Online publication date: 29 December 2009 PLEASE SCROLL DOWN FOR ARTICLE

Transcription

1 This article was downloaded by: On: 16 August 2010 Access details: Access Details: Free Access Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Hydrological Sciences Journal Publication details, including instructions for authors and subscription information: What is the distributed delayed Muskingum model? / Qu'est-ce que c'est le modèle de Muskingum distribué et avec retard? WITOLD G. STRUPCZEWSKI a ; JAROSLAW J. NAPIÓRKOWSKI a a Water Resources Department, Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland Online publication date: 29 December 2009 To cite this Article STRUPCZEWSKI, WITOLD G. and NAPIÓRKOWSKI, JAROSLAW J.(1990) 'What is the distributed delayed Muskingum model? / Qu'est-ce que c'est le modèle de Muskingum distribué et avec retard?', Hydrological Sciences Journal, 35: 1, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Ilydrological Sciences - Journal - des Sciences Hydrologiques, 35,1, 2/1990 What is the distributed delayed Muskingum model? WITOLD G. STRUPCZEWSKI & JAROSLAW J. NAPIÔRKOWSKI Water Resources Department, Institute of Geophysics, Polish Academy of Sciences, Ksiecia Janusza 64, Warsaw, Poland Abstract A gap between hydrodynamic modelling and conceptual modelling of flood routing has been filled by the introduction of conceptual elements. The conceptual model gives results equivalent to those of the complete linearized Saint Venant equations for Froude number equal to one and approximately so in that vicinity. It has been applied to the case of a uniform open channel with arbitrary cross section and friction law. Pure lag has been introduced to the Muskingum model and then such models with physically based parameters have been coupled in series forming a multiply delayed Muskingum model. The asymptotic case when, for a finite river reach, the number of submodels tends to infinity, gives the distributed delayed Muskingum model. Further, this model is a particular solution of the complete linearized Saint Venant equations known as the rapid flow model. The model not only originates from conceptual elements but its impulse response has a clear conceptual interpretation. Qu'est-ce que c'est le modèle de Muskingum distribué et avec retard? Résumé Une lacune entre les modèles hydrodynamique et conceptuel de propagation des crues est comblée par l'introduction des éléments conceptuels. Le modèle conceptuel donne des résultats équivalents à ceux du modèle de Saint Venant linéarisé pour un nombre de Froude égale à un. Cette proposition est appliquée pour un canal ouvert avec sections transversales quelconques et aussi avec une loi du frottement quelconque. Le retard du parcours est introduit au modèle de Muskingum. De tels modèles, avec paramètres déterminés sur des bases physiques, sont couplés et ils donnent finalement un modèle de Muskingum multiple avec retard. Pour le cas asymptotique, c'est à dire quand pour le bief du canal de longeur finie le nombre des modèlescomposants s'augmente à l'infini, nous obtenons le modèle de Muskingum distribué avec retard. D'autre part ce modèle est une solution particulière des équations de Saint Venant complètes et il est connu en tant que "Modèle d'écoulement rapide". Le modèle proposé est d'origine conceptuelle et sa réponse impulsionelle a aussi une interprétation conceptuelle claire. Open for discussion until 1 August

