%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This file has been autogenerated from the LaTeX part of the   %
% doxygen documentation; DO NOT EDIT IT! Change the model's .hh %
% file instead!!                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\-This model implements a $M$-\/phase flow of a fluid mixture composed of $N$ chemical species. \-The phases are denoted by lower index $\alpha \in \{ 1, \dots, M \}$. \-All fluid phases are mixtures of $N \geq M - 1$ chemical species which are denoted by the upper index $\kappa \in \{ 1, \dots, N \} $.

\-The standard multi-\/phase \-Darcy law is used as the equation for the conservation of momentum\-: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \boldsymbol{K} \left( \text{grad}\left(p_\alpha - \varrho_{\alpha} g\right) \right) \]

\-By inserting this into the equations for the conservation of the mass of each component, one gets one mass-\/continuity equation for each component $\kappa$ \[ \sum_{\kappa} \left( \phi \frac{\partial \varrho_\alpha x_\alpha^\kappa S_\alpha}{\partial t} - \mathrm{div}\; \left\{ \frac{\varrho_\alpha}{\overline M_\alpha} x_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \boldsymbol{K} \text{grad}\left( p_\alpha - \varrho_{\alpha} g\right) \right\} \right) = q^\kappa \] with $\overline M_\alpha$ being the average molar mass of the phase $\alpha$\-: \[ \overline M_\alpha = \sum_\kappa M^\kappa \; x_\alpha^\kappa \]

\-For the missing $M$ model assumptions, the model assumes that if a fluid phase is not present, the sum of the mole fractions of this fluid phase is smaller than $1$, i.\-e. \[ \forall \alpha: S_\alpha = 0 \implies \sum_\kappa x_\alpha^\kappa \leq 1 \]

\-Also, if a fluid phase may be present at a given spatial location its saturation must be positive\-: \[ \forall \alpha: \sum_\kappa x_\alpha^\kappa = 1 \implies S_\alpha \geq 0 \]

\-Since at any given spatial location, a phase is always either present or not present, one of the strict equalities on the right hand side is always true, i.\-e. \[ \forall \alpha: S_\alpha \left( \sum_\kappa x_\alpha^\kappa - 1 \right) = 0 \] always holds.

\-These three equations constitute a non-\/linear complementarity problem, which can be solved using so-\/called non-\/linear complementarity functions $\Phi(a, b)$ which have the property \[\Phi(a,b) = 0 \iff a \geq0 \land b \geq0 \land a \cdot b = 0 \]

\-Several non-\/linear complementarity functions have been suggested, e.\-g. the \-Fischer-\/\-Burmeister function \[ \Phi(a,b) = a + b - \sqrt{a^2 + b^2} \;. \] \-This model uses \[ \Phi(a,b) = \min \{a, b \}\;, \] because of its piecewise linearity.

\-These equations are then discretized using a fully-\/implicit vertex centered finite volume scheme (often known as 'box'-\/scheme) for spatial discretization and the implicit \-Euler method as temporal discretization.

\-The model assumes local thermodynamic equilibrium and uses the following primary variables\-:
\begin{itemize}
\item \-The component fugacities $f^1, \dots, f^{N}$
\item \-The pressure of the first phase $p_1$
\item \-The saturations of the first $M-1$ phases $S_1, \dots, S_{M-1}$
\item \-Temperature $T$ if the energy equation is enabled
\end{itemize}