# Mathematical Model of Ammonium Nitrogen Transport to Runoff with Different Slope Gradients under Simulated Rainfall

^{1}

^{2}

^{*}

## Abstract

**:**

_{m}) as a time-dependent parameter, which was not a constant value as in previous studies, and it was evaluated with a four-slope gradient and three rainfall intensities. The kinematic-wave equation for overland flow was solved by an approximately semi-analytical solution based on Philip’s infiltration model, while the diffusion-based mass conversation equation for overland nutrient transport was solved numerically. The results showed that the simulated runoff processes and ammonium nitrogen concentration transport to the overland flow agreed well with the experimental data. Further correlation analyses were made to determine the relationships between the slope gradient, rainfall intensity and the hydraulic and nutrient transport parameters. It turned out that these parameters could be described as a product of exponential functions of slope gradient and rainfall intensity. Finally, a diffusion-based model with a time-dependent mass transfer coefficient was established to predict the ammonium nitrogen transport processes at the experimental site under different slope gradients and rainfall intensities.

## 1. Introduction

^{32}P as a tracer to determine the degree of rainfall–soil interaction, and the results showed that the degree reached the maximum value at the soil surface and decreased very fast with soil depth. Thus, they assumed that the rainwater and soil water were mixed instantaneously and completely in the effective depth of interaction. However, the model did not perform well enough to simulate the solute transfer processes under high infiltration capacity conditions. Then, Ahuja and Lehman [15] proposed the incomplete mixing model, which meant that the runoff water does not mix completely with the soil water. Different from the mixing-layer model, the diffusion models have more clearly physical meanings which assume that the solute transfer between the underlying soil and the mixing layer is driven by the concentration gradient [19]. Then, the kinematic wave theory has been integrated into the solute mass balance equations in an attempt to figure out the mechanism of the solute exchange of the soil solution and exchange layer [16,20]. Gao, et al. [21] developed a physically based model that coupled the conventional convective–dispersion equation based on the mixing-layer theory and the Rose soil erosion model [22]. The model assumed that there existed an exchange layer which was equal to the shield layer of the Rose erosion model, and solute transport from the soil surface to overland flow was driven by raindrop dispersion. The model, for which all parameters could be measured or obtained from previous papers, fitted with the measured data very well [23,24]. Li, et al. [25] solved the model numerically under two rainfall intensities with field experiments. The raindrop dispersion-driven nutrient transport model fitted well with the experimental data.

_{m}) is a very important parameter to determine the peak values of nutrient concentration. For simplicity, the mass transfer coefficient is always treated as a constant during the whole rainfall experiment. However, the mass transfer coefficient will certainly be influenced by the increasing overland velocity and water height on the slope land. Based on the film theory, Wallach, et al. [26] put forward a method to calculate the mass transfer coefficient taking only diffusion into consideration. The equation indicates that the mass transfer coefficient is correlated with the water height of the overland flow. Therefore, this study aimed to establish a model based on Wallach’s assumption. The experimental study included (1) investigating the effects of different rainfall intensities and slopes on nutrient transport processes and (2) comparing the measured data with predicted data through an inverse estimate of model parameters to test the accuracy of the model.

## 2. Materials and Methods

#### 2.1. Soil and Slope Preparation

_{2}Cr

_{2}O

_{7}oxidation at 180 °C.

#### 2.2. Rainfall Simulation

#### 2.3. Experimental Procedure

^{3}to ensure the consistency of the physical and chemical soil properties. Then, the tested slope was levelled and left undisturbed until the beginning of the experiment at 50 mm/h. Before the start of the experiment at 25 mm/h, the above procedures were repeated. The experiments were carried out around 6 AM to eliminate the influences of wind. In order to ensure that the initial moisture content of each treatment was consistent, the slope was wetted with pre-wet treatment, which meant a constant rainfall intensity of 20 mm/h was applied on the experimental plot until the generation of runoff 12 h before the experiments. The soil moisture content of the surface soil (0–10 cm) was measured by the oven-dried method and the ammonium concentration of the surface soil was measured by indophenol-blue colorimetric methods using a spectrophotometer prior to the start of the experiment (Table 1).

#### 2.4. Theoretical Analysis

#### 2.4.1. Governing Equation

^{2}/min), h(x, t) is runoff depth (cm), x is the distance (cm) along the overland flow plane, t is time (min), i is infiltration rate (cm/min), and r is the rainfall intensity (cm/min).

^{1/3}), and $\mathrm{J}$ is the hydraulic gradient.

^{3}/cm

^{3}), ${\mathsf{\rho}}_{\mathrm{s}}$ is the soil bulk density (g/cm

^{3}), ${\mathrm{k}}_{\mathrm{s}}$ is solute adsorption coefficient (cm

^{3}/g), and ${\mathrm{k}}_{\mathrm{m}}$ is the mass transfer coefficient (cm/min).

#### 2.4.2. Solution of the Surface Runoff Equation

_{p}is the time of ponding (min), S is the sorptivity (cm min

^{−1/2}).t

_{p}and $\Delta \mathrm{t}$ can be solved by the modified Philip equation, ${\mathrm{t}}_{\mathrm{p}}=\frac{{\mathrm{S}}^{2}}{{2\mathrm{r}}^{2}}$ and $\Delta \mathrm{t}=\frac{{\mathrm{S}}^{2}}{{4\mathrm{r}}^{2}}$.

#### 2.4.3. Solution of the Solute Transport Equation

_{m}can be determined by the film theory. It is assumed that the overland flow can be divided into two regions: a near-surface laminar layer and a turbulent, completely mixed layer. The mass transfer is supposed to driven by molecular diffusion. Then, the mass transfer coefficient k

_{m}can be deduced by the hydraulic mechanics of the overland flow [26]. The mass transfer coefficient can be calculated by

^{2}/s), ρ is the density of water (kg/m

^{3}), R is the hydraulic radius (cm), approximately equal to the runoff height, μ is the water viscosity (kg/m), and D

_{w}is the solute diffusivity in water (cm

^{2}/h). Equation (11) indicates that the mass transfer coefficient k

_{m}is proportional to the h

^{1/3}. For simplicity, the solute concentration along the slope is assumed to be uniform. By substituting Equation (11) into Equations (6) and (7), we obtain the following:

#### 2.5. Data Analysis

^{2}) and root mean square error (RMSE) were applied to evaluate the agreement between the simulated results and experimental data in runoff and the nutrient transport process. R

^{2}and RMSE can be expressed as

## 3. Results and Discussions

#### 3.1. Effects of Slope Gradient and Rainfall Intensity on Runoff Process

^{3}under a rainfall intensity of 75 mm/h, respectively, indicating a higher runoff rate on a slope with a larger slope gradient. This phenomenon could also be observed under rainfall intensities of 50 mm/h and 25 mm/h. The results were in agreement with former research [29,30,31]. A possible explanation could be that as the slope steepens, more fine particles will be transported by overland flow, which may clog pores below the soil surface, thus reducing the infiltration capacity of steeper slopes [32]. The total runoff also increased with increasing rainfall intensities (Table 2), and runoff with larger rainfall intensities increased more quickly than for smaller rainfall intensities (Figure 2). From Table 2, it can be seen that the total runoff at a rainfall intensity of 75 mm/h was 6.9 times larger than that of 25 mm/h on a 5° slope, which was consistent with the results of Wei, et al. [33]. Generally, as rainfall intensity increased, the time to runoff was shorter. The infiltration rate more easily reached a stable state.

#### 3.2. Effects of Slope Gradient and Rainfall Intensity on the Transport of Ammonium Nitrogen

#### 3.3. Parameter Estimation

_{s}), the initial water content (θ

_{i}) and the saturated soil water content (θ

_{s}) are all listed in Table 1. The adsorption isotherm of ammonium nitrogen for the experimental soil was obtained based on the isothermal linear adsorption method [37], namely 1.74 cm

^{3}/g. The molecular diffusion coefficient in free water (D

_{w}) for ammonium nitrogen is assigned to be 0.063 cm

^{2}/h [38]. The Manning coefficient (n) is assigned to be 0.017 s/m

^{1/3}for the slope tested in this experiment. The water viscosity μ is 1.05 × 10

^{−3}kg/(m s) at 20 °C. The initial ammonium nitrogen concentrations of soil solution were measured directly and are shown in Table 1. The time to runoff, t

_{p}, was measured directly during the simulated rainfall (Table 3).

_{m}. The hydraulic parameters, S and c, could be estimated by the curve-fitting of the established runoff model (Equation (9)). The depth of the mixing layer under different rainfall intensities and slope gradient, h

_{m}, was determined by the curve-fitting method of the solute mass conversation model (Equations (12) and (13)). Parameters deduced by the curve-fitting method could be regarded as the best match between measured data and experimental data. However, these parameters could be influenced by many factors. For this experiment, rainfall intensity and slope gradient are the two dominant factors. In order to establish an ammonium nitrogen transportation model which can be applied to the experimental site, the effects of rainfall intensity and slope gradient on these parameters (t

_{p}, S, c, h

_{m}) were analyzed.

