% This file contains the paper:
% Preliminary Computer Modelling of Gas and Non-aqueous
% Tracers in Geothermal Reservoirs
% by Mark Trew, Michael O'Sullivan, etc
% 1999 All rights reserved.
\input{paper_header.tex}
\title{Modeling the Phase Partitioning Behavior of Gas Tracers Under
Geothermal Reservoir Conditions}
\author{Mark Trew\footnote{Corresponding author. Email: m.trew@auckland.ac.nz} \ \\ Michael O'Sullivan
\\
{\small Department of Engineering Science, University of Auckland, New
Zealand} \vspace{5mm} \\ Yoshio Yasuda \\ {\small JAPEX Research Center, Japan}}
\date{}
\begin{document}
\maketitle
%\doublespc
\hspace{-1em}\rule{\linewidth}{0.4pt}\newline
\begin{abstract}
A model of the liquid-vapor phase partitioning behavior of low
concentrations of gas tracers in water at geothermal temperatures and
pressures is presented. This model uses Henry's coefficient to describe the
variation of the gas tracer solubility with temperature and pressure. A new
method is described for the determination and representation of Henry's
coefficients. The method uses experimentally determined values of Henry's
coefficient and a theoretically predicted value of behavior at the critical
point of water to provide data which can be fitted by a semi-empirical
correlation. No assumptions regarding ideal behavior are necessary. The
semi-empirical correlation is a modified version of that presented by
\citeasnoun{paper:Harvey_96} and better accounts for high temperature and
non-ideal behavior. Sets of model coefficients are given for a range of
possible gas tracers. The resulting phase partitioning model is simple and
may be easily implemented in a numerical geothermal simulator. The use and
behavior of the model is illustrated by its application to a number of
idealised test problems.
\vspace{5mm}\hspace{-4.5mm}\emph{Keywords:} Henry's law coefficient; Phase partitioning; Fugacity; Ideal
gas; Peng-Robinson equation of state; Correlation.
\end{abstract}\rule{\linewidth}{0.4pt}
%\begin{table}[t] \centering
%\begin{tabular}{|>{\hspace{3mm}}p{40mm}p{110mm}|}\hline
%{} & {} \\
%{\bfseries Nomenclature}& {}\\
%{} & {} \\
%\end{tabular}
%\begin{tabular}{|>{\hspace{3mm}}p{20mm}p{130mm}|}
%{$A^*$,$B^*$,$C^*$} & {correlation coefficients for the modified Harvey
% correlation} \\
%{$\beta$} & {phase distribution coefficient for a gas tracer} \\
%{$C_H$} & {standard state Henry's coefficient (\Pa)} \\
%{$C^*_H$} & {modified Henry's coefficient (\Pa)} \\
%{$C_H^{\text{expt}}$} & {experimentally calculated Henry's coefficient (\Pa)} \\
%{$C^1_H$} & {initial fit of experimental Henry's coefficient (\Pa)} \\
%{$f$} & {fugacity} \\
%{$\hat{f}^k_i$} & {partial fugacity of component $i$ in phase $k$} \\
%{$f_i$} & {pure component fugacity of component $i$} \\
%{$f^s_i$} & {pure component fugacity of component $i$ at the standard pressure state} \\{$\gamma_i$} & {liquid phase activity coefficient of component $i$} \\
%{$\gamma_{ij}$} & {binary interaction coefficient between components $i$ and $j$} \\
%{$M_i$} & {molecular weight of component $i$ (\kgpmol)} \\
%{$P_i$} & {partial pressure of component $i$ (\Pa)} \\
%{$P^s_w$} & {water vapor pressure (\Pa)} \\
%{$P^c_i$} & {critical pressure of pure component $i$ (\Pa)} \\
%{$\hat{\phi}^k_i$} & {partial fugacity coefficient of component $i$ in phase $k$} \\
%{$R$} & {universal gas constant (8.314~\kJpkmolK)} \\
%{$\rho_i$} & {density of component $i$ (\kgpmcub)} \\
%{$T$} & {temperature (\dC)} \\
%{$T_a$} & {absolute temperature (\K)} \\
%{$T^c_i$} & {critical temperature of pure component $i$ (\dC)} \\
%{$T^r_i$} & {reduced temperature of component $i$, i.e. $T_a/T^c_i$} \\
%{$V$} & {mixture molar volume (\mcubpmol)} \\
%{$\hat{V}^k_i$} & {partial molar volume at infinite dilution and standard
% pressure state for component $i$ in phase $k$ (\mcubpmol)} \\
%{$w_i$} & {acentric factor of component $i$} \\
%{$X_i$} & {liquid phase mass fraction of component $i$} \\
%{$Y_i$} & {vapor phase mass fraction of component $i$} \\
%{$x_i$} & {liquid phase mole fraction of component $i$} \\
%{$y_i$} & {vapor phase mole fraction of component $i$} \\
%{$Z$} & {mixture compressibility} \\
%{} & {} \\ \hline
%\end{tabular}
%\end{table}
\section{Introduction}
Computer modelling of flows including geothermal tracers is motivated by the
widespread use of tracers in geothermal reservoir testing and management.
These tracer models are useful for both the design and interpretation of field
tests. The tracers used in geothermal fields may be broadly categorised into
gas tracers and liquid tracers. Gas tracers are defined as those tracer
chemicals that are highly volatile, sparingly soluble in the liquid phase and
are injected as a vapor. Examples of gas tracers include the noble gases,
refrigerants (e.g. R-23) and sulfur hexafluoride ($\text{SF}_{\text{6}}$). Liquid
tracers have low or moderate volatility and may be injected as liquids.
Examples include tritiated water, salt solutions (e.g. NaBr) and various
alcohols (e.g. methanol). Liquid tracers may exhibit liquid-vapor phase
partitioning behavior similar or quite different to that of the geothermal
water. As discussed in
\citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}, distinctive models
of phase partitioning behavior are required for each of these two tracer
categories. This paper describes recent advances in models of the phase
partitioning behavior of gas tracers.
The description of the liquid-vapor phase partitioning of gas tracers is based
on their solubility in water. Gas solubility data in water at low temperatures
are relatively abundant. However, they are scarce for the higher temperatures
encountered in geothermal reservoirs
\cite{paper:Schotte_85,paper:Japas_LeveltSengers_89}. Suitable methods for
extrapolating low temperature solubility data or relationships to high
temperatures must be considered. In this research only gas
tracers that are sparingly soluble are considered and, hence, the gas tracer is close to
infinite dilution in the liquid phase and its liquid-vapor partitioning
behavior can be modelled by Henry's law \cite{paper:Schotte_85}. Henry's law
has been used often in geothermal applications where gases are present; for
example: \citeasnoun{paper:OSullivan_Bodvarsson_Pruess_Blakeley_85},
\citeasnoun{paper:Mroczek_97} and
\citeasnoun{proc:Pruess_OSullivan_Kennedy_00}. Henry's law requires the
specification of Henry's coefficient which accounts for the solubility of the
gas tracer in water varying with temperature and perhaps pressure. The
representation of this coefficient is the key modeling consideration.
A geothermal flow simulator usually requires that a liquid-vapor phase
partitioning model provide the liquid and vapor phase mass fractions of tracer
for a given set of temperature and pressure conditions.
\citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00} describe the
calculations required to implement a liquid-vapor partitioning model for gas
tracers in the widely used TOUGH2 geothermal simulator program
\cite{report:Pruess_91,man:TOUGH2v2_99}. In developing the phase partitioning
model described in \citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}
it was assumed that gas behavior was ideal and that the concentration of gas
tracers in both the liquid and vapor phases was very small. Consequently,
given a temperature, $T$, and a partial gas pressure, $P_g$, the mass
fractions of tracer in the liquid phase, $X_g$, and in the vapor phase, $Y_g$,
were calculated as:
\begin{alignat}{1}
X_g &= \frac{P_g}{C_H(T)} \frac{M_g}{M_{\text{H}_2\text{O}}} \\
Y_g &= \frac{\rho_g(P_g,T)}{\rho_s(T)}
\end{alignat}
where $C_H(T)$ is Henry's coefficient as a function of temperature, $M_g$ is
the molecular weight of the gas, $M_{\text{H}_2\text{O}}$ is the molecular
weight of water, $\rho_g$ is the density of the gas and $\rho_s$ is the
density of steam. The gas density was calculated using an ideal gas law. The
value of Henry's coefficient is more strongly temperature dependent than
pressure dependent. In most cases a standard, or reference, pressure state of
water vapor pressure is adopted and the coefficient is given at the standard
state pressure.
In \citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00} the temperature
dependence of $C_H$ was represented by fitting discrete $C_H$ values at a
range of temperatures by the \citeasnoun{paper:Harvey_96} correlation. The
Harvey correlation is attractive for its simplicity and only contains three
unknown parameters that must be determined from experimental data. The
discrete $C_H$ values were determined for a given temperature, $T$, from
regressions of the gas distribution coefficients, $\beta$ (it was assumed that
$\beta=Y_g/X_g$), by using:
\begin{equation}
C_H(T) = \frac{\beta(T)R T_a \rho_s}{M_{\text{H}_2\text{O}}}
\end{equation}
where $R$ is the universal gas constant and $T_a$ is the absolute temperature.
The gas distribution coefficients were obtained from low temperature (i.e.
20\dC to 80\dC) regressions provided by Adams (pers. comm. 1999). These
regressions were based on published data for various gases from
\citeasnoun{paper:Wilhelm_Battino_Wilcock_77} and
\citeasnoun{paper:Wen_Muccitelli_79} and unpublished data from DuPont for the
refrigerant gas R-134a.
