矩形边界越流承压含水层中非完整井稳定流解析解

孙前林; 谭卫佳; 徐蓓艺; 王旭东

doi:10.21656/1000-0887.430398

矩形边界越流承压含水层中非完整井稳定流解析解

doi: 10.21656/1000-0887.430398

南京工业大学岩土工程研究所，南京 211816

基金项目:

国家自然科学基金青年科学基金项目 41807189

详细信息

作者简介:
孙前林(1997—)，男，硕士生(E-mail: 202061225023@njtech.edu.cn)

通讯作者:
王旭东(1963—)，男，教授(通讯作者. E-mail: cewxd@njtech.edu.cn)

中图分类号: TU470
计量
- 文章访问数: 251
- HTML全文浏览量: 80
- PDF下载量: 52
- 被引次数: 0
出版历程
- 收稿日期: 2022-12-21
- 修回日期: 2023-05-06
- 刊出日期: 2023-08-01

Analytical Solutions of Steady Flow Toward a Partially Penetrating Well in a Rectangular Leaky-Confined Aquifer

Institute of Geotechnical Engineering, Nanjing Tech University, Nanjing 211816, P.R.China

摘要

摘要: 针对矩形边界越流承压含水层中非完整井抽水引起的复杂地下水流动问题，建立了直角坐标系下越流承压含水层非完整井稳定流数学模型. 通过对地下水流动计算模型的有限Fourier变换和有限Fourier变换域降深函数的逆变换，提出了不同类型边界条件下越流承压含水层非完整井三维稳定流降深解析解. 在验证降深解析解正确性的基础上，通过降深解析解计算精度的分析，并结合非完整井抽水条件下含水层地下水流动特性，给出了降深解析解满足计算精度要求的计算项数取值. 探讨了含水层正交各向异性、抽水井完整性、井位布置等因素对含水层降深和地下水流动的影响规律，并利用工程案例阐明了降深解析解的工程适用性.
- 矩形边界承压含水层 /
- 非完整井 /
- 三维稳定流 /
- 直角坐标系 /
- 解析解
Abstract: For the complicated problem of groundwater flow to a partially penetrating well in a rectangular confined aquifer, a mathematical model describing the groundwater flow to a partially penetrating well pumped at a constant rate in a rectangular leaky-confined aquifer, was established. The analytical solutions of the 3D steady flow in the Cartesian coordinate system under different boundary conditions, were derived through the finite Fourier transform and the inverse transform. After the verification of the analytical solution of drawdown, the number of calculation items satisfying the calculation accuracy requirement was given, based on the analysis of the calculation accuracy of the analytical solution and the characteristics of the groundwater flow to a partially penetrating well. Moreover, the effects of orthotropy, well integrity and well location on the drawdown and seepage fields, were discussed. The engineering examples demonstrate the applicability of the proposed analytical method.
- rectangular confined aquifer /
- partially penetrating well /
- 3D steady flow /
- Cartesian coordinate system /
- analytical solution

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图 1 矩形边界越流承压含水层非完整井计算模型

Figure 1. The calculation model for the partially penetrating well in a rectangular leaky-confined aquifer

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图 2 降深分布曲线(越流与无越流)

Figure 2. Drawdown distribution curves(leakage and non-leakage)

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图 3 降深分布曲线(完整井与非完整井)

Figure 3. Drawdown distribution curves(fully penetrating well and partially penetrating well)

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图 4 降深及相对误差随计算项数的变化(完整井)

Figure 4. Variations of drawdowns and relative errors with the number of calculation items (fully penetrating well)

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图 5 降深及相对误差随计算项数的变化(非完整井)

Figure 5. Variations of drawdowns and relative errors with the number of calculation items (partially penetrating well)

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图 6 降深分布曲线

Figure 6. Drawdown distribution curves

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图 7 承压含水层非完整井水力梯度(剖面)

Figure 7. Gradients of the partially penetrating well in a confined aquifer(profile)

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图 8 降深等值线及水力梯度(平面)

Figure 8. Drawdown contours and gradients(plane)

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图 9 抽水井和观测井布置(单位: m)

Figure 9. The pumping well and the observation well layout(unit: m)

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图 10 地下水计算模型(单位: m)

Figure 10. The groundwater calculation model(unit: m)

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图 11 降深计算值与实测值对比

Figure 11. Comparison of calculated and measured values of the drawdown

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B1 不同类型边界条件下的特征函数、特征值、范数以及变换参数

B1. Characteristic functions, eigenvalues, norms and transformation parameters under different boundary conditions

coordinate axis	boundary condition	eigenvalue	characteristic function	norm	transformation parameter
x	$\left.s\right\|_{x=0}=\left.s\right\|_{x=L}=0$	$\alpha_m=\frac{m {\rm{ \mathsf{π} }} }{L}$	$\varphi_m\left(\alpha_m x\right)=\sin \left(\alpha_m x\right)$	$M\left(\alpha_m\right)=\frac{L}{2}(m \geqslant 1)$	$m \in[1, \infty)$
x	$\left.\frac{\partial s}{\partial x}\right\|_{x=0}=\left.s\right\|_{x=L}=0$	$\alpha_m=\frac{(2 m+1) {\rm{ \mathsf{π} }} }{2 L}$	$\varphi_m\left(\alpha_m x\right)=\cos \left(\alpha_m x\right)$	$M\left(\alpha_m\right)=\frac{L}{2}(m \geqslant 0)$	$m \in[0, \infty)$
y	$\left.s\right\|_{y=0}=\left.s\right\|_{y=B}=0$	$\beta_n=\frac{n {\rm{ \mathsf{π} }} }{B}$	$\psi_n\left(\beta_n y\right)=\sin \left(\beta_n y\right)$	$N\left(\beta_n\right)=\frac{B}{2}(n \geqslant 1)$	$n \in[1, \infty)$
	$\left.\frac{\partial s}{\partial y}\right\|_{y=0}=\left.\frac{\partial s}{\partial y}\right\|_{y=B}=0$	$\beta_n=\frac{n {\rm{ \mathsf{π} }} }{B}$	$\psi_n\left(\beta_n y\right)=\cos \left(\beta_n y\right)$	$N\left(\beta_n\right)=\left\{\begin{array}{l} B(n=0) \\ B / 2(n \geqslant 1) \end{array}\right.$	$n \in[0, \infty)$
	$\left.\frac{\partial s}{\partial y}\right\|_{y=0}=\left.s\right\|_{y=B}=0$	$\beta_n=\frac{(2 n+1) {\rm{ \mathsf{π} }} }{2 B}$	$\psi_n\left(\beta_n y\right)=\cos \left(\beta_n y\right)$	$N\left(\beta_n\right)=\frac{B}{2}(n \geqslant 0)$	$n \in[0, \infty)$
z	$\left.\frac{\partial s}{\partial z}\right\|_{z=0}=\left.\frac{\partial s}{\partial z}\right\|_{z=M}=0$	$\lambda_k=\frac{k {\rm{ \mathsf{π} }} }{M}$	$\chi_k\left(\lambda_k z\right)=\cos \left(\lambda_k z\right)$	$K\left(\lambda_k\right)=\left\{\begin{array}{l} M(k=0) \\ M / 2(k \geqslant 1) \end{array}\right.$	$k \in[0, \infty)$

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C1 降深解析解(case 1~3)

C1. Analytic solutions of the drawdown(case 1~3)

case	boundary condition	schematic plan	s(x, y, z)
1	4 fixed-head boundary		$\begin{aligned} s= & \frac{4 Q}{B L M}\left[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{\theta_{m n}}{\beta_{m n}^2} \sin \left(\frac{m {\rm{ \mathsf{π} }} x}{L}\right) \sin \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right)+\right. \\ & \left.2 \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{\xi_k \theta_{m n}}{\beta_{m n k}^2} \sin \left(\frac{m {\rm{ \mathsf{π} }} x}{L}\right) \sin \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right) \cos \left(\frac{k {\rm{ \mathsf{π} }} z}{M}\right)\right] \end{aligned}$
1	4 fixed-head boundary		$\theta_{m n}=\sin \left(\frac{m {\rm{ \mathsf{π} }} x_{\mathrm{w}}}{L}\right) \sin \left(\frac{n {\rm{ \mathsf{π} }} y_{\mathrm{w}}}{B}\right)$
2	4 impermeable boundary		$\begin{aligned} s= & \frac{Q}{B L M}\left[\sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \varepsilon_{m n} \frac{\theta_{m n}}{\beta_{m n}^2} \cos \left(\frac{m {\rm{ \mathsf{π} }} x}{L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right)+\right. \\ & \left.2 \sum_{k=1}^{\infty} \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \varepsilon_{m n} \frac{\xi_k \theta_{m n}}{\beta_{m n k}^2} \cos \left(\frac{m {\rm{ \mathsf{π} }} x}{L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right) \cos \left(\frac{k {\rm{ \mathsf{π} }} z}{M}\right)\right] \end{aligned}$
			$\theta_{m n}=\cos \left(\frac{m {\rm{ \mathsf{π} }} x_{\mathrm{w}}}{L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y_{\mathrm{w}}}{B}\right)$
			$\varepsilon_{m n}= \begin{cases}1, & m=0, n=0, \\ 2, & m=0, n \geqslant 1 \quad \text { or } \quad m \geqslant 1, n=0, \\ 4, \quad & m \geqslant 1, n \geqslant 1\end{cases}$
3	2 fixed-head boundary (parallel) 2 impermeable boundary		$\begin{aligned} s= & \frac{2 Q}{B L M}\left[\sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \varepsilon_n \frac{\theta_{m n}}{\beta_{m n}^2} \sin \left(\frac{m {\rm{ \mathsf{π} }} x}{L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right)+\right. \\ & \left.2 \sum_{k=1}^{\infty} \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \varepsilon_n \frac{\xi_k \theta_{m n}}{\beta_{m n k}^2} \sin \left(\frac{m {\rm{ \mathsf{π} }} x}{L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right) \cos \left(\frac{k {\rm{ \mathsf{π} }} z}{M}\right)\right] \end{aligned}$
			$\theta_{m n}=\sin \left(\frac{m {\rm{ \mathsf{π} }} x_{\mathrm{w}}}{L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y_{\mathrm{w}}}{B}\right)$
			$\varepsilon_n= \begin{cases}1, & n=0, \\ 2, & n \geqslant 1\end{cases}$
$\beta_{m n k}^2=K_x\left(\frac{m {\rm{ \mathsf{π} }} }{L}\right)^2+K_y\left(\frac{n {\rm{ \mathsf{π} }} }{B}\right)^2+K_z\left(\frac{k {\rm{ \mathsf{π} }} }{M}\right)^2+\frac{K_z^{\prime}}{M M^{\prime}}, \beta_{m n}^2=K_x\left(\frac{m {\rm{ \mathsf{π} }} }{L}\right)^2+K_y\left(\frac{n {\rm{ \mathsf{π} }} }{B}\right)^2+\frac{K_z^{\prime}}{M M^{\prime}}$

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C2 降深解析解(case 4)

C2. Analytic solutions of the drawdown(case 4)