#### 3.4. Modeling Runoff Processes and Ammonium Nitrogen Concentration in Overland Flow

#### 3.4.1. Modeling Runoff Processes

^{2}and RMSE ranged from 0.87 to 0.99 and 2.40 to 8.12 cm

^{2}/min, respectively. It could be concluded that the established runoff model based on Philip’s infiltration equation could accurately describe the runoff processes. In this runoff model, the time to runoff (t

_{p}), adsorptivity (S) and coefficient (c) are the three parameters which can determine the runoff processes with a given rainfall intensity and slope gradient. It could be shown that time to runoff (t

_{p}) decreased with an increase of rainfall intensity and slope gradient despite the fact that the t

_{p}at 20° was larger than that at 15° under a rainfall intensity of 75 mm/h. The results were mainly due to the smaller infiltration capacity of the larger gradient. A larger rainfall intensity would surely accelerate the saturation processes of the slope soil. The t

_{p}was then fitted with the exponential function. The R

^{2}was 0.997, indicating the relationship among t

_{p}and slope gradient, and the rainfall intensity could be described by the exponential function. The relationships could be expressed as

^{2}values were 0.895 and 0.678, respectively. The relationships could be expressed as

#### 3.4.2. Modelling Ammonium Nitrogen Concentrations in Runoff.

^{2}and RMSE ranged from 0.87 to 0.99 and 2.40 to 8.12 mg/L, respectively, indicating that the nutrient transport into the runoff could be simulated by the proposed model. The depth of the exchange layer (h

_{m}) and mass transfer coefficient were two important parameters to determine the accuracy of this model. It has been reported that the depth of the exchange layer ranged between 2 mm and 3 mm [13,41]. Li et al. [25] conducted a field experiment to simulate the ammonium nitrogen and nitrate nitrogen transport processes. The results showed that the depth of the exchange layer, which was influenced by the rainfall intensity and slope gradient, was between 0.4 cm and 0.6 cm. The results of Yang, et al. [42] indicated that the depth of the interaction zone increased with an increase of rainfall intensity and slope gradient, which was similar to our results. The mixing depth ranged from 0.08–0.38 cm, which increased with increasing slope gradients and rainfall intensities despite the fact that the mixing depth of 15° was smaller than that of 10° under a rainfall intensity of 25 mm/h. Then, the exponential function was applied to fit the results. The R

^{2}was 0.981. The relationships could be expressed as

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Unit discharge of runoff over rainfall time under different slope gradients and different rainfall intensities: (

**a**) 75 mm/h, (

**b**) 50 mm/h, (

**c**) 25 mm/h.

**Figure 3.**The ammonium nitrogen (NH

_{4}

^{+}-N) concentration in runoff over rainfall time under different slope gradients and different rainfall intensities: (

**a**) 75 mm/h, (

**b**) 50 mm/h (

**c**) 25 mm/h.

**Figure 4.**Comparison of the calculated and measured unit discharge of runoff under different slope gradients and different rainfall intensities: (

**a**) 75 mm/h, (

**b**) 50 mm/h, (

**c**) 25 mm/h.

**Figure 5.**Comparison of the calculated and experimental ammonium nitrogen (NH

_{4}

^{+}-N) concentration in the runoff under different slope gradients and different rainfall intensities: (

**a**) 75 mm/h, (

**b**) 50 mm/h, (

**c**) 25 mm/h.

**Figure 6.**The mass transfer coefficient changes over time under different slope gradients and different rainfall intensities: (

**a**) 75 mm/h, (

**b**) 50 mm/h, (

**c**) 25 mm/h.

**Figure 7.**Simulated versus observed values for the unit discharge of runoff and the ammonium nitrogen (NH

_{4}

^{+}-N) concentration in runoff: (

**a**) unit discharge of runoff, (

**b**) ammonium nitrogen concentration in runoff.

**Table 1.**Soil physical and chemical properties of soil samples collected from the study area (mean ± deviation standard).

Soil Type | Soil Texture (%) | Soil Bulk Density (g/cm^{3}) | pH | Organic Matter (g kg^{−1}) | Initial Soil Water Content (cm^{3}/cm^{3}) | Saturated Soil Water Content (cm^{3}/cm^{3}) | Initial Ammonium Nitrogen Concentration of Soil Solution (mg NH_{4}^{+}-N/L) | ||
---|---|---|---|---|---|---|---|---|---|

Sand/% (2.0–0.02 mm) | Silt/% (0.02–0.002 mm) | Clay/% (<0.002mm) | |||||||

Sandy | 89.55 ± 0.39 | 5.43 ± 0.43 | 5.02 ± 0.27 | 1.45 ± 0.09 | 8.40 ± 0.16 | 2.81 ± 0.07 | 0.207 ± 0.03 | 0.50 ± 0.07 | 45.6 ± 2.25 |

**Table 2.**The variation of total runoff and ammonium nitrogen losses under different rain intensities and slope gradients.