The gas tracer phase partitioning model described above and used by
\citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00} was easy to
implement because of its simplicity. However, there were a number of
limitations to the model, including: the assumption of ideal gas behavior; the
assumption of very small gas concentrations in all phases; the uncertain
extrapolation of low temperature relationships to geothermal temperatures; no
accounting for pressure effects in the specification of Henry's coefficient;
and occasional problems with the Harvey correlation of the $C_H$ values for
some gas tracers. These limitations are addressed in this paper.
A phase partitioning model is presented which contains some similarities to
that of \citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}. However,
the model described in this paper calculates mole fractions first and then
converts them to mass fractions and most significantly uses a modified
definition of Henry's coefficient that includes the effects of non-ideal
behavior and pressure dependence.
The experimental calculation and semi-empirical fitting of this modified form
of Henry's coefficient are the principal focus of this paper. The
determination and use of this coefficient does not assume ideal gas behavior
and incorporates the theoretical behavior of Henry's coefficients at the
critical point of water. A modified form of the semi-empirical
\citeasnoun{paper:Harvey_96} correlation for Henry's coefficient is proposed
that retains the simplicity of the original form but has improved fitting
characteristics. The final expression given for Henry's
coefficient correlates not only temperature effects, but also pressure
effects. The necessary correlation coefficients
are determined for a selection of gas tracers and finally some applications
are presented.
The effects of geothermal brine salinity on the phase partitioning behavior of
gas tracers is beyond the scope of this research. In general
it is expected that the solubility of the gas tracers in the liquid phase will
decrease with increasing electrolyte concentration
\cite{book:Walas_85,paper:Hermann_Dewes_Schumpe_95,paper:Smits_Peters_deSwaanArons_98,paper:Gao_Tan_Yu_99}.
\section{A Liquid-Vapor Phase Partitioning Model for Multiple Gas Tracers in Water} \label{sec:HenrysLawForModelling}
A liquid-vapor phase partitioning model that can be used in a geothermal flow
simulator will generally provide expressions for the liquid and vapor phase
mass fractions of gas tracer. However, the calculations are most easily performed to determine mole fractions with the mole fractions of tracer in each phase subsequently converted into mass fractions.
Consider a system with $N$ gas tracers and water. Given a low solubility, Henry's law \cite{sec:VanNess_Abbott_97} can be
used for tracer $i$ to relate the liquid mole fraction, $x_i$, to the
vapor phase mole fraction, $y_i$, the temperature, $T$, and the pressure, $P$,
as:
\begin{equation}
x_i = \frac{y_i P}{C^*_H(T,P)_i}.
\end{equation}
where $C^*_H(T,P)_i$ is a modified Henry's coefficient for tracer $i$ that
includes non-ideal behavior and accounts for the liquid phase solubility
varying with temperature and pressure. The vapor mole fraction and the
pressure can be conveniently combined into a single variable: ${P_g}_i=y_i P$.
This variable is identical to the partial pressure of an ideal gas component from
Dalton's law of partial pressures. Thus, if the partial pressure is used as a dependent variable along
with temperature and pressure, the liquid and vapor mole fractions of
the gas tracer $i$ are determined from:
\begin{alignat}{1}
x_i &= \frac{{P_g}_i}{C^*_H(T,P)_i} \label{HenryLawModifedHenryCoef1} \\
y_i &= \frac{{P_g}_i}{P}.
\end{alignat}
Once the values of $x$ and $y$ have been determined for the $N$ tracers, the liquid and
vapor phase water mole fractions are:
\begin{alignat}{1}
x_w &= 1-\sum^N_{i=1} x_i \\
y_w &= 1-\sum^N_{i=1} y_i.
\end{alignat}
The mass fraction values that are necessary for modelling the phase
partitioning of the gas tracer are calculated as:
\begin{alignat}{1}
X_i &= \frac{x_i M_i}{x_w M_w + \sum^N_{j=1} x_j M_j} \\
Y_i &= \frac{y_i M_i}{y_w M_w + \sum^N_{j=1} y_j M_j}.
\end{alignat}
where $M$ is the molecular weight, $X$ is the liquid phase mass fraction and
$Y$ is the vapor phase mass fraction. The use of these mass fractions in the
TOUGH2 geothermal simulator \cite{report:Pruess_91,man:TOUGH2v2_99} has been
described in \citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}. The
key to this phase partitioning model is the modified Henry's coefficient which
is discussed in the following sections.
\section{Experimental Calculation and Fitting of a Modified Henry's Coefficient}
\subsection{Henry's law and a modified form of Henry's coefficient}\label{sec:HenrysLawAndCoefficient}
This section shows how a modified form of Henry's coefficient can be obtained
from experimental data and used with Henry's law to calculate liquid mole
fractions of gas tracer in water for a general temperature and pressure state.
No assumption of ideal behavior is made. The only assumption is that the
solubility and, hence, liquid phase concentration of the gas tracer is
sufficiently small so as to be approximately infinitely dilute.
To account for a non-ideal component $i$ in a non-ideal mixture, the
quantities of fugacity, $f$, and the partial fugacity of component $i$,
$\hat{f}_i$, are necessary. The fugacity may be either that of the mixture or
that of the pure component $i$, i.e. $f_i$. The fugacity and partial fugacity
are defined as those quantities that replace pressure and partial pressure in
the ideal pressure, temperature and molar volume relationships in order to
maintain the validity of these relationships for non-ideal components and
mixtures \cite{book:Walas_85}. When multiple phases are
present, the partial fugacities of component $i$ in the liquid and vapor
phases become equal when the vapor and liquid phases are in equilibrium
\cite{sec:VanNess_Abbott_97}, i.e.:
\begin{equation}
\hat{f}^v_i = \hat{f}^l_i = \hat{f}_i.
\label{eqn:EquilibriumFugacities}
\end{equation}
%\begin{figure}[h] \centering
% \epsfig{figure=figs/HenrysLaw.eps,height=60mm}
% \caption{}\label{fig:HenrysLaw}
%\end{figure}
%\caption{Henry's law.}
Henry's law is based on the relationship between the gas tracer partial liquid
fugacity and its liquid mole fraction, $x$. This is shown in
\Figref{fig:HenrysLaw}. Henry's law defines a hypothetical linear liquid phase
partial fugacity and liquid mole fraction relationship for the gas tracer
which corresponds to the true state at infinite liquid dilution of the gas
tracer \cite{paper:Benson_Krause_89}. The slope of this relationship is the
value of Henry's coefficient, $C_H$. The formal definition of Henry's
coefficient is \cite{paper:Benson_Krause_89,sec:VanNess_Abbott_97}:
\begin{equation}
C_H = \lim_{x \rightarrow 0} \frac{\hat{f}^l_g}{x} \label{eqn:FormalDefnHenrysLaw}
\end{equation}
where $\hat{f}^l_g$ is the partial fugacity of the gas tracer in
the liquid phase.
Although Henry's coefficient is temperature and pressure dependent it is
usually given at a standard pressure state of water vapor pressure.
Pressure dependence is recovered by using a Poynting correction factor (PCF)
to reference the standard state Henry's coefficient at the water vapor
pressure, $C_H(T,P^{s}_w)$, to the general pressure state, $C_H(T,P)$
\cite{paper:Benson_Krause_89}:
\begin{equation}
C_H(T,P) = C_H(T,P^{s}_w) \exp{\left[ \frac{{\hat{V}^l_g} (P-P^{s}_w)}{RT_a}\right]}
\end{equation}
where ${\hat{V}^l_g}$ is the partial liquid molar volume at infinite dilution
and the standard pressure state, $R$ is the universal gas constant and $T_a$ is the absolute
temperature. The PCF is determined by integrating an expression for the rate
of change of fugacity with pressure, using the assumption that the partial
liquid molar volume varies negligibly with pressure
\cite{book:Walas_85,paper:Benson_Krause_89}.
The vapor phase partial fugacity coefficient of the gas tracer,
$\hat{\phi}^v_g$, is defined as \cite{sec:VanNess_Abbott_97}:
\begin{equation}
\hat{\phi}^v_g = \frac{\hat{f}^v_g}{y P} \label{eqn:VapPhasePartialFugCoefOfGas}
\end{equation}
where $\hat{f}^v_g$ is the partial fugacity of the gas tracer in the vapor
phase and $y$ is the mole fraction of the gas tracer in the vapor
phase. Rearrangement and division by the liquid phase mole fraction of tracer,
$x$, gives:
\begin{equation}
\frac{\hat{f}^v_g}{x} = \frac{\hat{\phi}^v_g y P}{x}
\label{eqn:FugacityLiquidMoleFractionRatio}
\end{equation}
Assuming that the solubility and mass of the gas tracer is sufficiently
small so that it is approximately infinitely dilute in the liquid phase,
\bref{eqn:EquilibriumFugacities}, \bref{eqn:FormalDefnHenrysLaw} and \bref{eqn:FugacityLiquidMoleFractionRatio} may be used to give the pressure and
temperature dependent Henry's coefficient as:
\begin{equation}
C_H(T,P) = \frac{\hat{\phi}^v_g y P}{x}
\end{equation}
A Poynting correction factor can then be used to express the standard state
Henry's coefficient as:
\begin{equation}
C_H(T,P^s_w) = \frac{\hat{\phi}^v_g y P}{x}\exp{\left[\frac{-\hat{V}^{l}_g(P-P^s_w)}{R
T_a}\right]} \label{eqn:xyHenryCoefRelationship}
\end{equation}
Experimental gas solubility data are usually available for gas-solvent binary
mixtures as the mole fraction of gas in the liquid solvent, $x$, at a given
pressure, $P$ and temperature, $T$. \Eqref{eqn:xyHenryCoefRelationship} can be
used to determine experimental values of $C_H$ using $T$, $P$, and $x$, if $y$,
$\hat{\phi}^v_g$ and $\hat{V}^{l}_g$ can be determined by auxiliary calculations.
