eos

Equations of State (MAM 11.Dec.2000) 1

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

mpv RT= (1.1)

van der Waals

( )2c

mm

ap v b RTv

⎡ ⎤+ − =⎢ ⎥

⎣ ⎦ (1.2)

Soave-Redlich-Kwong

( )( ) ( )c

mm m

a Tp v b RT

v v bα⎡ ⎤

+ − =⎢ ⎥+⎢ ⎥⎣ ⎦

(1.3)

Peng-Robinson

( )

( ) ( ) ( )cm

m m m

a Tp v b RT

v v b b v bα⎡ ⎤

+ − =⎢ ⎥+ + −⎢ ⎥⎣ ⎦

(1.4)


The parameters ca , ( )Tα , and b are empirically determined from

experimental data (for pure components, critical temperature and

pressure, and a specified point on the vapor pressure curve), ( )Tα

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.

We will focus on the two equations of most interest, the SRK and

PR EOSs, first writing the equations in the following common

form.

( )

( )( ) ( )1 2

cm

m m

a Tp v b RT

v f b v f bα⎡ ⎤

+ − =⎢ ⎥+ +⎢ ⎥⎣ ⎦

(1.5)

Using the following substitutions

mpvzRT

= (definition of gas deviation factor) (1.6)

( )2 2

ca T pA

R Tα

= (dimensionless)

bpBRT

= (dimensionless)

Equation 1.5 can be algebraically rewritten in the following form:

3 22 1 0 0z a z a z a− + − = (1.7)

where

( )0 1a B A wB B⎡ ⎤= + +⎣ ⎦

( )1a A B B u w u⎡ ⎤= − − +⎣ ⎦

( )2 1 1a B u= − −


SRK PR

1f = 1 1 2+ 2f = 0 1 2−

1 2u f f= + 1 2

1 2w f f= 0 -1 2 4u wσ = − 1 2 2

Note that even though Eq. 1.7 is written in terms of z , Eq. 1.6 can

always be used to determine the molar volume if desired.

The values of ca and b are determined by noting that at the

critical point,

0c

m T

pv∂

=∂

(1.8)

2

2 0c

m T

pv∂

=∂

(1.9)

Since ( ) 1cTα = , evaluating these expressions at the critical point

results in the following expressions

2 2

cc a

c

R Tap

= Ω (1.10)

cb

c

RTbp

= Ω (1.11)

with the following values determined from the various equations of

state.

SRK PR

aΩ = 0.427480 0.457236

bΩ = 0.086640 0.077796


cz = 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

( ) ( ) 21 1 rT m Tα ⎡ ⎤= + −⎣ ⎦ (1.12)

where,

21 2om m m mω ω= + − (1.13)

Values of the coefficients in Eq. 1.13 can be obtained from the

following table.

SRK PR

0m = 0.480 0.37464

1m = 1.574 1.54226

2m = 0.176 0.26992

The parameter ω is called the Pitzer accentric factor and is

empirically determined from the actual vapor pressure curve for a

substance according to the following relationship:

( )0.7log 1

rvr T

pω=

= − + (1.14)

where,

( )0.7

0.7r

v cvr T

c

p T Tp

p=

== (1.15)

and vp is vapor pressure.


The values of ω 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 z has

traditionally been used for gases, there is nothing restricting this to

be the case, since all of the quantities in Eq. 1.6 are defined for

liquids as well as gases. Of course the values of z for liquids will

be smaller than those for gases since liquids are denser.

To address the more complex question of “phase”, the equations of

state must be extended to include the concept of chemical

potential, sometimes called the Gibbs molar free energy (G ),

changes in which are defined by the following relationship.

mdG v dp= (1.16)

For an ideal gas, it can be shown that

( )lndG RT d p= (1.17)

For non-ideal fluids, a property called “fugacity” ( f ) is defined

having the following two properties.

