A fuzzy dynamic flood routing model for natural channels

Transcription

1 HYDROLOGICAL PROCESSES Hydrol. Process. (29) Published online in Wiley InterScience ( DOI:.2/hyp.73 A fuzzy dynamic flood routing model for natural channels R. Gopakumar and P. P. Mujumdar 2 * Department of Civil Engineering, Government Engineering College, Thrissur 9, India 2 Department of Civil Engineering, Indian Institute of Science, Bangalore 52, India Abstract: A fuzzy dynamic flood routing model (FDFRM) for natural channels is presented, wherein the flood wave can be approximated to a monoclinal wave. This study is based on modification of an earlier published work by the same authors, where the nature of the wave was of gravity type. Momentum equation of the dynamic wave model is replaced by a fuzzy rule based model, while retaining the continuity equation in its complete form. Hence, the FDFRM gets rid of the assumptions associated with the momentum equation. Also, it overcomes the necessity of calculating friction slope (S f ) in flood routing and hence the associated uncertainties are eliminated. The fuzzy rule based model is developed on an equation for wave velocity, which is obtained in terms of discontinuities in the gradient of flow parameters. The channel reach is divided into a number of approximately uniform sub-reaches. Training set required for development of the fuzzy rule based model for each sub-reach is obtained from discharge-area relationship at its mean section. For highly heterogeneous sub-reaches, optimized fuzzy rule based models are obtained by means of a neuro-fuzzy algorithm. For demonstration, the FDFRM is applied to flood routing problems in a fictitious channel with single uniform reach, in a fictitious channel with two uniform sub-reaches and also in a natural channel with a number of approximately uniform sub-reaches. It is observed that in cases of the fictitious channels, the s match well with those of an implicit numerical model (INM), which solves the dynamic wave equations using an implicit numerical scheme. For the natural channel, the s are comparable to those of the HEC-RAS model. Copyright 29 John Wiley & Sons, Ltd. KEY WORDS flood routing; dynamic wave; Saint Venant equations; monoclinal wave; fuzzy sets; fuzzy rules; fuzzy inference systems; neuro-fuzzy algorithms Received 2 June 2; Accepted 9 February 29 INTRODUCTION When a flood wave moves downstream a river, the wave configuration will be modified due to channel irregularities and roughness. Determination of this modification is called flood routing (Chow, 959). As the flood wave moves downstream, its characteristics such as the peak, the time to peak and the duration of the hydrograph changes, and also there will be attenuation (Singh, 2). Flood routing computations involve the determination of stage and discharge hydrographs at various locations along a river from the known flood hydrograph at the upstream end and based on specified initial and boundary conditions. Conventionally the flood routing is carried out by dynamic wave routing, which involves the solution of the following governing equations (Saint Venant equations) (Chow et al., 9): Continuity equation. A t C Q x D Momentum equation. Q t C [ ] Q 2 C ga h x A x gas C gas f D 2 * Correspondence to: P. P. Mujumdar, Department of Civil Engineering, Indian Institute of Science, Bangalore 52, India. pradeep@civil.iisc.ernet.in where A is flow cross-sectional area, h the flow depth, Q the discharge, S the bed slope, S f the friction slope, g the acceleration due to gravity, t the time variable and x the space variable. Here, it is assumed that lateral inflow is negligible. The Equations () and (2) are generally solved using implicit numerical models (INMs), to obtain variation of flow depth (h) and discharge (Q), with respect to time, at various locations along the channel. Gopakumar and Mujumdar (2) discussed the various difficulties involved in the application of INM in dynamic wave routing. A major difficulty mentioned by them is the uncertainties that are involved in the estimation of the friction slope (S f ) in the momentum Equation (2). The friction slope is usually computed with Manning s equation wherein the roughness coefficient (n) is estimated based on several channel characteristics such as surface roughness, size, shape, vegetation, irregularity, alignment, stage, discharge, etc. (Chow, 959). The roughness coefficient is one of the main variables used in calibrating the INM (US Army Corps of Engineers, 22). Slight error in estimated value of n can result in large errors in the computed stage and discharge values. But accurate estimation of n for natural channels is very difficult due to its dependence on the several factors stated above. Gopakumar and Mujumdar (2) have demonstrated that it is possible to overcome these difficulties by replacing the momentum Equation (2) by a fuzzy rule based model, while retaining the continuity Copyright 29 John Wiley & Sons, Ltd.

