opm-simulators/doc/handbook/ModelDescriptions/immisciblemodel.tex

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This model multi-\/phase flow of $M > 0$ immiscible fluids $\alpha$. By default, the standard multi-\/phase Darcy approach is used to determine the velocity, i.\-e. \[ \mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \;, \] although the actual approach which is used can be specified via the {\ttfamily Velocity\-Module} property. For example, the velocity model can by changed to the Forchheimer approach by
\begin{lstlisting}[style=eWomsCode]
 SET_TYPE_PROP(MyProblemTypeTag, VelocityModule,
      Ewoms::BoxForchheimerVelocityModule<TypeTag>);
\end{lstlisting}


The core of the model is the conservation mass of each component by means of the equation \[ \frac{\partial\;\phi S_\alpha \rho_\alpha }{\partial t} - \text{div} \left\{ \rho_\alpha \mathbf{v}_\alpha \right\} - q_\alpha = 0 \;. \]

These equations are discretized by a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization.

The model uses the following primary variables\-:
\begin{itemize}
\item The pressure $p_0$ in Pascal of the phase with the lowest index
\item The saturations $S_\alpha$ of the $M - 1$ phases that exhibit the lowest indices
\item The absolute temperature $T$ in Kelvin if energy is conserved via the energy equation
\end{itemize}