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It is common knowledge that the end user of stream flow data may necessarily not have any prior knowledge of the quality control measures applied in their generation, therefore, conclusions drawn most often times may not be effective as desired. Thus, this study is an attempt at providing an independent quality construct to boost the confidence in the use of stream flow data by developing regional flow duration curves for selected ungauged stations of the upper Niger River Basin, Nigeria. Toward this end, stream flow data for seven gauging stations cover some sub basins in the Basin were obtained; precisely, monthly stream flow data covering a range of eleven to fifty-three years period. The flow duration curves from the gauging stations were fitted with three probability distribution models;
i.e., logarithmic, power and exponential regression models. For the regionalisation, parameterisation was carried out in terms of the drainage area alone to allow for simplicity of models. Results obtained showed that the exponential regression model, in terms of Coefficient of Determination (R
^{2}) had the best fit. Though the regionalised model was simple, measurable agreement was obtained during the calibration and validation phases. However, considering the length of data used and probable variability in the stream flow regime, it is not possible to objectively generalise on the quality of the results. Against this backdrop, it suffices to take into cognisance the need to use an ensemble of catchment characteristics in the development of the flow duration curves and the overall regional models; this is important considering the implications of anthropogenic activities and hydro-climatic variations.

In Nigeria, the government has embarked on exploitation of alternative sources of energy based on domestic renewable resources, e.g., solar and hydropower. The geographic and climatic conditions of some regions in Nigeria endow them with a high potential for hydropower generation. The development of large hydropower schemes in Nigeria faces difficulties due to environmental and resettlement problems just like the case with many developing countries. Since most selected sites for small hydropower projects are normally located on small streams where flow records are rarely available, computation methods must be developed to estimate the streamflow and the power potential of the site.

The flow duration curve (FDC) is a common method to estimate the streamflow for small hydropower development. It is used to assess the anticipated availability of flow over time and consequently the power and energy on site. A typical example of this was that of [

Against the backdrop of the foregoing discussion, the objective of the study therefore, is to develop a simple model based on probability distributions to estimate the monthly FDC at ungauged sites in the Upper Niger River basin of Nigeria, which has a high potential to contribute to the development of small hydropower projects.

The Upper Niger River Basin, Nigeria consist of sub-basins (e.g., Gurara, Gbako, and Kaduna, among others) which lie in the intermediate zone between semi-arid climate in the north and sub-humid climate in the south; the climate is influenced by the seasonal movement of the Inter tropical Convergence Zone, which results in wet and dry seasons. Rain starts in April (early rains) or May and lasts till October, with the peak rainfall occurring in September. The dry season lasts between November and March with the mean annual rainfall of some locations in the Basin as follows: 1300 mm (Minna), 1500 mm (Abuja), 1600 mm (Kafanchan), 1250 mm (Kaduna) and 1400 mm (Jos). The mean monthly maximum and minimum temperatures in the basins are 37.3˚C and 19.7˚C, respectively, with the hottest months being February, March and April.

For this study, a total of 7 gauged sites (Kaduna, Shiroro, Kachia, Izom, Baro, Zungeru and Agaie) in the three selected rivers within the river basin controlling an area ranging from 900km^{2} to 6200 km^{2} were used. In this case, records of average monthly gauged flows for the respective rivers were obtained from Niger State and Kaduna State Water Boards as well as Power Holding Company of Nigeria (PHCN); these records were for gauging stations at Kachia and Izom (Gurara sub-basin), Kaduna, Shiroro and Zungeru (Kaduna sub-basin), and Agaie (Gbako sub-basin).

