## 3 Results and Discussion
### Dispersity Sand Column Test  
This dispersion test belongs to the horizontal one-dimensional flow plane two-dimensional dispersion test, and the longitudinal dispersion coefficient can be obtained through the point-by-point method. The calculation formula of the point-by-point method is simple, and the meaning is clear. It avoids complicated mathematical derivation using other methods. The sand is assumed to be isotropic, homogeneous, and with a one-dimensional stable flow field in the saturated sand, because the sand column is placed vertically. Therefore, the vertical flow direction is taken as the X axis; the horizontal flow direction is the Y and Z axes. When t = 0, 10 g of urea was added as the tracer at the origin, and the initial concentration was zero. If the Z term in the convection dispersion equation is set to zero, then the two-dimensional dispersion equation for the one-dimensional flow is obtained as follows [35]:  
\[\frac{\partial c}{\partial t}=\left(D_{L}/R_{d}\right)\frac{\partial^{2}c}{ \partial x^{2}}+\left(D_{T}/R_{d}\right)\frac{\partial^{2}c}{\partial y^{2}}- \left(u/R_{d}\right)\frac{\partial c}{\partial x}, \tag{10}\]  
The definite solution condition is as follows:  
\[\left\{ \begin{array}{l}C(x,y,0)=0\\ C(\pm\infty,y,t)=0\\ C(x,\pm\infty,t)=0\\ \int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}nCdxdy=m\end{array}  \right., \tag{11}\]  
\[\text{By transformation }\left\{ \begin{array}{l}Y=y\sqrt{\frac{D_{L}}{D_{T}}}\\ X=x-\left(u/R_{d}\right)t\end{array} \right.,\text{Formula (\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:  
Then, the expression of longitudinal dispersion coefficient \(D_{L}\) is  
\[D_{L}=\frac{R_{d}(t_{1}-t_{2})\Big{[}x^{2}-\big{(}u/R_{d}\big{)}^{2}t_{1}t_{2} \Big{]}}{4t_{1}t_{2}\ln\Big{(}\frac{Ct_{2}t_{2}}{c_{3}t_{3}}\Big{)}} \tag{18}\]  
where \(x\) is the coordinate of the observation hole, and the water velocity is \(u\) = 0.16 cm/min, being a known quantity. As long as concentrations \(C_{1}\) and \(C_{2}\) at \(t_{1}\) and \(t_{2}\) are measured and brought into Formulas (17) and (18), dispersion coefficient \(D\) and retardation coefficient \(R_{d}\) can be obtained by solving the equations simultaneously. The urea concentration values measured at A, B, and C at different time periods in the layer 16 fine sand from Shanghai are shown in Figure 13.  
The longitudinal dispersion and retardation coefficient of Shanghai layer 16 silty fine sand were calculated using Formulas (17) and (18), respectively, as shown in Table 3. Dispersion has an obvious size effect. Thus, the sensitivity of dispersion must be analyzed in actual conditions in a subsequent numerical simulation. It is well known that a size effect exists during dispersion. The dispersion observed in the laboratory was closely related to the scale of field observation for the actual dispersion in the field. The result of the current experiments shows that the dispersity observed in the sand column test was lower than that in the field test. The reason may be that the density of the soil column in the laboratory sand column test was larger than that used for the site sand, and the laboratory sand was more uniform than that on the site. The field dispersity test was performed in situ under primarily intact conditions; the structure and composition of the aquifer were not disturbed or damaged, and the obtained dispersion parameters were close to the actual values. Figure 13: Urea concentration values of three observation points at different time periods. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sand**} & **Water Velocity** & **Longitudinal** & **Retardation** \\  & **(cm/min)** & **Dispersion (cm)** & **Coefficient** \\ \hline Fine sand & 0.16 & 11.5 & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Longitudinal dispersion and block coefficient of fine sand in layer 16 of Shanghai. [PERSON] et al.