Published on *PNG 520: Phase Behavior of Natural Gas and Condensate Fluids* (https://www.e-education.psu.edu/png520)

Module Goal: To highlight the important properties used to characterize natural gas and condensate systems.

Module Objective: To present the most popular models for estimating properties of natural gas and condensate systems.

The *constant volume* heat capacity is defined by:

$${C}_{v}{\left(\frac{\partial U}{\partial T}\right)}_{V}$$

To see the physical significance of the constant volume heat capacity, let us consider a 1 lbmol of gas within a rigid-wall (constant volume) container. Heat is added to the system through the walls of the container and the gas temperature rises. It is evident that the temperature rise ($\Delta T$ ) is proportional to the amount of heat added,

$$Q\propto \Delta T$$

Introducing a constant of proportionality “cv”,

$$Q={c}_{v}\Delta T$$

In our experiment, no work was done because the boundaries (walls) of the system remained unchanged. Applying the first law of thermodynamics to this closed system, we have:

$$\Delta U={c}_{v}\Delta T$$

Therefore, for infinitesimal changes,

$${C}_{v}={\left(\frac{\partial U}{\partial T}\right)}_{V}$$

As we have seen, constant volume heat capacity is the amount of heat required to raise the temperature of a gas by one degree *while retaining its volume*.

Let us now consider the same 1 lbmol of gas confined in a piston-cylinder equipment (i.e., a system with non-rigid walls or boundaries). When heat is added to the system, the gas temperature rises and the gas expands so that *the pressure in the system remains the same at any time*. The piston displaces a volume $\Delta $ V and the gas increases its temperature in $\Delta $T degrees. Again, the temperature rise ($\Delta $T) is proportional to the amount of heat added, and the new constant of proportionality we use here is “c_{p}”,

$$Q={c}_{P}\Delta T$$

This time, some work was done because the boundaries (walls) of the system changed from their original position. Applying the first law of thermodynamics to this closed system, we have that:

$$\Delta U=Q-W$$

If the pressure remained the same both inside and outside the container, the system made some work against the surroundings in the amount of $W={P}_{\Delta}V$. Introducing (19.7) into (19.6),

$$\Delta U+P\Delta V={c}_{P}\Delta T$$

The left hand side of this equation represents the definition of *enthalpy change* ($\Delta H$ ) for a constant-pressure process. Therefore:

$$\Delta H={c}_{P}\Delta T$$

Finally, for infinitesimal changes,

$${c}_{P}={\left(\frac{\partial H}{\partial T}\right)}_{V}$$

The function “c_{p}” is called the *constant pressure heat capacity*. The constant pressure heat capacity is the amount of heat required to raise the temperature of a gas by one degree *while retaining its pressure*.

The units of both heat capacities are (Btu/lbmol-°F) and (cal/gr-°C). Their values are *never equal to each other*, not even for ideal gases. In fact, the ratio “c_{p}/c_{v}” of a gas is known as “k” — the heat capacity ratio — and it is never equal to unity. This ratio is frequently used in gas-dynamics studies.

$$k=\frac{{c}_{p}}{{c}_{v}}$$

Heat capacities can be calculated using equations of state. For instance, Peng and Robinson (1976) presented an expression for the departure enthalpy of a fluid mixture, shown below:

$$\ddot{H}=H-{H}^{\xb7}=RT(Z-1)+\frac{T\frac{d{(a\alpha )}_{m}}{dT}-{(a\alpha )}_{m}}{2\sqrt{2{b}_{m}}}\mathrm{ln}\left(\frac{Z+(\sqrt{2}+1)B}{Z-(\sqrt{2}-1)B}\right)$$

