From 3ca0d92724e297b9fdb9494af29c86c412660e2b Mon Sep 17 00:00:00 2001 From: Andreas Lauser Date: Tue, 21 Feb 2012 16:30:36 +0000 Subject: [PATCH] update the model descriptions for the box models in the handbook --- .../ModelDescriptions/1p2cboxmodel.tex | 11 ++++---- doc/handbook/ModelDescriptions/1pboxmodel.tex | 3 +-- .../ModelDescriptions/2p2cboxmodel.tex | 15 +++++------ .../ModelDescriptions/2p2cniboxmodel.tex | 13 +++++----- doc/handbook/ModelDescriptions/2pboxmodel.tex | 7 +++-- .../ModelDescriptions/2pniboxmodel.tex | 7 +++-- .../ModelDescriptions/3p3cboxmodel.tex | 26 +++++++++++++++++++ .../ModelDescriptions/3p3cniboxmodel.tex | 26 +++++++++++++++++++ .../ModelDescriptions/richardsboxmodel.tex | 7 +++-- .../ModelDescriptions/stokes2cmodel.tex | 16 ++++++++++++ .../ModelDescriptions/stokes2cnimodel.tex | 18 +++++++++++++ .../ModelDescriptions/stokesmodel.tex | 12 +++++++++ doc/handbook/models.tex | 21 ++++++++++++--- 13 files changed, 143 insertions(+), 39 deletions(-) create mode 100644 doc/handbook/ModelDescriptions/3p3cboxmodel.tex create mode 100644 doc/handbook/ModelDescriptions/3p3cniboxmodel.tex create mode 100644 doc/handbook/ModelDescriptions/stokes2cmodel.tex create mode 100644 doc/handbook/ModelDescriptions/stokes2cnimodel.tex create mode 100644 doc/handbook/ModelDescriptions/stokesmodel.tex diff --git a/doc/handbook/ModelDescriptions/1p2cboxmodel.tex b/doc/handbook/ModelDescriptions/1p2cboxmodel.tex index adf7432c7..37e324311 100644 --- a/doc/handbook/ModelDescriptions/1p2cboxmodel.tex +++ b/doc/handbook/ModelDescriptions/1p2cboxmodel.tex @@ -4,14 +4,13 @@ % file instead!! % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\-This model implements a one-\/phase flow of a compressible fluid, that consists of two components, using a standard \-Darcy approach as the equation for the conservation of momentum\-: \[ v_{D} = - \frac{\textbf K}{\mu} \left(\text{grad} p - \varrho {\textbf g} \right) \] -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: \[ v_{D} = - \frac{K}{\mu} \left(\text{grad} p - \varrho g \right) \] +\-Gravity can be enabled or disabled via the property system. \-By inserting this into the continuity equation, one gets \[ \Phi \frac{\partial \varrho}{\partial t} - \text{div} \left\{ \varrho \frac{\textbf K}{\mu} \left(\text{grad}\, p - \varrho {\textbf g} \right) \right\} = q \;, \] -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\-: \[ \Phi \frac{ \partial \varrho x}{\partial t} - \text{div} \left( \varrho \frac{{\textbf K} x}{\mu} \left( \text{grad}\, p - \varrho {\textbf g} \right) + \varrho \tau \Phi D \text{grad} x \right) = q. \] -The transport of the components is described by the following equation: \[ \Phi \varrho \frac{ \partial x}{\partial t} - \text{div} \left( \varrho \frac{K x}{\mu} \left( \text{grad} p - \varrho g \right) + \varrho \tau \Phi D \text{grad} x \right) = q. \] +\-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 fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method as time discretization. - -The primary variables are the pressure $p$ and the mole fraction of dissolved component $x$. +\-The primary variables are the pressure $p$ and the mole or mass fraction of dissolved component $x$. diff --git a/doc/handbook/ModelDescriptions/1pboxmodel.tex b/doc/handbook/ModelDescriptions/1pboxmodel.tex index 91b93512e..51bf61036 100644 --- a/doc/handbook/ModelDescriptions/1pboxmodel.tex +++ b/doc/handbook/ModelDescriptions/1pboxmodel.tex @@ -4,6 +4,5 @@ % file instead!! % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Adaption of the BOX scheme to the single phase isothermal flow model. Single phase compressible isothermal flow model, \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*} 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 in the problem file a fluid with constant density is chosen. +\-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. diff --git a/doc/handbook/ModelDescriptions/2p2cboxmodel.tex b/doc/handbook/ModelDescriptions/2p2cboxmodel.tex index fb378aa01..6e39c9d74 100644 --- a/doc/handbook/ModelDescriptions/2p2cboxmodel.tex +++ b/doc/handbook/ModelDescriptions/2p2cboxmodel.tex @@ -4,17 +4,16 @@ % file instead!! % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\-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\-: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \] -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 \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} -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. -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: +\-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 \-Two\-P\-Two\-C\-Indices\-::p\-Ws\-N or \-Two\-P\-Two\-C\-Indices\-::p\-Ns\-W. \-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