flood routing in a wide channel with a quadratic lag-and ... · flood routing in a wide channel...

Hydrological Sciences-Journal-des Sciences Hydrologiques, 42(2) April 1997 169

Flood routing in a wide channel with a quadratic lag-and-route method

PITER L. F. BENTURA & CLAUDE MICHEL Hydrology Division, CEMAGREF, BP 121, F-92185 Antony Cedex, France

Abstract In hydrological practice, flood routing is rarely achieved using the complete Saint-Venant equations. Instead, hydrological methods have been developed to provide users with simple methods that are robust yet efficient. The lag-and-route method has been widely used but its parameters have never been related to the physical parameters of channels. Moreover, the linear reservoir in this method is not well suited to reproduce effectively actual channel routing. A quadratic reservoir coupled with a lag component results in a far more accurate method while remaining numerically workable. This latter method depends on only two parameters which have been successfully related to physical channel features. This simple and efficient method enables one to gain insight into the main features of flood propagation in river channels.

Propagation de crue dans un canal large par une méthode combinant délai et stockage dans un réservoir quadratique Résumé Dans la pratique hydrologique, la propagation de crue est rarement effectuée au moyen des équations unidimensionnelles de Saint-Venant. Des méthodes propres à un usage hydrologique ont été élaborées afin de fournir aux utilisateurs des outils simples et robustes. La méthode qui combine délai et stockage par un réservoir est d'un emploi très répandu mais ses paramètres n'ont jamais été reliés aux seules caractéristiques physiques des canaux. De plus, le réservoir linéaire n'est pas très bien adapté à la propagation des crues. Le recours à un réservoir quadratique conduit à une méthode nettement plus précise, tout en restant numériquement simple. Cette méthode ne dépend que de deux paramètres qui ont pu être reliés avec succès aux caractéristiques des canaux empruntés par l'écoulement. Cette méthode simple et efficace permet de mieux cerner les principales caractéristiques de la propagation des écoulements.

INTRODUCTION

The so-called hydrological methods for flood routing have been widely used because basically such simple methods can easily cope with the paucity of data typically encountered in practical hydrology. The Muskingum method is most frequently used in practice and is still the subject of research (Hjelmfelt, 1985; Perumal, 1992a; Gill, 1992). Its main appeal comes from work by Cunge (1969) who demonstrated that this method could be considered as being numerically related to the Saint-Venant equations via the diffusion wave equation. Another widespread method is the one developed by Hay ami (1951) and which is tantamount to a unit hydrograph method. Another hydrological method is the lag-and-route method, attributed to Meyer (1941) according to Gill (1994). Nash (1959), in the conclusion of his paper, recommended

* Now at: Jurusan Teknik Sipil, Kampus ITS Sukolilo, Surabaya 60111, Indonesia.

Open for discussion until 1 October 1997

170 Piter L. F. Bentura & Claude Michel

to turn to the lag-and-route method to avoid difficulties arising from the use of the Muskingum method. The lag-and-route method has mostly been used with event-based rainfall-runoff models such as the Australian RORB model (Malone & Cordery, 1989). In this model, the linear routing reservoir, traditionally used in the lag-and-route method, was replaced by a nonlinear reservoir in order to achieve a better efficiency for the method. However, the exponent proposed in the RORB model was close to one. A good review of hydrological routing methods can be found in a number of papers (e.g., Laurenson & Mein, 1995; Weinmann & Laurenson, 1979). The lag-and-route method is in fact very appealing because the two components, i.e. translation with lag and attenuation with routing through a reservoir, are conceptual tools that are popular and acceptable among hydrologists. This is the reason why this model is the focus of the present paper.

The objectives of this study are to demonstrate the efficiency of using a quadratic reservoir and to find relationships for estimating the parameters of the lag-and-route model from channel physical parameters. In a quadratic reservoir, the reservoir outflow is proportional to the square of the storage in the reservoir. Such a quadratic reservoir was employed by the UK Institute of Hydrology as groundwater storage in the Probability-Distributed Model applied to flood forecasting (Moore, 1995). To start, the traditional linear lag-and-route model, which is a well-known reference, will be described. The corresponding lag-and-route models will be referred to as the linear and quadratic models, respectively. In the following two sections are set out the way these models have been calibrated (against which data) and validated.

