From 06c03717b865d51e7c17b9dd06d326ecfef18e27 Mon Sep 17 00:00:00 2001
From: Thomas Fetzer <thomas.fetzer@iws.uni-stuttgart.de>
Date: Thu, 7 Mar 2013 18:31:31 +0000
Subject: [PATCH] [ implicit ] added cc-method is usable for implicit model
 changed all \text{grad} to \textbf{grad} and all \textbf{div} \text{div}
 corrected component balances, that mole fraction gradients are used unified
 equations between the models, made K a tensor everywhere, D a scalar

git-svn-id: svn://svn.iws.uni-stuttgart.de/DUMUX/dumux/trunk@10353 2fb0f335-1f38-0410-981e-8018bf24f1b0
---
 dumux/implicit/1p/1pmodel.hh                  | 19 +++++++++----
 dumux/implicit/1p2c/1p2cmodel.hh              | 22 ++++++++-------
 dumux/implicit/2p/2pmodel.hh                  |  6 ++--
 dumux/implicit/2p2c/2p2cmodel.hh              | 13 +++++----
 dumux/implicit/2p2cni/2p2cnimodel.hh          | 17 +++++------
 dumux/implicit/2pni/2pnimodel.hh              | 14 +++++-----
 dumux/implicit/3p3c/3p3cmodel.hh              | 18 ++++++------
 dumux/implicit/3p3cni/3p3cnimodel.hh          | 28 +++++++++----------
 .../implicit/3p3cni/3p3cnivolumevariables.hh  |  2 +-
 .../mpnc/diffusion/volumevariables.hh         |  6 +++-
 dumux/implicit/richards/richardsmodel.hh      | 25 +++++++++--------
 11 files changed, 92 insertions(+), 78 deletions(-)

diff --git a/dumux/implicit/1p/1pmodel.hh b/dumux/implicit/1p/1pmodel.hh
index d4856cc0ef..edda3fa5c1 100644
--- a/dumux/implicit/1p/1pmodel.hh
+++ b/dumux/implicit/1p/1pmodel.hh
@@ -35,14 +35,21 @@ namespace Dumux
  * \ingroup OnePBoxModel
  * \brief A single-phase, isothermal flow model using the fully implicit scheme.
  *
- * Single-phase, isothermal flow model, which solves the mass
- * continuity equation
+ * Single-phase, isothermal flow model, which uses a standard Darcy approach as the
+ * equation for the conservation of momentum:
  * \f[
- \phi \frac{\partial \varrho}{\partial t} + \text{div} (- \varrho \frac{\textbf K}{\mu} ( \textbf{grad}\, p -\varrho {\textbf g})) = q,
+ v = - \frac{\textbf K}{\mu}
+ \left(\textbf{grad}\, p - \varrho {\textbf g} \right)
  * \f]
- * 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.
+ * 
+ * and solves the mass continuity equation:
+ * \f[
+ \phi \frac{\partial \varrho}{\partial t} + \text{div} \left\lbrace - \varrho \frac{\textbf K}{\mu} \left( \textbf{grad}\, p -\varrho {\textbf g} \right) \right\rbrace = q,
+ * \f]
+ * 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 supports compressible as well as incompressible fluids.
  */
 template<class TypeTag >
 class OnePBoxModel : public GET_PROP_TYPE(TypeTag, BaseModel)
diff --git a/dumux/implicit/1p2c/1p2cmodel.hh b/dumux/implicit/1p2c/1p2cmodel.hh
index adb997b8ef..174400ec0c 100644
--- a/dumux/implicit/1p2c/1p2cmodel.hh
+++ b/dumux/implicit/1p2c/1p2cmodel.hh
@@ -40,27 +40,29 @@ namespace Dumux
  * using a standard Darcy
  * approach as the equation for the conservation of momentum:
  \f[
- v_{D} = - \frac{\textbf K}{\mu}
- \left(\text{grad} p - \varrho {\textbf g} \right)
+ v = - \frac{\textbf K}{\mu}
+ \left(\textbf{grad}\, p - \varrho {\textbf g} \right)
  \f]
  *
  * Gravity can be enabled or disabled via the property system.
