CE322 Basic Hydrology
Jorge A. Ramirez
Muskingum Routing - Example
The inflow and outflow hydrographs of a river reach are tabulated below.
|
Time (h) |
Inflow (m3/s) |
Outflow (m3/s) |
|
1 |
93 |
85 |
|
2 |
137 |
91 |
|
3 |
208 |
114 |
|
4 |
320 |
159 |
|
5 |
442 |
233 |
|
6 |
546 |
324 |
|
7 |
630 |
420 |
|
8 |
678 |
509 |
|
9 |
691 |
578 |
|
10 |
675 |
623 |
|
11 |
634 |
642 |
|
12 |
571 |
635 |
|
13 |
477 |
603 |
|
14 |
390 |
546 |
|
15 |
329 |
479 |
|
16 |
247 |
413 |
|
17 |
184 |
341 |
|
18 |
134 |
274 |
|
19 |
108 |
215 |
|
20 |
90 |
170 |
A. Parameter Estimation
Use these observations to obtain the Muskingum routing parameters k and x for this river reach. The initial storage in the system is 715,000 m3.
Graphical Procedure:
The graphical procedure consists in generating graphs of [xI + (1-x)O] vs. S for different values of x, arbitrarily selected such that 0 < x < 0.5. The optimal value of x is selected as that which produces the narrowest and straightest loop graph of [xI + (1-x)O] vs. S. The slope of the least squares linear fit to the resulting points is the estimate of k.
a) Generate accumulated storage in the system. Use continuity equation as follows:
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
x=0.25 |
x=0.35 |
x=0.1 |
x=0.15 |
|||||
|
Inflow, I (m3/s) |
Outflow,O (m3/s) |
Ave. Inflow (m3/s) |
Ave. Outflow (m3/s) |
Storage (m3) |
Weighted Average Flux xI + (1-x)O (m3/s) |
|||
|
93 |
85 |
715000 |
87 |
87.8 |
85.8 |
86.21204 |
||
|
137 |
91 |
115 |
88 |
812200 |
102.5 |
107.1 |
95.6 |
97.96922 |
|
208 |
114 |
172.5 |
102.5 |
1064200 |
137.5 |
146.9 |
123.4 |
128.2414 |
|
320 |
159 |
264 |
136.5 |
1523200 |
199.25 |
215.35 |
175.1 |
183.3923 |
|
442 |
233 |
381 |
196 |
2189200 |
285.25 |
306.15 |
253.9 |
264.6645 |
|
546 |
324 |
494 |
278.5 |
2965000 |
379.5 |
401.7 |
346.2 |
357.634 |
|
630 |
420 |
588 |
372 |
3742600 |
472.5 |
493.5 |
441 |
451.816 |
|
678 |
509 |
654 |
464.5 |
4424800 |
551.25 |
568.15 |
525.9 |
534.6043 |
|
691 |
578 |
684.5 |
543.5 |
4932400 |
606.25 |
617.55 |
589.3 |
595.1201 |
|
675 |
623 |
683 |
600.5 |
5229400 |
636 |
641.2 |
628.2 |
630.8782 |
|
634 |
642 |
654.5 |
632.5 |
5308600 |
640 |
639.2 |
641.2 |
640.788 |
|
571 |
635 |
602.5 |
638.5 |
5179000 |
619 |
612.6 |
628.6 |
625.3037 |
|
477 |
603 |
524 |
619 |
4837000 |
571.5 |
558.9 |
590.4 |
583.9104 |
|
390 |
546 |
433.5 |
574.5 |
4329400 |
507 |
491.4 |
530.4 |
522.3653 |
|
329 |
479 |
359.5 |
512.5 |
3778600 |
441.5 |
426.5 |
464 |
456.2743 |
|
247 |
413 |
288 |
446 |
3209800 |
371.5 |
354.9 |
396.4 |
387.8502 |
|
184 |
341 |
215.5 |
377 |
2628400 |
301.75 |
286.05 |
325.3 |
317.2138 |
|
134 |
274 |
159 |
307.5 |
2093800 |
239 |
225 |
260 |
252.7894 |
|
108 |
215 |
121 |
244.5 |
1649200 |
188.25 |
177.55 |
204.3 |
198.789 |
|
90 |
170 |
99 |
192.5 |
1312600 |
150 |
142 |
162 |
157.8796 |
Columns 3 & 4 are the average inflow flux (Ii+1 + Ii)/2 and outflow flux (Oi+1 + Oi)/2, respectively.
Column 5 is the cumulative storage in the system obtained using the continuity equation below.
Columns 6 - 9 are the values of the weighted average flux [xI + (1-x)O] for different values of x. The graph of Columns 6 - 9 vs. Column 5 is shown below.
Based on these results, a value of x = 0.15 is selected. The best least squares fit to the corresponding points yields a value of k = 2.3 h.
