Areal Aspects of Drainage Basin | Watershed Management | Geography

Areal aspect of morphological study of drainage basin includes the description of arrangement of areal elements, i.e., law of stream areas, relationship between watershed area and the stream length, relationship between watershed area and the discharge, basin shape (outline form) etc., mainly.

Arrangement of Areal Elements:

In a watershed there are two forms of area encountered, in which one is connected to the stream and another is not with the stream, but contributing the surface flow directly to the stream of higher order than the first, called inter ‘basin-area’, as shown in Fig. 27.7. Thus, the total watershed area is the sum of area of stream basin and inter basin area.

Total Basin Area:

The area ‘A_u‘ of a basin of order ‘u’ is the total projected area al horizontal plane, contributing the overland flow to the stream segments of given order plus all tributaries of lower order, e.g. the area of a 4th-order basin would be the sum of areas of all Ist-order, IInd-order, and IIIrd-order basins plus all inter basin areas between them, given as under –

Law of Stream Areas:

The law of stream area states that the mean basin area of stream of each order approximate a direct geometric sequence.

It is expressed as –

Where,

A̅_u = mean area of basin of order ‘u’.

A̅₁ = mean area of the Ist-order basin.

R_a = area ratio, it is analogous to the length ratio (R_l).

This equation states that the mean drainage-basin areas of progressively higher order streams increase in geometric sequence in the same way as the stream lengths.

The values of a and b can be determined by fitting a straight line, using least-square method, i.e., by plotting the logarithm of A_u and their respective stream orders. The R_a is computed as antilog of ‘a’.

Relationship between Basin Area and Stream Length:

The basin area and stream length have the relationship in the form of power function, given as –

L = m Aⁿ …(27.21)

Where,

L = stream length

A = basin area

m and n = constant

The plot of logarithm of stream length (log L) and respective stream order ‘u’, yields linear relationship. The values of ‘m’ and ‘n’ are determined by plotting the stream lengths and corresponding drainage areas on log-log paper, keeping stream lengths on Y-axis and drainage areas on X-axis. Hack (1957) derived the follow­ing empirical relationship between stream length and basin area for the watersheds of Virginia and Maryland –

L = 1.4 A^0.6 … (27.22)

In which, L is the stream length (miles) and A is the drainage area (sq. mile).

Hack (1957) also developed the following relationship for basin area (A_u) of order ‘u’, by using the area and stream length in terms of Horton’s Law of drainage network composition, given as –

In which, R_Lb is the ratio of length ratio (R_L) to the bifurcation ratio (R_b), called Horton’s term.

Relationship between Drainage Area and Discharge:

The relationship between basin/drainage area and the discharge is found in the following form, which is power function –

Q = J. A^m … (27.24)

In which, Q is the discharge; A is the basin area and J and m are the constants, which are determined by fitting a regression line using the data on discharge and basin area. Normally, the value of exponent ‘m’ varies from 0.5 to 1.0.

Basin Shape:

It refers to the shape of boundary line of watershed or drainage basin. The basin shape is determined as the shape of projected surface on the horizontal plane of basin map. The evaluation of basin shape has importance to predict its effect on the stream-discharge relationship.

The quantitative expression of drainage-basin shape is predicted in different forms; the most common are outlined as under:

1. Form factor

2. Circulatory ratio; and

3. Elongation ratio.

1. Form Factor:

Horton (1932) described the basin shape using the term called form factor, which is defined as the ratio of basin area to the square of the basin length, given as under –

Where,

R_f = form factor, dimensionless

A_u = basin area, sq. km

L_b = basin length, km

2. Circulatory Ratio:

Miller (1935) used the term ‘circulatory ratio’ to indicate the basin shape, which is the ratio of basin area (A_u) to the area of a circle (A_c) which perimeter is the same to the perimeter of drainage-basin. It is given as –

The R_c is dimensionless. Miller reported that the value of R_c varies from 0.6 to 0.7. The value of R_c remains constant for Ist and IInd-order basins in homogeneous shales and dolomites, which indicates the tendency of small drainage basins with homogeneous geologic materials to preserve the geometrical similarity.

