Analytical Model of Pumping Tests with Depth-Dependent Hydraulic Conductivity in Leakage Aquifer System
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摘要: 传统的抽水试验模型假定含水层的渗透系数是恒定的，然而，由于地质作用等因素的影响，含水层渗透系数存在随深度衰减的现象.建立考虑越流、井储效应和随深度衰减的渗透系数的抽水试验数学模型，并推导了半解析解，其中，假设渗透系数随深度呈指数衰减.系统分析随深度衰减的渗透系数对抽水试验结果的影响.结果表明：含水层渗透系数随深度衰减系数（A）越大，井筒内地下水的降深越大；当含水层渗透系数随深度衰减时，滤水管的位置对抽水试验结果的影响比较显著，滤水管在含水层顶部时的井筒内降深比滤水管在含水层底部时的井筒水位降深要小；传统的抽水试验模型采用井筒水位降深数据反演的常数渗透系数与随深度衰减的渗透系数平均值近似.Abstract: The traditional pumping test model typically assumes that the aquifer is a constant, whereas the actual aquifer frequently exhibits depth-dependent hydraulic conductivity due to geological effects and other factors. In this paper, it develops a mathematical model of pumping tests that takes into account leakage, wellbore storage effect, and depth-dependent hydraulic conductivity, and solves the semi-analytical solution, where the depth-dependent hydraulic conductivity represented by an exponential function. The results show that: the greater the attenuation coefficient (A) of depth-dependent hydraulic conductivity, the greater the drawdown in the wellbore and the greater the range of landing funnel caused by pumping. When the hydraulic conductivity decays with depth, the location of the well screen has a significant impact on the pumping test results, the drawdown at the wellbore with the well screen at the upper aquifer location is smaller than the drawdown at the wellbore with the well screen at the lower aquifer location; the estimated hydraulic conductivity is an approximation of the depth-decaying hydraulic conductivity based on drawdown at the wellbore by traditional pumping test model.
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Key words:
- pumping test /
- numerical modeling /
- hydraulic conductivity /
- leakage /
- hydrogeology
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图 1 考虑越流和变化渗透系数的抽水试验模型
Fig. 1. Schematic diagram of the pumping test model considering the leakage and depth-decaying hydraulic conductivity
图 2 本研究模型和Yang et al. (2006)在不同$ r $处的水位降深对比
Fig. 2. Comparison of drawdown calculated by this study and Yang et al.(2006) at different locations
图 4 不同$ A $值在$ t=2\ \mathrm{d} $的降深等值线
Fig. 4. The drawdown contour map for different $ A $ when $ t=2\ \mathrm{d} $
图 5 井筒内不同$ \mathrm{越}\mathrm{流}\mathrm{系}\mathrm{数}R $下地下水降深
Fig. 5. The drawdown at the wellbore for different $ R $
图 7 不同滤水管位置下的井筒内的水位降深
Fig. 7. The drawdown at the wellbore for different locations of well screen
表 1 模型参数(据Zheng et al., 2019)
Table 1. Parameters used in this study (from Zheng et al., 2019)
参数 单位 取值 抽水流量($ Q $) $ {\mathrm{m}}^{3}/\mathrm{d} $ 83 井半径($ {r}_{w} $) $ \mathrm{m} $ 0.15 弹性贮水或释水率($ {S}_{s} $) $ 1/\mathrm{m} $ 0.000 5 含水层顶部渗透系数($ {K}_{0} $) $ \mathrm{m}/\mathrm{d} $ 4.0 衰减系数($ A $) $ 1/\mathrm{m} $ 0.11 弱透水层渗透系数($ {K}_{a} $) $ \mathrm{m}/\mathrm{d} $ 0.1 含水层厚度($ B $) $ \mathrm{m} $ 8.0 弱透水层厚度($ m $) $ \mathrm{m} $ 15 滤水管顶部位置($ d $) $ \mathrm{m} $ 2.0 滤水管底部位置($ l $) $ \mathrm{m} $ 6.0 
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表 2 不同$ A $下的$ {K}_{\mathrm{a}\mathrm{v}\mathrm{g}} $、$ {K}_{\mathrm{e}\mathrm{s}\mathrm{t}} $和$ E $值
Table 2. The values of $ {K}_{\mathrm{a}\mathrm{v}\mathrm{g}}, {K}_{\mathrm{e}\mathrm{s}\mathrm{t}} $ and $ E $ for different $ A $
$ A $(1/m) $ {K}_{\mathrm{a}\mathrm{v}\mathrm{g}} $(m/d) $ {K}_{\mathrm{e}\mathrm{s}\mathrm{t}} $(m/d) $ E $ 0.01 3.844 3.90 1.46% 0.10 2.753 2.74 0.47% 0.20 1.995 2.00 0.25% 0.30 1.516 1.51 0.40%
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