2 R. GOPAKUMAR AND P. P. MUJUMDAR Equation () in its complete form. While doing this they followed the procedure adopted by Bardossy and Disse (993) in modelling infiltration. Other similar studies have also been reported in the literature (Bardossy et al., 995; Dou et al., 999). Present study describes development of a fuzzy dynamic flood routing model (FDFRM) for natural channels, wherein the momentum Equation (2) is replaced by a fuzzy rule based model while retaining the continuity Equation () in its complete form. This study follows the same methodology as adopted by Bardossy and Disse (993). As will be seen in the remaining sections of this paper, the above fuzzy rule based model is constructed on physical characteristics of the channel such as its geometric characteristics (represented by Time ( t ) i-,,j+ 2 i,,j+ i,,j t i+,,j+ Distance (x) x N- N Figure. Computational domain for the FDFRM: N, number of nodes in the x direction; i, node number in the x direction; j, node number in the t direction A (m 2 ) 9 C 7 P O Q (m 3 /s) 9 D 7 J F A3 5 I A2 G 3 Mean curve 2 A H E Q (m 3 /s) A (m 2 ) Q Q 2 Q 3 Figure 2. Definition sketch for development of fuzzy rule based model for a uniform channel reach (downstream sub-reach of the channel under Case 2): Q A relationship; Q A relationship derived from the Q A relationship A (m 2 ) A (m 2 ) Mean curve Q (m 3 /s) Q (m 3 /s) Figure 3. Relationships obtained for the channel under Case, which are used for development of its fuzzy rule based model: Q A relationship; Q A relationship Membership degree Membership degree 9L L 3L L H 3H H 9H 2H A (m 2 ) 9L L 3L L H 3H H 9H 2H 2 2 Q (m 3 /s) Figure. Fuzzy membership functions for the channel under Case : input membership functions; output membership functions stage area relationships) and flow characteristics (represented by stage discharge relationships). It is developed on an equation for wave velocity, which does not directly depend on the various forces that appear in the momentum Equation (2). Hence the uncertainties associated with the estimation of these forces are avoided. It may be noted that even though hydrological routing methods such as the Muskingum method (Chow et al., 9) are available, which also do not require explicit specification of the roughness coefficient, the advantage of the present model is that it gives spatial distribution of the flow parameters also in addition to their temporal variation. The concept of fuzzy logic was first introduced by Zadeh (95). Details of fuzzy sets, fuzzy numbers, membership functions and fuzzy rules can be found in literature (Zimmermann, 99; Bardossy and Disse, DOI:.2/hyp

3 FDFRM FOR NATURAL CHANNELS 993; Ross, 995; Dou et al., 999). Fuzzy rule based models are suitable when information about the physical process is vague and data available are scarce. The physical information is used in development of the rules which are then tuned using the available data. These models are suitable for nonlinear input output mapping (Ross, 995). In fuzzy rule based modelling, a given input data undergo the processes of fuzzification, application of fuzzy operators, implication, aggregation and defuzzification, before finally forming into output from the model (Matlab, 995; Panigrahi and Mujumdar, 2). Good introduction to practical aspects of fuzzy rule based modelling is given by some authors (Russel and Campbell, 99; Kisi et al. 2). They highlighted the robustness of the fuzzy rule based models. Other advantages of these models include their transparent structure and simplicity in development. It has been reported in the literature that there exists a very high potential of the possibilities of applying fuzzy rule based models in water resources (Abebe et al., 2). Even though a significant research has already been carried out towards fuzzy rule based modelling of various hydrological processes, very little application of the technology has been found in flood routing. Gopakumar and Mujumdar (2) proposed a fuzzy dynamic wave routing model (FDWRM), the suitability of which has been demonstrated using prismatic channels, whereas natural channels are non-prismatic. Also, in that work, the training sets required for development of the fuzzy rule based model have been obtained from outputs of the INM. Application of the INM to natural channels is a tedious task due to the aforementioned difficulties associated with it. Hence the FDWRM can be expected to be more suitable for irrigation canals. Also, in general, gravity wave predominates in irrigation canal flows, whereas monoclinal wave predominates during floods in natural channels. Hence the procedure of development of the fuzzy rule based model for these two cases are different and that is the fundamental difference between the above FDWRM and the model being presented here, viz. FDFRM. The present study aims at development of the FDFRM to simulate flood movement in a natural channel, wherein nature of the flood wave can be approximated to a monoclinal wave (Chow, 959; Henderson, 9). When a flow change occurs at the upstream end of a channel, initial nature of the wave will be of inertia-dominated gravity type, which will gradually transform into noninertia monoclinal wave as the effects of bed slope and flow resistance increase with travel time and distance. Here it is assumed that back water effects are negligible. Therefore, a combination of the gravity wave and monoclinal wave models can provide a methodology for addressing the nonlinear dynamics involved in the flood wave propagation. Detailed analyses of these two wave forms are available in literature (Chow, 959; Henderson, 9; Ferrick, 25). The travel distance required for full development of a monoclinal wave corresponds to the length of the wave profile (Ferrick, 25). This Figure 5. Boundary conditions for flood routing in the channel under Case : stage hydrograph at upstream boundary; stage hydrograph at downstream boundary length, and correspondingly the required travel distance, depends on the ratio of normal depth (h n ) to bed slope (S ); the higher the ratio, more will be the required travel distance. Therefore, only relatively steep and shallow channels, with the ratio h n /S <5 (Ferrick, 25), are considered in the present study so that the monoclinal wave will be formed in short distances and nature of the wave can be assumed to remain the same over the entire channel reach. In case of uniform channels (i.e. channels having same cross section, bed slope and roughness for the entire reach), the monoclinal wave will propagate downstream with its shape remaining intact (Lighthill and Whitham, 955; Chow, 959; Henderson, 9; Agsorn and Dooge, 99). If the natural channel is not uniform, which is usually the case, then the wave configuration will be modified due to channel irregularities and roughness. In such cases, the channel reach can be divided into a number of approximately uniform sub-reaches and the monoclinal wave approximation can be applied to each individual sub-reach separately. The FDFRM developed in the present study is based on this principle. It may be noted that in the monoclinal wave, the inertia terms of the momentum Equation (2) are taken into account even DOI:.2/hyp