APPROACH USED TO ASSESS MODEL EFFICIENCY

Field data are never accurate enough to discriminate between close alternative models and thus the use of an undisputed hydraulic model is desirable. Consequently, it is postulated that the ideal performance for an approximate (hydrological) flood routing method is to emulate the routing effected by the Saint-Venant model, which is assumed to be an exact hydraulic model. Therefore the continuity equation and the full dynamic equation for the one-dimensional problem of flood routing in a wide channel is used as a reference. In this way, hypothetical stream flow data at the downstream end of a hypothetical channel can be generated. A hypothetical upstream hydrograph may then be routed along a channel reach according to the Saint-Venant equations, and the stream flow values at the outlet of the reach computed. These flow rates provide the ideal flow rates to be reproduced when using an approximate hydrological routing method. The Saint-Venant equations have been solved with the help of a numerical four-point implicit method along with a technique known as "double sweeping" with a time-weighting coefficient 8 equal to 0.75, i.e. the equations were solved for the point (x + Ax/2, t + QAt) where x and t are space and time coordinates, respectively, and Ax and At are corresponding increments of space and time.

To make sure that the main features of the flood routing were accurately captured, the upstream condition consisted of a long series of floods generated by

Flood routing in a wide channel with a quadratic lag~and~route method 171

successively adding single peak impulses, the Mi being described by the expression:

Pk(t) = Pd t-tt

exp 2 -2('-'J

(1) lmk J

where Pxk is the maximum ordinate of the flow impulse which begins at time tk and peaks at tk + tmk, while Pk is the stream flow increase induced by the Mi impulse at time t. In this expression all stream flows are actually discharges per unit width of channel. For each impulse the various parameter values were obtained at random from log-normal distributions as follows:

ln(Pxk): N(1A, 0.5) where PIk is expressed in m s ' ln(tmk): JV(10.2, 0.4) where tmk is expressed in seconds \n(tk-tkA): JV(11.2, 0.4) where tk is expressed in seconds and to 0

where the first figure inside the brackets is the mean and the second is the standard deviation.

A series of 22 successive floods (k = 1, 2, ..., 22), spread over a period of 12 600 two-minute time steps were generated at random and used on every channel reach tested. The complete hydrograph is described by the following summation:

P(t) = Qo+1tPk(t) (2)

In this expression, the baseflow, Q0, could be changed at each channel routing experiment. The hydrograph corresponding to equation (2) is outlined in Fig. 1. This complex hydrograph was used for all flood propagations, each one occurring in a

o

Fig. 1

24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408

TIME (Hours) The complete hydrograph inflow.


specific wide channel (except that Q0 was allowed to be changed in addition to changes of channel reach characteristics).

Generation of the downstream hydrograph was carried out using the Saint-Venant model along with an initial condition, an upstream boundary condition, and a supplementary downstream boundary condition. The propagation was taken as occurring in wide channels, i.e. flow rates were used as discharges per unit width. A hypothetical wide channel was described by four characteristics:

n: the Manning friction coefficient; s: the slope of the channel; LT: the length of the reach; and B: the width of the channel.

(This latter parameter does not appear explicitly if flow rates are expressed in m2 s'). The initial condition was an overall flow rate equal to the baseflow, Q

Different downstream boundary conditions were tested. One downstream boundary condition was described by a rating curve which corresponds to a normal depth of flow for a width slightly different from B, i.e. the downstream depth-flow relationship was:

w \ \ Q = -y3s* (3)

where y is depth of water, Q is flow rate, both at the downstream boundary, and w is the ratio of the outlet width to the channel width.

To avoid part of the influence of the two boundary conditions, the reach under study, (i.e. the reach where propagation was to be studied) was an inner sub-reach whose length was L = 0.75LT.

In order to test the two hydrological flood routing models, one hundred different reaches and boundary conditions were proposed. The characteristics of these conditions were drawn from log-normal distributions:

ln(n): iV(-3.5, 0.2) where n is the Manning roughness coefficient (m"1/3 s); ln(s): N(-l.2, 0.8) where s is the slope of the reach; ln(L): N(9.2, 0.2) where L is the length of the inner sub-reach (m); ln(Q0): N(-l.4, 0.2) where Q0 is the initial baseflow rate (m2s4); ln(w): N(0, 0.07)where w is the channel width ratio at the downstream end of the reach. When computing the flood propagation according to the Saint-Venant equations

with a double-sweep implicit method, Q(t), the hydrograph stream flow at the entrance of the inner sub-reach of length L was recorded along with the value of the stream flow at the exit section of this sub-reach: 0(t). A constant time step of 120 s was used with a space increment equal to LT/120, i.e. generally close to 83 m.

