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\-This model solves equations of the form

\[ \phi \frac{\partial (\rho_\alpha S_\alpha)}{\partial t} + \textbf{div}\, (\rho_\alpha \boldsymbol{v_\alpha}) = q_\alpha, \]

where $ S_\alpha $ is the saturation of phase alpha (wetting ( $ w $), non-\/wetting ( $ n $)) and $ \boldsymbol v_\alpha $ is the phase velocity defined by the multi-\/phase \-Darcy equation. \-If a phase velocity is reconstructed from the pressure solution it can be directly inserted in the previous equation. \-In the incompressible case the equation is further divided by the phase density $ \rho_\alpha $. \-If a total velocity is reconstructed the saturation equation is reformulated into\-:

\[ \phi \frac{\partial S_w}{\partial t} + f_w \textbf{div}\, \boldsymbol{v}_{t} + f_w \lambda_n \boldsymbol{K}\left(\textbf{grad}\, p_c + (\rho_n-\rho_w) \, g \, \textbf{grad} z \right)= q_\alpha, \] to get a wetting phase saturation or \[ \phi \frac{\partial S_n}{\partial t} + f_n \textbf{div}\, \boldsymbol{v}_{t} - f_n \lambda_w \boldsymbol{K}\left(\textbf{grad}\, p_c + (\rho_n-\rho_w) \, g \, \textbf{grad} z \right)= q_\alpha, \] if the non-\/wetting phase saturation is the primary transport variable.

\-The total velocity formulation is only implemented for incompressible fluids and $ f_\alpha $ is the fractional flow function, $ \lambda_\alpha $ is the mobility, $ \boldsymbol K $ the absolute permeability, $ p_c $ the capillary pressure, $ \rho $ the fluid density, $ g $ the gravity constant, and $ q $ the source term.

\-In the \-I\-M\-P\-E\-S models the default setting is\-:


\begin{itemize}
\item formulation\-: $ p_w-S_w $ (\-Property\-: {\ttfamily \-Formulation} defined as {\ttfamily \hyperlink{a00056_a04294fbcf0af5328016a160dbd8bfff9}{\-Decoupled\-Two\-P\-Common\-Indices\-::pw\-Sw}})
\item compressibility\-: disabled (\-Property\-: {\ttfamily \-Enable\-Compressibility} set to {\ttfamily false})
\end{itemize}


