工程科学与技术   2021, Vol. 53 Issue (3): 106-114

Analytical Solutions for Nonlinear Consolidation of Aquitard Induced by the Dropping of Groundwater Table
LI Chuanxun, JIANG Liuhui
Faculty of Civil Eng. and Mechanics, Jiangsu Univ., Zhenjiang 212013, China
Abstract: The dropping of groundwater table by pumping may cause consolidation settlement of aquitard underlying aquifer, and the nonlinear compressibility and permeability of soils have evident influences on the consolidation deformation. Therefore, the nonlinear compressibility and permeability of soils have to be considered in the nonlinear consolidation model of aquitard caused by groundwater pumping, while difficulties exist in solving this consolidation model. Based on the analysis of the existing nonlinear consolidation models, an approximate analytical solution for the nonlinear consolidation of aquitard was derived, in which by adopting the classical nonlinear relationships and assuming a constant initial effective stress at different depths, an approximate analytical solution for one-dimensional nonlinear consolidation of soils caused by large area pumping of phreatic water layer were obtained using the method of separation of variables when the ratio of compressibility index Cc to permeability index Ck was not equal to 1. When Cc/Ck→1, the solution can be reduced to the existing one-dimensional nonlinear consolidation solution for the case of Cc/Ck =1. The solution was applied to the settlement analysis of engineering case, and the comparisons between theoretical and actual measurement results further showed that the solution is reliable for actual problems. Finally, taking the single-stage and constant-rate dropping of groundwater table as an example, the consolidation settlement curves of aquitard under different factors were calculated by using the above solution and the non-linear consolidation behaviors were analyzed. The results showed that when the value of Ck is constant, the larger Cc is (resulting in larger Cc/Ck), the slower the consolidation rate is, but the greater both the settlement at the same time and the final settlement are. When the value of Cc remains constant, the smaller Ck is (resulting in larger Cc/Ck), the slower the consolidation rate is, and the smaller the settlement at the same time during the consolidation process is, however it does not affect the final settlement of aquitard. The faster the dropping rate of groundwater table is, the faster the consolidation rate of aquitard is, while the final settlements under different dropping rate are same. The settlement rate grows with the increase in the final value of groundwater table (hc), whereas the influence of Cc/Ck on the consolidation rate becomes more evident. The larger the specific weight of soils (γ) is, the larger the settlement at the same time factor is, and the slower the dissipation rate of the excess pore pressure is, which indicates that γ has no influence on the final value of the excess pore pressure.
Key words: dropping of groundwater table    nonlinear consolidation    variation of consolidation coefficient    analytical solution

Li[18]基于非线性应力应变关系推导了越流系统中，水位波动变化时，软土层1维非线性固结解析解，但其未能考虑土体渗透性的变化。张云等[19]考虑了土体的非线性变形及渗透特性，建立了以有效应力为变量的1维非线性地面沉降模型，用半解析法进行了求解。黄大中[20]在压缩系数与渗透系数同比例减小（即固结系数保持不变）的假定下得到了潜水层和承压层水位随时间大面积线性下降时的1维非线性固结解析解。但实际上土体在固结过程中压缩系数与渗透系数的变化往往并不成比例。目前，对由抽降水引发的、能反映土体非线性固结特性的弱透水层非线性固结解析解的研究仍较缺乏。

1 问题描述及固结控制方程建立 1.1 问题描述

 图1 潜水层水位下降引发的弱透水层1维固结示意图 Fig. 1 Schematic diagram of one-dimensional consolidation of aquitard caused by the dropping of groundwater table in phreatic aquifer

 图2 水位下降量与时间的关系 Fig. 2 Relationship between the dropping of groundwater table and time

 $h(t){\rm{ = }}{h_{\rm{c}}}$ (1)

 $h(t) = \left\{\!\!\!\! {\begin{array}{*{20}{c}} {\dfrac{{{h_{\rm{c}}}}}{{{t_{\rm{c}}}}}t,}\;{t \le {t_{\rm{c}}};} \\ {{h_{\rm{c}}},}\;{t > {t_{\rm{c}}}} \end{array}} \right.$ (2)