  * By inserting this into the continuity equation, one gets
  \f[
- \Phi \frac{\partial \varrho}{\partial t} - \text{div} \left\{
-   \varrho \frac{\textbf K}{\mu}  \left(\text{grad}\, p - \varrho {\textbf g} \right)
+ \phi\frac{\partial \varrho}{\partial t} - \text{div} \left\{
+   \varrho \frac{\textbf K}{\mu}  \left(\textbf{grad}\, p - \varrho {\textbf g} \right)
  \right\} = q \;,
  \f]
  *
- * The transport of the components is described by the following equation:
+ * The transport of the components \f$\kappa \in \{ w, a \}\f$ is described by the following equation:
  \f[
- \Phi \frac{ \partial \varrho x}{\partial t} - \text{div} \left( \varrho \frac{{\textbf K} x}{\mu} \left( \text{grad}\, p -
- \varrho {\textbf g} \right) + \varrho \tau \Phi D \text{grad} x \right) = q.
+ \phi \frac{ \partial \varrho X^\kappa}{\partial t}
+ - \text{div} \left\lbrace \varrho X^\kappa \frac{{\textbf K}}{\mu} \left( \textbf{grad}\, p -
+ \varrho {\textbf g} \right)
+ + \varrho D^\kappa_\text{pm} \frac{M^\kappa}{M_\alpha} \textbf{grad} x^\kappa \right\rbrace = q.
  \f]
  *
- * All equations are discretized using a fully-coupled vertex-centered
- * finite volume (box) scheme as spatial and
- * the implicit Euler method as time discretization.
+ * 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 primary variables are the pressure \f$p\f$ and the mole or mass fraction of dissolved component \f$x\f$.
  */
diff --git a/dumux/implicit/2p/2pmodel.hh b/dumux/implicit/2p/2pmodel.hh
index 45c972cdae..ef14dadb79 100644
--- a/dumux/implicit/2p/2pmodel.hh
+++ b/dumux/implicit/2p/2pmodel.hh
@@ -53,9 +53,9 @@ namespace Dumux
  \right\} - q_\alpha = 0 \;,
  \f]
  *
- * These equations are discretized by a fully-coupled vertex centered finite volume
- * (box) scheme as spatial and the implicit Euler method as time
- * discretization.
+ * 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 \f$p_c =
  * p_n - p_w\f$ and relative permeability \f$k_{r\alpha}\f$ and taking
diff --git a/dumux/implicit/2p2c/2p2cmodel.hh b/dumux/implicit/2p2c/2p2cmodel.hh
index 042ccf19f1..6d1f6cf1d0 100644
--- a/dumux/implicit/2p2c/2p2cmodel.hh
+++ b/dumux/implicit/2p2c/2p2cmodel.hh
@@ -39,7 +39,7 @@ namespace Dumux
  * approach is used as the equation for the conservation of momentum:
  * \f[
  v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K}
- \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right)
+ \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right)
  * \f]
  *
  * By inserting this into the equations for the conservation of the
@@ -49,16 +49,17 @@ namespace Dumux
  {\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\}
+ (\textbf{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 \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, g\}
  \f}
  *
- * This is discretized using a fully-coupled vertex
- * centered finite volume (box) scheme as spatial and
- * the implicit Euler method as temporal discretization.
