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.

Fluid properties are used to characterize the condition of a fluid at a given state. A reliable estimation and description of the properties of hydrocarbon mixtures is fundamental in petroleum and natural engineering analysis and design. Fluid properties are not independent, just as pressure, temperature, and volume are not independent of each other. Equations of State provide the means for the estimation of the P-V-T relationship, and from them many other thermodynamic properties can be derived. Compositions are usually required for the calculation of the properties of each phase. For a VLE system, using the tools we have discussed in the previous lectures, we are ready to predict some important properties of both the liquid (condensate) and vapor (natural gas) phases — by means of the known values of composition of both phases. Some of the most relevant are discussed next.

The molecular weight (MW) of each of the phases in a VLE system is calculated as a function of the molecular weight of the individual components (MW_{i}), provided that both the composition of the gas (y_{i}) and the liquid (x_{i}) are known:

$$M{W}_{g}={\displaystyle \sum _{i=1}^{n}{y}_{i}M{W}_{i}}$$

$$M{W}_{l}={\displaystyle \sum _{i=1}^{n}{x}_{i}M{W}_{i}}$$

Density is customarily defined as the amount of mass contained in a unit volume of fluid. Density is the single-most important property of a fluid, once we realize that most other properties can be obtained or related to density. Both specific volume and density — which are inversely proportionally related to each other — tell us the story of how far apart the molecules in a fluid are from each other. For liquids, density is high — which translates to a very high molecular concentration and short intermolecular distances. For gases, density is low — which translates to low molecular concentrations large intermolecular distances.

The question then is: Given this, how can we obtain this all-important property called density? This takes us back to Equations of State (EOS). Since very early times, there have been correlations for the estimation of density of the liquids (oil, condensates) and gases/vapors (dry gases, wet gases). In modern times, equations of state (EOS) are a natural way of obtaining densities. The density of the fluid ‘f’ is calculated using its compressibility factor (Z_{f}) as predicted by an appropriate equation of state. From the real gas law, the density can be expressed as:

$${\rho}_{f}=\frac{P}{RT}\left(\frac{M{W}_{f}}{{Z}_{f}}\right)$$

where: MW_{f} is the molecular weight of fluid ‘f’. Expression (18.2) is used for both the gas and liquid density. In either case, the proper value for MW_{f} (either MW_{g} or MW_{l}) and Z_{f} (either Z_{g} or Z_{o}) has to be used. This takes us back to the discussion of equations of state. From Equation (18.2) it is clear that all that we need is the Z-factor.

The all-important parameter to calculate density is the Z-factor, both for the liquid and vapor phases. The relation between liquid behavior and Z-factor is not obvious, because Z-factor has been traditionally defined for gases. However, we can get “Z” for liquids. “Z” is, indeed, a measure of departure from the ideal gas behavior. Fair enough, for defining “Z” for liquids, we still measure the departure of liquid behavior from ideal gas behavior. A “liquid state” is a tremendous departure from ideal-gas conditions, and as such, “Z” for a liquid is always very far from unity. Typical values of “Z” for liquids are small.

Equations of State have proven very reliable for the estimation of vapor densities, but they do not do as good a job for liquid densities. There is actually a debate among different authors about the reliability of Z-factor estimations for liquids using EOS. In fact, people still believe the EOS are not reliable for liquid density predictions and that we should use correlations instead. However, Peng-Robinson EOS provides fair estimates for vapor and liquid densities as long as we are dealing with natural gas and condensate systems.

Empirical correlations for Z-factor for natural gases were developed before the advent of digital computers. Although their use is in decline, they can still be used for fast estimates of the Z-factor. The most popular of such correlations include those of Hall-Yarboroug and Dranchuk-Abou-Kassem.

Chart look-up is another means of determining Z-factor of natural gas mixtures. These methods are invariably based on some type of corresponding states development. According to the theory of corresponding states, substances at corresponding states will exhibit the same behavior (and hence the same Z-factor). The chart of Standing and Katz is the most commonly used Z-factor chart for natural gas mixtures.

Methods of direct calculation using corresponding states have also been developed, ranging from correlations of chart values to sophisticated equation sets based on theoretical developments.

However, the use of equations of state to determine Z-factors has grown in popularity as computing capabilities have improved. Equations of state represent the most complex method of calculating Z-factor, but also the most accurate. A variety of equations of state have been developed to describe gas mixtures, ranging from the ideal EOS (which yields only one root for the vapor and poor predictions at high pressures and low temperatures), cubic EOS (which yields up to three roots, including one for the liquid phase), and more advanced EOS such as BWR and AGA8.

Hall K., and Yarborough, L. (1973), “A New Equation of State for Z-factor Calculations”, Oil and Gas Journal, June 1973, pp. 82-92.

Dranchuk, P. and Abou-Kassem, J. (1975), “Calculation of Z-factors for Natural Gases Using Equations-of-State”, JCPT, July-September 1975, p. 34-36.

Standing, M. and Katz, D. (1942), “Density of Natural Gases”, Trans. AIME, v. 146, pp. 140-149.