3. Elongation Ratio:

Schumm (1956) used the elongation ratio as an index to mark the shape of drainage basin. It is defined as the ratio of diameter of a circle which has the area same to the basin, to the maximum basin length, expressed as –

Where,

R_l = elongation ratio, dimensionless

D_c = diameter of circle which area is same to the given drainage basin.

L_bm = maximum basin length.

The R_l ranges between 0.6 and 1.0. It depends on the variations in climate and relief of the area. For the regions of very low relief, the value of R_l is very close to 1.0, while for the areas with strong relief and steep ground slope the R_l varies from 0.6 to 0.8.

Drainage Density:

It is an important parameter for expressing linear scale of land form in the stream-eroded topography. This term was used by Horton (1932). The drainage density (D_d) is defined as the ratio of total length of all stream segments (i.e., cumulative length of stream segments of all the orders) within the specified basin to the basin area projected on horizontal plane, expressed as –

Where,

D_d = drainage density

L_u = length of stream segment

A_u = basin area

K & N = trunk order of the stream segment and total number of streams, respectively.

The drainage density gets reduce with decrease in stream length, or increase in drainage basin area. Its unit is km per sq. km.

Horton (1945) also derived the following equation for determining the drainage density (D_d) of basin area of order ‘u’ by combining the laws of stream numbers and stream lengths, given as –

The prediction of drainage density is carried out with the help of basin map, using the instruments planimeter and chartometer. The planimeter measures the basin area from the map and chartometer measures the stream-length. The average drainage density of each order stream-segment is computed, separately.

All the factors, which affect the length and dimension of first-order basins, are very effective to control the drainage density of the watershed.

However, the main factors are written as under:

1. Nature of sub-surface materials

2. Vegetation

3. Relief etc.

A low drainage density is very common in the regions which are composed of highly resistant sub-soil strata, dense vegetation and low relief. On the contrast, a high drainage-density is there where weak sub-surface strata, sparse vegetations and strong relief are encountered.

Constant of Channel Maintenance:

It is used to describe the morphological property of drainage basin, introduced by Schumm in the year 1956. He defined the constant of channel maintenance as the inverse of drainage density, i.e.-

In which, C is denoted as the constant of channel maintenance. It is expressed as sq. km per km. The value of C gets increase as the area of land-form unit increases. Basically, the C indicates the magnitude of sq. km surface area of watershed required to sustain one linear ‘km’ length of stream-segment.

Strahler (1957) analysed the ‘C’ with the stream order ‘u’ by plotting the logarithm of basin area (log A_u) and logarithm of cumulative stream length (log ΣΣL_u), keeping them on ordinate and abscissa, respectively, as shown in Fig. 27.8.

The basin area and stream length are measured from the basin’s map. He found a straight line with 45°slope, which reveals to a linear relationship between them. The intercept point at log A_u axis indicates the value of ‘log C’. Thus ‘C’ is obtained by taking the antilog of intercept value, given as under –

In which, log C is the intercept point of straight line on log A_u axis. The value of C is thus equal to

C = antilog of intercept value

Stream Frequency:

Horton (1932) introduced this term to study the basin morphology, which is the number of stream-segments per unit basin area. It is expressed by the following relationship –

Melton (1956) also derived an empirical relationship between drainage density and the channel frequency, considering 156 drainage basins with vast range of scale, climate, relief, surface cover conditions and geologic types. The relationship is given as –

F = 0.694 D^d … (27.33)

The two hypothetical basins may have the same drainage density, but different stream-frequencies, or they may have same stream fre­quency but different drainage densities. In Fig. 27.9, basin-A and basin-B have same drainage density, but their stream frequencies are not the same. On the other hand the basin-C and basin-D even involve same stream-frequ­ency, but their drainage-den­sities are quite different.