\Secref{sec:VaporPhaseMoleFraction} describes the calculation of $y$ and sections
\ref{sec:PengRobinsonEOS} and \ref{sec:PartialMolarVolumes} describe the
calculation of $\hat{\phi}^v_g$ and $\hat{V}^{l}_g$ and other quantities
necessary to evaluate $y$. The experimentally derived
values of $C_H(T,P^s_w)$ are fitted by a semi-empirical relationship which
describes their variation with temperature.
When the semi-empirical relationship between $C_H(T,P^s_w)$ and temperature is defined, Hen-ry's
law is used to calculate the liquid mole fractions at a given temperature and
pressure. \Eqref{eqn:xyHenryCoefRelationship} can be rearranged to:
\begin{equation}
x = \frac{P_g}{\frac{C_H(T)}{\hat{\phi}^v_g(T)}} \exp{\left[\frac{-\hat{V}^{l}_g(P-P^s_w)}{R
T_a}\right]}
\end{equation}
where $P_g=yP$ is the partial gas pressure. This expression for $x$ accounts for
non-ideal behavior as well as temperature and pressure effects. It can be
observed that if this equation is further rearranged to:
\begin{equation}
x = \frac{P_g}{C^*_H(T) \exp{\left[\frac{\hat{V}^{l}_g(P-P^s_w)}{R
T_a}\right]}} \label{eqn:HenryLawModifedHenryCoef2}
\end{equation}
then the modified standard state Henry's coefficient, $C^*_H(T)$, implicitly
contains the partial fugacity coefficient necessary to account for non-ideal
behavior. \Eqref{eqn:HenryLawModifedHenryCoef2} is identical to \bref{HenryLawModifedHenryCoef1}. It is proposed then that the modified Henry's coefficient $C^*_H =
C_H(T,P^s_w)/\hat{\phi}^v_g(T,P^s_w)$ rather then $C_H$ be fitted using a semi-empirical
fit. This approach has the attractive feature that once $C^*_H$ is fitted, the
partial fugacity coefficient does not need to be calculated every time the
liquid mole fraction of the gas tracer, $x$, at
a given temperature and pressure is required.
\subsection{The vapor phase mole fraction of gas tracer} \label{sec:VaporPhaseMoleFraction}
This section describes the derivation of the calculations required to
determine the vapor phase mole fraction of gas tracer, $y$, given $TPx$
experimental data. The derivation assumes that the experimental data are for a
binary mixture of gas tracer and water and that the solubility and mass of the
gas tracer are sufficiently small so as to be approximately infinitely dilute
in the liquid phase. This means that the activity coefficient of the water
component is very close to unity for all temperatures and does not need to be
modelled \cite{paper:FernandezPrini_Crovetto_85,paper:Mroczek_97}. In order to take advantage of this, the expression for the vapor
phase mole fraction of gas tracer is based on relationships involving the
water component.
The liquid phase activity coefficient
for water, $\gamma_w$, is defined as\cite{sec:VanNess_Abbott_97}:
\begin{equation}
\gamma_w = \frac{\hat{f}^l_w}{x_w f_w}
\end{equation}
The vapor phase partial fugacity coefficient, $\hat{\phi}^v_w$, is \cite{sec:VanNess_Abbott_97}:
\begin{equation}
\hat{\phi}^v_w = \frac{\hat{f}^v_w}{y_w P} \label{eqn:VapPhasePartialFugCoef}
\end{equation}
where $y_w$ is the vapor phase mole fraction of water.
At equilibrium:
\begin{equation}
\hat{f}^l_w = \hat{f}^v_w
\end{equation}
With rearrangement, and using the fact that the activity coefficient of water
is one, the vapor phase mole fraction of water is:
\begin{equation}
y_w = \frac{x_w f_w}{\hat{\phi}^v_w P}
\end{equation}
By definition the pure component fugacity at the standard pressure state,
$P^s_w$, is: $f^s_w=\phi^s_w P^s_w$. The pure component fugacity of water may be
referred to a general pressure state through the use of a PCF:
\begin{equation}
f_w = \phi^s_w P^s_w \exp{\left[\frac{\hat{V}^{l}_w(P-P^s_w)}{R T_a}\right]}
\end{equation}
Assuming that $x$ and $y$ refer to the liquid and vapor mole fractions of gas
tracer respectively, the expression relating the liquid mole fraction of gas
tracer to the vapor mole fraction of gas tracer in a binary mixture with
water at a given temperature is:
\begin{equation}
1 - y = \frac{\gamma_w (1-x)\phi^s_w P^s_w }{\hat{\phi}^v_w
P}\exp{\left[\frac{\hat{V}^{l}_w(P-P^s_w)}{R T_a}\right]} \label{eqn:VapourLiquidMoleFractionRelationship}
\end{equation}
An important feature of \bref{eqn:VapourLiquidMoleFractionRelationship} is
its non-linearity. The value of $\hat{\phi}^v_w$ also depends on the unknown vapor
mole fraction of tracer $y$, therefore $y$ must be determined iteratively. The
calculations required for the auxiliary values: $\phi^s_w$, $\hat{\phi}^v_w$ and $\hat{V}^{l}_w$; are described in sections \ref{sec:PengRobinsonEOS} and \ref{sec:PartialMolarVolumes}.
\subsection{Calculating the Henry coefficient, $C_H(T,P^s_w)$ when only
$\beta$ is known} \label{sec:HenryCoefficientFromBeta}
For the gas tracers R-134a, R-124 and R-125 considered in this research,
experimental $TPx$ data were not available. However, regressions of the gas
distribution coefficient, $\beta$, with temperature were provided by Adams (pers. comm. 1999). The gas distribution coefficient represents the
distribution of the gas tracer between the liquid and vapor phases and is the
ratio of the vapor phase concentration to the liquid phase concentration. If
this ratio is expressed in units of molality, it can be rearranged to give a
distribution ratio in terms of the vapor and liquid phase mole fractions of
the gas tracer:
\begin{equation}
\beta = \frac{y(1-x)}{x(1-y)}
\end{equation}
This ratio can be rearranged to give an expression for the liquid phase mole
fraction of gas tracer:
\begin{equation}
x = \frac{y}{\beta + y (1-\beta)}
\end{equation}
The expression for $x$ can be used together with:
\begin{equation}
1 - y = \frac{\gamma_w (1-x)\phi^s_w P^s_w }{\hat{\phi}^v_w
P}\exp{\left[\frac{V^{l}_w(P-P^s_w)}{R T_a}\right]}
\end{equation}
to iteratively determine $x$ and $y$ values from $\beta$ values derived at low
temperatures from a $\beta$ regression. The $x$ and $y$ values can then be
used to determine the standard state Henry's coefficients using
\bref{eqn:xyHenryCoefRelationship}.
\subsection{The Peng-Robinson equation of state} \label{sec:PengRobinsonEOS}
A relationship between the pressure, temperature and molar volume of a gas
tracer and water mixture is necessary to determine successfully the auxiliary
values ($\hat{\phi}^v_g$, $\hat{V}^{l}_g$, $\phi^s_w$, $\hat{\phi}^v_w$ and $\hat{V}^{l}_w$) necessary for calculating $y$, $C_H$ and $C^*_H$. One such
relationship is the theoretical Peng-Robinson equation of state
\cite{paper:Peng_Robinson_76}. This equation of state has been widely used in
chemical engineering applications, and was used by
\citeasnoun{paper:Mroczek_97} in his experimental work with $\text{SF}_{\text{6}}$ at
geothermal temperatures and pressures.
For a mixture, the Peng-Robinson equation of state (PREOS) \cite{paper:Peng_Robinson_76} is:
\begin{equation}
P = \frac{RT_a}{V-b} - \frac{a(T_a)}{V(V+b) + b(V-b)}
\end{equation}
where $P$ is the mixture pressure, $T_a$ is the absolute temperature, $V$ is the
mixture molar volume and $a$ and $b$ are mixture parameters depending on
the pure component critical
parameters ($T^c_i$, $P^c_i$), the acentric factor ($w_i$)\footnote{$w_i$ was
originally defined to represent the acentricity or nonsphericity of a
molecule. At present it is used as a measure of the complexity of a molecule
with respect to its geometry and polarity \cite{book:Reid_Prausnitz_Sherwood_77}.} and the temperature. For
component $i$ in a multi-component mixture, $a_i$
and $b_i$ are defined as:
\begin{alignat}{1}
a_i(T) &= 0.45724 \frac{R^2 {T^c_i}^2}{P^c_i} \alpha(T^r_i,\omega_i)\\
b_i(T) &= 0.07780 \frac{R {T_c}_i}{{P_c}_i}
\end{alignat}
where $T^r_i$ is the temperature reduced by the critical temperature of component
$i$, i.e. $T^r_i=T_a/T^c_i$.
