Three different nonlinear models – an exponential function model, a logarithmic function model, and a power function model – were evaluated as possible candidates for the storage–outflow relationship of a reservoir. Also, the storage coefficient was derived by analyzing the impulse response function of the reservoir. Additionally, by applying the theoretical results, the storage coefficients of four dam reservoirs in the Han River Basin, Korea, were estimated and evaluated with respect to the dam reservoir and basin characteristics. Summarizing the results, first, only the exponential function model was found to provide a realistic storage coefficient for a reservoir. Second, the storage coefficient was found to be strongly and linearly proportional to the flood control volume divided by the channel length.

## INTRODUCTION

An important purpose of a dam is to reduce flood runoff (Mitchell 1962; Yoshikawa *et al*. 2010). That is, by storing some portion of flood runoff in the space prepared by a dam, one can decrease the peak flow as well as the runoff volume. The stored water in a dam reservoir is released by following a rule specifically made for the dam, or simply the dam operation rule. This role of a dam is called the storage effect, which can be quantified by the storage coefficient. The storage effect is very dependent on the storage capacity, which is obtained using the reservoir's hypsometric curve, or the elevation–capacity curve. However, the storage effect is also dependent on various factors such as the dam operation rule, relative size of the storage capacity to the basin area, and inflow volume per flood event.

The reservoir characteristics are quantified ultimately by a curve of the storage–outflow relationship. For example, in the simplest case, one can imagine a reservoir where the water is released freely through the spillway if the water stage is higher than the spillway crest. The storage–outflow relation curve can be derived easily by considering the hypsometric curve of the reservoir and the dimensions of the spillway. For a given water stage, the storage volume can be estimated using the hypsometric curve, and the outflow using the weir formula. The reservoir storage–outflow relationship is to be used for reservoir flood routing, and the storage effect of the reservoir is fully dependent on this storage–outflow relationship (Richter & Thomas 2007).

Quantification of the storage effect of a dam reservoir is always required for evaluating the flood control ability of the dam itself, as well as for evaluating the role of the dam in the entire river basin. However, as the reservoir storage–outflow relationship is dependent on both the morphological characteristics of the reservoir and the dam operation rule, it generally becomes nonlinear. Differently from the linear reservoir case, it is not easy to quantify the storage effect or to define the storage coefficient. Simulation of dam operation for a given inflow is the only way one can consider to evaluate the ability of a dam (Kwon & Shim 1998; Gul *et al*. 2010; Eum *et al*. 2012).

In this paper, the authors seek to quantify the storage effect of a dam reservoir by solving the continuity equation along with nonlinear reservoir models. Three different nonlinear models – a power function model, a logarithmic function model, and an exponential function model – are evaluated and compared with the linear reservoir model. The storage coefficient of the dam reservoir can then be derived by analyzing the outflow hydrograph, based on Sabol (1988). Finally, by applying the derived theoretical results, the dam storage coefficients of four dams in the Han River Basin, Korea, are derived and evaluated with respect to the dam reservoir and basin characteristics.

This paper is composed of a total of five sections including the introduction and conclusions. The following section deals with the theoretical background of nonlinear reservoir models, and the third section covers the storage and lag characteristics of a reservoir based on the three nonlinear reservoir models reviewed in the second section. Finally, the fourth section covers the application example of the four dam reservoirs in the Han River Basin, Korea.

## RESERVOIR FLOOD ROUTING BASED ON NONLINEAR RESERVOIR MODELS

### Three nonlinear reservoir models

There have been many studies that have considered the power function for relating the storage and the outflow. For example, Laurenson (1964) expressed the storage effect of a river basin using a power function. The storage function method (Kimura 1961) is another example that relates the storage in a basin or channel to the outflow by the power function. Porter (1975) and Boyd & Bufill (1989) also commented about using the power function to represent the storage effect of a basin. This nonlinear model has also been extended to reservoir flood routing, such as in Mein *et al*. (1974), Kidd & Lowing (1979), Foroud & Broughton (1981), Pirt (1983) and Eyre & Crees (1984). The power function has also been used in many recent studies for modeling the flow in the hill slope, low flow and base flow (Ali *et al*. 2013; Charron & Ouarda 2015; Eris & Wittenberg 2015).

### Outflow from a nonlinear reservoir

*t*> 0, one can getwhere it is assumed that . This result indicates that the outflow from a reservoir is inversely proportional to time.

## STORAGE AND LAG CHARACTERISTICS OF RESERVOIRS

### Linear reservoir case

*K**) is derived by dividing the outflow by its first derivative at the inflection point (Sabol 1988). That is,Here, it should be mentioned that the definitions above of the concentration time and storage coefficient are based on linear system assumptions. By applying the linear channel concept, the inflection point becomes located at the time point when the water drop having the longest flow path reaches the basin outlet. All the direct runoff has entered the artificial linear reservoir, and from this time point, only the reservoir plays the role of deciding the outflow (or runoff) characteristics.