The value of the enthalpy of the fluid (H) is obtained by adding up this enthalpy of departure (shown above) to the ideal gas enthalpy (H*). Ideal enthalpies are sole functions of temperature. For hydrocarbons, Passut and Danner (1972) developed correlations for ideal gas properties such as enthalpy, heat capacity and entropy as a function of temperature. Therefore, an analytical relationship for “c_{p}” can be derived taking the derivative of (19.12), as shown below:

$$\begin{array}{l}{C}_{P}={C}_{P}^{\xb7}+R\left(T{\left(\frac{\partial Z}{\partial T}\right)}_{P}+Z-1\right)+\frac{T\frac{d{(a\alpha )}_{m}}{dT}-{(a\alpha )}_{m}}{2\sqrt{2{b}_{m}}}\\ \left[\frac{{\left(\frac{\partial Z}{\partial T}\right)}_{P}+2.414{\left(\frac{\partial B}{\partial T}\right)}_{P}}{Z+2.414B}-\frac{{\left(\frac{\partial Z}{\partial T}\right)}_{P}-0.414{\left(\frac{\partial B}{\partial T}\right)}_{P}}{Z-0.414B}\right]+\frac{T\frac{{d}^{2}{\left(a\alpha \right)}_{m}}{d{T}^{2}}}{2\sqrt{2{b}_{m}}}\mathrm{ln}\left(\frac{Z+2.414B}{Z-414B}\right)\end{array}$$

where:

${{\displaystyle C}}_{P}^{\xb7}={\left(\frac{\partial {H}^{\xb7}}{\partial T}\right)}_{P}$
= ideal gas C_{P},

also found in the work of Passut and Danner (1972).

The second derivative of ${\left(a\alpha \right)}_{m}$ with respect to temperature can be calculated through the expression:

$$\begin{array}{l}\frac{{d}^{2}}{d{T}^{2}}{\left(a\alpha \right)}_{m}=-\frac{0.45724{R}^{2}}{2\sqrt{T}}\underset{i}{{{\displaystyle \sum}}^{\text{}}}{\displaystyle \sum}_{j}{c}_{i}{c}_{j}\left(1-{k}_{ij}\right)\\ \left[f\left({w}_{j}\right)\left(\frac{{\alpha}_{i}^{0.5}{T}_{ci}}{{P}_{ci}^{0.5}}\right){\left(\frac{{T}_{cj}}{{P}_{cj}}\right)}^{0.5}{\psi}_{i}+f\left({w}_{i}\right)\left(\frac{{\alpha}_{j}^{0.5}{T}_{cj}}{{P}_{cj}^{0.5}}\right){\left(\frac{{T}_{ci}}{{P}_{ci}}\right)}^{0.5}{\psi}_{j}\right]\end{array}$$

where,

$${\psi}_{i}=-\frac{f\left({w}_{i}\right)}{2\sqrt{{T}_{ci}T{\alpha}_{i}}}-\frac{1}{2T}$$

For the evaluation of expression (19.13), the derivative of the compressibility factor with respect to temperature is also required. Using the cubic version of Peng-Robinson EOS, this derivative can be written as:

$${\left(\frac{\partial Z}{\partial T}\right)}_{P}=-\left(\frac{{\left(\frac{\partial {\Omega}_{2}}{\partial T}\right)}_{P}{Z}^{2}+{\left(\frac{\partial {\Omega}_{3}}{\partial T}\right)}_{P}Z+{\left(\frac{\partial {\Omega}_{4}}{\partial T}\right)}_{P}}{3{Z}^{2}+2{\Omega}_{2}Z+{\Omega}_{3}}\right)$$

