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Characteristic of compositional multiphase models is that the phases
are not only matter of a single chemical substance. Instead, their
composition in general includes several species, and for the mass transfer,
the component behavior is quite different from the phase behavior. In the following, we
give some basic definitions and assumptions that are required for the
formulation of the model concept below. As an example, we take a
three-phase three-component system water-NAPL-gas
\cite{A3:class:2002a}. The modification for other multicomponent
systems is straightforward and can be found, e.\ g., in
\cite{A3:bielinski:2006,A3:acosta:2006}.
The term {\it component} stands for constituents of the phases which
can be associated with a unique chemical species, or, more generally, with
a group of species exploiting similar physical behavior. In this work, we
assume a water-gas-NAPL system composed of the phases water (subscript
$\text{w}$), gas ($\text{g}$), and NAPL ($\text{n}$). These phases are
composed of the components water (superscript $\text{w}$), air
($\text{a}$), and the organic contaminant ($\text{c}$) (see Fig.
\ref{fig:phaseMassEnergyTransfer}).
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\item[Equilibrium:]
For the non-isothermal multiphase processes in porous media under
consideration, we state that the assumption of local thermal
equilibrium is valid since flow velocities are small. We neglect
chemical reactions and biological decomposition and assume chemical
equilibrium. Mechanical equilibrium is not valid in a porous medium,
since discontinuities in pressure can occur across a fluid-fluid
interface due to capillary effects.
\item[Notation:]
The index $\alpha \in \{\text{w}, \text{n}, \text{g}\}$ refers
to the phase, while the superscript $\kappa \in \{\text{w}, \text{a}, \text{c}\}$ refers
to the component.
\end{description}
\begin{table}
\begin{tabular}{llll}
$p_\alpha$ & phase pressure & $\phi$ & porosity \\
$T$ & temperature & $K$ & absolute permeability tensor \\
$S_\alpha$ & phase saturation & $\tau$ & tortuosity \\
$x_\alpha^\kappa$ & mole fraction of component $\kappa$ in phase $\alpha$ & $\boldsymbol{g}$ & gravitational acceleration \\
$X_\alpha^\kappa$ & mass fraction of component $\kappa$ in phase $\alpha$ & $q^\kappa_\alpha$ & volume source term of $\kappa$ in $\alpha$ \\
$\varrho_{\text{mol},\alpha}$ & molar density of phase $\alpha$ & $u_\alpha$ & specific internal energy \\
$\varrho_{\alpha}$ & mass density of phase $\alpha$ & $h_\alpha$ & specific enthalpy \\
$M$ & molar mass of a phase or component & $c_\text{s}$ & specific heat enthalpy \\
$k_{\text{r}\alpha}$ & relative permeability & $\lambda_\text{pm}$ & heat conductivity \\
$\mu_\alpha$ & phase viscosity & $q^h$ & heat source term \\
$D_\alpha^\kappa$ & diffusivity of component $\kappa$ in phase $\alpha$ & $\boldsymbol{v}_{a,\alpha}$ & advective velocity \\
$\boldsymbol{v}_\alpha$ & velocity (Darcy or free flow)& & \\
\end{tabular}
\caption{Notation list for most of the variables and indices used in \Dumux.}
\todo[inline]{Diese Liste macht ohne ausführliche Beschreibung der Modelle eigentlich keinen Sinn mehr.
