q_t =
 B_w \, q_W
+ \frac{B_o - R_v B_g}{1 - R_v R_s} \, q_O
+ \frac{B_g - R_s B_o}{1 - R_v R_s} \, q_G 

q_t  =
 B_w \, q_W
+ B_o \, q_O \, \frac{1 - R_{vn}}{1 - R_{vn} R_{sn}} 
+ B_g  \, q_G \, \frac{1 - R_{sn}}{1 - R_{vn} R_{sn}}

B_w = B_w(p_{\rm ref}), \ B_o = B_o(p_{\rm ref}), \ B_g = B_g(p_{\rm ref})

R_{vn} = \frac{R_v B_o}{B_g} \bigg| _{P_{\bf ref}} \ , \quad 
R_{sn} = \frac{R_s B_g}{B_o} \bigg| _{P_{\bf ref}} \ , \quad
R_{vp} = \frac{\dot R_v B_o}{B_g} \bigg| _{P_{\bf ref}} \ , \quad 
R_{sp} = \frac{\dot R_s B_g}{B_o} \bigg| _{P_{\bf ref}}

s(\mathbf{r}) = \{ s_w(\mathbf{r}), \ s_o(\mathbf{r}), \ s_g(\mathbf{r})  \}

c_t(s) = c_r (1 + R_{sn} s_o + R_{vn} s_g) + c_w s_w + c_o s_o (1+R_{sn}) + c_g s_g (1 + R_{vn}) + R_{sp} s_o + R_{vp} s_g

с_r = - \frac{1}{\phi} \frac{d \phi}{dP} \bigg| _{P_{\bf ref}},  \quad 
с_w = \frac{1}{\rho_w} \frac{d \rho_w}{dP} \bigg| _{P_{\bf ref}}, \quad 
с_o = \frac{1}{\rho_o} \frac{d \rho_o}{dP} \bigg| _{P_{\bf ref}}, \quad 
с_g = \frac{1}{\rho_g} \frac{d \rho_g}{dP} \bigg| _{P_{\bf ref}} 

\beta = \phi \, c_t \bigg| _{P_{\bf ref}}

\alpha(s) = \Big \langle \frac{k} {\mu} \Big \rangle = \alpha_w(s) + \alpha_o(s) \big( 1 + R_{sn} \big) + \alpha_g(s) \big( 1 + R_{vn} \big)

\alpha_w = k_a \alpha_{rw}(s), \quad \alpha_o = k_a \alpha_{ro}(s), \quad \alpha_g = k_a \alpha_{rg}(s)

k_a(\mathbf{r}) \bigg| _{P_{\bf ref}}

\alpha_{rw}(s) = \frac{k_{rw}(s)}{\mu_w}, \quad \alpha_{ro}(s) = \frac{k_{ro}(s)}{\mu_o}, \quad \alpha_{rg}(s) = \frac{k_{rg}(s)}{\mu_g}

\mu_w = \mu_w(P_{\bf ref}), \quad  \mu_o = \mu_o(P_{\bf ref}), \quad \mu_g = \mu_g(P_{\bf ref})

\rho_{\alpha} = \frac{ \alpha_{rw} \rho_w + \alpha_{ro} \rho_o (1 + R_{sn})  + \alpha_{rg} \rho_g (1+R_{vn}) }{ \alpha_{rw}  + \alpha_{ro}  (1 + R_{sn})  + \alpha_{rg}  (1+R_{vn}) }

\rho_w = \rho_w(P_{\rm ref}), \ \rho_o = \rho_o(P_{\rm ref}), \ \rho_g = \rho_g(P_{\rm ref})

 g = 9.81 \ \textrm{m} / \textrm{s}^2

 \chi = \frac{\alpha}{\beta} = \Big \langle \frac{k} {\mu} \Big \rangle \frac{1}{\phi c_t} \bigg| _{P_{\bf ref}}

\sigma = \alpha \ h = \Big \langle \frac{k} {\mu} \Big \rangle \ h \bigg| _{P_{\bf ref}}

h

q_t =  B_w \, q_W + B_o \, q_O + B_g \, ( q_G - R_s \, q_O)

c_t(s) = c_r (1 + R_{sn} s_o ) + c_w s_w + c_o s_o (1+R_{sn}) + c_g s_g  + R_{sp} s_o 

\alpha(s) = \Big \langle \frac{k} {\mu} \Big \rangle = \alpha_w(s) + \alpha_o(s) \big( 1 + R_{sn} \big) + \alpha_g(s) 

\rho_{\alpha} = \frac{ \alpha_{rw} \rho_w + \alpha_{ro} \rho_o (1 + R_{sn})  + \alpha_{rg} \rho_g  }{ \alpha_{rw}  + \alpha_{ro}  (1 + R_{sn})  + \alpha_{rg}  }

q_t = B_w \, q_W + B_o  q_O

c_t(s) = c_r  + c_w s_w + c_o s_o

\alpha(s) = \Big \langle \frac{k} {\mu} \Big \rangle = \alpha_w(s) + \alpha_o(s)

\rho_{\alpha} = \frac{ \alpha_{rw} \rho_w + \alpha_{ro} \rho_o   }{ \alpha_{rw}  + \alpha_{ro}}

\sigma = \Big \langle \frac{k} {\mu} \Big \rangle \, h = k \, h\,  \Bigg[ \frac{k_{rw}}{\mu_w} + \frac{k_{ro}}{\mu_o} \Bigg]