Commit 8e3e4617 by Alexander Kissinger

### - Updated the model section in the handbook. The nonisothermal model...

- Updated the model section in the handbook. The nonisothermal model description is added and the 2pni, 2p2cni, 3p3cni are removed.
- Updated models.tex accordingly
- Generated documentation for the non-isothermal model

git-svn-id: svn://svn.iws.uni-stuttgart.de/DUMUX/dumux/trunk@13437 2fb0f335-1f38-0410-981e-8018bf24f1b0
parent fb2b307a
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 non-\/isothermal two-\/phase flow of two compressible and partly miscible fluids $\alpha \in \{ w, n \}$. Thus each component $\kappa \in \{ w, a \}$ can be present in each phase. Using the standard multiphase Darcy approach a mass balance equation is solved\-: \begin{eqnarray*} && \phi \frac{\partial (\sum_\alpha \varrho_\alpha X_\alpha^\kappa S_\alpha )}{\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} (\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\}\\ &-& \sum_\alpha \text{div} \left\{ D_{\alpha,\text{pm}}^\kappa \varrho_{\alpha} \frac{M^\kappa}{M_\alpha} \textbf{grad} x^\kappa_{\alpha} \right\} - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{w, a\} \, , \alpha \in \{w, n\} \end{eqnarray*} For the energy balance, local thermal equilibrium is assumed which results in one energy conservation equation for the porous solid matrix and the fluids\-: \begin{eqnarray*} && \phi \frac{\partial \left( \sum_\alpha \varrho_\alpha u_\alpha S_\alpha \right)}{\partial t} + \left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ &-& \text{div} \left( \lambda_\text{pm} \textbf{grad} \, T \right) - q^h = 0 \qquad \alpha \in \{w, n\} \end{eqnarray*} All equations are discretized using a vertex-\/centered finite volume (box) or cell-\/centered finite volume scheme as spatial and the implicit Euler method as time discretization. By using constitutive relations for the capillary pressure $p_c = p_n - p_w$ and relative permeability $k_{r\alpha}$ and taking advantage of the fact that $S_w + S_n = 1$ and $X^\kappa_w + X^\kappa_n = 1$, the number of unknowns can be reduced to two. If both phases are present the primary variables are, like in the nonisothermal two-\/phase model, either $p_w$, $S_n$ and temperature or $p_n$, $S_w$ and temperature. The formulation which ought to be used can be specified by setting the {\ttfamily Formulation} property to either {\ttfamily Two\-P\-Two\-C\-Indices\-::p\-Ws\-N} or {\ttfamily Two\-P\-Two\-C\-Indices\-::p\-Ns\-W}. By default, the model uses $p_w$ and $S_n$. In case that only one phase (nonwetting or wetting phase) is present the second primary variable represents a mass fraction. The correct assignment of the second primary variable is performed by a phase state dependent primary variable switch. The phase state is stored for all nodes of the system. The following cases can be distinguished\-: \begin{itemize} \item Both phases are present\-: The saturation is used (either $S_n$ or $S_w$, dependent on the chosen formulation). \item Only wetting phase is present\-: The mass fraction of air in the wetting phase $X^a_w$ is used. \item Only non-\/wetting phase is present\-: The mass fraction of water in the non-\/wetting phase, $X^w_n$, is used. \end{itemize}
 ... ... @@ -18,7 +18,7 @@ In the I\-M\-P\-E\-S models the default setting is\-: \begin{itemize} \item formulation\-: $p_w-S_w$ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00095_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}}) \item formulation\-: $p_w-S_w$ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00099_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}}) \item compressibility\-: disabled (Property\-: {\itshape Enable\-Compressibility} set to {\itshape false}) \end{itemize} ... ...
 ... ... @@ -16,7 +16,7 @@ The total velocity formulation is only implemented for incompressible fluids and In the I\-M\-P\-E\-S models the default setting is\-: formulation\-: $p_w$ -\/ $S_w$ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00095_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}}) formulation\-: $p_w$ -\/ $S_w$ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00099_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}}) compressibility\-: disabled (Property\-: {\itshape Enable\-Compressibility} set to {\itshape false}) ... ...