Each propagation corresponded to a situation described by five parameters: three parameters for the channel reach (s, n, L), one parameter for the initial condition (Q0), and one parameter for the downstream boundary condition (w). Thus, there were 100 sets of the two hydrographs, Q(f), Oit), at the ends of the inner sub-reach, L. Each hydrograph was composed of twenty-two impulses corresponding to the "exact" routing of one upstream hydrograph, P(t). The hydrograph outflow 0(f) was

Flood routing in a wide channel with a quadratic lag-and-route method 173

Saint-Venant routing(Q,0]

—f-l/12—f-

-4-Fig. 2 Flood routing along a channel reach by the Saint-Venant model.

the one against which the iag-and-route model, generating R(t), was checked. A schematic representation of a channel reach flood routing, using the foregoing notations, is displayed in Fig. 2.

THE LINEAR LAG-AND-ROUTE MODEL

This model has two components which can be used in either order. For example, the upstream flow is first routed through a reservoir whose outflow is related to its storage value by a linear relationship. Let Q(t) be the flow rate at the upstream end of the reach under study, with t being time. Let R(t) be the outflow from a reservoir whose storage is V(t). The relationship between R(t) and V(f) is as follows:

R(t) = V(t)/S

where S is the time constant of the reservoir and is the first parameter of the model. From continuity dV/dt = Q{t) - R(t). Replacing R this relationship becomes:

dVldt = Q(t) - V(t)IS

The foregoing differential equation has to be solved in order to get an explicit expression for V(t), and consequently for R(t). This equation can be solved when the upstream flow is described by straight line segments between data ordinates which is the usual format of flow data. During one time increment, At, the inflow at time t from the start of At is:

G(Ï) = Ô,+(Ô2-ÔI) I At.

(4)

where £?, and Q2 are the flow rates at the beginning and end of the time step, respectively. Let R{ be the value of the outflow computed at the end of the previous time interval. With such boundary conditions, it is shown in Appendix 1 that the


solution for R2 at the end of At is:

R2=C0R1+ClQ1+C2Q2 (5)

where:

C 0 = e x p ( - ^ ) Cl=±(l-C0)-CQ C 2 = l - A ( i _ c 0 ) (6)

Next consider, the lag component. The flow at the end of the channel reach will be R(t + T), evolving from Rl at time T to R2 at time At + T, where T is the second parameter of the model and has units of time.

THE QUADRATIC LAG-AND-ROUTE MODEL

As before, the upstream flow is first routed through a reservoir, one whose outflow is now related to its storage value by a quadratic relationship. With the same notation, the relationship between R(t) and V(t) for the quadratic model is as follows:

R(t) = [V(t)/Sf

or, more generally:

R(t) = [V(t)/S\a (7)

Unlike the previous use of the same parameter notation, the dimension of the parameter S now depends on the value of the exponent a. Throughout the paper, SI units will be used. The continuity equation for the reservoir again reads:

dV(t)/dt = Q(t) - R(t)

Substituting for R this relationship becomes:

dV/dt = Q(t) - [V(t)/S]a

Henceforth, a will be equal to 2. For this quadratic case the differential equation can be integrated analytically if the upstream inflow is taken to be constant for each time step. Therefore, if the inflow hydrograph evolves linearly from g, to Q2 during the time step At, it is necessary to split At into N sub-intervals et, assuming an approximately constant flow rate in each of these subintervals. For each sub-interval of duration 8t, one may denote the approximately constant incoming flow as Q*. In Appendix 2, it is shown that to secure good accuracy it is sufficient (relative error less than 0.001) to choose N to be greater than or equal to 11 (AtQ*05)/S. The following expression is obtained:

R.(t + 8t) =

5 Q*&t R(tr+ s

Ritf'bt 1 + -

(8) o/

S

Note that this latter expression remains valid even for Q* = 0. The analytical computa-


tions are similar to those carried out by Singh & Scarlatos (1987) for a nonlinear Muskingum flood routing method. Equation (8) has to be iterated N times to yield R2

(Appendix 2). As for the linear model, allowance has to be made for a lag T, and so R(t + At + T) = R2, where T is the second parameter of the model and has units of time.

CALIBRATION OF THE TWO LAG-AND-ROUTE MODELS

The calibration of parameters S and T was carried out with a steepest descent optimization method for each channel routing experiment. The initialization was made using R(O) = Q0. The criterion used for the calibration for each channel experiment was:

F = 100 i=i 12600

I(a-o,)2 (9)

where Q is the upstream flow rate, O is the Saint-Venant downstream flow rate at the end of the modelled channel reach, and R is the downstream flow rate as computed by the chosen lag-and-route model, and the subscript / refers to values at (t/At) +1. The criterion F compares the sum of squared errors of the model tested to that of the zero-level model which would accept the upstream flow rate as the required downstream flow rate. For each model, a set of 100 values of F was obtained. To simplify the comparisons, only the first two moments of the two sets of criteria values are given:

linear model: u. = 92.13, a = 0.88

quadratic model: u. = 97.73, CJ = 0.65

With the latter model, the relative root mean square error, equal to ^l-u/100, is 15%.

