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q_O = \frac { q_o/B_o + R_v \cdot q_g/B_g } {1 - R_s R_v}

q_G =\frac{q_g/B_g + R_s \cdot q_o/B_o}{1-R_s R_v}

q_W =  \frac{q_w}{B_w}

q_L =  q_O + q_W

q_o = B_o \cdot ( q_O - R_v \, q_G)

q_g = B_g \cdot ( q_G - R_s \, q_O)

q_w = B_w \cdot q_W

q_t = q_o + q_g + q_w

\dot m_O = \rho_O \cdot q_O

\dot m_G = \rho_G \cdot q_G

\dot m_W = \rho_W \cdot q_W

\dot m = \dot m_O + \dot m_G + \dot m_GW 

\dot m_o = \rho_O \cdot q_o/B_o = \rho_O \cdot (q_O - R_v \, q_G))

\dot m_g = \rho_G \cdot q_g/B_g = \rho_O \cdot (q_G - R_s \, q_O)

\dot m_w = \rho_W \cdot q_w/B_w

\dot m = \dot m_Oo + \dot m_Gg + \dot m_Gw 

\rho_o = \rho_O/B_o

\rho_g = \rho_G/B_g

\rho_w = \rho_W/B_w

\rho_t = \dot m_t/q_t

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}

\rho = \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 }

\rho = s_o \, \rho_o + s_g \, \rho_g + s_w \, \rho_w

c = s_o \, c_o + s_g \, c_g + s_w \, c_w