4 R. GOPAKUMAR AND P. P. MUJUMDAR (c).3 (d) Figure. Simulation results of flood routing in the channel under Case : discharge hydrograph at 2 km; discharge hydrograph at km; (c) stage hydrograph at 2 km; (d) stage hydrograph at km though their effects are negligible (Ferrick and Goodman, 99; Ferrick, 25). Thus the monoclinal wave represents the complete dynamic wave and therefore the diffusion or kinematic wave approximations are not suitable in the present case. Also, the kinematic wave routing does not permit attenuation, whereas there will be attenuation of the flood wave in real situations. Hence, theoretically, one has to carry out complete dynamic wave routing in order to simulate the movement of flood wave in natural channels. The FDFRM is not a replacement of the complete dynamic wave routing model, rather it can be used as a subsequent step for simplifying the complicated model. Advantages of the fuzzy logic based model are that it requires fewer parameters and is not very sensitive to parameter changes (Bardossy and Disse, 993). As mentioned earlier, the momentum Equation (2) is replaced by a fuzzy rule based model. This is done based on the principle that during unsteady flow, the disturbances in the form of discontinuities in the gradient of the flow parameters, which appear in the governing Equations () and (2), will propagate along the characteristics with a velocity equal to that of velocity of the shallow water wave (V w ) (Cunge et al., 9). The discontinuities in the gradient of the flow parameters A and Q are obtained as the difference between their final steady flow gradient and present gradient. At any node (i, j C ) in the computational domain (Figure ), these discontinuities are computed by adopting the following discretization: Discontinuity in the MODEL DEVELOPMENT A uniform channel reach is considered for development of the model. The continuity Equation () is discretized at node (i, j), in the x t computational domain (Figure ), using the Lax diffusive scheme (Chaudhry, 993), which is an explicit finite difference scheme, as follows: t [AjC i Ð5 ð A j i C Aj ic ] C 2x [Qj ic Qj i ] D 3 where i is the node number in the x direction, and j is the node number in the t direction. gradient of area A D AjC i x D [AF.S i Discontinuity in the A F.S i ] [AjC i A jc i ] x gradient of discharge Q D QjC i x D [QF.S i D [QjC i Qi F.S ] [QjC i Q jc i ] x x Q jc i ] 5 DOI:.2/hyp

5 FDFRM FOR NATURAL CHANNELS Figure 7. Boundary conditions for flood routing in the channel under Case 2: stage hydrograph at upstream boundary; stage hydrograph at downstream boundary Note that Qi F.S D Qi F.S during the final steady flow condition. In Equations () and (5), the notations with superscript F.S indicate the values of the flow parameters corresponding to the final steady condition. The ratio of discontinuity in the gradient of Q and discontinuity in the gradient of A, at the node (i, j C ), will give the wave velocity (V w ) at this node (Appendix) i.e. wave velocity V w at the node i, j C D QjC i A jc i If it is assumed that the wave velocity (V w ) corresponding to a certain Q remains constant over the range of flow depths being considered, then a set of crisp rules can be written, relating A and Q and based on Equation (), as IF A is low THEN Q is low, IF A is medium THEN Q is medium, IF A is high THEN Q is high etc.wherelow, medium, high etc. are quantitative variables. It will be shown in the next paragraph that such crisp relations do not exist between A and Q. Therefore the rules become fuzzy under the concept that the V w remains approximately constant over the range of flow depths. Then low, medium, high etc. become linguistic variables. Also, as monoclinal wave is being considered, the velocity of which does not change while moving forward, these rules will be applicable for the entire channel reach. The fuzzy rule based model is developed on this principle. As the flood wave is being approximated to monoclinal wave, the wave velocity (V w ) can also be obtained from discharge (Q) area (A) relationship, based on the Kleitz- Seddon principle (Chow, 959). The Q A relationship for a cross section of a typical uniform channel reach is shown as the curve O P C in Figure 2. As the channel reach is uniform, the initial and final steady conditions will correspond to uniform flow and this Q A relationship will hold good for all cross sections. Hence, from Figure 2 and Equation (), it can be realized that A corresponding to the same Q will be different at different flow depths, and therefore crisp relations do not exist between A and Q. All values of A corresponding to each Q are determined as the training sets and the fuzzy rule based model is developed by adopting the following procedure. Between any two cross sections, for every Q all possible corresponding A values are obtained from the aforementioned Q A relationship (Figure 2). The result is shown in Figure 2. In this figure, the coordinates of the maximum point D are same as the coordinates of the maximum point C of the Q A curve. This corresponds to a case when Q or A is a maximum at one section and zero at the other or vice versa. The minimum point E corresponds to the case where the co-ordinates (Q, A) are same at both the sections. The co-ordinates in between D and E are obtained by analysing all possible combinations of the co-ordinates (Q, A) between the two sections. It has been found that all these co-ordinates lie within the looped curve E F D G E (Figure 2). Co-ordinates of the curve E F D are same as the coordinates of the Q A curve O P C. Ordinates of the curve E G D correspond to reversed ordinates of the curve E F D. Mean values of the upper bound ordinates on E F D and lower bound ordinates on E G D lie on the curve E H I J D. As mentioned previously, the fuzzy rules are developed relating the antecedent (A) and the consequent (Q). Development of the fuzzy rule based model involves the determination of the membership function parameters and construction of the fuzzy rule base. This is achieved using training sets, as described by Bardossy and Disse (993). These training sets are obtained from the looped Q A curve (Figure 2) by adopting the following procedure. The universe of discourse of the consequent Q is divided into a number of equal partitions. The midpoint of each partition is taken as its prototype point, i.e. a membership equal to unity. Also, the value of membership function corresponding to a prototype point is zero at all other prototype points. This procedure of partitioning is explained in Ross (995). Additionally, the membership functions are assumed to be triangular. As accuracy of fuzzy rule based models has been reported to DOI:.2/hyp

