Commit a6f910d7 authored by Andreas Lauser's avatar Andreas Lauser
Browse files

handbook update model descriptions

We now use the doxygen detailed description for the immiscible
decoupled models, as approved by Markus.

git-svn-id: svn://svn.iws.uni-stuttgart.de/DUMUX/dumux/trunk@7863 2fb0f335-1f38-0410-981e-8018bf24f1b0
parent 384f96e7
......@@ -4,5 +4,5 @@
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\-Single-\/phase compressible isothermal flow model, \begin{align*} \phi \frac{\partial \varrho}{\partial t} + \text{div} (- \varrho \frac{\textbf K}{\mu} ( \text{grad}\, p -\varrho {\textbf g})) = q, \end{align*} discretized using a vertex-\/centered finite volume (box) scheme as spatial and the implicit \-Euler method as time discretization. \-Of course, the model can also be used for incompressible single phase flow modeling, if a fluid with constant density is chosen in the problem file.
\-Single-\/phase, isothermal flow model, which solves the mass continuity equation \[ \phi \frac{\partial \varrho}{\partial t} + \text{div} (- \varrho \frac{\textbf K}{\mu} ( \textbf{grad}\, p -\varrho {\textbf g})) = q, \] discretized using a vertex-\/centered finite volume (box) scheme as spatial and the implicit \-Euler method as time discretization. \-The model supports compressible as well as incompressible fluids.
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\-This model solves equations of the form \[ \textbf{div}\, \boldsymbol v = q. \] \-The velocity $ \boldsymbol v $ is the single phase \-Darcy velocity\-: \[ \boldsymbol v = -\frac{1}{\mu} \boldsymbol K \left(\textbf{grad}\, p + \rho \, g \, \textbf{grad}\, z\right), \] where $ p $ is the pressure, $ \boldsymbol K $ the absolute permeability, $ \mu $ the viscosity, $ \rho $ the density, and $ g $ the gravity constant, and $ q $ is the source term. \-At the boundary, $ p = p_D $ on $ \Gamma_{Dirichlet} $, and $ \boldsymbol v \cdot \boldsymbol n = q_N$ on $ \Gamma_{Neumann} $.
......@@ -4,9 +4,11 @@
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\-This model implements two-\/phase flow of two completely immiscible fluids $\alpha \in \{ w, n \}$ using a standard multiphase \-Darcy approach as the equation for the conservation of momentum\-: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \textbf{K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} {\textbf g} \right) \]
\-This model implements two-\/phase flow of two immiscible fluids $\alpha \in \{ w, n \}$ using a standard multiphase \-Darcy approach as the equation for the conservation of momentum, i.\-e. \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \textbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} {\textbf g} \right) \]
\-By inserting this into the equation for the conservation of the phase mass, one gets \[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \right\} - q_\alpha = 0 \;, \] discretized by a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit \-Euler method as time discretization.
\-By inserting this into the equation for the conservation of the phase mass, one gets \[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \right\} - q_\alpha = 0 \;, \]
\-This equations are by a fully-\/coupled vertex centered finite volume (box) 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 $S_w + S_n = 1$, the number of unknowns can be reduced to two. \-Currently the model supports choosing either $p_w$ and $S_n$ or $p_n$ and $S_w$ as primary variables. \-The formulation which ought to be used can be specified by setting the {\ttfamily \-Formulation} property to either {\ttfamily \-Two\-P\-Common\-Indices\-::p\-Ws\-N} or {\ttfamily \-Two\-P\-Common\-Indices\-::p\-Ns\-W}. \-By default, the model uses $p_w$ and $S_n$.
The model implements the decoupled equations of two-phase flow of two completely immiscible fluids. These equations can be derived from the two-phase flow equations shown for the two-phase box model (\doxyref{TwoPBoxModel}{p.}{classDune_1_1TwoPBoxModel}). The first equation to solve is a pressure equation of elliptic character. The second one is a saturation equation, which can be hyperbolic or parabolic.
This model allows different combinations of primary variables, which can be $p_w$-$S_w$, $p_w$-$S_n$, $p_n$-$S_w$, $p_n$-$S_n$, or $p$-$S_w$ and $p$-$S_n$, where $p$ is no phase pressure but a global pressure.
As the equations are only weakly coupled they do not have to be solved simultaneously but can be solved sequentially. First the pressure equation is solved implicitly, second the saturation equation can be solved explicitly. This solution procedure is called \doxyref{IMPES}{p.}{classDune_1_1IMPES} algorithm (IMplicit Pressure Explicit Saturation).
In comparison to a fully coupled model, different discretization methods can be applied to the different equations. So far, the pressure equation is discretized using a cell centered finite volume scheme (optionally with multi point flux approximation), a mimetic finite difference scheme or a finite element scheme. The saturation equation is discretized using a cell centered finite volume scheme. Default time discretization scheme is an explicit Euler scheme.
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\-This model solves equations of the form \[ \phi \left( \rho_w \frac{\partial S_w}{\partial t} + \rho_n \frac{\partial S_n}{\partial t}\right) + \textbf{div}\, \boldsymbol{v}_{total} = q. \] \-The definition of the total velocity $\boldsymbol{v}_{total}$ depends on the choice of the primary pressure variable. \-Further, fluids can be assumed to be compressible or incompressible (\-Property\-: {\ttfamily \-Enable\-Compressibility}). \-In the incompressible case a wetting ( $ w $) phase pressure as primary variable leads to
\[ - \textbf{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_w + f_n \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \]
a non-\/wetting ( $ n $) phase pressure yields \[ - \textbf{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] and a global pressure leads to \[ - \textbf{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_{global} + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q. \] \-Here, $ p_\alpha $ is a phase pressure, $ p_ {global} $ the global pressure of a classical fractional flow formulation (see e.\-g. \-P. \-Binning and \-M. \-A. \-Celia, ''\-Practical implementation of the fractional flow approach to multi-\/phase flow simulation'' , \-Advances in water resources, vol. 22, no. 5, pp. 461-\/478, 1999.), $ p_c = p_n - p_w $ is the capillary pressure, $ \boldsymbol K $ the absolute permeability, $ \lambda = \lambda_w + \lambda_n $ the total mobility depending on the saturation ( $ \lambda_\alpha = k_{r_\alpha} / \mu_\alpha $), $ f_\alpha = \lambda_\alpha / \lambda $ the fractional flow function of a phase, $ \rho_\alpha $ a phase density, $ g $ the gravity constant and $ q $ the source term.
\-For all cases, $ p = p_D $ on $ \Gamma_{Dirichlet} $, and $ \boldsymbol v_{total} \cdot \boldsymbol n = q_N $ on $ \Gamma_{Neumann} $.
\-The slightly compressible case is only implemented for phase pressures! \-In this case for a wetting ( $ w $) phase pressure as primary variable the equations are formulated as \[ \phi \left( \rho_w \frac{\partial S_w}{\partial t} + \rho_n \frac{\partial S_n}{\partial t}\right) - \textbf{div}\, \left[\lambda \boldsymbol{K} \left(\textbf{grad}\, p_w + f_n \, \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] and for a non-\/wetting ( $ n $) phase pressure as \[ \phi \left( \rho_w \frac{\partial S_w}{\partial t} + \rho_n \frac{\partial S_n}{\partial t}\right) - \textbf{div}\, \left[\lambda \boldsymbol{K} \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] \-In this slightly compressible case the following definitions are valid\-: $ \lambda = \rho_w \lambda_w + \rho_n \lambda_n $, $ f_\alpha = (\rho_\alpha \lambda_\alpha) / \lambda $ \-This model assumes that temporal changes in density are very small and thus terms of temporal derivatives are negligible in the pressure equation. \-Depending on the formulation the terms including time derivatives of saturations are simplified by inserting $ S_w + S_n = 1 $.