The results show that a strongly nonlinear reservoir (a = 2) provided by far the better model. In contrast, in the current literature, a is generally taken between 1 and 1.5. The linear reservoir was maintained in the following further studies due to its enlightening simplicity. Moreover, an explanation of its parameters had never been properly achieved, and remained to be made.

A by-product of the foregoing analysis was an ensemble of two sets of a hundred pairs of parameter values corresponding to the two lag-and-route models and the hundred routing experiments, each carried out with a given channel reach and the given multi-peak upstream hydrograph. It would have been cumbersome to display these results. Instead relationships between these parameters and the characteristics of the channel reaches were sought, i.e. the length L, the slope s, the Manning friction coefficient n, and occasionally the downstream boundary coefficient w and the base flow Q0. For the case when the method is to be used in flood forecasting, it


is not advisable to introduce hydrograph characteristics such as the peak flow rate of the upstream hydrograph or its time to peak, although this has been done in many studies (Mein et al., 1974; Pilgrim, 1976; Boyd et al., 1979; Lettenmaier & Wood, 1993). Obviously, these characteristics are not always known in the actual context of forecasting. Parameters were not allowed to vary with the current value of the inflow rate (Kohler, 1958) as has often been done for the Muskingum method (Becker & Kundzewicz, 1987; Guang-Te & Singh; 1992; Perumal, 1992b) and which could come across as risky in the sense that such a method improves the results in ordinary situations but could worsen them in extreme situations or in the vicinity of peaks.

For the Hay ami model and for another lumped model, Baptista & Michel (1989, 1990) found it very difficult to relate model parameters to physical characteristics of channel reaches as did Bravo et al. (1994) for the Muskingum model. When a hydro-logical model can be analytically derived from physical equations as was done by Cunge (1969) for the Muskingum method, expressions can be established along the same physical derivation after linearization around a given level of discharge value (reference conditions). This solution was proposed by Dooge et al. (1982). However, the choice of the reference discharge is generally problematic. The main reasons for the difficulty in predicting the values of the parameters S and T appeared to be firstly, the intricate role of slope, and secondly, the overlapping of the two components of the lag-and-route methods since the reservoir routing causes some part of the delaying process in addition to that caused by the lag component itself (parameter interaction). The results are presented below for the linear and the quadratic lag-and-route methods.

PARAMETER ASSESSMENT FOR THE LINEAR LAG-AND-ROUTE METHOD

Expressions for estimating the S and T parameters from physical features of channels cannot be derived using an analytical approach as was done, for instance, for the Hayami model, since the lag-and-route models cannot be traced back to the Saint-Venant model. The only possible course of action therefore is to look, empirically and a posteriori, for the relationships that must exist between the physical parameters of the channels and the lumped parameters of the present models. The first solution was to attempt a linear regression relationship between the logarithms of the parameters of the lag-and-route method and the logarithms of the physical parameters of the channel reach under study. Such predictive models were evaluated as follows:

ln(5)= -11.05 - 1.94 \n{s) +1.83 ln(n) + 0.26 ln(w) +0.88 ln(g0) 1.15 ln(I) (5.36) (0.12) (0.49) (1.54) (0.48) (0.56)

standard error 0.90.

ln(7) = - 3.46 + 0.23 ln(s) - 0.19 ln(n) + 0.41 ln(w) + 0.12 ln(g0) + 1.36 ln(L) (2.56) (0.06) (0.23) (0.73) (0.30) (0.27)

standard error 0.43.


The numbers within brackets are the standard errors of estimate corresponding to the regression coefficients situated just above. Scatter plots illustrate the poor quality of these models (Figs 3 and 4). This poor result underscores the difficulty of explaining the model parameters. Nevertheless one interesting result was that the initial and boundary conditions had no sensitive influence on the parameters since the relevant coefficients were not significant. One needs to look for better explanations of the S and T parameters. When calibrating these two parameters of the model, it was found that there was a strong correlation between them. The best solution

9 -

u. 8-tu \-LU 7 -

2

PA

RA

Q UJ S -

< IAII

h -

« m 3 -

2 -

1 -

R2

o

= 0.80 st.er. = 0.90 o •

Qo

. cPo .-•o % g

o

o

0 c ° ..,-"' a o r> °^ ° o

o "

° ° o, o° o ,,-"" °

o

4 5 6 7 OBSERVED PARAMETER

Fig. 3 Regression of parameter ln(S) for the linear model.