6 R. GOPAKUMAR AND P. P. MUJUMDAR (c).3 (d) Figure. Simulation results of flood routing in the channel under Case 2: discharge hydrograph at 2 km; discharge hydrograph at km; (c) stage hydrograph at 2 km; (d) stage hydrograph at km Figure 9. Boundary conditions for flood routing in the Critical Creek (Case 3): stage hydrograph at upstream boundary; stage hydrograph at downstream boundary be less sensitive to shape of their membership functions (Sugeno and Yasukawa, 993), this assumption is considered reasonable. Let (Q,Q 2,Q 3 )beatypical fuzzy subset developed accordingly (Figure 2), where (Q,Q 3 ) is the support and Q 2, which is the mean of Q and Q 3, is the prototype point. The corresponding fuzzy subset of the antecedent, viz. (A,A 2,A 3 ), is obtained as shown in Figure 2. Here A is the lower bound ordinate corresponding to Q,A 2 is the mean ordinate corresponding to Q 2 and A 3 is the upper bound ordinate corresponding to Q 3. Similarly all corresponding fuzzy subsets of the antecedent and consequent are determined. Then they are assigned linguistic names such as low, medium, high etc. and are related through the fuzzy rules mentioned earlier. The Mamdani implication method of inference (Mamdani and Assilian, 975) and the centroid method of defuzzification (Ross, 995) are used within the fuzzy rule based model. If the channel reach is not exactly uniform, then it is divided into a number of uniform sub-reaches and the fuzzy rule based model for each sub-reach is developed separately. As natural channels are under consideration, the sub-reaches will be only approximately uniform. Therefore, the Q A relationship for the mean cross section of each sub-reach is utilized for development of fuzzy rule based model for that sub-reach. In case the heterogeneity of the sub-reach, with respect to its cross sections, is very high, and then an optimized fuzzy rule based model, based on Q A relationships of different cross sections, is obtained by means of a neuro-fuzzy algorithm, explained subsequently through an application. DOI:.2/hyp

7 FDFRM FOR NATURAL CHANNELS (c) (c) Figure. Simulation results of flood routing in the Critical Creek (Case 3): discharge hydrograph at cross section no. ; discharge hydrograph at cross section no. 7; (c) discharge hydrograph at cross section no. 2 METHOD OF COMPUTATION Values of flow depth (h) and discharge (Q) at beginning of the time step are to be specified at all the computational nodes, along the channel, as initial conditions. The three boundary conditions required by the FDFRM are the inflow discharge hydrograph at the upstream Figure. Simulation results of flood routing in the Critical Creek (Case 3): stage hydrograph at cross section no. ; stage hydrograph at cross section no. 7; (c) stage hydrograph at cross section no. 2 boundary, the stage hydrograph at the upstream boundary and the stage hydrograph at the downstream boundary. The data in terms of h are to be converted in terms of area (A), while applying the FDFRM. This means that the h A relationships should be available at all the computational nodes. Also, for every subreach, the stage discharge relationships should be available at the boundary nodes and at the mean section. Stage discharge relationships at the other computational nodes are obtained by interpolation. In case of a channel with a single uniform reach, the following are the steps involved in the flood routing computations: Step. From the known initial conditions and using the discretized continuity Equation (3), the values of A at the end of the time step (t) (Figure ) are obtained at all the nodes. As Equation (3) is based on an explicit finite difference scheme, the time step should be chosen satisfying the C.F.L (Courant-Frederich-Levy) condition (Chaudhry, 97). Step 2. For the considered inflow flood discharge into the reach during the present time step, the corresponding final steady water surface profile along the reach is obtained from the stage discharge relationships at the computational nodes. DOI:.2/hyp

8 R. GOPAKUMAR AND P. P. MUJUMDAR Figure 2. The MATLAB/SIMULINK model developed for optimization of the fuzzy rule based models of various sub-reaches of the Critical Creek (Case 3), by means of the neuro-fuzzy algorithm NEFCON Step 3.Using the A values obtained in Step along with its values corresponding to the final steady profile, and applying the upstream and downstream boundary stage hydrograph ordinates, the A values at the nodes are calculated using the Equation (), proceeding from upstream to downstream. Step. Corresponding Q values at the nodes are obtained by running the fuzzy rule based model. Step 5. The values of Q at each node at the end of the time step are obtained from the Q values obtained in Step, using the Equation (5) and the upstream boundary discharge hydrograph ordinates. Step. Thus values of Q and A at the end of the time step are obtained at all the nodes. They are used as the new initial conditions and the Steps () (5) are repeated for the next time step. This procedure is continued until the entire time period is covered. For the case of a channel with two uniform subreaches, the steps involved in the computations are as follows: Step. From the known initial conditions and using the discretized continuity Equation (3), the values of A at the end of the time step (t) are obtained at all the internal nodes of both the sub-reaches. The flow depth/stage at the junction node of the subreaches is obtained by interpolation between the flow depth/stage values of the last internal node of the upstream sub-reach and the first internal node of the downstream sub-reach. Step 2. For the considered inflow flood discharge during the present time step, the corresponding final steady water surface profile along the upstream sub-reach is obtained from the stage discharge relationships at the computational nodes of this sub-reach. The values of A, Q as well as Q at all nodes of this sub-reach, corresponding to the end of the time step (t), are obtained by adopting the same computational procedure as that described previously for the single-reach channel. Step 3. The value of Q at the junction of the subreaches is taken as the inflow flood discharge to the downstream sub-reach. Corresponding final steady profile for this sub-reach is obtained and DOI:.2/hyp

