Effect of Mixing Effect and Scale-Dependent Dispersion for Radial Solute Transport near the Injection Well
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摘要: 径向弥散是指溶质在径向流场下的迁移规律，被广泛用于描述含水层修复领域中污染物的迁移过程.然而，在现有描述径向弥散的模型中，往往忽略了井内混合效应对溶质径向弥散的影响.建立新的注入井附近溶质径向运移动力学模型，同时考虑井内混合效应与弥散度的尺度效应.采用Laplace变换推导该模型的半解析解，利用Stehfest数值逆变换获取溶质在实数空间的解.通过与不考虑混合效应的模型对比研究混合效应对溶质径向弥散的影响，并利用室内渗流槽中的溶质径向弥散实验数据验证模型的合理性与适用性.结果表明：混合效应和尺度效应对注水井附近溶质径向弥散有显著影响.具体地讲，井内的混合效应越显著，在井壁处及含水层中的穿透曲线越低，溶质浓度达到峰值所需时间越长，与不考虑混合效应模型的差异越明显；随尺度效应的增强，溶质提前穿透且扩散范围变大，溶质浓度达到峰值所需时间越长；与前人的模型相比，本研究模型能更好地模拟注水井附近的溶质径向弥散问题.Abstract: Radial solute transport refers to a dispersive transport process of a solute under a radial flow field, which has been widely used to describe the solute transport around the well in aquifer remediation. However, in the previous studies, the mixing effect in the wellbore has often been ignored, assuming that the wellbore is infinitely small or the concentration of the wellbore is constant during the injection period. In this study, a new mathematical model to describe radial solute transport of the injection well is proposed considering both the mixing effect and scale effect. The analytical solution is derived by the Laplace transform and Stehfest numerical inversion method. The linear dispersion model with mixing effect (LDM) is compared with the linear dispersion model with no-mixing effect (LDNM) to illustrate the mixing effect and the scale effect. Moreover, the robustness of the new model is tested using the experimental data. The results show that mixing effect and scale effect have a great influence on radial solute transport. Specifically, the greater mixing effect results in the lower the breakthrough curves (BTCs) in both wellbore and aquifer, the longer time for the solute concentration to reach its peak. With the well radius increasing, the difference between the models with and without mixing effect is more obvious. Additionally, with the increase of scale-dependent dispersion, the arrival time of the BTC peak values decreases. LDM (linear dispersion model with mixing effect) is more reasonable than LDNM (linear dispersion model with no-mixing effect) in describing radial solute transport.
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图 1 井壁处(a)和r=12.5 cm处(b)在不同井半径下穿透曲线的对比
Fig. 1. Comparison of BTCs for different well radii at wellbore (a) and at r=12.5 cm(b)
图 2 井壁处(a)和r=12.5 cm处(b)在不同井半径下相对误差error随时间的变化
Fig. 2. Comparison of error for different well radii at wellbore (a) and at r=12.5 cm(b)
图 3 不同k值下r=12.5 cm处LDM和LDNM的穿透曲线的比较
Fig. 3. Comparison of BTCs between LDM and LDNM for different k values at r=12.5 cm
图 4 井壁处及r=15 cm(a)和r=20 cm(b)处LDM和LDNM穿透曲线与观测值的拟合
Fig. 4. Fitness of the observed BTCs by LDM and LDNM at the wellbore at r=15 cm(a) and r=20 cm (b)
表 1 溶质不同注入条件下的特定解析解
Table 1. The specific solution under different boundary conditions
溶质注入边界条件 LDM LDNM Laplace空间下的解 浓度通量连续 $C_{1}=\frac{C_{0} /(s \beta+1)}{s \varepsilon k r_{\mathrm{w}}^{\gamma+1} K_{\gamma+1}\left(\varepsilon r_{\mathrm{w}}\right)}$ $C_{1}=\frac{C_{0}}{s \varepsilon k r_{\mathrm{w}}^{\gamma+1} K_{\gamma+1}\left(\varepsilon r_{\mathrm{w}}\right)}$ $\bar{C}_{r}=C_{1} r^{\gamma} K_{\gamma}(\varepsilon r)$ 瞬时注入 $C_{1}=\frac{M /(s \beta+1)}{Q \varepsilon k r_{\mathrm{w}}^{\gamma+1} K_{\gamma+1}\left(\varepsilon r_{\mathrm{w}}\right)}$ $C_{1}=\frac{M}{Q \varepsilon k r_{\mathrm{w}}^{\gamma+1} K_{\gamma+1}\left(\varepsilon r_{\mathrm{w}}\right)}$ $\bar{C}_{r}=C_{1} r^{\gamma} K_{\gamma}(\varepsilon r)$ 
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