+ * 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 \f$p_c =
  * p_n - p_w\f$ and relative permeability \f$k_{r\alpha}\f$ and taking
diff --git a/dumux/implicit/2p2cni/2p2cnimodel.hh b/dumux/implicit/2p2cni/2p2cnimodel.hh
index 7e2e3659c2..e48867d83b 100644
--- a/dumux/implicit/2p2cni/2p2cnimodel.hh
+++ b/dumux/implicit/2p2cni/2p2cnimodel.hh
@@ -32,7 +32,7 @@ namespace Dumux {
  * \brief Adaption of the fully implicit 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
- * \f$\alpha \in \{ w, n \}\f$. Thus each component \f$\kappa \{ w, a \}\f$ can be present in
+ * \f$\alpha \in \{ w, n \}\f$. Thus each component \f$\kappa \in \{ w, a \}\f$ can be present in
  * each phase.
  * Using the standard multiphase Darcy approach a mass balance equation is
  * solved:
@@ -40,8 +40,9 @@ namespace Dumux {
  && \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\}
+ (\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\}
  *     \f}
@@ -51,16 +52,16 @@ namespace Dumux {
  && \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}\,
+ \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\,
  p_\alpha
  - \varrho_\alpha \mathbf{g} \right) \right\} \\
- &-& \text{div} \left( \lambda_{pm} \text{grad} \, T \right)
+ &-& \text{div} \left( \lambda_\text{pm} \textbf{grad} \, T \right)
  - q^h = 0 \qquad \alpha \in \{w, n\}
  \f}
  *
- * This is discretized using a fully-coupled vertex
- * centered finite volume (box) scheme as spatial and
- * the implicit Euler method as temporal discretization.
+ * 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 \f$p_c =
  * p_n - p_w\f$ and relative permeability \f$k_{r\alpha}\f$ and taking
diff --git a/dumux/implicit/2pni/2pnimodel.hh b/dumux/implicit/2pni/2pnimodel.hh
index c190f74043..51ec8dcb8c 100644
--- a/dumux/implicit/2pni/2pnimodel.hh
+++ b/dumux/implicit/2pni/2pnimodel.hh
@@ -38,12 +38,12 @@ namespace Dumux {
  * multiphase Darcy approach, the mass conservation equations for both
  * phases can be described as follows:
  * \f[
- \phi \frac{\partial \phi \varrho_\alpha S_\alpha}{\partial t}
+ \phi \frac{\partial \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)
+ \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\}
@@ -54,7 +54,7 @@ namespace Dumux {
  * matrix and the fluids: 
  
  \f{align*}{
- \frac{\partial \phi \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t}
+ \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}
  - 
@@ -72,9 +72,9 @@ namespace Dumux {
  * p_\alpha/\varrho_\alpha\f$ 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.
+ * 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 \f$p_w\f$, \f$S_n\f$
  * and \f$T\f$ or \f$p_n\f$, \f$S_w\f$ and \f$T\f$ as primary
diff --git a/dumux/implicit/3p3c/3p3cmodel.hh b/dumux/implicit/3p3c/3p3cmodel.hh
index 3063d796f8..849fd3cbec 100644
--- a/dumux/implicit/3p3c/3p3cmodel.hh
+++ b/dumux/implicit/3p3c/3p3cmodel.hh
@@ -41,29 +41,29 @@ namespace Dumux
  * approach is used as the equation for the conservation of momentum:
  * \f[
  v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K}
- \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right)
+ \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right)
  * \f]
  *
  * By inserting this into the equations for the conservation of the
  * components, one transport equation for each component is obtained as
  * \f{eqnarray*}
- && \phi \frac{\partial (\sum_\alpha \varrho_{\text{mol}, \alpha} x_\alpha^\kappa
+ && \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_{\text{mol}, \alpha} x_\alpha^\kappa \mbox{\bf K}
- (\text{grad}\, p_\alpha - \varrho_{\text{mass}, \alpha} \mbox{\bf g}) \right\}
+ \varrho_\alpha x_\alpha^\kappa \mbox{\bf K}
+ (\textbf{grad}\, p_\alpha - \varrho_\alpha \mbox{\bf g}) \right\}
  \nonumber \\
  \nonumber \\
- && - \sum\limits_\alpha \text{div} \left\{ D_{pm}^\kappa \varrho_{\text{mol},
- \alpha } \text{grad}\, x_\alpha^\kappa \right\}
+ && - \sum\limits_\alpha \text{div} \left\{ D_\text{pm}^\kappa \varrho_\alpha \frac{M^\kappa}{M_\alpha}
+ \textbf{grad} x^\kappa_{\alpha} \right\}
  - q^\kappa = 0 \qquad \forall \kappa , \; \forall \alpha
  \f}
  *
  * Note that these balance equations are molar.
