Comparación de dos tipos de hidrograma unitario de Clark


Victor M. Ponce y Nicole R. Nuccitelli

21 Octubre 2013

HEC-1 CLARK UNIT HYDROGRAPH


 T* = Ti / Tc

 A* = Ai / A

 A* = 1.414 T*1.5                      [0 ≤ T* ≤ 0.5] 

 A* = 1 - 1.414 (1 - T*)1.5            [0.5 ≤ T* ≤ 1] 

(1)


Table 1.  HEC's default time-area histogram for A = 1,000 km2 and Tc = 6 hr.
[1] [2] [3] [4]
Histogram increment i Cumulative time Ti at the end of increment i (hr) Cumulative area Ai at the end of cumulative time Ti
(km2)		 Incremental area ΔAi
(km2)
1 1 96.2 96.2
2 2 272.1 175.9
3 3 500 227.9
4 4 727.9 227.9
5 5 903.8 175.9
6 6 1,000 96.2


Effective rainfall hyetograph

Fig. 1  Calculated time-area histogram.

TIME-AREA METHOD

Time-area method: (a) isochrone delineation; (b) time-area histogram

Fig. 2  Time-area method:  (a) isochrone delineation; (b) time-area histogram.


Effective rainfall hyetograph

Fig. 3  Effective rainfall hyetograph.

Table 2.  Time-area method of watershed routing.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Time
(hr)		 Time-area histogram subareas
(km2)		 Partial flows (km2-cm/hr)
for indicated rainfall increments		 Outflow
(km2-cm/hr)		 Outflow
(m3/s)
0.5
cm/hr		 1.0
cm/hr		 2.0
cm/hr		 1.5
cm/hr		 1.0
cm/hr		 0.5
cm/hr
0 - 0 - - - - - 0 0
1 10 5 0 - - - - 5 13.9
2 30 15 10 0 - - - 25 69.4
3 20 10 30 20 0 - - 60 166.7
4 40 20 20 60 15 0 - 115 319.4
5 - 0 40 40 45 10 0 135 375
6 - - 0 80 30 30 5 145 402.8
7 - - - 0 60 20 15 95 263.9
8 - - - - 0 40 10 50 138.9
9 - - - - - 0 20 20 55.6
10 - - - - - - 0 0 0
Sum 100 - - - - - - 650 -

 Tb = Tc + tr 

(2)

LINEAR RESERVOIR


O2 = C0 I2 + C1 I1 + C2 O1


C0 = (Δt/K) / [2 + (Δt/K)]

C1 = C0

 C2 = [2 - (Δt/K)] / [2 + (Δt/K)] 

(3)

Table 3.  Routing of a time-area hydrograph through a linear reservoir.
[1] [2] [3] [4] [5] [6] [7]
Time
(hr)		 Time-area hydrograph ordinates
(km2-cm/hr)		 Partial flows (Eq. 3) (km2-cm/hr) Outflow from the linear reservoir
(km2-cm/hr)		 Outflow
(m3/s)
C0 I2 C1 I1 C2 O1
0 0 - - - 0 0
1 5 1 0 0 1 2.78
2 25 5 1 0.6 6.6 18.33
3 60 12 5 3.96 20.96 58.22
4 115 23 12 12.58 47.58 132.17
5 135 27 23 28.55 78.55 218.19
6 145 29 27 47.13 103.13 286.47
7 95 19 29 61.88 109.88 305.22
8 50 10 19 65.93 94.93 263.69
9 20 4 10 56.96 70.96 197.11
10 0 0 4 42.58 46.58 129.37
11 0 0 0 27.95 27.95 77.64
12 0 0 0 16.77 16.77 46.58
13 0 0 0 10.06 10.06 27.94
14 0 0 0 6.04 6.04 16.77
15 0 0 0 3.62 3.62 10.07
16 0 0 0 2.17 2.17 6.03
17 0 0 0 1.30 1.30 3.62
18 0 0 0 0.78 0.78 2.17
19 0 0 0 0.47 0.47 1.30
20 0 0 0 0.28 0.28 0.78
21 0 0 0 0.17 0.17 0.47
22 0 0 0 0.10 0.10 0.28
23 0 0 0 0.06 0.06 0.17
24 0 0 0 0.04 0.04 0.10
25 0 0 0 0.02 0.02 0.07
Sum 650 - - - 650.00 -


Time-area method: (a) isochrone delineation; (b) time-area histogram

Fig. 4  Effect of routing the time-area hydrograph through a linear reservoir.