Sollen wir sie auch irgendwie auf das doxygen packen?}
\end{table}
\begin{tikzpicture} [>=latex,scale=0.6, every node/.style={transform shape}]
% Ellipse 1 solid
\coordinate (A) at (1,-0.5);
\draw [fill=black!70](A) rectangle(3.5,2) node at(2.25,-0.9) {solid phase (porous matrix)};
% Ellipse 2 water
\coordinate (B) at (-3,5);
\draw [fill=black!20](B) circle(1.5cm);
\node [yshift=5mm]at(-3,2.5){water phase $(w)$};
\draw[fill=white] (B)--+(1.5,0)arc(0:-55:1.5cm)--(B);
\draw[fill=black!40] (B)--+(-55:1.5cm)arc(-55:-105:1.5cm)--(B);
% Ellipse 3 gas
\coordinate (C) at (6,5);
\draw [](C) circle (1.5cm);
\node[yshift=5mm]at(6,2.5){gas phase $(g)$};
\draw [fill=black!40](C)--+(1.5,0)arc(0:-75:1.5cm)--(C);
\draw [fill=black!20] (C)--+(-75:1.5cm)arc(-75:-150:1.5cm)--(C);
% Ellipse 4 napl
\coordinate (D) at (2,10);
\draw [fill=black!40](D) circle (1.5cm);
\node[yshift=5mm]at(2,11.4){NAPL phase $(n)$};
\draw [fill=white](D)--+(1.5,0)arc(0:-45:1.5cm)--(D);
\draw [fill=black!20] (D)--+(0,-1.5)arc(-90:-45:1.5cm)--(D);
% arrows
%A-B
\draw [<->,white](0.5,1.8)--(-1.6,3.6) node[black,above,sloped,pos=0.5]{adsorption};
\draw [<->](0.5,1.8)--(-1.6,3.6) node[below,sloped,pos=0.5]{desorption};
\draw[<-](-1,5.4)--(4,5.4)node[above,sloped,pos=0.5]{condensation, dissolution};
\draw[->](-1,4.5)--(4,4.5)node[above,sloped,pos=0.5]{evaporation, degassing};
%B-D
\draw[<->](-2,6.7)--(0.3,8.7)node[above,sloped,pos=0.5]{dissolution};
%D-C
\draw[->](3.6,8.9)--(5.2,7)node[above,sloped,pos=0.5]{evaporation};
\draw[rotate around={-51:(4,6.8)}](3.35,7.95) ellipse (1.5cm and 0.45cm); %Ellipse um evaporation
\draw[<-](3,8.3)--(4.5,6.5)node[above,sloped,pos=0.55]{condensation};
% thermal energy
\filldraw [black!40](8.5,9.5)rectangle(11,8.5);
\draw (8.5,9.5)rectangle(11,11.5);
\draw (8.5,9.5)--(8.5,8.5);
\draw (11,9.5)--(11,8.5);
\draw [decorate,decoration={bent,aspect=0.4,amplitude=6},fill=black!40](11,8.5)--(8.5,8.5);
\foreach \x in {8.75,9,...,10.8}
\draw [->,decorate,decoration={snake,post length=2mm},thick](\x,7)--(\x,8.15);
\foreach \x in {8.8,9.4,10,10.6}
\draw [->,dotted,decorate,decoration={snake,post length=2mm}](\x,9.5)--(\x,10.7);
\node at(9.75,11){gas};
\node at(9.75,9){NAPL};
\node at(9.85,6.7){thermal energy $(h)$};
% legende
\node at (7.5,1.7){Mass components};
\draw[](6,1)rectangle +(0.3,0.3) node at(6.3,1.15) [right]{Air};
\filldraw[black!20](6,0.4) rectangle +(0.3,0.3) node at (6.3,0.55)[black,right]{Water};
\filldraw[black!40](6,-0.2) rectangle +(0.3,0.3) node at (6.3,-0.1)[right,black]{Organic contaminant (NAPL)};
\filldraw[black!70](6,-0.8) rectangle +(0.3,0.3) node at (6.3,-0.75)[right,black]{Solid phase};
\end{tikzpicture}
\caption{Mass and energy transfer between the phases}
\label{fig:phaseMassEnergyTransfer}
\todo{modelliste einfügen, auf doxygen verweisen, Text überarbeiten (Christoph)}
The following description of the available models is automatically extracted
from the Doxygen documentation.
The fully-implicit models described in this section are using the box or the
cell centered finite volume method as described in section \ref{box} and \ref{cc}
for spatial and the implicit Euler
method as temporal discretization. The models themselves are located in
subdirectories of \texttt{dumux/implicit} of the \Dumux distribution.
The basic idea the so-called decoupled models have in common is to reformulate the
equations of multi-phase flow (e.g. Eq. \ref{A3:eqmass1}) into one equation for
pressure and equations for phase-/component-/etc. transport. The pressure equation
is the sum of the mass balance equations and thus considers the total flow of the
fluid system. The new set of equations is considered as decoupled (or weakly coupled)
and can thus be solved sequentially. The most popular decoupled model is the so-called
fractional flow formulation for two-phase flow which is usually implemented applying
an IMplicit Pressure Explicit Saturation algorithm (IMPES).
In comparison to a fully implicit model, the decoupled structure allows the use of
different discretization methods for the different equations. The standard method
used in the decoupled models is a cell centered finite volume method. Further schemes,
so far only available for the two-phase pressure equation, are cell centered finite
volumes with multi-point flux approximation (MPFA O-method) and mimetic finite differences.
An $h$-adaptive implementation of both decoupled models is provided for two dimensions.