3 Witold G. Strupczewski&Jaroslaw J. Napiôrkowski 66 INTRODUCTION In recent years the relationships between various types of models used in hydrology have been thoroughly studied. Within the variety of approximate flood routing models there is the important class of linear stationary models which covers hydrodynamic, lumped conceptual and black box models. While hydrodynamic models are of a distributed type, i.e. they are described by means of partial differential equations, conceptual models are usually represented by a set of ordinary differential equations. Linear conceptual models are built of simple conceptual elements such as a linear reservoir, or a Muskingum model, or a linear channel which provides pure translation. In fact, the linear channel is both a hydrodynamic and a conceptual model. It is the simplest hydrodynamic model and is known as the linear kinematic wave solution. The natural step in linking both structures requires a replacement of derivatives with respect to distance by the relevant finite differences. Due to the necessity of broad simplifications such an approach reduces the simplicity of conceptual structures rather than consolidates the confidence in the physical background of such structures. Therefore in order to reconcile both approaches it seems reasonable to look for complex conceptual structures which may be closer to hydrodynamics than their particular elements. This paper considers a model built from a series of Muskingum elements with pure lag models. As for all other linear models the complete linearized Saint Venant equations will be used as the standard thus enabling a physical evaluation of the model parameters. The conceptual Muskingum model, which seemed to be purely empirical, has been shown to be linked with models based on convective diffusion equations. Cunge (1969) found a similarity of the difference schemes for both models and derived relationships between their parameters. A more direct connection between the model and physics was shown by Strupczewski & Kundzewicz (1980) and Napiôrkowski et al. (1981). A nonlinear convective diffusion model has been lumped under the assumption of linear changes of water level along the river reach and linearized around the steady state. Dooge et al. (1982), using the method of inverse order, obtained the Muskingum model as the first term of a Volterra series and derived results applicable to any shape of cross section and to any type of friction law. The Muskingum model fails if the length of the flood wave is small compared to the length of the channel reach. In such cases a series of Muskingum models (a multiple Muskingum model) can be applied as was first proposed by Cunge (1969). Strupczewski & Napiôrkowski (1986) showed that, for a finite river reach, the infinite multiple Muskingum model with physically based parameters is equivalent to the kinematic diffusion model. Therefore the best analogue of the Muskingum lumped model in the domain of distributed models is the kinematic diffusion model (not the convective diffusion model) and, as a conceptual model, it can be called (Strupczewski et al., 1988) the distributed Muskingum model. However, it fits satisfactorily the solution of the linearized St Venant equations only for small Froude numbers and slow rising waves. The aim of this paper is to introduce a pure

4 67 What is the distributed delayed Muskingum model? lag to the Muskingum model and to examine the properties of infinite series of such models. DELAYED SINGLE AND MULTIPLE MUSKINGUM MODELS One of the most popular approaches to the mathematical description of open channel flow is the Muskingum method, which was first proposed by McCarthy (1939). In this method, an outflow from the river reach responds immediately for any change of an inflow to the reach. Connecting a linear channel model (a pure translatory system which reflects the time of propagation of a perturbation along the positive characteristic) with the Muskingum model (responsible for the attenuation of the system response) gives the delayed Muskingum model (DM) which is described by the continuity and dynamic equations: S(0 = Q x (t - T) - Q 2 (t) (1) S(t) = K[a Q x {t - T) + (1 - a) Q 2 (t)] where Q x is the inflow to the reach, Q 2 is the outflow from the reach, 5 is the storage of the DM and K, a and T are model parameters. The initial condition is given by S(t) = S"(t) for t e [-T, 0]. The inflow-outflow transfer function of the DM model in Laplace transform domain (Doetsch, 1961) and the impulse response in the time domain are given respectively by: H DM (s) a 1 - a 1 + K(l - a)s exp(-ts) (3) (2) h DM (t) = - hit - T) + 1 K(l - af exp f t-r 1 (4) where 6(f) denotes the Dirac delta function and 1(f) the unit step function. For longer river reaches one can use n identical DM models connected in series, i.e. the multiple delayed Muskingum model (MDM model), with the transfer function: HMDM (S) = [Hj^s)]" = - a a 1 - a 1 + X(l - a)s n exp(- nrs) (5)

