Numerical model of two-region flow by constant-head pumping in an unconfined aquifer
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摘要:
为了揭示在非承压含水层中定水头抽水试验引起的达西−非达西两区流动机理，提出基于有限差分法的地下水定水头抽水井流数值模型。该模型根据抽水的流态特征将含水层分为2个区域：靠近抽水井的有限非达西渗流区域和远离抽水井的半无限达西渗流区域，其中非达西流区域流态的模拟基于Izbash方程实现。通过与COMSOL Multiphysics的有限元数值解进行比较，验证了所提出数值解的可靠性。最后，研究有限非达西流效应对水头和抽水井抽水速率的影响以及井内水头对抽水井抽水速率的影响。研究表明：在抽水试验中非达西区域的影响不可忽略，湍流会分别导致两区流中水头较纯非达西流和纯达西流的水头偏大和偏小，且随抽水时间的增加逐渐变大；通过减小抽水井井内水头或增大非达西系数可提高抽水速率，但该影响会随抽水时间的增加而逐渐减弱；断面流量随径向距离的增大而不断减小，断面流量与径向距离曲线下降速率不断减小，且在转换界面处会出现转折点。该模型为定量研究在非达西流和达西流耦合作用下抽水井附近的井流水头特征提供了一种简洁的方法，并为调查定水头抽水测试期间的抽水速率提供理论依据。
Abstract:In order to reveal the mechanism of two-region Darcian and non-Darcian flow induced by constant-head pumping tests in unconfined aquifers, a numerical model of well flow by constant-head pumping underground based on finite difference solution was proposed. In the model, it is assumed that the aquifer is divided into two regions according to the pumping flow characteristics: the finite non-Darcian flow region near the pumping well and the semi-infinite Darcian flow region far away from the pumping well. Specifically, the flow regime in the non-Darcian flow region was simulated by the Izbash’s equation, and the reliability of the proposed solution was verified by comparison with finite element numerical solutions by COMSOL Multiphysics. Finally, the influence of finite non-Darcian effect on hydraulic head and pumping rate, as well as the influence of hydraulic head in the pumping well on pumping rate, was especially studied. The results show that the influence of non-Darcian flow region in the pumping test cannot be ignored. The turbulent flow makes the hydraulic head of the two-region flow larger than that of the pure non-Darcian flow and smaller than that of pure Darcian flow. Besides, such difference of the hydraulic head is increased with the pumping time. The pumping rate can be increased by reducing the hydraulic head in the pumping well or increasing the non-Darcian coefficient, but the effect is gradually decreased as the pumping continues. The flow rate at the cross-section is decreased with the increase of radial distance, the gradient of the flow rate-radial distance curve is decreased with time, and a turning point will appear at the conversion interface between the two regions. The proposed model provides a simple method for quantitative studies on the characteristics of hydraulic head near the pumping well under the coupling of non-Darcian and Darcian effects, and provides a theoretical basis for determining the pumping rate during constant-head pumping test.
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图 1 考虑两区转换的非承压含水层定水头抽水模型
rw—抽水井有效半径，m；hw—抽水井中水头，m；R—N区和D区转换界面到抽水井中心的距离，m；b—含水层厚度，m
Fig. 1 Schematic of the constant-head test in an unconfined aquifer considering the two-region transform
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图 2 有限差分解与Jacob和Lohman解对比
Fig. 2 Comparison between the finite difference solution with Jacob and Lohman solution
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图 3 不同R值下有限差分解与数值解的水头−时间曲线对比
Fig. 3 Comparison of head-time curves between finite difference solution and numerical solution for different R
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图 4 有限差分解界面距离−时间曲线与数值解的比较
Fig. 4 Interface distance-time curve of finite difference solution vs numerical solution
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图 6 泵井内不同水头下流量的动态变化
Fig. 6 Dynamic development of the flow rates with different hydraulic head inside the pumping well
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图 7 不同非达西湍流系数对泵送流量的影响
Fig. 7 Effects of different non-Darcy coefficients on pumping flow
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图 8 不同时间下径向距离对断面流量的影响
Fig. 8 Influence of radial distance on cross-section flow at different time
表 1 假设案例的参数值
Table 1 Parameters of each hypothetical cases
案例 $ {h}_{\mathrm{w}}/\mathrm{m} $ ${K}_{1}/{(\mathrm{m}\cdot\mathrm{d}^{-1})}^{ {{n} }_{1} }$ ${K}_{2}/(\mathrm{m}\cdot \mathrm{d}^{-1})$ $ {n}_{1} $ $ {S}_{1} $ $ {S}_{2} $ $ b/\mathrm{m} $ ${r}_{{\rm{w}}}/\mathrm{m}$ $ r/\mathrm{m} $ $ d/\mathrm{m}\mathrm{m} $ A 9 1.1 1.1 1.1 0.01 0.01 12 0.2 3 0.3 B 7,8,9 1.1 1.1 1.2 0.01 0.01 10 0.2 5 0.3 C 9 1.1 1.1 1.0,1.2,1.4 0.01 0.01 12 0.2 2 0.3 D 9 1.1 1.1 1.2 0.01 0.01 12 0.2 2 0.3 
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