handbook: regenerated model descriptions from the doxygen sources

doc/handbook/ModelDescriptions/1p2cboxmodel.tex

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-Adaption of the BOX scheme to the one-\/phase two-\/component flow model. This model implements an one-\/phase flow of an incompressible fluid, that consists of two components, using a standard Darcy approach (neglecting gravitation) as the equation for the conservation of momentum: \[ v_{D} = - \frac{K}{\mu} \left(\text{grad} p - \varrho g \right) \]
+Adaption of the BOX scheme to the one-\/phase two-\/component flow model. This model implements an one-\/phase flow of an incompressible fluid, that consists of two components, using a standard Darcy approach as the equation for the conservation of momentum: \[ v_{D} = - \frac{K}{\mu} \left(\text{grad} p - \varrho g \right) \]
 
-By inserting this into the continuity equation, one gets \[ - \text{div} \left\{ \varrho \frac{K}{\mu} \left(\text{grad} p - \varrho g \right) \right\} = q \;, \]
+Gravity can be enabled or disabled via the Property system. By inserting this into the continuity equation, one gets \[ - \text{div} \left\{ \varrho \frac{K}{\mu} \left(\text{grad} p - \varrho g \right) \right\} = q \;, \]
 
 The transport of the components is described by the following equation: \[ \Phi \varrho \frac{ \partial x}{\partial t} - \text{div} \left( \varrho \frac{K x}{\mu} \left( \text{grad} p - \varrho g \right) + \varrho \tau \Phi D \text{grad} x \right) = q. \]
 

doc/handbook/ModelDescriptions/2p2cboxmodel.tex

@@ -5,16 +5,16 @@
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-Adaption of the BOX scheme to the two-\/phase two-\/component flow model. This model implements two-\/phase two-\/component flow of two compressible and partially miscible fluids $\alpha \in \{ w, n \}$ composed of the two components $\kappa \in \{ w, a \}$. The standard multiphase Darcy approach is used as the equation for the conservation of momentum: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} K \left(\text{grad} p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \]
+Adaption of the BOX scheme to the two-\/phase two-\/component flow model. This model implements two-\/phase two-\/component flow of two compressible and partially miscible fluids $\alpha \in \{ w, n \}$ composed of the two components $\kappa \in \{ w, a \}$. The standard multiphase Darcy approach is used as the equation for the conservation of momentum: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \]
 
-By inserting this into the equations for the conservation of the components, one gets one transport equation for each component \begin{eqnarray*} && \phi \frac{\partial (\sum_\alpha \varrho_\alpha X_\alpha^\kappa S_\alpha )} {\partial t} - \sum_\alpha \nabla \cdot \left\{ \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} ({\bf \nabla} p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\} \nonumber \\ \nonumber \\ &-& \sum_\alpha \nabla \cdot \left\{{\bf D_{pm}^\kappa} \varrho_{\alpha} {\bf \nabla} X^\kappa_{\alpha} \right\} - \sum_\alpha q_\alpha^\kappa = \quad 0 \qquad \kappa \in \{w, a\} \, , \alpha \in \{w, g\} \end{eqnarray*}
+By inserting this into the equations for the conservation of the components, one gets one transport equation for each component \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} \mbox{\bf K} (\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\} \nonumber \\ \nonumber \\ &-& \sum_\alpha \text{div} \left\{{\bf D_{\alpha, pm}^\kappa} \varrho_{\alpha} \text{grad}\, X^\kappa_{\alpha} \right\} - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{w, a\} \, , \alpha \in \{w, g\} \end{eqnarray}
 
 This is discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as temporal 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. The used primary variables are, like in the two-\/phase model, either $p_w$ and $S_n$ or $p_n$ and $S_w$. The formulation which ought to be used can be specified by setting the {\ttfamily Formulation} property to either TwoPTwoCIndices::pWsN or TwoPTwoCIndices::pNsW. By default, the model uses $p_w$ and $S_n$. Moreover, the second primary variable depends on the phase state, since a primary variable switch is included. The phase state is stored for all nodes of the system. 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 {\ttfamily Formulation}), as long as $ 0 < S_\alpha < 1$.
-\item Only wetting phase is present: The mass fraction of, e.g., air in the wetting phase $X^a_w$ is used, as long as the maximum mass fraction is not exceeded ($X^a_w<X^a_{w,max}$)
-\item Only non-\/wetting phase is present: The mass fraction of, e.g., water in the non-\/wetting phase, $X^w_n$, is used, as long as the maximum mass fraction is not exceeded ($X^w_n<X^w_{n,max}$)
+\item Only wetting phase is present: The mass fraction of, e.g., air in the wetting phase $X^a_w$ is used, as long as the maximum mass fraction is not exceeded ( $X^a_w<X^a_{w,max}$)
+\item Only non-\/wetting phase is present: The mass fraction of, e.g., water in the non-\/wetting phase, $X^w_n$, is used, as long as the maximum mass fraction is not exceeded ( $X^w_n<X^w_{n,max}$)
 \end{itemize}
 

