flood routing

Hydrological Sciences-Journal-des Sciences Hydrologiques, 46(3) June 2001 349

Reservoir attenuation of floods from ungauged basins

KWAN TUN LEE, CHIN-HSIN CHANG, MING-SANG YANG Department of River and Harbor Engineering, National Ocean University, Keelung 202, Taiwan, ROC e-mail: [email protected]

WEI-SHENG YU Computer Division, Chung Yu Junior College of Business Administration, Keelung 202, Taiwan, ROC

Abstract In a typical reservoir routing problem, the givens are the inflow hydro-graph and reservoir characteristic functions. Flood attenuation investigations can be easily accomplished using a hydrological or hydraulic routing of the inflow hydro-graph to obtain the reservoir outflow hydrograph, unless the inflow hydrograph is unavailable. Although attempts for runoff simulation have been made in ungauged basins, there is only a limited degree of success in special cases. Those approaches are, in general, not suitable for basins with a reservoir. The objective of this study is to propose a procedure for flood attenuation estimation in ungauged reservoir basins. In this study, a kinematic-wave based geomorphic IUH model was adopted. The reservoir inflow hydrograph was generated through convolution integration using the rainfall excess and basin geomorphic information. Consequently, a fourth-order Runge-Kutta method was used to route the inflow hydrograph to obtain the reservoir outflow hydrograph without the aid of recorded flow data. Flood attenuation was estimated through the analysis of the inflow and outflow hydrographs of the reservoir. An ungauged reservoir basin in southern Taiwan is presented as an example to show the applicability of the proposed analytical procedure. The analytical results provide valuable information for downstream flood control work for different return periods.

Key words reservoir routing; kinematic-wave based GIUH; Runge-Kutta method

Laminage de crues de bassins versants non jaugés par un réservoir Résumé Classiquement, dans un problème de laminage par un réservoir, les données sont l'hydrogramme d'entrée et les fonctions caractéristiques du réservoir. L'étude du laminage de la crue peut être menée aisément en appliquant un transfert hydrologique ou hydraulique à l'hydrogramme d'entrée pour estimer l'hydro-gramme de sortie, sauf si l'hydrogramme d'entrée est indisponible. Bien que des tentatives de simulation de débit aient été réalisées sur des bassins versants non jaugés, elles ne sont satisfaisantes que dans certains cas. Ces approches ne sont en général pas applicables à des bassins versants présentant un réservoir. L'objectif de cette étude est de proposer une procédure d'estimation du laminage de crue en bassins versants non jaugés présentant un réservoir. Nous nous sommes appuyés sur un modèle de type HUI géomorphologique à base d'onde cinématique. L'hydrogramme entrant dans le réservoir a été généré par convolution de la pluie nette et de l'information geomorphologique caractérisant le bassin versant. Puis une méthode de Runge-Kutta du quatrième ordre a été utilisée pour transférer l'hydrogramme entrant en hydrogramme sortant du réservoir, sans recours à des données observées. Le laminage a été estimé à partir de l'analyse des hydrogrammes entrant et sortant du réservoir. La procédure analytique ainsi proposée est mise en œuvre sur un bassin versant non jaugé présentant un réservoir au sud de Taiwan. Les résultats analytiques fournissent une information pertinente pour le contrôle des crues à l'aval, et ce pour différentes périodes de retour.

Mots clefs laminage en réservoir; GIUH à base d'onde cinématique; méthode de Runge-Kutta

Open for discussion until 1 December 200]

mailto:[email protected]

