PROBLEMS IN REVERSE ROUTING

Transcription

1 A C T A G E O P H Y S I C A P O L O N I C A Vol. 53, no. 4, pp PROBLEMS IN REVERSE ROUTING James DOOGE and Michael BRUEN Centre for Water Resources Research, University College Dublin Earlsfort Terrace, Dublin 2, Ireland; michael.bruen@ucd.ie Abstract In contrast to direct downstream routing of flood flows, the complementary problem of reverse upstream routing is ill-posed and the calculations are subject to instability. However, the reverse routing procedure is useful both in irrigation management and in the control of flash floods by upstream reservoirs. Systematic study of reverse routing in uniform channels reveals a number of features that could prove of value in optimising such procedures. Key words: flood routing, reverse routing, instability. 1. INTRODUCTION In a number of problems of practical flow management the performance can be improved or optimised by the use of reverse routing of unsteady open channel flow. Procedures based on reverse routing have been used in hydraulic practice in recent decades. The main focus in this paper is the problem of the use of reverse routing to avoid downstream flooding by the modification of the releases from an upstream reservoir. The essential basis of such a procedure involves the following steps: (a) The forecast of reservoir inflow in good time; (b) The computation of reservoir outflow according to the routine procedure of reservoir operation; (c) The direct downstream routing of this reservoir outflow to the point of potential local flood damage; (d) The modification of the upper part of the predicated flood hydrograph at this danger point to a shape which would avoid flood damage;

2 358 J. DOOGE and M. BRUEN (e) The reverse routing of this modified downstream hydrograph in an upstream direction to obtain the required hydrograph of reservoir outflow for damage avoidance; (f) Confirmation of this desired outflow hydrograph by downstream routing of the hydrograph (possibly smoothed) to the downstream location of potential damage. We are concerned in this paper with the feasibility of step (e) in the above procedure. Basic mathematical theory would dismiss the possibility of accuracy in such a procedure since the problem is not well posed. Heuristically, since normal flood routing generally involves attenuation or damping of a wave profile, then reverse routing should involve amplification or intensification of a wave and that any errors in the stage or discharge values will also be amplified during the calculation. The same objection can be made to the problem of unit hydrograph derivation (i.e. deconvolution) but this has not deterred hydrologists from using the unit hydrograph approach over the past 60 years. Special care must be taken in such an approach but reliable results of sufficient accuracy for practical purposes can be obtained if such care is taken. The same is true of reverse routing. This paper explores the problem and seeks to set limits to the region of acceptable solutions. 2. BASIC EQUATIONS OF FLOOD ROUTING The basic formulation of unsteady one-dimensional flow in open channels is due to St.-Venant (1871). He wrote the continuity equation as Q x A + = t 0, (1) where Q(x,t) is the discharge and A(x,t) the area of flow. He wrote the momentum equation in terms of elevation, z, and velocity, u, i.e. z 1 u u u τ P x g t g x γ A = 0, (2) where z(x,t) is the elevation of the water surface, u(x,t) is the velocity of flow, A(x,t) is the cross-sectional area of flow and P(x,t) its wetted perimeter, τ 0 (x,t) is the boundary shear, g is the acceleration due to gravity, and γ is the weight density of water. For a prismatic channel, the momentum equation can be written in terms of discharge, Q, and cross-sectional area A, i.e. in the 2-dimensional state space form, as A 1 Q 1 Q n + + = S 0 2 4/3 B x ga x A ga t A R Q Q, (3)

