Commit 06c03717 authored by Thomas Fetzer's avatar Thomas Fetzer
Browse files

[ 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
parent 320c5813
......@@ -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)
......
......@@ -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$x\f$.
*/
......
......@@ -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
......
......@@ -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
......
......@@ -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
......
......@@ -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
......
......@@ -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
......
......@@ -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
......
......@@ -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;
......
......@@ -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
{
......
......@@ -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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