  *
- * The equations are discretized using a fully-coupled vertex
- * centered finite volume (BOX) scheme as spatial scheme and
- * the implicit Euler method as temporal discretization.
+ * 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
  * \f$S_w + S_n + S_g = 1\f$ for the saturations and
diff --git a/dumux/implicit/3p3cni/3p3cnimodel.hh b/dumux/implicit/3p3cni/3p3cnimodel.hh
index cff31e7491..1e92c8dc16 100644
--- a/dumux/implicit/3p3cni/3p3cnimodel.hh
+++ b/dumux/implicit/3p3cni/3p3cnimodel.hh
@@ -33,26 +33,26 @@ namespace Dumux {
  * \brief Adaption of the fully implicit scheme to the non-isothermal three-phase three-component flow model.
  *
  * This model implements three-phase three-component flow of three fluid phases
- * \f$\alpha \in \{ water, gas, NAPL \}\f$ each composed of up to three components
- * \f$\kappa \in \{ water, air, contaminant \}\f$. The standard multiphase Darcy
+ * \f$\alpha \in \{ \text{water, gas, NAPL} \}\f$ each composed of up to three components
+ * \f$\kappa \in \{ \text{water, air, contaminant} \}\f$. The standard multiphase Darcy
  * approach is used as the equation for the conservation of momentum:
  * \f[
  v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K}
- \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right)
+ \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right)
  * \f]
  *
  * By inserting this into the equations for the conservation of the
  * components, one transport equation for each component is obtained as
  * \f{eqnarray*}
- && \phi \frac{\partial (\sum_\alpha \varrho_{\text{mol}, \alpha} x_\alpha^\kappa
+ && \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_{\text{mol}, \alpha} x_\alpha^\kappa \mbox{\bf K}
- (\text{grad}\; p_\alpha - \varrho_{\text{mass}, \alpha} \mbox{\bf g}) \right\}
+ \varrho_\alpha X_\alpha^\kappa \mbox{\bf K}
+ (\textbf{grad}\; p_\alpha - \varrho_\alpha \mbox{\bf g}) \right\}
  \nonumber \\
  \nonumber \\
- && - \sum\limits_\alpha \text{div} \left\{ D_{pm}^\kappa \varrho_{\text{mol},
- \alpha } \text{grad} \; x_\alpha^\kappa \right\}
+ && - \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
  \f}
  *
@@ -64,18 +64,16 @@ namespace Dumux {
  && \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}\,
+ \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\,
  p_\alpha
  - \varrho_\alpha \mathbf{g} \right) \right\} \\
- &-& \text{div} \left( \lambda_{pm} \text{grad} \, T \right)
+ &-& \text{div} \left( \lambda_{pm} \textbf{grad} \, T \right)
  - q^h = 0 \qquad \alpha \in \{w, n, g\}
  \f}
  *
-
- *
- * The equations are discretized using a fully-coupled vertex
- * centered finite volume (BOX) scheme as spatial scheme and
- * the implicit Euler method as temporal discretization.