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 non-\/isothermal two-\/phase flow for two immiscible fluids $\alpha \in \{ w, n \}$. Using the standard multiphase Darcy approach, the mass conservation equations for both phases can be described as follows\-: $\phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \textbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \right\} - q_\alpha = 0 \qquad \alpha \in \{w, n\}$ For the energy balance, local thermal equilibrium is assumed. This results in one energy conservation equation for the porous solid matrix and the fluids\-: \begin{align*} \phi \frac{\partial \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t} & + \left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\,p_\alpha - \varrho_\alpha \mbox{\bf g} \right) \right\} \\ & - \text{div} \left(\lambda_{pm} \textbf{grad} \, T \right) - q^h = 0, \qquad \alpha \in \{w, n\} \;, \end{align*} where $h_\alpha$ is the specific enthalpy of a fluid phase $\alpha$ and $u_\alpha = h_\alpha - p_\alpha/\varrho_\alpha$ is the specific internal energy of the phase. All equations are discretized using a vertex-\/centered finite volume (box) or cell-\/centered finite volume scheme as spatial and the implicit Euler method as time discretization. Currently the model supports choosing either $p_w$, $S_n$ and $T$ or $p_n$, $S_w$ and $T$ as primary variables. The formulation which ought to be used can be specified by setting the {\ttfamily Formulation} property to either {\ttfamily Two\-P\-N\-I\-Indices\-::p\-Ws\-N} or {\ttfamily Two\-P\-Indices\-::p\-Ns\-W}. By default, the model uses $p_w$, $S_n$ and $T$.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 three-\/phase three-\/component flow of three fluid phases $\alpha \in \{ \text{water, gas, NAPL} \}$ each composed of up to three components $\kappa \in \{ \text{water, air, contaminant} \}$. The standard multiphase Darcy approach is used as the equation for the conservation of momentum\-: $v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right)$ By inserting this into the equations for the conservation of the components, one transport equation for each component is obtained as \begin{eqnarray*} && \phi \frac{\partial (\sum_\alpha \varrho_\alpha X_\alpha^\kappa S_\alpha )}{\partial t} - \sum\limits_\alpha \text{div} \left\{ \frac{k_{r\alpha}}{\mu_\alpha} \varrho_\alpha X_\alpha^\kappa \mathbf{K} (\textbf{grad}\; p_\alpha - \varrho_\alpha \mbox{\bf g}) \right\} \nonumber \\ \nonumber \\ && - \sum\limits_\alpha \text{div} \left\{ D_{\alpha,\text{pm}}^\kappa \varrho_\alpha \frac{M^\kappa}{M_\alpha} \textbf{grad} x^\kappa_{\alpha} \right\} - q^\kappa = 0 \qquad \forall \kappa , \; \forall \alpha \end{eqnarray*} Note that these balance equations above are molar. In addition to that, a single balance of thermal energy is formulated for the fluid-\/filled porous medium under the assumption of local thermal equilibrium \begin{eqnarray*} && \phi \frac{\partial \left( \sum_\alpha \varrho_\alpha u_\alpha S_\alpha \right)}{\partial t} + \left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ &-& \text{div} \left( \lambda_{pm} \textbf{grad} \, T \right) - q^h = 0 \qquad \alpha \in \{w, n, g\} \end{eqnarray*} All equations are discretized using a vertex-\/centered finite volume (box) or cell-\/centered finite volume scheme as spatial and the implicit Euler method as time discretization. The model uses commonly applied auxiliary conditions like $S_w + S_n + S_g = 1$ for the saturations and $x^w_\alpha + x^a_\alpha + x^c_\alpha = 1$ for the mole fractions. Furthermore, the phase pressures are related to each other via capillary pressures between the fluid phases, which are functions of the saturation, e.\-g. according to the approach of Parker et al. The used primary variables are dependent on the locally present fluid phases. An adaptive primary variable switch is included. The phase state is stored for all nodes of the system. The following cases can be distinguished\-: \begin{itemize} \item All three phases are present\-: Primary variables are two saturations $(S_w$ and $S_n)$, a pressure, in this case $p_g$, and the temperature $T$. \item Only the water phase is present\-: Primary variables are now the mole fractions of air and contaminant in the water phase $(x_w^a$ and $x_w^c)$, as well as temperature and the gas pressure, which is, of course, in a case where only the water phase is present, just the same as the water pressure. \item Gas and N\-A\-P\-L phases are present\-: Primary variables $(S_n$, $x_g^w$, $p_g$, $T)$. \item Water and N\-A\-P\-L phases are present\-: Primary variables $(S_n$, $x_w^a$, $p_g$, $T)$. \item Only gas phase is present\-: Primary variables $(x_g^w$, $x_g^c$, $p_g$, $T)$. \item Water and gas phases are present\-: Primary variables $(S_w$, $x_w^g$, $p_g$, $T)$. \end{itemize}
