**6.2 Drainage area (DA)**

Drainage basin area is an important component of hydrological analysis which controls sediment supply from provenance to sedimentary basin. Ref. [47] is of the opinion that key information of drainage can be obtained from fluvial deposits, as which are mainly transported from the provenance by fluvial channels. In recent years, a number of studies have investigated river data such as river channel dimensions, bankfull discharge and drainage area generally follow power laws in which bankfull thickness is positively correlated with the drainage area [28, 48]. The power law equations has been used to predict the drainage area of ancient Middle Triassic fluvial deposits of southwest Utah [44], ref. [47] measured bankfull channel-belt thickness of Early Miocene in the Gulf Of Mexico Basin to estimate the drainage areas. Whereas [49, 50] reconstructed drainage basin and sediment routing for the Cretaceous and Paleocene of Gulf of Mexico and Middle Jurassic of northern Qaidam Basin, northwest China respectively by using fluvial scaling relationship. The power law relationship is as:

$$D\_{\rm A} = 249.0 \,\text{D}\_{\rm c}^{2.44} \tag{37}$$

Where Dc = bankfull channel-belt thickness (bankfull thickness) and DA = Drainage area. Using this Eq. (37), the drainage area comes out to be 107.4 � <sup>10</sup><sup>3</sup> km<sup>2</sup> , 86.5 � <sup>10</sup><sup>3</sup> km<sup>2</sup> , 60.52 � <sup>10</sup><sup>3</sup> km<sup>2</sup> and 42.2 � <sup>10</sup><sup>3</sup> km<sup>2</sup> respectively for the Brahmini, Pachwara Chuperbhita, and Hurra coal basins of Rajmahal Gondwana basin.

Ref. [51] estimate a rivers drainage area based on a power-law relationship between drainage area and peak discharge by the equation:

$$\mathbf{Q\_w} = k \, D\_A{}^m \tag{38}$$

Where Qw is water discharge in m<sup>3</sup> /sec, DA is catchment area (drainage area) in km<sup>2</sup> , *k* constant equal to 0.075 and an *m* exponent of 0.80. Subsequent work by ref. [52] has shown that, the constant and exponent variables of power law relationship (38) change based on different climatic zones as shown in ref. [52] (**Table 1**). As humid climate have been visualized by many workers during deposition of Barakar formation thereby the *k* constant equal to 0.0161 and exponent *m* of 0.9839 have been used. Putting these values in Eq. (38)

$$\mathbf{D}\_{\mathbf{A}} = \mathbf{0}.\mathbf{01}\mathbf{61} \,\mathbf{Q}\_{\mathbf{w}} \,^{0.9839} \tag{39}$$

Where DA = drainage area and Qw = mean annual discharge. Substituting the estimated values of Qw in the above power equation the drainage area was approximately 136.2 � 103 sq. km in Brahmini coal basin and decreased to 28.5 � 103 sq. km to Hurra coal basin as sedimentation progressed in Rajmahal master Gondwana basin.

Barakar sandstone (Permian) drainage networks appear to have been broadly similar in scale to modern intra-basinal rivers draining active mountain belts and the continental-scale drainages, like the Ganga (India-861.4 � <sup>10</sup><sup>3</sup> km<sup>2</sup> ), the Indus (India-321.2 � <sup>10</sup><sup>3</sup> km<sup>2</sup> ), Brahmaputra (Bangladesh-194.4 � 103 km<sup>2</sup> ), the Po (Italy-70.0 � <sup>10</sup><sup>3</sup> km<sup>2</sup> ), and the Rhone (France- 96.2 � <sup>10</sup><sup>3</sup> km<sup>2</sup> ).

#### **6.3 Stream length (L)**

The length of a stream refers to the total length of stream channels in the drainage basin and therefore has units with dimension (L). Theories postulated shows that drainage area and stream length are related in a power function ([53], for references therein). If works done by these workers are correct, a power relation equation can be then derived which would be in form:

$$\mathbf{L} = \mathbf{a} \left( \mathbf{D}\_{\mathbf{A}} \right)^{\mathbf{b}} \tag{40}$$

Where L = stream length, DA = drainage area and a, b are constants. Ref. [54] derived the values of constants as: a = 1.109 and b = 0.545 while studying relationship between drainage basin area and length of stream for River Gongola, Nigeria. Substituting the constant values in Eq. (40) we have power relation as;

$$\mathbf{L} = \mathbf{1}.\mathbf{109} \left(\mathbf{D}\_{\mathbf{A}}\right)^{0.545} \tag{41}$$

When previously determined value of DA from Eq. (37) is substituted, Eq. (41) indicates that the length of the Late Paleozoic Barakar River was about 703 km in

Brahmini sub basin and gradually decreasing as channel sinuosity increases through Chuperbhita sub basin (525 km) to Hurra sub basin to about 427 km in Rajmahal Gondwana master basin.