350 Kwan Tun Lee et al.

INTRODUCTION

Flood waves passing through a reservoir are delayed and attenuated while they enter and spread over the reservoir storage area. Excess water during storms can be temporarily stored in the reservoir to alleviate downstream flooding. Numerical solutions of the continuity and momentum equations are usually applied to investigate flood propagation through a reservoir (Garcia-Navarro & Zorraquino, 1993). Linearized equations (Singh & Li, 1993) can also retain the essential dynamic features of the flood waves in a reservoir. If the reservoir is not excessively long and the inflow hydrograph does not rapidly change with time, the flow can be approximated using a simple technique known as level-pool routing (Linsley et al, 1982; Fread, 1993). In this technique, the reservoir is assumed always to have a horizontal water surface throughout its area. Graphical techniques such as the storage indication method (Lawler, 1964) were used in previous works. Currently, direct computational techniques, without the aid of graphs are more acceptable. Numerical methods, such as the Runge-Kutta algorithm (Chow et al, 1988; Kessler & Diskin, 1991; Bedient & Huber, 1992) and the iterative trapezoidal integration algorithm (Fread, 1993), are used to generate the outflow hydrograph in each time interval. However, both hydraulic and hydrological routing methods require the upstream inflow hydrograph of the reservoir. This is considered difficult for flood routing in an ungauged reservoir basin.

Runoff simulation in ungauged basins is one of the most difficult problems in water resources engineering. Although many methods provide runoff estimation in ungauged basins (Snyder, 1938; Gray, 1961; SCS, 1986), they are limited to small basins and under specific geomorphic conditions. The renewed researches for runoff routing in ungauged areas are TOPMODEL (Beven & Kirkby, 1979; Takeuchi et al, 1999) and the geomorphic instantaneous unit hydrograph (GIUH). In the GIUH approach (Rodriguez-Iturbe & Valdes, 1979; Gupta et al., 1980), rainfall excess is assumed to follow different runoff paths to reach the basin outlet according to the drainage pattern. Recently, Lee & Yen (1997) developed a kinematic-wave based GIUH model (KW-GIUH) to estimate the mean travel times on overland areas and in channels. The drawback of the runoff travel-time determination based on empirical equations in the original GIUH is then solved. In the KW-GIUH model, the basin rainfall-runoff relationship can be obtained using only the rainfall excess and the geomorphic information of the basin. Therefore, the KW-GIUH is applicable to basins without discharge records for model parameter calibration.

In this study, flood routing in a reservoir basin was accomplished using the KW-GIUH model to estimate the inflow hydrograph of the reservoir. The Runge-Kutta method was adopted for storage routing to obtain the outflow hydrograph. The routing procedure specifications were the rainfall excess, the basin geomorphic information, and the reservoir characteristic functions. Flood attenuation and reservoir storage efficiency were investigated through the analysis of the inflow and outflow hydrographs of the reservoir. The proposed analytical procedure was applied to an ungauged reservoir basin in southern Taiwan. The amount of flood water released from the reservoir was analysed for flood protection work carried out further downstream for different return periods.

RESERVOIR INFLOW SIMULATION

In traditional hydrology, the basin hydrological response function is, in general, based on hydrological records, which are used to determine the shape of the hydrograph or to

Reservoir attenuation of floods from ungauged basins 351

calibrate model parameters. For basins without streamflow records, attempts have been made to link the hydrological response function to basin geomorphic characteristics.

Geomorphic runoff modelling

Based on the Strahler ordering scheme, a basin of order Q can be divided into different states. Let xoi denote the /th-order overland flow regions, and xt denote the /th-order channels, in which / = 1, 2,..., Q.. For an instantaneous rainfall, the hydrograph is only influenced by the geomorphic characteristics of the basin. Thus, the instantaneous unit hydrograph of the basin can be expressed as (Rodriguez-Iturbe & Valdes, 1979):

<t)=l[fJ&*f^)*f*St>---*fJt)\rp(w) (D

where weW, W is the path space given as W = {xoi, x,, xj,..., JCQ); u(t) is the basin hydrological response function; fx(t) is the travel-time probability-density

function in state Xj with a mean value of Tx ; * denotes as a convolution integral; and

P(w) is the probability of a raindrop adopting path w.