 ... ... @@ -6,7 +6,7 @@ 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 momentum approximation can be selected via \char\"{}\-Base\-Flux\-Variables\char\"{}\-: Darcy (\hyperlink{a00270}{Implicit\-Darcy\-Flux\-Variables}) and Forchheimer (\hyperlink{a00271}{Implicit\-Forchheimer\-Flux\-Variables}) relations are available for all Box models. The momentum approximation can be selected via \char\"{}\-Base\-Flux\-Variables\char\"{}\-: Darcy (\hyperlink{a00278}{Implicit\-Darcy\-Flux\-Variables}) and Forchheimer (\hyperlink{a00279}{Implicit\-Forchheimer\-Flux\-Variables}) relations are available for all Box models. By inserting this into the equations for the conservation of the mass of each component, one gets one mass-\/continuity equation for each component $\kappa$ $\sum_{\kappa} \left( \phi \frac{\partial \left(\varrho_\alpha x_\alpha^\kappa S_\alpha\right)}{\partial t} + \mathrm{div}\; \left\{ v_\alpha \frac{\varrho_\alpha}{\overline M_\alpha} x_\alpha^\kappa \right\} \right) = q^\kappa$ with $\overline M_\alpha$ being the average molar mass of the phase $\alpha$\-: $\overline M_\alpha = \sum_\kappa M^\kappa \; x_\alpha^\kappa$ ... ...
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 generic energy balance for single and multi-\/phase transport problems. Currently the non-\/isothermal model can be used on top of the 1p2c, 2p, 2p2c and 3p3c models. Comparison to simple analytical solutions for pure convective and conductive problems are found in the 1p2c test. Also refer to this test for details on how to activate the non-\/isothermal model. For the energy balance, local thermal equilibrium is assumed. This results in one energy conservation equation for the porous solid matrix and the fluids\-: \begin{align*} \phi \frac{\partial \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t} & + \left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\,p_\alpha - \varrho_\alpha \mbox{\bf g} \right) \right\} \\ & - \text{div} \left(\lambda_{pm} \textbf{grad} \, T \right) - q^h = 0. \end{align*} where $h_\alpha$ is the specific enthalpy of a fluid phase $\alpha$ and $u_\alpha = h_\alpha - p_\alpha/\varrho_\alpha$ is the specific internal energy of the phase.
 ... ... @@ -8,7 +8,7 @@ This model implements laminar Stokes flow of a single fluid, solving the momentu and the mass balance equation $\frac{\partial \varrho_g}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_g {\boldsymbol{v}}_g\right) - q_g = 0.$ By setting the property {\ttfamily Enable\-Navier\-Stokes} to {\ttfamily true} the Navier-\/\-Stokes equation can be solved. In this case an additional term $\varrho_g \left(\boldsymbol{v}_g \boldsymbol{\cdot} \boldsymbol{\nabla} \right) \boldsymbol{v}_g$ is added to the momentum balance equation. By setting the property {\ttfamily Enable\-Navier\-Stokes} to {\ttfamily true} the Navier-\/\-Stokes equation can be solved. In this case an additional term $+ \varrho_g \left(\boldsymbol{v}_g \boldsymbol{\cdot} \boldsymbol{\nabla} \right) \boldsymbol{v}_g$ is added to the momentum balance equation. This is discretized by a fully-\/coupled vertex-\/centered finite volume (box) scheme in space and by the implicit Euler method in time.
 ... ... @@ -10,7 +10,7 @@ Momentum Balance\-: $\frac{\partial \left(\varrho_g {\boldsymbol{v}}_g\right)} Mass balance equation\-: \[ \frac{\partial \varrho_g}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_g {\boldsymbol{v}}_g\right) - q_g = 0$ \hyperlink{a00084}{Component} mass balance equations\-: $\frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \boldsymbol{\nabla} X_g^\kappa \right) - q_g^\kappa = 0$ \hyperlink{a00088}{Component} mass balance equations\-: $\frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \boldsymbol{\nabla} X_g^\kappa \right) - q_g^\kappa = 0$ This is discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method in time.
 ... ... @@ -10,9 +10,9 @@ Momentum Balance\-: $\frac{\partial \left(\varrho_g {\boldsymbol{v}}_g\right)} Mass balance equation\-: \[ \frac{\partial \varrho_g}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_g {\boldsymbol{v}}_g\right) - q_g = 0$ \hyperlink{a00084}{Component} mass balance equation\-: $\frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \boldsymbol{\nabla} x_g^\kappa \right) - q_g^\kappa = 0$ \hyperlink{a00088}{Component} mass balance equation\-: $\frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \boldsymbol{\nabla} x_g^\kappa \right) - q_g^\kappa = 0$ Energy balance equation\-: $\frac{\partial (\varrho_g u_g)}{\partial t} + \boldsymbol{\nabla} \left( \boldsymbol{\cdot} \varrho_g h_g {\boldsymbol{v}}_g - \lambda_g \boldsymbol{\nabla} T \right) - q_T = 0$ Energy balance equation\-: $\frac{\partial (\varrho_g u_g)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g h_g {\boldsymbol{v}}_g - \sum_\kappa \left[ h^\kappa_g D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \nabla x^\kappa_g \right] - \lambda_g \boldsymbol{\nabla} T \right) - q_T = 0$ This is discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as temporal discretization.
 ... ... @@ -219,20 +219,12 @@ subdirectories of \texttt{dumux/implicit} of the \Dumux distribution. \subsubsection{The two-phase model: TwoPModel} \input{ModelDescriptions/2pimplicitmodel} \subsubsection{The non-isothermal two-phase model: TwoPNIModel} \input{ModelDescriptions/2pniimplicitmodel} \subsubsection{The two-phase, two-component model: TwoPTwoCModel} \input{ModelDescriptions/2p2cimplicitmodel} \subsubsection{The CO2 model: CO2Model} \input{ModelDescriptions/co2implicitmodel} \subsubsection{The non-isothermal two-phase, two-component model: TwoPTwoCNIModel} \input{ModelDescriptions/2p2cniimplicitmodel} \subsubsection{The non-isothermal CO2 model: CO2NIModel} \input{ModelDescriptions/co2niimplicitmodel} \subsubsection{The three-phase model: ThreePModel} \input{ModelDescriptions/3pimplicitmodel} ... ... @@ -240,8 +232,8 @@ subdirectories of \texttt{dumux/implicit} of the \Dumux distribution. \subsubsection{The three-phase, three-component model: ThreePThreeCModel} \input{ModelDescriptions/3p3cimplicitmodel} \subsubsection{The non-isothermal three-phase, three-component model: ThreePThreeCNIModel} \input{ModelDescriptions/3p3cniimplicitmodel} \subsubsection{The non-isothermal model: NIModel} \input{ModelDescriptions/nonisothermalimplicitmodel} \subsubsection{The $M$-phase, $N$-component model: MpNcModel} \input{ModelDescriptions/mpncimplicitmodel} ... ...
 ... ... @@ -19,7 +19,7 @@ /*! * \file * * \brief the implicit non-isothermal model * \brief The implicit non-isothermal model. */ #ifndef DUMUX_NI_MODEL_HH #define DUMUX_NI_MODEL_HH ... ... @@ -29,12 +29,38 @@ namespace Dumux { /*! * \ingroup NIModel * * * This model implements a non-isothermal flow for single and multi-phase * transport problems * \brief The implicit non-isothermal model. * * This model implements a generic energy balance for single and multi-phase * transport problems. Currently the non-isothermal model can be used on top of * the 1p2c, 2p, 2p2c and 3p3c models. Comparison to simple analytical solutions * for pure convective and conductive problems are found in the 1p2c test. Also refer * to this test for details on how to activate the non-isothermal model. * * For the energy balance, local thermal equilibrium is assumed. This * results in one energy conservation equation for the porous solid * matrix and the fluids: \f{align*}{ \phi \frac{\partial \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t} & + \left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\,p_\alpha - \varrho_\alpha \mbox{\bf g} \right) \right\} \\ & - \text{div} \left(\lambda_{pm} \textbf{grad} \, T \right) - q^h = 0. \f} * where \f$h_\alpha\f$ is the specific enthalpy of a fluid phase * \f$\alpha\f$ and \f$u_\alpha = h_\alpha - * p_\alpha/\varrho_\alpha\f$ is the specific internal energy of the * phase. * * \todo The equations have to be added here */ template class NIModel : public GET_PROP_TYPE(TypeTag, IsothermalModel) { ... ...