# Interpreting the Manning Roughness Coefficient in Overland Flow Simulations with Coupled Hydrological-Hydraulic Distributed Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Aggregated Hydrological Modelling

^{2}basin located in Mexico, was firstly analysed with an aggregated hydrological modelling approach through the HEC-HMS software [43]. A standard methodology was applied, based on the local administration guidelines, to discretize the basin. The unit hydrograph (UH) methodology was used for the rainfall-runoff transformation processes, which was obtained from the dimensionless unit hydrograph of the Soil Conservation Service [16,44]. The ${t}_{lag}$ parameter and the value of the time of concentration (${t}_{c}$) were estimated by means of the Kirpich formula. Precipitation losses were accounted for using the SCS-CN method [44,45], also referred to as the NRCS-CN method [46] after the U.S. Soil Conservation Service was renamed as Natural Resources Conservation Service. All parameters of the numerical model are detailed in the Supplementary Material.

#### 2.2. Distributed Hydrological Modelling

## 3. Case Studies

#### 3.1. Case Study 1: Adjustment of the Roughness Coefficient Based on the Time of Concentration

#### 3.2. Case Study 2: Adjustment of the Roughness Coefficient Based on the Peak Time and Discharge from Aggregated Hydrological Models

^{2}and its mean slope is 0.75%. Land uses are characterized by large extensions of secondary vegetation (48.51%) and forested (24.23%) areas. Other areas are covered by rivers (10.70%), pasturelands (10.60%), and rainforest agriculture extents (5.42%). According to the World Reference Base for Soil Resources [67], the basin edaphology is mainly composed of large areas of regosol (71%) and other types of soil such as phaeozem (11.3%) and cambisol (9.6%). From the land uses classification, a Curve Number (CN) of 76.7 was estimated as an area weighted average value in the whole basin.

#### 3.3. Case Study 3: Adjustment of the Roughness Coefficient Based on Observed Storm Events

^{2}and a reservoir at the outlet of 61 hm

^{3}of storage capacity. Weather conditions of the basin are characterized by a wide variability of the rainfall regime, which is influenced by marine conditions with small thermal as well as rain variations [72]. Extreme weather conditions—such as heavy rain events with high precipitation intensities and water accumulations concentrating in a few days or hours, and water scarcity associated to long dry-weather periods—are typical of the area [73,74].

## 4. Results

#### 4.1. Case Study 1

^{1/3}, in intervals of 0.01 s/m

^{1/3}. The purpose was to obtain the best approximation for the minimum, the maximum, and the mean times of concentration (${t}_{c}$) calculated with the empirical formulas (Table 2). The maximum discharge generated with the aforementioned rainfall intensity is, in this case, a constant discharge. For the estimation of ${t}_{c}$, the discharge was considered to be constant when its variation between two consecutive results time steps was lower than 0.1% of the discharge. Results time steps of 60 s were used.

^{1/3}) was needed for the S-curve to fit with the minimum ${t}_{c}$, which, in this case, corresponds to the Kirpich formula. In Trinity (Figure 2c) and Brazos (North Elm Creek) (Figure 2d) basins, it was not possible to reproduce the minimum ${t}_{c}$ without considering any friction since the hydrograph needed more time than the minimum ${t}_{c}$ to stabilize. For this reason, they are not plotted in Figure 2. This induces the thought that the Kirpich formula, the one that provides such minimum values, is quite unrealistic for these basins, as water propagation is slower even with no friction. This is in agreement with the assertion of Michailidi et al. [11] that “the Kirpich formula is a special case of a very general expression that is valid under very limited conditions” and that this formula was obtained for basins where channel flow was predominant.

^{1/3}and between 3 and 5 h for $n$ = 0.14 s/m

^{1/3}) after an initial rapid discharge increase, which can be attributed to the effect of small basins having reached their maximum discharges.

^{1/3}in order to achieve similar responses of the basin to those obtained by common time of concentration formulas and an aggregated model. As expected, a low value of the Manning coefficient provides faster response, while high values in a slower runoff advance. It is worth mentioning that a unique roughness coefficient value was used all over the basin; thus, a more detailed roughness discretisation would result in higher values at hillsides and areas outside of the hydrographic network, and lower ones in rivers and floodplains.

#### 4.2. Case Study 2

^{1/3}, corresponding to urban settlements, to 0.207 s/m

^{1/3}corresponding to forested area. All these values widely exceed the traditionally recommended values for hydraulic computations in artificial or natural channels and floodplains areas [38,39,40,81,82,83].

#### 4.3. Case Study 3

^{1/3}with steps of 0.02 s/m

^{1/3}, resulting in different hydrological responses. The computed water elevation in the reservoir at the basin outlet was compared with the field observations along the events and at the end of each one. Figure 4 shows the simulated water elevation for the four rainfall events analysed.

^{1/3}—one that achieves the best adjustment (0.005% of error). For this event, significant differences on the arrival time of the flood can be observed, with a time gap of around 12 h between the hydrographs corresponding to the minimum and maximum $n$ values. The evolution of the water elevation during the second-half of day 2 clearly shows faster hydrological responses for low values of $n$.

^{1/3}.

^{1/3}, the water elevation at the end of the event shows a good adjustment with relative error being less than 0.16%.

^{1/3}). In general, the hydrological response of the model is more sensible for low values of $n$ than for high ones, especially from the point of view of water resources. As expected, low values of $n$ produce larger overland flow discharges in shorter times.

^{1/3}. The precision of the results is aligned with the assumption of a unique Manning roughness coefficient value for the whole catchment, which in this case, was appropriate for hydrological purposes. On the other hand, for values of $n$ in the lower range, oscillations on the water surface elevation appear.

^{2}). Table 5 summarizes the performance of the model for the four rainfall events in terms of these indicators. A good adjustment is seen for 20130304_3d and 20131115_3d events, with low values of MAE and RMSE, but also with high values of R

^{2}. The performance of 20141128_2d event is, in general, the poorest—the simulated water resources are clearly underestimated during almost all the event. In contrast with the other events, as the hydrological response is slower in comparison with the observations, the indicators show the best adjustments are for low values of $n$. Finally, the 20150320_6d event presents the worst R

^{2}values. A peer-to-peer results analysis (RMSE and MAE) reveals that the model has, in general, a good adjustment, especially for values of $n$ greater than 0.06 s/m

^{1/3}.

## 5. Discussion

**On the roughness coefficient values and their effect on the hydrological response**

**On the role of the domain discretisation in the basin hydrological response**

^{3}/s is observed for calculation meshes with more than 1.9 million elements. This peak discharge increment is probably due to the better representation of the topography, in general, as well as the better definition of the river-channel, in particular, which improve the flow propagation representation. This last domain discretisation (1.9 M elements) represents approximately 19 elements per ha, which is still far from the values of flood studies where more than 1000 elements per ha is quite common [95].

**On the numerical approach**

## 6. Conclusions

^{1/3}, which would correspond to the hydraulic value for very smooth material, much different from those existing in natural basins.

## Supplementary Materials

^{3}/s) for the four basins of the Case Study 1 estimated by the Rational Method for a constant rainfall intensity of 25 mm/h, Table S2: Characteristics of the discretization of the Marquelia basin and parameters of the aggregated numerical approach, Table S3: Spatial characteristics of the land uses of the Marquelia basin and Manning roughness coefficient best fit to the 50-years return period synthetic hydrograph.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of the three case studies and the analysed basins: four in USA (Case Study 1), one in Mexico (Case Study 2), and one in Spain (Case Study 3).

**Figure 2.**Calculated S-type hydrographs related to the minimum (red line), maximum (blue line), and mean (green line) time of concentration (${t}_{c}$) calculated with empirical formulas, and each corresponding Manning coefficient ($n$): (

**a**) Brazos basin, Cow Bayu; (

**b**) San Antonio basin, Escondido Creek; (

**c**) Trinity basin, North Creek; and (

**d**) Brazos basin, North Elm Creek.

**Figure 3.**Results of Case Study 2: (

**a**) hydrograph resulting from the aggregated approach (dashed line), and from the distributed approach, the best fit of the roughness coefficient resulted (continuous line); (

**b**) first 24 h of the hydrograph from the distributed approach and hydrographs considering ± 20% of the roughness coefficients (dot and dashed lines).

**Figure 4.**Water elevation evolution in the reservoir for the events 20130304_3d (

**a**), 20131115_3d (

**b**), 20141128_3d (

**c**), and 20150320_6d (

**d**). Representation of the observed data (dotted line) and the results of the simulations using different Manning roughness coefficients (coloured lines).

**Figure 5.**Sensitivity analysis in Case Study 2 (Marquelia basin): (

**a**) variation of the peak discharge with the mesh size; (

**b**) variation of the time of concentration with the rainfall intensity.

Characteristic | Brazos Basin (Cow Bayu) | San Antonio Basin (Escondido Creek) | Trinity Basin (North Creek) | Brazos Basin (North Elm Creek) |
---|---|---|---|---|

Area (km^{2}) | 13.08 | 22.80 | 58.96 | 119.46 |

Mean slope (%) | 5.90 | 2.90 | 5.20 | 1.40 |

Main channel length (km) | 7.09 | 8.00 | 18.02 | 33.34 |

Formula | Time of Concentration (h) | |||
---|---|---|---|---|

Brazos Basin (Cow Bayu) | San Antonio Basin (Escondido Creek) | Trinity Basin (North Creek) | Brazos Basin (North Elm Creek) | |