9 FDFRM FOR NATURAL CHANNELS A, Q as well as Q at all nodes of this subreach, corresponding to the end of the time step (t), are computed in the same way as that for the upstream sub-reach. Step. Thus values of Q and A at the end of the time step are obtained at all the nodes. They are used as the new initial conditions and the Steps () (3) are repeated for the next time step. This procedure is continued until the entire time period is covered. The above procedure of computation, for the channel with two sub-reaches, can be extended to a channel with any number of sub-reaches. MODEL APPLICATION To demonstrate the potential of the FDFRM, regarding its capability to produce outputs comparable to those provided by the INM, it is applied to flood routing problems Elevation (m) Station (m).7 Legend Max WS Ground Bank Station Discharge cumecs (c) Elevation (m) (c) Elevation (m) Station (m) Station (m).7.7 Legend Max WS Ground Bank Station Legend Max WS Ground Bank Station Figure 3. Cross sections within the heterogeneous sub-reach of the Critical Creek (Case 3), along with maximum water surface levels during HEC-RAS flood routing: cross section no. ; cross section no. Ð5; (c) cross section no Figure. Simulation results of flood routing in the Critical Creek, after optimizing the fuzzy rule based models using NEFCON: discharge hydrograph at cross section no. ; discharge hydrograph at cross section no. 7; (c) discharge hydrograph at cross section no. 2 in (i) a fictitious channel with a single uniform reach, (ii) a fictitious channel with two uniform sub-reaches and (iii) a natural channel with a number of approximately uniform sub-reaches. The finite difference INM described by Chow et al. (9), which makes use of the Preissmann scheme (Abbott, 979) for discretization, has been used in the present study for comparison of results. DOI:.2/hyp

10 R. GOPAKUMAR AND P. P. MUJUMDAR (c) Figure 5. Simulation results of flood routing in the Critical Creek, after optimizing the fuzzy rule based models using NEFCON: stage hydrograph at cross section no. ; stage hydrograph at cross section no. 7; (c) stage hydrograph at cross section no. 2 Case : Flood routing in a channel with a single uniform reach Flood routing in a -km-long trapezoidal channel with bed slope (S ) equal to Ð3 is considered for study. Bed width of the channel is 5Ð m, with side slope Ð5. Friction slope is predicted using Manning s equation for use in the INM, with roughness coefficient n equal to Ð25. The channel reach is uniform with respect to its cross section, bed slope and roughness. The fuzzy rule based model for the channel is developed in accordance with the procedure explained under the section model development. For this, the Q A relationship for the channel is obtained and is shown in Figure 3. The Q A relationship is derived from this plot and is shown in Figure 3. The membership functions for the antecedent (A) and the consequent (Q) are obtained based on the training sets derived from this Q A relationship. A total of 2 membership functions are derived for both A and Q, and they are shown in Figure. In this figure, L implies low values and H implies high values. The higher the coefficient of L and H, the higher will be their gradation towards low and high values, respectively. Also, as mentioned previously, development of the fuzzy rules is based on the concept that the wave velocity (V w ), obtained from Equation () and corresponding to each Q, remains approximately constant over the range of flow depths being considered. As such, these rules are derived in the form IF A is a THEN Q is a where a take values in ascending order from 9L to 2H (Figure ). Hence a total of 2 rules are included in the fuzzy rule base. Uniform flow exists initially with a base flow (Q b )of 5Ð m 3 /s at a flow depth Ð m. Flood routing has been done using both the INM and the FDFRM. The flood hydrograph applied at the upstream end is given by ( Q t D Q b C Ð2 Q b cos t ) t p for t t p ( Q t D Q b C Ð2 Q b cos t ) b t t b t p for t p <t t b 7 where t b is the time base, taken equal to 3 2 s (2 h), and t p is the time to peak taken equal to t b /2. For the INM, the following normal depth condition has been applied at the downstream boundary: Q end D n A end R 2/3 end S/2 where R is the hydraulic radius and the notations with the subscript end indicate the values of the respective flow parameters at the downstream end. The downstream end is extended for km in order to avoid effect of this Figure. Definition sketch for computation of discontinuities in the gradient of discharge (Q) and flow cross-sectional area (A), for the case of a positive wave moving downstream, in the presence of back water DOI:.2/hyp