The study is primarily patterned after some studies such as [

1) Development of Regular Flow Duration Curves and Regional Models

In the development of the FDC, catchment area was taken as the major characteristics in all the models; this choice was informed by the lack of available information on other catchment characteristics of interest. The FDCs were constructed by re-assembling the flow time series values in the decreasing order of magnitude assigning flow values to class intervals and counting the number of occurrences (time steps) within each class intervals. Accumulated class frequencies were then calculated and expressed as a percentage of the total number of time steps in the record. The lower limit of every discharge class interval was plotted against the percentage points and then the discharges of exceedance percentage Q_{P%} (p = 1, 5, 10, …, 99) for each catchment were calculated using specific FDC. Based on the submissions of [

Mathematical models of the flow duration curves were developed based on the logarithmic, power, and exponential transformation framework. Resulting from this, the following equations were employed corresponding to the respective framework; that is, Equations (1)-(6)

where, a?f are the coefficients, Q is the discharge; D is the Duration and

By using regression analysis, the models as represented by Equations (1)-(3) and (4)-(6) were fitted to each set of paired values of Q versus D and ^{2}) closest to 1 were considered best fit with values of R^{2} of Equations (1)-3), being statistically equal to that of Equations (4)-(6), respectively. ^{2} for the models for each station.

S/No. | Location | Logarithmic Equations (1)-(3) | Power Equations (2) and (5) | Exponential Equations (3) and (6) |
---|---|---|---|---|

1. | Kaduna | 0.94 | 0.75 | 0.99 |

2. | Shiroro | 0.96 | 0.76 | 0.96 |

3. | Kachia | 0.96 | 0.74 | 0.99 |

4. | Izom | 0.96 | 0.71 | 0.98 |

5. | Baro | 0.97 | 0.70 | 0.98 |

Average | 0.97 | 0.73 | 0.98 |

2) Regional Flow Duration Models

Three sub-basins of the Upper Niger River basin were selected for the study. These sub-basins, for the purposes of this study were (1) Kaduna, (2) Gbako, (3) and Gurara sub-basins. Details of stations (name, area and the period of the records) are as given in

Since the logarithmic and exponential models gave the highest values of average R^{2}, model development was created from the data in _{1}, a_{2} and d_{1}, d_{2} from the logarithmic models in Equations (1) and (4) and that of coefficients c_{1}, c_{2} and f_{1}, f_{2} from the exponential models in Equations (3) and (6), respectively for each of the study locations. The plots for the determination of the regionalised parameters are as shown in Figures 3-6; these parameters so determined were then employed in the regionalisation framework.

S/No. | Station | Data Length | Catchment Area (km^{2}) | Characteristics |
---|---|---|---|---|

1. | Kaduna | 28 yrs | 1647 | Calibration |

2. | Shiroro | 11 yrs | 3500 | Calibration |

3. | Kachia | 25 yrs | 4020 | Calibration |

4. | Izom | 24 yrs | 6200 | Calibration |

5. | Baro | 53 yrs | 5300 | Validation |

6. | Zungeru | 17 yrs | 1750 | Validation |

7. | Agaie | 26 yrs | 900 | Validation |

Location | Equation (1) | Equation (3) | Equation (4) | Equation (6) | |||||
---|---|---|---|---|---|---|---|---|---|

S/No. | Station Name | ||||||||

1. | Kaduna | 916.02 | −197.8 | 870.68 | −0.043 | 4.4983 | −0.971 | 4.2706 | −0.043 |

2. | Shiroro | 1558.20 | −349.0 | 1381.60 | −0.049 | 5.2739 | −1.18 | 4.5955 | −0.048 |

3. | Kachia | 783.89 | −175.0 | 835.42 | −0.056 | 5.2132 | −1.163 | 5.273 | −0.054 |

4. | Izom | 123.90 | −273.8 | 1356.80 | −0.053 | 4.983 | −1.10 | 5.3873 | −0.052 |

5. | Baro | 1047.00 | −230.9 | 1254.40 | −0.056 | 4.9108 | −1.082 | 5.549 | −0.054 |

In doing so, the drainage area (A) and the coefficients were plotted to identify the relationships; The relationships were simply established by methods of least squares (i.e.,

The straight-line coefficients (j_{1} to j_{4}, k_{1} to k_{4}, l_{1} to l_{4} and m_{1} to m_{4}) were further determined using regression analysis.

