfill \\ \sqrt {\frac{{QT}}{{{\text{π }}L{n_{\text{e}}}}} + r_{\text{g}}^{\text{2}}} \hfill \\ \end{gathered}$
上限解	$\begin{gathered} H_{{\text{g2}}}^{\text{c}} = \frac{{{\mu _{{\text{g0}}}}Q}}{{2{\text{π }}KL{\mu _{\text{w}}}T}}\left[ {\int_0^T {\exp (at){\text{ln}}{{\bar R}^{\text{c}}}(t)} {\text{d}}t - \begin{array}{{20}{c}} {} \\ {} \end{array}} \right. \hfill \\ \left. {\begin{array}{{20}{c}} {} \\ {} \end{array}\frac{{\exp (aT) - 1}}{a}{\text{ln}}{r_{\text{g}}}} \right] \hfill \\ \end{gathered}$	$\begin{gathered} {{\bar R}^{\text{c}}}(t) = \hfill \\ \sqrt {\frac{{Qt}}{{{\text{π }}L{n_{\text{e}}}}} + r_{\text{g}}^{\text{2}}} \hfill \\ \end{gathered}$

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