Kinematic-wave based GIUH

According to equation (1) of the GIUH model, the hydrological response function of a basin is the summation of the travel time convolution in each flow path that can be described based on basin geomorphology. However, the runoff velocity varies spatially and temporally in basins rendering the model complicated and impracticable (Rodriguez-Iturbe & Valdes, 1979). Basin empirical formulas for travel time determination were usually adopted in previous GIUH models (Rodriguez-Iturbe & Valdes, 1979; Gupta et al, 1980; Agnese et al, 1988). Lee & Yen (1997) considered a sub-basin consisting of two identical rectangular overland-flow planes as a V-shaped model (Wooding, 1965). The overland-flow planes contribute lateral flow into a channel of uniform cross-section and slope. For an /th-order sub-basin with a mean overland length L , the time

needed for water to travel through the overland plane obtained by means of kinematic wave approximation is (Henderson & Wooding, 1964):

i

T, = -^/2.m_i

(2)

where n0 is the roughness coefficient for overland planes, S(> is the mean /th-order

overland slope, m is an exponent, which can be recognized as 5/3 from Manning's equation, and ie is the rainfall excess intensity. The mean /th-order overland length can be expressed as (Lee & Yen, 1997):

Lo, =^5^ (3) 2NiLi:,

where A is the basin total area, P0A is the initial state probability, which is equal to the

ratio of the /th-order overland areas to the total basin area, TV, is the number of


z'th-order channels, and Lt, is the mean /th-order channel length. In addition, the time

needed for water to travel through the fth-order channel can be written as (Lee & Yen, 1997):

Tr =• B:

2i„L„, hi +

2iencL„,Lc

^WB~ -K (4)

where nc is the roughness coefficient for channels, Sc is the mean fth-order channels

slope, Bi is the width of the fth-order channel, and hco is the inflow depth of the fth-

order channel due to water transported from upstream reaches. The value of hco is

equal to zero for i = 1 because no channel flow is transported from upstream. For 1 < i < Q,, hco can be expressed as (Lee & Yen, 1997):

h.„ = e c \ Ï iniN.A-AP^

N;B;S 1/2 (5)

where A, is the mean fth-order drainage area. By using equations (2), (3), (4) and (5), the travel time for different states can be estimated analytically from overland and channel hydraulics instead of relying on basin empirical formulas.

The probability-density function of travel time in equation (1) was proposed as an exponential or a uniform distribution (Rodriguez-Iturbe & Valdes, 1979; Gupta et al., 1980). To simplify the mathematical operation, an exponential distribution is adopted herein. Thus, the hydrological response of a basin can be treated conceptually as a combination of linear reservoirs in series and/or in parallel, depending on the stream network of the basin. Hence, equation (1) can be analytically obtained from:

"W=S t t t a e x p ( _ ) + b exp(-—) + b e x p ( - — ) + •

+ èQexp(-—-) KL

(6)

where aoi, bt, bj,..., £>Q are coefficients. The coefficients can be determined by comparing coefficients in partial fractions after applying the Laplace transformation. The output of the basin, which is determined using the convolution integral of the rainfall excess input and the hydrological response function u(t), can be expressed as:

ô( f) = J0'»e(T)"(f-T)d>: (7)

where Q is the direct runoff at the basin outlet, ie is the rainfall excess, and x is a dummy variable.

By applying equations (l)-(7), the upstream inflow hydrograph of the reservoir basin can be obtained based on rainfall excess and the geomorphic information of the basin. It should be emphasized that the hydrological response function u(t) in


equation (7) varies temporally with the rainfall hyetograph because the runoff travel time Tx in the hydrological response function is a function of rainfall excess

intensity, ie. Because of this, a linear model can describe nonlinear effect (Kundzewicz & Napiorkowski, 1986), The dynamic nature of the basin hydrological response function is considered the major merit of the KW-GIUH model.

Calculation of geomorphic parameters and estimation of roughness coefficients

Most of the geomorphic information in the KW-GIUH can be obtained from topographic maps. Techniques are now available for extracting slope properties, drainage divides, drainage areas, and stream networks based on digital elevation models analysis (Band, 1986; Jenson & Domingue, 1988; Lee, 1998) with less effort. However, a field investigation is still required to determine the channel width B„ a parameter required in equations (4) and (5).