The calculated basin values were inserted into Equations (7) and (8) for the dimensioned logarithmic and exponential models and (12) to (13) for the dimensionless Logarithmic and exponential models in order to compute the discharges (Q) corresponding to percent of time (D) at intervals increasing 1% each time up to 100%; for each station, these were computed and compared between the results from the Logarithmic and exponential models to find the best fitted model. Based on the re-parameterisation, Equations (15)-(18) were obtained; i.e., for the respective logarithmic and exponential schema.

The estimation of each sub-basin’s representative average flow (Q) in Equations (20) and (21) was performed by analysing the relationship between mean annual flow and drainage area, as in Equation (19)

where A is drainage area in km^{2}; “a” as well as “b” are constants, their values are as presented in

Coefficients | Basins | Model | |||
---|---|---|---|---|---|

j_{1} | j_{2} | R^{2} | Dimensioned Logarithmic | ||

523.48 | 0.1417 | 0.671 | |||

j_{3} | j_{4} | R^{2} | |||

−114.69 | −0.0316 | 0.6436 | |||

k_{1} | k_{2} | R^{2} | Dimensionless Logarithmic | ||

4.2775 | 0.0002 | 0.9246 | |||

k_{3} | k_{4} | R^{2} | |||

−0.9114 | −0.00005 | 0.9239 | |||

l_{1} | l_{2} | R^{2} | Dimension exponential | ||

583.79 | 0.1345 | 0.7797 | |||

l_{3} | l_{4} | R^{2} | |||

−0.0389 | −0.000003 | 0.9286 | |||

m_{1} | m_{2} | R^{2} | Dimensionless exponential | ||

3.8389 | 0.0003 | 0.8114 | |||

m_{3} | m_{4} | R^{2} | |||

−0.0390 | −0.000003 | 0.9067 |

Station | Drainage Area (km^{2}) | a | B | ^{3}/sec) |
---|---|---|---|---|

Kaduna | 1647 | 4.1849 | 0.4795 | 145.91 |

Shiroro | 3500 | 209.44 | ||

Kachia | 4020 | 223.83 | ||

Izom | 6200 | 275.51 | ||

Baro | 5300 | 255.55 |

where l_{1} to l_{4}, m_{1} to m_{4}, and a, b, are constants, respectively

3) Model Calibration and validation for flow prediction

To ascertain the adequacy or otherwise of the regional models, model calibration and validation were carried out. The accuracy of regional models was examined by using the measured discharges. In order to do this, measured and simulated results were compared in terms of root mean square relative error (E_{R})

where D is the percent of time between 1% to 100%, Q_{Dc} is the computed discharge at any percent of time; Q_{Dm} is the measured discharge at any percent of time. Using the coefficients J_{1} to J_{4}, K_{1} to K_{4}, l_{1} to l_{4} and m_{1} to m_{4}, the constants a, b and the drainage area of each station, the predicted discharges at 1% to 100% with interval 1% for each step or percentage of time (D) were determined from Equations (15)-(18).

It is evident from the development of the regular FDCs that both the logarithmic and exponential probability distribution models were appropriate for the understanding of the flow dynamics of the Basin. The findings here are not in accord with that of [

S/No. | Station | Low flow index | |
---|---|---|---|

1. | Kaduna | 4.56 | 4.50 |

2. | Shiroro | 6.39 | 3.60 |

3. | Kachia | 6.60 | 7.14 |

4. | Izom | 4.80 | 10.00 |

5. | Baro | 4.20 | 14.29 |

6. | Zungeru | 1.41 | 1.73 |

7. | Agaie | 5.33 | 4.00 |

E_{R} of Model Validation | ||||
---|---|---|---|---|

Station | FDC Model | Dimensionless Model | ||

Logarithmic, Equation (7) | Exponential, Equation (8) | Logarithmic, Equation (12) | Exponential Equation (13) | |

