## Parameter Assignment For Equations Of State

In practice, vapor/liquid reservoir phase behavior is calculated by an equation of state (EOS). The two most common EOSs that have been used for oil-recovery solvent-injection processes are the Peng-Robinson EOS^{[1]} and the Soave-Redlick-Kwong EOS.^{[2]}

Of the two, the Peng-Robinson EOS seems to be the one most often cited in the literature and is the one discussed in some detail. The Soave-Redlick-Kwong EOS is used in a similar manner to predict solvent/oil phase behavior.

## Calculating phase behavior with equations of state (EOS)

Peng and Robinson originally proposed the two-parameter EOS shown next for a pure component:

....................(1)

....................(2)

....................(3)

and ....................(4)

where

*ω*= the component acentric factor*T*_{c}= component critical temperature- and
*p*_{c}= component critical pressure.

For heavier components, where *ω* > 0.49, the following equation is recommended:

....................(5)

The constants in **Eqs. 2** and **4** are often designated Ω_{a} and Ω_{b}.

**Eq. 1** represents continuous fluid behavior from the solvent to liquid state, and it can be rewritten as

....................(6)

where ....................(7)

and ....................(8)

Jhaveri and Youngren^{[3]} adapted a procedure used by Peneloux *et al.*^{[4]} and modified the original **Eq. 1** to include a third parameter to allow more-accurate volumetric predictions, which is recommended for solvent/oil simulations. The third parameter does not change the vapor/liquid equilibrium conditions determined by the unmodified, two-parameter equation. Instead, it modifies the phase volumes by making a translation along the volume axis. **Eqs. 9** and **10** give the modified three-parameter equation:

....................(9)

....................(10)

where *s* is the volumetric shift parameter.

For mixtures:

....................(11)

....................(12)

....................(13)

and ....................(14)

In **Eq. 12**, *∂*_{ij} is the binary interaction coefficient that characterizes the binary formed by components *i* and *j*. **Eqs. 10** through **13** apply both to pure components and to lumped pseudocomponents that represent two or more pure components in complex mixtures.

The following expression derived from thermodynamic relationships and the EOS allows calculation of the fugacity, *f*_{j}, of component *j* in a mixture:

....................(15)

Thus, by satisfying the equilibrium condition , vapor/liquid equilibrium ratios can be calculated, and flash calculations can be made to calculate the compositions of vapor and liquid in equilibrium, molar splits, and volumes.

Solution of the EOS does not calculate phase viscosities directly. This is done from some external calculation once the phase compositions and densities are known. A commonly used calculation for liquid-mixture viscosity is the Lohrenz-Bray-Clark method, which requires the critical volumes of each component or pseudocomponent in the mixture.^{[5]} Refer to Oil viscosity and Gas viscosity for more information on calculating viscosities.

## Characterizing the fluid system

To use **Eqs. 1** through **15** for calculating the phase behavior and properties of solvent compositional processes in oil recovery, the following steps must be taken to "characterize" the fluid system in question:

- Analyze the oil composition. This can be done by distillation or chromatographic methods. An extended analysis through at least C
_{25+}is preferred. The advantage of distillation is that molecular weight, boiling point, and density can be measured on the distillation cuts. - Represent the multicomponent reservoir fluid by an appropriate division into pure components and pseudocomponents. Pure components through C 5 plus three to five pseudocomponents usually will suffice. It may be possible to reduce the number of pure components and pseudocomponents further by combining similar components.
- Make an initial assignment of critical pressure and temperature, acentric factor, critical volume (or critical compressibility), volumetric shift parameter, and interaction parameters for each component and pseudocomponent.
- Tune the above properties for the pseudocomponents by comparing predicted phase behavior and properties with suitable experimental data.
- Methods for dividing into pseudocomponents and estimating critical properties, shift parameters, and binary interaction coefficients are described in detail in Whitson and Brule.
^{[6]}

Because of the approximations inherent in an EOS as well as the approximations required to represent a multicomponent reservoir fluid in a tractable form, it should be expected that phase-behavior properties and equilibrium compositions predicted with an EOS will depart from measured values over the range of composition and pressure conditions anticipated in a reservoir simulation. For this reason, additional adjustment of EOS parameters will be required for predictions to represent experimental measurements adequately. These adjustments usually are made by regression.