A number of choices of the $\alpha(T^r_i,\omega_i)$ function have been
presented in the literature \cite{paper:Peng_Robinson_76,paper:Tsai_Chen_98},
derived by fitting pure component vapor pressure values and finding the values
of $\alpha$ such that the liquid and vapor phase fugacities match along the
saturation curve. In this research a generalised $\alpha$ function given by
\citeasnoun{paper:Twu_Coon_Cunningham_95} has been used. The $\alpha$ function
for each component is expanded as a power series in the acentric factor and
the unknown functions fitted from vapor pressure values. The results are
\cite{paper:Twu_Coon_Cunningham_95}:
\begin{alignat}{1}
\alpha &= \alpha^{(0)} + \omega \left(\alpha^{(1)} - \alpha^{(0)} \right) \\
T^r \leq 1 & \begin{cases}
\alpha^{(0)} = {T^r}^{-0.171813} e^{0.125283\left(1-{T^r}^{1.77634}\right)} & \\
\alpha^{(1)} = {T^r}^{-0.607352} e^{0.511614\left(1-{T^r}^{2.20517}\right)} &
\end{cases} \\
T^r > 1 & \begin{cases}
\alpha^{(0)} = {T^r}^{-0.792615} e^{0.401219\left(1-{T^r}^{-0.992615}\right)} & \\
\alpha^{(1)} = {T^r}^{-1.98471} e^{0.024955\left(1-{T^r}^{-9.98471}\right)} &
\end{cases}
\end{alignat}
The mixture values of $a$ and $b$ are \cite{paper:Peng_Robinson_76}:
\begin{alignat}{1}
a &= \sum_i \sum_j z^k_i z^k_j a_{ij} \label{eqn:MixingRulea}\\
a_{ij} &= (1-\gamma_{ij}) \sqrt{a_i a_j} \\
b &= \sum_i z^k_i b_i \label{eqn:MixingRuleb}
\end{alignat}
where $z^k_i$ is the mole fraction of component $i$ in phase $k$ of the
mixture. The binary interaction component, $\gamma_{ij}$, characterises
the binary formed by components $i$ and $j$. Its value was set to 0.5 in this
work, following \citeasnoun{paper:Mroczek_97}. In addition to these original
mixture rules for $a$ and $b$ given by \citeasnoun{paper:Peng_Robinson_76} other rules have been proposed, including those derived from
combining Gibbs free energy models with an equation of state such as the PREOS
\cite{paper:Fischer_Gmehling_96,paper:Dahl_Michelsen_90,paper:Gupte_Rasmussen_Fredenslund_86,paper:Dahl_Fredenslund_Rasmussen_91,paper:Larsen_Rasmussen_Fredenslund_87}.
The PREOS may be expressed as a cubic equation in the mixture compressibility
factor, $Z$:
\begin{equation}
Z^3 - (1-B)Z^2 + (A-3B^2-2B)Z - (AB - B^2-B^3) = 0
\end{equation}
where:
\begin{alignat}{1}
A &= \frac{a_m P}{R^2 T_a^2} \\
B &= \frac{b_m P}{RT_a} \\
Z &= \frac{PV}{RT_a}. \label{eqn:DefnOfZ}
\end{alignat}
The real positive roots of this equation of the compressibilities of the
current phase state. For example, in a two-phase state, the largest positive
root is the vapor phase compressibility and the smallest positive root is the
liquid phase compressibility. The partial
fugacity coefficient for component $i$ in phase $k$, necessary for further
calculations, may be calculated as:
\begin{multline}
\ln{\phi^k_i} = \frac{b_i}{b} (Z^k - 1) - \ln{(Z^k-B)} - \\
\frac{A}{2\sqrt{2}B} \left(\frac{2\sum_j z^k_j a_{ji}}{a} - \frac{b_i}{b}\right)
\ln{\left(\frac{Z^k + (1+\sqrt{2})B}{Z^k + (1-\sqrt{2})B}\right)}
\end{multline}
\subsection{Partial molar volumes}\label{sec:PartialMolarVolumes}
The partial molar volumes are necessary for the determination of the Poynting
correction factors, i.e. pressure dependence, in the calculation of the vapor phase mole fraction and
the standard state Henry coefficient. The molar volume of phase $k$ of a
mixture is related to the mixture compressibility for that phase (see \Eqref{eqn:DefnOfZ}) as:
\begin{alignat}{1}
V^k &= \frac{Z^k R T}{P} \\
\Rightarrow \quad \delby{V^k}{z^k} &= \frac{RT}{P} \delby{Z^k}{z^k} \label{eqn:dVdxdZdx}
\end{alignat}
where $z^k$ is the mole fraction of gas tracer in phase $k$. For an $N$
component mixture, the partial molar volume for component $i$ is found from
the mixture volume as \cite{book:Walas_85}:
\begin{equation}
\hat{V}^k_i = V^k - \sum^N_{j \neq i} z^k_j \delby{V^k}{z^k_j}
\end{equation}
For a binary mixture, if the liquid mole fraction of the gas tracer is $x$,
then the water and gas component partial molar volumes in the liquid phase are:
\begin{alignat}{1}
\hat{V}^l_w &= V^l - x \delby{V^l}{x} \\
\hat{V}^l_g &= V^l + (1-x) \delby{V^l}{x} \label{eqn:PartialMolarVolumeLiquidPhaseGas}
\end{alignat}
The mixture molar volume derivative with respect to the gas tracer mole
fraction is found from \Eqref{eqn:dVdxdZdx} and the PREOS cubic equation for
the compressibility:
\begin{alignat}{1}
&\delby{}{x} \left({Z^l}^3-(1-B){Z^l}^2+\left(A-3B^2-2B\right){Z^l} - \left(AB-B^2-B^3\right)\right) = 0 \\
\Rightarrow \quad &\delby{{Z^l}}{x} = \frac{-{Z^l}^2 \delby{B}{x} - {Z^l}\left(\delby{A}{x} -
2(1+3B)\delby{B}{x} \right) + B \delby{A}{x} + \left(A-2B-3B^2\right)\delby{B}{x}}{\left(3{Z^l}^2 - 2{Z^l}(1-B) + A - 2B-3B^2\right)}
\end{alignat}
To evaluate $\delby{{Z^l}}{x}$, the PREOS must be solved to find $Z^l$ (the liquid phase root of the cubic PREOS), and mixture values for $A$, $B$, $\delby{A}{x}$ and
$\delby{B}{x}$ are required. Using a single coefficient mixing rule (equations \bref{eqn:MixingRulea} to \bref{eqn:MixingRuleb}) these mixture values are:
\begin{alignat}{1}
A &= \frac{P}{R^2 T^2} \left( (1-x)^2 a_1 + 2x(1-x)(1-\gamma_{12})\sqrt{a_1
a_2} + x^2 a_2 \right) \\
\Rightarrow \quad \delby{A}{x} &= \frac{P}{R^2 T^2} \left( -2(1-x) a_1 + 2(1-2x)(1-\gamma_{12})\sqrt{a_1
a_2} + 2x a_2 \right)
\end{alignat}
and
\begin{alignat}{1}
B &= \frac{P}{RT}((1-x)b_1 + x b_2) \\
\Rightarrow \quad \delby{B}{x} &= \frac{P}{RT} (b_2 - b_1)
\end{alignat}
\subsection{The modified Harvey correlation for Henry's coefficients} \label{sec:ModifiedHarveyCorrelation}
The Harvey semi-empirical correlation for Henry's coefficients
\cite{paper:Harvey_96,paper:Harvey_98} used by
\citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00} has the form:
\begin{equation}
\ln{C_H} = \ln{P^s_w} + \frac{A}{T^r_w} + B\frac{(1-T^r_w)^{0.355}}{T^r_w} + C
(T^r_w)^{-0.41}e^{(1-T^r_w)} \label{eqn:HarveyCorrelation}
\end{equation}
where $A$, $B$ and $C$ are correlation coefficients determined by fitting
experimental data. This correlation was based on a number of previous works \cite{paper:Japas_LeveltSengers_89,paper:Harvey_LeveltSengers_90} and
was found to be effective for both interpolation and extrapolation of Henry's
coefficients for a number of gases over a wide temperature range \cite{paper:Harvey_96,paper:Mroczek_97,paper:Harvey_98}.
The term $B(1-T^r_w)^{0.355}/T^r_w$ is important as it correctly reproduces
the divergent behavior of $\ln{C_H}$ at the solvent (water) critical point, i.e. \cite{paper:Schotte_85}:
\begin{equation}
\leftopen \dby{\ln{C_H}}{T}\right|_{T=T^c_w} = - \infty.
\end{equation}
However, it is also known that at the solvent (water) critical point \cite{paper:Beutier_Renon_78,paper:Schotte_85,paper:Japas_LeveltSengers_89}:
\begin{equation}
C_H(T^c_w,P^c_w) = \hat{\phi}^v_g(T^c_w,P^c_w) P^c_w \label{eqn:LimitingCHValue}
\end{equation}
In this research it has been found through experimentation that the limiting
value of $C_H$ is not represented by the Harvey correlation in the form of
\bref{eqn:HarveyCorrelation}, although the divergent behavior of $\ln{C_H}$ is
correctly represented. To represent the limiting value, given in
\bref{eqn:LimitingCHValue}, it is necessary that: $A+C =
\ln{\hat{\phi}^v_g(T^c_w,P^c_w)}$. In practice, this constraint has been found
to give a poor fit to the predominantly low temperature experimentally derived
$C_H$ values.
It can, however, be observed that if
$C_H(T,P^s_w)/\hat{\phi}^v_g(T,P^s_w)$ is to be fitted by a
correlation similar to that of \citeasnoun{paper:Harvey_96}, then the final
three terms must disappear as the solvent critical point is reached, i.e. $T^r_w
\rightarrow 1$. A modified Harvey correlation (MHC) with these characteristics is proposed here:
\begin{multline}
\ln{\frac{C_H}{\hat{\phi}^v_g(T,P^s_w)}} = \ln{P^s_w} +
A^*\frac{(1-T^r_w)^{0.8}}{T^r_w} + B^*\frac{(1-T^r_w)^{0.355}}{T^r_w} + \\ C^*
(1-T^r_w)^{0.8}(T^r_w)^{-0.41}e^{(1-T^r_w)} \label{eqn:ModifiedHarveyCorrelation}
\end{multline}
The MHC retains the simplicity of only three unknown coefficients, $A^*$, $B^*$ and
$C^*$, that must be fitted. The exponent of $0.8$ has been arrived at by experimentation and gives good behavior. At the solvent
critical point:
\begin{equation}
\ln{C_H} = \ln{\hat{\phi}^v_g(T^c_w,P^c_w)} + \ln{P^c_w}
\end{equation}
as required.