5 Witold G. Strupczewski&Jaroslaw J. Napiôrkowsld 68 and the impulse response: hmdm {t) = [_ _ n 6(r - /IT) Ll - a J n-lft _ -rv'-l f t _ nt "> * ' l(f - nr) K(l- a) + I?! If J exp (1 - a) n+i KXi - 1)! As with all types of models, it is necessary to find the best values of model parameters. Strupczewski & Kundzewicz (1980) derived the relationship between the DM model parameters and hydraulic characteristics of a rectangular channel with Chezy friction by matching the first three cumulants of the DM impulse response with those of the Linear Channel Response (LCR), i.e. the solution of the linearized St Venant equations for a semi-infinite channel. In the present paper, the physical estimates of the submodels parameters (i.e. a, K and T) obtained by the same technique for any compact shape of cross section and any friction law are used. They are valid for disturbances around a steady state. CUMULANTS OF IMPULSE RESPONSES The use of cumulants to study the properties of a linear model impulse response was introduced by Dooge & Harley (1967). The cumulants of impulse response are generated by the logarithm of the Laplace transform of the impulse response function: klh(t)] = (-iy {In [#<*)]},. o (? ) d/ where r is the order of cumulant. Cumulants of the multiply delayed Muskingum model Cumulants of the impulse response of the MDM model can be obtained from equation (5) by evaluating the rth derivative at s = 0. k^ M = n (K + T) (8a) kmdm = (r _ j), r (1 _ ay _ (_ fl) r] Kr r > 1 (gb)

6 69 What is the distributed delayed Muskingum model? Cumulants of the linear channel response The linearized St Venant equation for one-dimensional unsteady flow in a uniform channel is hyperbolic in form. Accordingly there are two real characteristics along which the discontinuities in the derivatives of the solution will propagate. For a Froude number less than one the secondary characteristic direction will be in an upsteam direction. Hence, to solve the linear two-point boundary problem, both upstream and downstream boundary condition (with the appropriate upstream and downstream transfer functions) have to be taken into account (Dooge & Napiôrkowski, 1986). The occurrence of an upstream movement (the effect of a downstream boundary condition) can be neglected only by an assumption of a semi-infinite channel. The cumulants of the Linear Channel Response (LCR) for a semi-infinite channel, any shape of cross-section and any friction law, have a general form (Dooge et al, 1987): klcr = QjCjnf) Zr /D r - 1 (9) where z = xlc k (10) is the passage time of a kinematic wave through the channel reach; D = Stfly (11) is the dimensionless length of the channel reach; x is the distance from the upstream boundary; c k is the kinematic wave speed; m is the ratio of the kinematic wave speed to the average velocity of flow (for wide rectangular channel m = 3/2 for Chezy friction and m = 5/3 for Manning friction); F is the Froude number; S^ is the bottom slope; and y is the hydraulic mean depth. Values of F, m, c k and y~ are taken at the reference conditions used for linearization. The i r functions for the first four cumulants are: *i = 1 (12) tfi 2 = [1 - (m - l) 2 F 2 ]/m (13) [1 + (m - l)f 2 }lm (14) (J> 4 = 15 <l> 2 [1-2(Q.lm 2 - m + \)F 2 + (m - if^m 2 (15) Denoting:

7 Wiiold G. Strupczewski & Jaroslaw J. Napiôrkowski 70 <J>3 z z a = = [1 H + (JW (m - \) If? 2 F 2 } J n (16) * 2 D ml) (m - 1) 2 F 2 X = D o = -- 2 o.5/n 0.5/n ^ -^ D (17) *j [1 + (m - 1)F 2 ] 2 A = - ax (18) one can express the first four cumulants of the LCR in convenient form in terms of a, X, and A as: klcr = A + ax (19) klcr =2cc2 x (20) klcr =6<X3 X (21) k^cr = 18 a 4 X (22) Strupczewski & Napiôrkowski (1988) showed that equations (19 to (22) hold also for Froude number equal to one. PHYSICAL ESTIMATION OF MODEL PARAMETERS In order to evaluate the physically based parameters of the MDM model the cumulant matching technique will be used. For the purpose of this study, the parameter n of the MDM model is assumed to be a given value. Therefore there are three parameters left for physical evaluation. Matching the first three cumulants of the MDM model and those of the LCR it is easy to express the LCR parameters (a, X, A) in terms of the MDM model parameters (a. K, T): 2 1-3a(l - a) a = K (23) 3 1-2a 9 X = (1-2a) i L 3 (24) 8 [1-3a(l - a)] 2

8 71 What is the distributed delayed Muskingum model? n(k + T) - «X (25) It remains to express (a, K, T) in terms of (a, X, A). One can see from equation (10) that to evaluate a as a function of X a fourth order polynomial must be solved, and that there are two real solutions for a (one positive and one negative) if: 8 X «1 9 n The exact solution for the parameter a is given by: (26) where: '1,2 2 (X/n) 1/3 3 W {2IXf± [2(2X)* - X 2 f (27) X = ' f8 X] z - VS-,1/3 1-9 n Substitution of equation (27) into equation (23) gives: 3l/3 K 1,2 7* f8 Xl 2 1 Vi> V3 9 n. (28) a (X/n) w {X*± [2(2/J*0* - Z] M } (29) and finally, from equation (25): A + ax 1,2 - K 1.2 (30) Putting n = 1 into equation (26) and taking into account equation (17), the upper limit of a river reach length for the DM model is obtained: 2i2 D max 9 9[1 + (w - 1)F Z ] 4 m[l - (m - 1) 2.F 2 ] (31) Equation (31) corresponds to zero value of the parameter s, Le. it defines the optimum length (in terms of cumulant fitting) of a reach for the single reservoir model with pure delay. As an example, the region of applicability of the DM model defined by equation (31) and the a-isolines (for positive and negative values of a) for a wide rectangular channel with the Manning friction law (m = 5/3) are shown

9 Witold G. Strupczewski & Jaroslaw J. Napiôrkowski 72 on Figs 1(a) and 1(b). It can be seen that as the dimensionless length, D, goes to zero the positive value of a approaches 0.5 and the negative value -<*>. Since a physical system is not an anticipating one, the pure delay, T, in equations (1) and (2) should have a positive value. This requirement adds the second constraint of the region for a > 0 and in particular it shows that a very short length of reach is outside the region of applicability of the DM model with a positive value of a. Note that m = 5/3 applies only in the limit of a very wide channel. For real rivers (lower values of m) the a-isolines are shifted to the left according to equation (26). Since within the whole region of applicability two solutions exist for the DM model parameters corresponding to a positive value of a {a v K v Tj) and a negative value of a (a 2, K^, T 2 ), the question arises of the choice of solution which fits the LCR model better. This can be answered by comparison of the fourth cumulants of the respective impulse responses or of the pure lags. Matching additionally for n - 1 the fourth cumulants given by equations (8b) and (22): M 4 (a) = Af M (a) - kf R = 0 (32) provides the optimum length of the DM river reach in terms of the Froude number; this is drawn on Fig. 2 for m = 5/3 and a wide rectangular channel. The A-D line represents all points for which Ak 4 (a > 0) = 0, the area EAD contains all points for which àk 4 (a > 0) > 0, and the area DCA contains all points for which Afc 4 (a > 0) < 0. The A-B line represents points for which Afc 4 (a < 0) = 0, the area EAB contains points for which Ak 4 (a < 0) > 0, and the area BAG contains points for which Ak 4 (a < 0) < 0. Note that the point A represents the flow conditions (F = 0.49) and the length of reach (D = 2.04) when the model of a single reservoir with pure delay agrees with the CLE model to an accuracy of as many as four cumulants; furthermore, that for negative value of a, the Froude number and the value of a go to one and minus infinity respectively as the optimum length approaches zero. Figure 2 shows additionally that a negative a value is better in terms of the fourth cumulants then a positive one except for the small wedge EAC In particular, a negative value of a should be applied for a very short reach. Figure 3, drawn in terms of pure lag agreement for the same friction law and channel slope, leads to similar conclusions. The pure delay of the LCR model, expressed by: F ALCR = m ( A + ax N / 33) V 1 +F is compared with the pure lags r x and T 2 of the DM model. Figure 3 shows that a positive value of a is better in terms of pure delay only in the vicinity of the line HJ representing the condition T x = A LCR, namely in the area bounded by lines: Ffl (a = 0 isoline), HK representing the condition Ti _ ALCË\ = j T 2 _ ALC^ ) and JQ (j^ = 0) Note that the point H