Reservoir oils usually are subjected to routine pressure/volume/temperature (PVT) experiments that give the volumetric and phase-behavior information necessary for predicting conventional recovery methods such as solution solvent drive or waterflooding. Experiments such as constant-composition expansion, differential liberation, constant-volume depletion, and separator tests provide black-oil properties. Other PVT experiments are more specific for solvent injection. These include swelling tests and multiple-contact experiments.

The swelling experiment is sometimes called a pressure-composition diagram determination. Injection solvent is added to reservoir oil in increments to give mixtures that contain increasing amounts of injection solvent. After each addition of solvent, the saturation pressure is measured at reservoir temperature. Overall composition of these mixtures ranges from that of black oil to compositions up to and beyond near-critical conditions (i.e., overall compositions that traverse a range from bubblepoint to dewpoint mixtures at reservoir temperature). Thus, the swelling experiment provides some PVT and phase-equilibrium information on mixture ranges that might reflect compositions as solvent displaces oil through the reservoir. It provides information on the saturation pressure of injection-solvent/oil mixtures, the swelling or increase in oil formation volume factor as solvent is added, the composition of the critical mixture, and the liquid saturation vs. pressure in the two-phase region of the diagram.

Multiple-contact tests seek to simulate the solvent/oil multiple contacting that occurs in a reservoir. A forward multicontact experiment tries to simulate multicontacting in a vaporizing-solvent drive. A reverse multicontact experiment tries to simulate the multicontacting that occurs in a purely condensing-solvent drive. The experiments give information concerning equilibrium-phase volumes and compositions.

In a reverse-contact experiment, the PVT cell is charged with the reservoir fluid at the desired pressure and temperature, and an increment of injection solvent is added sufficient to form a two-phase mixture (or a three-phase mixture in some tests). The phases are allowed to equilibrate, and phase volumes are measured. The solvent phase is then displaced from the cell, and oil and solvent compositions are measured. The procedure is repeated, with injection of a new increment of injection solvent introduced into the cell to contact equilibrium oil left after the first contact.

In the forward-contact experiment, the oil phase is displaced after the first contact, and the remaining equilibrium-solvent phase in the cell is contacted with a fresh increment of reservoir oil.

The objective of tuning is to ensure that the EOS predicts fluid properties and phase equilibrium compositions accurately over the range of pressure, temperature (if this varies), and composition that one expects to encounter in a simulation. If the simulation is for a solvent compositional process, then at a minimum the EOS should predict properties and phase equilibrium for the range of injection-solvent/oil mixtures and pressures encountered in the simulation study. It also should predict adequately for any black-oil conditions expected in the simulation (e.g., waterflooding or pressure depletion before solvent injection) and for the separator conditions expected.

Pedersen *et al.*^{[7]} observed that an EOS tuned to match a specific set of data may not give reliable predictions for other data not included in the tuning process. However, when both sets of data are included in the tuning process, the prediction for either one may not be quite as good as for tuning against these data individually.

It seems prudent that at a minimum, there should be differential depletion data, separator tests, and swell data to tune an EOS against for making solvent-compositional simulations. Swelling tests are necessary when near-critical compositions are expected in the simulation, and it is necessary for the swell tests to explore this composition region. Swell tests with several different injected-solvent compositions might be warranted if optimization of the solvent composition is an objective of the simulation study.