\subsection{An empirical correlation for the liquid phase partial molar volumes of gas tracers} \label{sec:EmpiricalCorrelationLiquidPhasePartialMolarVolume}
As shown in \bref{eqn:HenryLawModifedHenryCoef2}, if pressure effects are to
be included in the liquid-vapor partitioning model, then the partial molar gas
volume at any given temperature is required to determine the Poynting
correction factor. The calculations required for partial molar volumes given
in \Secref{sec:PartialMolarVolumes} are overly cumbersome to use in an
implementation of the partitioning model in a geothermal flow simulator. In
addition, the effect of the Poynting correction factor is usually small.
Therefore, it was decided to investigate a correlation for the liquid phase
partial molar volumes in terms of temperature and molecular weight. A
collection of gas tracers identical to those considered later in
\Secref{sec:ModifiedHenrysCoefficientsForSectionOfGasTracers} ($\text{SF}_{\text{6}}$,
R-13, R-14, R-22, R-23, R-116, R-C318, R-134a, R-124 and R-125) were used to
develop the correlation.
\Eqref{eqn:PartialMolarVolumeLiquidPhaseGas} was used to calculate the partial molar volumes of the gas tracers
under consideration. The results over a temperature range from 0\dC to
approximately 370\dC are shown in \Figref{fig:InfDilPartialMV}(a). The
expected behavior is seen. For example, the molar volume of
$\text{SF}_{\text{6}}$ at its normal boiling point (-64\dC) is estimated to be
$0.73\times 10^{-4}\mcubpmol$ \cite{paper:Mroczek_97}, so a simple
extrapolation suggests that the partial
molar volumes for $\text{SF}_{\text{6}}$ derived from
\Eqref{eqn:PartialMolarVolumeLiquidPhaseGas} and the PREOS are quantitatively correct, at least
at low temperatures. Furthermore, it has been shown that the partial
molar volume of a gas solute diverges at the critical temperature of
the solvent \cite{paper:Schotte_85}. So the partial molar volume
behavior of the gas tracers is at least qualitatively correct at the
critical point of water. A study of various infinite dilution partial
molar volume models, including the PREOS, has shown that the models
are only suitable for qualitative predictions around the solvent
critical point \cite{paper:Liu_Macedo_95}.
The mean partial molar volumes for the 10 gas tracers under
consideration have been sampled at a large number of temperature
sample points and have been fitted by the non-linear empirical
function:
\begin{equation}
\hat{V}^l_g(T^r_w) \approx \frac{-6.7\times 10^{-4} T^r_w + 8.1\times 10^{-4}}{15(1-T^r_w)^{1.28}} \label{eqn:VlgMeanFit}
\end{equation}
This is shown in \Figref{fig:InfDilPartialMV}(b).
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Computed values.]
% {\epsfig{figure=HenryGraphs/InfDilPartialMV_bw.eps,width=75mm}}
% \subfigure[Fitted mean value.]
% {\epsfig{figure=HenryGraphs/InfDilPartialMV_MeanFit_bw.eps,width=75mm}}}
% \caption{}\label{fig:InfDilPartialMV}
%\end{figure}
%\caption{Infinite dilution partial molar volumes for gas tracers. Also shown
% are the mean values for each temperature.}
The fit of the mean partial molar volumes by a function of $T^r_w$ has been
further moderated by a linear function of the molecular weight of the gas,
$M_g$, to account for
deviations from the mean values. The coefficients of the moderating function
have been fitted to all the data in a least squares sense. The final form of
the empirical partial molar volume function is:
\begin{equation}
\hat{V}^l_g(T^r_w,M_g) \approx (6.20M_g+0.34)\frac{-6.7\times 10^{-4} T^r_w +
8.1\times 10^{-4}}{15(1-T^r_w)^{1.28}} \label{eqn:VlgMgFit}
\end{equation}
This moderation provides improved representation at no extra cost of parameter
specification.
The quality of the fit to the mean values and the molecular weight moderated
fit is shown in \Figref{fig:InfDilPartialMV_Fitting}. In both plots, the
fitted partial molar volumes are plotted against the values calculated using
the Peng-Robinson equation of state. A perfect fit would lie along the
diagonal line shown in \Figref{fig:InfDilPartialMV_Fitting}.
\Figref{fig:InfDilPartialMV_Fitting}(a) clearly indicates that a significant
improvement in the empirical correlation for these gas tracers has been
achieved through the use of the molecular weight, at almost negligible
extra computational expense. In particular, the fit is quite good up to
350\dC. \Figref{fig:InfDilPartialMV_Fitting}(b) shows some interesting
results for $\text{CO}_2$ and $\text{O}_2$. Although \bref{eqn:VlgMeanFit}
was not determined using $\text{CO}_2$ or $\text{O}_2$, the final correlation
\bref{eqn:VlgMgFit} is good at predicting their partial molar volumes.
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[$\text{SF}_{\text{6}}$, R-13, R-14, R-22, R-23, R-116, R-C318, R-134a,
% R-124 and R-125.]
% {\epsfig{figure=HenryGraphs/InfDilPartialMV_FitTempValues_bw.eps,width=75mm}}
% \subfigure[$\text{O}_2$ and $\text{CO}_2$.]
% {\epsfig{figure=HenryGraphs/InfDilPartialMV_FitTempValues_O2CO2_bw.eps,width=75mm}}}
% \caption{}\label{fig:InfDilPartialMV_Fitting}
%\end{figure}
%\caption{Characteristics of the mean partial molar volume fit and the
% successful moderation by the molecular weight.}
\subsection{Summary of the steps required to fit the modified Henry's coefficients}\label{sec:FittingProcess}
This section presents an outline and summary of the steps required to
determine the modified Harvey correlation coefficients.
\begin{enumerate}
\item As described in sections \ref{sec:VaporPhaseMoleFraction} to \ref{sec:PartialMolarVolumes}, experimental $TPx$ or $TP\beta$ data are used to calculate $\hat{V}^l_w$,
$\phi^v_w$, $y$ (and $x$ if unknown), $\hat{V}^l_g$, $\hat{\phi}^v_w$ and
$\hat{\phi}^v_g$. The calculation of $y$ is non-linear as $y$ and
$\hat{\phi}^v_w$ are inter-dependent. These values are determined
iteratively. The experimental values of Henry's coefficient at the standard
state, $C_H(T,P^s_w)$, are determined from:
\begin{equation}
C^{\text{expt}}_H(T,P^s_w) = \frac{\hat{\phi}^v_g y P}{x} \exp{\left[ \frac{-\hat{V}^l_g
(P-P^s_w)}{R T_a} \right]}.
\end{equation}
\item Due to the sparsity of experimental data, it has been found best to
scale $C^{\text{expt}}_H(T,P^s_w)$ by the partial fugacity coefficient of
the gas tracer at the critical point and fit these values in a least squares
sense by the modified Harvey correlation (MHC) described in
\Secref{sec:ModifiedHarveyCorrelation}. The result of this fit is the
intermediate quantity: $C^1_H(T,P^s_w)$.
\item The $C^1_H(T,P^s_w)$ function is evaluated for a number of temperatures from 0\dC to 374\dC and a modified Henry coefficient is calculated as:
\begin{equation}
{C^1_H}^*(T,P^s_w) = \frac{C^1_H(T,P^s_w)}{\hat{\phi}^v_g(T,P^s_w)}
\hat{\phi}^v_g(T^c_w,P^c_w).
\end{equation}
\item In order to obtain a function for modeling purposes, the ${C^1_H}^*(T,P^s_w)$ values are fitted in a least-squares sense
using the MHC described in \Secref{sec:ModifiedHarveyCorrelation}. The
resulting correlation gives $C^*_H(T,P^s_w)$ which is a standard state
Henry's coefficient modified to account for non-ideality through the
inclusion of the standard state fugacity coefficient.
\item The Poynting correction factor using the empirical correlation for the liquid phase partial molar volumes (\Secref{sec:EmpiricalCorrelationLiquidPhasePartialMolarVolume}) is added to give a representation of a
modified Henry's coefficient applicable to a general temperature and pressure
state:
\begin{multline}
\ln{C^*_H(T,P)} = \ln{P^s_w} + A^* \frac{(1-T^r_w)^{0.8}}{T^r_w} + B^*
\frac{(1-T^r_w)^{0.355}}{T^r_w} + \\ C^*(1-T^r_w)^{0.8}(T^r_w)^{-0.41} \exp{(1-T^r_w)} + \\
(6.20M_g + 0.34)\frac{(-6.7\times 10^{-4} T^r_w + 8.1\times
10^{-4})}{15(1-T^r_w)^{1.28}T^c_w T^r_w}(P - P^s_w). \label{eqn:ModHarveyPressCorrect}
\end{multline}
This is the modified Henry's coefficient which is used in
\bref{HenryLawModifedHenryCoef1} for calculating the gas tracer phase
partitioning.
\end{enumerate}
\section{Modified Henry's Coefficients for a Selection of Gas Tracers}
\label{sec:ModifiedHenrysCoefficientsForSectionOfGasTracers}
The selection of gases that have been specifically considered in this work
are: $\text{SF}_{\text{6}}$, R-13, R-14, R-22, R-23, R-116, R-C318, R-134a, R-124 and
R-125. Of these, R-23 and R-134a are currently used as tracers in The Geysers
geothermal field (Adams pers. comm. 2001). R-13, R-22 and R-124 are listed as
controlled substances by the Montreal Protocol (1987) and have a non-zero
ozone-depletion potential, rendering them unacceptable as tracer chemicals.