Rainfall Intensity (mm/h) | Slope Gradient | Total Runoff (m^{3}) | Total Ammonium Nitrogen Losses (mg NH_{4}^{+}-N) |
---|---|---|---|

75 | 5 | 1.82 | 2768 |

10 | 2.01 | 5563 | |

15 | 2.11 | 7485 | |

20 | 2.18 | 9356 | |

50 | 5 | 1.09 | 1468 |

10 | 1.18 | 2252 | |

15 | 1.21 | 3341 | |

20 | 1.28 | 4141 | |

25 | 5 | 0.23 | 457 |

10 | 0.28 | 868 | |

15 | 0.35 | 1162 | |

20 | 0.37 | 1584 |

**Table 3.**The initial time to runoff (t

_{p}), adsorptivity (S), R

^{2}and RMSE obtained by curve-fitting the results of the runoff model under different rainfall intensities and slope gradients.

Designed Rainfall Intensities | Slope Gradient | t_{p} (min) | S (cm/min^{1/2}) | c | R^{2} | RMSE (cm^{2}/min) |
---|---|---|---|---|---|---|

75 | 5 | 2.15 | 0.26 | 0.15 | 0.94 | 8.12 |

10 | 1.60 | 0.22 | 0.12 | 0.96 | 6.89 | |

15 | 1.28 | 0.20 | 0.10 | 0.96 | 6.52 | |

20 | 1.45 | 0.21 | 0.06 | 0.99 | 3.93 | |

50 | 5 | 4.57 | 0.25 | 0.10 | 0.98 | 2.95 |

10 | 3.17 | 0.21 | 0.11 | 0.93 | 5.32 | |

15 | 3.28 | 0.21 | 0.05 | 0.95 | 4.58 | |

20 | 2.70 | 0.19 | 0.08 | 0.92 | 5.99 | |

25 | 5 | 18.15 | 0.25 | 0.13 | 0.87 | 2.75 |

10 | 15.35 | 0.23 | 0.10 | 0.92 | 2.55 | |

15 | 12.50 | 0.21 | 0.05 | 0.89 | 3.44 | |

20 | 11.20 | 0.20 | 0.06 | 0.94 | 2.40 |

**Table 4.**The mixing depth (h

_{m}), R

^{2}and RMSE obtained by curve-fitting the results of the solute transport model under different rainfall intensities and slope gradients.

Designed Rainfall Intensity (mm/h) | Slope Gradient | h_{m} (cm) | R^{2} | RMSE (mg/L) |
---|---|---|---|---|

75 | 5 | 0.20 | 0.89 | 0.92 |

10 | 0.30 | 0.96 | 1.04 | |

15 | 0.31 | 0.92 | 1.63 | |

20 | 0.38 | 0.96 | 1.14 | |

50 | 5 | 0.14 | 0.95 | 0.89 |

10 | 0.18 | 0.99 | 0.73 | |

15 | 0.20 | 0.92 | 1.73 | |

20 | 0.22 | 0.92 | 1.56 | |

25 | 5 | 0.07 | 0.90 | 1.67 |

10 | 0.10 | 0.96 | 1.07 | |

15 | 0.08 | 0.94 | 2.45 | |

20 | 0.12 | 0.96 | 1.78 |

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**MDPI and ACS Style**

Xing, W.; Yang, P.; Ao, C.; Ren, S.; Xu, Y.
Mathematical Model of Ammonium Nitrogen Transport to Runoff with Different Slope Gradients under Simulated Rainfall. *Water* **2019**, *11*, 675.
https://doi.org/10.3390/w11040675

**AMA Style**

Xing W, Yang P, Ao C, Ren S, Xu Y.
Mathematical Model of Ammonium Nitrogen Transport to Runoff with Different Slope Gradients under Simulated Rainfall. *Water*. 2019; 11(4):675.
https://doi.org/10.3390/w11040675

**Chicago/Turabian Style**

Xing, Weimin, Peiling Yang, Chang Ao, Shumei Ren, and Yao Xu.
2019. "Mathematical Model of Ammonium Nitrogen Transport to Runoff with Different Slope Gradients under Simulated Rainfall" *Water* 11, no. 4: 675.
https://doi.org/10.3390/w11040675