**5.1 Boundary shear stress (τb)**

The boundary shear stress (τb) acting on the bed of the channel can be calculated as:

$$
\pi\_{\mathsf{b}} = \mathfrak{p} \,\, \mathsf{g} \,\, \mathsf{D}\_{\mathsf{c}} \,\, \mathsf{S}\_{\mathsf{c}} \,\, \tag{18}
$$

where τ<sup>b</sup> = the boundary shear stress, ρ = the fluid density, g = gravitational acceleration, Dc = averaged channel flow depth, and Sc is averaged water-surface paleoslope. Both field and laboratory experiments have shown that initial motion of bed materials in coarse-medium grained rivers typically occurs at a transport stage between 1 and 3 [31]. This relationship between the flow and its container can be applied to all natural channels with some error and has been recently applied in ancient fluvial deposits [32] of the Sharon Formation, USA, ref. [33] for the Parthenon Sandstone USA. Using previous estimates of Dc and Sc in Eq. (16) the boundary shear stress is estimated to be in between 10.68–5.24 N/m2 for the Barakar Rivers in Rajmahal Gondwana master basin.

The critical shear stress (τcr) represents the necessary boundary shear to move the bed-load materials, based upon their grain size, grain shape, effective density, and roughness. Therefore, the formulation to express critical shear stress (τcr) for noncohesive sand is provided by Shield [29] and is given as:

$$
\boldsymbol{\pi}\_{\rm cr} = \boldsymbol{\pi}^\* \left( \rho\_{\rm s} - \rho\_{\rm w} \right) \mathbf{g} \, \mathbf{D}\_{\rm 50} \tag{19}
$$

Where τcr = the critical shear stress; τ\* = the Shield number for the given particle non-dimensional critical shear stress; ρ<sup>s</sup> = the grain density (assumed to be quartz with a density of 2650 kg/cm<sup>3</sup> ; ρ<sup>w</sup> = fluid density 1000 kg/m3 ; g = the acceleration due to gravity in m/sec<sup>2</sup> and D50 = median particle size in meter. Critical shear stress is calculated from solving Eq. (19) when the transport stage was set to initial motion i.e., 1, using paleohydrological data for bankfull depth, and grain size data which comes out in between 0.169–0.204 N/m<sup>2</sup> or Pa.

Sediment mobility for a given particle size occurs when the boundary shear stress exceed the critical shear stress, in other words τ<sup>b</sup> > τcr. This relationship has been observed in the Barakar sandstones of present study.
