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

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

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

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

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

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

q_w = B_w \cdot q_W

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

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

q_t = q_o + q_g + q_w

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

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

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

\rho_t = (\dot m_O + \dot m_G + \dot m_G)/q_t