11 FDFRM FOR NATURAL CHANNELS boundary condition on variation of the flow parameters within the study reach. Distance interval (x) of m and time interval (t) of s have been used in the simulation. Stage hydrograph outputs from the INM, at the upstream and downstream ends of the study reach, are shown in Figure 5. They are used as the boundary conditions for the FDFRM, along with the discharge hydrograph, as in Equation (7). Flood routing using the FDFRM has been carried out with the computational procedure explained under the Section on Method of Computation. Stage and discharge hydrographs obtained at sections with 2- and -km distance from the upstream end of the channel, by applying both the FDFRM and the INM, are shown in Figure. From the figure it can be seen that the s match very well with those of the INM. Case 2: Flood routing in a channel with two uniform sub-reaches In this case, the channel reach consists of two subreaches; each of them is uniform with respect to its cross section, bed slope and roughness. Lengths of the subreaches are km each. Geometric parameters as well as the fuzzy rule based model of the upstream sub-reach are same as those of the single-reach channel discussed under Case. Geometric parameters of the downstream sub-reach are same as those of the upstream sub-reach with the only difference that the bed width is increased to 5Ð5 m.theq A and Q A relationships for the downstream sub-reach are shown in Figure 2 and, respectively. Membership functions of the fuzzy rule based model for this sub-reach are derived from the training sets obtained from this Q A relationship. It has been found that A ranges from to 9Ð5 m 2, whereas Q ranges from to 2 m 3 /s. Natures of the membership functions and of the rules, of the fuzzy rule based model for this sub-reach, are similar to those of the fuzzy rule based model for the upstream sub-reach. Uniform flow exists initially with a base flow (Q b )of 5Ð m 3 /s and at a flow depth Ð m for the upstream sub-reach and Ð5 m for the downstream sub-reach. Flood routing is carried out using both the INM and the FDFRM. The same flood hydrograph (Equation (7)) is applied at the upstream boundary. For the INM, the same normal depth condition (Equation ()) is applied at the downstream boundary. Similar to the Case, in order to avoid the effect of this downstream boundary condition on the flow parameters of the study reach, the downstream end of the channel is extended for km. Also, for the INM, the flow depth and discharge at the junction node are taken to be equal for both the subreaches. The upstream and downstream boundary stage hydrographs required for the FDFRM are obtained from the outputs of the INM, and they are shown in Figure 7. The FDFRM computations are performed in accordance with the procedure explained under the section method of computation. Stage and discharge hydrographs obtained at sections with 2- and -km distance from the upstream end, by applying both the FDFRM and the INM, are shown in Figure. From the figure it can be seen that, for this case also, the s match well with those of the INM. Also from Figure (c) and (d), it can be concluded that the attenuation of the stage hydrograph, as the flood wave moves through the channel, is predicted properly by the FDFRM. Case 3: Flood routing in a natural channel Flood routing in the channel Critical Creek, as described in HEC-RAS applications guide (US Army Corps of Engineers, 22), has been taken up for study. The channel consists of only one heterogeneous reach named upper reach. Length of the channel is around 2 m with bed slope Ð. The Manning s roughness coefficient (n) for the main channel is Ð and for the flood plain, it is Ð. Thus the hydraulics involved in the flood movement within the main channel is different from that over the flood plain and they are taken as two separate channels in HEC-RAS. Therefore, theoretically speaking, the FDFRM should include separate fuzzy rule based models for the main channel and flood plain. Such a case is beyond the scope of present study, which aims at only developing the fundamentals of the FDFRM. The procedure of equivalent Manning s n for the entire cross section has not been tried because HEC-RAS adopts this method for the main channel only. The sole purpose of this application is to verify whether the FDFRM is capable of producing results comparable to those of HEC-RAS, for this natural channel, by taking heterogeneity in shape of its cross sections into account. Therefore, the entire cross section is considered as a single channel section with a uniform roughness coefficient of Ð7, which is the average of the roughness coefficients of the main channel and flood plain. Both HEC-RAS and FDFRM models are run under this condition. Also, expansion/contraction losses are considered negligible. Twelve cross sections have been taken along the channel reach, which are numbered serially from to 2, cross section no. 2 being the most upstream one. These cross sections are placed wherever there is significant change in shape of the cross section. Hence, the sub-reach between any two adjacent cross sections can be assumed to be approximately uniform. A mean cross section is interpolated in every sub-reach, between adjacent cross sections, and they are numbered in the routine way followed in HEC-RAS. For example, the mean cross section interpolated between the cross sections 2 and has been numbered as Ð5, the one interpolated between cross sections and has been numbered as Ð5 etc. For a field case, the stage area as well as discharge area relationships for all these cross sections have to be obtained from direct measurements. In the present study, these relationships are obtained from steady flow analysis using HEC-RAS. This analysis has been performed for a series of discharge values ranging from 2 to 3 m 3 /s. Flow in the channel is under sub-critical DOI:.2/hyp