( )lndG RT d f= (1.18)

0

limp

f p→

= (1.19)

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, ϕ as


0

1exppf z dp

p pϕ

⎡ ⎤−= = ⎢ ⎥

⎢ ⎥⎣ ⎦∫ (1.20)

results in the following expression

1ln 1 lnmv

mm

RTz z p dvRT v

ϕ∞

⎛ ⎞= − − + −⎜ ⎟

⎝ ⎠∫ (1.21)

The terms in Eq. 1.21 can, of course, be evaluated using an EOS,

resulting in the following.

( ) 2ln 1 ln ln

2

uz BAz z B uB z B

σ

ϕ σσ

+⎡ ⎤+⎢ ⎥= − − − − ⎢ ⎥−⎢ ⎥+

⎣ ⎦

(1.22)

1.1 Solutions

In general an EOS can be solved for either one of the three

parameters z (or mv ), p , or T . First we’ll deal with what’s called

a “flash” calculation, solving for z when pressure and temperature

are known.

Equation 1.7 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 z

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. 1.7.

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. 1.22. Use the z-factor from the lowest fugacity.

4. Calculate densities, pMzRT

ρ = , and/or molar volumes,

mzRTv

p= , 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.


1.2 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 jz (not to be confused with the z-factor), where j 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 jx and the mole fractions of the gaseous phase jy . By

definition, 1j j jj j j

x y z= = =∑ ∑ ∑ .

Often compositions of the liquid and gaseous phases are expressed

in terms of equilibrium rations, jK , defined by

jj

j

yK

x= (1.23)

Note that jK 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 jK 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, gn% . 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 jz , jK , and gn% then fully define the composition of a two-

phase mixture. Liquid and gas compositions can be determined by

molar balances using the following relationships.

( )1 1j

jg j

zx

n K=

+ −% (1.24)

( )1 1j j

j j jg j

K zy K x

n K= =

+ −% (1.25)

Note that a bubble points, j jx z= , while at dew points, j jy z= .

Dealing with mixtures also requires that chemical potentials and

fugacities must be calculated for each component, defined by the

following relationships.

( )lnj jdG RT d f= (1.26)

0

lim j jp

f y p→

= (1.27)

Equation 1.27 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., gj ljf f= for all components. We can also define

the fugacity coefficient for each component as

jj

j

fy p

ϕ = (1.28)

With these definitions, it then turns out that


ljj

gj

Kϕϕ

= (1.29)

To apply the equations of state for mixtures, effective mixture

coefficients for Eq. 1.7 must be determined using some sort of

“mixing rules”. For both the PR and SRK equations of state, the

following mixing rules are used to obtain the effective mixture

values of b and ca α .

( )1c j cj j i ci i ijj i

a y a y aα α α δ= −∑ ∑ (1.30)

j jj

b y b=∑ (1.31)

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

one components values of ca α and b . Also, even though these

two equations are based on gas mixtures, they can also be applied

to liquid mixtures by substituting the jx for the jy . Equation 1.30

is a double summation, with both indexes i and j going over the

entire number of components in the system.

The ijδ are called “binary interaction coefficients” and are

empirical measures of the attractive and repulsive forces between

molecules of unlike size. Note that 0ii jjδ δ= = and 0ij jiδ δ= = .

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.

( ) ( ) ( ) 2ln 1 ln ln

2

j j j j

uz BAz B z B A B uB z B

σ

ϕ σσ

+⎡ ⎤+⎢ ⎥′ ′ ′= − − − − − ⎢ ⎥−⎢ ⎥+

⎣ ⎦

(1.32)

where,

( )2 1cj j i ci i ij

ij

c

a y aA

a

α α δ

α

−′ =

∑ (1.33)

jj

bB

b′ = (1.34)

Note that Eq. 1.32 reverts to Eq. 1.22 where there is only one

component present.

For two-phase mixtures, Eq. 1.7 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 above equations can be simplified using the following

definitions.

cjbj

cj

Tf

p= (1.35)

( ) ( )21 21 1cj

aj o j jcjcj

T Tf T m m mTp

ω ω⎡ ⎤⎛ ⎞⎢ ⎥= + + − −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(1.36)


Then,

( )2 1a j aj i ai ijj i

pA y f y fT

δ= Ω −∑ ∑ (1.37)

b j bjj

pB y fT

= Ω ∑ (1.38)

( )( )

12

1

aj i ai iji

jj aj i ai ij

j i

f y fA

y f y f

δ

δ

−′ =

−

∑∑ ∑

(1.39)

bjj

j bjj

fB

y f′ =∑

(1.40)

The procedure for doing a flash calculation on a mixture then is the

following.

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

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

3. Find a value of gn% which maintains molar balances,

i.e., that ensures 1j jj j

x y= =∑ ∑ (more on this later).

4. Use the jx in the mixing equations to find the coefficients and solve the EOS for the liquid. Select the smallest root. Likewise use the jy to solve the EOS for the gas phase. Select the largest root.

5. Calculate fugacity coefficients for each component in each phase using Eq. 1.32. Use these to determine the

jK from Eq. 1.29.


6. Repeat from Step 2, using the calculated jK 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.

( )2T C

j jT Cj j

K KError

K K

−=∑ (1.41)

7. Iterations should be stopped when the Eq. 1.41 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 gn% . The following procedure is typically used.

When the jz and jK are known (as in Step 3 above), the correct

value of gn% is the one that ensures molar balances. Although either

the jx or jy equation may be used for this purpose, here we will

focus on the jx equation, Eq. 1.24. Using Newton-Raphson

iteration to solve this equation results in the following.

( ) ( )11 1

jg

j g j

zf n

n K= −

+ −∑%%

(1.42)

( ) ( )( ) 2

1

1 1

j jg

jg j

z Kf n

n K

−′ =

⎡ ⎤+ −⎣ ⎦∑%

% (1.43)

Recall that Newton-Raphson iteration involves successive

guessing, with the “new” guess calculated from the “old” one by

( )( )

1kgk k

g g kg

f nn n

f n+ = −

′

%% %

% (1.44)

One of the roots of Eq. 1.42 is always 0gn =% , so the initial guess

should start well away from this value. Outside the two-phase


envelope, the value of gn% 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 1j jj

z K <∑ (1.45)

Gaseous phase 1j

j j

zK

<∑ (1.46)

Bubble point 1j jj

z K =∑ (1.47)

Dew point 1j

j j

zK

=∑ (1.48)

Two-phase 1j jj

z K >∑ and 1j

j j

zK

>∑ (1.49)

The above procedure can also be used to calculate bubble and dew

point pressures, by simply iterating on pressure until the

appropriate Eq. 1.47 or 1.48 is true. Again, the Excel Solver is a

good way to do this type of calculation.

1.3 Exact Values of EOS Parameters

van der Waals

2 2 2 227 0.421875

64c c

cc c

R T R Tap p

= =

1 0.1258

c c

c c

RT RTbp p

= =


3 0.3758cz = =

Soave-Redlich-Kwong

2 2 2 21 3 2 3

2

1 2 2 0.4274809

c cc

c c

R T R Tap p

⎛ ⎞+ += =⎜ ⎟⎝ ⎠

1 32 1 0.086640

3c c

c c

RT RTbp p

⎛ ⎞−= =⎜ ⎟⎝ ⎠

1 0.3333333cz = =

Peng-Robinson

( ) ( )

( ) ( )

2 2 2 2

2 3 1 3

1 3 2 3

1715112 80 216 2 13

1 2205 1134 0.4572361536 16 2 13 16 2 13

162 16 2 13 45 16 2 13

c cc

c c

R T R Tap p

⎡ ⎤− + −⎢ ⎥

−⎢ ⎥⎢ ⎥

= + + =⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥− − + −⎢ ⎥⎣ ⎦

( )( )1 3

1 3

34316 2 3516 2 131 0.07779610081536 144 16 2 13

16 2 13

c c

c c

RT RTbp p

⎡ ⎤− +⎢ ⎥−⎢ ⎥= =⎢ ⎥+ + −⎢ ⎥−⎢ ⎥⎣ ⎦

( )

( )1 3

1 31 711 16 2 13 0.30740132 16 2 13

cz⎡ ⎤⎢ ⎥= − + − =⎢ ⎥

−⎢ ⎥⎣ ⎦

1.4 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).

eos

Documents