1.2 固结相关土工参数的推导

 $e = {e_0} - {C_{\rm{c}}}\lg \left({{{\sigma '} / {{{\sigma_0 '}}}}} \right)$ (3)
 $e = {e_0} + {C_{\rm{k}}}\lg \left({{{{k_{\rm{v}}}} / {{k_{{\rm{v0}}}}}}} \right)$ (4)

 ${m_{\rm{v}}} = - \frac{1}{{1 + {e_0}}}\frac{{\partial e}}{{\partial \sigma '}} = {m_{\rm{v}}}_0\frac{{{{\sigma_0 '}}}}{{\sigma '}}$ (5)

 ${k_{\rm{v}}} = {k_{{\rm{v}}0}}{\left({\frac{{{{\sigma_0 '}}}}{{\sigma '}}} \right)^{{{{C_{\rm{c}}}} / {{C_{\rm{k}}}}}}}$ (6)

 ${C_{\rm{v}}} = {C_{{\rm{v}}0}}{\left({\frac{{{{\sigma_0 '}}}}{{\sigma '}}} \right)^{{{{C_{\rm{c}}}} / {{C_{\rm{k}}} - 1}}}}$ (7)

1.3 固结控制方程及求解条件

 $\frac{1}{{{\gamma _{\rm{w}}}}}\frac{\partial }{{\partial {\textit{z}}}}\left({{k_{\rm{v}}}\frac{{\partial u}}{{\partial {\textit{z}}}}} \right) = \frac{1}{{1 + {e_0}}}\frac{{\partial e}}{{\partial t}}$ (8)

 $\frac{{\partial e}}{{\partial t}} = - \frac{{{C_{\rm{c}}}}}{{\sigma '\ln\; 10}}\frac{{\partial \sigma '}}{{\partial t}} = - \frac{{{C_{\rm{c}}}}}{{\sigma '\ln\; 10}}\left(\frac{{\partial \sigma }}{{\partial t}} - \frac{{\partial u}}{{\partial t}}\right)$ (9)

 $\sigma ' = {\sigma '_0} + q\left(t \right) - u$ (10)

 ${\;\;\;\;\;\;\;\;C_{{\rm{v}}0}}\frac{{\sigma '}}{{{{\sigma_0 '}}}}\frac{\partial }{{\partial {\textit{z}}}}\left[ {{{\left({\frac{{{{\sigma_0 '}}}}{{\sigma '}}} \right)}^{{{{C_{\rm{c}}}} / {{C_{\rm{k}}}}}}}\frac{{\partial u}}{{\partial {\textit{z}}}}} \right] = \frac{{\partial u}}{{\partial t}} - \frac{{{\rm{d}}q}}{{{\rm{d}}t}}$ (11)

 $q = (\gamma - {\gamma _{{\rm{sat}}}})h(t)$ (12)

 $u\left({0,t} \right) = - {\gamma _{\rm{w}}}h(t),\;t > 0$ (13)
 $\frac{{\partial u}}{{\partial {\textit{z}}}} = 0,\;{\textit{z}} = H$ (14)

 ${\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;u} = {q_0} = (\gamma - {\gamma _{{\rm{sat}}}}){h_0},\;t = 0$ (15)

2 固结模型的近似解析解 2.1 变量代换后的控制方程及求解条件

 ${C_{\rm{v}}}_0w\frac{{{\partial ^2}v}}{{\partial {{\textit{z}}^2}}} = \frac{{\partial v}}{{\partial t}} - R(t)$ (16)

 ${C_{\rm{v}}}_0{w_0}\frac{{{\partial ^2}v}}{{\partial {{\textit{z}}^2}}} = \frac{{\partial v}}{{\partial t}} - R(t)$ (17)

 $v = 0,\;{\textit{z}} = 0$ (18)
 $\frac{{\partial v}}{{\partial {\textit{z}}}} = 0,\;{\textit{z}} = H$ (19)
 ${\;\;\;\;\;\;\;\;\;v} = {\left[ {{{\left({{\sigma _0'} + {q_0} + {p_0}} \right)} / {{\sigma _0'} }}} \right]^{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} - 1,\;\;t = 0$ (20)

2.2 超静孔隙水压力解答

 $v = \sum\limits_{m = 1}^\infty {\sin \left(\frac{{M{\textit{z}}}}{H}\right)} {{\rm{e}}^{ - {\beta _m}t}}\left[ {{B_m} + {C_m}{T_m}(t)} \right]$ (21)

