From bb5d0f36e97845e798c8475b7726fa628e7059f9 Mon Sep 17 00:00:00 2001 From: Andreas Lauser Date: Fri, 5 Jun 2009 08:28:42 +0000 Subject: [PATCH] improve documentation of the two-phase model, make the boundary conditions of the lens problem readable --- .../ModelDescriptions/1p2cboxmodel.tex | 2 +- doc/handbook/ModelDescriptions/1pboxmodel.tex | 2 +- doc/handbook/ModelDescriptions/2pboxmodel.tex | 6 +++- .../ModelDescriptions/richardsboxmodel.tex | 6 +++- doc/handbook/models.tex | 36 +++++++++++-------- 5 files changed, 33 insertions(+), 19 deletions(-) diff --git a/doc/handbook/ModelDescriptions/1p2cboxmodel.tex b/doc/handbook/ModelDescriptions/1p2cboxmodel.tex index c313d9159..8b1378917 100644 --- a/doc/handbook/ModelDescriptions/1p2cboxmodel.tex +++ b/doc/handbook/ModelDescriptions/1p2cboxmodel.tex @@ -1 +1 @@ -Here comes the detailed documentation. + diff --git a/doc/handbook/ModelDescriptions/1pboxmodel.tex b/doc/handbook/ModelDescriptions/1pboxmodel.tex index c313d9159..70d29be0c 100644 --- a/doc/handbook/ModelDescriptions/1pboxmodel.tex +++ b/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. diff --git a/doc/handbook/ModelDescriptions/2pboxmodel.tex b/doc/handbook/ModelDescriptions/2pboxmodel.tex index 217c6e7b3..fd6fb609e 100644 --- a/doc/handbook/ModelDescriptions/2pboxmodel.tex +++ b/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 $S_n$. diff --git a/doc/handbook/ModelDescriptions/richardsboxmodel.tex b/doc/handbook/ModelDescriptions/richardsboxmodel.tex index c313d9159..1f5dd989d 100644 --- a/doc/handbook/ModelDescriptions/richardsboxmodel.tex +++ b/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. diff --git a/doc/handbook/models.tex b/doc/handbook/models.tex index aa4c29442..21867c4a1 100644 --- a/doc/handbook/models.tex +++ b/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: