\-This model solves equations of the form \[\phi\left(\rho_w \frac{\partial S_w}{\partial t}+\rho_n \frac{\partial S_n}{\partial t}\right)+\textbf{div}\,\boldsymbol{v}_{total}= q. \]\-The definition of the total velocity $\boldsymbol{v}_{total}$ depends on the choice of the primary pressure variable. \-Further, fluids can be assumed to be compressible or incompressible (\-Property\-: {\ttfamily\-Enable\-Compressibility}). \-In the incompressible case a wetting $(w)$ phase pressure as primary variable leads to
\[-\textbf{div}\,\left[\lambda\boldsymbol K \left(\textbf{grad}\, p_w + f_n \textbf{grad}\, p_c +\sum f_\alpha\rho_\alpha\, g \,\textbf{grad}\, z\right)\right]= q, \]
a non-\/wetting ( $ n $) phase pressure yields \[-\textbf{div}\,\left[\lambda\boldsymbol K \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c +\sum f_\alpha\rho_\alpha\, g \,\textbf{grad}\, z\right)\right]= q, \] and a global pressure leads to \[-\textbf{div}\,\left[\lambda\boldsymbol K \left(\textbf{grad}\, p_{global}+\sum f_\alpha\rho_\alpha\, g \,\textbf{grad}\, z\right)\right]= q. \]\-Here, $ p_\alpha$ is a phase pressure, $ p_{global}$ the global pressure of a classical fractional flow formulation (see e.\-g. \-P. \-Binning and \-M. \-A. \-Celia, ''\-Practical implementation of the fractional flow approach to multi-\/phase flow simulation'', \-Advances in water resources, vol. 22, no. 5, pp. 461-\/478, 1999.), $ p_c = p_n - p_w $ is the capillary pressure, $\boldsymbol K $ the absolute permeability, $\lambda=\lambda_w +\lambda_n $ the total mobility depending on the saturation ( $\lambda_\alpha= k_{r_\alpha}/\mu_\alpha$), $ f_\alpha=\lambda_\alpha/\lambda$ the fractional flow function of a phase, $\rho_\alpha$ a phase density, $ g $ the gravity constant and $ q $ the source term.
\-The slightly compressible case is only implemented for phase pressures! \-In this case for a wetting $(w)$ phase pressure as primary variable the equations are formulated as \[\phi\left(\rho_w \frac{\partial S_w}{\partial t}+\rho_n \frac{\partial S_n}{\partial t}\right)-\textbf{div}\,\left[\lambda\boldsymbol{K}\left(\textbf{grad}\, p_w + f_n \,\textbf{grad}\, p_c +\sum f_\alpha\rho_\alpha\, g \,\textbf{grad}\, z\right)\right]= q, \] and for a non-\/wetting ( $ n $) phase pressure as \[\phi\left(\rho_w \frac{\partial S_w}{\partial t}+\rho_n \frac{\partial S_n}{\partial t}\right)-\textbf{div}\,\left[\lambda\boldsymbol{K}\left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c +\sum f_\alpha\rho_\alpha\, g \,\textbf{grad}\, z\right)\right]= q, \]\-In this slightly compressible case the following definitions are valid\-: $\lambda=\rho_w \lambda_w +\rho_n \lambda_n $, $ f_\alpha=(\rho_\alpha\lambda_\alpha)/\lambda$\-This model assumes that temporal changes in density are very small and thus terms of temporal derivatives are negligible in the pressure equation. \-Depending on the formulation the terms including time derivatives of saturations are simplified by inserting $ S_w + S_n =1$.
\-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})
