2.2.1. Mathematical Model

• Water Movement Model.

The mathematical model of water movement in the water level fluctuation zone can be expressed as the Richards equation, as follows:

$$\mathcal{L}\mathcal{L}(h)\frac{\partial\theta}{\partial t} = \frac{\partial}{\partial z}\Big[K(h)[\frac{\partial h}{\partial t} - \cos(\alpha)]\Big] - S(z,t) \tag{1}$$

where *C*(*h*) is the water capacity (cm−1); *K*(*h*) is the hydraulic conductivity (cm/d); *h* is the negative pressure (cm); *z* represents the position coordinates in the parallel water flow direction (cm); *t* is the time (d); *α* is the angle between the water flow direction and the vertical (◦); *θ* is the volumetric water content (cm3/cm3); and *S*(*z*, *t*) is the water absorption strength of plant roots (cm3/cm3·d<sup>−</sup>1).

• Solute transport model in the Vadose zone.

The solute transport model only considers the behavioral characteristics of convection, diffusion, adsorption, degradation, etc., and uses the traditional convection–diffusion equation to describe the transport process. The equation is expressed as:

$$\frac{\partial}{\partial t}(\theta \mathcal{C}) = \frac{\partial}{\partial z}(\theta D\_L \frac{\partial \mathcal{C}}{\partial z} - v \mathcal{C}) - \rho\_b \frac{\partial (\rho\_s K\_L \mathcal{C})}{\partial t} - \mathcal{C}\_0 \exp(-kt) \tag{2}$$

where *C* is the solute concentration in the liquid phase (mg/L); *DL* is the longitudinal dispersion coefficient (cm/d); *v* is the Darcy flow velocity (cm/d); *ρ<sup>b</sup>* is the soil bulk density (mg/cm); *ρ<sup>s</sup>* is the soil bulk density (mg/cm3); *KL* is the adsorption distribution coefficient (cm3/g); *<sup>C</sup>*<sup>0</sup> exp(−*kt*) is the source-sink term (cm3/cm3·d<sup>−</sup>1); the others are the same as above.

• Nitrogen migration and transformation model.

The transport model of nitrogen in the soil varies according to the different forms of nitrogen. The transport and transformation process of NH4 <sup>+</sup> is mainly subjected to adsorption and nitrification. The equation is as follows:

$$\begin{cases} \begin{array}{ll} \frac{\partial \langle \theta c\_{1} \rangle}{\partial t} = \frac{\partial}{\partial z} \Big( \theta D\_{L} \frac{\partial c\_{1}}{\partial z} \Big) - \frac{\partial}{\partial z} (V c\_{1}) - \rho\_{b} \frac{\partial \langle \rho\_{b} k\_{c} c\_{1} \rangle}{\partial t} - k\_{1} \theta c\_{1} \\ c\_{1}(Z, 0) = c\_{10}(Z) & 0 \le Z \le L, t = 0 \\ - \left( \theta D\_{L} \frac{\partial c\_{1}}{\partial z} - q c\_{1} \right) = \varepsilon(t) c\_{0} & Z = 0, t = 0 \\ c\_{1}(L, t) = c\_{1}L & Z = L, t > 0 \end{array} \tag{3} \\ \begin{array}{ll} \text{(3)} \\ \end{array} \end{cases} \tag{4}$$

Meanwhile NO2 − and NO3 − are mainly affected by nitrification and denitrification, and the equation is as follows:

$$\begin{cases} \frac{\partial(\theta c\_i)}{\partial t} = \frac{\partial}{\partial z} \left( \theta D\_L \frac{\partial c\_i}{\partial z} \right) - \frac{\partial}{\partial z} (v\_i c\_i) - k\_i \theta c\_i\\ c\_i(Z, 0) = c\_{i0}(Z) & 0 \le Z \le L, t = 0\\ c\_i(0, t) = c\_{i0}(t) & Z = 0, t > 0\\ c\_i(L, t) = c\_i L & Z = L, t > 0 \end{cases} \tag{4}$$

where *θ* is the soil volume water content (cm3/cm3); *c*<sup>1</sup> is the soil solution NH4 <sup>+</sup> concentration (mg/L); *DL* is the longitudinal diffusion coefficient (cm/d); *ke* is the adsorption distribution coefficient (cm3/g) for NH4 <sup>+</sup> in soil; *k*1, *k*<sup>2</sup> are the denitrification rate constants (d<sup>−</sup>1) for NH4 <sup>+</sup> and NO2 <sup>−</sup>, respectively; *c*10(*t*) is the soil NH4 <sup>+</sup> initial concentration (mg/L); *c*<sup>0</sup> is the inlet solution NH4 <sup>+</sup> concentration (mg/L); *c*1*<sup>L</sup>* is the diving NH4 <sup>+</sup> concentration (mg/L); *c*2,*c*<sup>2</sup> are the NO2 − and NO3 − concentration (mg/L), respectively; *k*<sup>3</sup> is the denitrification rate constant (d<sup>−</sup>1); *c*20(*z*), *c*30(*z*) are the soil NO2 − and NO3 − initial concentration (mg/L), respectively; and *c*2*L*,*c*3*<sup>L</sup>* are the diving NO2 − and NO3 − concentration (mg/L), respectively. No nitrogen input through the water.

2.2.2. Initial Conditions and Boundary Conditions

• Initial conditions.

At the initial moment, the water level was set to 30 cm, and the initial concentration of pollutants is shown in Table 2.



• Boundary conditions.

1. Water transport and boundary conditions.

According to the experimental model, the upper boundary was in direct contact with the atmosphere and was set as the atmospheric boundary. The lower boundary was set as the variable head boundary due to the rise and fall of the water level.

2. Solute transport boundary.

According to the model, the upper boundary condition was the pollutant concentration boundary, and the lower boundary condition was the zero concentration gradient boundary. In the model setting, the primary considerations were adsorption and desorption, as well as nitrification and denitrification.
