An Approximate Analytical Solution for Two-Region Non-Darcian Flow Toward a Well in a Leaky Aquifer
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摘要: 构建了越流含水层中抽水井附近非达西流动的两区模型, 即距离抽水井较近的区域为非达西流, 而相对较远区域为达西流, 两区之间的临界半径可根据临界雷诺数确定.采用线性化近似方法和Laplace变换相结合分别得到了非达西流区域和达西流区域的水位降深在拉氏空间下的解析解, 应用数值Laplace逆变换—Stehfest方法得到其在实空间下的水位降深, 并与相应的全达西模型和全非达西模型进行了比较, 结果表明: 在抽水初期不同临界半径情况下非达西流区域的水位降深曲线互相重合, 并与全非达西模型所得到的结果相吻合; 在抽水后期的结果与全非达西流模型存在明显的差异.在抽水初期, 非达西渗透系数kD越大, 非达西流区域和达西流区域的水位降深越大; 在抽水后期, kD越大, 非达西流区域水位降深越小, 而kD的变化对达西流区域的水位降深影响较小.越流补给在非达西流情况下对水位降深的影响与达西流情况下的结果基本类似, 且只存在于抽水后期.考虑井储影响后, 不同kD和越流补给因子BD情况下抽水初期井中的水位降深在双对数坐标表现为直线且相互重合.Abstract: In this paper, we propose a two-region non-Darcian flow model near a pumping well in a leaky aquifer. The flow near the pumping well is assumed to be non-Darcian, with the area nearby defined as non-Darcian flow region, while the flow far away from the pumping well can be regarded as Darcian flow. The critical distance distinguishing the non-Darcian flow region and Darcian flow region can be determined by the critical Renolds number. We have used a linearization procedure coupled with Laplace transform to solve such a two-region non-Darcian flow model. The drawdowns both in the non-Darcian flow region and Darcian flow region have been obtained by using the so-called Stefest numerical Laplace inversion method. We have compared our results with those for the one-region Darcian flow model and the one-region non-Darcian flow model. The results indicate that: (1) The drawdowns in the non-Darcian flow region of different critical distances approach the same asymptotic value at early stages, as well as the result for the one-region non-Darcian flow model; while at late stages, significant difference has been found between the drawdowns obtained in this study; (2) A larger "non-Darcian hydraulic conductivity" kD results in a greater drawdown in the entire aquifer at early stages, while leads to a smaller drawdown in the non-Darcian flow region at late stages and has little impact on the drawdowns in the Darcian flow region; (3) The leakage effect on the drawdown is similar to that for the Darcian flow case, and it only exists at late stages; (4) When the wellbore storage is considered, all the drawdowns inside the well for different kD and dimensionless leakage parameter BD values approach the same asymptotic value at early stages and are straight lines in double logarithmic paper at early stages.
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Key words:
- pumping well /
- leaky aquifer /
- non-Darcian flow /
- environment engineering /
- hydrogeology
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图 2 非达西流区域水位降深与Wen et al.(2008a)全非达西流动模型比较
Fig. 2. Comparison of the solution in this study with Wen et al. (2008a)
图 3 不同kD(a)和不同BD(b)非达西流区域水位降深规律
Fig. 3. Drawdowns in non-Darcian flow region for different kD values (a) and BD values (b)
图 4 不同kD(a)和不同BD(b)达西流区域水位降深规律
Fig. 4. Drawdowns in Darcian flow region for different kD values (a) and BD values (b)
图 5 考虑井储影响后不同kD(a)和不同BD(b)井中水位降深曲线
Fig. 5. Drawdowns inside the well for different kD values (a) and BD values (b) with wellbore storage
表 1 无量纲变量
Table 1. Dimensionless variables used in this study
${r_D} = \frac{r}{m}$ ${r_{wD}} = \frac{{{r_w}}}{m}$ ${r_{cD}} = \frac{{{r_c}}}{m}$ ${R_{cD}} = \frac{{{R_c}}}{m}$ ${q_D} = - \frac{{4\pi {m^2}}}{Q}q$ ${s_D} = \frac{{4\pi {k_2}m}}{Q}s$ ${k_D} = \frac{{{k_1}}}{{{k_2}}}{\left[ {\frac{Q}{{4\pi {m^2}}}} \right]^{1 - n}}$ ${t_D} = \frac{{{k_2}t}}{{Sm}}$ ${B_D} = \frac{{B{k_2}}}{{{m^2}}}$ 
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