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

The linear reservoir model is not generally applied to the flood routing problem of a reservoir. This is simply because the relationship between the storage of a reservoir and the outflow is rarely linear. In general, the reservoir storage is much bigger than the outflow. Also, as the depth increases, the increase of the storage becomes much bigger than that of the outflow. Thus, the relationship between the two cannot be represented by a line, but a curve. This behavior may be well represented by a power function. The basic form of a power function is as follows:
1
Here, and refer to constants that reflect the storage effect of a nonlinear reservoir model.

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).

Although it may not seem so realistic for the relationship between the storage and outflow, there is also a logarithmic function model represented by the following function:
2
This model is, in fact, unrealistic for application to the reservoir storage–outflow relationship, because the outflow should be greater than the storage. However, there are many application examples where this nonlinear relationship was assumed. For example, Lambert (1969) used this logarithmic function to derive a rainfall-runoff model, and Green (1979) also tried a similar approach. Beven & Kirkby (1979) used this logarithmic function model as a part of the TOPMODEL concept. Moore (1997) also reviewed this function for storage–outflow modeling of stream flow recession.
In this study, the authors additionally propose an exponential function form of the nonlinear reservoir model:
3
where and are also constants. In fact, the behavior of this exponential function model is very similar to that of the power function model. By controlling these two parameters, it is easy to mimic the behavior of the storage–outflow relationship.

Outflow from a nonlinear reservoir

With the given relationship between the storage and outflow, it is possible to derive the outflow from a reservoir by solving the continuity equation. First, as the simplest case, the outflow for the case of the logarithmic function model can be derived as follows. That is, by substituting the storage of the continuity equation by Equation (2) above, one can get the following differential equation for the outflow, :
4
By considering the instantaneous inflow to the nonlinear reservoir (in fact, this is to derive the impulse response function of a reservoir), one can solve the above continuity equation. That is, for t > 0, one can get
5
where it is assumed that . This result indicates that the outflow from a reservoir is inversely proportional to time.
The same procedure can also be applied to the power function case, whose result is expressed as follows:
6
This result is more complex than the previous case, but this also shows that the outflow has a decreasing pattern with respect to time. The only difference is the role of the power applied to the time. If , the outflow decays very fast, but if , the outflow decays more slowly. If , it becomes a linear reservoir. However, the result becomes invalid when , as the outflow increases as time increases.
The procedure to derive the outflow for the case of the exponential function is also similar. When considering the instantaneous inflow to the nonlinear reservoir, one can derive the following equation:
7
Unfortunately an analytical solution of Equation (7) is not available. The maximum number of terms on the right-hand-side of Equation (7) is only two in the case that the analytical interpretation is favored. In this case, the solution can be derived by introducing the Wright function (Corless & Jeffrey 2002):
8
where and is defined by , satisfying . Also, in this equation, the subscript 1 + 2 indicates that only the first and second terms of Equation (7) are considered in deriving the outflow hydrograph.

STORAGE AND LAG CHARACTERISTICS OF RESERVOIRS

Linear reservoir case

Deriving the storage and lag (or delay) characteristics in reservoir flood routing requires determining the storage coefficient and concentration time of a reservoir using the impulse response function of the reservoir, or using the flood routing results along with the instantaneous inflow. In the case that the impulse response function of a reservoir is given in a functional form, it is easy then to derive the storage coefficient and concentration time. First, the concentration time is defined as the time to the inflection point of the impulse response function. Using this concentration time, it is possible to evaluate the lag characteristic of a reservoir. Second, the storage coefficient (K*) is derived by dividing the outflow by its first derivative at the inflection point (Sabol 1988). That is,
9
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.
However, as the authors are considering a real reservoir, the inflection point of an impulse response function becomes located at the origin. This is simply because the inflow to the reservoir is assumed to be instantaneous. If the reservoir is linear, the outflow can be expressed as follows:
10
It is easy to show that the inflection is located at the origin (i.e., the concentration time is zero) and the storage coefficient .

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

By applying the same definitions as in the previous section, one can readily derive the concentration time and storage coefficient of a nonlinear reservoir. It is straightforward to derive the storage coefficients of the logarithmic and power function models of nonlinear reservoirs, which are
11
12
In both cases, the storage coefficient is expressed as a function of time. It is interesting that the storage coefficients derived are not expressed by all the parameters used for relating the storage and outflow. Simply applying the inflection point, i.e., the origin, the storage coefficient becomes zero. This result indicates that these two nonlinear reservoir models are not practical for quantifying the storage characteristics of a reservoir.
The procedure is also the same for the exponential function model. However, it is rather complicated to analytically derive the concentration time and storage coefficient using the full equation given in Equation (7). The only way available is to consider the two terms of the left-hand side of Equation (7), or to use the outflow hydrograph given in Equation (8). The storage coefficient is now derived as follows:
13
where can be estimated using the relations and by the definition of the function (Corless & Jeffrey 2002). As , and is estimated to be about 0.567. Now the storage coefficient, based on Equation (13), is determined to be .