It should also be mentioned that, if the reservoir is nonlinear, the storage coefficient estimated by Equation (9) may only be an approximation. Differently from the linear reservoir case, the storage coefficient estimated is no longer a proportional coefficient between the storage and outflow. It simply explains the ratio between the outflow and its change in time at the origin. This ratio may not be the exact storage coefficient as in the linear reservoir, but the authors believe it can be used as a good approximation.

### Nonlinear reservoir case

As mentioned earlier, an analytical derivation of the storage coefficient considering more terms in Equation (7) is not available. Thus, in this study, further analysis was done numerically by increasing the number of terms in Equation (7). In all the cases considered, the inflection point was found to be the origin; that is, the concentration time is zero. It was also found that the storage coefficient converged as the number of terms in Equation (7) increased. This trend is consistent regardless of the value of , and the storage coefficient is decided to be higher than by about 72%.

## APPLICATIONS

### Study basin and dams

In this study, the authors evaluated dam reservoirs in the Han River Basin in Korea. The Han River Basin is the largest river basin in Korea, located at the center of the Korean Peninsula. The basin area is 34,674.0 km^{2} and its channel length (CL) is 459.3 km. There are more than ten dams in the Han River Basin, among which only four dams – Chungju Dam, Soyanggang Dam, Hwacheon Dam, and Hoengseong Dam – have flood control ability. The other dams are mostly single-purpose dams for electric power generation (MLTM 2009).

Dam . | Basin characteristics . | Dam reservoir characteristics . | |||||||
---|---|---|---|---|---|---|---|---|---|

Area (km^{2})
. | Length (km) . | Slope (%) . | Restricted water level RWL (EL. m)
. | High water level HWL (EL. m)
. | Flood water level FWL (EL. m)
. | Gross storage capacity (10^{6}m^{3})
. | Flood control volume (10^{6}m^{3})
. | Reservoir operation method (ROM) . | |

Hwacheon Dam | 4,092 | 171.8 | 53.8 | 175.0 | 181.0 | 183.0 | 1,018.4 | 213 | Empirical method |

Soyanggang Dam | 2,703 | 145.0 | 48.8 | 190 | 193.5 | 198 | 2,900 | 770 | SRD ROM |

Hoengseong Dam | 209 | 37 | 41.9 | 178.2 | 180 | 180 | 86.9 | 9.5 | Rigid ROM |

Chungju Dam | 6,648 | 252.7 | 51.2 | 138 | 141 | 145 | 2,750 | 616 | SCR ROM |

Dam . | Basin characteristics . | Dam reservoir characteristics . | |||||||
---|---|---|---|---|---|---|---|---|---|

Area (km^{2})
. | Length (km) . | Slope (%) . | Restricted water level RWL (EL. m)
. | High water level HWL (EL. m)
. | Flood water level FWL (EL. m)
. | Gross storage capacity (10^{6}m^{3})
. | Flood control volume (10^{6}m^{3})
. | Reservoir operation method (ROM) . | |

Hwacheon Dam | 4,092 | 171.8 | 53.8 | 175.0 | 181.0 | 183.0 | 1,018.4 | 213 | Empirical method |

Soyanggang Dam | 2,703 | 145.0 | 48.8 | 190 | 193.5 | 198 | 2,900 | 770 | SRD ROM |

Hoengseong Dam | 209 | 37 | 41.9 | 178.2 | 180 | 180 | 86.9 | 9.5 | Rigid ROM |

Chungju Dam | 6,648 | 252.7 | 51.2 | 138 | 141 | 145 | 2,750 | 616 | SCR ROM |

SCR ROM = Spillway Discharging Rule ROM, SRD ROM = Scheduled Release Discharge ROM, Empirical method = method to maximize electric power generation.

### Dam storage–discharge relations

The storage–outflow relation curves of the Chungju Dam and Soyanggang Dam were derived from the stage-storage and stage-discharge curves provided in the ‘Practice Manual for Dam Operation’ (K-water 2009). Also, the same for the Hwacheon Dam and Hoengseong Dam were derived from the stage-storage and stage-discharge curves in the ‘Master Plan for Channel Maintenance in the Han River Basin’ (MOCT 2002).

### Curve fitting and storage coefficients

*R*

^{2}, of the four models are also compared in Table 2 for all four dams considered in this study. As can be seen in this table (also in the figure), the linear model and exponential function model look better than the logarithmic and power function models. Among the linear and exponential function models, the exponential function model was found to be the best one with the highest

*R*

^{2}value.

Model . | Determination coefficient (R^{2}). | |||
---|---|---|---|---|

Hwacheon Dam . | Soyanggang Dam . | Hoengseong Dam . | Chungju Dam . | |

Linear | 0.9913 | 0.9972 | 0.9964 | 0.9977 |

Exponential | 0.9927 | 0.9973 | 0.9982 | 0.9980 |

Logarithmic | 0.9385 | 0.9823 | 0.9902 | 0.9832 |

Power | 0.9540 | 0.9882 | 0.9933 | 0.9936 |

Model . | Determination coefficient (R^{2}). | |||
---|---|---|---|---|

Hwacheon Dam . | Soyanggang Dam . | Hoengseong Dam . | Chungju Dam . | |

Linear | 0.9913 | 0.9972 | 0.9964 | 0.9977 |

Exponential | 0.9927 | 0.9973 | 0.9982 | 0.9980 |

Logarithmic | 0.9385 | 0.9823 | 0.9902 | 0.9832 |

Power | 0.9540 | 0.9882 | 0.9933 | 0.9936 |

Dam . | Hwacheon Dam . | Soyanggang Dam . | Hoengseong Dam . | Chungju Dam . |
---|---|---|---|---|