Least Squares Procedure
|
Inflow (m3/s) |
Outflow (m3/s) |
Storage |
O2 (m3/s)2 |
I2 (m3/s)2 |
OI (m3/s)2 |
SO (m6/s) |
SI (m6/s) |
|
93 |
85 |
715000 |
7225 |
8649 |
7905 |
60775000 |
66495000 |
|
137 |
91 |
812200 |
8281 |
18769 |
12467 |
73910200 |
111271400 |
|
208 |
114 |
1064200 |
12996 |
43264 |
23712 |
121318800 |
221353600 |
|
320 |
159 |
1523200 |
25281 |
102400 |
50880 |
242188800 |
487424000 |
|
442 |
233 |
2189200 |
54289 |
195364 |
102986 |
510083600 |
967626400 |
|
546 |
324 |
2965000 |
104976 |
298116 |
176904 |
960660000 |
1618890000 |
|
630 |
420 |
3742600 |
176400 |
396900 |
264600 |
1571892000 |
2357838000 |
|
678 |
509 |
4424800 |
259081 |
459684 |
345102 |
2252223200 |
3000014400 |
|
691 |
578 |
4932400 |
334084 |
477481 |
399398 |
2850927200 |
3408288400 |
|
675 |
623 |
5229400 |
388129 |
455625 |
420525 |
3257916200 |
3529845000 |
|
634 |
642 |
5308600 |
412164 |
401956 |
407028 |
3408121200 |
3365652400 |
|
571 |
635 |
5179000 |
403225 |
326041 |
362585 |
3288665000 |
2957209000 |
|
477 |
603 |
4837000 |
363609 |
227529 |
287631 |
2916711000 |
2307249000 |
|
390 |
546 |
4329400 |
298116 |
152100 |
212940 |
2363852400 |
1688466000 |
|
329 |
479 |
3778600 |
229441 |
108241 |
157591 |
1809949400 |
1243159400 |
|
247 |
413 |
3209800 |
170569 |
61009 |
102011 |
1325647400 |
792820600 |
|
184 |
341 |
2628400 |
116281 |
33856 |
62744 |
896284400 |
483625600 |
|
134 |
274 |
2093800 |
75076 |
17956 |
36716 |
573701200 |
280569200 |
|
108 |
215 |
1649200 |
46225 |
11664 |
23220 |
354578000 |
178113600 |
|
90 |
170 |
1312600 |
28900 |
8100 |
15300 |
223142000 |
118134000 |
|
SO2 = 3514348 |
SI2 = 3804704 |
SIO = 3472245 |
SSO = 29062547000 |
SSI = 29184045000 |
Using the above equations yields:
A = 1255.626164 s
B = 7029.100513 s
k = A+B = 8284.726677 s = 2.3 h
x = A/(A + B) = 0.151559154
Observe that these results for k and x are the same as those of the graphical procedure. For comparison purposes, the observed outflow hydrograph and that predicted using the estimated values of k and x are graphed below.
B. Muskingum Routing
Use the Muskingum routing procedure to route the hydrograph tabulated below through the same river reach of Part A
Select a Dt = 1 h, as suggested by the inflow data. However, check that with the selected Dt, parameter values meet restrictions:
x < 0.5 Dt/k < 1 - x
For this case: 0.1515 < (0.5) (3600)/8284.73 < 1 - 0.1515 Thus, OK. Proceed with routing, by obtaining Co, C1, and C2.
This yields: Co = 0.061787; C1 = 0.346074; and C2 = 0.592139. Using these values in the Muskingum routing equation:
obtain the outflow hydrograph as tabulated below. The resulting hydrographs are also graphed below.
.
|
Time (h) |
Inflow (m3/s) |
Co x Ii+1 (m3/s) |
C1 x Ii (m3/s) |
C2 x Oi (m3/s) |
Outflow (m3/s) |
|
1 |
50 |
50 |
|||
|
2 |
100 |
6.1787 |
17.3037 |
29.60695 |
53.08935 |
|
3 |
200 |
12.3574 |
34.6074 |
31.43627 |
78.40107 |
|
4 |
325 |
20.08078 |
69.2148 |
46.42433 |
135.7199 |
|
5 |
450 |
27.80415 |
112.4741 |
80.36505 |
220.6433 |
|
6 |
600 |
37.0722 |
155.7333 |
130.6515 |
323.457 |
|
7 |
700 |
43.2509 |
207.6444 |
191.5315 |
442.4268 |
|
8 |
780 |
48.19386 |
242.2518 |
261.9782 |
552.4238 |
|
9 |
790 |
48.81173 |
269.9377 |
327.1117 |
645.8611 |
|
10 |
775 |
47.88493 |
273.3985 |
382.4396 |
703.723 |
|
11 |
750 |
46.34025 |
268.2074 |
416.7018 |
731.2494 |
|
12 |
680 |
42.01516 |
259.5555 |
433.0013 |
734.572 |
|
13 |
590 |
36.45433 |
235.3303 |
434.9687 |
706.7534 |
|
14 |
500 |
30.8935 |
204.1837 |
418.4962 |
653.5734 |
|
15 |
420 |
25.95054 |
173.037 |
387.0063 |
585.9938 |
|
16 |
350 |
21.62545 |
145.3511 |
346.9898 |
513.9663 |
|
17 |
300 |
18.5361 |
121.1259 |
304.3395 |
444.0015 |
|
18 |
250 |
15.44675 |
103.8222 |
262.9106 |
382.1796 |
|
19 |
225 |
13.90208 |
86.5185 |
226.3034 |
326.724 |
|
20 |
200 |
12.3574 |
77.86665 |
193.466 |
283.6901 |
