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\lhead{\textsc{PeGn624a}}
\chead{Compositional Modeling - Application to Enhanced Oil Recovery }
\rhead{Spring 2009}
\lfoot{}
\cfoot{\thepage\ of \pageref{LastPage}}
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%========  v TitlePage v =======================
{\huge\textbf{\textsc{PeGn624} Homework \# 18}}
\begin{itemize}
\item Assigned: Wednesday, March 25, 2009
\item Due: Thursday, April 2, 2009 in class
\end{itemize}


%========  v Definitions v =======================

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%========  v Main Paper v =======================


\section{Numerical Values}

Calculate the numerical values of the matrix equations at the initial conditions for the project.


The primary variables are $P_o$, $S_o$, $T$.
When $T<T_s$, $S_g=0$ and $T_s$ is used as the fourth primary variable.
When $T\geq T_s$, $S_g$ is used as the fourth primary variable.

\begin{landscape}

%--------------------------------------------------------------------------
\subsection{Part (a): Initial Form, $T<T_s$}

Calculate the numerical values for $A_{11}$, $A_{12}$, $R_{11}$, $R_{12}$, $R_{13}$, $R_{14}$, and $R_{15}$ defined
in \Eqref{mat1}.  Use the initial conditions of $P_o=500\units{psia}$ and $T_{R}=100 \degrees \units{F}$.

\begin{multline}
\label{mat1}
  \overbrace{
{\scriptsize
\begin{array}{c||cccc||cccc||cccc||}
\hhline{~#====#====#====#}
  W:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  O:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  E:  & X & X & 0 & 0   & X & X & 0 & 0   & 0 & 0 & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  O:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  E:  & X & X & 0 & 0   & X & X & 0 & 0   & X & X & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  O:  & 0 & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  E:  & 0 & 0 & 0 & 0   & X & X & 0 & 0   & X & X & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
\end{array}
}
  }^{A_{11} = \textsf{LHS}}
%
\times
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
\end{array}
  }^{\delta}
  -
  \overbrace{
{\scriptsize
\begin{array}{c||cccc||cccc||cccc||}
\hhline{~#====#====#====#}
  W:  & X & X & X & 0   & 0& 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  O:  & X & X & X & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  E:  & X & X & X & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  G:  & X & 0 & X & X   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0   & X & X & X & 0   & 0 & 0 & 0 & 0 \\
  O:  & 0 & 0 & 0 & 0   & X & X & X & 0   & 0 & 0 & 0 & 0 \\
  E:  & 0 & 0 & 0 & 0   & X & X & X & 0   & 0 & 0 & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & X & 0 & X & X   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & X & X & 0 \\
  O:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & X & X & 0 \\
  E:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & X & X & 0 \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & 0 & X & X \\
\hhline{~#====#====#====#}
\end{array}
}
  }^{A_{12} = \textsf{RHS}}
\times
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
\end{array}
  }^{\delta}
  = \\
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{11}=LHS_{P_c}}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{12}=LHS_{D}}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{13}=LHS_{\text{other}}}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{14}=RHS^n}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  X \\
\hhline{#=#}
  X \\
  X \\
  X \\
  X \\
\hhline{#=#}
  X \\
  X \\
  X \\
  X \\
\hhline{#=#}
\end{array}
  }^{R_{15}=RHS^\ell}
\end{multline}

%--------------------------------------------------------------------------
\clearpage
\subsection{Part (b): Revised Form, $T<T_s$}

Eliminate zeros in $A_{12}$ and adjust all terms in the appropriate rows.
Calculate the numerical values for $A_{31}$, and $R_{31}$ defined
in \Eqref{mat2}. The $X^{\star}$ represents adjusted values based on eliminating zeros.
The $0^{\star}$ represents the values which have been eliminated. Use the initial conditions of $P_o=500\units{psia}$ and $T_{R}=100 \degrees \units{F}$.