740,480,321.7 | 2,212,231,081.0 | 51,738,704.8 | 450,944.7 | |

0.000000007797 | 0.00000001129 | 0.00000006842 | 0.00002909 | |

0.98 | 0.99 | 0.99 | 0.99 | |

5.8 | 25.0 | 3.5 | 13.1 | |

9.9 | 42.8 | 6.1 | 22.5 |

Dam . | Hwacheon Dam . | Soyanggang Dam . | Hoengseong Dam . | Chungju Dam . |
---|---|---|---|---|

740,480,321.7 | 2,212,231,081.0 | 51,738,704.8 | 450,944.7 | |

0.000000007797 | 0.00000001129 | 0.00000006842 | 0.00002909 | |

0.98 | 0.99 | 0.99 | 0.99 | |

5.8 | 25.0 | 3.5 | 13.1 | |

9.9 | 42.8 | 6.1 | 22.5 |

The storage coefficient of a dam reservoir is proportional to the multiplication of the two parameters of the nonlinear reservoir model, and , as in Equations (8) and (13). Roughly, is 72% higher than . Table 3 also summarizes the storage coefficients estimated for the four dams considered here. As can be seen in Table 3, the biggest storage coefficient, 42.8 hours, was estimated for the Soyanggang Dam and the smallest, 6.1 hours, for the Hoengseong Dam. Those for the Chungju Dam and Hwacheon Dam were estimated at about 22.5 hours and 9.9 hours, respectively.

The estimated storage coefficients seem to be proportional to the flood control volume. However, the storage coefficient of the Soyanggang Dam was estimated to be about twice that of the Chungju Dam even though their flood control storages are similar. Also, the difference between the storage coefficients of the Hwacheon Dam and the Hoengseong Dam (9.9 vs 6.1 hours) seems too small compared with the difference in their flood control volumes (213 vs 9.5 million tons). That is, simply the flood control volume does not explain the difference between the storage coefficients among dams. To find the answer to the question ‘What decides the storage coefficient of a dam?’ the authors explored the possible relationships between the dam storage coefficient and various factors of dam reservoirs and basins.

### Dominant factors for dam storage effect

As primary factors, authors considered the basin area (A, km^{2}), channel length (CL) (km), basin slope (S, %), gross storage volume (GV, 10^{6} m^{3}), flood control volume (FV, 10^{6} m^{3}), and surface area of a dam reservoir (SA, km^{2}). These factors were selected arbitrarily as those assumed to be related to the storage coefficient of a dam reservoir. Scatter plots were made between the storage coefficients and these primary factors, but did not show any strong linear dependency. Only the flood control volume showed a weak linear relationship with the storage coefficient of a dam reservoir. However, this does not explain the difference between the Soyanggang Dam and the Chungju Dam, or that between the Hwacheon Dam and the Hoengseong Dam.

## SUMMARY AND CONCLUSION

In this study, three different nonlinear reservoir models – a power function model, a logarithmic function model, and an exponential function model – were evaluated as possible candidates to explain the storage–outflow relationship of a dam reservoir. The authors also derived the impulse response function of a reservoir by solving the continuity equation along with the given nonlinear storage–outflow relationship under the condition of an instantaneous unit inflow. The storage coefficient could be derived by analyzing the impulse response function derived. Additionally, by applying the theoretical results, the storage coefficients of four dam reservoirs in the Han River Basin, Korea, were estimated and evaluated with respect to the reservoir and basin characteristics. The authors summarize the results as follows.

First, the impulse response function of a reservoir could be derived analytically by solving the continuity equation along with the nonlinear reservoir models considered in this study. Among them, both the power function and exponential function models were found to provide realistic results.

The storage coefficient of a dam reservoir could also be derived for all three nonlinear models by analyzing the impulse response function, based on Sabol (1988). However, it was found that only the exponential function model provided a realistic storage coefficient for a reservoir.

In application to four dam reservoirs in the Han River Basin, Korea, it was found that the exponential function model fits very well with the storage–outflow data collected at the dam reservoirs. The parameters of the exponential function model were estimated for the zone of flood control volume during the wet season between the RWL and FWL, which were also used to estimate the storage coefficient of the dam reservoirs. The storage coefficient of the dam reservoirs estimated varied from 6.1 to 42.8 hours by dam.

Finally, the authors sought to find dominant factors of the storage coefficient of a dam reservoir. However, it was found that the storage coefficient could not be explained fully by any single characteristic value of the dam reservoir or basin. Interestingly, the storage coefficient was found to be strongly and linearly proportional to the flood control volume divided by the CL.

## ACKNOWLEDGEMENTS

This research was supported by a grant (14AWMP-B082564-01) from the Advanced Water Management Research Program funded by the Ministry of Land, Infrastructure and Transport of the Korean government.