Specific gravity is defined as the ratio of fluid density to the density of a reference substance, both defined at the same pressure and temperature. These densities are usually defined at standard conditions (14.7 psia and 60°F). For a **condensate**, oil or a liquid, the reference substance is *water*:

$${\gamma}_{o}=\frac{{({\rho}_{0})}_{sc}}{{({\rho}_{w})}_{sc}}$$

The value of water density at standard conditions is 62.4 lbm/ft^{3} approximately. For a **natural gas**, or any other gas for this matter, the reference substance is *air*:

$${\gamma}_{g}=\frac{{({\rho}_{g})}_{sc}}{{({\rho}_{air})}_{sc}}$$

Or, equivalently, substituting Equation (18.2) evaluated at standard conditions (${Z}_{sc}\approx 1$ for most gases),

$${\gamma}_{g}=\frac{M{W}_{g}}{M{W}_{air}}$$

where the value of the molecular weight for air is MW_{air} = 28.96 lbm/lbmol. Specific gravity is nondimensional because both numerator and denominator have the same units.

Petroleum and Natural Gas Engineers also use another gravity term which is called API gravity. It is used for liquids (e.g., condensates) and is defined as:

$$\xb0API=\frac{141.5}{{\gamma}_{o}|{}_{sc}}-131.5$$

By definition (see Equation 18.3), the specific gravity of water is unity. Therefore, water has an API gravity of 10. The API gravity of 10 is associated with very heavy, almost asphaltic, oils. Light crude oils have an API greater than or equal to 45°. *Condensate* gravities range between 50° and 70° API. Liquid condensates are normally light in color.

Due to the dramatically different conditions prevailing at the reservoir when compared to the conditions at the surface, we do not expect that 1 barrel of fluid at reservoir conditions could contain the same amount of matter as 1 barrel of fluid at surface conditions. Volumetric factors were introduced in petroleum and natural gas calculations in order to readily relate the *volume* of fluids that are obtained at the surface (stock tank) to the volume that the fluid actually occupied when it was compressed in the reservoir.

For example, the volume that a *live oil* occupies at the reservoir is *more* than the volume of oil that leaves the stock tank at the surface. This may be counter-intuitive. However, this is a result of the evolution of gas from oil as pressure decreases from reservoir pressure to surface pressure. If an oil had no gas in solution (i.e., a *dead oil*), the volume that it would occupy at reservoir conditions is less than the volume that it occupies at the surface. In this case, only liquid compressibility plays a role in the change of volume.

**The formation volume factor of a natural gas** (B_{g}) relates the volume of 1 lbmol of gas at reservoir conditions to the volume of the same lbmol of gas at standard conditions, as follows:

$${B}_{g}=\frac{\text{Volumeof1lbmolofgasatreservoirconditions,RCF}}{\text{Volumeof1lbmolgasatstandardconditions,SCF}}$$

Those volumes are, evidently, the specific molar volumes of the gas at the given conditions. The reciprocal of the specific molar volume is the molar density, and thus, Equation (18.5) could be written:

$${B}_{g}=\frac{{\tilde{v}}_{g}/{}_{res}}{{\tilde{v}}_{g}/{}_{sc}}=\frac{{\overline{\rho}}_{g}/{}_{sc}}{{\overline{\rho}}_{g}/{}_{res}}=\frac{({\rho}_{g}/M{W}_{g}){|}_{sc}}{({\rho}_{g}/M{W}_{g}){|}_{res}}$$

Introducing the definition for densities in terms of compressibility factor,

$${B}_{g}=\frac{\frac{{P}_{sc}}{R{T}_{sc}{Z}_{sc}}}{\frac{P}{RTZ}}$$

Therefore, recalling that ${Z}_{sc}\approx 1$ ,

$${B}_{g}=\frac{{P}_{sc}}{{T}_{sc}}\frac{ZT}{P}=0.005035{\frac{ZT}{P}}_{\left[RCF/SCF\right]}$$

Gas formation volume factors can be also expressed in terms of [RB/SCF]. In such a case, 1 RB = 5.615 RCF and we write:

$${B}_{g}=0.005035{\frac{ZT}{P}}_{\left[RCF/SCF\right]}$$

The formation volume factor of an **oil or condensate** (B_{o}) relates the volume of 1 lbmol of liquid at reservoir conditions to the volume of that liquid once it has gone through the surface separation facility.

