opm-simulators/doc/handbook/ModelDescriptions/blackoilmodel.tex
Andreas Lauser 3e55945ce5 change namespace from Dumux to Ewoms
eWoms hereby declares full independence. Humor aside, the main
technical advantage of this is, that it is now possible to easily
install both, Dumux and eWoms on a system using a package management
system without bad tricks.
2012-11-18 16:58:22 +01:00

42 lines
2.8 KiB
TeX

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This file has been autogenerated from the LaTeX part of the %
% doxygen documentation; DO NOT EDIT IT! Change the model's .hh %
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The black-\/oil model is a three-\/phase, three-\/component model widely used for oil reservoir simulation. The phases are denoted by lower index $\alpha \in \{ w, g, o \}$ (\char`\"{}water\char`\"{}, \char`\"{}gas\char`\"{} and \char`\"{}oil\char`\"{}) and the components by upper index $\kappa \in \{ W, G, O \}$ (\char`\"{}\-Water\char`\"{}, \char`\"{}\-Gas\char`\"{} and \char`\"{}\-Oil\char`\"{}). The model assumes partial miscibility\-:
\begin{itemize}
\item Water and the gas phases are immisicible and are assumed to be only composed of the water and gas components respectively-\/
\item The oil phase is assumed to be a mixture of the gas and the oil components.
\end{itemize}
The densities of the phases are determined by so-\/called {\itshape formation volume factors}\-:
\[ B_\alpha := \frac{\varrho_\alpha(1\,\text{bar})}{\varrho_\alpha(p_\alpha)} \]
Since the gas and water phases are assumed to be immiscible, this is sufficint to calculate their density. For the formation volume factor of the the oil phase $B_o$ determines the density of {\itshape saturated} oil, i.\-e. the density of the oil phase if some gas phase is present.
The composition of the oil phase is given by the {\itshape gas formation factor} $R_s$, which defined as the volume of gas at atmospheric pressure that is dissolved in saturated oil at a given pressure\-:
\[ R_s := \frac{x_o^G(p)\,\varrho_{mol,o}(p)}{\varrho_g(1\,\text{bar})}\;. \]
This allows to calculate all quantities required for the mass-\/conservation equations for each component, i.\-e.
\[ \sum_\alpha \frac{\partial\;\phi c_\alpha^\kappa S_\alpha }{\partial t} - \sum_\alpha \text{div} \left\{ c_\alpha^\kappa \mathbf{v}_\alpha \right\} - q^\kappa = 0 \;, \] where $\mathrm{v}_\alpha$ is the filter velocity of the phase $\alpha$.
By default $\mathrm{v}_\alpha$ is determined by using the standard multi-\/phase Darcy approach, i.\-e. \[ \mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \;, \] although the actual approach which is used can be specified via the {\ttfamily Velocity\-Module} property. For example, the velocity model can by changed to the Forchheimer approach by
\begin{lstlisting}[style=eWomsCode]
SET_TYPE_PROP(MyProblemTypeTag, VelocityModule,
Ewoms::BoxForchheimerVelocityModule<TypeTag>);
\end{lstlisting}
The primary variables used by this model are\-:
\begin{itemize}
\item The pressure of the phase with the lowest index
\item The two saturations of the phases with the lowest indices
\end{itemize}