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\chapter[Models]{Physical and numerical models}

\section{Physical and mathematical description} 

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}.

\subsection{Basic Definitions and Assumptions for the Compositional
  Model Concept}
\textbf{Components:}
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{A3:fig:mundwtrans}).
%
\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\linewidth]{EPS/masstransfer}
  \caption{Mass and energy transfer between the phases}
  \label{A3:fig:mundwtrans}
\end{figure}

\textbf{Equilibrium:}
For the nonisothermal 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.

\textbf{Notation:} The index $\alpha \in \{\text{w}, \text{n}, \text{g}\}$ refers 
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to the phase, while the superscript $\kappa \in \{\text{w}, \text{a}, \text{c}\}$ refers 
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to the component. \\
\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 \\
$k_{\text{r}\alpha}$ & relative permeability & $c_\text{s}$ & specific heat enthalpy \\
$\mu_\alpha$ & phase viscosity & $\lambda_\text{pm}$ & heat conductivity \\
$D_\alpha^\kappa$ & diffusivity of component $\kappa$ in phase $\alpha$ & $q^h$ & heat source term \\
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$\boldsymbol{v}_\alpha$ & Darcy velocity & $\boldsymbol{v}_{a,\alpha}$  & advective velocity
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\end{tabular}


\subsection{Balance Equations}
For the balance equations for multicomponent systems, it is in many
cases convenient to use a molar formulation of the continuity
equation. Considering the mass conservation for each component allows
us to drop source/sink terms for describing the mass transfer between
phases. Then, the
molar mass balance can be written as:
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\begin{multline}
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  \label{A3:eqmass1}
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 \phi \frac{\partial (\sum_\alpha \varrho_{\text{mol}, \alpha}
    x_\alpha^\kappa S_\alpha )}{\partial t}  
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 - \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
        \alpha}}{\mu_\alpha} \varrho_{\text{mol}, \alpha}
    x_\alpha^\kappa K (\grad p_\alpha -
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    \varrho_{\alpha} \boldsymbol{g}) \right)  \\
  % 
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  %
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 - \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mol},
      \alpha} \grad x_\alpha^\kappa \right)  
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 - q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}.
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\end{multline}
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The component mass balance can also be written in terms of mass fractions 
by replacing molar densities by mass densities and mole by mass fractions.
To obtain a single conserved quantity in the temporal derivative, the total 
concentration, representing the mass of one component per unit volume, is defined as
\begin{displaymath}
C^\kappa = \sum_\alpha \phi S_\alpha \varrho_{\text{mass},\alpha} X_\alpha^\kappa \; .
\end{displaymath}
Using this definition, the component mass balance is written as:

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\begin{multline}
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  \label{A3:eqmass2}
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    \frac{\partial C^\kappa}{\partial t} = 
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  \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
        \alpha}}{\mu_\alpha} \varrho_{\text{mass}, \alpha}
    X_\alpha^\kappa K (\grad p_\alpha +
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    \varrho_{\text{mass}, \alpha} \boldsymbol{g}) \right)  \\
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  %
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   + \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mass},
      \alpha} \grad X_\alpha^\kappa \right)  
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 + q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}.
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\end{multline}
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In the case of non-isothermal systems, we further have to balance the
thermal energy. We assume fully reversible processes, such that entropy
is not needed as a model parameter. Furthermore, we neglect 
dissipative effects and the heat transport due to molecular
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diffusion. The energy balance can then be
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formulated as:
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\begin{multline}
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  \label{A3:eqenergmak1}
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  \phi \frac{\partial \left( \sum_\alpha \varrho_{\alpha}
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      u_\alpha S_\alpha \right)}{\partial t} + \left( 1 -
    \phi \right) \frac{\partial \varrho_{\text{s}} c_{\text{s}}
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    T}{\partial t}  
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 - \Div \left( \lambda_{\text{pm}} \grad T \right)
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   \\
   - \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
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        \alpha}}{\mu_\alpha} \varrho_{\alpha} h_\alpha
    K \left( \grad p_\alpha - \varrho_{\alpha}
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      \boldsymbol{g} \right) \right)  
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 - q^h \; = \; 0.
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\end{multline}
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In order to close the system, supplementary constraints for capillary pressure, saturations and mole
fractions are needed, \cite{A3:helmig:1997}. 
According to the Gibbsian phase rule, the number of degrees of freedom
in a non-isothermal compositional multiphase system is equal to the
number of components plus one. This means we need as many independent
unknowns in the system description. The
available primary variables are, e.\ g., saturations, mole/mass
fractions, temperature, pressures, etc.

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\input{box}
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\section{Available models} 
The following description of the available models is automatically extracted 
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from the Doxygen documentation.
% \textbf{TODO}: Unify notation. 
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\subsection{Fully-implicit models} 

The fully-implicit models described in this section are using the box
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scheme as described in Section \ref{box} for spatial and the implicit Euler
method as temporal discretization. The models themselves are located in
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subdirectories of \texttt{dumux/boxmodels} of the \Dumux distribution.
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\subsubsection{The single-phase model: OnePBoxModel} 
\input{ModelDescriptions/1pboxmodel}

\subsubsection{The single-phase, two-component model:  OnePTwoCBoxModel} 
\input{ModelDescriptions/1p2cboxmodel}

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\subsubsection{The two-phase model using the Richards assumption: RichardsBoxModel} 
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\input{ModelDescriptions/richardsboxmodel}

\subsubsection{The two-phase model: TwoPBoxModel}
\input{ModelDescriptions/2pboxmodel}

\subsubsection{The non-isothermal two-phase model: TwoPNIBoxModel} 
\input{ModelDescriptions/2pniboxmodel}

\subsubsection{The two-phase, two-component model: TwoPTwoCBoxModel} 
\input{ModelDescriptions/2p2cboxmodel}

\subsubsection{The non-isothermal two-phase, two-component model: TwoPTwoCNIBoxModel} 
\input{ModelDescriptions/2p2cniboxmodel}

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\subsubsection{The three-phase, three-component model: ThreePThreeCBoxModel}
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\input{ModelDescriptions/3p3cboxmodel}

\subsubsection{The non-isothermal three-phase, three-component model: ThreePThreeCNIBoxModel} 
\input{ModelDescriptions/3p3cniboxmodel}

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\subsubsection{The $M$-phase, $N$-component model: MpNcBoxModel} 
\input{ModelDescriptions/mpncboxmodel}

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\subsubsection{The Stokes model: StokesModel} 
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\input{ModelDescriptions/stokesmodel}

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\subsubsection{The isothermal two-component Stokes model: Stokes2cModel} 
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\input{ModelDescriptions/stokes2cmodel}

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\subsubsection{The non-isothermal two-component Stokes model: Stokes2cniModel} 
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\input{ModelDescriptions/stokes2cnimodel}
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\subsection{Decoupled models}
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%
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.
%
\subsubsection{The one-phase model}
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\input{ModelDescriptions/1pdecoupledmodel}

\subsubsection{The two-phase model}

\paragraph{Pressure model}
\input{ModelDescriptions/2pdecoupledpressuremodel}
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\paragraph{Saturation model}
\input{ModelDescriptions/2pdecoupledsaturationmodel}
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\subsubsection{The two-phase, two-component model}
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\input{ModelDescriptions/2p2cdecoupledpressuremodel}
\input{ModelDescriptions/2p2cdecoupledtransportmodel}
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