The total volume occupied by 1 lbmol of liquid at reservoir conditions (V_{o})_{res} can be calculated through the compressibility factor of that liquid, as follows:

Upon separation, some gas is going to be taken out of the liquid stream feeding the surface facility. Let us call “n_{st}” the moles of liquid leaving the stock tank per mole of feed entering the separation facility. The volume that 1 lbmol of reservoir liquid is going to occupy after going through the separation facility is given by:

$${({V}_{o})}_{res}={\left(\frac{{n}_{st}{Z}_{o}RT}{P}\right)}_{sc}\text{wheren=1lbmol,}$$

Here we assume that the last stage of separation, the stock tank, operates at standard conditions. Introducing Equations (18.12) and (18.11) into (18.10), we end up with:

$${B}_{o}=\frac{{\left(\frac{n{Z}_{o}RT}{P}\right)}_{res}}{{\left(\frac{{n}_{st}{Z}_{0}RT}{P}\right)}_{sc}}$$

or,

$${B}_{o}=\frac{1}{{n}_{st}}\frac{{\left({Z}_{o}\right)}_{res}}{{\left({Z}_{o}\right)}_{sc}}\frac{T}{P}{\frac{{P}_{sc}}{{T}_{sc}}}_{\left[RB/STB\right]}$$

Please notice that (Z_{o})_{sc} — unlike Z_{sc} for a gas — is never equal to one. Oil formation volume factor can be also seen as the volume of reservoir fluid required to produce one barrel of oil in the stock tank.

The isothermal compressibility of a fluid is defined as follows:

$${c}_{f}=-\frac{1}{V}{\left(\frac{\partial V}{\partial \rho}\right)}_{T}$$

This expression can be also given in term of fluid density, as follows:

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

For **liquids**, the value of isothermal compressibility is very small because a unitary change in pressure causes a very small change in volume for a liquid. In fact, for slightly compressible liquid, the value of compressibility (c_{o}) is usually assumed independent of pressure. Therefore, for small ranges of pressure across which c_{o} is nearly constant, Equation (18.16) can be integrated to get:

$${c}_{o}\left(p-{p}_{b}\right)=\mathrm{ln}\left(\frac{{\rho}_{o}}{{\rho}_{ob}}\right)$$

In such a case, the following expression can be derived to relate two different liquid densities $({\rho}_{o}\text{,}\rho \text{ob)}$
at two different pressures (p, p_{b}):

$${\rho}_{o}={\rho}_{ob}\left[1+{c}_{o}(p-{p}_{b})\right]$$

The Vasquez-Beggs correlation is the most commonly used relationship for c_{o}.

For **natural gases,** isothermal compressibility varies significantly with pressure. By introducing the real gas law into Equation (18.16), it is easy to prove that, for gases:

$${c}_{g}=\frac{1}{P}-\frac{1}{Z}{\left(\frac{\partial Z}{\partial P}\right)}_{r}$$

Note that for an ideal gas, c_{g} is just the reciprocal of the pressure. “c_{g}” can be readily calculated by graphical means (chart of Z versus P) or by introducing an equation of state into Equation (18.19).

Surface tension is a measure of the surface free energy of liquids, i.e., the extent of energy stored at the surface of liquids. Although it is also known as interfacial force or interfacial tension, the name *surface tension* is usually used in systems where the liquid is in contact with gas.

Qualitatively, it is described as the force acting on the surface of a liquid that tends to minimize the area of its surface, resulting in liquids forming droplets with spherical shape, for instance. Quantitatively, since its dimension is of force over length (lbf/ft in English units), it is expressed as the force (in lbf) required to break a film of 1 ft of length. Equivalently, it may be restated as being the amount of surface energy (in lbf-ft) per square feet.

Katz *et al.* (1959) presented the Macleod-Sudgen equation for surface tension ($\sigma $) calculations in dynes/cm for hydrocarbon mixtures:

$${\sigma}^{1/4}={\displaystyle \sum _{i=1}^{n}Pc{h}_{i}}\left[\frac{{\rho}_{i}}{62.4(M{W}_{l})}{x}_{i}-\frac{{\rho}_{g}}{62.4(M{W}_{g})}{y}_{i}\right]$$

where:

Pch_{i} = Parachor of component “i”,

x_{i} = mole fraction of component “i” in the liquid phase,

y_{i} = mole fraction of component “i” in the gas phase.

In order to express surface tension in field units (lbf/ft), multiply the surface tension in (dynes/cm) by 6.852177x10^{-3}. The parachor is a temperature independent parameter that is calculated experimentally. Parachors for pure substances have been presented by Weinaug and Katz (1943) and are listed in Table 18.1.

Component | Parachor |
---|---|

CO_{2} |
78.0 |

N_{2} |
41.0 |

C_{1} |
77.0 |

C_{2} |
108.0 |

C_{3} |
150.3 |

iC_{4} |
181.5 |

nC_{4} |
189.9 |

iC_{5} |
225.0 |

nC_{5} |
231.5 |

nC_{6} |
271.0 |

nC_{7} |
312.5 |

nC_{8} |
351.5 |

Weinaug and Katz (1943) also presented the following empirical relationship for the parachor of hydrocarbons in terms of their molecular weight:

$$Pc{h}_{i}=-4.6148734+2.558855M{W}_{i}+3.404065\cdot {10}^{-4}M{W}_{i}^{2}+\frac{3.767396\cdot {10}^{3}}{M{W}_{i}}$$

- This correlation may be used for pseudo-components or for those hydrocarbons not shown in Table 18.1.

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

*Please note:*

- Your answer 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.

- Which of all the properties we have studied so far would you use to compare and distinguish among wet natural gas, dry natural gas, and gas-condensate systems? Explain why.