They have been included in this study for comparison purposes only.
$\text{SF}_{\text{6}}$ has characteristics that may contribute to its relative lack of
success as a tracer in vapor-dominated systems (Mroczek 1997, Adams pers.
comm. 2001). However, extensive experimental $\text{SF}_{\text{6}}$ solubility data
are available for a wide range of temperatures. This was useful for
developing and testing the Henry's coefficient correlations.
The key properties of gas tracer considered here and the modified Harvey
correlation coefficients resulting from the fitting process described in
\Secref{sec:FittingProcess} are given in \Tabref{tab:GasTracerData}.
\Figref{fig:CHComparisons} compares the \citeasnoun{paper:Harvey_96}
correlation of Henry's coefficients (as used in
\cite{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}) with the modified Harvey
correlation for $\text{SF}_{\text{6}}$, R-134a and R-23. These examples illustrate that
the improved theoretical foundations and empirical fitting process for Henry's
coefficients presented in this research are clearly superior to those of
\cite{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00} at the temperatures found
in geothermal reservoirs.
%\begin{table}[h] \centering
% \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
% {\emph{Compound}} & {$M$(\kgpmol)} & {$T^c$(\dC)} & {$P^c$(\MPa)} & {$\omega$}
% & {$A^*$} & {$B^*$} & {$C^*$} \\ \hline \hline
% {$\text{H}_2\text{O}$} & {0.018016} & {374} & {22.06} & {0.742} & {-} & {-} & {-} \\ \hline
% {$\text{SF}_{\text{6}}$} & {0.146056} & {45.54} & {3.76} & {0.215} & {-42.0138} & {16.1840} & {30.0582} \\ \hline
% {R-13} & {0.104459} & {28.81} & {3.946} & {0.180} & {-60.0736} & {11.1937} & {51.7480} \\ \hline
% {R-14} & {0.088005} & {-45.65} & {3.739} & {0.186} & {-66.1391} & {4.7463} & {66.3009} \\ \hline
% {R-22} & {0.086468} & {96.15} & {4.97} & {0.221} & {-56.3330} & {4.5208} & {53.9459} \\ \hline
% {R-23} & {0.070014} & {25.74} & {4.836} & {0.267} & {-82.3196} & {8.7996} & {73.3207} \\ \hline
% {R-116} & {0.138012} & {19.85} & {3.06} & {0.28} & {-79.9564} & {5.1976} & {79.3443} \\ \hline
% {R-C318} & {0.200031} & {115.35} & {2.78} & {0.356} & {-86.7454} & {6.0113} & {84.0756} \\ \hline
% {R-134a} & {0.102031} & {106.85} & {3.690} & {0.239} & {-59.0986} & {6.8705} & {54.0940} \\ \hline
% {R-124} & {0.13648} & {126.75} & {3.72} & {0.281} & {-84.7575} & {8.5418} & {76.0311} \\ \hline
% {R-125} & {0.120022} & {68.85} & {3.44} & {0.259} & {-127.7211} & {11.4972} & {112.2099} \\ \hline
% \end{tabular}
%\caption{}\label{tab:GasTracerData}
%\end{table}
%\caption{Key properties and MHC coefficient values.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[$\text{SF}_{\text{6}}$.]
% {\epsfig{figure=figs/CompareSF6CHValues.eps,width=50mm}}
% \subfigure[R-134a.]
% {\epsfig{figure=figs/CompareR134aCHValues.eps,width=50mm}}
% \subfigure[R-23.]
% {\epsfig{figure=figs/CompareR23CHValues.eps,width=50mm}}}
% \caption{}\label{fig:CHComparisons}
%\end{figure}
% \caption{Comparison of Henry's coefficients calculated using the
% \protect\citeasnoun{paper:Harvey_96} correlation and the modified Harvey correlation.}
Figures \ref{fig:ModHarveyFitSF6} to \ref{fig:ModHarveyFitR125} show the
temperature and pressure variations of the Henry coefficient correlations.
Also shown is the available experimental data and the fit at each step in the
process. The results indicate that non-ideal behavior becomes more significant
for temperatures in excess of approximately 200\dC. In addition, it can be observed that in general the
variation in the modified Henry's coefficient with pressure from the standard
state pressure is small. The exception to this is near the critical
temperature of water where the partial molar volume of the gas tracer in
mixture with water diverges (see \Secref{sec:EmpiricalCorrelationLiquidPhasePartialMolarVolume}). It may be expected that for most practical simulations
the pressure correction term in \bref{eqn:ModHarveyPressCorrect} is not
necessary, however, given that it adds negligible computational overhead it
can be retained for completeness.
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/SF_6_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/SF_6_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitSF6}
%\end{figure}
%%\caption{$\text{SF}_{\text{6}}$.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-13_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-13_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR13}
%\end{figure}
%%\caption{R-13.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-14_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-14_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR14}
%\end{figure}
%%\caption{R-14.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-22_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-22_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR22}
%\end{figure}
%%\caption{R-22.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-23_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-23_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR23}
%\end{figure}
%%\caption{R-23.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-116_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-116_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR116}
%\end{figure}
%%\caption{R-116.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-C318_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-C318_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitRC318}
%\end{figure}
%%\caption{R-C318.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-134a_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-134a_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR134a}
%\end{figure}
%%\caption{R-134a.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-124_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-124_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR124}
%\end{figure}
%%\caption{R-124.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Standard state.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-125_lnCH_bw.eps,width=75mm}}
% \subfigure[Temperature and pressure variation.]
% {\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-125_lnCHTP_bw.eps,width=75mm}}}
% \caption{}\label{fig:ModHarveyFitR125}
%\end{figure}
%%\caption{R-125.}
\section{Applications} \label{sec:Applications}
\subsection{Two-phase stream-tube flow}
The comparative liquid and vapor partitioning behavior of gas tracers is best observed in
a simple stream-tube test problem. The problem and its
specifications are shown in \Figref{fig:StreamTube}.
%\begin{figure}[h] \centering
% \epsfig{figure=figs/StreamTube.eps,height=60mm}
% \caption{}\label{fig:StreamTube}
%\end{figure}
%\caption{The stream-tube test problem specifications.}
Fluid is injected with an enthalpy of $\text{1.4}\times
\text{10}^{\text{6}}~\Jpkg$, generating a two-phase flow regime. The
steady-state pressure, vapor pressure, temperature and vapor saturation
distributions are shown in \Figref{fig:SteadyStateFlow}.
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Pressure.]
% {\epsfig{figure=figs/SSPressure.eps,width=50mm}}
% \subfigure[Temperature.]
% {\epsfig{figure=figs/SSTemperature.eps,width=50mm}}
% \subfigure[Vapor saturation.]
% {\epsfig{figure=figs/SSVaporSat.eps,width=50mm}}}
% \caption{}\label{fig:SteadyStateFlow}
%\end{figure}
%\caption{Two-phase flow conditions at steady state.}
Three gas tracers are considered: $\text{SF}_{\text{6}}$, R-134a and R-23. The standard
pressure state variations of Henry's coefficients with temperature for these
tracers are given in \Figref{fig:CHComparisons}. This figure shows that at the
temperatures considered in this problem (100\dC to 300\dC) the $C_H$ values
for $\text{SF}_{\text{6}}$ obtained using the modified Harvey correlation are similar
to those of the \citeasnoun{paper:Harvey_96} correlation using the
coefficients given in
\citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}. It is expected
then that the phase partitioning model presented here and that of
\citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00} will give similar
results since the key differences lie in the determination of Henry's
coefficients.
The gas tracers are each injected at a rate of 0.1~\kgps for 20 minutes. The
liquid and vapor phase tracer return curves at the production
well are shown for the two-phase flow in Figures \ref{fig:GasTracerResultsSF6}
to \ref{fig:GasTracerResultsR23}. Phase partitioning behavior calculated using the
\citeasnoun{paper:Harvey_96} correlation of Henry's coefficients is compared to the
model presented in this paper. The implementation of the partitioning
models in TOUGH2 has been validated by comparison with known analytic
solutions for single phase transport and convergence analysis for multiple
phases.
As expected from the low gas tracer solubility, the production well
concentrations of the gas tracers in the liquid phase are at least an order of
magnitude less than those in the vapor phase. Figures
\ref{fig:GasTracerResultsSF6}(a) and \ref{fig:GasTracerResultsSF6}(b) show the
expected minor differences between partitioning behavior calculated using the
\citeasnoun{paper:Harvey_96} correlation of Henry's coefficients and the
modified Harvey correlation of this paper. The differences lie in the peak
predicted tracer concentration and not in a time shift of the tracer pulse,
thus suggesting that these arise due to the emergence of non-ideal behavior of
the $\text{SF}_{\text{6}}$ tracer as the concentration increases. In the case
of R-134a and R-23 gas tracers, the \citeasnoun{paper:Harvey_96} correlation
of Henry's coefficients predicts lower solubilities at higher temperatures
than those of the modified Harvey correlation and, particularly in the case of
R-23, this is clearly erroneous behavior. Figures
\ref{fig:GasTracerResultsSF6} to \ref{fig:GasTracerResultsR23} show that the
impact of the new models is to retard the breakthrough times of the gas
tracers.
In the single phase region, \Figref{fig:SteadyStateFlow}(a) shows that the
difference $P-P^s_w$ varies from close to zero at the production to over
100~\Pbar at the injection well. Although the value of $P-P^s_w$ is not
negligible, its impact on the final results through the Poynting correction
factor remains negligible and for most practical simulations the influence of
pressure variations on the phase partitioning models may be neglected.
