HW-Chapter 9
Naoki Mizukami
9-1. Detention basin with 700ft x 700ft with 4:1 side slope. Top of the berm is at elevation 9 ft.
Inflow hydrograph is a triangular shape with a peak discharge of 125 cfs at a time of 8 hr and
with a base time of 24 hr. The detention basin is initially empty. Route the inflow hydrograph
through the detention basin and determine the peak outflow and stage.
Stage-storage relationship is computed as follows
z
z
0
0
V ( z )   A( z )dz   700  4 z  2dz 
8 z  700 3  700 3
24
Stage-outflow relationship is given and summarized along with storage in the table below.
Table. Stage-storage-outflow relationships based on given outlet structures and detention basin geometry
Z[ft]
O[cfs]
0
1
2
3
4
5
6
7
8
9
S[cf]
0
2.1
4.6
6
7.2
8.2
9.1
9.9
36.9
85.5
0
495621
1002571
1520976
2050965
2592667
3146208
3711717
4289323
4879152
2S/dt+O [cfs]
0.0
277.4
561.6
851.0
1146.6
1448.6
1757.0
2072.0
2419.9
2796.1
Table. reservoir routing using storage-indication method
T
hr
I
cfs
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0
15.6
31.3
46.9
62.5
78.1
93.8
109.4
125
117.2
109.4
101.6
93.8
85.9
78.1
70.3
62.5
54.7
46.9
39.1
31.3
23.4
15.6
7.8
0
2S/dt-O
cfs
0
15.4
61.3
137.4
243.1
377.6
540.5
732.8
954.2
1181.7
1392.1
1585.8
1762.9
1923.5
2065.3
2171.9
2248.8
2300.5
2331.1
2343.7
2341.2
2325.8
2299.6
2264.2
2221.2
2S/dt+O
cfs
0
15.6
62.3
139.5
246.8
383.7
549.5
743.7
967.2
1196.4
1408.3
1603.1
1781.2
1942.6
2087.5
2213.7
2304.7
2366
2402.1
2417.1
2414.1
2395.9
2364.8
2323
2272
O
cfs
0
0.1
0.5
1.1
1.9
3
4.5
5.5
6.5
7.4
8.1
8.7
9.2
9.6
11.1
20.9
28
32.7
35.5
36.7
36.5
35
32.6
29.4
25.4
S
cf
0.00E+00
2.79E+04
1.11E+05
2.49E+05
4.41E+05
6.85E+05
9.81E+05
1.33E+06
1.73E+06
2.14E+06
2.52E+06
2.87E+06
3.19E+06
3.48E+06
3.74E+06
3.95E+06
4.10E+06
4.20E+06
4.26E+06
4.28E+06
4.28E+06
4.25E+06
4.20E+06
4.13E+06
4.04E+06
Z [ft]
ft
0
0.1
0.2
0.5
0.9
1.4
2
2.6
3.4
4.2
4.9
5.5
6.1
6.6
7
7.4
7.7
7.8
7.9
8
8
7.9
7.8
7.7
7.6
HW-Chapter 9
Naoki Mizukami
The peak outflow is 36.7 cfs at 19 hr. The maximum storage is 8 ft at 19 hr.