where,

$$\begin{array}{l}{\left(\frac{\partial {\Omega}_{2}}{\partial T}\right)}_{P}={\left(\frac{\partial B}{\partial T}\right)}_{P}\\ {\left(\frac{\partial {\Omega}_{3}}{\partial T}\right)}_{P}={\left(\frac{\partial A}{\partial T}\right)}_{P}-6B{\left(\frac{\partial B}{\partial T}\right)}_{P}-2{\left(\frac{\partial B}{\partial T}\right)}_{P}\\ {\left(\frac{\partial {\Omega}_{4}}{\partial T}\right)}_{P}=-\left[A{\left(\frac{\partial B}{\partial T}\right)}_{P}+B{\left(\frac{\partial A}{\partial T}\right)}_{P}-2B{\left(\frac{\partial B}{\partial T}\right)}_{P}-3{B}^{2}{\left(\frac{\partial B}{\partial T}\right)}_{P}\right]\\ {\left(\frac{\partial A}{\partial T}\right)}_{P}=\frac{A}{{\left(a\alpha \right)}_{m}}\frac{d{\left(a\alpha \right)}_{m}}{dT}-2\frac{A}{T}\\ {\left(\frac{\partial B}{\partial T}\right)}_{P}=-\frac{B}{T}\end{array}$$

“c_{p}” and “c_{v}” values are thermodynamically related. It can be proven that this relationship is controlled by the P-V-T behavior of the substances through the relationship:

$${c}_{p}-{c}_{v}=T{\left(\frac{\partial V}{\partial T}\right)}_{P}{\left(\frac{\partial P}{\partial T}\right)}_{V}$$

For ideal gases, $PV=nRT$ and Equation (18.28) collapses to:

$${c}_{p}^{\xb7}-{c}_{v}^{\xb7}=R$$

One remarkable difference between flow of condensate (or liquid) and natural gases through a pipeline is that of the effect of pressure drop on temperature changes along the pipeline. This is especially true when heat losses to the environment do not control these temperature variations. Natural gas pipelines usually cool with distance (effect commonly called ‘Joule–Thomson cooling’), while oil lines heat. The reason for such dissimilarity pertains to the different effect that pressure drop has on the entropy of a natural gas than on the entropy of an oil mixture. Katz (1972) and Katz and Lee (1990) presented a very enlightening discussion on this regard.

Whether or not a gas cools upon expansion or compression — that is, when subjected to pressure changes — depends on the value of its Joule–Thomson coefficient. This is not only important for natural gas pipeline flow, but also for the recovery of condensate from wet natural gases. In the cryogenic industry, turboexpanders are used to subject a wet gas to a sudden expansion (sharp pressure drop) in order to cool the gas stream beyond its dew point and recover the liquid dropout.

Thermodynamically, the Joule–Thomson coefficient is defined as the isenthalpic change in temperature in a fluid caused by a unitary pressure drop, as shown:

$$\eta ={\left(\frac{\partial T}{\partial P}\right)}_{H}$$

Using thermodynamic relationships, alternative expressions can be written. For example, using the cycling rule we may write:

$${\left(\frac{\partial H}{\partial P}\right)}_{T}{\left(\frac{\partial P}{\partial T}\right)}_{H}{\left(\frac{\partial T}{\partial H}\right)}_{P}=-1$$

or,

$${\left(\frac{\partial H}{\partial P}\right)}_{T}=-{\left(\frac{\partial H}{\partial T}\right)}_{P}{\left(\frac{\partial T}{\partial P}\right)}_{H}$$

$${\left(\frac{\partial H}{\partial P}\right)}_{T}=-{c}_{P}\eta $$

We have also seen that we can express enthalpy changes in terms of pressure, temperature and volume changes:

$${\left(\frac{\partial H}{\partial P}\right)}_{T}=\left[\tilde{v}-T{\left(\frac{\partial \tilde{v}}{\partial T}\right)}_{P}\right]$$

Additionally, the following identity can be derived:

$$\eta =\frac{R{T}^{2}}{P{c}_{P}}{\left(\frac{\partial Z}{\partial T}\right)}_{P}$$

All together, we have several ways of calculating the Joule–Thompson coefficient for a fluid, as shown next:

$$\eta ={\left(\frac{\partial T}{\partial P}\right)}_{H}=\frac{1}{{c}_{P}}\left[T{\left(\frac{\partial \tilde{v}}{\partial T}\right)}_{P}-\tilde{v}\right]=-\frac{1}{{c}_{P}}{\left(\frac{\partial H}{\partial P}\right)}_{T}=\frac{R{T}^{2}}{P{c}_{P}}{\left(\frac{\partial Z}{\partial T}\right)}_{P}$$

Once the constant pressure specific heat “c_{p}” is calculated as discussed in the previous lecture, all the entries in the previous expression are known and the Joule–Thomson coefficient can be analytically calculated. An interesting observation from all above expressions for “$\eta $ ” is that the Joule–Thompson coefficient of an ideal gas is identically equal to zero. However, real fluids take positive or negative Joule–Thompson values.