12 R. GOPAKUMAR AND P. P. MUJUMDAR regime, and normal depth condition has been applied at the downstream boundary. For this case also, in order to avoid the effect of this downstream boundary condition on variation of flow parameters within the study reach, the downstream end of the channel has been extended for 3 km. The stage area and discharge area relationships for all the cross sections have been developed from the s corresponding to the steady flow analysis. As mentioned previously, the fuzzy rule based model for each sub-reach has been derived from the Q A relationship for the respective mean cross section. The range of values of A and Q of the fuzzy rule based model for each sub-reach is given in Table I. A total of 3 membership functions are derived for both A and Q and these two variables are related through 3 rules, in each fuzzy rule based model. For flood routing, the following discharge hydrograph, which is based on Log Pearson Type III distribution (Murty et al., 23), is applied at the upstream boundary Q t D Q b C Q p Q b e t t p / t g t p ( ) t tp / t g t p t p 9 where Q b is the initial base flow existing in the channel, taken equal to 2 m 3 /s, Q p the peak flow rate, taken as 3 m 3 /s, and t p the time to peak, taken equal to 2 s ( h). t g is the time to centroid of the hydrograph, which is obtained from the relation t p /t g D Ð95. The time interval used in the simulation is s, and the total simulation period is s (3 h). The flood routing is carried out using both the HEC-RAS and the FDFRM. Flood routing using the HEC-RAS is based on its unsteady flow analysis capability, which has been developed on the same principles as that of the INM. The same normal depth downstream boundary condition, as that used for the steady flow analysis, is applied in the unsteady flow analysis also. The HEC- RAS stage hydrograph outputs, at the upstream and Table I. Range of input variable (A) and output variable (Q) of the fuzzy rule based model for each sub-reach of the Critical Creek River (Case 3) Sub-reach Range of input variable (A) Minimum (m 2 ) Maximum (m 2 ) Range of output variable (Q) Minimum (m 3 /s) Maximum (m 3 /s) 2 2Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 3 downstream ends of the study reach, are shown in Figure 9. They are used as the boundary conditions for flood routing using the FDFRM, in addition to the upstream discharge hydrograph (Equation (9)). The s (discharge and stage hydrographs) for three typical cross sections are shown in Figures and, along with the corresponding s. It is observed that the s are comparable to those of the HEC-RAS, except for the cross sections, 7 and 2. This is due to the significant heterogeneity in cross sections of the sub-reaches, 7 and 3 2, and therefore the fuzzy rule based models, which were originally derived on the assumption that the subreaches are approximately uniform, are not able to provide satisfactory results. Thus there is a need for deriving optimum fuzzy rule based models for these subreaches, which will be capable of taking the heterogeneity into account. This is achieved by means of the neurofuzzy algorithm NEFCON (Nürnberger et al., 999), as described below. Derivation of optimum fuzzy rule based models using the neuro-fuzzy algorithm NEFCON Optimization of membership function parameters of fuzzy systems are usually done by trial and error, which is error prone and highly time consuming. Therefore, neurofuzzy algorithms are used for optimizing these parameters automatically. These algorithms combine fuzzy systems with learning techniques derived from neural networks. ANFIS (Matlab, 995) is a commonly used neuro-fuzzy algorithm, but it can deal with only Sugeno-type fuzzy systems and also requires huge input output data sets. NEFCON (Nürnberger et al., 999) is another neurofuzzy algorithm that can deal with Mamdani-type fuzzy system, and requires less data as compared to ANFIS. It works under MATLAB/SIMULINK environment and is adopted in the present study. In NEFCON, any fuzzy system is represented in a three-layer neural-network-like architecture. The nodes in the network use t-norms or t-conorms instead of the activation functions usually used in neural networks. The first layer represents input variables, the middle layer the fuzzy rules and the third layer the output variables. Fuzzy sets are encoded as connection weights; input fuzzy sets being connection weights between the first and the middle layers, and output fuzzy sets being connection weights between the middle and the third layers. The optimization of the connection weights (membership function parameters) are based on reinforcement learning and back propagation algorithm, and it is achieved by widening or reducing the support of the antecedents and shifting the consequents. Further details of NEFCON can be found in Nürnberger et al. (999). The NEFCON-SIMULINK model developed for the present study is shown in Figure 2. The various cross sections, within the sub-reach, are shown in Figure 3, along with their maximum water surface levels during the HEC-RAS flood routing. In order to optimize the fuzzy rule based model for the sub-reach DOI:.2/hyp

13 FDFRM FOR NATURAL CHANNELS, the fuzzy system developed at the mean section Ð5 is used within the NEFCON model. Then separate optimum fuzzy systems are derived for the sub-reaches Ð5 and Ð5. For deriving the optimum fuzzy system for the sub-reach Ð5, the (Q, A) pairs corresponding to the cross section are used, in a form similar to that of a look up table, in the Subsystem and Subsystem 2 blocks (Figure 2). Various A values corresponding to the cross section Ð5 are provided as the model input from the NEFCON SigGen block, in the form of a repeating signal. As the fuzzy system developed at the section Ð5 is used, the NEFCON Fuzzy block will output Q values corresponding to this section. These Q values are taken as the inputs to the Subsystem and Subsystem 2 blocks, and they will output A values corresponding to the cross section. The NEFCON algorithm optimizes the membership functions in such a way that the difference between these output A values and the corresponding A values originally input from the NEFCON SigGen is a minimum. The same procedure is adopted to derive the optimum fuzzy system for the other sub-reach, Ð5. It may be noted that the basic methodology of flood routing is not changing here; the only difference is that separate optimized fuzzy rule based models are used for each sub-reach Ð5 and Ð5. By adopting the above procedure, the optimum fuzzy rule based models for the other heterogeneous sub-reaches, i.e. 7 and 3 2, are also similarly derived. The flood routing is carried out using these optimized fuzzy rule based models, and the resulting discharge and stage hydrographs, at cross sections, 7 and 2, are shown in Figures and 5. These figures indicate that the FDFRM can produce results comparable to those of the HEC-RAS model for heterogeneous sub-reaches also, by making use of the optimized fuzzy rule based models. In order to quantify the improvement achieved by means of the optimized fuzzy rule based models, the root mean square error (RMSE) values of the s from the s are calculated, corresponding to the stage and discharge hydrographs at sections, 7 and 2, using the following formula: [ HEC RAS output ] 2 RMSE D No. of data points The RMSE values obtained are shown in Table II. These values indicate that significant improvements in the simulated outputs are achieved by making use of optimized fuzzy rule based models within the FDFRM. CONCLUSIONS Limitations of the conventional dynamic wave model, relating to its flood routing application in natural channels, can be overcome by means of the FDFRM presented in this paper. In the FDFRM, the momentum equation of Table II. Root mean square error values obtained from FDFRM and s of Critical Creek (Case 3) FDFRM and HEC-RAS outputs Discharge hydrographs at section Discharge hydrographs at section 7 Discharge hydrographs at section 2 Stage hydrographs at section Stage hydrographs at section 7 Stage hydrographs at section 2 Root mean square error Without optimized fuzzy rule based models With optimized fuzzy rule based models Ð32 m 3 /s Ð2337 m 3 /s Ð59 m 3 /s Ð29 m 3 /s Ð79 m 3 /s Ð359 m 3 /s Ð5 m Ð5 m Ð2 m Ð29 m Ð5 m Ð37 m the dynamic wave model is replaced by a fuzzy rule based model while retaining the continuity equation in its complete form. By replacing the momentum equation by the robust fuzzy rule based model, the associated assumptions are eliminated, thus making the FDFRM applicable to any general channel. The fuzzy rule based model is constructed on physical characteristics of the channel such as its geometric characteristics (represented by stage area relationships) and flow characteristics (represented by stage discharge relationships). It is developed on an equation for wave velocity, which does not directly depend on the various forces that appear in the momentum equation. As there is no need of estimating the friction slope, the model gets rid of the involved uncertainties. The FDFRM is based on the concept that the flood wave in a natural channel can be approximated to a monoclinal wave (Chow, 959; Henderson, 9). The channel is discretized into a number of approximately uniform sub-reaches and the monoclinal wave approximation is applied to each individual sub-reach separately. The fuzzy rule based model for each sub-reach is developed form training sets (Bardossy and Disse, 993), which are obtained from discharge area relationship at its mean cross section. In case of highly heterogeneous sub-reaches, optimized fuzzy rule based models have been derived using a neuro-fuzzy algorithm. A MAT- LAB/SIMULINK model, based on the neuro-fuzzy algorithm NEFCON (Nürnberger et al., 999), is developed for this purpose. For demonstration, the FDFRM is applied to flood routing in a fictitious channel with a single uniform reach, in a fictitious channel with two uniform sub-reaches and also in a natural channel with a number of approximately uniform sub-reaches, which is the Critical Creek River described in HEC-RAS applications guide (US Army Corps of Engineers, 22). It is observed that in case of the fictitious channels, the s match very well with those of an INM, which solves the dynamic DOI:.2/hyp