 ${T_m}(t) = \int_0^t {{{\rm{e}}^{{\beta _m}\tau }}R\left(\tau \right){\rm{d}}\tau }$ (22)

 \begin{aligned}[b] u =& {\sigma '_0} + q - {\sigma '_0} \left[ {\left(\dfrac{{{{\sigma_0 '}} + q + {\gamma _{\rm{w}}}h(t)}}{{{{\sigma_0 '}}}}\right)^{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} - \right.\\ &\left.\displaystyle \sum\limits_{m = 1}^\infty {\sin \left(\dfrac{{M{\textit{z}}}}{H}\right)} {{\rm{e}}^{ - {\beta _m}t}}\left[ {{B_m} + {C_m}{T_m}(t)} \right] \right]^{\frac{{{C_{\rm{k}}}}}{{{C_{\rm{k}}} - {C_{\rm{c}}}}}}\end{aligned} (23)

 ${U_{\rm{p}}} = \dfrac{{\displaystyle \int_0^H {(\sigma ' - {{\sigma_0 '}}){\rm{d}}{\textit{z}}} }}{{\displaystyle \int_0^H {({{\sigma' _f}} - {{\sigma_0 '}}){\rm{d}}{\textit{z}}} }}=\dfrac{{qH - \displaystyle \int_0^H {u{\rm{d}}{\textit{z}}} }}{{({p_{\rm{c}}} + {q_{\rm{c}}})H}}$ (24)

t时刻土层发生的沉降变形值St为：

 ${S\!_{\rm{t}}} = \int_0^H {\frac{{{e_0} - {{e}}}}{{1 + {e_0}}}} {\rm{d}}{\textit{z}} = \frac{{{C_{\rm{c}}}}}{{1 + {e_0}}}\int_0^H {\lg \frac{{\sigma '}}{{{{\sigma_0 '}}}}} {\rm{d}}{\textit{z}}$ (25)

 ${S\!_\infty } = \frac{{{C_{\rm{c}}}}}{{1 + {e_0}}}\int_0^H {\lg \frac{{{{\sigma'_f }}}}{{{{\sigma'_0 }}}}} {\rm{d}}{\textit{z}}{\rm{ = }}\frac{{{C_{\rm{c}}}\lg\; {N_{\rm{q}}}}}{{1 + {e_0}}}H$ (26)

 ${U_{\rm{s}}} = \frac{{{S\!_{\rm{t}}}}}{{{S\!_\infty }}} = \frac{{\displaystyle \int_0^H {\lg \dfrac{{\sigma '}}{{{{\sigma'_0 }}}}} {\rm{d}}{\textit{z}}}}{{\lg\; {{N_{\rm{q}}}} \cdot H}}$ (27)
3 特殊降水模式下的解答 3.1 水位瞬时下降模式

 $q = \left({\gamma - {\gamma _{{\rm{sat}}}}} \right){h_{\rm{c}}}{\rm{ = }}{q_{\rm{c}}}$ (28)
 ${\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\{\!\!\!\! \begin{array}{l} u = - {\gamma _{\rm{w}}}{h_{\rm{c}}} = - {p_{\rm{c}}},\;{\textit{z}} = 0; \\ \dfrac{{\partial u}}{{\partial {\textit{z}}}} = 0,\;{\textit{z}} = H; \\ u = {q_{\rm{c}}},\;t = 0 \end{array} \right.}$ (29)

${B_m} = \dfrac{2}{M}\left({{N_{\rm{q}}^{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} - 1} \right)$ ${T_m}(t) = 0$

$B_m$ $C_m$ $\;\beta_m$ 代入式（23）得到弱透水层中超静孔隙水压力为：

 {\;\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}[b]u =& {\sigma '_0} + {q_{\rm{c}}} - {\sigma '_0}\bigg[ {N_{\rm{q}}^{1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}}} - \left({{N_{\rm{q}}^{1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}}} - 1} \right)\cdot\bigg. \\ &\bigg.\displaystyle \sum\limits_{m = 1}^\infty {\dfrac{2}{M}} \sin \left(\dfrac{{M{\textit{z}}}}{H}\right){{\rm{e}}^{ - {M^2}{w_0}{T_{\rm{v}}}}} \bigg]^{\frac{{{C_{\rm{k}}}}}{{{C_{\rm{k}}} - {C_{\rm{c}}}}}}\end{aligned}} (30)