In gauged basins, the overland and channel roughness coefficients in equations (2), (4), and (5) can be obtained using an optimization technique with the help of observed hydrographs. Nevertheless, the roughness coefficients determination in ungauged basins can be based only on field surveys. References for the channel roughness coefficient determination can be found in general open-channel flow books (Chow, 1959; Chaudhry, 1993). A possible way to determine the overland roughness coefficient is based on land cover analysis. References, which provide a link between overland surface conditions and the roughness coefficient, can be found in the Hydrologie Engineering Center (1990) for practical applications. Hence, the overland roughness coefficient in ungauged basins in this study was determined by field surveys and remote sensing images analysis.

RESERVOIR ROUTING FORMULATION

An alternative to the storage indication method applied in previous engineering work was the Runge-Kutta numerical method to solve the continuity equation. The Runge-Kutta method does not require the computation of the special storage-outflow function, and it is more closely related to the hydraulics of flow through the reservoir (Chow et al, 1988).

The continuity equation of a reservoir is expressed as:

^r = QiB(t)-QoJH) (8) at

where Sr is the reservoir storage in volume, Qm(t) is the reservoir inflow as a function of time t, and Q0ut(H) is the reservoir outflow as a function of the water elevation H. Since the reservoir water surface Ar is a function of the water elevation, the change in storage dSr due to a change in elevation is equal to Ar(H)dH. Thus, the continuity equation for the reservoir flow can be rewritten as:

df Ar(H) = f(t,H) (9)

where t is the independent variable, and H is the dependent variable. The solution for

354 Kwan Tun Lee et ai

equation (9) is extended forward in small increments of time t, using known values of the water elevation H. Various orders of the Runge-Kutta algorithm have been derived (Carnahan et al, 1969). A fourth order scheme was adopted in this work. The water elevation H at the (n + i)th time interval can be expressed as (Bedient & Huber, 1992):

Hn+1=H„+7[*1+2*,+2fc3 + *4]Af (10) 6

where Hn is the water elevation at the nth time interval, At is the small time increment, and the coefficients ki,k2,h, and k4 are determined using:

kl = f(tn,Hn) (11)

k2=f{tn+At/2,Hn+^âa) (12)

k3=f{tn+At/2,Hn+ik2At) (13)

k4 = f{tn+At,Hn+k3At) (14)

Using equations (10)-(14) and the gout vs H curve of the reservoir outlet device, the outflow hydrograph of the reservoir can be calculated in each time interval.

Using the KW-GIUH model and the Runge-Kutta method, the flowchart for flood attenuation analysis in an ungauged reservoir basin is shown in Fig. 1. The input hydrological data for a storm event simulation is only the measured rainfall hyetograph of the storm, and historical rainfall records are required for a design work to conduct rainfall frequency analyses.

MODEL APPLICATION

An ungauged reservoir basin was used as an example in this study to demonstrate the applicability of the proposed procedure. The Akung-Tien reservoir is a small reservoir located in southern Taiwan. The purpose of this reservoir is for downstream flood control. A portion of the storage water is conveyed to neighbouring farms for irrigation usage. There is only one rainfall gauging station in this area. Flow record for the reservoir basin was unavailable. Recently, floods have frequently occurred in the downstream area. The detention capability of the reservoir must be determined in order to improve the flood prevention work downstream.

Geomorphic conditions and reservoir characteristic functions

The reservoir basin area is 32 km2. The height of the main dam is 31 m, and the elevation of the top of the main dam is 42.0 m. The results of the SPOT images analysis show that the land cover is 49.3% forest, 35.6% grass and farm field, 5% building and road, 1.3% bare soil, and 8.8% water body and some inadequately classified regions. As shown in Fig. 2, due to the complicated stream network, the basin is divided into three sub-basins for runoff simulation. In sub-basin I, the area is 9.07 km2; the mean overland length is 0.65 km; and the mean overland slope is 0.131. The geomorphic parameters of sub-basins II and III are listed in Table 1. Reservoir sedimentation is a severe problem in the Akung-Tien basin. Seventy-one percent


START

Input • basin geomorphic information • reservoir characteristic functions • basin rainfall records

Storm event simulation Design work

' ' Measured rainfall hyetograph

" Frequency analysis of historical rainfall

records

Deduction of rainfall losses Generation of design hyetograph by alternating block method