ce LU H LU

< 7 ce < CL

o LU

S h-« LU

R2 = 0.39 st.er. = 0.43

OBSERVED PARAMETER Fig. 4 Regression of parameter ln(7) for the linear model.


seemed to be to define a celerity corresponding to L/(T + S), denoted C, which was regressed against s and n to yield:

ln(C) = 0.94 + 0.59 Id — \n )

(0.0013) (0.003)

In this equation, the standard error was 0.013 (for natural logarithms), with C in ms'1, T and S in seconds and L in metres. The coefficient of correlation between the two regression coefficients was -0.20. One can write equivalently:

C = 2.57 f r,\059

(10)

When looking at equation (10), it seems tempting to write y-Js/nj and to identify

the exponent 0.6 with the inverse of the exponent 5/3 in equation (3) (Thirriot, personal communication, 1995). To check whether this presumption was likely, the study was continued with exponents other than 5/3 in equation (3). The results seemed to favour the groupingVs/w and the connection between the exponent in equation (10) and in the Manning equation. For instance, with the Chezy relationship, where 2/3 is replaced by 1/2, (and therefore 1 + 2/3 by 1 + 1/2 (equation (3))) and the exponent 3/5 by 2/3, the following expression for C was found:

C = 2.17 — V n J

As for the fraction T/(T + S), the following expression was first derived:

0.0007 0.016 4s £>=L23- —~~+ -0 .30— (11)

5 + 0.0003 n n (0.031) (0.00002) (0.00088) (0.022)

when 0 < D< 1, then T/(T + S) = D and T + S = L/C, when D < 0, then T = 0 and S = L/C, when D > 1, then 5 = 0 and T = L/C.

The standard error in equation (11) was 0.044. Regressions for the linear model are shown in Figs 5 and 6.

Equation (11) is very interesting because it stresses which component is important for the channel reach under study. For example, when 5 = 0, which is a rather frequent situation, the propagation reduces to a simple translation (kinematic wave). Conversely, when D < 0, then T is equal to zero, and there is less possibility for proper flood forecasting since the actual lead time offered by the model is, strictly speaking, equal to T. Thus, the expression at the right-hand side of equation (11) allows one to check whether or not there is a substantial lag allowing use of the model for flood forecasting at the end of a given reach from available information at the upstream end of the channel reach.


1.4-

t£. UJ1.2-

Lil lAIV

OC 1.0-< Q. D IU 0.8 -h-< S feo.6-LLI

0.4 -

0.2 -O-''

R2 =

•9

0.99

0 - ""

— — T - ~ -

st.er. = 0.01 . 6

0

* ® ° _5

« P

oq®

*̂P o ^

O G

0 ' O

O..-Ô

, . . , , ] . ———f ; • •• • ; — r

0.2 0.4 0.6 0.8 1.0 1.2

OBSERVED PARAMETER Fig. 5 Regression of ln(Q for the linear model.

1 . ^

1.0 -

DC UJ

U J 0 . 8 -2 < ce < û - 0 . 6 -Q UJ H < 5 0.4 -H-W UJ

0.2 -

0 .0 -

R2

0

= 0.97

0 " 0 0

st.er. = 0.04

c 0 .

0

°8

cP .. '

I • i

° H' 0 °?°'"'s§

o„

0 .o"o

i I I \ • -,

0.4 0.6 0.8

OBSERVED PARAMETER Fig. 6 Regression of D for the linear model.

PARAMETER ASSESSMENT FOR THE QUADRATIC LAG-AND-ROUTE METHOD

The quadratic model is more efficient than the linear one, but the possibility to balance out the two components cannot be achieved in the simple way done above because 5" and T do not have the same physical dimension. Expressions therefore


were worked out directly for the two parameters S and T. Since the quadratic model is the one recommended for practical use based on its clear superiority, a search was made for as efficient relationships as possible. To allow some complexity in these relationships it was necessary to use a larger sample of channel reaches than the one used for comparing the two models. Some 600 channel reaches were selected at random along the previous lines increasing substantially the size of the sample. With the resulting data the following relationship was found to explain S:

0.0003 In(5) = 2.0 +

0.0067 0.30w + 0.711n(I) + s+0.0002 n v ' 4~s

(0.035) (0.000005) (0.00024) (0.0036) (0.0068)

(12)

with the correlation matrix of the regression coefficients:

1.00 0.24 -0.27 -0.95 -0.25

0.24 1.00 -0.88 0.02 -0.97

-0.27 -0.88 1.00 -0.03 0.90

-0.95 0.02 -0.03 1.00 -0.03

-0.25 -0.97 0.90 -0.03 1.00

where S is in m s1/2, L is in metres and n in m1/3 s. Standard errors of the estimates are within brackets underneath the corresponding coefficients in equation (12), and the standard error 0.018. A scatter plot appears in Fig 7. For the second parameter (7), the following relationship was obtained:

\n(T) = -4.17-0.721n(j+0.00015) + 1211n(n) + 1271n(i.)-1.29/(Ls)-0.54«/>/s (13)

(0.07) (0.007) (0.02) (0.009) (0.033) (0.014)

m i -UJ 9.6

< ce < Q- 9.0

a H < H 8.5 H W 111

R2 = 0.99 st.er. = 0.02

OBSERVED PARAMETER Fig. 7 Regression of parameter ln(S) for the quadratic model.


with the correlation matrix between the coefficients:

1.00

-0.07

-0.18

-0.79

-0.05

-0.06

-0.07

1.00

0.71

-0.04

-0.12

0.76

-0.18

0.71

1.00

-0.34

-0.64

0.91

-0.79

-0.04

-0.34

1.00

0.54

-0.37

-0.05

-0.12

-0.64

0.54

1.00

-0.69

-0.06

0.76

0.91

-0.37

-0.69

1.00

where T is expressed in seconds, L in metres, and n in m1/3s. The figures within brackets again are the standard deviations of the regression coefficients, with the standard error of the error term 0.037. Introduction of two terms depending on the slope, s, one combined with the length of the reach, L, and the second combined with the Manning coefficient proved necessary to get the above small standard error. It is evidence of the complex role of slope in flow routing. A scatter plot is shown in Fig. (8).

w8.o UJ

< a: 7.5 < o. Û UJ7.0

<

UJ

R2 = 0.989 st.er. = 0.037

"

_ °LMÊÊ

q^JPra^

0 .••-"

t 1 I

mm^

K o

X°

7.0 7.5 OBSERVED PARAMETER

Fig. 8 Regression of parameter ln(7) for the quadratic model.

APPLICATION OF THE PROPOSED METHOD

Although the reach length is integral to the above relationships, one has to address the problem discussed by Diskin & Ding (1994) about the independence of the method of reach subdivision. This property was called "consistency" by Ponce & Theurer (1982). Actually the models cannot rigorously yield the same results when the reach under study considered as a whole or is subdivided into any number of reaches. Since the experiments were carried out with a median length of 10 000 metres, it is suggested that the actual channel reach be divided into a number of sub-


reaches which each exhibit a common length as closely as possible, in a geometrical sense, to 10 000. Choosing M sub-reaches means that the relative error would be greater than were the reach subdivided into M + 1 sub-reaches. That is:

(L/M)/10 000 < 10 000/[L/(M + 1)]

This inequality states that the relative error between LI M and 10 000 is smaller than that between 10 000 and LI{M + 1). A similar inequality holds, for a smaller number of sub-reaches, i.e.:

[L/(M - 1)]/10 000 > 10 000/(L/M)

Putting these inequalities together, M has to verify the double inequality:

^ ¥ ^ < ï ^ 5 o - ^ M ( M + 1 )

For each sub-reach the parameters 5 and T are to be calculated using equations (10) and (11) for the linear model, and using equations (12) and (13) for the quadratic one. The hydrograph at the upstream end of the sub-reach is then transformed employing equations (4) to (6) for the linear model and equation (8) for the quadratic model. The procedure is then repeated for the next sub-reach. There may be a small numerical diffusion effect because of the broken line description of the inflow hydrograph used with these models. The validity of both models is limited to the domain described by the log-normal distributions used for the generation of hypothetical channels and flood impulses. If the flow rates were altogether outside the foregoing ranges, the results might depart from the Saint-Venant solution for flood propagation.

As a form of validation of the two models, a new inflow hydrograph was used as input to a new channel reach having the following characteristics: L = 43 000 m, n = 0.05 m4'3 s, s - 0.0005, Q0 = 0.2 m2 s"1. The results derived from a four-fold (four sub-reaches of 10, 750 m) application of the two models appear in Figs 9(a) and 9(b). Only the first hundred hours of the hydrographs have been drawn. The linear and quadratic models yielded F criteria values of 90.50% and 98.70%, respectively. It is interesting to see that the lag time between the peaks seems to depend on the amplitude of the preceding peak flow. This result has been effected with the quadratic lag-and-route model without introducing variation in the parameters.

