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# Equations of state for miscible processes

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
• Tc = component critical temperature
• and pc = 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 Ωa and Ωb.

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, fj, 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:

1. Analyze the oil composition. This can be done by distillation or chromatographic methods. An extended analysis through at least C25+ is preferred. The advantage of distillation is that molecular weight, boiling point, and density can be measured on the distillation cuts.
2. 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.
3. 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.
4. Tune the above properties for the pseudocomponents by comparing predicted phase behavior and properties with suitable experimental data.
5. 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 Nca = capillary number, dimensionless Δpg = 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 Tc = critical temperature, ° R Tr = reduced temperature, T/Tc v = volume, cubic ft vxs = fluid velocity along a streamline, ft/sec μoi = oil viscosity, cp

## References

1. 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
2. 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
3. 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
4. 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
5. 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
6. Whitson, C.H. and Brule, M.R. 2000. Phase Behavior, Vol. 20. Richardson, Texas: Monograph Series, SPE.
7. 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