EFFECT OF SPATIAL AND TEMPORAL DISCRETIZATIONS ON THE SIMULATIONS USING CONSTANT-PARAMETER AND VARIABLE-PARAMETER MUSKINGUM METHODS

Transcription

1 INDIAN INSTITUTE OF TECHNOLOGY ROORKEE EFFECT OF SPATIAL AND TEMPORAL DISCRETIZATIONS ON THE SIMULATIONS USING CONSTANT-PARAMETER AND VARIABLE-PARAMETER MUSKINGUM METHODS Muthiah Perumal and C. Madhusudana Rao Professor, Department of Hydrology, Indian Institute of Technology Roorkee Assist. Professor, Department of Civil Engineering, National Institute of Technology Jamshedpur

2 Introduction Flood wave movement is a nonlinear process Many models used for catchment runoff simulations still employ linear theory based models for runoff simulations FLOOD ROUTING METHODS CONSTANT-PARAMETER MUSKINGUM METHOD (MRM) (as used in the SWAT model based on the Muskingum-Cunge (Cunge1969) approach) VARIABLE-PARAMETER MUSKINGUM METHOD ()

3 Objective is: In focus The focus of this study is concerned only with one of the component processes, namely, flood routing in main channels using the Muskingum method. To explore the impact of spatial and temporal discretizations on the routing simulations of the constant-parameter and the variable-parameter Muskingum methods 3

4 CONSTANT-PARAMETER MUSKINGUM METHOD Frame work of Muskingum Routing Method (MRM) (as used in the SWAT model based on the Muskingum-Cunge (Cunge1969) approach) Routing Equation Q C1I C I1 C 3Q1 Coefficients Parameters where K C k 1 L k C Q A k 5 3 V respectively are the channel length (km) and celerity of the flood wave (m/s) is the channel flow velocity (m/s). c C C C 1 3 K t K 1 t K t K 1 t K 1 t K 1 t is weighing factor (-.5); To avoid the numerical instability and negative outflow computation, the following condition is recommended in the SWAT model K t K 1 4

5 The constant parameter-muskingum method by estimating its two constant parameters using a reference discharge. Q ref = (Q pave + Q b )/ Q pave = (Q pinf + qpout)/ Y ref (from Newtorn Raphson method) t If < k, the Muskingum coefficient C o = - Ve Generally negative values of coefficients are avoided For best results, the t should be so chosen that k > t >k 5

6 VARIABLE-PARAMETER MUSKINGUM METHOD () Frame Work of the Method The Variable Parameter McCarthy-Muskingum () method proposed by Perumal and Price [13] directly derived from the Saint-Venant Equations. for routing flood waves in semi-infinite rigid bed prismatic channels having any cross-sectional shape/follows either Mannings friction law. during steady flow depth relationship. in the channel reach there exist discharge and during unsteady flow in the channel reach, the discharge observed at any section has its corresponding normal depth at upstream section. Allows the Simultaneous computations of stage hydrograph corresponding to a given inflow/routed hydrograph. 6

7 Frame Work of Method Saint-Venant equations Continuity equation Q x A t (1) S f y v v 1 v S x g x g t Momentum equation () S f = friction slope; S o = bed slope; y/x = water surface slope; (v/g)(v/x)= convective acceleration; (1/g)(v/t)=local acceleration. Magnitudes of various terms in eqn. () are usually small in comparison with S o [Henderson, 1966; NERC, 1975]. 7

8 Parameter estimation using the method 1 y u I M 3 Q M Q 3 j+1 t t t/ M 3 Δx/ y M y 3 Δx L y d O j 1 x/ x L x Fig. 1 Definition Sketch of the Routing Reach Fig. Numerical grid adopted for application in synchronization with Fig.1 8

9 Parameter estimation using the method Muskingum Routing Equation Q C Q C Q C Q d, j 1 t 1 u, j1 t u, jt 3 d, jt Where Q The estimated downstream d, j 1 t discharges at time j 1 t Q Q u, j 1 t Q t u, j t d, j t The observed upstream discharges at time The observed upstream discharges at time The estimated downstream discharges at time The routing time interval j 1 jt jt t C C 1 C 3 Coefficients t. K. t. K. 1, j1 t, j1 t, j 1 t, j1 t t. K. t. K. 1, jt, jt, j 1 t, j1 t t. K. 1 t. K. 1, jt, jt, j 1 t, j1 t 9

10 Parameter estimation using the method K Routing parameters, j 1, j1 t t V x M o, j 1 t, j1 t 4 PdR dy Q3, 1 1 F j t 9 da dy 1. S. B. c. x o M, j 1 t Mo, j1 t 1 Q3, j1 t. S. B. c. x M, j1 t o M, j 1 t Mo, j1 t ( By neglecting inertial terms) The governing finite difference equation of the routing method dq P dr c 1 v da 3 B dy Q3, j 1 t Q I, 1 I 3,, j t O, 1 O j t j t j t, jt x t V, 1 VMo, j t Mo j t 1

11 Numerical Experiments Twelve numerical test runs have been conducted in this study by changing space ( x) and routing time ( t) intervals to explore the effect of spatial and temporal discretizations on the routing simulations of the constant-parameter and the variableparameter Muskingum methods. In this study, a uniform prismatic trapezoidal channel with a bed slope, S o =., Manning s roughness coefficient, n =.4 and bed width of 5m contained by dykes with a slope ratio (1 V : 5 Horizontal (z)) has been used. Test Run No. Space step (km) Time step (h) 1 4 (Single Reach) 1. 4 (Single Reach) (Single Reach) (Single Reach) (1 Sub-reaches) 1. y 6 4 (1 Sub-reaches) 3. b 7 4 (1 Sub-reaches) 6. Trapezoidal channel 8 4 (1 Sub-reaches) (4 Sub-reaches) (4 Sub-reaches) (4 Sub-reaches) (4 Sub-reaches) z

12 Inflow Hydrograph and Benchmark solution details Pearson type-iii distribution expressed as 1 1 t 1 t t p Q( t) Qb ( Q p Qb ) exp t p 1 The benchmark solutions were obtained by model (USACE, 1) by routing the inflow hydrograph for a reach length of 4km in prismatic Trapezoidal channel using the Space step = 1m and Time step = 3sec **Flow characteristics Q b = 1 m 3 /sec Q p = 3 m 3 /sec T p = 19 h (8 days), Shape factor 1.5 1

13 Results and Discussions Table 1 Summary of performance criteria showing reproduction of pertinent characteristics of the results by the method for routing in Trapezoidal channel reaches using spatial and temporal discretization on the simulations. Test Run Space Time µ q µ y Discharge Routing Stage Computation No. step (km) step (h) (%) (%) EVOL (%) η q (%) q per (%) t pqer (h) η y (%) y per (%) t pyer (h)

14 3 16 Discharge (m 3 /sec) Space step = 4 km Time step = 1. hour Inflow Stage (m) Space step = 4 km Time step =1. hour Input stage Time (h) Figure 1. Typical simulated discharge hydrograph of the method for a space step of 4km and a time step of 1.hour in a Time (h) Figure. Typical computed stage hydrographs of the method for a space step of 4km and a time step of 1.hour in a Discharge (m 3 /sec) Space step = 4 km Time step = 4 hour Inflow Time (Days) Figure 3. Typical simulated discharge hydrograph of the method for a space step of 4km and a time step of 4.hour in a Stage (m) Space step = 4 km Time step = 4 hour Input stage Time (Days) Figure 4. Typical computed stage hydrographs of the method for a space step of 4km and a time step of 4.hour in a 14

15 3 16 Discharge (m 3 /sec) Space step = 4 km Time step = 1. hour Inflow Stage (m) Space step = 4 km Time step =1. hour Input stage Time (h) Figure 5. Typical simulated discharge hydrograph of the method for a space step of 4km and a time step of 1.hour in a Time (h) Figure 6. Typical computed stage hydrographs of the method for a space step of 4km and a time step of 1.hour in a Discharge (m 3 /sec) Space step = 4 km Time step = 4. hour Inflow Stage (m) Space step = 4 km Time step = 4. hour Input stage Time (Days) Time (Days) Figure 7. Typical simulated discharge hydrograph of the method for a space step of 4km and a time step of 4.hour in a Figure 8. Typical computed stage hydrographs of the method for a space step of 4km and a time step of 4.hour in a 15

16 3 16 Discharge (m 3 /sec) Space step = 1 km Time step = 1. hour Inflow Stage (m) Space step = 1 km Time step =1. hour Input stage Time (h) Figure 9. Typical simulated discharge hydrograph of the method for a space step of 1km and a time step of 1.hour in a Time (h) Figure 1. Typical computed stage hydrographs of the method for a space step of 1km and a time step of 1.hour in a 16 Discharge (m 3 /sec) Space step = 1 km Time step = 4. hours Inflow Stage (m) Space step = 1 km Time step = 4. hours Input stage Time (Days) Figure 11. Typical simulated discharge hydrograph of the method for a space step of 1km and a time step of 4.hour in a Time (Days) Figure 1. Typical computed stage hydrographs of the method for a space step of 1km and a time step of 4.hour in a 16

17 Discharge (m 3 /sec) Space step = 4km Time step = 4.hour Inflow Outflow Observed () Time (Days) Figure 13. Typical computed discharge hydrographs of the constant-parameter method for a space step of 4km and a time step of 4.hour in a trapezoidal channel using reference discharge

18 Conclusions A preliminary investigation carried out with the considered objective shows that the routing solution obtained using a longer routing time interval induces significant numerical diffusion of the routed hydrograph leading to over attenuation of the inflow flood peak and, thereby, resulting in poor reproduction of the benchmark solution. 18

19 19