Theoretical evaluation of concentration time and storage coef ﬁ cient with their application to major dam basins in Korea

This study theoretically evaluated the basin concentration time and storage coef ﬁ cient with their empirical formulas available worldwide. The evaluation results were also validated in the application to major dam basins in Korea. The ﬁ ndings are summarized as follows. As a result of analytical analysis, the concentration time was found to be proportional to the main channel length under laminar ﬂ ow conditions and to the square of it under turbulent ﬂ ow conditions, but inversely proportional to the channel slope. It was also found that the storage coef ﬁ cient and the concentration time are linearly but loosely related. Most empirical formulas for the concentration time concurred with the basic equation form, but just a few for the storage coef ﬁ cient. Applications to major dam basins in Korea also showed that the concentration time agrees well with the result of theoretical analysis. However, the behavior of the storage coef ﬁ cient varied much, basin by basin, indicating that additional factors may be needed to explain it.


INTRODUCTION
The runoff characteristics of a basin can be quantified by interpreting the components of a flood hydrograph. T c and K are critical components when determining the peak flow and peak time of a flood hydrograph. The Clark instantaneous unit hydrograph (IUH), which is generally used for basin flood routing in Korea, can also be derived by these two parameters, T c and K (Clark ). That is, under the assumption of linear system theory, these two parameters can sufficiently represent the characteristics of the rainfall-runoff process in a basin.
However, even though only two parameters are involved in the construction of the Clark IUH, it is not simple to estimate them. Actually, it may be impossible to determine the unique set of parameters that can be applied to various rainfall-runoff events. This is simply because the nonlinear basin system is assumed to be linear. It is well known that the rainfall-runoff process in a basin is nonlinear (Kundzewicz & Napiórkowski ; Sinha et al. ). Furthermore, as these two parameters are correlated with each other, their estimation procedure can be very complex. In practice, even for an observed rainfall-runoff event, it is practically impossible to estimate a unique set of T c and K.
etc., are diverse. Nevertheless, it has been argued that the components of those formulas are very similar each other (Yoo ). This finding indicates that there exist major contributing factors to determine T c and K, as well as their basic equation forms. If this assumption is true, the regionalization for the consistent application of an empirical formula to a large basin will be practical. However, major contributing factors of empirical formulas for T c and K have not been investigated in depth by previous researchers.
The main objective of this study is to evaluate the fundamental factors of T c and K with their theoretical background and empirical formulas available worldwide. Ultimately this study is going to show if a valid basic equation form exists for T c and K. The result will then be applied to major dam basins in Korea to evaluate its applicability.

Theory for concentration time
A theoretical background for T c can be found in Singh (), where an equation for T c was derived by analyzing surface flow using kinematic wave theory. Even though the equation was derived under some simplifying assumptions like rectangular and converging cross-section, it provided a basic idea about T c in a basin. The equation is: where T c is the concentration time, L is the longest flow length of a basin, i is the rainfall intensity, η is the kinematic wave friction-related parameter varying in space (friction parameter), and m is a constant.  (2)) and turbulent flow conditions (Equation (3)): where S is the channel slope and β, v, g, k, and M are all constants and ε is the equivalent roughness. As can be seen in Equations (2) and (3) the relation between T c and K. Nash () assumed that a basin can be represented by serially connecting x linear reservoirs with its storage coefficient K* (here, it should be noticed that K is different from K*) and the resulting Nash IUH can be expressed as follows: where Q x is the outflow at time t, x is the number of linear reservoirs and Γ( ) is the gamma function.
The advantage of using the Nash IUH is its availability for theoretical analysis. That is, the K and T c of the Nash IUH can be derived simply by applying the definition of T c (from the end of the effective rainfall to the deflection point of the falling limb of the runoff hydrograph) and the definition of K given by Sabol (). T c and K of the Nash IUH are derived as follows (Yoo et al. ): It is also possible to derive the relation between Equations (5) and (6) as follows: The above relation is especially significant as it is expressed as a function of only one parameter, the number of linear reservoirs x. If x is assumed to be a unique value over a given basin, the ratio between T c and K remains the same at any location in the basin.
However, if x varies by the location in the basin, the relationship between T c and K also varies.

Study basins
A total of five dam basins in Chungju, Namgang, Andong, Imha and Hapcheon were considered. Among these five Each basin contains many stream gauge stations, among which this study selected only those less affected by dam or other hydraulic structures to secure the accurate rainfallrunoff data for reasonable parameter estimation. As a result, 18 stream gauge stations within Chungju, nine within Namgang, six within Andong, four within Imha and four within Hapcheon were selected. Table 1 summarizes the topographic characteristics of the sub-basins considered in this study. In Table 1, A is the basin area (km 2 ), L is the channel length (km), S is the basin slope.