Johnstone and Cross [64] | 1.98 | 3.05 | 3.35 | 9.07 |

DPW [65] | 1.63 | 2.40 | 4.13 | 10.02 |

NCRS [15] | 3.72 | 6.33 | 8.93 | 24.28 |

Giandotti [65] | 4.77 | 8.20 | 9.23 | 17.77 |

Kirpich [18] | 0.92 | 1.35 | 1.95 | 5.40 |

Viparelli [66] | 1.37 | 1.60 | 3.42 | 6.58 |

Témez [17] | 2.28 | 2.85 | 4.73 | 9.70 |

Minimum | 0.92 | 1.35 | 1.95 | 5.40 |

Mean | 2.38 | 3.68 | 5.10 | 11.83 |

Maximum | 4.77 | 8.20 | 9.23 | 24.28 |

Event | Cumulated Rainfall (mm) | Max. Intensity in 5-min (mm/h) | CN |
---|---|---|---|

20130304_3d | 181.3 | 30.0 | 81 |

20131115_3d | 123.2 | 54.0 | 50 |

20141128_2d | 150.9 | 61.2 | 65 |

20150320_6d | 197.4 | 67.2 | 50 |

**Table 4.**Manning roughness coefficient according to the land uses discretisation shown in Figure S3b that provides the best adjustment in the calibration process.

Land Use | $\mathit{n}$ |
---|---|

Rainforest agriculture | 0.138 |

Urban settlements | 0.051 |

Pine and oak forest | 0.207 |

Reservoir | 0.069 |

Pastureland | 0.069 |

River | 0.083 |

Savanna | 0.138 |

**Table 5.**Coefficient of determination (R2), mean absolute error (MAE), and root mean square error (RMSE) for each Manning roughness coefficient ($n$) evaluated.

Event | Statistic | Manning Coefficient (s/m^{1/3}) | |||||||
---|---|---|---|---|---|---|---|---|---|

0.02 | 0.04 | 0.06 | 0.08 | 0.10 | 0.12 | 0.14 | 0.16 | ||

20130304_3d | RMSE | 0.512 | 0.361 | 0.263 | 0.258 | 0.236 | 0.297 | 0.352 | 0.352 |

MAE | 0.262 | 0.130 | 0.069 | 0.067 | 0.056 | 0.088 | 0.124 | 0.124 | |

R2 | 0.953 | 0.982 | 0.990 | 0.993 | 0.993 | 0.991 | 0.988 | 0.984 | |

20131115_3d | RMSE | 0.214 | 0.145 | 0.095 | 0.073 | 0.070 | 0.078 | 0.088 | 0.099 |

MAE | 0.046 | 0.021 | 0.009 | 0.005 | 0.005 | 0.006 | 0.008 | 0.010 | |

R2 | 0.903 | 0.964 | 0.979 | 0.973 | 0.962 | 0.945 | 0.925 | 0.902 | |

20141128_2d | RMSE | 0.485 | 0.521 | 0.580 | 0.630 | 0.669 | 0.703 | 0.732 | 0.758 |

MAE | 0.235 | 0.272 | 0.336 | 0.397 | 0.447 | 0.494 | 0.536 | 0.575 | |

R2 | 0.767 | 0.756 | 0.734 | 0.714 | 0.699 | 0.685 | 0.672 | 0.660 | |

20150320_6d | RMSE | 0.511 | 0.374 | 0.305 | 0.282 | 0.269 | 0.261 | 0.255 | 0.252 |

MAE | 0.261 | 0.140 | 0.093 | 0.080 | 0.072 | 0.068 | 0.065 | 0.063 | |

R2 | 0.255 | 0.282 | 0.298 | 0.303 | 0.306 | 0.309 | 0.311 | 0.314 |

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## Share and Cite

**MDPI and ACS Style**

Sanz-Ramos, M.; Bladé, E.; González-Escalona, F.; Olivares, G.; Aragón-Hernández, J.L.
Interpreting the Manning Roughness Coefficient in Overland Flow Simulations with Coupled Hydrological-Hydraulic Distributed Models. *Water* **2021**, *13*, 3433.
https://doi.org/10.3390/w13233433

**AMA Style**

Sanz-Ramos M, Bladé E, González-Escalona F, Olivares G, Aragón-Hernández JL.
Interpreting the Manning Roughness Coefficient in Overland Flow Simulations with Coupled Hydrological-Hydraulic Distributed Models. *Water*. 2021; 13(23):3433.
https://doi.org/10.3390/w13233433

**Chicago/Turabian Style**

Sanz-Ramos, Marcos, Ernest Bladé, Fabián González-Escalona, Gonzalo Olivares, and José Luis Aragón-Hernández.
2021. "Interpreting the Manning Roughness Coefficient in Overland Flow Simulations with Coupled Hydrological-Hydraulic Distributed Models" *Water* 13, no. 23: 3433.
https://doi.org/10.3390/w13233433