What other properties are we interested in? We are interested in flow properties. Whether you are interested in flow in pipes or in porous media, one of the most important transport properties is *viscosity*. Fluid viscosity is a measure of its internal resistance to flow. The most commonly used unit of viscosity is the centi-poise, which is related to other units as follows:

1 c_{p} = 0.01 poise = 0.000672 lbm/ft-s = 0.001 Pa-s

**Natural gas viscosity** is usually expected to increase both with pressure and temperature. A number of methods have been developed to calculate gas viscosity. The method of Lee, Gonzalez and Eakin is a simple relation which gives quite accurate results for typical natural gas mixtures with low non-hydrocarbon content. Lee, Gonzalez and Eakin (1966) presented the following correlation for the calculation of the viscosity of a natural gas:

$${\mu}_{g}=1\cdot {10}^{-4}{k}_{v}EXP\left({x}_{v}{\left(\frac{{\rho}_{g}}{62.4}\right)}^{{y}_{v}}\right)$$

where:

$${k}_{v}=\frac{\left(9.4+0.02M{W}_{g}\right){T}^{1.5}}{209+19M{W}_{g}+T}$$

$${y}_{v}=2.4-0.2{x}_{v}$$

$${x}_{v}$$

In this expression, temperature is given in (°R), the density of the fluid (${\rho}_{g}$
) in lbm/ft^{3} (calculated at the pressure and temperature of the system), and the resulting viscosity is expressed in centipoises (c_{p}).

The most commonly used **oil viscosity** correlations are those of Beggs-Robinson and Vasquez-Beggs. Corrections must be applied for under-saturated systems and for systems where dissolved gas is present in the oil. However, in compositional simulation, where both gas and condensate compositions are known at every point of the reservoir, it is customary to calculate ** condensate viscosity** using Lohrenz, Bray & Clark correlation. It this type of simulation, it is usual to calculate

Lohrenz, Bray and Clark (1964) proposed an empirical correlation for the prediction of the viscosity of a liquid hydrocarbon mixture from its composition. Such expression, originally proposed by Jossi, Stiel and Thodos (1962) for the prediction of the viscosity of dense-gas mixtures, is given below:

$$\mu ={\mu}^{\xb7}+{\xi}_{m}^{-1}{\left(0.1023+0.023364{\rho}_{r}+0.058533{\rho}_{r}^{2}-0.040758{\rho}_{r}^{3}+0.0093724{\rho}_{r}^{4}\right)}^{4}-1\cdot {10}^{-4}$$

where:

$\mu $
= fluid viscosity (c_{p}),

${\mu}^{\ast}$
= viscosity at atmospheric pressure (c_{p}),

${\xi}_{m}$
= mixture viscosity parameter (cp^{-1}),

${\rho}_{r}$
= reduced liquid density (unitless),

Lohrenz *et al.* original paper presents a typographical error in Equation (19.26). Here it is written as originally proposed by Jossi, Stiel and Thodos (1962). All four parameters listed above have to be calculated as a function of critical properties in order to apply Equation (19.26). Lohrenz *et al.* original paper uses scientific units, here we present the equivalent equations in field (English) units.