Simulation of reservoir upstream of basin rainfall-runoff by KW-GIUH model

Reservoir flood routing by Runge-Kutta method

Flood attenuation analysis

Ç END J

Fig. 1 Flowchart of the analytical procedure.

of the available reservoir storage capacity has been filled with sediment. The reservoir storage capacity has decreased from 20.5 X 106 m3 after dam construction in 1953 to 5.9 X 106 m3 in 1993. The stage-storage relationship measured in 1993 was:

Sr = 178871(f/-30)2 + 59638(#-30) + 112261 (15)

where Sr is the reservoir storage (m ), and H is the water elevation (m). Differentiating equation (15) with respect to H yields:

A r(iï) = 357742(#-30) + 59638/ï (16)

where A,(H) is the area of the water surface (m2). In this earthfill dam, a 1.5 m concrete pipe is used to convey water for irrigation,

and the maximum capacity of the pipe is 5 m3 s"1. The main drainage facility is the Morning Glory spillway to drain the water past the dam during floods. The elevation of the crest of the Morning Glory spillway is 35.5 m, and its outflow function is:


Qom = 74.375(H - 35.5)1'7881 for H < 36.6 m (17)

<20ut = 2.2934(# - 35.5) + 77.5 for H > 36.6 m (18)

where gout is the discharge (m3 s"1), and H is the water elevation (m).

Storm event simulation

The KW-GIUH model has been tested in several gauged basins in Taiwan (Lee & Yen, 1997) as well as in the United States (Yen & Lee, 1997) and shown good results. In the Akung-Tien basin, the discharge record is unavailable. The Chu-Tzu-Chiao rainfall station located near the dam site is the only hydrological gauging station in this area. To perform a storm runoff simulation using the KW-GIUH model, the hydrographs resulting from those three sub-basins (as shown in Fig. 2) are summed to obtain the inflow hydrograph of the reservoir. The outflow hydrograph of the reservoir can then be calculated using the Runge-Kutta method.

Reservoir storage area

Sub-basin I

Sub-basin II

Sub-basin m

* i v .•••>

•i

N

0 2 km

Fig. 2 Map of the Akung-Tien basin.

As stated previously, in ungauged basins the roughness coefficients nc and n0 can only be determined from field surveys and land cover analysis using remote sensing images. Based on references from Chow (1959) and HEC (1990) as well as the authors' previous experience (Lee & Chiang, 1997; Lee, 1998), the prior roughness values were chosen as nc = 0.04 and n0 = 0.6. The IUHs for the three sub-basins were determined. Figure 3 shows an example of the IUHs for sub-basin III. In this figure, the peaks of IUHs vary proportionally with the rainfall excess intensity.


0 8

Fi!

4 Time (h)

>. 3 Variation of IUH with rainfall excess rate (sub-basin III)

40

38

36 —

34 —

32

30

0

"T

24 48

1994 August

recorded simulated

120

Fig

72 96 Time (h)

. 4 Recorded and simulated water stage for August 1994 event.

144

Over the past 40 years, there was only one unofficial stage record at the reservoir, due to a storm which occurred in August 1994. This record was measured at different time intervals. The proposed procedures were applied and the simulated and recorded stages at the dam site are shown in Fig. 4. The rainfall excess hyetograph used in this simulation was determined by deducting the abstractions from the rainfall record assuming a constant loss / = 1.0 mm h"1. This value is adopted by the Taiwan Water Conservancy Agency in mountain areas to represent saturated hydraulic conductivity because soil is usually wetted by antecedent rainfalls before the main storm. The tendency of the stage hydrograph, shown in Fig. 4, was captured by using the proposed routing technique. The deviation between the recorded and simulated water stages results possibly from the precision of the observed stage data, the accuracy of the stage-storage relationship, and the model simulation error. The rainfall excess rate, the simulated reservoir inflow discharge, and the simulated reservoir outflow discharge are also shown in Fig. 5. In this figure, flood attenuation and reservoir storage capacity are explicitly demonstrated in this storm event. By using the roughness condition nc = 0.04 and n0 - 0.6, simulation for another event (July 1997 storm) is presented in Fig. 6. Although no stage record can provide model verification work, the hydrographs show the same tendency for flood peak attenuation in this reservoir.