3.2 水位单级等速下降模式

 $q=\left\{\!\!\!\! {\begin{array}{cc}\dfrac{{q}_{\rm{c}}}{{t}_{\rm{c}}}t, \; t\le {t}_{\rm{c}}; \\ {q}_{\rm{c}}, \; t>{t}_{\rm{c}}\end{array}} \right.$ (31)
 ${\;\;\;\;\;\;\;\;\;\;\;\;\;\left\{ \!\!\!\!\begin{array}{l} u = \left\{ \!\!\!\!{\begin{array}{*{20}{c}} { - \dfrac{{{p_{\rm{c}}}}}{{{t_{\rm{c}}}}}t,}\;{t \le {t_{\rm{c}}},}\;{\textit{z}} = 0\\ { - {p_{\rm{c}}}},\;{t > {t_{\rm{c}}},}\;{\textit{z}} = 0 \end{array}}; \right.\\ \dfrac{{\partial u}}{{\partial {\textit{z}}}} = 0,\;{\textit{z}} = H;\\ u = 0,\;t = 0 \end{array} \right.}$ (32)

 $u = {\sigma '_0} + q - {\sigma '_0}{\left[ {\left( {\frac{{{{\sigma '}_0} + q + {\gamma _{\rm{w}}}h(t)}}{{{{\sigma '}_0}}}} \right)^{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} - \sum\limits_{m = 1}^\infty {\frac{2}{M}} \sin \left(\frac{{M\textit{z}}}{H}\right){{\rm{e}}^{ - {M^2}{w_0}{T_{\rm{v}}}}}{T_m}(t) \right]^{\frac{{{C_{\rm{k}}}}}{{{C_{\rm{k}}} - {C_{\rm{c}}}}}}}$ (33)
 ${T_m}(t)=\left\{\!\!\!\! {\begin{array}{*{20}{l}} {\left(1 - \dfrac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}\right){{\rm{e}}^{\frac{{ - {M^2}{w_0}{T_{{\rm{vc}}}}}}{{{N_{\rm{q}}} - 1}}}}\left[ \dfrac{{{{\left(1 + \dfrac{{{N_{\rm{q}}} - 1}}{{{T_{{\rm{vc}}}}}}{T_{\rm{v}}}\right)}^{1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}}} - 1}}{{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} + \displaystyle \sum\limits_{k = 1}^\infty {\dfrac{{{{\left({\dfrac{{{M^2}w{}_0{T_{{\rm{vc}}}}}}{{{N_{\rm{q}}} - 1}}} \right)}^k}}}{{k!}}} \dfrac{{{{\left(1 + \dfrac{{{N_{\rm{q}}} - 1}}{{{T_{{\rm{vc}}}}}}{T_{\rm{v}}}\right)}^{k + 1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}}} - 1}}{{k + 1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} \right],}\;{{T_{\rm{v}}} \le {T_{{\rm{vc}}}};} \\ {\left(1 - \dfrac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}\right){{\rm{e}}^{\frac{{ - {M^2}{w_0}{T_{{\rm{vc}}}}}}{{{N_{\rm{q}}} - 1}}}}\left[ \dfrac{{{N_{\rm{q}}}^{1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}} - 1}}{{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} + \displaystyle \sum\limits_{k = 1}^\infty {\dfrac{{{{\left({\dfrac{{{M^2}w{}_0{T_{{\rm{vc}}}}}}{{{N_{\rm{q}}} - 1}}} \right)}^k}}}{{k!}}} \dfrac{{{N^{k + 1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}}_{\rm{q}}} - 1}}{{k + 1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} \right],}\;{{T_{\rm{v}}} > {T_{{\rm{vc}}}}} \end{array}} \right.$ (34)

