Scientific Procedures Applied to the Planning, Design and Management of Water Resources Svstems (Proceedings of the Hamburg Symposium, August 1983). IAHS Publ. no. 147.

Hydrodynamic determination of parameters of linear flood routing models

ZBIGNIEW KUNDZEWICZ Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland

ABSTRACT In this paper different methods of hydrodynamic determination of parameters of approximate hydraulic and hydrological flood routing models are analysed and com­pared with respect to severity of assumptions and the final equations for parameters. It is shown that the values of parameters obtained by several methods of physical interpretation are close to each other. Thus the problem of the choice of method for hydrodynamic determination of parameters is not critical. The physical interpretation of parameters of approximate flood routing models is cheaper than other methods of identification. Thus, in cases where accuracy requirements are not stringent the simplifications discussed in the paper can be considered useful and practical.

Détermination hydrodynamique des paramètres de modèles linéaires de propagation des variations de l'onde de crue le long d'une rivière RESUME Dans cette communication on analyse différentes méthodes de détermination hydrodynamique du comportement hydraulique approché et des modèles hydrologiques de propagation de l'onde de crue et on les compare en examinant particulièrement la sévérité des hypothèses et les équations finales des paramètres. On montre que les valeurs des paramètres obtenues par plusieurs méthodes d'interprétation physique sont proche les unes des autres. Aussi le problème du choix de la méthode pour la détermination hydrodynamique des paramètres n'est pas un problème critique. L'interprétation physique des paramètres de modèles approximatifs de propagation de l'onde de crue est moins coûteuse que les autres méthodes d'identification. Aussi pour le cas où les besoins d' exactitude ne seraient pas très pressants les simplifi­cations étudiées dans cette communication peuvent être considérées comme utiles et pratiques.

INTRODUCTION

The hydrodynamic equations of open channel flow formulated by de Saint Venant in 1871 have been widely accepted as a faithful repre­sentation of the process. However, this model puts severe demands on quantity and quality of data as well as on computational requirements. That is why the applicability of the complete nonlinear equations is restricted and numerous approximate flood routing models are being

149

150 Zbigniew Kundzewicz

developed. The analysis performed in this paper pertains to the important

class of linear stationary models, whose parameters depend neither on process variables (postulate of linearity) nor on time (postulate of stationarity).

One of the crucial steps in the process of modelling is the determination of model parame-ters. In the case of hydrodynamic models the parameters have a physical sense and can be either measured or assessed in the field. The parameters of approximate conceptual or black box system models are usually identified from inflow and outflow data. There is a large number of techniques for identification of model parameters at one's disposal. However, due to the ill-posedness of the inverse problem and badly shaped topo­graphy of the criterion function of the optimization task, identifi­cation of the model parameters is not simple.

It may also happen in practice, that no historical inflow-outflow records are available, as in the case of ungauged rivers or in design projects of flow regulation. Then the model parameters must be determined from morphological channel parameters (length of the reach, geometry of cross sections, bed slope, roughness co­efficient) and from aggregated characteristics of the process of flow (reference depth, reference flow rate).

Therefore it is very useful to establish the relationships between the parameters of linear flood routing models and the hydrodynamic parameters with full physical significance.

In the hydrology literature of the last decades several methods were developed for determination of conceptual parameters based on their physical sense. These approaches can be grouped in the following classes of methods:

(a) direct interpretation; (b) matching the model in question with pattern hydrodynamic

models, using impulse responses (matching by moments or cumulants) or using finite difference schemes;

(c) correlation and regression techniques (of particular import­ance in rainfall-runoff modelling).

DIRECT PHYSICAL INTERPRETATION

Hydrodynamic linear models

Linear dynamic wave There is a number of linear channel flow models of hydrodynamic origin, whose parameters have a direct physical interpretation. The principal linear hydrodynamic model is the linearized version of the dynamic wave model (Sant Venant equations with negligible lateral inflow). It can be written in the form of the following linear second order partial differential equation of hyperbolic type:

32Q ,32Q 32Q 90 3Q a — - + b — — + c — - + d-^ + e-^ = 0 (1) 3x2 3x3t 3t2 3x 3t

All the coefficients used in this equation have a direct physical

Hydrodynamic determination of parameters 151

relevance. If linearization was performed for small disturbances around a steady uniform flow in a wide uniform channel with rectan­gular cross section and Chezy friction law, the values of these coefficients can be obtained from the following equations (after Dooge & Harley, 1967a):

a = v2 - gy (2) o o

b = 2v (3) o

c = 1 (4)

d = 3gl (5)

e = 2gl/v (6) o

where v0 is reference flow velocity, y0 is reference depth, and I is bottom slope.

