**4.5 Width of channel belt (Wcb)**

There are a number of empirical relationships between channel width and channel belt width, for example, from ref. [25] (Eq. (12)) and [1] (Eq. (13)).

$$W\_{\rm cb} = 59.9 \,\text{J} \,\text{D}\_{\rm c}^{1.80} \tag{12}$$

$$\mathcal{W}\_{\mathbf{c}} = \left(\mathcal{W}\_{\mathbf{c}\mathbf{b}} / 6.89\right)^{1.01} \tag{13}$$

Taking the mean bankfull channel depth of 15.00 m–11.78 m gives a range of channel-belt widths from 6780 m–2205 m. The value in these calculations is that they give a range of possible channel and channel-belt sizes that compare favorably with outcrop observations where the Barakar channels are confined, or where channel margin are observable.

Sedimentological descriptions of the fluvial deposits of the study area show a predominance of dune-scale cross-stratified, pebbly, coarse to medium grained sandstone, fining upward into fine grained sandstone and shale, shaly coal and thick coal seams characterized by asymmetrical fining upward cycles [11]. The abundant shale and fine clastics deposited in the floodplain demonstrate that the Barakar Rivers carried a significant shaly load in suspension, as well as a sandy bed load moving primarily as dune-scale bedform.

#### **4.6 Paleo-channel slope (Sc)**

Slope affects river plan form (i.e. braiding is favored by steeper slope and meandering by gentler slope) and facies boundaries [13, 28]. Paleoslope in ancient river system is calculated using physics-based methods, such as Shield's [29] threshold for bed-load movement, especially where grain size, channel depth and channel width of a paleo-river are known [4, 16]. Paleoslope is estimated using grain size and density of sediment grains following the empirical equation [22, 29];

$$\left(\mathsf{T}^\*\right)\_{\mathsf{bf}\,\mathsf{50}} = (D\_{\mathsf{c}}\mathsf{S}\mathsf{c})/(\mathsf{RD}\_{\mathsf{50}})\_{\mathsf{i}}$$

on re-arranging we get

$$\text{Sc} = \text{\textbullet}^\* \text{\textbullet}^\* \text{RD}\_{50}/D\_\text{c} \tag{14}$$

Where *Sc* is the paleoslope, τ\* bf50 is the bankfull Shields number for dimensionless shear stress, *Dc* is the mean bankfull channel flow depth, R is the submerged dimensionless density of sand-gravel sediment (ρs–ρw), ρ<sup>s</sup> = the grain density (assumed to be quartz, with density of 2.65 g/cm3 ), ρ<sup>w</sup> = the fluid density; and *D*<sup>50</sup> is the median grain size, τ\* bf50 is assumed to be 1.86 [22]. When working with ancient fluvial systems, *D*<sup>c</sup> is measureable from outcrop or estimated by empirical formula (Eqs. (7) and (8)), and *D*<sup>50</sup> for this study was estimated from observations of rocks in outcrop using magnifying lens 10� and/or from grain size analysis. When previously determined values of channel depth and median grain size are substituted, Eq. (13) indicate that paleochannel slope or gradient was about 0.000162 for Brahmini sub basin and decreased to 0.000116 in northern Hurra sub basin as sedimentation progressed in the area.

Another estimate of the paleoslope of the Early Permian Barakar River can be derived using the model [23] of which is based on Bayesian regression analysis relating paleoslope to bankfull channel depth, *D*c, and median grain size *D*<sup>50</sup> that takes the form:

$$\text{Log } \mathbb{S}\_{\mathbb{C}} = \mathfrak{a}\_0 + \mathfrak{a}\_1 \text{ log } D\_{\mathbb{S}0} + \mathfrak{a}\_2 \text{ log } D\_{\mathbb{C}} \tag{15}$$

Where α0, α1, and α<sup>2</sup> are empirical constant parameters with associated uncertainties are given by �02.08 for α0, 0.254 for α1, and � 1.09 for α<sup>2</sup> respectively [30] (**Table 1**). Empirical paleoslope reconstruction, such as the one applied here for the Barakar rivers; typically involve approximately an order-of-magnitude variation fitted to Trampush's model. Ultimately, this may have a significant control on the uncertainty of any quantitative estimation of past flow conditions. The sediment grain sizes of the Brahmini and Pachwara channel rages from very coarse to coarse grained sand (≈ 0.72–0.57 mm) with the fine grained sand (≈ 0.25 mm). The grain sizes of remaining two sub basins (Chuperbhita and Hurra) yielded a D50 grain size value between 0.45 to 0.35 mm, categorized as medium to fine grained sandstone. By substituting values for D50 (0.72, 0.57, 0.45 and 0.35 mm), Dc (13.63–9.28 m) in Eq. (15), the channel paleoslope is calculated to between 0.000176 and 0.000138.

Additional approximation of the paleoslope can be derived using the Manning equation for the open channel flow systems. This same relationship has been used to determine the paleochannel slope in many ancient fluvial rivers [5, 8].

$$\mathbf{V\_{c}} = \mathbf{1.486} \left( \mathbf{R^{2/3} S\_{c}^{1/2}} \right) / n \tag{16}$$

In this equation, Vc is the average flow velocity, R is the hydraulic radius, approximately equal to the channel depth, Dc, Sc is the channel paleoslope, and *n* is the Manning roughness coefficient (For metric calculations (CGS), 1.486 is replaced by 1

*Analyzing Sedimentary Rocks to Evaluate Paleo Dimensions and Flow Dynamics of Permian… DOI: http://dx.doi.org/10.5772/intechopen.106994*

because Manning equation employed FPS system). Rearranging Eq. (16) and putting R=Dc and 1 in place of 1.486, the formula becomes:

$$\mathbf{Sc}^{\natural} = \mathfrak{n} \mathbf{V}\_{\mathbf{c}} / \mathbf{D}\_{\mathbf{c}} \tag{17}$$

Paleoslope of the Barakar channels estimated using Eq. (17) ranges from 0.00038 to 0.00065 (**Table 1**), which suggests a low paleoslope comparable to slope ranges for the Amazon, Mississippi and Niger Rivers (slope range 0.00002–0.0005; 21). The estimated moderately high paleoslope from Eqs. (15)–(17) for the Brahmini and Pachwara sub basins are an order of magnitude equivalent to the estimated slope of the low sinuosity channel, whereas, estimated lower slope for Chuperbhita and Hurra sub basin channels agrees with the dominant fine clastics observed in the sand body. These values are within the low end of the range of values recorded from modern, humid-tropical climate, fluvial systems that transported mixed load sediments.