\-In the \-I\-M\-P\-E\-S models the default setting is\-:
\begin{itemize}
\item formulation\-: $ p_w-S_w $ (\-Property\-: {\ttfamily \-Formulation} defined as {\ttfamily \hyperlink{a00056_a04294fbcf0af5328016a160dbd8bfff9}{\-Decoupled\-Two\-P\-Common\-Indices\-::pw\-Sw}})
\item compressibility\-: disabled (\-Property\-: {\ttfamily \-Enable\-Compressibility} set to {\ttfamily false})
\end{itemize}
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\-This model solves equations of the form
\[ \phi \frac{\partial (\rho_\alpha S_\alpha)}{\partial t} + \textbf{div}\, (\rho_\alpha \boldsymbol{v_\alpha}) = q_\alpha, \]
where $ S_\alpha $ is the saturation of phase alpha (wetting ( $ w $), non-\/wetting ( $ n $)) and $ \boldsymbol v_\alpha $ is the phase velocity defined by the multi-\/phase \-Darcy equation. \-If a phase velocity is reconstructed from the pressure solution it can be directly inserted in the previous equation. \-In the incompressible case the equation is further divided by the phase density $ \rho_\alpha $. \-If a total velocity is reconstructed the saturation equation is reformulated into\-:
\[ \phi \frac{\partial S_w}{\partial t} + f_w \textbf{div}\, \boldsymbol{v}_{t} + f_w \lambda_n \boldsymbol{K}\left(\textbf{grad}\, p_c + (\rho_n-\rho_w) \, g \, \textbf{grad} z \right)= q_\alpha, \] to get a wetting phase saturation or \[ \phi \frac{\partial S_n}{\partial t} + f_n \textbf{div}\, \boldsymbol{v}_{t} - f_n \lambda_w \boldsymbol{K}\left(\textbf{grad}\, p_c + (\rho_n-\rho_w) \, g \, \textbf{grad} z \right)= q_\alpha, \] if the non-\/wetting phase saturation is the primary transport variable.
\-The total velocity formulation is only implemented for incompressible fluids and $ f_\alpha $ is the fractional flow function, $ \lambda_\alpha $ is the mobility, $ \boldsymbol K $ the absolute permeability, $ p_c $ the capillary pressure, $ \rho $ the fluid density, $ g $ the gravity constant, and $ q $ the source term.
\-In the \-I\-M\-P\-E\-S models the default setting is\-:
\begin{itemize}
\item formulation\-: $ p_w-S_w $ (\-Property\-: {\ttfamily \-Formulation} defined as {\ttfamily \hyperlink{a00056_a04294fbcf0af5328016a160dbd8bfff9}{\-Decoupled\-Two\-P\-Common\-Indices\-::pw\-Sw}})
\item compressibility\-: disabled (\-Property\-: {\ttfamily \-Enable\-Compressibility} set to {\ttfamily false})
\end{itemize}
......@@ -4,7 +4,11 @@
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\-This model implements a non-\/isothermal two-\/phase flow of two completely immiscible fluids $\alpha \in \{ w, n \}$. \-Using the standard multiphase \-Darcy approach, the mass conservation equations for both phases can be described as follows\-: \begin{eqnarray*} && \phi \frac{\partial (\varrho_\alpha S_\alpha )}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} (\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\} - q_\alpha^\kappa = 0 \qquad \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} \mbox{\bf K} \left( \text{grad} \, p_\alpha - \varrho_\alpha \mbox{\bf g} \right) \right\} \\ &-& \text{div} \left( \lambda_{pm} \text{grad} \, T \right) - q^h = 0, \qquad \alpha \in \{w, n\}. \end{eqnarray*}
\-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\-: \[ \phi \frac{\partial \phi \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathrm{K} \left( \textrm{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \right\} - q_\alpha = 0 \qquad \alpha \in \{w, n\} \]
\-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*} \frac{\partial \phi \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.
\-The equations are discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit \-Euler method as time discretization.
......
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\-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 $\kappa$ \[ \sum_{\kappa} \left( \phi \frac{\partial \varrho_\alpha x_\alpha^\kappa S_\alpha}{\partial t} - \mathrm{div}\; \left\{ \frac{\varrho_\alpha}{\overline M_\alpha} x_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \boldsymbol{K} \mathbf{grad}\left( p_\alpha - \varrho_{\alpha} g\right) \right\} \right) = q^\kappa \] 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, the 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}
......@@ -4,9 +4,9 @@
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\-In the unsaturated zone, \-Richards' equation is frequently used to calculate the water distribution above the groundwater level. \-It can be derived from the twophase equations, i.\-e. \[ \frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t} - \text{div} \left\{ \rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; {\textbf K} \text{grad}\left[ p_\alpha - g\rho_\alpha \right] \right\} = q_\alpha, \] where $\alpha \in \{w, n\}$ is the fluid phase, $\rho_\alpha$ is the fluid density, $S_\alpha$ is the fluid saturation, $\phi$ is the porosity of the soil, $k_{r\alpha}$ is the relative permeability for the fluid, $\mu_\alpha$ is the fluid's dynamic viscosity, $K$ is the intrinsic permeability, $p_\alpha$ is the fluid pressure and $g$ is the potential of the gravity field.
\-In the unsaturated zone, \-Richards' equation is frequently used to approximate the water distribution above the groundwater level. \-It can be derived from the two-\/phase equations, i.\-e. \[ \frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t} - \text{div} \left\{ \rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \textbf{grad}\left[ p_\alpha - g\rho_\alpha \right] \right\} = q_\alpha, \] where $\alpha \in \{w, n\}$ is the fluid phase, $\rho_\alpha$ is the fluid density, $S_\alpha$ is the fluid saturation, $\phi$ is the porosity of the soil, $k_{r\alpha}$ is the relative permeability for the fluid, $\mu_\alpha$ is the fluid's dynamic viscosity, $\mathbf{K}$ is the intrinsic permeability, $p_\alpha$ is the fluid pressure and $g$ is the potential of the gravity field.
\-In contrast to the full twophase model, the \-Richards model assumes gas as the non-\/wetting fluid and that it exhibits a much lower viscosity than the (liquid) wetting phase. (\-For example at atmospheric pressure and at room temperature, the viscosity of air is only about $1\%$ of the viscosity of liquid water.) \-As a consequence, the $\frac{k_{r\alpha}}{\mu_\alpha}$ term typically is much larger for the gas phase than for the wetting phase. \-For this reason, the \-Richards model assumes that $\frac{k_{rn}}{\mu_n}$ tends to infinity. \-This implies that the pressure of the gas phase is equivalent to a static pressure and can thus be specified externally and that therefore, mass conservation only needs to be considered for the wetting phase.
\-In contrast to the full two-\/phase model, the \-Richards model assumes gas as the non-\/wetting fluid and that it exhibits a much lower viscosity than the (liquid) wetting phase. (\-For example at atmospheric pressure and at room temperature, the viscosity of air is only about $1\%$ of the viscosity of liquid water.) \-As a consequence, the $\frac{k_{r\alpha}}{\mu_\alpha}$ term typically is much larger for the gas phase than for the wetting phase. \-For this reason, the \-Richards model assumes that $\frac{k_{rn}}{\mu_n}$ is infinitly large. \-This implies that the pressure of the gas phase is equivalent to the static pressure distribution and that therefore, mass conservation only needs to be considered for the wetting phase.
\-The model thus choses the absolute pressure of the wetting phase $p_w$ as its only primary variable. \-The wetting phase saturation is calculated using the inverse of the capillary pressure, i.\-e. \[ S_w = p_c^{-1}(p_n - p_w) \] holds, where $p_n$ is a given reference pressure. \-Nota bene that the last step is assumes that the capillary pressure-\/saturation curve can be inverted uniquely, so it is not possible to set the capillary pressure to zero when using the \-Richards model!
\-The model thus choses the absolute pressure of the wetting phase $p_w$ as its only primary variable. \-The wetting phase saturation is calculated using the inverse of the capillary pressure, i.\-e. \[ S_w = p_c^{-1}(p_n - p_w) \] holds, where $p_n$ is a given reference pressure. \-Nota bene, that the last step is assumes that the capillary pressure-\/saturation curve can be uniquely inverted, so it is not possible to set the capillary pressure to zero when using the \-Richards model!