case	boundary condition	schematic plan	s(x, y, z)
4	2 fixed-head boundary 2 impermeable boundary		$\begin{aligned} s= & \frac{4 Q}{B L M}\left[\sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{\theta_{m n}}{\beta_{m n}^2} \cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x}{2 L}\right) \cos \left(\frac{(2 n+1) {\rm{ \mathsf{π} }} y}{2 B}\right)+\right. \\ & \left.2 \sum_{k=1}^{\infty} \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{\xi_k \theta_{m n}}{\beta_{m n k}^2} \cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x}{2 L}\right) \cos \left(\frac{(2 n+1) {\rm{ \mathsf{π} }} y}{2 B}\right) \cos \left(\frac{k {\rm{ \mathsf{π} }} z}{M}\right)\right] \end{aligned}$
4	2 fixed-head boundary 2 impermeable boundary		$\theta_{m n}=\cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x_{\mathrm{w}}}{2 L}\right) \cos \left(\frac{(2 n+1) {\rm{ \mathsf{π} }} y_{\mathrm{w}}}{2 B}\right)$
$\beta_{m n k}^2=K_x\left(\frac{(2 m+1) {\rm{ \mathsf{π} }} }{2 L}\right)^2+K_y\left(\frac{(2 n+1) {\rm{ \mathsf{π} }} }{2 B}\right)^2+K_z\left(\frac{k {\rm{ \mathsf{π} }} }{M}\right)^2+\frac{K_z^{\prime}}{M M^{\prime}}, \beta_{m n}^2=K_x\left(\frac{(2 m+1) {\rm{ \mathsf{π} }} }{2 L}\right)^2+K_y\left(\frac{(2 n+1) {\rm{ \mathsf{π} }} }{2 B}\right)^2+\frac{K_z^{\prime}}{M M^{\prime}}$

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C3 降深解析解(case 5, 6)

C3. C3 Analytic solutions of the drawdown(case 5, 6)

case	boundary condition	schematic plan	s(x, y, z)
5	1 fixed-head boundary 3 impermeable boundary		$\begin{aligned} s= & \frac{2 Q}{B L M}\left[\sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \varepsilon_n \frac{\theta_{m n}}{\beta_{m n}^2} \cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x}{2 L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right)+\right. \\ & \left.2 \sum_{k=1}^{\infty} \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \varepsilon_n \frac{\xi_k \theta_{m n}}{\beta_{m n k}^2} \cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x}{2 L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right) \cos \left(\frac{k {\rm{ \mathsf{π} }} z}{M}\right)\right] \end{aligned}$
			$\theta_{m n}=\cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x_{\mathrm{w}}}{2 L}\right) \cos \left(\frac{n {\rm{ \mathsf{π} }} y_{\mathrm{w}}}{B}\right)$
			$\varepsilon_n= \begin{cases}1, & n=0 \\ 2, & n \geqslant 1\end{cases}$
6	3 fixed-head boundary 1 impermeable boundary		$\begin{aligned} s= & \frac{4 Q}{B L M}\left[\sum_{n=1}^{\infty} \sum_{m=0}^{\infty} \frac{\theta_{m n}}{\beta_{m n}^2} \cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x}{2 L}\right) \sin \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right)+\right. \\ & \left.2 \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \sum_{m=0}^{\infty} \frac{\xi_k \theta_{m n}}{\beta_{m n k}^2} \cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x}{2 L}\right) \sin \left(\frac{n {\rm{ \mathsf{π} }} y}{B}\right) \cos \left(\frac{k {\rm{ \mathsf{π} }} z}{M}\right)\right] \end{aligned}$
6	3 fixed-head boundary 1 impermeable boundary		$\theta_{m n}=\cos \left(\frac{(2 m+1) {\rm{ \mathsf{π} }} x_{\mathrm{w}}}{2 L}\right) \sin \left(\frac{n {\rm{ \mathsf{π} }} y_{\mathrm{w}}}{B}\right)$
$\beta_{m n k}^2=K_x\left(\frac{(2 m+1) {\rm{ \mathsf{π} }} }{2 L}\right)^2+K_y\left(\frac{n {\rm{ \mathsf{π} }} }{B}\right)^2+K_z\left(\frac{k {\rm{ \mathsf{π} }} }{M}\right)^2+\frac{K_z^{\prime}}{M M^{\prime}}, \beta_{m n}^2=K_x\left(\frac{(2 m+1) {\rm{ \mathsf{π} }} }{2 L}\right)^2+K_y\left(\frac{n {\rm{ \mathsf{π} }} }{B}\right)^2+\frac{K_z^{\prime}}{M M^{\prime}}$

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