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 Bernd Flemisch committed Jul 16, 2010 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 \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 Philipp Nuske committed Dec 10, 2010 45 to the phase, while the superscript $\kappa \in \{\text{w}, \text{a}, \text{c}\}$ refers Bernd Flemisch committed Jul 16, 2010 46 47 48 49 50 51 52 53 54 55 56 57 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 \\ Philipp Nuske committed Feb 23, 2012 58 $\boldsymbol{v}_\alpha$ & Darcy velocity & $\boldsymbol{v}_{a,\alpha}$ & advective velocity Bernd Flemisch committed Jul 16, 2010 59 60 61 62 63 64 65 66 67 68 69 \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: % Philipp Nuske committed Feb 23, 2012 70 \begin{multline} Bernd Flemisch committed Jul 16, 2010 71 \label{A3:eqmass1} Philipp Nuske committed Feb 23, 2012 72 73 \phi \frac{\partial (\sum_\alpha \varrho_{\text{mol}, \alpha} x_\alpha^\kappa S_\alpha )}{\partial t} Bernd Flemisch committed Jul 16, 2010 74 75 76 - \sum\limits_\alpha \Div \left( \frac{k_{\text{r} \alpha}}{\mu_\alpha} \varrho_{\text{mol}, \alpha} x_\alpha^\kappa K (\grad p_\alpha - Philipp Nuske committed Feb 23, 2012 77 78 \varrho_{\alpha} \boldsymbol{g}) \right) \\ % Bernd Flemisch committed Jul 16, 2010 79 % Philipp Nuske committed Feb 23, 2012 80 81 - \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mol}, \alpha} \grad x_\alpha^\kappa \right) Bernd Flemisch committed Jul 16, 2010 82 - q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}. Philipp Nuske committed Feb 23, 2012 83 \end{multline} Bernd Flemisch committed Jul 16, 2010 84 85 86 87 88 89 90 91 92 93 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: Philipp Nuske committed Feb 23, 2012 94 \begin{multline} Bernd Flemisch committed Jul 16, 2010 95 \label{A3:eqmass2} Philipp Nuske committed Feb 23, 2012 96 \frac{\partial C^\kappa}{\partial t} = Bernd Flemisch committed Jul 16, 2010 97 98 99 \sum\limits_\alpha \Div \left( \frac{k_{\text{r} \alpha}}{\mu_\alpha} \varrho_{\text{mass}, \alpha} X_\alpha^\kappa K (\grad p_\alpha + Philipp Nuske committed Feb 23, 2012 100 \varrho_{\text{mass}, \alpha} \boldsymbol{g}) \right) \\ Bernd Flemisch committed Jul 16, 2010 101 102 % % Philipp Nuske committed Feb 23, 2012 103 104 + \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mass}, \alpha} \grad X_\alpha^\kappa \right) Bernd Flemisch committed Jul 16, 2010 105 + q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}. Philipp Nuske committed Feb 23, 2012 106 \end{multline} Bernd Flemisch committed Jul 16, 2010 107 108 109 110 111 112 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 Philipp Nuske committed Feb 23, 2012 113 diffusion. The energy balance can then be Bernd Flemisch committed Jul 16, 2010 114 115 formulated as: % Philipp Nuske committed Feb 23, 2012 116 \begin{multline} Bernd Flemisch committed Jul 16, 2010 117 \label{A3:eqenergmak1} Philipp Nuske committed Feb 23, 2012 118 \phi \frac{\partial \left( \sum_\alpha \varrho_{\alpha} Bernd Flemisch committed Jul 16, 2010 119 120 u_\alpha S_\alpha \right)}{\partial t} + \left( 1 - \phi \right) \frac{\partial \varrho_{\text{s}} c_{\text{s}} Philipp Nuske committed Feb 23, 2012 121 T}{\partial t} Bernd Flemisch committed Jul 16, 2010 122 - \Div \left( \lambda_{\text{pm}} \grad T \right) Philipp Nuske committed Feb 23, 2012 123 124 \\ - \sum\limits_\alpha \Div \left( \frac{k_{\text{r} Bernd Flemisch committed Jul 16, 2010 125 126 \alpha}}{\mu_\alpha} \varrho_{\alpha} h_\alpha K \left( \grad p_\alpha - \varrho_{\alpha} Philipp Nuske committed Feb 23, 2012 127 \boldsymbol{g} \right) \right) Bernd Flemisch committed Jul 16, 2010 128 - q^h \; = \; 0. Philipp Nuske committed Feb 23, 2012 129 \end{multline} Bernd Flemisch committed Jul 16, 2010 130 131 132 133 134 135 136 137 138 139 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. Bernd Flemisch committed Feb 01, 2012 140 \input{box} Bernd Flemisch committed Jul 16, 2010 141 142 143 144 \section{Available models} The following description of the available models is automatically extracted Andreas Lauser committed Feb 21, 2012 145 146 from the Doxygen documentation. % \textbf{TODO}: Unify notation. Bernd Flemisch committed Jul 16, 2010 147 148 149 150 \subsection{Fully-implicit models} The fully-implicit models described in this section are using the box Philipp Nuske committed Jul 20, 2012 151 152 scheme as described in Section \ref{box} for spatial and the implicit Euler method as temporal discretization. The models themselves are located in 153 subdirectories of \texttt{dumux/boxmodels} of the \Dumux distribution. Bernd Flemisch committed Jul 16, 2010 154 155 156 157 158 159 160 \subsubsection{The single-phase model: OnePBoxModel} \input{ModelDescriptions/1pboxmodel} \subsubsection{The single-phase, two-component model: OnePTwoCBoxModel} \input{ModelDescriptions/1p2cboxmodel} Christoph Grueninger committed Feb 24, 2012 161 \subsubsection{The two-phase model using the Richards assumption: RichardsBoxModel} Bernd Flemisch committed Jul 16, 2010 162 163 164 165 166 167 168 169 170 171 172 173 174 175 \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} Andreas Lauser committed Feb 22, 2012 176 \subsubsection{The three-phase, three-component model: ThreePThreeCBoxModel} Andreas Lauser committed Feb 21, 2012 177 178 179 180 181 \input{ModelDescriptions/3p3cboxmodel} \subsubsection{The non-isothermal three-phase, three-component model: ThreePThreeCNIBoxModel} \input{ModelDescriptions/3p3cniboxmodel} Andreas Lauser committed Feb 22, 2012 182 183 184 \subsubsection{The $M$-phase, $N$-component model: MpNcBoxModel} \input{ModelDescriptions/mpncboxmodel} Christoph Grueninger committed Feb 24, 2012 185 \subsubsection{The Stokes model: StokesModel} Andreas Lauser committed Feb 21, 2012 186 187 \input{ModelDescriptions/stokesmodel} Christoph Grueninger committed Feb 24, 2012 188 \subsubsection{The isothermal two-component Stokes model: Stokes2cModel} Andreas Lauser committed Feb 21, 2012 189 190 \input{ModelDescriptions/stokes2cmodel} Christoph Grueninger committed Feb 24, 2012 191 \subsubsection{The non-isothermal two-component Stokes model: Stokes2cniModel} Andreas Lauser committed Feb 21, 2012 192 \input{ModelDescriptions/stokes2cnimodel} Bernd Flemisch committed Jul 16, 2010 193 194 \subsection{Decoupled models} Markus Wolff committed Feb 22, 2012 195 196 197 198 199 % 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} Andreas Lauser committed Feb 22, 2012 200 201 202 203 204 205 \input{ModelDescriptions/1pdecoupledmodel} \subsubsection{The two-phase model} \paragraph{Pressure model} \input{ModelDescriptions/2pdecoupledpressuremodel} Bernd Flemisch committed Jul 16, 2010 206 Andreas Lauser committed Feb 22, 2012 207 208 \paragraph{Saturation model} \input{ModelDescriptions/2pdecoupledsaturationmodel} Bernd Flemisch committed Jul 16, 2010 209 Markus Wolff committed Feb 22, 2012 210 \subsubsection{The two-phase, two-component model} Melanie Darcis committed Oct 04, 2012 211 212 \input{ModelDescriptions/2p2cdecoupledpressuremodel} \input{ModelDescriptions/2p2cdecoupledtransportmodel} Bernd Flemisch committed Jul 16, 2010 213 214 215 216 217 %%% Local Variables: %%% mode: latex %%% TeX-master: "dumux-handbook" %%% End: