CIVL253 - HYDROLOGY
Suggested Exercises #1
1. The world population in 1980 has been estimated at about 4.5 billions. The annual
population increase during the preceding decade was about 2 percent. At this rate of
population growth, predict the year when there will be a shortage of fresh-water resources
if everyone in the world enjoyed the present highest living standard, for which freshwater use is bout 6.8 m3/day (1800 gal/day) per capita including public water supplies and
water withdrawn for irrigation and industry. Assume that 47,000 km3 of surface and
subsurface runoff is available for use annually.
2. The equation
k
dQ(t )
dt  Q(t )  I (t )
has been used to describe the response of streamflow to a constant rate of precipitation
continuing indefinitely on a watershed. In this case, let I(t)=1 for t>0, and Q(t)=0 for t=0.
Solve the differential equation and plot the values of I(t) and Q(t) over a 10-hour period if
k=2hr.
3. Compute the constant draft from a 500-hectare reservoir for a 30-day period during which
the reservoir level dropped half a meter despite an average upstream inflow of 200,000
m3/day. During the period, the total seepage loss was 2 cm, the total precipitation was
10.5 cm, and the total evaporation was 8.5 cm.
4. From the hydrologic records of over 50 years on a drainage basin of area 500 km2, the
average annual rainfall is estimated as 90 cm and the average annual runoff as 33 cm. A
reservoir in the basin, having an average surface area of 1700 hectares, is planned at the
basin outlet to collect available runoff for supplying water to a nearby community. The
annual evaporation over the reservoir surface is estimated as 130 cm. There is no
groundwater leakage or inflow to the basin. Determine the available average annual
withdrawal from the reservoir for water supply.
5. The consecutive monthly flows into and out of a reservoir in a given year are the
following, in relative units:
Month
Inflow
Outflow
J
3
6
F
5
8
M A
4
3
7 10
M J J A
4 10 30 15
6 8 20 13
S
6
4
O
4
5
N
2
7
D
1
8
The reservoir contains 60 units at the beginning of the year. How many units of water are
in the reservoir at the middle of August? At the end of the year?
6. The following table contains weekly precipitation and interception amounts from April to
October 1972 by Lam (1974) in Hong Kong. Plot the percentage of rainfall interception
against the weekly rainfall amount and propose a plausible mathematical model
(including determining model parameters) to describe their relationship.
Week
Rainfall
(mm)
Interception (mm)
Week
Rainfall
(mm)
Interception (mm)
Week
Rainfall
(mm)
Interception (mm)
1
87.4
5.3
10
9.8
2.6
19
77.2
3.0
2
0.5
0.4
11
52.8
12.2
20
49.6
1.8
3
0.2
0.2
12
51.2
4.9
21
58.2
5.0
4
34.2
6.4
13
16.3
5.9
22
18.7
9.3
5
96.4
2.0
14
28.2
2.0
23
22.5
6.5
6
18.2
5.5
15
96.4
2.6
24
46.1
5.2
7
90.2
5.0
16
24.8
4.1
25
52.8
9.1
8
15.2
6.9
17
45.9
3.0
26
5.8
1.1
9
11.5
4.9
18
44.0
3.6
27
13.3
1.1
Problem-1
Problem-2
Problem-3
Problem- 4
Problem-5
Problem-6
Week
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Rainfall
(mm)
87.4
0.5
0.2
34.2
96.4
18.2
90.2
15.2
11.5
9.8
52.8
51.2
16.3
28.2
96.4
24.8
45.9
44.0
77.2
49.6
58.2
18.7
22.5
46.1
52.8
5.8
13.3
Intercept
(mm)
5.3
0.4
0.2
6.4
2.0
5.5
5.0
6.9
4.9
2.6
12.2
4.9
5.9
2.0
2.6
4.1
3.0
3.6
3.0
1.8
5.0
9.3
6.5
5.2
9.1
1.1
1.1
% Interception
6.06
80.00
100.00
18.71
2.07
30.22
5.54
45.39
42.61
26.53
23.11
9.57
36.20
7.09
2.70
16.53
6.54
8.18
3.89
3.63
8.59
49.73
28.89
11.28
17.23
18.97
8.27
Based on the interception percentage (see column 4) calculated in the above table, a plot of
“intercept%” vs. “rainfall amount” is shown below. The plot clearly reveals non-linear
relation, more likely to be in the form of exponential decay, between the “intercept%” and
“rainfall amount”.
100
Intercept %
80
60
40
20
0
0
20
40
60
80
100
120
Rainfall amount (mm)
Then, by changing “percentage%” in the log-scale, a linear relation between “logpercentage%” and “rainfall amount” is visible.
Intercept %
100
10
1
0
20
40
60
80
100
120
Rainfall amount (mm)
Hence, we could propose the following model
Ln(Y) =  +  × X + 
where Y = “intercept%”; X = “rainfall amount”;  = error term; and Ln() = natural
logarithmic transform.
Through least square regression procedure, the best-fit equation is
Ln(Intercept%) = 3.78  0.0286 × Rainfall
The corresponding standard error and coefficient of determination are
S = 0.5896
R-Sq = 68.4%
The regression equation shows that “intercept%” decreases with “rainfall amount”. The fitted
“intercept%”, along with the observed values, are plotted below.
Intercept %
100
10
1
0
20
40
60
80
Rainfall amount (mm)
100
120