CLARK UNIT HYDROGRAPH


Isochrones for the Appomattox river at Petersburg, Virginia

Fig. 5  Isochrones for the Appomattox river at Petersburg, Virginia (Clark 1945).


 O2 = 2 C0 I2 + + C2 O1 

(4)


Table 4.  Calculation of Clark's Appomatox River example.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Time
(d)		 Time
(hr)		 Time-area histogram percentages
(%)		 Increment
of area
(mi2) 		12 Inflow
(mi2-in/hr)		 Inflow
(mi2-in/hr)		 Partial flows (mi2-in/hr) Outflow
(mi2-in/hr)		 Outflow (UH ordinates)
(cfs)
2 C0 I2 C2 O1
0 0 0 0 0 0 0 0 0 0
0.5 12 1.8 24.030 24.030 2.003 1.121 0.000 1.121 723.673
1 24 3.8 50.730 50.730 4.228 2.367 0.493 2.861 1846.170
1.5 36 6.9 92.115 92.115 7.676 4.299 1.259 5.557 3586.395
2 48 10.8 144.180 144.180 12.015 6.728 2.445 9.174 5920.052
2.5 60 19.1 254.985 254.985 21.429 11.899 4.036 15.936 10283.798
3 72 7.6 101.460 101.460 8.455 4.735 7.012 11.747 7580.380
3.5 84 6.5 86.775 86.775 7.231 4.050 5.168 9.218 5948.631
4 96 5.5 73.425 73.425 6.119 3.247 4.056 7.482 4828.621
4.5 108 9.0 120.150 120.150 10.013 5.607 3.292 8.899 5742.958
5 120 14.0 186.900 186.900 15.575 8.722 3.916 12.638 8155.470
5.5 132 9,5 126.825 126.825 10.569 5.919 5.561 11.479 7407.792
6 144 5.5 73.425 73.4250 6.119 3.427 5.051 8.477 5470.652
6.5 156 0 0 0 0 0 3.730 3.730 2407.087
7 168 0 0 0 0 0 1.641 1.641 1059.118
7.5 180 0 0 0 0 0 0.722 0.722 466.012
8 192 0 0 0 0 0 0.318 0.318 205.045
8.5 204 0 0 0 0 0 0.140 0.140 90.220
9 216 0 0 0 0 0 0.062 0.062 039.697
9.5 228 0 0 0 0 0 0.027 0.027 17.467
10 240 0 0 0 0 0 0.012 0.012 7.685
10.5 252 0 0 0 0 0 0.005 0.005 3.382
11 264 0 0 0 0 0 0.002 0.002 1.488
11.5 276 0 0 0 0 0 0.001 0.001 0.655
12 288 0 0 0 0 0 0.00044 0.00044 0.288
Sum - 100.00 3315.000 - - - - 111.249 71786.924


Table 5.  Calculation of the Clark unit hydrograph.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
Time
(hr)		 Time-area histogram subareas
(km2)		 Partial flows (km2-cm/hr)
for indicated rainfall increments 		Sum
(discrete
unit-runoff
hyetograph)
(km2-cm/hr)		 Partial flows (Eq. 4) (km2-cm/hr) Outflow from the linear reservoir
(km2-cm/hr)		 Outflow
(m3/s)
0.5
cm/hr		 0.5
cm/hr		 2 C0 I2 C2 O1
0 - - - - - - 0 0
1 10 5 - 5 2 0 2 5.56
2 30 15 5 20 8 1.2 9.2 25.56
3 20 10 15 25 10 5.52 15.52 43.11
4 40 20 10 30 12 9.31 21.31 59.19
5 - - 20 20 8 12.79 20.79 57.75
6 - - - - 0 12.47 12.47 34.65
7 - - - - 0 7.48 7.48 20.78
8 - - - - 0 4.49 4.49 12.47
9 - - - - 0 2.69 2.69 7.48
10 - - - - 0 1.61 1.61 4.488
11 - - - - 0 0.97 0.97 2.688
12 - - - - 0 0.58 0.58 1.62
13 - - - - 0 0.35 0.35 0.978
14 - - - - 0 0.21 0.21 0.58
15 - - - - 0 0.13 0.13 0.358
16 - - - - 0 0.08 0.08 0.22
17 - - - - 0 0.05 0.05 0.13
18 - - - - 0 0.03 0.03 0.08
19 - - - - 0 0.02 0.02 0.05
20 - - - - 0 0.01 0.01 0.03
21 - - - - 0 0.006 0.006 0.016
22 - - - - 0 0.004 0.004 0.011
Sum 100 - - - - 0 100.00 -

PONCE'S CLARK UNIT HYDROGRAPH


Table 6.  Calculation of the Clark unit hydrograph according to Ponce (1989).
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Time
(hr)		 Histogram subareas
(km2)		 Partial flows (km2-cm/hr)
for indicated rainfall increments		 Sum
(continuous
unit
hydrograph)
(km2-cm/hr)		 Partial flows (Eq. 3) (km2-cm/hr) Outflow from the linear reservoir
(km2-cm/hr)		 Outflow
(m3/s)
0.5
cm/hr		 0.5
cm/hr		 C0 I2 C1 I1 C2 O1
0 - - - 0 - - - 0 0
1 10 5 - 5 1 0 0 1 2.78
2 30 15 5 20 4 1 0.6 5.6 15.55
3 20 10 15 25 5 4 3.36 12.36 34.33
4 20 20 10 30 6 5 7.42 18.42 51.17
5 - - 20 20 4 6 11.05 21.05 58.47
6 - - - 0 0 4 12.63 16.63 46.19
7 - - - - 0 0 9.98 9.98 27.72
8 - - - - 0 0 5.99 5.99 16.64
9 - - - - 0 0 3.59 3.59 9.98
10 - - - - 0 0 2.15 2.15 5.98
11 - - - - 0 0 1.29 1.29 3.58
12 - - - - 0 0 0.78 0.78 2.17
13 - - - - 0 0 0.47 0.47 1.30
14 - - - - 0 0 0.28 0.28 0.78
15 - - - - 0 0 0.17 0.17 0.47
16 - - - - 0 0 0.10 0.10 0.28
17 - - - - 0 0 0.06 0.06 0.17
18 - - - - 0 0 0.04 0.04 0.11
19 - - - - 0 0 0.02 0.02 0.06
20 - - - - 0 0 0.01 0.01 0.03
21 - - - - 0 0 0.006 0.006 0.016
22 - - - - 0 0 0.004 0.004 0.011
Sum 100 - - - - - 0 100.00 -


Table 7.  Comparison of Clark's original unit hydrograph with Ponce's version.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
Time
(hr)		 Partial flows (Eq. 3) (km2-cm/hr) Clark's original
unit hydrograph
(km2-cm/hr)		 Partial flows (Eq. 3) (km2-cm/hr) Ponce's Clark
unit hydrograph
(km2-cm/hr)
C0 I2 C1 I1 C2 O1 C0 I2 C1 I1 C2 O1
0 - - - - - - - 0
1 1 1 0 2 1 0 0 1
2 4 4 1.2 9.2 4 1 0.6 5.6
3 5 5 5.52 15.52 5 4 3.36 12.36
4 6 6 9.31 21.31 6 5 7.42 18.42
5 4 4 12.79 20.79 4 6 11.05 21.05
6 0 0 12.47 12.47 0 4 12.63 16.63
7 0 0 7.48 7.48 0 0 9.98 9.98
8 0 0 4.49 4.49 0 0 5.99 5.99
9 0 0 2.69 2.69 0 0 3.59 3.59
10 0 0 1.61 1.61 0 0 2.15 2.15
11 0 0 0.97 0.97 0 0 1.29 1.29
12 0 0 0.58 0.58 0 0 0.78 0.78
13 0 0 0.35 0.35 0 0 0.47 0.47
14 0 0 0.21 0.21 0 0 0.28 0.28
15 0 0 0.13 0.13 0 0 0.17 0.17
16 0 0 0.08 0.08 0 0 0.10 0.10
17 0 0 0.05 0.05 0 0 0.06 0.06
18 0 0 0.03 0.03 0 0 0.04 0.04
19 0 0 0.02 0.02 0 0 0.02 0.02
20 0 0 0.01 0.01 0 0 0.01 0.01
21 0 0 0.006 0.006 0 0 0.006 0.006
22 0 0 0.004 0.004 0 0 0.004 0.004
Sum - - - 100.00 - - 0 100.00

Time-area method: (a) isochrone delineation; (b) time-area histogram

Fig. 6  Comparison of Clark's original and Ponce's Clark unit hydrographs.
131023 09:30