CE322 Basic Hydrology
Jorge A. Ramirez
Homework No.6 - Solution
1. Do problem 10.3 from your textbook.
Drainage density:
where LT
is the total length of streams. From the data given, LT = 44 mi; and Abasin
= 6400 acres.
Thus, Dd
= (44 mi)/[(6400 acres) (43560 ft2/acre))/(5280
ft/mi)2 = 4.4 mi-1
2. For the basin of Problem 10.3:
a)
Use the
Strahler ordering scheme to order the drainage network and determine the order
of the basin, W.
Therefore,
Basin Order W = 3.
b)
Estimate
the Bifurcation Ratio, RB,
and the Length Ratio, RL.
Use
the same procedure outlined below in the solution to Problem 3.
|
Order |
Number of
Streams |
Average
Length (mi) |
log(N) |
log(L) |
|
1 |
9 |
1.777778 |
0.954243 |
0.249878 |
|
2 |
4 |
4.75 |
0.60206 |
0.676694 |
|
3 |
1 |
9 |
0 |
0.954243 |
The linear regression analysis for the
pairs (w,
log(Nw)) returns a slope m = -0.47712. Thus,
RB = 3.
The linear regression analysis for the
pairs pairs (w, log(Lw)) returns a slope m = 0.352182. Thus,
RL = 2.25.
c)
Estimate
the average length of overland flow, Lo.
Average length of overland flow:
Thus, Lo
= 1/2/4.4 mi-1 = 0.113636 mi
3. Obtain estimates of the Bifurcation
Ratio, RB, the Length
Ratio, RL, and the Area Ratio,
RA, using the data
tabulated below. Also, for the same data, a) determine the area of the basin,
b) the total length of streams, c) the drainage density, Dd, and d) the average length of overland flow, Lo.
|
Order, w |
Number of Streams |
Average Length (km) |
Average Area (km2) |
|
1 |
50 |
1.5 |
4 |
|
2 |
13 |
4 |
12 |
|
3 |
8 |
9 |
36 |
|
4 |
3 |
23 |
108 |
|
5 |
1 |
59 |
324 |
The Horton law of stream numbers states that there exists a geometric relationship between the number of streams of a given order Nw and the corresponding order, w. The parameter of this geometric relationship is the Bifurcation Ratio, RB.
(1)
The Horton law of stream lengths states that there exists a geometric relationship between the average length of streams of a given order and the corresponding order, w. The parameter of this relationship is the so-called Length Ratio, RL.
(2)
The Horton law of stream areas states that there exists a geometric relationship between the average area drained by streams of a given order and the corresponding order w. The parameter of this relationship is the so-called Area Ratio, RA.
(3)
In the equations above, W is the order of the basin, and the over-bar indicates the average value of the corresponding variable.
a) Taking logarithms of each of the above equations leads to:
(4)
(5)
(6)
These equations are linear in w. Thus, estimates of the RB, RL, and RA, can be obtained by linear regression of:
,
,
,
respectively. Denoting by m the slopes of the corresponding fits, the above estimates are obtained as:
For the problem at hand:
|
Order |
Number of
Streams |
Average
Length |
Average Area |
log(N) |
log(L) |
log(A) |
|
1 |
50 |
1.5 |
4 |
1.69897 |
0.176091 |
0.60206 |
|
2 |
13 |
4 |
12 |
1.113943 |
0.60206 |
1.079181 |
|
3 |
8 |
9 |
36 |
0.90309 |
0.954243 |
1.556303 |
|
4 |
3 |
23 |
108 |
0.477121 |
1.361728 |
2.033424 |
|
5 |
1 |
59 |
324 |
0 |
1.770852 |
2.510545 |
A) Law of Stream Numbers and Bifurcation
Ratio:
The linear regression analysis returns a
slope m = -0.40348. Thus, RB
= 2.532073.
Law of Stream Lengths and Length Ratio:
The linear regression analysis returns a
slope m = 0.394919. Thus, RL
= 2.48267.
Law of Stream Areas and Area Ratio:
The linear regression analysis returns a
slope m = 0.477121. Thus, RA
= 3.0.
The total length of streams can be
calculated as:
Using the above equation leads to LT = 327 km.
Drainage density:
Thus, Dd
= (437 km)/(324 km2) = 1.009259 km-1
Average length of overland flow:
Thus, Lo
= 0.495413 km = 495.41 m