improve documentation of the two-phase model, make the boundary conditions of the lens problem readable

doc/handbook/ModelDescriptions/1p2cboxmodel.tex

@@ -1 +1 @@
-Here comes the detailed documentation. 
+

doc/handbook/ModelDescriptions/1pboxmodel.tex

@@ -1 +1 @@
-Here comes the detailed documentation. 
+Single phase isothermal flow model is implemented for compressible flow. \begin{align*} \phi \frac{\partial \varrho}{\partial t} + \vec{\nabla} \cdot (- \varrho \frac{\bar{\bar{K}}}{\mu} ( \nabla p -\varrho \vec{g})) = q \end{align*} However, the model can also be used for incompressible single pahse flow modeling, when in problem file a fluid with constant density is chosen. 

doc/handbook/ModelDescriptions/2pboxmodel.tex

@@ -1 +1,5 @@
-\doxyref{TwoPBoxModel}{p.}{classDune_1_1TwoPBoxModel} describes the box discretization of an isothermal twophase flow model. The underlying equations are obtained after inserting Darcy's law into the mass balance equation for each phase, yielding \begin{align*} \phi \frac{\partial (\varrho_{\text{mass,w}} S_\text{w})}{\partial t} -\Div \left( \lambda_\text{w} \varrho_{\text{mass,w}} K \left(\grad p_\text{w} - \varrho_{\text{mass,w}}\boldsymbol{g} \right)\right) - q_\text{w} &= 0, \\ \phi \frac{\partial (\varrho_{\text{mass,n}} S_\text{n})}{\partial t} - \Div \left( \lambda_\text{n} \varrho_{\text{mass,n}} K\left( \grad p_\text{n} - \varrho_{\text{mass,n}}\boldsymbol{g} \right)\right) - q_\text{n} &= 0. \end{align*} You can pick the formulation by setting the \char`\"{}Formulation\char`\"{} property. The default is $p_\text{w}$-$S_\text{n}$. 
+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: \[ \vec{v_\alpha} = - \frac{k_{r\alpha}}{\mu_\alpha} K \left(\grad p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \]
+
+By inserting this into the equation for the conservation of the phase mass, one gets \[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \Div \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} K \left(\grad p_\alpha - \varrho_{\alpha} \boldsymbol{g} \right) \right\} = q_\alpha \;, \] which is discretized by this model using the 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 permability $k_{r\alpha}$ 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 chosing 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 {\tt Formulation} property to either either {\tt \doxyref{TwoPIndices::pWsN}{p.}{structDune_1_1TwoPIndices_7bab78f10df58319eafe8b79a1d28553}} or {\tt \doxyref{TwoPIndices::pNsW}{p.}{structDune_1_1TwoPIndices_b2b28761c782605cf080f9b6f9b618b5}}. By default, the model uses $p_w$ and $S_n$. 

doc/handbook/ModelDescriptions/richardsboxmodel.tex

@@ -1 +1,5 @@
-Here comes the detailed documentation. 
+In the unsaturated zone Richards` equation can be used. Gas has resistance against the water flow in porous media. However, viscosity of air is about 1\% of the viscosity of water, which makes it highly mobile compared to the water phase. Therefore, in Richards` equation only water phase with capillary effects are considered, where pressure of the gas phase is set to a reference pressure (${p_n}_{ref}$).
+
+\begin{align*} \varrho \hspace{1mm} \phi \hspace{1mm} \frac{\partial S_w}{\partial p_c} \frac{\partial p_c}{\partial t} - \nabla \cdot (\frac{kr_w}{\mu_w} \hspace{1mm} \varrho_w \hspace{1mm} K \hspace{1mm} (\nabla p_w - \varrho_w \hspace{1mm} \vec{g})) \hspace{1mm} = \hspace{1mm} q \\ ,where \hspace{1mm} p_w = {p_n}_{ref} - p_c \end{align*} Here $ p_w $, $ p_c $, $ p_n $ denotes water pressure, capillary pressure, non-wetting phase reference pressure repectively.
+
+To overcome convergence problem $ \frac{\partial S_w}{\partial p_c} $ is taken from old iteration step. 