......@@ -178,12 +178,15 @@ subdirectories of \texttt{dumux/boxmodels} of the \Dumux distribution.
\subsubsection{The non-isothermal two-phase, two-component model: TwoPTwoCNIBoxModel}
\input{ModelDescriptions/2p2cniboxmodel}
\subsubsection{The three-phase, three-component model: ThreePThreeCNIBoxModel}
\subsubsection{The three-phase, three-component model: ThreePThreeCBoxModel}
\input{ModelDescriptions/3p3cboxmodel}
\subsubsection{The non-isothermal three-phase, three-component model: ThreePThreeCNIBoxModel}
\input{ModelDescriptions/3p3cniboxmodel}
\subsubsection{The $M$-phase, $N$-component model: MpNcBoxModel}
\input{ModelDescriptions/mpncboxmodel}
\subsubsection{The \textsc{Stokes} model: StokesModel}
\input{ModelDescriptions/stokesmodel}
......@@ -199,9 +202,15 @@ The basic idea the so-called decoupled models have in common is to reformulate t
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}
\input{ModelDescriptions/1pdecoupledmodel}
\subsubsection{The two-phase model}
\paragraph{Pressure model}
\input{ModelDescriptions/2pdecoupledpressuremodel}
\subsection{The two-phase model}
\input{ModelDescriptions/2pdecoupledmodel}
\paragraph{Saturation model}
\input{ModelDescriptions/2pdecoupledsaturationmodel}
\subsubsection{The two-phase, two-component model}
\input{ModelDescriptions/decoupled2p2c}
......
......@@ -24,7 +24,6 @@
*
* \brief Adaption of the BOX scheme to the non-isothermal twophase flow model.
*/
#ifndef DUMUX_2PNI_MODEL_HH
#define DUMUX_2PNI_MODEL_HH
......@@ -56,7 +55,7 @@ namespace Dumux {
* results in one energy conservation equation for the porous solid
* matrix and the fluids:
\f{eqnarray*}
\f{align*}{
\frac{\partial \phi \sum_alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t}
& +
\left( 1 - \phi \right) \frac{\partial (\varrho_s c_s T)}{\partial t}
......@@ -66,7 +65,7 @@ namespace Dumux {
\varrho_\alpha h_\alpha
\frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K}
\left( \textbf{grad}\,p_\alpha - \varrho_\alpha \mbox{\bf g} \right)
\right\} \ \
\right\} \\
& - \text{div} \left(\lambda_{pm} \textbf{grad} \, T \right)
- q^h = 0, \qquad \alpha \in \{w, n\} \;,
\f}
......
......@@ -39,14 +39,14 @@ namespace Dumux
* composed of \f$N\f$ chemical species. The phases are denoted by
* lower index \f$\alpha \in \{ 1, \dots, M \}\f$. All fluid phases
* are mixtures of \f$N \geq M - 1\f$ chemical species which are
* denoted by the upper index \f$\kappa \in \{ 1, \dot, N \} \f$.
* denoted by the upper index \f$\kappa \in \{ 1, \dots, N \} \f$.
*
* The standard multi-phase Darcy law is used as the equation for
* the conservation of momentum:
* \f[
v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \boldsymbol{K}
\left(
\text{grad} p_\alpha - \varrho_{\alpha} \boldsymbol{g}
\text{grad}\left(p_\alpha - \varrho_{\alpha} g\right)
\right)
\f]
*
......@@ -57,11 +57,11 @@ namespace Dumux
\sum_{\kappa} \left(
\phi \frac{\partial \varrho_\alpha x_\alpha^\kappa S_\alpha}{\partial t}
-
\nabla \cdot
\mathrm{div}\;
\left\{
\frac{\varrho_\alpha}{\overline M_\alpha} x_\alpha^\kappa
\frac{k_{r\alpha}}{\mu_\alpha} \boldsymbol{K}
(\nabla p_\alpha - \varrho_{\alpha} \boldsymbol{g})
\mathbf{grad}\left( p_\alpha - \varrho_{\alpha} g\right)
\right\}
\right)
= q^\kappa
......
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