\begin{equation}
\label{mat2}
  \overbrace{
{\scriptsize
\begin{array}{c||cccc||cccc||cccc||}
\hhline{~#====#====#====#}
  W:  & X^{\star} & X^{\star} & 0^{\star} & 0   & X^{\star}& 0 & 0 & 0            & 0 & 0 & 0 & 0 \\
  O:  & X^{\star} & X^{\star} & 0^{\star} & 0   & X^{\star} & 0 & 0 & 0           & 0 & 0 & 0 & 0 \\
  E:  & X & X & X & 0                           & X & X & 0 & 0                   & 0 & 0 & 0 & 0 \\
  G:  & X & 0 & X & X                           & 0 & 0 & 0 & 0                   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0   & X^{\star} & 0 & 0 & 0 \\
  O:  & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0   & X^{\star} & 0 & 0 & 0 \\
  E:  & X & X & 0 & 0                   & X & X & X & 0                           & X & X & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0                   & X & 0 & X & X                           & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0           & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0 \\
  O:  & 0 & 0 & 0 & 0           & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0 \\
  E:  & 0 & 0 & 0 & 0           & X & X & 0 & 0                   & X & X & X & 0 \\
  G:  & 0 & 0 & 0 & 0           & 0 & 0 & 0 & 0                   & X & 0 & X & X \\
\hhline{~#====#====#====#}
\end{array}
}
  }^{A_{31}}
\times
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta T_s \\
\hhline{#=#}
\end{array}
  }^{\delta}
  = \\
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X^{\star} \\
  X^{\star} \\
  X \\
  X \\
\hhline{#=#}
  X^{\star} \\
  X^{\star} \\
  X \\
  X \\
\hhline{#=#}
  X^{\star} \\
  X^{\star} \\
  X \\
  X \\
\hhline{#=#}
\end{array}
  }^{R_{31}}
\end{equation}

%--------------------------------------------------------------------------
\subsection{Part (c): Extract Temperature and Pressure, $T<T_s$}

Extract the pressure and temperatures from $A_{41}$.
Calculate the numerical values for $A_{41}$, and $R_{41}$ defined
in \Eqref{mat3}. Use the initial conditions of $P_o=500\units{psia}$ and $T_{R}=100 \degrees \units{F}$.

\begin{equation}
\label{mat3}
  \overbrace{
{\scriptsize
\begin{array}{||cc||cc||cc||}
\hhline{#==#==#==#}
  X & X & X & 0 & 0 & 0 \\
  X & X & X & 0 & 0 & 0 \\
\hhline{#==#==#==#}
  X & 0 & X & X & X & 0 \\
  X & 0 & X & X & X & 0 \\
\hhline{#==#==#==#}
  0 & 0 & X & 0 & X & X \\
  0 & 0 & X & 0 & X & X \\
\hhline{#==#==#==#}
\end{array}
}
  }^{A_{41}}
\times
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  \delta P_o \\
  \delta T \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
\hhline{#=#}
\end{array}
  }^{\delta}
  = \\
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
\hhline{#=#}
  X \\
  X \\
\hhline{#=#}
  X \\
  X \\
\hhline{#=#}
\end{array}
  }^{R_{41}}
\end{equation}

%--------------------------------------------------------------------------
\clearpage
\subsection{Part (d): Initial Form, $T\geq T_s$}

Calculate the numerical values for $A_{11}$, $A_{12}$, $R_{11}$, $R_{12}$, $R_{13}$, $R_{14}$, and $R_{15}$ defined
in \Eqref{mat4}.  Use the conditions of $P_o=500\units{psia}$ and $T=500 \degrees \units{F}$.

\begin{multline}
\label{mat4}
  \overbrace{
{\scriptsize
\begin{array}{c||cccc||cccc||cccc||}
\hhline{~#====#====#====#}
  W:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  O:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  E:  & X & X & 0 & 0   & X & X & 0 & 0   & 0 & 0 & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  O:  & X & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  E:  & X & X & 0 & 0   & X & X & 0 & 0   & X & X & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  O:  & 0 & 0 & 0 & 0   & X & 0 & 0 & 0   & X & 0 & 0 & 0 \\
  E:  & 0 & 0 & 0 & 0   & X & X & 0 & 0   & X & X & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
\end{array}
}
  }^{A_{11} = \textsf{LHS}}
%
\times
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
\end{array}
  }^{\delta}
  -
  \overbrace{
{\scriptsize
\begin{array}{c||cccc||cccc||cccc||}
\hhline{~#====#====#====#}
  W:  & X & X & X & X   & 0& 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  O:  & X & X & X & 0   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  E:  & X & X & X & X   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
  G:  & X & X & X & X   & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0   & X & X & X & X   & 0 & 0 & 0 & 0 \\
  O:  & 0 & 0 & 0 & 0   & X & X & X & 0   & 0 & 0 & 0 & 0 \\
  E:  & 0 & 0 & 0 & 0   & X & X & X & X   & 0 & 0 & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0   & X & X & X & X   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & X & X & X \\
  O:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & X & X & 0 \\
  E:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & X & X & X \\
  G:  & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & X & X & X & X \\
\hhline{~#====#====#====#}
\end{array}
}
  }^{A_{12} = \textsf{RHS}}
\times
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
\end{array}
  }^{\delta}
  = \\
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{11}=LHS_{P_c}}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{12}=LHS_{D}}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{13}=LHS_{\text{other}}}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
  X \\
  X \\
  X \\
  0 \\
\hhline{#=#}
\end{array}
  }^{R_{14}=RHS^n}
  +
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X \\
  X \\
  X \\
  X \\
\hhline{#=#}
  X \\
  X \\
  X \\
  X \\
\hhline{#=#}
  X \\
  X \\
  X \\
  X \\
\hhline{#=#}
\end{array}
  }^{R_{15}=RHS^\ell}
\end{multline}

%--------------------------------------------------------------------------
\clearpage
\subsection{Part (e): Revised Form, $T\geq T_s$}

Eliminate zeros in $A_{12}$ and adjust all terms in the appropriate rows.
Calculate the numerical values for $A_{31}$, and $R_{31}$ defined
in \Eqref{mat5}. The $X^{\star}$ represents adjusted values based on eliminating zeros.
The $0^{\star}$ represents the values which have been eliminated.
Use the conditions of $P_o=500\units{psia}$ and $T=500 \degrees \units{F}$.

\begin{equation}
\label{mat5}
  \overbrace{
{\scriptsize
\begin{array}{c||cccc||cccc||cccc||}
\hhline{~#====#====#====#}
  W:  & X^{\star} & X^{\star} & 0^{\star} & 0^{\star}   & X^{\star}& 0 & 0 & 0            & 0 & 0 & 0 & 0 \\
  O:  & X^{\star} & X^{\star} & 0^{\star} & 0           & X^{\star} & 0 & 0 & 0           & 0 & 0 & 0 & 0 \\
  E:  & X^{\star} & X^{\star} & X^{\star} & 0^{\star}   & X^{\star} & X^{\star} & 0 & 0   & 0 & 0 & 0 & 0 \\
  G:  & X & X & X & X                                   & 0 & 0 & 0 & 0                   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0^{\star}   & X^{\star} & 0 & 0 & 0 \\
  O:  & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0           & X^{\star} & 0 & 0 & 0 \\
  E:  & X^{\star} & X^{\star} & 0 & 0   & X^{\star} & X^{\star} & X^{\star} & 0^{\star}   & X^{\star} & X^{\star} & 0 & 0 \\
  G:  & 0 & 0 & 0 & 0                   & X & X & X & X                                   & 0 & 0 & 0 & 0 \\
\hhline{~#====#====#====#}
%
  W:  & 0 & 0 & 0 & 0           & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0^{\star} \\
  O:  & 0 & 0 & 0 & 0           & X^{\star} & 0 & 0 & 0           & X^{\star} & X^{\star} & 0^{\star} & 0 \\
  E:  & 0 & 0 & 0 & 0           & X^{\star} & X^{\star} & 0 & 0   & X^{\star} & X^{\star} & X^{\star} & 0^{\star} \\
  G:  & 0 & 0 & 0 & 0           & 0 & 0 & 0 & 0                   & X & X & X & X \\
\hhline{~#====#====#====#}
\end{array}
}
  }^{A_{31}}
\times
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
  \delta P_o \\
  \delta T \\
  \delta S_o \\
  \delta S_g \\
\hhline{#=#}
\end{array}
  }^{\delta}
  = \\
  \overbrace{
\scriptsize
\begin{array}{||c||}
\hhline{#=#}
  X^{\star} \\
  X^{\star} \\
  X^{\star} \\
  X \\
\hhline{#=#}
  X^{\star} \\
  X^{\star} \\
  X^{\star} \\
  X \\
\hhline{#=#}
  X^{\star} \\
  X^{\star} \\
  X^{\star} \\
  X \\
\hhline{#=#}
\end{array}
  }^{R_{31}}
\end{equation}

\subsection{Part (f): Extract Temperature and Pressure, $T\geq T_s$}

Extract the pressure and temperatures from $A_{41}$.
Calculate the numerical values for $A_{41}$, and $R_{41}$ defined
in \Eqref{mat3}.
Use the conditions of $P_o=500\units{psia}$ and $T=500 \degrees \units{F}$.


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\end{document}