140
120
Q [cfs]
100
80
60
40
20
0
0
5
10
15
20
25
20
25
Time [hr]
Figure. Inflow and outflow hydrograph
10
9
8
7
Z [ft]
6
5
4
3
2
1
0
0
5
10
15
Time [hr]
Figure. Temporal change in stagee, Z
HW-Chapter 9
Naoki Mizukami
9-3. Rout the inflow hydrograph given using Muskingum method with Δt = 2 hr, Δx = 25000 ft,
θ = 3.6 hr, X = 0.2. Plot the inflow and outflow hydrographs and determine the percent reduction
in the inflow peak as well as the travel time of the peak
Paramters
3.6 hr
C0
0.2
C1
2 hr
C2
25000 ft
theta=
X=
dt=
dx=
t
hr
I
cfs
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
0.072
0.443
0.485
C0xI2
C1xI1
C2xO1
O2
cfs
cfs
cfs
cfs
100 100.0
500
36.1
44.3
48.5
128.9
1500
108.2
221.6
62.4
392.3
2500
180.4
664.9
190.1
1035.5
5000
360.8
1108.2
501.7
1970.8
11000
793.8
2216.5
954.9
3965.2
22000
1587.6
4876.3
1921.3
8385.2
28000
2020.6
9752.6
4062.9
15836.1
28500
2056.7
12412.4
7673.2
22142.3
26000
1876.3
12634.0
10728.7
25239.0
22000
1587.6
11525.8
12229.2
25342.6
17500
1262.9
9752.6
12279.4
23294.9
14000
1010.3
7757.7
11287.2
20055.3
10000
721.6
6206.2
9717.5
16645.3
7000
505.2
4433.0
8065.3
13003.4
4500
324.7
3103.1
6300.6
9728.5
2500
180.4
1994.8
4713.8
6889.0
1500
108.2
1108.2
3338.0
4554.5
1000
72.2
664.9
2206.8
2943.9
500
36.1
443.3
1426.4
1905.8
100
7.2
221.6
923.4
1152.3
Peak time Peak Q
hr
cfs
Inflow
16
28500
Outflow
20
25342.6
Peak reduction
11.1%
travel time of the peak
4.0
HW-Chapter 9
Naoki Mizukami
30000
25000
Q, cfs
20000
Inflow
15000
Outflow
10000
5000
0
0
5
10
15
20
25
30
35
Time, hr
Figure. Inflow and outflow hydrograph
40
45
HW-Chapter 9
Naoki Mizukami
Q, cfs
9-16. Using Muskingum-Cunge method to route the inflow hydrograph and river in example 9.6
for S0 = 0.001.
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0
2
Time, hr
4
6
8
Figure. Inflow hydrograph
Average Q to compute normal depth, y0.
Q
500 * 7  (4500  500 ) * 6 / 2
1T
 2500 cfs
 Q(t )dt 
T 0
7
Using average Q, find average velocity, flow area with manning’s equation
5
3
1
1
Kn A
Kn  y 0  b  m  y 0  3
2 
2
Q

S


S
0
0
2
n 23
n
P
b  2 y0 1  m 2 3
5


 y 0  100  2 y 0 3
5
1.49
2500 

0.025
100  2 y
0
1 2

2
2 3
 0.001
1
2
Solving for y0, y0 = 4.65 ft,
2500
= 4.91 ft/s
4.65  (100  2  4.65)
So wave celerity is calculated
5
5
c k  V   4.91  8.19
3
3
V  Q/A 
Δt = 0.5 hr is chosen so that 5 discretized points are in rising part of inflow hydrograph
1
2
x   ck t 
Q
BS 0 ck
 1
2500

   8.19  0.5  60  60 
  8659 ft
118.6  0.001  8.19 
 2
So select Δx = 6000 ft
Compute Courant number, Cn as
t
0.5  60  60
Cn  ck
 8.19 
 2.46
x
6000
HW-Chapter 9
Naoki Mizukami
One more parameter, Muskingum weighting parameter, X is computed as
 1
Q
1
2500

  1 
X  1 
  0.29
2
BS 0 ck x  2  118.6  0.001  8.19  6000 
Using parameters, X and Cn, M-C coefficients are computed as
0.5C n  X
0.5  2.46  0.29

 0.488
1  X  0.5C n 1  0.29  0.5  2.46
0.5C n  X
0.5  2.46  0.29
C1 

 0.785
1  X  0.5C n 1  0.29  0.5  2.46
1  0.5C n  X 1  0.5  2.46  0.29
C2 

 0.274
1  X  0.5C n 1  0.29  0.5  2.46
C0 
Outflow is computed using the Muskingum routing equations written by
Qi j 11  C 0  Qi j 1  C1  Qi j  C 2  Qi j 1
where the Q i j refers to the flow rate at space j and time i. The result of routing is as follows
Table. Muskingum-Cunge routing with constant parameter.
t
hr
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
x= 6000
I
C0xI2 C1xI1 C2xO1
cfs
cfs
cfs
cfs
500
1500
732.7
392.6 -136.8
2500 1221.2 1177.7 -270.5
3500 1709.7 1962.9 -582.5
4500 2198.2 2748.1 -845.6
4000 1954.0 3533.2 -1122.2
3500 1709.7 3140.7 -1194.5
3000 1465.5 2748.1 -1000.5
2500 1221.2 2355.5 -879.3
2000
977.0 1962.9 -738.2
1500
732.7 1570.3 -602.5
1000
488.5 1177.7 -465.4
500
244.2
785.2 -328.6
500
244.2
392.6 -191.8
500
244.2
392.6 -121.8
500
244.2
392.6 -140.9
x= 1200
O
C0xI2 C1xI1 C2xO1
cfs
cfs
cfs
cfs
500
988.5
482.9
392.6 -136.8
2128.5 1039.7
776.1 -202.1
3090.2 1509.5 1671.2 -441.6
4100.6 2003.1 2426.3 -749.6
4365.0 2132.3 3219.7 -1007.0
3655.9 1785.9 3427.3 -1189.0
3213.1 1569.6 2870.5 -1101.2
2697.4 1317.7 2522.8 -913.7
2201.7 1075.5 2117.9 -800.9
1700.6
830.7 1728.7 -654.7
1200.9
586.6 1335.2 -521.2
700.8
342.3
942.9 -383.3
445.1
217.4
550.2 -246.8
515.0
251.6
349.4 -142.5
495.9
242.2
404.4 -125.5
x=1800
O
C0xI2 C1xI1 C2xO1
cfs
cfs
cfs
cfs
500
738.6
360.8
392.6 -136.8
1613.7
788.3
579.9 -168.7
2739.1 1338.0 1267.1 -328.3
3679.8 1797.6 2150.7 -623.1
4345.0 2122.5 2889.3 -910.0
4024.1 1965.7 3411.5 -1122.5
3338.8 1631.0 3159.6 -1164.3
2926.8 1429.7 2621.5 -992.3
2392.5 1168.7 2298.0 -837.1
1904.7
930.4 1878.5 -719.6
1400.6
684.2 1495.5 -571.8
901.9
440.6 1099.7 -440.0
520.8
254.4
708.2 -301.1
458.5
224.0
408.9 -181.0
521.2
254.6
360.0 -123.7
O
cfs
500
616.6
1199.5
2276.8
3325.2
4101.8
4254.8
3626.2
3058.9
2629.7
2089.3
1607.9
1100.3
661.5
451.9
490.9
HW-Chapter 9
Naoki Mizukami
5000
4500
4000
Q, cfs
3500
Inflow
x=6000ft
x=1200ft
x=1800ft
3000
2500
2000
1500
1000
500
0
0
2
4
6
8
Time, hr
Figure. Inflow and outflow hydrographs from Muskingum-Cunge routing.
HW-Chapter 9
Naoki Mizukami
9-18. Write the code for Muskingum-Cunge routing using the 4 point variable parameter method
and apply it to Example 9.6.
Matlab Code
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Muskingum-Cunge with variable coefficient method
%Coefficients X, and ck (wave celerity) are dynamic in time and space due to
%varying Q, resulting in variable C0, C1, and C2.
%
% brief description of 4 point variable parameter method
%the 4 point method computes an average Q using inflow and outflow at the first step and next step and
%use the average Q to estimate M-C parameters. This computation is performed every time outflow at the
%next time step is computed. Since outflow at the next time step is unknown, three point Q are used for
%first guess for average Q. Iteration is necessary to make average Q converged at a certain error
%tolerance.
%
%
clear all
close all
%Reading inflow hydrograph
data1 = load('inhydro18.dat');
t = data1(:,1) % time [hr]
Q(:,1) = data1(:,2) %inflow [cfs]
dt = (t(2)-t(1))*60*60;% time interval [hr]
%River geometry
L = 18000; %river reach length [ft]
S0 = 0.0005; %bed slope [ft/ft]
b = 100; %bottom width [ft]
m = 2; %side slope
n = 0.025; %manning n
%some constant
Kh = 1.49; %manning equation
%define routing reach length, dx [ft]
dx = 9000;
%compute distance x where hydrographs are computed
x(1) = 0; %begining of river reach
for i = 2:L/dx+1
x(i) = x(i-1)+dx;
end
%initial condition at x
for j = 1:length(x)
Q(1,j) = Q(1,1);
end
%compute Muskingum-Cunge routing
%go thru all time steps and spatial step
HW-Chapter 9
Naoki Mizukami
for i = 2:length(t)
for j = 2:length(x)
%initial guess for Q using three points (inflow at 1st time step and 2nd time step
%and outflow at 1st time step.
Qave = (Q(i-1,j-1)+Q(i,j-1)+Q(i-1,j))/3;
%iteration till 4 point average Q are converged at 0.1 of error tolerance
while 1
%Define manning equation
func = @(y) Kh/n*(y*(b+2*y))^(5/3)/(b+2*y*sqrt(1+m^2))^(2/3)*sqrt(S0)-Qave;
%compute depth, y0, based on manning equation. use bisect method with 0.01 error tolerance
y0 = fbisect(func,0,10,0.01);
%compute flow area, A, velocity, V, and wave celerity, ck, top width, TOP, given depth y0
A = y0*(b+2*y0);
V= Qave/A;
ck = 5/3*V;
TOP = b+2*m*y0;
%Compute coefficients for Muskingum-Cunge equation
X = 0.5*(1-Qave/TOP/S0/ck/dx);
Cn = ck*dt/dx;
C0=(0.5*Cn-X)/(1-X+0.5*Cn);
C1=(0.5*Cn+X)/(1-X+0.5*Cn);
C2=(1-0.5*Cn-X)/(1-X+0.5*Cn);
%Compute outflow at next time step
Q(i,j)=C0*Q(i,j-1)+C1*Q(i-1,j-1)+C2*Q(i-1,j);
%Take average of 4 points
Qave1=(Q(i-1,j-1)+Q(i,j-1)+Q(i-1,j)+Q(i,j))/4;
%If difference between current 4 point average and previous one is less than 0.1
%leave iteration loop, otherwise use new average to compute new estimate of
%outflow at next time step
if abs(Qave-Qave1)< 0.1; break; end
Qave = Qave1;
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of code
HW-Chapter 9
Naoki Mizukami
Results (applied this code to example 9.6)
Table. Muskingum-Cunge routing (inflow and routed hydrograph at x =9000 and x = 18000)
T [hr]
x=
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
Q [cfs]
0
x= 9000
x=18000
500.0
500.0
500.0
1500.0
596.3
501.5
2500.0
1587.4
692.1
3500.0
2724.1
1749.2
4500.0
3823.2
2998.9
4000.0
4220.2
3898.0
3500.0
3816.4
4018.9
3000.0
3338.8
3643.4
2500.0
2860.2
3190.6
2000.0
2379.8
2728.1
1500.0
1913.9
2278.8
1000.0
1454.8
1839.0
500.0
1016.2
1419.9
500.0
626.7
1036.5
500.0
533.4
715.2
500.0
509.0
579.0
5000
infow
x=9000 ft
x=18000 ft
4500
4000
3500
Q [cfs]
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
Time [hr]
5
Figure. Inflow and outflow hydrographs
6
7
8