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q_o = \frac{ B_o \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}

\rho_o = \frac{
\dot m_o}{q_o}= \frac{\rho_O + \rho_G \, R_s}{B_o}

\dot m_o = \rho_o \cdot q_o = (\rho_O + \rho_G \, R_s) \cdot \frac{ q_o}{B_o} = 
\frac{(\rho_O + \rho_G \, R_s) \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}

q_g = \frac{ B_g \cdot ( q_G - R_s \, q_O)}{1- R_v \, R_s}

\rho_g = \frac{\dot m_g}{q_g}= \frac{\rho_G + \rho_O \, R_v}{B_g}

\dot m_g = \rho_g \cdot q_g = (\rho_G + \rho_O \, R_v) \cdot \frac{q_g }{B_g} =
\frac{ (\rho_G + \rho_O \, R_v) \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}

q_w = B_w \cdot q_W

\rho_w =\frac{\dot m_w}{q_w}=  \frac{\rho_W}{B_w}

\dot m_w = \rho_w \cdot q_w =   \rho_W \cdot \frac{q_w}{B_w} = \rho_W \cdot q_W

q_t = q_o + q_g + q_w

q_t = \frac{B_o - B_g \, R_s}{1-R_v \, R_s} \cdot q_O 

+\frac{B_g - B_o \, R_v}{1-R_v \, R_s} \cdot q_G 

+ B_w \cdot q_W

q_t = \frac{B_o - B_g \, R_s}{1-R_v \, R_s } \cdot \frac{\dot m_O }{\rho_O}

+\frac{B_g - B_o \, R_v}{1-R_v \, R_s } \cdot \frac{\dot m_G }{\rho_G}

+ B_w\cdot \frac{\dot m_W}{\rho_W}

\dot m = \dot m_o + \dot m_g + \dot m_w = \dot m_O + \dot m_G + \dot m_G 

\rho_t = \frac{\dot m}{q_t} = \frac{\dot m_O + \dot m_G + \dot m_G}{
\frac{B_o - B_g \, R_s}{1-R_v \, R_s } \cdot \frac{\dot m_O }{\rho_O}

+\frac{B_g - B_o \, R_v}{1-R_v \, R_s } \cdot \frac{\dot m_G }{\rho_G}

+ B_w\cdot \frac{\dot m_W}{\rho_W}
}

s_o = \frac{q_o}{q_t} = \frac{ B_o \, (q_O - R_v \, q_G)}{(B_o - B_g \, R_s) \, q_O + (Bg - B_o \, R_v) \, q_G + B_w \, (1- R_v \, R_s) \, q_W }

s_g = \frac{q_g}{q_t} = \frac{ B_g \, (q_G - R_s \, q_O)}{(B_o - B_g \, R_s) \, q_O + (Bg - B_o \, R_v) \, q_G + B_w \, (1- R_v \, R_s) \, q_W }

s_w = \frac{q_w}{q_t} = \frac{ B_w \, (1- R_v \, R_s) \, q_W}{(B_o - B_g \, R_s) \, q_O + (Bg - B_o \, R_v) \, q_G + B_w \, (1- R_v \, R_s) \, q_W }