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\partial_t \bigg [  \phi \ \rho_W  \bigg ]  + \nabla \bigg (     \rho_w \ \mathbf{u}_w     \bigg )      =  q_{mW}(\mathbf{r}) 

\partial_t \bigg [  \phi \ \rho_O \bigg ]  + \nabla \bigg (   {\tilde m}_{Oo} \ \rho_o \ \mathbf{u}_o 

+  {\tilde m}_{Og} \ \rho_{g} \  \mathbf{u}_g    \bigg )       =  q_{mO}(\mathbf{r})

\partial_t \bigg [  \phi \ \rho_G  \bigg ]  +  \nabla  \bigg (  {\tilde m}_{Go} \ \rho_{o} \ \mathbf{u}_o

+    {\tilde m}_{Gg} \ \rho_g \ \mathbf{u}_g  \bigg )     =  q_{mG}(\mathbf{r}) 

\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w  \mathbf{g} )

\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o   \mathbf{g} )

\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g  \mathbf{g} )

P_o - P_w = P_{cow}(s_w)

P_o - P_g = P_{cog}(s_g)

s_w + s_o + s_g = 1