3 PROBLEMS IN REVERSE ROUTING 359 A where B = is the top-width of the water surface and z is its elevation. z Equations (1), (2) and (3) or their equivalents are the basic equations for both direct routing and reverse routing. In the standard direct routing problem for tranquil flow (i.e., Froude numbers less than 1), there will be one upstream boundary condition which will be either: or Q(0, t) = Q ( t) (4a) 1 A(0, t) = A( t) (4b) and one downstream boundary condition which may take any of the three forms: or or 1 QLt (, ) = Q( t) (5a) 2 A( Lt, ) = A( t) (5b) 2 [ ALt] QLt (, ) = Φ (, ). (5c) In the case of eq. (5c) the function Φ [ ] may be any prescribed relationship between discharge and cross-sectional area of flow, e.g. steady-state rating curve, kinematic rating curve, etc. In the case of unsteady flow, the relationship between Q(x,t) and A(x,t) is not single-valued but a looped rating curve (Henderson, 1966). In practice, the downstream boundary condition is often taken as a steady-state rating curve at a section some distance downstream of the section of interest and thus will be a close approximation to the true relationship. When this is done, the relationship will be of the appropriate looped form at the section of interest. The specification of the problem is completed by the inclusion of two initial conditions given by and Qx (,0) = Q( x) (6) 0 0 A( x,0) = A ( x). (7) The solution of eqs. (1) and (3) subject to the boundary conditions of eqs. (4) and (5) and the initial conditions of eqs. (6) and (7) is a well-posed problem. Because of the non-linear form of the momentum equation (2) or (3), no analytical solution of the St.-Venant equations is possible. In hydrology, the earliest approach to the solution of this problem was on the basis of lumped conceptual models. If eq. (1) is integrated along the direction of flow (x) from Section 1 (upstream) to Section 2 (downstream), then we obtain

4 360 J. DOOGE and M. BRUEN Q t ds dt 1, 2 () Q () t =, (8) 1 2 where, S 1,2 (t) is the storage in the channel reach between Sections 1 and 2. The linear conceptual models used in hydrology replace the momentum equation (2) or (3) by an assumed relationship between S 1,2 (t) and the two end flows Q 1 (t) and Q 2 (t). The earlier models, e.g. Muskingum (McCarthy, 1939), lag and route (Meyer, 1941) and Kalinin -Milyukov (1957), all used linear relationships but non-linear models have since been developed. Other approaches have been to simplify the momentum equation (2) or (3) to enable an analytical solution to be found. Among these are the kinematic wave solution which reproduces translation but not subsidence of the flood wave and thus can match the lag but not the higher moments of the channel response and the diffusion analogy which can match the first and second moments but not the third and higher moments. Attention will be concentrated here on the simplification obtained by linearisation of the complete St.-Venant equation (2). This approach which has an analytical solution for uniform channels has been generalised over the past thirty years (Dooge and Harley, 1967a, b; Napiórkowski, 1992) and reproduces all the phenomena of nonlinear unsteady flow except shock waves. The linear approach can also be used in lumped form, i.e. by integrating along the reach, and written as Q () t = Q () t h() t, (9) 2 1 where denotes the operation of convolution and h(t) is the linear channel response which corresponds to the unit hydrograph which is the (apparent) linear response of a catchment to an impulse inflow. All the techniques and experience of the unit hydrograph approach can be applied to the determination of the linear channel response (LCR). The approach can be applied to natural channels with varying cross-sections but the assumption of linearity will be a source of error when the LCR derived from one set of inflow-outflow data is used to predict the value of the downstream outflow for an upstream inflow substantially different in magnitude or shape from the inflow used to calibrate the LCR. The full non-linear St.-Venant equations can be solved by numerical simulation using any of the large number of methods available in the literature (e.g., Abbott, 1979; Cunge et al., 1980). Because the problem of direct flood routing is well posed, a good deal of information is available on the stability and accuracy of these various methods. It should be remembered, however, that the theoretical analysis of stability and of amplitude error and phase error is made on the basis of linearised versions of the equation and often on simplified versions of the linear equations. The Preissmann (1961) scheme is used in this report for the numerical solution of all forms of the partial differential equations.

5 PROBLEMS IN REVERSE ROUTING NATURE OF THE INVERSE PROBLEM The process of deriving the downstream hydrograph on the basis of the upstream hydrograph and of channel properties can be expressed as 1 [ ] Q() t = H Q ( x,) t i i 1, (10) where H is an operator which transforms Q i (t) to Q 1 (x,t) for the given parameter set of channel characteristics. In particular at the downstream end of the channel 2[ ] Q () t = H Q () t. (11) 2 1 The inverse to this problem is the solution of Q () t = H Q () t, (12) 1 2 [ ] 1 2 where H -1 is a general inverse operator. For the case where the behaviour is assumed to be linear, eqs. (8) (11) takes the form of eq. (9) above: Q () t = Q () t h() t, (13) 2 1 where h(t) which is the linear channel response is independent of the input Q 1 (t) and depends only on the channel properties (Dooge and Harley, 1967a, b). Any method of deconvolution used to derive h(t) in eq. (9) from known functions Q 1 (t) and Q 2 (t) can be applied to the determination of Q 1 (t) for known functions h(t) and Q 2 (t). For the case where Q 1 (t) is modified to have a lower peak, the accuracy of such unit hydrograph approaches depends on the degree of change between the unmodified inflow and outflow set Q 1 and Q 2, and the modified set Q 1 and Q 2. In the case where the change is significant the difficulty could be overcome by subjecting the modified upstream inflow set to successive downstream routing and upstream routing. In the case of channels of uniform slope and cross-section, the theoretical linear channel response given by the analytical solution (Napiórkowski, 1972) could be used. One of the earliest applications of reverse routing was in the control of irrigation canals of limited length (Bodley and Wylie, 1978; Liu et al., 1992; Bautista et al., 1997). The present practice in the use of such valve-stroking has recently been reviewed (Clemens, 1998). The short lengths of channel involved make the transfer of experience in that area to flood routing in long channels somewhat hazardous. Franchini and Todini (1986) used the analytical solution of the diffusion analogy version of the basic equation (Hayami, 1951; Dooge, 1973) for the case of a semi-infinite channel to an irrigation canal 4.5 km in length with good results. In developing finite difference schemes for reverse routing, two discretisation parameters must be specified, i.e. θ for the time step and ψ for the space step (Fig. 1). Eli et al. (1974) used the

6 362 J. DOOGE and M. BRUEN j+1 1- θ P time θ j i 1- ψ ψ i+1 distance Fig. 1. Discretisation for 4-point schemes. implicit four-point difference scheme for a channel of length 37 miles (60 km) with θ = 0.5 and ψ = 0.5. They had good results except for periods of low flow. Szymkiewicz (1993) used the more general form of the Preissmann scheme for a channel length of 49 km using the same boundary conditions as Eli et al. (1974). These were a double initial condition, i.e. both discharge and depth specified at the downstream end of the channel, at x = L. and QLt (, ) = Q( t) (14a) 2 2 ylt (, ) = y( t) (14b) together with a single, constant, condition at the initial time, i.e. either or Qx (,0) yx (,0) and a single condition at the final time, T, i.e. either or QxT (, ) yxt (, ) = Q (15a) 0 0 = y (15b) = Q (16a) T T = y. (16b) Szymkiewicz found optimum results for θ = 0.4 and ψ = 0.85, but it depended on the smoothness of the input. In a later paper the same author analysed the numerical stability of the approach on the basis that the roles of the x-axis and the t-axis were interchanged in the reverse routing problem (Szymkiewicz, 1996). This general approach may be described as the 90 rotation method since the solution area and the

7 PROBLEMS IN REVERSE ROUTING 363 governing boundary conditions are rotated by 90 in an anti-clockwise direction. A series of numerical simulations were carried out under the present project using this 90 0 rotation method with various values of θ and Ψ for both the linearised St.-Venant equations and the original non-linear equations as described in the next section of the report. Another application was the development of a suitable and reliable technique for the solution of the practical problem of reservoir control of the Dunajec River, a tributary of the Upper Vistula, by Nachlik and Witt (1990; 1993). In their work the 90 rotation method was abandoned and the single boundary condition at t = 0 was replaced by a second boundary condition at the final time t = T. They got more reliable results when this change was made and values of θ in the range 0.45 < θ < ACCURACY OF DIRECT ROUTING The first requirement for a reliable method of flow control is that the initial downstream routing (step c in Sect. 1 above) is within reasonable bands of accuracy. This can be tested for the case of the linearised equations since an analytical solution is available for any prescribed upstream flow. Accordingly such a comparison is made for the linear channel response to a prescribed sinusoidal inflow. The linear channel response to an infinite train of sinusoids can de derived analytically and its characteristics determined, such as the degree of attenuation, amount of phase shift, and the shape of the looped rating curve (Kundzewicz and Dooge, 1989) The degree of attenuation can be clearly shown to depend on the frequency of the input train and on the Froude number of the reference conditions. Figure 2, taken from their paper, Fig. 2. Amplitude attenuation as a function of frequency (Kundzewicz and Dooge, 1989).