CONCLUSION

A new version of the standard lag-and-route method has been proposed, the routing reservoir storage characteristic being quadratic instead of linear. A seemingly efficient solution has been described to the problem of parameter explanation in terms of the physical parameters of the channel reach under study. The relative root mean square error of the quadratic method is 15% of the one achieved when using the zero-level model which identifies the modelled outflow with the inflow. The


0 12 24 36 48 60 72 84 98 108

TIME (Hours)

Q Upstream -—--Q Downstream Saint-Venant ~ « ~ Q Downstream Lag-and-Route

Fig. 9 Flood routing (a) with the linear model, and (b) with the quadratic model for the time period between 0 and 108 h (validation).

difficulties in the search for suitable relationships between model parameters and channel characteristics underscored the subtlety of the interplay of the two model operations the delaying (lag) and the peak smoothing (reservoir) components.


Although the quadratic model clearly outperforms the linear model, the latter remains appealing due to its simplicity and the possibility to highlight, in a very simple way, the interacting roles of the two components of the lag-and-route method. Moreover, it is not certain that the data dealt with in practical applications justify the use of the quadratic model. The complex role of the channel slope has been emphasised, and the implication for flood forecasting being hindered due to the possible fading away of the lag component has been clearly demonstrated. Applications of the methods using actual inflow-outflow data have already been carried out and will be reported later.

Acknowledgements Comments by R. J. Moore, UK Institute of Hydrology, are gratefully acknowledged.

REFERENCES

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Baptista, M. & Michel, C. (1990) Influence des caractéristiques des biefs sur la propagation des pointes de crue (Influence of channel characteristics on peak flow propagation). La Houille Blanche 2, 141-148.

Becker, A. & Kundzewicz, Z. W. (1987) Non-linear flood routing with multilinear models. Wat. Resour. Res. 23(6), 1043-1048.

Boyd, M., J., David, H., Pilgrim & Cordery, I. (1979) A storage routing model based on catchment geomorphology. J. Hydrol. 42, 209-230.

Bravo, R., Dow, D. A. & Rogers, J. R. (1994) Parameter determination for the Muskingum-Cunge flood routing method. Wat. Resour. Bull. AWRA 30(5), 891-899.

Cunge, J. A. (1969) On the subject of a flood propagation computation method (Muskingum method). J. Hydraul. Res. 7(2), 205-230.

Diskin, M. H. & Ding, Y. (1994) Channel routing independent of length subdivision. Wat. Resour. Res. 30(5), 1529-1534.

Dooge, J. C. I., Strupczewski, W. G., & Napiorkowski, J. J. (1982) Hydrodynamic derivation of storage parameters of the Muskingum model. / . Hydrol. 54, 371-387.

Gill, M. A. (1992) Numerical solution of Muskingum equation. / . Hydraul. Engng Div. ASCE 118(5), 804-809. Gill, M. A. (1994) Discussion of "Step-function response of Muskingum reach" by J. C. I. Dooge, M. Perumal &

Q. J. Wang. / . Irrig. Drain. Engng 120(3), 695-697. Guang-Te, W. & Singh, V. P. (1992) Muskingum method with variable parameters for flood routing in channels. J.

'Hydrol. 134, 57-~76. Hayami, S. (1951) On the propagation of flood waves. Bull. 1, Disaster Prev. Res. Inst., Kyoto Univ., Japan, 1-16. Hjelmfelt, A. T. (1985) Negative outflows from Muskingum flood routing. /. Hydraul. Engng Div. ASCE 111(6), 1010-

1014. Kohler, M. A. (1958) Mechanical analogs aid graphical flood routing. J. Hydraul. Div. ASCE 84 (HY2), paper 1585. Laurenson, E. M. & Mein, R. G. (1995) Hydrograph synthesis by runoff routing. In: Computer Models of Watershed

Hydrology, ed. by V. P. Singh, 151-164. Water Resources Publications, Colorado, USA. Lettenmaier, D. P. & Wood, E. F. (1993) Hydrological forecasting. Chapter 26 in: Handbook of Hydrology, ed. by

D. R. Maidment. McGraw-Hill, New York, USA. Malone, T. A. & Cordery, I. (1989) An assessment of network models in flood forecasting. In: New Directions of

Surface Water Modelling (Proc. Baltimore Symp., May 1989), ed. by M. L. Kavvas, 115-124. IAHS Publ. no. 181.

Mein, R. G., Laurenson, E. M. & McMahon, T. A. (1974) Simple non-linear model for flood estimation. Proc. Am. Soc. Civ. Engng 100(HY11), 1507-1518.

Meyer, O. H. (1941) Simplified flood routing. /. Civ. Engng Div. ASCE 11(5), 306-307. Moore, R. J. (1985) The probability-distributed principle and runoff production at point and basing scales. Hydrol. Sci.

J. 30(2), 273-287. Morel-Seytoux, H. J. (1993) Les modèles analytiques de surface. HYDROWAR Report # 93.9, Hydrology Days

Publications, 57Selby Lane, Atherton, California 94027, USA. Nash, J. E. (1959) A note on the Muskingum flood routing method. J. Geophys. Res. 64, 1053-1056. Perumal, M. (1992a) The cause of negative initial outflow with the Muskingum method. Hydrol. Sci. J. 37(4), 391-402.

Flood routing in a wide channel with a quadratic lag-and~route method 185

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Wat. Resour. Res. 12, 487-496. Ponce, V. M. & Theurer, F. D. (1982) Accuracy criteria in diffusion routing. J. Hydraul. Div. ASCE 108(HY6), 747-

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ASCE 105(12), 1521-1535.


APPENDIX 1

Routing through a linear reservoir

Adopting the same notation as in the foregoing text, and recalling the relationship R = VIS, during a time interval At let the upstream flow rate evolve as a straight line from gi to Q2. Starting with a downstream flow rate equal to i?, then the flow rate at the downstream end of the reach at the end of the time interval, R2, is found by solving the linear differential equation:

dV/dt = Q(t) - R{f)

given that:

Off) = & + (Ô2 - Qùt'àî for O < t < At

and:

R(t) = V(t)/S

Eliminating Vyields:

S dR/dt + R = Ô, + (Q2- Ql)t/At

The solution is the sum of the general solution of the linear equation with a zero right-hand side plus a particular solution of the complete equation. After manipulation, R(t) is given by the expression:

m = M - a + (a - Qx)si t] expH/s) + (a - aw ? + a - (a - a)» t R2 is then obtained by putting t = At. Denoting exp(-Ar/S) by © for the sake of notation simplicity gives:

R2 = ©#, + a t ( l ~ ©)S/Af - to] + Q2[l - (1 - (ù)S/At]


APPENDIX 2

Routing through a quadratic reservoir

In order to obtain an analytical solution, it is necessary to consider a time interval (80 small enough so as the incoming discharge can be considered as approximately constant (Q*) within St. To this end, the initial time interval, At, is split into N sub-intervals. For each of these smaller time intervals, the differential equation is:

, 2 dV

d? Q-

which by separating the variables, gives:

dV

Q*-V

= df (14)

Two cases have to be considered. If Q* = 0 then:

dV _ _d£

~ V2 ~ S1

which gives:

\v(t+8t)]-i=[v{t)r+8t

s2

and hence:

R(t + ât) = R(t) -0.5 8t

+ — S

On the other hand, when Q* > 0 and denoting V/[SQ *0 5 ) by x, equation (14) yields:

àx Q*03

which gives, for 0 < x < 1:

tantr'Oc) = tanh^1(x.) + : Q-

S -(t-tx)

Transforming both members by hyperbolic tangent, the model equation becomes:

x, +tanh S

-At

x = -

1 + x, tanh Q>

-At

It can be verified that this expression is also valid when x > 1.


In terms of V:

S

V(t +g*05tanh

/-Q*0.5g^

1 + ( vit) ^

SQ*°\ tanh /-Q*0.5g^

and using R = [ — j on the right-hand side:

R(t + 8t) =

R(t)05 + Q *°5 tanh 5 S A Q*u>St

1+ - p - | tanh rQ*°s5t-\

(15)

The approximation Q(t) » Q* is valid if ôf is small enough. A numerical check showed that the error in R2 is less than (g, + Rt + g2)/1000 provided that of < S/(ll<2*05), where St = MN mi N* U(AtQ*03 IS). Under this condition, tanh(x) « x and therefore equation (15) can be simplified to become:

.„ (9*S/12

i?(0 +J

(16) R(t + 6t) = S

1-Rjty'èt

s Note that this latter expression is valid even for Q* = 0. This was not the case for equation (15). The computation involved using equation (16) has to be reiterated N

Q |

+ ^ At

At

-5>t

Fig. 10 Breakdown into time sub-steps for application of the quadratic model.


times to produce the required result at the end of the time step At. It seems that equation (16) can be generalized for any parameter a with the following expression:

R(t + 8t) = R(t)"la+(a-\)

Q *2(l-l/cx) 8t

l + (a-l) R(ty-Ua?>t

s

a/(a-l)

(17)

Since At has been broken down into N sub intervals, equation (16) has to be repeated N times as shown in Fig (10). In this case, for the ith application, it can be shown that:

Q* = [i _ (/ _ O.S)/N\Q1 + [(/ - 0.5)/N]Q2

Received 11 March 1996; accepted 2 August 1996

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