doc/handbook/models.tex

@@ -49,12 +49,13 @@ $p_\alpha$ & 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 $\alpha$ & $q^\kappa$ & volume source term \\
+$X_\alpha^\kappa$ & mass fraction of component $\kappa$ in phase $\alpha$ & $q^\kappa_\alpha$ & volume source term of component $\kappa$ in phase $\alpha$ \\
 $\varrho_{\text{mol},\alpha}$ & molar density of phase $\alpha$ & $u_\alpha$ & specific internal energy \\
-$\varrho_{\text{mass},\alpha}$ & mass density of phase $\alpha$ & $h_\alpha$ & specific enthalpy \\
+$\varrho_{\alpha}$ & 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 
+$D_\alpha^\kappa$ & diffusivity of component $\kappa$ in phase $\alpha$ & $q^h$ & heat source term \\
+$v_\alpha$ & Darcy velocity & $v_{a,\alpha}$  & advective velocity
 \end{tabular}
 
 
@@ -73,7 +74,7 @@ molar mass balance can be written as:
  - \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
         \alpha}}{\mu_\alpha} \varrho_{\text{mol}, \alpha}
     x_\alpha^\kappa K (\grad p_\alpha -
-    \varrho_{\text{mass}, \alpha} \boldsymbol{g}) \right) \nonumber \\
+    \varrho_{\alpha} \boldsymbol{g}) \right) \nonumber \\
   %
   \nonumber \\
   %
@@ -116,8 +117,8 @@ formulated as:
 %
 \begin{eqnarray}
   \label{A3:eqenergmak1}
-  && \phi \frac{\partial \left( \sum_\alpha \varrho_{\text{mass},
-        \alpha} u_\alpha S_\alpha \right)}{\partial t} + \left( 1 -
+  && \phi \frac{\partial \left( \sum_\alpha \varrho_{\alpha}
+      u_\alpha S_\alpha \right)}{\partial t} + \left( 1 -
     \phi \right) \frac{\partial \varrho_{\text{s}} c_{\text{s}}
     T}{\partial t} \nonumber 
  - \Div \left( \lambda_{\text{pm}} \grad T \right)
@@ -126,8 +127,8 @@ formulated as:
   \nonumber \\
   %
   && - \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
-        \alpha}}{\mu_\alpha} \varrho_{\text{mass}, \alpha} h_\alpha
-    K \left( \grad p_\alpha - \varrho_{\text{mass}, \alpha}
+        \alpha}}{\mu_\alpha} \varrho_{\alpha} h_\alpha
+    K \left( \grad p_\alpha - \varrho_{\alpha}
       \boldsymbol{g} \right) \right) \nonumber 
  - q^h \; = \; 0.
 \end{eqnarray}
@@ -148,25 +149,25 @@ fractions, temperature, pressures, etc.
 
 \subsection{Fully coupled models} 
 
-\subsubsection{OnePBoxModel} 
+\subsubsection{The single-phase model: OnePBoxModel} 
 \input{ModelDescriptions/1pboxmodel}
 
-\subsubsection{OnePTwoCBoxModel} 
+\subsubsection{The single-phase, two-component model:  OnePTwoCBoxModel} 
 \input{ModelDescriptions/1p2cboxmodel}
 
-\subsubsection{RichardsBoxModel} 
+\subsubsection{The two-phase model using Richards' assumption: RichardsBoxModel} 
 \input{ModelDescriptions/richardsboxmodel}
 
-\subsubsection{TwoPBoxModel} 
+\subsubsection{The two-phase model: TwoPBoxModel} 
 \input{ModelDescriptions/2pboxmodel}
 
-\subsubsection{TwoPNIBoxModel} 
+\subsubsection{The non-isothermal two-phase model: TwoPNIBoxModel} 
 \input{ModelDescriptions/2pniboxmodel}
 
-\subsubsection{TwoPTwoCBoxModel} 
+\subsubsection{The two-phase, two-component model: TwoPTwoCBoxModel} 
 \input{ModelDescriptions/2p2cboxmodel}
 
-\subsubsection{TwoPTwoCNIBoxModel} 
+\subsubsection{The non-isothermal two-phase, two-component model: TwoPTwoCNIBoxModel} 
 \input{ModelDescriptions/2p2cniboxmodel}
 
 
@@ -177,3 +178,8 @@ fractions, temperature, pressures, etc.
 \input{models_decoupled2p2c}
 
 
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: "dumux-handbook"
+%%% End: