# Optimal In-Stream Structure Design through Considering Nitrogen Removal in Hyporheic Zone

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Model

#### 2.1.1. Hydrologic Engineering Center’s River Analysis System (HEC-RAS) for Surface Water Simulation

_{1}and Z

_{2}are elevations of the main channel inverts (m), Y

_{1}and Y

_{2}are depths of water at cross sections (m), V

_{1}and V

_{2}are average velocities at cross sections (m/s), c

_{1}and c

_{2}are velocity weighting coefficients, g is gravitational acceleration (m/s

^{2}), and h

_{e}is energy head loss (m).

#### 2.1.2. COMSOL Multiphysics for Hyporheic Flow Simulation

^{3}), u is the flow velocity (m/s), t is the time (s), p is the pressure (Pa), μ is the dynamic viscosity (Pa·s), μ

_{e}is the effective viscosity (Pa·s), k is the intrinsic permeability (m

^{2}), ε is the Brinkman porosity, and μu/k is the Darcy term (Pa/m).

#### 2.1.3. Nitrogen Transport/Removal Calculation

_{0}is the initial concentration of the nitrogen (i.e., 0.307 mol/m

^{3}in this study), C

_{RT}is the concentration of the nitrogen (mol/m

^{3}) with the reaction time of RT, C is the change in the nitrogen concentration in the time duration of RT, and r is the reaction rate constant (day

^{−1}). It is worth noting that the unit of the RT in Equation (3) should be day. Thus, the nitrogen removal amount (NRA, mol) for each flow path can be expressed as follows:

_{max}):

^{3}) in the HZ at the RTmean (i.e., the mean RT) and the total initial nitrogen in the HZ. Thus, the NRR can be computed as follows:

#### 2.1.4. Modeling Scenarios

^{3}/s, and depth to bedrock of 5 m. It is worth noting that, in this study, the h values used for determining the optimal h lie within the range of the modeling scenarios (i.e., [0.3, 0.7] m), but with equal intervals of 0.01 m.

#### 2.2. Optimal In-Stream Structure Design

#### 2.2.1. Framework

#### 2.2.2. Relevant Indicators

_{max}) can be related to the height of the weir (h). Therefore, the L

_{max}can be expressed as follows:

_{max}. It means that particle tracing is first used in the numerical simulations and then the RT values for the subsurface flow paths are used to develop the coefficients of the regression equations. This study assumes that (1) the RT and the hyporheic flux in the upstream (Q

_{u}) can be expressed as functions of x using the regression method, where x is the distance measured from the left boundary of the domain, and (2) the coefficients of the regression equations can be related to h. Previous studies have shown that the regression equations in the form of power law can be the best to describe the RT and Q

_{u}[36,37] and, therefore, the RT and Q

_{u}can be expressed as follows:

_{1}(h) and b

_{1}(h) are the coefficients of the regression equation for the RT, and a

_{2}(h) and b

_{2}(h) are the coefficients of the regression equation for the Q

_{u}. Based on the simulation results under different scenarios with different h values, the forms of these coefficients can be determined (see Section 3 for details).

#### 2.2.3. Objective Function

_{j}and NRR

_{j}are the CNRA and NRR values corresponding to the j-th h value, respectively; NCNRA

_{j}and NNRR

_{j}are the j-th normalized CNRA and NRR values; M is the data length, which is equal to the total number of h values.

#### 2.3. Validation and Sensitivity Analysis Methods

^{2}·s), c

_{NO3-}is the concentration of the nitrate (mol/m

^{3}), and k

_{d}is the reaction rate constant of denitrification (s

^{−1}).

_{i}is the i-th height (m) and q

_{HZ}is the hyporheic flux (m

^{2}/s). This index considers the HZ as a single reaction tank, and the RT

_{max}is the maximum value of the denitrification reaction time. The other one is as follows:

_{max}(h

_{i})).

## 3. Results and Discussion

#### 3.1. Regression Equations

#### 3.1.1. The Maximum Upstream Distance in the Subsurface Flow Influenced by the Weir (L_{max})

_{max}; moreover, a higher weir should lead to a larger L

_{max}. In this study, the simulation results show that the L

_{max}increases along with the increase of the height of the weir (h) (Figure 2). In order to evaluate the accuracy of the regression equation, the coefficient of determination (R

^{2}) is selected as the assessment criterion. As the R

^{2}value is very high (i.e., 0.98), the relationship between L

_{max}and h can be expressed as follows:

_{max}. Moreover, the relationship is simple and linear, which could be partly due to the homogeneity of the streambed.