+ * 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
  * \f$S_w + S_n + S_g = 1\f$ for the saturations and
diff --git a/dumux/implicit/3p3cni/3p3cnivolumevariables.hh b/dumux/implicit/3p3cni/3p3cnivolumevariables.hh
index 499ffb6392..85b971fc7d 100644
--- a/dumux/implicit/3p3cni/3p3cnivolumevariables.hh
+++ b/dumux/implicit/3p3cni/3p3cnivolumevariables.hh
@@ -41,7 +41,7 @@ namespace Dumux
 template <class TypeTag>
 class ThreePThreeCNIVolumeVariables : public ThreePThreeCVolumeVariables<TypeTag>
 {
-    //! \cond 0
+    //! \cond false
     typedef ThreePThreeCVolumeVariables<TypeTag> ParentType;
 
     typedef typename GET_PROP_TYPE(TypeTag, Scalar) Scalar;
diff --git a/dumux/implicit/mpnc/diffusion/volumevariables.hh b/dumux/implicit/mpnc/diffusion/volumevariables.hh
index d16f3f2ae8..f1a311d353 100644
--- a/dumux/implicit/mpnc/diffusion/volumevariables.hh
+++ b/dumux/implicit/mpnc/diffusion/volumevariables.hh
@@ -20,7 +20,7 @@
  * \file
  *
  * \brief This file contains the diffusion module for the vertex data
- *        of the fully coupled two-phase N-component model
+ *        of the fully coupled MpNc model
  */
 #ifndef DUMUX_MPNC_DIFFUSION_VOLUME_VARIABLES_HH
 #define DUMUX_MPNC_DIFFUSION_VOLUME_VARIABLES_HH
@@ -30,6 +30,10 @@
 
 namespace Dumux {
 
+/*!
+ * \brief Variables for the diffusive fluxes in the MpNc model within 
+ *        a finite volume.
+ */
 template<class TypeTag, bool enableDiffusion>
 class MPNCVolumeVariablesDiffusion
 {
diff --git a/dumux/implicit/richards/richardsmodel.hh b/dumux/implicit/richards/richardsmodel.hh
index cb7c76862c..7f305d3155 100644
--- a/dumux/implicit/richards/richardsmodel.hh
+++ b/dumux/implicit/richards/richardsmodel.hh
@@ -40,14 +40,14 @@ namespace Dumux
  *
  * In the unsaturated zone, Richards' equation
  \f[
- \frac{\partial\;\phi S_w \rho_w}{\partial t}
+ \frac{\partial\;\phi S_w \varrho_w}{\partial t}
  -
- \text{div} \left(
- \rho_w \frac{k_{rw}}{\mu_w} \; \mathbf{K} \;
- \text{\textbf{grad}}\left(
- p_w - g\rho_w
- \right)
+ \text{div} \left\lbrace
+ \varrho_w \frac{k_{rw}}{\mu_w} \; \mathbf{K} \;
+ \left( \text{\textbf{grad}}
+ p_w - \varrho_w \textbf{g}
  \right)
+ \right\rbrace
  =
  q_w,
  \f]
@@ -56,18 +56,19 @@ namespace Dumux
  * 
  * It can be derived from the two-phase equations, i.e.
  \f[
- \frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t}
+ \phi\frac{\partial S_\alpha \varrho_\alpha}{\partial t}
  -
- \text{div} \left(
- \rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \;
- \text{\textbf{grad}}\left(
- p_\alpha - g\rho_\alpha
- \right)
+ \text{div} \left\lbrace
+ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \;
+ \left( \text{\textbf{grad}}
+ p_\alpha - \varrho_\alpha \textbf{g}
  \right)
+ \right\rbrace
  =
  q_\alpha,
  \f]
  * where \f$\alpha \in \{w, n\}\f$ is the fluid phase,
+ * \f$\kappa \in \{ w, a \}\f$ are the components,
  * \f$\rho_\alpha\f$ is the fluid density, \f$S_\alpha\f$ is the fluid
  * saturation, \f$\phi\f$ is the porosity of the soil,
  * \f$k_{r\alpha}\f$ is the relative permeability for the fluid,
-- 
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