Linear diffusion analogy The most useful simplified flood routing model originating from hydrodynamics seems to be the diffu­sion analogy method. Unlike the hyperbolic linear dynamic wave model, the second order partial differential equations of diffusion analogy is of the parabolic type.

(7)

The above equation can be developed from the dynamic wave model in at least three ways:

(a) By neglecting acceleration terms in the nonlinear model of the dynamic wave and linearizing (Hayami, 1951).

(b) By neglecting acceleration terms in the linear model of dynamic wave (Lighthill & Whitham, 1955) .

(c) By introducing kinematic wave approximations of derivatives 3 Q/3x3t and 3 Q/3t in the linearized dynamic wave model (Dooge & Harley, 1967b).

The value of the celerity coefficient c, for Chezy friction law and a wide rectangular channel reads in each of the above develop­ments :

c = 1.5 v (8) o

Each of the methods mentioned yields a different equation for diffusivity coefficient, D, in terms of system characteristics, as illustrated in Table 1.

Conceptual models

The method of direct physical interpretation of conceptual models is developed by introducing simplifying assumptions to the hydrodynamic laws and producing structures similar to the conceptual model in question.

152 Zbigniew Kundzewicz

TABLE 1 Physical interpretation of parameters of linear flood routing models

FIYDRODYNAMIC DIFFUSION ANALOGY Parameter

c 1.5 v_

Methods of direct interpretation

(i) (H) (Hi)

Moment matching

D

v y o• o 21

v y o o 21

1 v y

o o 21

like (Hi)

KALININ-MILJUKOV METHOD

Direct interpretation Moment matching

K 1 yo 9 v I

o

2 M

4

9

1 F1 \

6 LI

v I o

(4 - FQ)yc

MUSKINGUM METHOD

K 1.5 v

o 3IL

+ F 2 U

1_ _ !_ I 9_\_o_

2 3 \ 4 J IL i_ o.3 -2-2 IL

4 2 * 1 Fz

9 o,

* Matching difference schemes (Cunge, 1969; Koussis, 1978).

t Moment matching (Dooge, 1973); Lumping by difference of variations for Chezy friction (Dooge et a l . , 1982).

§ Lumping by difference of variations for Manning friction (Dooge et a l . , 1982).

Kalinin-Miljukov method The idea of the Kalinin-Miljukov method (mathematically equivalent to the cascade of linear reservoirs) is to subdivide the channel reach modelled into a number of so called characteristic reaches, for which linear relations between storages and outflows are valid. Under the assumptions of linear longitudinal profile of free water surface and mutually unique relationships between outflow from the reach and stage in the half-length of the reach, the length of the elementary channel reach reads:

1 = v By o Jo

dQ 1/2 (9)

where B is water surface width,

Hgdrodynamic determination of parameters 153

The physical interpretation of model parameters for uniform reach with rectangular cross section and subject to Chezy friction law is given in Table 1.

Muskingum model Strupczewski & Kundzewicz (1980a) developed a novel derivation of the Muskingum method from the simplified hydro-dynamic equations of open channel flow.

It can be shown, that neglecting the acceleration terms of the dynamic equation of flow (for assessment of magnitude of terms, cf. Henderson, 1966; Kuchment, 1972; and Natural Environment Research Council, 1975), one gets

l

Q = Q (l - T I 1 ) (10> su \ I dx /

where

Q = i B I ° - 5 y 5 / 3 (ID su n

is the steady and uniform reference flow and n is Manning's co­efficient.

Under the assumption of linear longitudinal profile of the free water surface, the equation of storage in the reach reads (after Strupczewski & Kundzewicz, 1980a)

2 a b

where Qj is inflow to the reach; Q2 is outflow from the reach;

a = -BI (13) n

b = l - ± | * ( 1 4 )

I dx

The value of b can be determined in terms of Qj, Q2 , a, L, I as a solution of the following equation:

1 (J>.6 „0.6\ 0.3/ \ ,.,_ - ^ - Q2 -Q1 = b (l - b ) (15) a LI

Another storage equation of the type similar to (12) was intro­duced by Tsunematsu (1978), who assumed a parabolic free surface profile and used the perturbation method:

„ 1 L n 0" 6 (B + 2 yQ) /„0.6 0.6\ (16) S _ 2 0.3 lQl + Q2 I

I

Linearized versions of these equations produce the physical inter­pretation of the Muskingum model parameters (cf. Strupczewski & Kundzewicz, 1980a).

154 Zbigniew Kundzewicz

Conceptual diffusion analogy Becker & Sosnowski (1969) assumed that the rating curve for unsteady flow can be approximated by a linearized curve for steady and uniform flow:

Q s u = V (17)

plus a correction factor that for a uniform rectangular channel reads :

AQ = - k Q B ^ (18) 2

that is

Q = Q + AQ = k v - k B|£ (19) su 1 2 dx

Differentiating after time and making use of the continuity equation one gets the following diffusion analogy model:

| a = - c|a + D ^ (20) dt dx

where c = k^/B and D = k2-However, the approximation (17)-(19) is not supported by hydro-

dynamic laws. Making use of the Manning friction law and Jones formula for a looped rating curve one gets the following relation:

Q = Q + AQ * n I0"5 B y 5 / 3 - 0.5 n l"0"5 B y 5 / 3 |* (21) su dx

Since this condition differs from (19) , the diffusion analogy model introduced by Becker & Sosnowski (1969) can be regarded as a concep­tual one.

MATCHING IMPULSE RESPONSES

The relationships between parameters of physical (hydrodynamic) and conceptual models can be also obtained by analysis of similarity of impulse responses or transfer functions of linear models. The method of matching impulse responses by moments (or cumulants) was introduced by Dooge (1973) and supported by Strupczewski & Kundzewicz (1979). It is easy to prove that the moment matching method is mathematically equivalent to the method of matching power series developments of transfer functions (Laplace transforms of impulse responses) introduced by Marr (1977).

By formulating equations for the first two initial moments of impulse responses of the linear dynamic wave model and the two-parameter conceptual model in question, the relationships between the parameters of both models can be obtained. Solution of these equations yields an interpretation of conceptual parameters in terms of hydrodynamic parameters. The theoretical background of this pro­cedure and assessment of the degree of similarity of performance of both models was analysed by Kundzewicz (1978) and Strupczewski & Kundzewicz (1979).

In these references it was shown, that matching the first N

Hydrodynamic determination of parameters 155

moments of impulse responses of two linear models in question yields: (a) equivalence of all moments of outflows from both models if

the inflow signal common to both models is a time polynomial of the N-th order; and

(b) equivalence of the first N moments of responses of both models to arbitrary inflows, under the assumption that both models were initially relaxed.

Instead of the method of matching by moments the theoretically equivalent method of matching by cumulants can be used. The latter method is advantageous due to simplicity of calculations. The equations for physical interpretation of conceptual parameters obtained with the help of moment (cumulant) matching of impulse responses of the linearized dynamic wave model and several conceptual models are given in Table 1.

MATCHING DIFFERENCE SCHEMES

The method for finding physical interpretation of conceptual para­meters by matching difference schemes of hydrodynamic and conceptual models was introduced by Cunge (1969).

The linear kinematic wave model written in an implicit finite difference scheme reads:

i(x «T -c

Y ( Q n + ; -+ \ J + 1

-*>)

n + 1 - Q.

J

+

) •

( • •

At -*)Kl-

• ' ) ( « ; • . -

- Q° )

. ~ (22) Ax

The above equation with K = Ax/c and Y = 0,5 is identical to the classical version of the Muskingum model:

Qn:; = C 0 n + C O

n + 1 + C Q" (23)

J+1 1 J 2 J 3 J+1 where

C = K X + A t / 2 (24) S K(l - X) + At/2 K }

c - ^ Z . 2 - K X _ (25) C2 K(l - X) + At/2 (^>

c = K (1 - X) - At/2 3 K (1 - X) + At/2 *• '

that is

C1 + C 2 + C 3 = 1 (27)

The difference equation (22) approximates the linear kinematic wave equation and the linear diffusion equation. The analysis of degrees of approximation was given by Cunge (1969) and Miller &

156 Zbigniew Kundzewicz

Cunge (1975). The Muskingum model in the version (23) is the best approximation

(of second order) of the equation of diffusion analogy, if the values of parameters K and X are taken according to the following equations :

K = Ax/c (28)

X = 0.5[1 - Q /(BclAx)] 0 é X < 0.5 (29) o

A different approach was introduced by Koussis (1978) who formulated the following Muskingum model equations with continuous time for a short reach:

| | = Q(x,t) - Q(x + Ax,t) (30)

S = K [X Q(x,t) + (1 - X) Q(x + Ax,t)] (31)

If the function Q(x + Ax,t) in these equations is developed into a Taylor series with respect of x and all the terms of the order higher than two are neglected, then the following diffusion analogy equation can be obtained:

_9Q Ax 3£ 3t K 3?

c(l - X)Ax - ^ ( A x ) 2 )2Q — (32) N..2

By comparing the coefficients of equation (32) with physically significant parameters of the diffusion analogy, the formulae (28)~<29) for hydrodynamic interpretation of conceptual Muskingum parameters can be obtained.

Yet another approach was considered in an earlier work of Koussis (1976). This approach, however, yielded interpretation of conceptual Muskingum parameters in terms of numerical parameter At as well. The time interval At subject to the choice by the modeller did not occur in the original mathematical model.

Dooge et al. (1982) suggested a more versatile scheme of physi­cally interpreting the Muskingum parameters. The dynamic wave model was linearized around a reference trajectory and the following lumped formula for variations in flow area was accepted:

3 Ô A2 - Ô A1 JL (6A) = _^L^ i (33) dx L

The general formula for parameter X, as developed by Dooge et al. (1982) reads:

X = 0.5 +

where

^ \ T T ^ [1 - (m - I ) 2 F2] (34) 9S-/3A A L o f o

3S^ 33^ A m = —1 / - A _° (35) m 3A 7 3Q Q

o

Hydrodynamic determination of parameters 157

and F 0 is the Froude number. In the case of a rectangular channel equation (34) takes the form:

X = 0 - 5 - | ï ï (X--ïFo) (36)

X = 0 . 5 - 0 . 3 ^ ( l - f F^) (37)

for the Chezy friction law and Manning friction law, respectively.

COMPARISON OF METHODS OF INTERPRETATION

The present author (Kundzewicz, 1982) compared the three types of methods of physical interpretation as discussed above with respect to simplifying assumptions accepted and with respect to resulting equations for conceptual parameters in terms of physical system characteristics. Brief corollaries from this comparison are given in Table 2.

It should be generally mentioned that since the conceptual models analysed are linear, one has also to consider the limitations which the pattern linear hydrodynamic models are subjected to. In the method of matching impulse responses or matching finite difference schemes the assumption of the ideal pattern hydrodynamic model (linear dynamic wave or linear diffusion analogy) is necessary. In the real case the hydrodynamic model can be a faithful description of the actual process if small deviations from the steady state are of concern.

The material given above enables a rough approximation of values of conceptual model parameters to be obtained. The accuracy of simulation with the help of conceptual parameters determined from physical interpretation is sufficient for many practical purposes.

Exact optimizational identification of conceptual parameters which requires data on inflow to and outflow from the reach to be known, can produce better accuracy of parameter determination but at a significantly higher cost.

It has been shown, that different methods of physical interpre­tation starting from different assumptions yield essentially similar results.

Analysis in the present paper is in fact restricted to the class of two parameter flood routing models. The problem of determination of three parameter conceptual flood routing models was approached by Strupczewski & Kundzewicz (1980b, 1981).

One should bear in mind that in the discussed methods of physical interpretation of parameters the system was assumed to be highly idealized (rectangular geometry of cross sections, uniformity. constant roughness coefficient and constant bottom slope). This is seldom the case in nature, but even a river of complex topography can be roughly described by effective constant parameters of the modelled reach representing mean geometry and roughness.

Another fundamental assumption of the linear theory of hydrologi-cal systems is that signals occur as small disturbances from a steady and uniform reference state. This assumption is never actually fulfilled. That is why a mean flow from the particular

158 Zbigniew Kundzewicz

flood wave used to be taken as reference flow in the equations for physical interpretation of model parameters. Although such a pro­cedure is biased with a systematic error, the results produced in a very simple way can be acceptable if the accuracy demands are not too severe.

TABLE 2 Advantages and disadvantages of different methods of physical interpretation of parameters of linear flood routing models

Advantages Disadvantages

DIRECT INTERPRETATION

The degree of justification of neglecting or simplification of particular terms of hydrodynamic models can be assessed

Assumption of linear water surface along the reach (linear reservoir, Muskingum) restricts the applicability of the method to short reaches. For longer reaches multiple models in series (cascades) necessary

MATCHING IMPULSE RESPONSES BY MOMENTS

Criterion of similarity of per­formance of the models (equivalence of first Initial moments of Impulse responses) enable assessment of properties of outputs from the models to be performed. Relations hold for arbitrary lengths of uniform reaches

Assumption of pattern linear hydrodynamic model necessary

MATCHING FINITE DIFFERENCES

Criterion of similarity-approximation properties of the pattern hydrodynamic model of the known order

Assumption of pattern linear hydrodynamic model (usually in a simplified form, e.g. diffusion analogy) necessary

Matching ordinary differential equations (conceptual model) and partial differential equations requires lumping the latter (hydrodynamic pattern model) to one molecule

Assumption of small reach lengths can be necessary

The equations for physical interpretation can contain purely numerical parameters (space and/or time increments) of the difference scheme

Hydro dynamic determination of parameters 159

ACKNOWLEDGEMENTS The research reported in this paper was performed in part within the framework of Governmental Research Problem PR-7 in the Institute of Geophysics, Polish Academy of Sciences, under scientific supervision of Professor Witold Strupczewski and in part during the author's fellowship of the Alexander von Humboldt Foundation at the Institute Wasserbau III, University of Karlsruhe, Federal Republic of Germany under scientific supervision of Professor Erich Plate.

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