Kaduna | 25.43 | 8.81 | 31.45 | 27.74 |

Shiroro | 36.75 | 17.95 | 35.15 | 31.58 |

Kachia | 20.88 | 49.49 | 49.20 | 69.19 |

Izom | 56.34 | 16.88 | 26.70 | 27.63 |

Baro | 24.69 | 27.45 | 32.64 | 35.52 |

Average | 32.81 | 24.11 | 35.02 | 38.33 |

E_{R} of Model Calibration | ||||
---|---|---|---|---|

station | FDC Model | Dimensionless Model | ||

Logarithmic, Equation (7) | Exponential, Equation (8) | Logarithmic, Equation (12) | Exponential Equation (13) | |

Zungeru Agaie | 77.97 32.22 | 78.11 18.99 | 79.02 47.65 | 76.82 48.18 |

Average | 55.09 | 48.55 | 63.33 | 62.50 |

S/No | Station | Logarithmic | Exponential | ||
---|---|---|---|---|---|

R_{max} (%) | R_{min} (%) | R_{max} (%) | R_{min} (%) | ||

1. | Kaduna | 95.81 | 65.8 | 95.57 | 52.65 |

2. | Shiroro | 70.31 | 56.7 | 76.41 | 79.30 |

3. | Kachia | 76.98 | 13.53 | 72.26 | 81.00 |

4. | Izom | 63.73 | 86.6 | 60.84 | 95.60 |

5. | Baro | 67.08 | 79.67 | 64.61 | 11.85 |

Average | 74.0 | 60.0 | 73.0 | 64.00 |

S/No | Station | Logarithm | Exponential | ||
---|---|---|---|---|---|

R_{max} (%) | R_{min} (%) | R_{max} (%) | R_{min} (%) | ||

1. | Zungeru | 23.7 | 5.02 | 24.12 | 7.96 |

2. | Agaie | 73.98 | 6.95 | 76.83 | 11.47 |

Average | 48.84 | 5.99 | 9.72 | 50.48 |

phases in terms of correlation metrics. It can be seen that the exponential model using FDC parameter gives reasonably well estimations of the FDC for the stations considered. However, in view of the short length of data used for the study, the model that gives the smallest root mean square relative error (E_{R}) value can be used to predict flow at the ungauged site within the Sub-basins. The contrasts between these values (E_{R} and R^{2}) should not be related since E_{R} is used to measure the prediction error of the proposed models whereas the values of R^{2} indicate how strong linear correlation existed between the model coefficients (i.e., a_{1}, a_{2}, e_{1}, e_{2}, f_{1}, f_{2}, j_{1}, j_{2}) and the drainage area (A).

Despite the extent of correlation or relative error margins, it is pertinent to point that using just a representative catchment characteristic may not be sufficient to reflect the hydrologic variation of a particular catchment as evidenced in _{max} and R_{min} [

Based on the findings of the study it is imperative to note that the regionalisation of flow duration curves is an effective approach for stream flow generation and extension, especially in the face of data scarcity. In addition, it is clearly evident that the use of catchment characteristics as input parameters, to a large extent for model development whether conceptual or statistical model in essential. However, results obtained by employing only the drainage area in the overall regionalised model indicate that it is not representative enough thus considering the length of data used and the attendant problem of flow variability, it suffices to note that an ensemble of catchment characteristics may be imperative. It is also important to take into consideration the ability of the models to reproduce the flow signatures; in this case, the prediction of extreme values is critical. Both the logarithmic and exponential function models employed in the FDCs portrayed different characteristics in the calibration and validation stages. The exponentially regionalised models overwhelmingly performed better than the logarithmic as shown in the validation phase. Considering the results obtained in the overall, objective generalisations may not be possible though, the parameters obtained for the models in the regionalisation procedure were largely optimised. Hence, effective conclusion on the suitability of using the drainage area as a representative parameter in a copious attempt to understand the overall behaviour of a hydrologic response unit should be done with cautious optimism; especially taking cognisance of the implications of geologic characteristics of drainage basins in affecting low flows.

Martins Yusuf Otache,Muhammad Abdullahi Tyabo,Iyanda Murtala Animashaun,Lydia Pam Ezekiel, (2016) Application of Parametric-Based Framework for Regionalisation of Flow Duration Curves. Journal of Geoscience and Environment Protection,04,89-99. doi: 10.4236/gep.2016.45009