However, given that its inclusion in the phase partitioning model presents
negligible computational overhead, the term can be quite easily retained for
completeness.
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Liquid phase.]
% {\epsfig{figure=figs/NewOldCoefLiquidPhase2Phase.eps,width=75mm}}
% \subfigure[Vapor phase.]
% {\epsfig{figure=figs/NewOldCoefVapourPhase2Phase.eps,width=75mm}}}
% \caption{Tracer breakthrough time history for the liquid and vapor
% phases. The predicted breakthroughs for the original partitining model given in \protect \citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}
% are compared to the model presented in this research.}\label{fig:GasTracerResults}
%\end{figure}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Liquid phase.]
% {\epsfig{figure=figs/SF6Liquid.eps,width=75mm}}
% \subfigure[Vapor phase.]
% {\epsfig{figure=figs/SF6Vapor.eps,width=75mm}}}
% \caption{}\label{fig:GasTracerResultsSF6}
%\end{figure}
%\caption{Tracer breakthrough time history for $\text{SF}_{\text{6}}$.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Liquid phase.]
% {\epsfig{figure=figs/R134aLiquid.eps,width=75mm}}
% \subfigure[Vapor phase.]
% {\epsfig{figure=figs/R134aVapor.eps,width=75mm}}}
% \caption{}\label{fig:GasTracerResultsR134a}
%\end{figure}
%\caption{Tracer breakthrough time history for R-134a.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Liquid phase.]
% {\epsfig{figure=figs/R23Liquid.eps,width=75mm}}
% \subfigure[Vapor phase.]
% {\epsfig{figure=figs/R23Vapor.eps,width=75mm}}}
% \caption{}\label{fig:GasTracerResultsR23}
%\end{figure}
%\caption{Tracer breakthrough time history for R-23.}
\subsection{An idealised three-dimensional reservoir}
The utility and value of gas tracer phase partitioning models coupled with a
geothermal reservoir simulator is illustrated by the following problem.
Steady-state two-phase convective flow is formed in an idealised reservoir of
volume 1~\kmcub. The reservoir material is homogeneous with a porosity of 0.1
and a permeability of $\text{10}^{\text{-14}}~\msq$. The boundary blocks below
the central column have a fixed internal energy of
$\text{1.85}\times\text{10}^{\text{4}}~\Jpkg$. The steady state temperature and
vapor saturation iso-surfaces are shown in \Figref{fig:CubeSSIsoSurfaces}.
100~\kg of $\text{SF}_{\text{6}}$ is introduced into the center of the reservoir. The
predicted transport over time of the gas tracer in the liquid and vapor phases
is shown in \Figref{fig:CubeSpreadOfSF6}. The sparingly soluble $\text{SF}_{\text{6}}$
tracer is preferentially transported in the vapor phase, however, the flow is
two-phase and local equilibrium exists between the phases at every point.
Consequently, $\text{SF}_{\text{6}}$ is also detected in the liquid phase farther from
the injection point than would be the case for a non-volatile tracer. For this
example the cut-off detection point has been nominally set to a mass fraction
of $10^{-12}$. This problem shows how models of tracer transport in geothermal
reservoirs can be used as predictive and correlative tools to enhance both the
design and the interpretation of field tracer tests.
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[200\dC iso-surface.]
% {\epsfig{figure=figs/CubeT200_bw_med.eps,width=50mm}}
% \subfigure[10\pcent vapor saturation iso-surface.]
% {\epsfig{figure=figs/Cube10vs_bw_med.eps,width=50mm}}}
% \caption{}\label{fig:CubeSSIsoSurfaces}
%\end{figure}
%%\caption{Temperature and vapor saturation iso-surfaces.}
%\begin{figure}[h] \centering
% \mbox{
% \subfigure[Soon after injection.]
% {\epsfig{figure=figs/CubeSF61_bw_med.eps,width=50mm}}
% \subfigure[Approximately 50 days.]
% {\epsfig{figure=figs/CubeSF62_bw_med.eps,width=50mm}}
% \subfigure[Approximately 100 days.]
% {\epsfig{figure=figs/CubeSF63_bw_med.eps,width=50mm}}}
% \caption{}\label{fig:CubeSpreadOfSF6}
%\end{figure}
%%\begin{figure}[h] \centering
%% \mbox{
%% \subfigure[Soon after injection.]
%% {\epsfig{figure=figs/CubeSF61.eps,width=50mm}}
%% \subfigure[Approximately 50 days.]
%% {\epsfig{figure=figs/CubeSF62.eps,width=50mm}}
%% \subfigure[Approximately 100 days.]
%% {\epsfig{figure=figs/CubeSF63_bw_big.eps,width=50mm}}}
%% \caption{}\label{fig:CubeSpreadOfSF6}
%%\end{figure}
%%\caption{Predicted spread of $\text{SF}_{\text{6}}$ in the vapor and liquid phases over 100
%% days.}
%%\begin{figure}[h] \centering
%% \mbox{
%% \subfigure[200\dC iso-surface.]
%% {\epsfig{figure=figs/CubeT200_email.ps,width=50mm}}
%% \subfigure[10\pcent vapor saturation iso-surface.]
%% {\epsfig{figure=figs/Cube10VS_email.ps,width=50mm}}}
%% \caption{Temperature and vapor saturation iso-surfaces.}\label{fig:CubeSSIsoSurfaces}
%%\end{figure}
%%\begin{figure}[h] \centering
%% \mbox{
%% \subfigure[Soon after injection.]
%% {\epsfig{figure=figs/CubeSF61_email.ps,width=50mm}}
%% \subfigure[Approximately 50 days.]
%% {\epsfig{figure=figs/CubeSF62_email.ps,width=50mm}}
%% \subfigure[Approximately 100 days.]
%% {\epsfig{figure=figs/CubeSF63_email.ps,width=50mm}}}
%% \caption{Predicted spread of $\text{SF}_{\text{6}}$ in the vapor and liquid phases over 100
%% days.}\label{fig:CubeSpreadOfSF6}
%%\end{figure}
\section{Summary and Conclusions}
A model of liquid-vapor phase partitioning of gas tracers in water at
geothermal temperatures and pressures has been presented. The key feature of
this model is the definition and determination of the Henry's coefficients
that are used in Henry's law to model the liquid phase mole fraction of gas
tracers. This coefficient has been derived from first principles and the
assumption of ideal gas behavior has not been necessary. The definition of
Henry's coefficient has been modified so that non-ideal behavior can be easily
included in Henry's law. Experimental values of Henry's coefficient derived
from solubility data have been fitted in two steps by a modified
semi-empirical Harvey correlation that accounts for the theoretical behavior
at the critical point of water. The semi-empirical correlation also accounts
for pressures other than the vapor pressure of water, which is the standard
state pressure. The new methods have been used to determine correlation
coefficients for a range of gas tracers and the liquid-vapor phase
partitioning models have been applied to some test problems.
The models and experimental reduction methods described in this research
retain the simplicity and attractions of those presented in previous research
by \citeasnoun{proc:Trew_OSullivan_Harvey_Anderson_Pruess_00}. However, this
new research has reduced levels of uncertainty in the process of determining
the experimental values of Henry's coefficients and has proposed a new
correlation of Henry's coefficients which is satisfactory for geothermal temperatures and pressures.
\section*{Acknowledgements}
This study has been funded in part by the New Energy \& Industrial Technology
Development Organisation of Japan (NEDO). We also acknowledge the input of
Mike Adams (EGI Utah), who provided gas solubility data and helpful comments.
\bibliographystyle{geothermics}
\bibliography{../../lit_research/TracerRefs.bib}
\newpage
\thispagestyle{empty}
\section*{Figure Captions}
\setlength{\parindent}{0mm} % Paragraph indentation and separation
\setlength{\parskip}{\baselineskip}
Fig. \ref{fig:HenrysLaw}. The definition of Henry's law as a hypothetical
linear relationship of slope $C_H$ between the liquid phase partial gas
fugacity, $\hat{f}^l_g$, and the liquid mole fraction of tracer, $x$.
Fig. \ref{fig:InfDilPartialMV}. Theoretically calculated infinite dilution
partial molar volumes, $\hat{V}^l_g$, for a selection of gas tracers. (A)
Maximum, minimum and mean $\hat{V}^l_g$ values. (B) Mean values and the
empirical fit to the mean values without including molecular weight.
Fig. \ref{fig:InfDilPartialMV_Fitting}. The quality of the empirical fit to
the theoretically calculated infinite dilution
partial molar volumes, $\hat{V}^l_g$. The fit improves when the molecular
weight is included. (A) Fitting the mean values determined
from data for:
$\text{SF}_{\text{6}}$, R-13, R-14, R-22, R-23, R-116, R-C318, R-134a, R-124
and R-125. (B) Using the same mean values to empirically represent the
behavior of $\text{O}_{\text{2}}$ and $\text{CO}_{\text{2}}$.
Fig. \ref{fig:CHComparisons}. A comparison of Henry's law coefficients calculated
using the \protect\citeasnoun{paper:Harvey_96} correlation and the modified
Harvey correlation. (A) $\text{SF}_{\text{6}}$. (B) R-134a. (C) R-23.
Fig. \ref{fig:ModHarveyFitSF6}. Henry's law coefficient fit for
$\text{SF}_{\text{6}}$. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR13}. Henry's law coefficient fit for R-13. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR14}. Henry's law coefficient fit for R-14. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR22}. Henry's law coefficient fit for R-22. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR23}. Henry's law coefficient fit for R-23. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR116}. Henry's law coefficient fit for R-116. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitRC318}. Henry's law coefficient fit for R-C318. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR134a}. Henry's law coefficient fit for R-134a. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR124}. Henry's law coefficient fit for R-124. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:ModHarveyFitR125}. Henry's law coefficient fit for R-125. (A) At standard state. (B) The variation with temperature
and pressure.