3.3 Cc/Ck→1条件下超静孔压解的退化

Cc/Ck→1的条件下，水位单级等速下降时的超静孔压解可退化为文献[20]中相同条件下Cc/Ck=1时解的形式。超静孔压解u的退化过程为：

 \begin{aligned}[b] \mathop {\lim }\limits_{{C_{\rm{c}}}/{C_{\rm{k}}} \to 1} u \!=\!& \mathop {\lim }\limits_{{C_{\rm{c}}}/{C_{\rm{k}}} \to 1} {{\sigma' _0}} + q -\\ &{{\sigma _0'}}{\left[ {{{\left( {\frac{{{{\sigma' _0}} + q + {\gamma _{\rm{w}}}h(t)}}{{{{\sigma '_0}}}}} \right)}^{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} - v} \right]^{\frac{1}{{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}}}}=\\ &{{\sigma' _0}} \!+\! q \!-\! {{\sigma '_0}}{{\rm{e}}^{{{\mathop {\lim }\limits_{{C_{\rm{c}}}/{C_{\rm{k}}} \to 1} }}\frac{{{{\left( {\frac{{{{\sigma '_0}} + q + {\gamma _{\rm{w}}}h(t)}}{{{{\sigma '_0}}}}} \right)}^{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}} \!-\! 1 \!-\! v}}{{1 - {C_{\rm{c}}}/{C_{\rm{k}}}}}}}=\\ &{{\sigma' _0}} + q - {{\sigma '_0}}{{\rm{e}}^w} \\[-10pt] \end{aligned} (35)

4 工程实例对比

 图5 Ck值对沉降曲线的影响 Fig. 5 Influence of Ck on settlement curves

5 算例及固结性状分析

5.1 Cc/Ck对固结性状的影响

 图3 Cc/Ck对平均固结度Up曲线的影响 Fig. 3 Influence of Cc/Ck on the curves of average consolidation degree Up

 图4 Cc值对沉降曲线的影响 Fig. 4 Influence of Cc on settlement curves

5.2 水位下降速率对固结性状的影响

 图6 水位下降速率对沉降曲线的影响 Fig. 6 Influence of the dropping rate of groundwater table on settlement curves

5.3 不同Cc/Ck下水位下降终值对固结性状的影响

 图7 Cc/Ck下水位下降终值对 ${{{\sigma '}}}$ / ${{{\sigma_{\bf{0}} '}}}$ 沿深度分布的影响 Fig. 7 Influence of the final dropping values of groundwater table on distribution of ${{{\sigma '}}}$ / ${{{\sigma_{\bf{0}} '}}}$ along depth with different Cc/Ck

 图8 Cc/Ck下水位下降终值对沉降曲线的影响 Fig. 8 Influence of the final dropping values of groundwater table on settlement curves with different Cc/Ck

5.4 砂土层自然重度 ${{\gamma}}$ 对固结速率的影响

 图9 砂土层自然重度 ${{\gamma}}$ 对超静孔压消散的影响 Fig. 9 Influence of specific density of sand layer on dissipation of excess pore pressure

 图10 砂土层自然重度对超静孔压沿深度分布的影响 Fig. 10 Influence of specific density of sand layer on distribution of excess pore pressure with depth

 图11 砂土层自然重度对Cv/Cv0沿深度分布的影响 Fig. 11 Influence of specific density of sand layer on distribution of Cv/Cv0 with depth

 图12 砂土层自然重度 ${{\gamma}}$ 对沉降曲线的影响 Fig. 12 Influence of specific density of sand layer on settlement curves

6 结　论

1）基于弱透水层初始有效应力沿深度均匀分布的假定，推导了考虑土体非线性压缩和渗透特性的由潜水层水位随时间大面积均匀下降引发的弱透水层1维非线性固结近似解析解，且当Cc/Ck→1时，超静孔压解可退化为相同模型下Cc/Ck=1时的现有解。

2）Cc/Ck越大，弱透水层固结速率越缓慢。Ck不变时，Cc值越大，固结速率越慢，但沉降速率越快；Cc不变时，Ck越小，固结速率越慢，固结过程中沉降速率越慢，但不影响最终沉降量。

3）水位下降越快，沉降速率越快，但最终沉降量相同。

4）水位下降终值hc越大，沉降速率越快，Cc/Ck对固结速率的影响越显著。

5）Cc/Ck<1时，砂土层自然重度γ越大，相同时间因子下固结系数越大，固结速率越快，相应的沉降量越大。Cc/Ck>1时，砂土层自然重度γ越大，相同时间因子下固结系数越小，固结速率越慢，相应的沉降量越大。

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