8 364 J. DOOGE and M. BRUEN shows the relationship of the attenuation per unit length as a function of the frequency of the infinite train of sinusoids for an infinitely wide rectangular channel with Chezy friction using the diffusion analogy with parameters based on matching the first and second moments of the complete linear channel response. It shows that substantial attenuation in downstream peak flow is largely due to a severe reduction in the high frequency component of the upstream inflow. Consequently, reverse routing involves the restoration of these high frequency components. In numerical routing this process is naturally susceptible to contamination by high frequency components arising from the approximation involved in finite difference equations. In the present study the upstream inflow is taken as consisting of three parts: (1) a steady flow of 500 m 3 /s for 12 hours, (2) a sinusoidal wave rising to 4500 m 3 /s in 6 hours and returning to 500 m 3 /s after a further 6 hours, and (3) a further period with a constant flow of 500 m 3 /s. The channel was assumed to be a 100 m wide rectangular channel with Manning friction (n = 0.025). The reference discharge for linearization was taken as 2500 m 3 /s. The computations were made for a steep slope of S 0 = (F 0 = 0.70), for a moderate slope, S 0 = (F 0 = 0.51) and for a mild slope of S 0 = (F 0 = 0.17). The first step is to compute the downstream hydrograph computed by the 4-point Preissmann scheme for the linearised equations scheme and to compare the results with those predicted by the theoretical analysis. The most convenient method of doing this is to compare the first few cumulants which together characterise the properties of the linear channel response. The first cumulant which is equal to the first moment about the origin characterises the lag in the linear flood routing process. The second cumulant which is equal to the second moment about the origin characterises the dispersion of the process. The third cumulant which is equal to the third moment about the centre characterises the increase in skewness. The fourth cumulant which is equal to the fourth moment about the centre minus 3 times the square of the second cumulant. The cumulants of the analytical solution can be determined by appropriate analysis (Romanowicz et al., 1988). Effect of channel slope on direct routing Table 1 steep Bed slope case moderate mild Bed slope, S Froude no., F Peak Q 2 (linear) Peak Q 2 (nonlinear)

9 PROBLEMS IN REVERSE ROUTING 365 While the use of the linearised eq uations is appropriate for comparison with the analytical values of the cumulants and shape factors, the question arises of the errors arising from linearization. For a 100 km length of the standard channels the differences between the two cases as computed by the prescribed numerical computation scheme are shown on Table 1. It is clear from the second column that the reduction in peak for the steep channel (F 0 = 0.7) is only just over 1 per cent and hence reverse routing for flow control is not appropriate in this case. Accordingly, consideration of reverse routing in this paper is confined to the cases of moderate (F 0 = 0.51) and mild (F 0 = 0.17) slopes. The next step was to apply reverse routing to the downstream conditions obtained by the computations described in Sect. 4. This was done by using the 4-point Preissmann scheme. The stability of such computations was analysed by Szymkiewicz (1993; 1996) who deduced that the procedure would be unconditionally stable for θ 0.5 and ψ EXAMPLES OF REVERSE ROUTING RESULTS In reverse routing the computation proceeds up the channel step by step from a down- reverse stream section at which both discharge and area are prescribed until the values are derived at the original upstream section at which flow is controlled. In the reverse routing procedure both depth (y 2 ) and the discharge (Q 2 ) are prescribed at the downstream end of the channel and flow rate (Q) is prescribed along the length of the channel for both the initial time (t = 0) and the final time (t = 100 hours, in this paper). The results for various values of the parameters (θ, ψ) for the linearised estimates of (a) the original peak flow and (b) the length for instability collapse are shown in Table 2 for the mild slope and in Table 3 for the moderate slope. In these Tables, an entry between 1 and 99 indicates the reverse routing failed at that distance (in km) upstream from the downstream boundary. A larger value (exceeding 4000 in all cases) indicates that the reverse routing reached the upstream boundary and the value in the table is the estimated peak flow (m 3 /s) of the reverse routed wave. For both moderate and mild slopes the best results are obtained for the higher values of ψ and for the medium and lower values of θ. The nonlinear is less stable than the linear case, particularly for the moderate slope (F 0 = 0.51) case. However, as with the linear case, the best results are for the higher values of ψ and the lower values of θ. The results for the case of a mild slope are given in Table 4 and for the moderate slope in Table 5. A comparison of these two tables reveals that the range of successful parameter combinations is considerably less in the case of the moderate slope (F 0 = 0.51) than for the mild slope (F 0 = 0.17). For the non-linear, moderate slope, case, the calculation using θ = 0.5 and ψ = 0.5 does not reach beyond 28 km upstream from the downstream boundary before it collapses due to instability. However, for the other three cases (linear with moderate slope and