The value added by multiple-contact tests is unclear. These are the most difficult and expensive of the experiments discussed earlier, yet they provide direct measurements of vapor/liquid equilibrium compositions and molar splits for a composition path that at least crudely mimics the development of compositions at the leading or trailing edges of the solvent/oil transition zone, which is, of course, what the simulator is trying to calculate. However, for the condensing/vaporizing process, multiple-contact experiments do not give compositions that are very near the critical point.

Although they are difficult and expensive to run, slimtube tests give a direct verification of the ability of the EOS to predict minimum miscibility pressure (MMP) or minimum miscibility enrichment (MME). If the EOS after regression of parameters does not predict slimtube MMP or MME, further adjustment of parameters is required.

## Nomenclature

a | = | constant |

b | = | constant |

i | = | component i |

j | = | component j |

k | = | permeability, md |

ω | = | component acentric factor, dimensionless |

Z | = | fluid compressibility factor, dimensionless |

n | = | number of components |

N_{ca} | = | capillary number, dimensionless |

Δp_{g} | = | pressure gradient through the displacing phase, psi |

R | = | universal gas constant, units consistent with other equation parameters |

s | = | volumetric shift parameter, dimensionless |

T | = | temperature, ° R |

T_{c} | = | critical temperature, ° R |

T_{r} | = | reduced temperature, T/T_{c} |

v | = | volume, cubic ft |

v_{xs} | = | fluid velocity along a streamline, ft/sec |

μ_{oi} | = | oil viscosity, cp |

## References

- ↑Peng, D.Y. and Robinson, D.B. 1976. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 15 (1): 59-64. http://dx.doi.org/10.1021/i160057a011
- ↑Soave, G. 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 27 (6): 1197-1203. http://dx.doi.org/http://dx.doi.org/10.1016/0009-2509(72)80096-4
- ↑Jhaveri, B.S. and Youngren, G.K. 1984. Three-Parameter Modification of the Peng-Robinson Equation of State to Improve Volumetric Predictions. SPE Res Eval & Eng 3 (3): 1033–1040. SPE-13118-PA. http://dx.doi.org/10.2118/13118-PA
- ↑Péneloux, A., Rauzy, E., and Fréze, R. 1982. A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilib. 8 (1): 7-23. http://dx.doi.org/http://dx.doi.org/10.1016/0378-3812(82)80002-2
- ↑Lohrenz, J., Bray, B.G., and Clark, C.R. 1964. Calculating Viscosities of Reservoir Fluids From Their Compositions. J Pet Technol 16 (10): 1171–1176. SPE-915-PA. http://dx.doi.org/10.2118/915-PA
- ↑Whitson, C.H. and Brule, M.R. 2000. Phase Behavior, Vol. 20. Richardson, Texas: Monograph Series, SPE.
- ↑Pedersen, K.S., Thomassen, P., and Fredenslund, A. 1984. Thermodynamics of petroleum mixtures containing heavy hydrocarbons. 1. Phase envelope calculations by use of the Soave-Redlich-Kwong equation of state. Ind. Eng. Chem. Process Des. Dev. 23 (1): 163-170. http://dx.doi.org/10.1021/i200024a027

## Noteworthy papers in OnePetro

Mohebbinia, S., Sepehrnoori, K., & Johns, R. T. (2013, July 29). Four-Phase Equilibrium Calculations of Carbon Dioxide/Hydrocarbon/Water Systems With a Reduced Method. Society of Petroleum Engineers. doi:10.2118/154218-PA

Larson, L. L., Silva, M. K., Taylor, M. A., & Orr, F. M. (1989, February 1). Temperature Dependence of L1/L2/V Behavior in CO2/Hydrocarbon Systems. Society of Petroleum Engineers. doi:10.2118/15399-PA

Okuno, R., Johns, R., & Sepehrnoori, K. (2010, June 1). A New Algorithm for Rachford-Rice for Multiphase Compositional Simulation. Society of Petroleum Engineers. doi:10.2118/117752-PA

