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\phi_t = \phi_e  + \phi_t s_{wb} 

s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}

Rock volume

V = V_m + V_t = (1-\phi_t) V + \phi_t V

where

\phi_t = \frac{V_t}{V}

Total pore volume 

V_t = \phi_t V = V_e + V_{wb} = \phi_e V + s_{wb} V_t

where

V_e = V_t  (1 - s_{wb})

and therefore:

\phi_e = \phi_t  (1 - s_{wb})

Total volume of water is a sum of shale-bound water 

V_{wt} = V_{wb} + V_{wf}

and relates to 

s_{wt} V_t = s_{wb} V_t + s_w V_e = s_{wb} V_t + s_w V_t  (1 - s_{wb})

or

s_{wt} = s_{wb} + s_w (1 - s_{wb})

which gives an explicit formula for formation water saturation:

s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}

\frac{1}{R_t} = \phi_t^m s_{wt}^n \, \Big[ \frac{1}{R_w} + \frac{s_{wb}}{s_{wt}} \Big( \frac{1}{R_{wb}} - \frac{1}{R_w} \Big)  \Big] \quad \Rightarrow \quad s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}

s_{wb} = \zeta_{wb} V_{sh}

\frac{1}{R_{sh}} = B \cdot Q_V