10 366 J. DOOGE and M. BRUEN Results for linearised reverse routing mild channel slope case (S 0 = ) Table 2 ψ θ Results for linearised reverse routing moderate channel slope case (S 0 = 0.001) Table 3 ψ θ both mild slope cases) the reverse routing reaches the upstream boundary (a distance of 100 km) for the same parameter values. In contrast, for the parameter values θ = 0.6 and ψ = 0.5, all four cases break down quite early in the calculation. Table 6 shows the distance (in km) traveled up-

11 PROBLEMS IN REVERSE ROUTING 367 Results for nonlinear reverse routing mild channel slope case (S 0 = ) Table 4 ψ θ Results for nonlinear reverse routing moderate channel slope case (S 0 = 0.001) Table 5 ψ θ stream before the calculations break down. Examination of a number of such unstable cases seems to suggest that instability arises towards the end of the recession, at its transition to steady flow, and then contaminates the recession limb and ultimately the peak of the estimated upstream hydrograph. Figure 3 shows the shape of the hydro-

12 368 J. DOOGE and M. BRUEN Table 6 Example of stability limits for reverse routing. (The table gives the distances over which the reverse routing travelled before the calculation breaks down due to instability. L = 100 km and θ = 0.6 and ψ = 0.5) Reverse routing Channel slope case moderate mild Linear 35 km 30 km Nonlinear 15 km 24 km Fig. 3. Example of instability for nonlinear reverse routing with moderate channel slope. graph at 15 km from the downstream boundary for the nonlinear, moderate slope case, with θ = 0.6 and ψ = 0.5. Figure 4 shows the shape of the hydrograph at 24 km for the corresponding mild slope case. These sample calculations show a systematic pattern and indicate that a more extensive set of numerical experiments would provide practical rules for the use of reverse routing in flash flood control in open channels by managing upstream reservoir releases (Bruen and Dooge, 1993).

13 PROBLEMS IN REVERSE ROUTING 369 Fig. 4. Example of instability for nonlinear reverse routing with mild channel slope. 6. CONCLUSIONS The tests carried out, of which the examples given above are typical, lead to a number of provisional conclusions. The most important of these are: (1) despite the ill-posed nature of the reverse routing problem, the process is possible for surprisingly long lengths of channel: (2) the stability of the process is highly sensitive to the values of the discretisation parameters (θ and ψ) used in the finite difference process; and (3) despite the lower downstream attenuation for moderate and high Froude numbers, instability can occur over a wide range of channel slopes. The tests carried out under the study were confined to a single shape of inflow hydrograph and a single frequency. The results of these tests are such as to indicate that a wider series of similar tests could provide useful practical values for the use of reverse routing in practical flood control. Acknowledgement. This work was done as part of the TELFLOOD research project which was financially supported by the European Union (Contract no. ENV6-CT ).