Gorucu, S. E., & Johns, R. T. (2013, February 18). Comparison of Reduced and Conventional Phase Equilibrium Calculations. Society of Petroleum Engineers. doi:10.2118/163577-MS

Stalkup, F. I., & Yuan, H. (2005, January 1). Effect of EOS Characterization on Predicted Miscibility Pressure. Society of Petroleum Engineers. doi:10.2118/95332-MS

## External links

Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro

## See also

Miscible flooding

Phase diagrams of miscible processes

Compositional simulation of miscible processes

Equations of state

PEH:Miscible_Processes

For the use of this in cosmology, see Equation of state (cosmology). For the use of this concept in optimal control theory, see Optimal control § General method.

In physics and thermodynamics, an **equation of state** is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (**PVT**), or internal energy.^{[1]} Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.

## Overview[edit]

The most prominent use of an equation of state is to correlate densities of gases and liquids to temperatures and pressures. One of the simplest equations of state for this purpose is the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. However, this equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid. Therefore, a number of more accurate equations of state have been developed for gases and liquids. At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions.

Measurements of equation-of-state parameters, especially at high pressures, can be made using lasers.^{[2]}^{[3]}^{[4]}

In addition, there are also equations of state describing solids, including the transition of solids from one crystalline state to another. There are equations that model the interior of stars, including neutron stars, dense matter (quark–gluon plasmas) and radiation fields. A related concept is the perfect fluidequation of state used in cosmology.

In practical context, the equations of state are instrumental for PVT calculation in process engineering problems and especially in petroleum gas/liquid equilibrium calculations. A successful PVT model based on a fitting equation of state can be helpful to determine the state of the flow regime, the parameters for handling the reservoir fluids, piping and sizing.

## Historical[edit]

### Boyle's law (1662)[edit]

Boyle's Law was perhaps the first expression of an equation of state.^{[citation needed]} In 1662, the Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:

The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.

### Charles's law or Law of Charles and Gay-Lussac (1787)[edit]

In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:

### Dalton's law of partial pressures (1801)[edit]

Dalton's Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.

Mathematically, this can be represented for *n* species as:

### The ideal gas law (1834)[edit]

In 1834 Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the *ideal gas law*. Initially, the law was formulated as *pV _{m}* =

*R*(

*T*+ 267) (with temperature expressed in degrees Celsius), where

_{C}*R*is the gas constant. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K, giving:

### Van der Waals equation of state (1873)[edit]

In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules.^{[5]} His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich-Kwong.

## Major equations of state[edit]

For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form:

In the following equations the variables are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.

- = pressure (absolute)
- = volume
- = number of moles of a substance
- = =
**molar volume**, the volume of 1 mole of gas or liquid - = absolute temperature
- = ideal gas constant (8.3144621 J/(mol·K))
- = pressure at the critical point
- = molar volume at the critical point
- = absolute temperature at the critical point

### Classical ideal gas law[edit]

The classical ideal gas law may be written:

In the form shown above, the equation of state is thus

.

If the calorically perfect gas approximation is used, then the ideal gas law may also be expressed as follows

where is the density, is the adiabatic index (ratio of specific heats), is the internal energy per unit mass (the "specific internal energy"), is the specific heat at constant volume, and is the specific heat at constant pressure.

## Cubic equations of state[edit]

Cubic equations of state are called such because they can be rewritten as a cubic function of V_{m}.

### Van der Waals equation of state[edit]

The Van der Waals equation of state may be written:

where is molar volume. The substance-specific constants and can be calculated from the critical properties and (noting that is the molar volume at the critical point) as:

Also written as

Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation is called the attraction parameter and the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.

The van der Waals equation may be considered as the ideal gas law, "improved" due to two independent reasons:

- Molecules are thought as particles with volume, not material points. Thus cannot be too little, less than some constant. So we get () instead of .
- While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write ( something) instead of . To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~, the force acting on the whole element is ~~

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