10 73 What is the distributed delayed Muskingum model? (a) 1.0 OS [ } m "z, O o OS V* \ \"* \ * * V \ \ \ \ \ \ \ V i t i i lit l/d l/d Fig. 1 The region of applicability of the DM model and the a-isolines for (a) negative and (b) positive values of a. 10.0

11 Witold G. Strupczewski & Jaroslav/ J. Napiôrkowski 74 E D C Fig. 2 The optimum length of a DM river reach as a function of the Froude number. represents the flow conditions (F = 0.32) and the length of reach (D = 1.62) when in addition to the first three cumulants, also the pure delays of the single linear reservoir model with linear channel and of the LCR model are equal. One can see from Figs 2 and 3 that for positive values of the parameter a, a good fit for the fourth cumulant (the area EAC on Fig. 2) excludes a good fit for pure delays (the area HLFK on Fig. 3), because these sets have no common points. However for a < 0, as is shown on Fig. 4, a good fit for four cumulants follows a good fit for pure lags, and vice versa The line expressed by equation (32) and the line: T 2 - A LCK_ 0 (34) are very close despite different starting points. Therefore the MDM with an appropriate value of the parameter n can be considered as a good approximation of the LCR both in terms of the fourth cumulant and pure lag for a Froude number greater than 0.4 in the case of rectangular channel with Manning friction.

12 75 What is the distributed delayed Muskingum model? \1 1»T >0 \r S> 1 o 1 LCR T 2 =i Tj<0 ce w pa -i H I o D!s w Q O œ - s - K A i^g. 3.0 ASYMPTOTIC CASE - «I.CR I \ \ \ * \, J \ /D Comparison of the DM pure lag with that of the CLE. Taking equation (5), describing the transfer function of the MDM, with the physically based parameters given by equations (27), (29) and (30), consider its limiting case when n tends to infinity. This implies modelling a river reach of a finite length, x, by an infinite number of physically based DM models. For an infinitely short subreach, the values of the parameters corresponding to the negative solution for the parameter a are adopted, namely (a 2, JL^ T 2 ). Note that for large n the MDM with parameters (a v K v Tj) is outside the region of applicability. One can see from equations (23) to (25) (which can be verified by calculating the limiting cases for equations (27) to (30)) that for large values of n the model parameters can be approximated by: a s - n/x K s a\ln TSi/fl (35) (36) (37) Replacing the parameters a, K and T in equation (5) by values given by equations (35) to (37) one gets an approximation of the transfer function of

13 Witold G. Strupczewski & Jaroslaw J. Napiôrkowski u m z td Q D o Ak (a<0)=0 4 _i 1 i 1 I il i I I.0 l.e E+00 l.e+01 i/d Fig. 4 Comparison of fourth cumulants and pure lags for a < 0. the MDM for large value of n: Jf^1DM (^) : \/n n/\ 1 + as + <x\s/n exp(-as) (38) From l'hopital's theorem one gets the transfer function of the multiply delayed Muskingum model for the asymptotic case of the number of subreaches tending to infinity: H(&) urn n-kx> HMUM ^ = exp - As - X as (39) It is interesting to note that the above equation is the solution in the Laplace transform domain of a partial differential equation originating from the linearized St Venant equations. Strupczewski & Napiôrkowski (1988), in order to filter out the downstream boundary conditions, used the kinematic wave solution to approximate the "diffusion term" (d 2 Q/dx?). The hydrodynamic model so obtained (the rapid flow model) provides the exact solution of the linearized St Venant equations for a Froude number equal to one (and a good approximation in the vicinity of F = 1) and is described by

14 77 What is the distributed delayed Muskingum model? the transfer function of equation (39). Its impulse response in the time domain is a Poisson-Gamma mixture with pure lag, i.e. the unit volume of the Dirac delta function entering the system is delayed by a linear channel with a time lag A and then divided between parallel series of linear reservoirs with a time constant a according to the Poisson distribution with mean X. Strupczewski & Napiôrkowski (1988) discussed properties of the rapid flow model and provided physical estimates of its parameters, which are conformable to equations (16) to (18). Therefore the name distributed Muskingum model for the asymptotic case of the multiply delayed Muskingum model described by equation (25) seems to be appropriate. CONCLUSIONS A linkage of the structures of conceptual and hydrodynamic flood routing models has been established by derivation of the linear hydrodynamic model known as the rapid flow model from conceptual blocks such as Muskingum and linear channel models. It was done for a uniform channel of any shape of cross section and for any friction law. An infinite series of such blocks defined for a finite river reach with physically based parameters produces an impulse response function consistent with that of the rapid flow model. Also physical estimates of all parameters (a, X, A) remain the same. The rapid flow model as a conceptual model can be called the distributed delayed Muskingum model. The dual character adds attractiveness to the model which is simple in structure, free of backwater effects and easy to expand for more complex cases such as tributary and lateral inflow or surface runoff modelling. The authors hope that this study will contribute to overcome the division between followers of conceptual and hydrodynamic models. REFERENCES Cunge, J. A. (1969) On the subject of a method for calculating the propagation of flood waves (Muskingum method). /. HydrauL Res. 2, Doetsch, G. (1961) Guide to Applications of Laplace Transfonns. Van Nostrand, New York - London. Dooge, J. C. I. & Harley, B. M. (1967) Linear theory of open channel flow. Unpublished memorandum, Dep. Civ. Eng., University College Cork. Dooge, J. C. I. & Napiôrkowski, J. J. (1986) The effect of the downstream boundary condition in the linearized St Venant equations. Quart J. Math. Appl. Mech. 40 (2), Dooge, J. C. L, Napiôrkowski, J. J. & Strupczewski, W. G. (1987) Properties of the generalized downstream channel response. Acta Geophys. Pol. 35 (4), Dooge, J. C. I., Strupczewski, W. G. & Napiôrkowski, J. J. (1982) Hydrodynamic derivation of storage parameters of the Muskingum model. /. HydroL 54, McCarthy, G. T. (1939) The unit hydrograph and flood routing. US Corps Engrs, Providence, Rhode Island, USA. Napiôrkowski, J. J., Strupczewski, W. G. & Dooge, J. C. I. (1981) Lumped nonlinear flood routing model and its simplification to the Muskingum model. Proc. Int. Symp. on Rainfall- Runoff Modelling, May 18-21, Mississippi, Strupczewski, W. G. & Kundzewicz, Z. W. (1980) Choice of a linear three-parametric conceptual flood routing model and evaluation of its parameters. Acta Geophys. Pol 28 (2), Strupczewski, W. G. & Napiôrkowski, J. J. (1986) Asymptotic behaviour of physically based

15 Witold G. Strupczewski&Jaroslaw J. NapiôrkowsM 78 multiple Muskingum model. Proc. 4th Int. Hydroi. Symp, Fort Collins Colorado, USA, July 1985, éd. Shen H. W. et a/, Strupczewski, W. G. & NapiôrkowsM, J. J. (1988) Linear flood routing model for rapid flow. Presented at Int Conf. on Fluvial Hydraulics, Budapest, 30 May-3 June. Strupczewski, W. G., Napiôrkowski, J. J. & Dooge, J. C. I. (1988) Properties of the distributed Muskingum model. Submitted to /. HydroL Received 11 November 1988; accepted 9 June 1989.