14 R. GOPAKUMAR AND P. P. MUJUMDAR wave equations using an implicit numerical scheme. For the natural channel, the s are comparable to those of the HEC-RAS. From these results, it is concluded that the FDFRM can be used effectively for flood routing in natural channels. The main limitations of this model are the time constraint as well as the diffusion error introduced by the numerical scheme (Lax diffusive scheme) used for discretizing the continuity equation (Chaudhry, 97). APPENDIX: EQUATION FOR WAVE VELOCITY (V W ) The equation for wave velocity is derived on the assumption that the disturbances form shallow water waves of small amplitude (Cunge et al., 9). Figure shows the case of a positive wave moving in the downstream direction, in the presence of back water. The initial discharge is Q and corresponding steady water surface profile is a n p. The discharge at the upstream end is then increased to Q 2, which results in the formation of the positive wave mentioned above. m n is the position of the wave front at any time t. The wave velocity at m, at time t, isv w. o p is the position of the wave front at time t C t, after it moves through a small distance x. Here the control volume approach (Chow et al., 9; Chaudhry, 993) is used to derive the expression for V w. In this approach, t! (Chow et al., 9), so that x is infinitesimally small and therefore the wave shape remain intact during the above motion. b m o is the final steady water surface profile corresponding to the discharge Q 2. Discontinuity in gradient of the flow parameter (Q or A) at point m and at time t can be determined as the difference between its gradient corresponding to the final steady condition, which the flow will attain under the effect of the disturbance and its gradient corresponding to the condition at time t, both gradients evaluated at point m. Accordingly, the discontinuity in the gradients of flow cross-sectional area (A) and discharge (Q) are computed as follows: Discontinuity in gradient of A at m, at time t [ ] [ ] A A D D A x x m o m n Discontinuity in gradient of Q at m, at time t [ ] [ ] Q Q D D Q 2 Q x m o x m n D Q 2 where A is the area between the final and initial steady profiles at m, and is the length of the wave m n, which is measured in the x direction. As per the Reynolds transport theorem (Chow et al., 9), the continuity equation can be written, with reference to the control volume shown in Figure, as d dt c.v. dc V Ð da D c.s. 3 where is water density, c.v indicates control volume and c.s the control surface. The distance (x) moved by the wave front, in time t, can be written in terms of V w as x D V w ð t. Also in Equation (3), d is the change in volume of water within the control volume during time t. d may be written as, d Dx ð A D V w ð t ð A. Hence Equation (3) can be written in discrete form, for the control volume, as t [V w ð t ð A] C Q Q 2 D But Q 2 Q D Q Therefore, V w D Q 5 A Equations (), (2) and (5) imply that the dynamic wave velocity at any point (m) at a certain time (t) canbe determined as the ratio of discontinuity in the gradient of discharge and discontinuity in the gradient of area, both computed at that point and at the specified time. Hence the general expression for dynamic wave velocity can be written as V w t m D Discontinuity in gradient of Q t m Discontinuity in gradient of A t m D Q t m A t m In order to demonstrate the accuracy of the derived Equation (), and its associated discretized Equations Table III. Dynamic wave velocity (V w ) computed using different methods Location (distance from upstream end in kilometres) Time of arrival of wave front (h) Dynamic wave velocity (m/s) Method (dynamic wave routing) Method 2 (analytical method) Method 3 (Equations (), () and (5)).5 Ð27 Ð99 Ð339 Ð3. Ð39 Ð37 Ð39 Ð25.5 Ð972 Ð2 Ð35 Ð7 2. Ð333 Ð77 Ð35 Ð5 2.5 Ð722 Ð327 Ð3 Ð35 DOI:.2/hyp