#### 3.1.2. Residence Time (RT)

_{max}increases along with the increase of the h. According to the assumptions in Section 2.2.2, the RT can be expressed as the function of x in the form of a power law using the regression method. Therefore, the coefficients of the regression equations for all h values can be obtained, which are listed in Table 2. The R

^{2}values of the regression equations are also shown in this table, and they are all higher than 0.91, indicating that the regression equations fit the simulation results very well.

_{1}decreases with the increase of the h while b

_{1}increases with the increase of the h. Thus, the relationships between the two coefficients and h can be expressed as follows:

_{1}(i.e., Ln(a

_{1})) are used for establishing the regression equation because the range of a

_{1}is large. The R

^{2}values are high, i.e., 0.89 for Ln(a

_{1}) and 0.88 for b

_{1}, respectively, which indicate that these two coefficients can be represented well by the h.

#### 3.1.3. Hyporheic Flux in the Upstream (Q_{u})

_{u}and the distance from the left boundary of the domain under the five modeling scenarios with different h values. It can be observed that the maximum Q

_{u}increases along with the increase of the h. In order to better match the simulation results, Equation (9) is adjusted as follows:

_{u}under the five modeling scenarios with different h values. The R

^{2}values of the regression equations are all higher than 0.71, indicating that the regression equations can basically fit the simulation results.

_{u}and h. It is observed that a

_{2}increases with the increase of the h while b

_{2}decreases with the increase of the h. Thus, the relationships between the two coefficients and h can be expressed as follows:

^{2}value for a

_{2}is equal to 1.00, which indicates the perfect fit; the R

^{2}value for b

_{2}is also high, i.e., 0.87. As a result, these two coefficients can be closely related to the h.

#### 3.2. Objective Function Calculation

#### 3.2.1. Cumulative Nitrogen Removal Amount (CNRA) and Nitrogen Removal Ratio (NRR)

_{u}values for each designated h value can be firstly calculated based on Equations (8) and (17)–(19), and then, the corresponding CNRA values can be calculated based on Equations (3)–(5). Similarly, the NRR values for each designated h value can be calculated based on Equations (6), (8) and (17). Figure 7 shows the relationships of the CNRA and NRR with the h. As h increases, the CNRA increases but the NRR decreases. It is clear that there will be an optimal h which strikes a balance between the two indicators.

#### 3.2.2. Optimal Height of the Weir

#### 3.2.3. Validation

#### 3.3. Sensitivity Analysis

^{3}/s (Table 1), two other river discharges (i.e., 0.5 m

^{3}/s and 1.5 m

^{3}/s) are used to investigate the change of the optimal h when river discharge increases/decreases by 50%. The denitrification simulation method mentioned in Section 2.3 is adopted and M1

_{den}is employed to evaluate the performance of the weir in the nitrogen removal. For each of the three river discharges, simulations under three different heights of the weir (i.e., 0.53 m, 0.54 m and 0.55 m) are conducted in order to detect the generic trends of the optimal h.

^{3}/s to 0.5 m

^{3}/s, the maximum M1

_{den}value, which indicates the best performance in the nitrogen removal, is associated with the height of 0.53 m (see column 1 in Table 6), and the actual optimal height might be lower for the discharge of 0.5 m

^{3}/s. Even 0.53 m might not be the optimal h under the river discharge of 0.5 m

^{3}/s, and it is smaller than the optimal h under the river discharge of 1.0 m

^{3}/s (i.e., 0.54 m). Therefore, it can be concluded that the optimal h decreases when the river discharge decreases. Conversely, if the river discharge increases from 1.0 m

^{3}/s to 1.5 m

^{3}/s, the maximum M1

_{den}value is associated with the height of 0.55 m (see column 3 in Table 6), which indicates that the optimal h increases when the river discharge increases. As a result, the optimal h can be generally sensitive to the river discharge. It is not surprising, as the larger river discharge may bring more nitrogen from surface water to the HZ; if the nitrogen loading is below the HZ nitrate removal capacity, the weir should be higher in order to improve the performance in the nitrogen removal because the higher weir can enhance the interaction between surface water and groundwater and the denitrification in the HZ. However, if the nitrogen loading is above the nitrogen removal capacity of the HZ, the water should be treated first instead of increasing the height of the weir.

_{u}), are shown in Figure 9. It is observed that hyporheic flux increases when the slope increases. In contrast, the influence of depth to bedrock on hyporheic flux is not significant, which indicates that the active zone of the HZ can be limited to a narrow range.

#### 3.4. Limitations and Future Studies

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Numerical model and results. The illustration on the top right corner shows the cross-section for surface water simulation. The pink lines in the bottom figure indicate the flow paths and the red arrows indicate the hyporheic flux.

**Figure 2.**Relationship between the maximum upstream distance in the subsurface flow influenced by the weir (L

_{max}) and the height of the weir (h).

**Figure 3.**Relationship between the residence time (RT) and the location along the stream (x) under the five modeling scenarios with different heights of the weir (h).

**Figure 4.**Relationship between the two coefficients of the regression equation for calculating the residence time (RT) and the height of the weir (h).

**Figure 5.**Relationship between the hyporheic flux (Q

_{u}) and the location along the stream (x) under the five modeling scenarios with different heights of the weir (h).

**Figure 6.**Relationship between the two coefficients of the regression equation for calculating the hyporheic flux (Q

_{u}) and the height of the weir (h).

**Figure 7.**Relationship of the cumulative nitrogen removal amount (CNRA) and nitrogen removal ratio (NRR) with the height of the weir (h).

**Figure 8.**Comparison of relationships between the value of the objective function and the height of the weir (h) obtained from different h ranges.

Variables to be Held Constant | Variable to be Changed |
---|---|

Slope = 0.01 River discharge = 1 m ^{3}/sDepth to bedrock = 5 m | Height of the weir = 0.3, 0.4, 0.5, 0.6, and 0.7 m |

**Table 2.**The coefficients of the regression equations for calculating the residence time (RT) under the five modeling scenarios with different heights of the weir (h), and their R

^{2}values.

h (m) | h = 0.3 | h = 0.4 | h = 0.5 | h = 0.6 | h = 0.7 |
---|---|---|---|---|---|

a_{1} | 1.44 × 10^{12} | 2.46 × 10^{9} | 3.40 × 10^{7} | 1.01 × 10^{7} | 2.57 × 10^{6} |

Ln(a_{1}) | 28.00 | 21.62 | 17.34 | 16.13 | 14.76 |

b_{1} | −10.14 | −7.26 | −5.27 | −4.79 | −4.17 |

R^{2} | 0.97 | 0.94 | 0.91 | 0.93 | 0.92 |

**Table 3.**The coefficients of the regression equations for calculating the hyporheic flux (Q

_{u}) under the five modeling scenarios with different heights of the weir (h), and their R

^{2}values.

h (m) | h = 0.3 | h = 0.4 | h = 0.5 | h = 0.6 | h = 0.7 |
---|---|---|---|---|---|

a_{2} | 0.0099 | 0.016 | 0.022 | 0.029 | 0.035 |

b_{2} | −1.51 | −1.69 | −1.73 | −1.80 | −1.82 |

R^{2} | 0.83 | 0.71 | 0.79 | 0.78 | 0.84 |

h (m) | h = 0.5 | h = 0.54 | h = 0.6 |
---|---|---|---|

CNRA | 2.81 | 3.15 | 3.43 |

NRR | 14.2% | 20% | 14.5% |

CNRA·NRR | 0.40 | 0.63 | 0.50 |

h (m) | h = 0.5 | h = 0.53 | h = 0.54 | h = 0.55 | h = 0.6 |
---|---|---|---|---|---|

RT_{max} (day) | 2.98 | 2.96 | 2.89 | 2.58 | 2.33 |

M1_{den} (mol) | 1.07 × 10^{−5} | 1.08 × 10^{−5} | 1.12 × 10^{−5} | 9.77 × 10^{−6} | 7.94 × 10^{−6} |

M2_{den} (mol) | 1.09 × 10^{−5} | 1.10 × 10^{−5} | 1.17 × 10^{−5} | 1.15 × 10^{−5} | 1.01 × 10^{−5} |

M1_{den} (mol) | River Discharge (m^{3}/s) | |||
---|---|---|---|---|

0.5 | 1.0 | 1.5 | ||

Height of the in-stream structure (m) | 0.53 | 1.01 × 10^{−5} | 1.08 × 10^{−5} | 9.70 × 10^{−6} |

0.54 | 9.43 × 10^{−6} | 1.12 × 10^{−5} | 1.04 × 10^{−5} | |

0.55 | 8.08 × 10^{−6} | 9.77 × 10^{−6} | 1.06 × 10^{−5} |

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Liu, S.; Chui, T.F.M.
Optimal In-Stream Structure Design through Considering Nitrogen Removal in Hyporheic Zone. *Water* **2020**, *12*, 1399.
https://doi.org/10.3390/w12051399

**AMA Style**

Liu S, Chui TFM.
Optimal In-Stream Structure Design through Considering Nitrogen Removal in Hyporheic Zone. *Water*. 2020; 12(5):1399.
https://doi.org/10.3390/w12051399

**Chicago/Turabian Style**

Liu, Suning, and Ting Fong May Chui.
2020. "Optimal In-Stream Structure Design through Considering Nitrogen Removal in Hyporheic Zone" *Water* 12, no. 5: 1399.
https://doi.org/10.3390/w12051399