doc/handbook/ModelDescriptions/2p2cniboxmodel.tex

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-Adaption of the BOX scheme to the non-\/isothermal two-\/phase two-\/component flow model. This model implements a non-\/isothermal two-\/phase flow of two compressible and partly miscible fluids $\alpha \in \{ w, n \}$. Thus each component $\kappa \{ w, a \}$ 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} \mbox{\bf K} (\text{grad} p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\}\\ &-& \sum_\alpha \text{div} \left\{{\bf D_{\alpha, pm}^\kappa} \varrho_{\alpha} \text{grad} X^\kappa_{\alpha} \right\} - \sum_\alpha q_\alpha^\kappa = \quad 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( \text{grad}\: p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ &-& \text{div} \left( \lambda_{pm} \text{grad} \: T \right) - q^h \qquad = \quad 0 \qquad \alpha \in \{w, n\} \end{eqnarray*}
+Adaption of the BOX scheme to the non-\/isothermal two-\/phase two-\/component flow model. This model implements a non-\/isothermal two-\/phase flow of two compressible and partly miscible fluids $\alpha \in \{ w, n \}$. Thus each component $\kappa \{ w, a \}$ 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} \mbox{\bf K} (\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\}\\ &-& \sum_\alpha \text{div} \left\{{\bf D_{\alpha, pm}^\kappa} \varrho_{\alpha} \text{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( \text{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ &-& \text{div} \left( \lambda_{pm} \text{grad} \, T \right) - q^h = 0 \qquad \alpha \in \{w, n\} \end{eqnarray*}
 
 This is discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as temporal 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 TwoPTwoIndices::pWsN} or {\ttfamily TwoPTwoCIndices::pNsW}. 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 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}

doc/handbook/ModelDescriptions/2pboxmodel.tex

@@ -5,9 +5,9 @@
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-Adaption of the BOX scheme to the twophase flow model. 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: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} K \left(\text{grad} p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \]
+Adaption of the BOX scheme to the twophase flow model. 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: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} \left(\text{grad} p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \]
 
-By inserting this into the equation for the conservation of the phase mass, one gets \[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} K \left(\text{grad} p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \right\} = q_\alpha \;, \] 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 \[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \right\} - q_\alpha = 0 \;, \] discretized by a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization.
 
 By 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 TwoPCommonIndices::pWsN} or {\ttfamily TwoPCommonIndices::pNsW}. By default, the model uses $p_w$ and $S_n$.
 

doc/handbook/ModelDescriptions/2pniboxmodel.tex

@@ -5,7 +5,7 @@
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-Adaption of the BOX scheme to the non-\/isothermal twophase flow model. 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 = \quad 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} \mathbf{K} \left( \text{grad} \: p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} \\ &-& \text{div} \left( \lambda_{pm} \text{grad} \: T \right) - q^h \qquad = \quad 0, \qquad \alpha \in \{w, n\}. \end{eqnarray*}
+Adaption of the BOX scheme to the non-\/isothermal twophase flow model. 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*}
 
 The equations are discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization.
 

doc/handbook/ModelDescriptions/richardsboxmodel.tex

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-This model implements a variant of the Richards equation for quasi-\/twophase flow. 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. \[ \frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t} - \mathbf{div} \left\{ \frac{k_{r\alpha}}{\mu_\alpha}\;K \mathbf{grad}\left[ p_\alpha - g\rho_\alpha \right] \right\} = q_\alpha, \] 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.
+This model implements a variant of the Richards equation for quasi-\/twophase flow. 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. \[ \frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t} - \mathbf{div} \left\{ \rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\;K \mathbf{grad}\left[ p_\alpha - g\rho_\alpha \right] \right\} = q_\alpha, \] 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 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.