Fig. \ref{fig:StreamTube}. The stream-tube test problem specifications.
Fig. \ref{fig:SteadyStateFlow}. The two-phase flow conditions at steady
state in the stream-tube test problem. The distance is from the production well. (A) Pressure, $P$, and water vapor pressure, $P^s_w$, variation. (B)
Temperature variation. (C) Vapor saturation variation.
Fig. \ref{fig:GasTracerResultsSF6}. The tracer return curves for
$\text{SF}_{\text{6}}$ in the stream-tube test problem. The time is from injection. (A) Liquid phase. (B) Vapor phase.
Fig. \ref{fig:GasTracerResultsR134a}. The tracer return curves for R-134a in the stream-tube test problem. The time is from injection. (A) Liquid phase. (B) Vapor phase.
Fig. \ref{fig:GasTracerResultsR23}. The tracer return curves for R-23 in the stream-tube test problem. The time is from injection. (A) Liquid phase. (B) Vapor phase.
Fig. \ref{fig:CubeSSIsoSurfaces}. The Temperature and vapor saturation
iso-surfaces in an idealised three-dimensional reservoir. (A) 200\dC
iso-surface. (B) 10\pcent vapor saturation iso-surface.
Fig. \ref{fig:CubeSpreadOfSF6}. The predicted liquid and vapor phase mass fractions of $\text{SF}_{\text{6}}$
in an idealised three-dimensional reservoir over 100 days. (A) Following
injection. (B) After 50 days. (C) After 100 days.
\newpage
\pagestyle{empty}
\section*{Table Captions}
Table \ref{tab:GasTracerData}\newline The molecular weight ($M$), critical
temperature ($T^c$), critical pressure ($P^c$), acentric factor ($\omega$), and
modified Harvey correlation coefficients ($A^*$, $B^*$ and $C^*$) for water
and a selection of gas tracers.
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\epsfig{figure=figs/HenrysLaw.eps,width=150mm}
\caption{}\label{fig:HenrysLaw}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=HenryGraphs/InfDilPartialMV_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=HenryGraphs/InfDilPartialMV_MeanFit_bw.eps,width=150mm}}}
\caption{}\label{fig:InfDilPartialMV}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=HenryGraphs/InfDilPartialMV_FitTempValues_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=HenryGraphs/InfDilPartialMV_FitTempValues_O2CO2_bw.eps,width=150mm}}}
\caption{}\label{fig:InfDilPartialMV_Fitting}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=figs/CompareSF6CHValues.eps,width=80mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/CompareR134aCHValues.eps,width=80mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/CompareR23CHValues.eps,width=80mm}}}
\caption{}\label{fig:CHComparisons}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/SF_6_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/SF_6_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitSF6}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-13_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-13_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR13}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-14_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-14_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR14}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-22_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-22_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR22}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-23_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-23_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR23}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-116_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-116_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR116}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-C318_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-C318_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitRC318}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-134a_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-134a_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR134a}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-124_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-124_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR124}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-125_lnCH_bw.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=/usr/people/trew/postdoc/Matlab/NewGraphs/R-125_lnCHTP_bw.eps,width=150mm}}}
\caption{}\label{fig:ModHarveyFitR125}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\epsfig{figure=figs/StreamTube.eps,width=150mm}
\caption{}\label{fig:StreamTube}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=figs/SSPressure.eps,width=80mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/SSTemperature.eps,width=80mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/SSVaporSat.eps,width=80mm}}}
\caption{}\label{fig:SteadyStateFlow}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=figs/SF6Liquid.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/SF6Vapor.eps,width=150mm}}}
\caption{}\label{fig:GasTracerResultsSF6}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=figs/R134aLiquid.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/R134aVapor.eps,width=150mm}}}
\caption{}\label{fig:GasTracerResultsR134a}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=figs/R23Liquid.eps,width=150mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/R23Vapor.eps,width=150mm}}}
\caption{}\label{fig:GasTracerResultsR23}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=figs/CubeT200_bw_med.eps,width=100mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/Cube10vs_bw_med.eps,width=100mm}}}
\caption{}\label{fig:CubeSSIsoSurfaces}
\end{figure}
\newpage
\pagestyle{empty}
\begin{figure}[H] \centering
\mbox{
\subfigure
{\epsfig{figure=figs/CubeSF61_bw_med.eps,width=80mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/CubeSF62_bw_med.eps,width=80mm}}}
\mbox{
\subfigure
{\epsfig{figure=figs/CubeSF63_bw_med.eps,width=80mm}}}
\caption{}\label{fig:CubeSpreadOfSF6}
\end{figure}
\newpage
\pagestyle{empty}
\begin{table}[H] \centering
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
{\emph{Compound}} & {$M$(\kgpmol)} & {$T^c$(\dC)} & {$P^c$(\MPa)} & {$\omega$}
& {$A^*$} & {$B^*$} & {$C^*$} \\ \hline
{$\text{H}_2\text{O}$} & {0.018016} & {374} & {22.06} & {0.742} & {-} & {-} & {-} \\ \hline
{$\text{SF}_{\text{6}}$} & {0.146056} & {45.54} & {3.76} & {0.215} & {-42.0138} & {16.1840} & {30.0582} \\ \hline
{R-13} & {0.104459} & {28.81} & {3.946} & {0.180} & {-60.0736} & {11.1937} & {51.7480} \\ \hline
{R-14} & {0.088005} & {-45.65} & {3.739} & {0.186} & {-66.1391} & {4.7463} & {66.3009} \\ \hline
{R-22} & {0.086468} & {96.15} & {4.97} & {0.221} & {-56.3330} & {4.5208} & {53.9459} \\ \hline
{R-23} & {0.070014} & {25.74} & {4.836} & {0.267} & {-82.3196} & {8.7996} & {73.3207} \\ \hline
{R-116} & {0.138012} & {19.85} & {3.06} & {0.28} & {-79.9564} & {5.1976} & {79.3443} \\ \hline
{R-C318} & {0.200031} & {115.35} & {2.78} & {0.356} & {-86.7454} & {6.0113} & {84.0756} \\ \hline
{R-134a} & {0.102031} & {106.85} & {3.690} & {0.239} & {-59.0986} & {6.8705} & {54.0940} \\ \hline
{R-124} & {0.13648} & {126.75} & {3.72} & {0.281} & {-84.7575} & {8.5418} & {76.0311} \\ \hline
{R-125} & {0.120022} & {68.85} & {3.44} & {0.259} & {-127.7211} & {11.4972} & {112.2099} \\ \hline
\end{tabular}
\caption{}\label{tab:GasTracerData}
\end{table}
% Nomenclature
\begin{table}[t] \centering
\begin{tabular}{>{\hspace{3mm}}p{40mm}p{110mm}}
{} & {} \\
{\bfseries Nomenclature}& {}\\
{} & {} \\
\end{tabular}
\begin{tabular}{>{\hspace{3mm}}p{20mm}p{130mm}}
{$A^*$,$B^*$,$C^*$} & {correlation coefficients for the modified Harvey
correlation} \\
{$\beta$} & {phase distribution coefficient for a gas tracer} \\
{$C_H$} & {standard state Henry's coefficient (\Pa)} \\
{$C^*_H$} & {modified Henry's coefficient (\Pa)} \\
{$C_H^{\text{expt}}$} & {experimentally calculated Henry's coefficient (\Pa)} \\
{$C^1_H$} & {initial fit of experimental Henry's coefficient (\Pa)} \\
{$f$} & {fugacity} \\
{$\hat{f}^k_i$} & {partial fugacity of component $i$ in phase $k$} \\
{$f_i$} & {pure component fugacity of component $i$} \\
{$f^s_i$} & {pure component fugacity of component $i$ at the standard pressure state} \\{$\gamma_i$} & {liquid phase activity coefficient of component $i$} \\
{$\gamma_{ij}$} & {binary interaction coefficient between components $i$ and $j$} \\
{$M_i$} & {molecular weight of component $i$ (\kgpmol)} \\
{$P_i$} & {partial pressure of component $i$ (\Pa)} \\
{$P^s_w$} & {water vapor pressure (\Pa)} \\
{$P^c_i$} & {critical pressure of pure component $i$ (\Pa)} \\
{$\hat{\phi}^k_i$} & {partial fugacity coefficient of component $i$ in phase $k$} \\
{$R$} & {universal gas constant (8.314~\kJpkmolK)} \\
{$\rho_i$} & {density of component $i$ (\kgpmcub)} \\
{$T$} & {temperature (\dC)} \\
{$T_a$} & {absolute temperature (\K)} \\
{$T^c_i$} & {critical temperature of pure component $i$ (\dC)} \\
{$T^r_i$} & {reduced temperature of component $i$, i.e. $T_a/T^c_i$} \\
{$V$} & {mixture molar volume (\mcubpmol)} \\
{$\hat{V}^k_i$} & {partial molar volume at infinite dilution and standard
pressure state for component $i$ in phase $k$ (\mcubpmol)} \\
{$w_i$} & {acentric factor of component $i$} \\
{$X_i$} & {liquid phase mass fraction of component $i$} \\
{$Y_i$} & {vapor phase mass fraction of component $i$} \\
{$x_i$} & {liquid phase mole fraction of component $i$} \\
{$y_i$} & {vapor phase mole fraction of component $i$} \\
{$Z$} & {mixture compressibility} \\
{} & {} \\
\end{tabular}
\end{table}
\end{document}