14 370 J. DOOGE and M. BRUEN References Abbott, M.B., 1979: Computational Hydraulics, Pitman Publ., London. Bautista, E., A.J. Clemmens and T. Strelkoff, 1997: Comparison of numerical procedures for gate stroking, ASCE J. Irrig. and Drain. Engrg. 123, pp Bodley, W.E., and F.B. Wylie, 1978: Control of transients in series of channels and gates, J. Hydraul. Div. ASCE 104, HY10, pp Bruen, M., and J.C.I.Dooge, 1993: Flash flood control using reverse routing, Technical Report on TELFLOOD Project, pp. 48, Centre for Water Resources Research, University College Dublin. Clemmens, A.J. (ed.), 1998: Canal Automation, Special issue of ASCE J. Irrig. and Drain. Engrg. 124 (1), pp Cunge, J.A., F.M. Holly and A. Verwey, 1980: Practical Aspects of Computational River Hydraulics, Pitman Publ., London, pp Dooge, J.C.I., 1973: Linear Theory of Hydrologic Systems, Tech. Bull. 1468, USDA-ARS, pp Dooge, J.C.I., and B.M. Harley, 1967a: Linear routing in uniform open channels, Proc. Intern. Hydrology Symposium, Fort Collins, Colorado, 6-8 Sept. 1, pp Dooge, J.C.I., and B.M. Harley, 1967b: Linear routing in uniform open channels, Proc. Intern. Hydrology Symposium, Fort Collins, Colorado, 6-8 Sept. 2, pp Dooge, J.C.I., J.J. Napiórkowski and W.G. Strupczewski, 1987: The linear downstream response of a generalized uniform channel, Acta Geophys. Pol. 35 (3), pp Eli, R.N., J.M. Wiggert and D.N. Contractor, 1974: Reverse flow routing by the implicit method, Water Resources Res. 10 (3), pp Franchini, M., and E. Todini, 1986: PAB. Uno schema di calcelo per la simulazione di partrobaziase propagantezi cantro corrente in canali a pelo libero, Energia Elettrica 63 (6) (in Italian). Hayami, S., 1951: On the propagation of flood waves, Disaster Prevent. Res. Inst. Bull. 1, pp Henderson, F M., 1966: Open Channel Flow, Macmillan Company, New York. Kalinin, G.P., and P.I. Milyukov, 1957: On the computation of unsteady flow in open channels, Meteor. Gidrol. 10, pp (in Russian). Kundzewicz, Z.W., and J.C.I. Dooge, 1989: Attenuation and phase shift in linear flood routing, Hydrolog. Sci. J. 34 (1), pp Liu, F., J. Feyen, and J. Berlamant, 1992: Computation methods for regulating unsteady flow in open channels, ASCE J. Irrig. and Drain. Engrg. 118, pp McCarthy, G.T., 1939: The unit hydrograph and flood routing, Unpublished paper presented at the Conference of the North Atlantic Division of U.S. Corps of Engineers, Providence, Rhode Island, June 1938 (revised March 1939). Meyer, O.K., 1941: Simplified flood routing, Civil Engineering 11 (5), pp

15 PROBLEMS IN REVERSE ROUTING 371 Nachlik, E., and M. Witt, 1990: Application of hydrodynamic models of flood wave propagation in the upper Vistula river systems in flood protection of the basin, Report No. CPBR , Technical University of Cracow (unpublished report, in Polish). Nachlik, E., and M. Witt, 1993: Hydraulic models of outflow control in upper Vistula system, Proc. Intern. Conf. Hydrosci. Engineering, Washington. Napiórkowski, J.J., 1992: Linear theory of open channel flow. In: J.P. O Kane (ed.), Advances in Theoretical Hydrology, Elsevier, Amsterdam, pp Preissmann, A., 1961: Propagation des intumescences dans les canaux et revieres. 1er Congres de l Assoc. Francaise de Calcul, Grenoble, pp Romanowicz, R.J., J.C.I. Dooge and Z.W. Kundzewicz, 1988: Moments and cumulants of linearised St. Venant equation, Adv. Water Resour. 11, pp St.-Venant, B. de, 1871: Théorie de liquids non-permanent des eaux avec application aux cave des rivieres et à l introduction des marées dans leur lit (Theory of the unsteady movement of water with application to river floods and the introduction of tidal floods in their river beds), Acad. Sci. Paris, Comptes Rendus 73, pp and 33, pp Szymkiewicz, R., 1993: Solution of the inverse problem for the Saint Venant equations, J. Hydrol. 147, pp Szymkiewicz, R., 1996: Numerical stability of implicit four-point scheme applied to inverse linear flood routing, J. Hydrol. 176, pp Thomas, H.A., 1977: Hydraulics of Flood Movement in Rivers, Carnegie Institute of Applied Technology, Pittsburgh. Weizmann, P.E., and E.M. Lawrenson, 1979: Approximate flood routing methods: a review, ASCE J. Hydraulic Res. 105 (2), pp Received 6 June 2005 Accepted 17 August 2005