For the *viscosity of the mixture at atmospheric pressure* (${\mu}^{\ast}$
), Lohrenz *et al.* suggested using the following Herning & Zipperer equation:

$${\mu}^{\xb7}=\frac{{\displaystyle \sum _{i}{z}_{i}{\mu}_{i}^{\xb7}\sqrt{M{W}_{i}}}}{{\displaystyle \sum _{i}{z}_{i}\sqrt{M{W}_{i}}}}$$

where:

z_{j} = mole composition of the i-th component in the mixture,

MW_{i} = molecular weight of the i-th component (lbm/lbmol)

${\mu}_{i}^{\xb7}$
= viscosity of the i-th component at low pressure (c_{p}):

${\mu}_{i}^{\xb7}=\frac{34\cdot {10}^{-5}{T}_{ri}^{0.94}}{{\xi}_{i}}$
[ if T_{ri} ≤ 1.5 ]

${\mu}_{i}^{\xb7}=\frac{17.78\cdot {10}^{-5}{\left(4.5{T}_{ri}-1.67\right)}^{0.625}}{{\xi}_{i}}$
[ if T_{ri} > 1.5

where:

T_{ri} = reduced temperature for the i-th component (T/T_{ci}),

MW_{i} = viscosity parameter of the i-th component, given by: $${\xi}_{i}=\frac{5.4402{T}_{ci}^{1/6}}{\sqrt{M{W}_{i}}{P}_{ci}^{2/3}}$$

For the *mixture viscosity parameter *($\xi m$
), Lohrenz *et al.* applied an equivalent expression to that shown above but using pseudo-properties for the mixture:

$$\xi m=\frac{5.4402{T}_{pc}^{1/6}}{\sqrt{M{W}_{l}}{P}_{pc}^{2/3}}$$

where:

T_{pc} = pseudocritical temperature (^{o}R),

P_{pc} = pseudocritical pressure (psia),

MW_{l} = liquid mixture molecular weight (lbm/lbmol).

The *reduced density of the liquid mixture* (${\rho}_{r}$
) is calculated as:

$${\rho}_{r}=\frac{{\rho}_{l}}{{\rho}_{pc}}=\left(\frac{{\rho}_{l}}{M{W}_{l}}\right){V}_{pc}$$

${\rho}_{pc}$ = mixture pseudocritical density (lbm/ft

V

All mixture pseudocritical properties are calculated using Kay’s mixing rule, as shown:

$${T}_{pc}={\displaystyle \sum {z}_{i}{T}_{ci}}$$

$${P}_{pc}={\displaystyle \sum {z}_{i}{P}_{ci}}$$

$${V}_{pc}={\displaystyle \sum {z}_{i}{V}_{ci}}$$

“z_{i}” pertains to the fluid molar composition, T_{ci} is given in ^{o}R, P_{ci} in psia, and V_{ci} in ft^{3}/lbmol. When the critical volumes are known in a mass basis (ft^{3}/lbm), each of them is to be multiplied by the corresponding molecular weight. In the case of lumped C_{7+} heavy fractions, Lorentz *et al.* (1969) presented a correlation for the estimation C_{7+} critical volumes.

Lee, A., Gonzalez, M., Eakin, B. (1966), “The Viscosity of Natural Gases”, SPE Paper 1340, Journal of Petroleum Technology, vol. 18, p. 997-1000.

Lohrenz, J., Bray, B.G., Clark, C.R. (1964), “Calculating Viscosities of Reservoir Fluids from their compositions”, SPE Paper 915, Journal of Petroleum Technology, p. 1171-1176.

Answer the following problem, and submit your answer to the drop box in Canvas that has been created for this module.

*Please note:*

- Your answers must be submitted in the form of a Microsoft Word document.
- Include your Penn State Access Account user ID in the name of your file (for example, "module2_abc123.doc").
- The due date for this assignment will be sent to the class by e-mail in Canvas.
- Your grade for the assignment will appear in the drop box approximately one week after the due date.
- You can access the drop box for this module in Canvas by clicking on the Lessons tab, and then locating the drop box on the list that appears.

- Gas metering is a most important activity in the natural gas business. Among the properties we have studied, which one would you emphasize in terms of accuracy for most common gas meters?