u

Ç 40 — E E ^ 80 —

400 —

320 —

240 —

C 160 — E

O 8 0 -

Ttffl

r\f 'N 1 / 1 *» j 1

n B 0

1 1 i

-yy- i r «*• | i i

1 /J' 1 ' 1

r

1994 August

Inflow

- — - Outflow

"Is

1 1 " 1 '

0 24 96 120 48 72 Time (h)

Fig. 5 Simulated inflow and outflow hydrographs for the August 1994 event

144

0 30 90 60

Time (h) Fig. 6 Simulated inflow and outflow hydrographs for the July 1997 event.

120

Flood attenuation analysis for design work

For flood control work further downstream, hydrograph analysis for specified return periods is required. Since flow records are unavailable, a synthetic hyetograph combined with the proposed KW-GIUH model and the Runge-Kutta routing technique is one of the acceptable alternatives to obtain the design hydrographs at the reservoir


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outlet. Rainfall frequency analysis indicates that the records of the Chu-Tzu-Chiao station follow a log Pearson Type III distribution. The rainfall intensity-duration functions for different return periods are:

1447 for T= 10 yrs (19) (td + 20)u

2554

(frf+35)a

4687

(td+55f

12588

(td+95f

for T= 25 yrs

for T= 50 yrs

for T= 100 yrs

(20)

(21)

(22)

where / is the rainfall intensity (mm h" ); td is the rainfall duration (min); and Tis the return period (years).

To account for the temporal rainfall characteristics, the alternating block method (Chow et al., 1988) is adopted to derive the design hyetograph. By using equations (19)-(22), the design hyetographs for different return periods are shown in Fig. 7. The duration of the design hyetographs was chosen as 24 h, which can produce the highest

10 Time (h)

Fig. 7 Design hyetograph of the Akung-Tien basin.


0.20-

0 . 1 5 -

0.10-

Flood attenuation ratio

Flood storage ratio

400

- 0 . 7 S

- 0 . 6

•0.5

1000 600 800 Inflow peak discharge (m3/s)

Fig. 8 Flood attenuation ratio and flood storage ratio vs reservoir inflow peak discharge.

runoff peak rate for the given temporal rainfall distribution for different return periods. Using the design hyetographs as input conditions, the results of the design inflow peak and the design outflow peak for the reservoir basin simulated by the KW-GIUH model and the Runge-Kutta method are listed in Table 2.

To perform a quantitative analysis in the reservoir basin, two ratios were used as criteria to evaluate flood attenuation and reservoir storage efficiency during a storm. The flood attenuation ratio is defined as the outflow peak divided by the inflow peak, and the storage ratio is defined as the maximum storage divided by the total volume of the inflow hydrographs (Basha, 1994). As shown in Table 2 and Fig. 8, the flood attenuation ratio was from 0.11 to 0.17 and the storage ratio was from 0.54 to 0.67 for these four return periods. The flood attenuation ratio decreased with the inflow peak increase. On the contrary, the storage ratio increased with the inflow peak increase. Hydroeconomic analysis will be performed in the future to select the optimum design return period to be used to determine the downstream flood prevention work.

CONCLUSION

The procedure described herein offers an efficient approach for flood attenuation analysis in ungauged reservoir basins. The main advantage of the proposed procedure is that hydrograph analysis can be carried out without the use of any data. The procedure is based on basin geomorphic information and reservoir characteristics only. There is no empirical equation involved in the analytical procedure. The geomorphic parameters in the KW-GIUH model can be obtained by applying current geographic information systems (GIS). The computational works of the KW-GIUH model and the Runge-Kutta method are easily managed in a PC operation.

Acknowledgements This study is part of a research work supported by the National Science Council, Taiwan, ROC, under grant NSC 88-2625-Z-019-002. Financial support from the National Science Council is gratefully acknowledged. The hydrological data from the Akung-Tien Administration Bureau is highly appreciated.


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Engng ASCE 2(1), 1-9. Received 28 February 2000; accepted 7 December 2000

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