Preparation of storm event data
Hourly rainfall data were used in this study. The observation period of the data varies from 1 year to 36 years depending on the rain gauge station. In cases where several rain gauges were available, the basin average rainfall data were prepared by the Thiessen polygon method. Major storm events were then separated by applying some conditions like the mean rainfall intensity 10 mm/hr or maximum rainfall intensity 50 mm/hr. Additionally, only those storm events satisfying the AMC (Antecedent Moisture Condition) III condition were selected to make sure of enough runoff volume. As a result, each stream gauge station could secure storm events from a minimum 2 to a maximum 47, and they were used for the estimation of T c and K.

EVALUATION OF EMPIRICAL FORMULAS Concentration time
Various empirical formulas for T c have been proposed globally and are summarized in Table 2. In the table, T c is the concentration time (hours), A is the basin area (km 2 ), L is the channel length (km), S is the basin slope or channel slope, N is the retardance coefficient, V is the mean velocity     The empirical formulas for T c summarized in Table 2 are compared with those in the theoretical analysis. Most of the empirical formulas are expressed as a function of both L and S. As, theoretically, T c should be represented by a function of L 2 /S for turbulent flow and by a function of L/S for laminar flow, those empirical formulas in Table 2 can be compared effectively on the plane of exponents of L and S, as shown in Figure 2

Storage coefficient
The empirical formulas for K identified in this study are summarized in Table 3, where K represents the storage coefficient (hour), C is the runoff coefficient, L is the channel length (km), A is the basin area (km 2 ), S is the channel slope, and b and a are the correction factors.
Based on the theoretical analysis in the previous section, it was found that K can be proportional to T c . Thus, the forms of empirical formulas can be analyzed similarly to those for T c . Like Figure 2(a), Figure 2(b) compares empirical formulas of K in Table 3  This study estimated T c and K of an observed storm event using the method proposed by Yoo et al. (). This method is a recursive approach considering the structure of the Nash IUH (Nash ). Yoo et al. () showed that the Nash IUH has a distinct relationship between T c and K, as in Equation (7). Also, as shown in Equations (5) and (6), T c and K are correlated nonlinearly. In fact, T c shows more sensitivity to x than K does. This difference causes the strong nonlinear behavior of these two parameters when they are estimated from rainfall-runoff measurements. K can show rather stable behavior without significant variation, but T c cannot. However, the method by Yoo et al. () was found to overcome this problem when estimating T c and K using the observed rainfallrunoff data.

Results and discussion
Empirical equations for T c and K were derived for each basin using those estimated using the observed data. Figure 3 compares the resulting empirical equations on log-log paper.
As can be seen in Figure 3 especially the slopes and intercepts are found to be different basin by basin. That is, K seems to vary much basin by basin. In fact, this phenomenon was also found in the previous section where various forms of empirical formulas were analyzed. Differently from T c , more various forms of empirical formulas exist, and they also consider other basin characteristics like A and shape factor. Simply L and S may not be enough to explain K.
Finally, this study evaluated the relationship between T c and K, which was found to be linear. The coefficients of determination of those regression lines are from 0.932 to 0.995.
However, it also true that each line is different from the others. The difference may be quantified by the Russell parameter (Russell et al. ), which is the relation between K and T c or simply the slope of the line in this figure. The highest Russell parameter was found in Hapcheon dam basin, which is 1.262, and in Chungju dam basin it was the smallest at 0.986.
Even though the Russell parameter in Hapcheon dam basin is a bit higher than 1.2, all basins are included in the range 0.8 ∼ 1.2 of natural basins in Korea (Jeong & Yoon ).

CONCLUSIONS
This study tried the theoretical evaluation of T c and K with more than 20 empirical formulas available worldwide. The evaluation result was also confirmed in the application to major dam basins in Korea. The findings of this study are as follows: (1) T c could be expressed as a function of channel characteristics: L and S. Especially, T c was found to be expressed as a function of L/S under laminar flow conditions and L 2 /S under turbulent flow conditions. (2) Most empirical formulas for T c are found to follow the basic equation form considered in this study, however, those for K are found to vary considerably. It seems that K may not be explained fully by L 2 /S. Additional factors may be needed to effectively explain the storage effect of a basin.
(3) In the application to several dam basins in Korea, it was found that both T c and K could be well modeled by the factor L 2 /S. Especially, the relationship between T c and L 2 /S was very consistent in all basins, but the behavior of K varied much basin by basin.