**5.2 Paleoflow velocity (Vc)**

Under the assumption of steady, uniform (i.e. constant channel depth) flow in a channel that has a rigid boundary (i.e. flow conditions up to bed motion), the threshold mean velocity, Vt, can be computed based on resistance coefficients and channel geometry by two commonly and widely used methods. The first employs the Manning roughness coefficient, *n*:

$$\mathbf{V\_{c}} = \left(\mathbf{R}^{0.67}\mathbf{S}\mathbf{c}^{0.50}\right) / n \tag{20}$$

And the second uses the Darcy-Weisbach friction factor, *f*:

$$\mathbf{V\_c = \left[ (\mathbf{8 g R Sc/f}) \right]^{0.50}} \tag{21}$$

Where R is the hydraulic radius is approximately equal to the depth *D*c, Sc is the channel slope and *n* is the Manning roughness coefficient. The threshold mean flow velocity is a function of the size of bed particles that have been on the way, as evident in the equation for determining the resistance coefficient *n* and *f* below [Eqs. (21) and (22)].

Unlike the Manning empirical equation, the Darcy-Weisbach equation uses a dimensionless friction factor, has a sound theoretical basis, and exact accounts for the acceleration from gravity; moreover, the relative bed roughness does not influence the exponents of hydraulic radius and channel slope. For these reasons, the Darcy-Weisbach equation is preferred over the Manning approach as discussed [34]. Many algebraic manipulations has been used to calculate Manning roughness (*n*) and Darcy-Weisbach friction factor (*f*) and then various relationships have been proposed between resistance coefficients (*n, f*) and sediment grain size [35] as:

$$n = 0.039 \,\text{D}\_{50} \,\text{^{0.167}} \tag{22}$$

Ref. [34] derived a roughness coefficient encompassed a range of straight, braided and meandering, bed sediment sizes (sand and gravel), and river sizes (small scale to large river), as well as perennial rivers:

$$(8/f)^{0.50} = 2.2 \left(D\_{\rm c}/D\_{\rm 50}\right)^{-0.055} \text{Sc}^{-0.275} \tag{23}$$

Paleoflow velocities in the Barakar Rivers can be estimated using the method [36]. Their method is based on specific bed forms (e.g. ripples and dunes) are stable within specific ranges of grain size, flow velocity (**Figure 2**).

According to ref. [14] the height of a dune is strongly dependent on flow depth, therefore dune-scale cross sets is particularly useful in determining both flow velocity and water depth. If above assertion is correct than a flow velocity of between 65 and 160 cm/sec is estimated based on grain size (fine sand to coarse sand) using bed forms, grain size, and water depth and flow velocity plot [36]. Using this Eq. (23), the roughness coefficient (f) comes out to be between 0.0372–0.0334 (**Table 1**).

#### **5.3 Rouse number (Z)**

To determine dominant mode of sediment transport, the non-dimensional scale parameter Rouse number, Z, was calculated as

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

**Figure 2.**

*Fine and coarse-medium grained bedform phase diagrams [37]. Inferred range of velocity for Hurra coal basin is 80–125 cm/sec. Pachwara coal basin is 60–150 cm/sec. Chuperbhita coal basin 58–160 cm/sec. Brahmini coal basin 150–200 cm/sec.*

$$\mathbf{Z} = \mathbf{W}\_s / \text{βκ } \mathbf{U}\_\* \tag{24}$$

Where β is a constant taken as 1, and κ is the von Karman constant, taken as 0.40, U\* is the boundary shear velocity, and sediment settling velocity, *W*s, was calculated as a function of grain size following [38] as;

$$\mathbf{W}\mathbf{s} = \mathbf{R} \,\mathbf{g} \,\mathrm{D}\_{50}\,^2 / \mathrm{C}\_1 \,\mathrm{v} + \left( \mathrm{0.75} \,\mathrm{C}\_2 \,\mathrm{R} \,\mathrm{g} \,\mathrm{D}\_{50} \,^3 \right)^2 \tag{25}$$

Where g is the Earth's gravitational acceleration, D50 is the median diameter of a particle, v is the kinematic viscosity of water (1 � <sup>10</sup>�<sup>6</sup> for water at 20<sup>o</sup> C and C1 = 18 and C2 = 1 are constants associated with grain sphericity and roundness. And boundary shear velocity U\* determined as.

$$\mathbf{U}\_{\*} = \sqrt{\mathfrak{r}\_{\mathbf{b}}/\mathfrak{p}\_{\mathbf{w}}}$$

Where τ<sup>b</sup> is the boundary shear of the fluid and ρ <sup>w</sup> is the mass density of the fluid. With Z, dominant mode of sediment transport in alluvial system is typically bed load for Z > 2.5, 50% suspended load (i.e. mixed load) for 1.2 < Z < 2.5. Setting previously estimated values of R, Ws (sediment fall velocity), D50, and kinematic viscosity of water, Eq. (24) indicates that the Rouse Number (Z) was about 3.58 and 3.46 for the Brahmini and Pachwara and decreased to 2.01 and 1.71 in the northern Chuperbhita and Hurra sub-basins. These estimated values for Z is characteristic of a bed load channel in the southern part and mixed load (50% suspended load) for the northern part of the Rajmahal Gondwana master basin. These estimated values are in agreement with Schumm sediment load parameter (M) calculated by Eq. (11) suggests convincingly that the southern part of the basin predominantly deposited by bed load channels whereas the northern part mostly deposited by mixed load channel.
