EOS

1 EQUATIONS OF STATE

When the effects of complex phase behavior cannot be accurately calculated using simple

approaches, it is often desirable to utilize some sort of equation of state (EOS). An EOS

approach is often recommended when dealing with volatile oils and retrograde condensate gases.

The two most common equations of state utilized in petroleum engineering applications are the

Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) equations, which historically were

derived from van der Waals’ (vdW) equation. These three equations are called “cubic” because

they result in a cubic representation for the molar volume. The basic equations are the following:

Ideal Gas

van der Waals

Soave-Redlich-Kwong

Peng-Robinson

The parameters , , and are empirically determined from experimental data

(for pure components, critical temperature and pressure, and a specified point on the vapor

pressure curve), being a function of temperature and having a value of one at the critical

temperature. Note that the parameters do not have the same value in each equation.

Equations of State (MAM 04.Feb.2000) 1

We will focus on the two equations of most interest, the SRK and PR EOSs, first writing

the equations in the following common form.

Using the following substitutions

(definition of gas deviation factor)

(dimensionless)

(dimensionless)

Equation can be algebraically rewritten in the following form:

where

SRK PR

1

0

1 2

0 -1

1

Note that even though Eq. is written in terms of , Eq. can always be used to determine

the molar volume if desired.


The values of and are determined by noting that at the critical point,

Since , evaluating these expressions at the critical point results in the following

expressions

With the following values determined from the various EOSs.

SRK PR

0.427480 0.457236

0.086640 0.077796

0.333333 0.307401

Note that the value of the z-factor at the critical point is also given. The numbers above

can be expressed analytically, as given at the end of this document.

For both the SRK and PR equations, the temperature-dependent parameter is expressed

as

where,


Values of the coefficients in Eq. can be obtained from the following table.

SRK PR

0.480 0.37464

1.574 1.54226

0.176 0.26992

The parameter is called the Pitzer accentric factor and is empirically determined from

the actual vapor pressure curve for a substance according to the following relationship:

where,

and is vapor pressure.

The values of are typically determined from published tables, along with critical

temperature and pressure.

It should be noted that these cubic forms of the equation of state can apply to liquids as

well as gases. Even though has traditionally been used for gases, there is nothing restricting

this to be the case, since all of the quantities in Eq. are defined for liquids as well as gases. Of

course the values of for liquids will be smaller than those for gases since liquids are denser.

To address the more complex question of “phase”, the EOSs must be extended to include

the concept of chemical potential, sometimes called the Gibbs molar free energy ( ), changes

in which are defined by the following relationship.

For an ideal gas, it can be shown that


For non-ideal fluids, a property called “fugacity” ( )is defined having the following two

properties:

Note that the fugacity can be thought of as the potential for transfer between phases. This

means that molecules will tend to move phase-wise, so as to minimize the fugacity.

Further defining a fugacity coefficient, as

results in the following expression

The terms in Eq. can, of course, be evaluated using an EOS, resulting in the following:

2 SOLUTIONS

In general an EOS can be solved for either one of the three parameters (or ), , or

. First we’ll deal with what’s called a “flash” calculation, solving for when pressure and

temperature are known.

Equation is a cubic equation. In general the following possibilities are possible for the

roots (solutions) of this equation:


1. Three real roots, some or all possibly repeated.

2. One real root and two imaginary roots.

For pure components, away from the vapor pressure line, there is generally one real root

and two imaginary roots. The value of will be relatively large for gases and relatively small

for liquids. Near the vapor pressure line, however, there will be three real roots. It is in this

region that fugacities are needed to determine phase.

When there are three real roots, the largest root represents the z-factor of the gaseous

phase, while the smallest root represents the z-factor of the liquid phase. The middle root is non-

physical. A determination of which phase is present can be made by comparing the fugacities for

both phases. The phase with the lowest fugacity is the one present.

On the vapor pressure line, of course, there are two phases simultaneously present. When

two phases are present concurrently, this means that their fugacities must be equal, since

molecules can be in either phase. In fact, the way vapor pressure is usually determined at a given

pressure is to iterate on the above equations, seeking the pressure that results in equal liquid and

gas fugacities.

For pure components, then, the procedure for determining the z-factor (and thus the

specific molar volume and/or density) is the following:

1. For a given pressure and temperature, determine the coefficients in Eq. .

2. Solve this equation for the largest and smallest real roots using a cubic root solver.

3. Calculate both liquid and gaseous fugacity coefficients (if there are two) using Eq. . Use the z-factor from the lowest fugacity.

4. Calculate densities, , and/or molar volumes, , as desired.

If it is desired to find a vapor pressure at a given temperature, iterate on the above

procedure, until a pressure is selected so that the fugacities calculated in Step 3 are equal. The

Excel Solver works quite well for this.


3 MIXTURES

For mixtures an additional consideration must be taken into account, composition. With

so-called flash calculations, the composition of the total system (mixture) is usually taken as a

“known”. The mole fractions of each component are given the symbols (not to be confused

with the z-factor), where is the component index, spanning the total number of components in

the system. Likewise the mole fractions of the liquid phase are given the symbols and the

mole fractions of the gaseous phase . By definition, .

Often compositions of the liquid and gaseous phases are expressed in terms of

equilibrium rations, , defined by

Note that will be small (but never zero) for components that prefer to be in the liquid

phase, and much greater than one (but never infinite) for components that prefer to be in the

gaseous phase. The values of can be determined from correlations, but the more modern

approach is to determine them through EOSs.

The final composition variable needed is the total mole fraction of the mixture that is in

the gaseous phase, . This variable is, of course, not defined outside the two-phase envelope,

and has a value ranging from zero (at a bubble point) to one (at a dew point) inside the envelope.

The , , and then fully define the composition of a two-phase mixture. Liquid

and gas compositions can be determined by molar balances using the following relationships.


Note that a bubble points, , while at dew points, .

Dealing with mixtures also requires that chemical potentials and fugacities must be

calculated for each component, defined by the following relationships.

Equation states that fugacities must approach partial pressures (ideal behavior) as

pressure approaches zero.

At equilibrium, each component’s fugacity must be the same in both phases, i.e.,

for all components. We can also define the fugacity coefficient for each component as

With these definitions, it then turns out that

To apply the EOSs for mixtures, effective mixture coefficients for Eq. must be

determined using some sort of “mixing rules”. For both the PR and SRK EOSs, the following

mixing rules are used to obtain the effective mixture values of and .

Note that for a single component, these two equations will yield one components values

of and . Also, evn though these two equations are based on gas mixtures, they can also be


applied to liquid mixtures by substituting the for the . Equation is a double summation,

with both indexes i and j going over the entire number of components in the system.

The are called “binary interaction coefficients” and are empirical measures of the

attractive and repulsive forces between molecules of unlike size. Note that and

. There are many ways that binary interaction coefficients are characterized (Ahmed,

1989), but in general they increase as the relative difference between molecular weights increase.

When no data is available, values of zero are sometimes use. Often binary interaction

coefficients are used to “history match” EOS calculations against actual PVT experiments.

EOS calculations of the fugacity coefficients for each phase are then made using the

following equation.

where,

Note that Eq. reverts to Eq. where there is only one component present.

For two-phase mixtures, Eq. must be solved twice, once for the liquid phase and once

for the gas phase. When two phases exist, both EOS calculations will yield three real roots. The

smallest root of the liquid equation should be taken as the z-factor for the liquid phase, and the

largest root of the gas equation should be taken as the gas z-factor.

The procedure for doing a flash calculation on a mixture the is the following.


1. Calculate the and for each component. These are functions of pressure and temperature, but not composition.

2. Guess values for the . Various correlations are available for estimating these values (e.g., McCain, 1990).

3. Find a value of which maintains molar balances, i.e., that ensures

(more on this later).

4. Use the in the mixing equations to find the coefficients and solve the EOS for the

liquid. Select the smallest root. Likewise use the to solve the EOS for the gas phase. Select the largest root.

5. Calculate fugacity coefficients for each component in each phase using Eq. . Use these to determine the from Eq. .

6. Repeat from Step 2, using the calculated as the new trial values. Stop when the calculated values are near the trial values. McCain (1990) suggests the following possible error calculation to determine convergence.

Iterations should be stopped when the Eq. results in a value less than some specified tolerance.

Step 3 is usually done with a Newton-Raphson type iteration to find the value of . The

following procedure is typically used.

When the and are known (as in Step 3 above), the correct value of is the one

that ensures molar balances. Although either the or equation may be used for this purpose,

here we will focus on the equation, Eq. . Using Newton-Raphson iteration to solve this

equation results in the following.


Recall that Newton-Raphson iterative involves successive guessing, with the “new” guess

calculated from the “old” one by

One of the roots of Eq. is always , so the initial guess should start well away from

this value. Outside the two-phase envelope, the value of is undefined and may take on non-

physical values. The following relationships define how to determine whether the calculation is

being done inside the two-phase envelope or not.

Liquid phase

Gaseous phase

Bubble point

Dew point

Two-phase and

The above procedure can also be used to calculate bubble and dew point pressures, by

simply iterating on pressure until the appropriate Eq. or is true. Again, the Excel Solver is a

good way to do this type of calculation.


4 EXACT VALUES OF EOS PARAMETERS

van der Waals

Soave-Redlich-Kwong

Peng-Robinson


5 REFERENCES

Ahmed, T.: Hydrocarbon Phase Behavior, Gulf Publishing Co., Houston (1989).

McCain, W.D., Jr.: The Properties of Petroleum Fluids, PennWell Publishing Co., Tulsa (1990).


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