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 km2 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).

This study considered the four dams that have flood control ability. Three of these four dams are operated by K-water and only the Hwacheon Dam is operated by Korea Hydro & Nuclear Power. Originally, the Hwacheon Dam was constructed as a single-purpose dam for hydropower generation, but was changed to have flood control ability in 1973 (KOWACO 2004). Figure 1 shows the Han River Basin in the Korean Peninsula and the locations of the four dams considered in this study. Their basin and reservoir characteristics are summarized in Table 1. As can be seen in Table 1, the Chungju Dam has the largest basin area, and the Hoengseong Dam the smallest. The gross storage volume and the flood control volume are largest in the Soyanggang Dam, and smallest in the Hoengseong Dam.
Table 1

Characteristics of the basins and dam reservoirs considered in this study

DamBasin characteristics
Dam reservoir characteristics
Area (km2)Length (km)Slope (%)Restricted water level RWL (EL. m)High water level HWL (EL. m)Flood water level FWL (EL. m)Gross storage capacity (106m3)Flood control volume (106m3)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
DamBasin characteristics
Dam reservoir characteristics
Area (km2)Length (km)Slope (%)Restricted water level RWL (EL. m)High water level HWL (EL. m)Flood water level FWL (EL. m)Gross storage capacity (106m3)Flood control volume (106m3)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.

Figure 1

Han River Basin and location of dams considered in this study.

Figure 1

Han River Basin and location of dams considered in this study.

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

The storage–outflow curves obtained are shown in Figure 2. In this figure, the restricted water level (RWL) indicates the water level for flood control during the wet season (July 21–September 20) in Korea (Jeong & Yoon 2009). As the RWL is about 3 m lower than the high water level (HWL), a larger volume of flood control storage can be secured. That is, only the volume above the RWL is related to flood control. The gross storage volume is not directly related to flood control.
Figure 2

Observed storage–outflow relations for the four dams considered in this study.

Figure 2

Observed storage–outflow relations for the four dams considered in this study.

Figure 3 shows the curve-fitting results of all four models (three nonlinear models and one linear model) for the Hwacheon Dam as an example. The parameters of the exponential function model in this study were estimated by the method of least squares. The estimated coefficients of determination, R2, 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 R2 value.
Table 2

Comparison of coefficients of determination (R2) for the four models considered in this study (above the RWL)

ModelDetermination coefficient (R2)
Hwacheon DamSoyanggang DamHoengseong DamChungju 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
ModelDetermination coefficient (R2)
Hwacheon DamSoyanggang DamHoengseong DamChungju 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
Figure 3

Comparison of curve-fitting results for the data above the RWL (Hwacheon Dam).

Figure 3

Comparison of curve-fitting results for the data above the RWL (Hwacheon Dam).

Figure 4 also compares the difference in the fitting of the exponential function model to the entire data and those above the RWL. As confirmed in Figure 3, the exponential function model fits the data above the RWL much better. This result is natural because the storage–outflow relationship below the RWL is more closely related to the water supply and hydropower generation. The water level of the RWL looks like the inflection point of the storage–outflow curve separating it into two parts for water use and flood control. Table 3 summarizes the parameters of the exponential function model fitted to the data above the RWL and their coefficients of determination. The coefficient of determination is very high, confirming that the exponential function model considered in this study is appropriate to explain the relationship between the flood control storage and outflow of dams.
Table 3

Parameters of the exponential function model derived for each dam ( and storage coefficient are also calculated for each dam)

DamHwacheon DamSoyanggang DamHoengseong DamChungju 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
DamHwacheon DamSoyanggang DamHoengseong DamChungju 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
Figure 4

Fitting results of the exponential function model to the entire data (thin line) and those above the RWL (thick line).

Figure 4

Fitting results of the exponential function model to the entire data (thin line) and those above the RWL (thick line).

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, km2), channel length (CL) (km), basin slope (S, %), gross storage volume (GV, 106 m3), flood control volume (FV, 106 m3), and surface area of a dam reservoir (SA, km2). 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.

Additionally, the authors assessed secondary factors based on the primary factors considered above. There were, in total, eight cases: GV/A, GV/CL, GV/S, GV/SA, FV/A, FV/CL, FV/S and FV/SA. The scatter plots made are provided in Figure 5. As can be seen in this figure, a rather consistent linear relationship could be found when considering GV/CL and FV/CL. Among these two, as the storage coefficient was derived for the flood control volume zone, the authors selected the case with FV/CL. This factor FV/CL was found to be valid to explain the difference between the Soyanggang Dam and the Chungju Dam, as well as that between the Hwacheon Dam and the Hoengseong Dam. That is, it was found that the relative size of the flood control volume with respect to the CL decides the storage coefficient of a dam reservoir.
Figure 5

Scatter plots between storage coefficient and secondary factors of dam reservoir and basin.

Figure 5

Scatter plots between storage coefficient and secondary factors of dam reservoir and basin.

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.

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