PETE 310
Lectures # 36 to 37
Cubic Equations of State
…Last Lectures
Instructional Objectives
 Know the data needed in the EOS to evaluate fluid
properties
 Know how to use the EOS for single and for
multicomponent systems
 Evaluate the volume (density, or z-factor) roots
from a cubic equation of state for
 Gas phase (when two phases exist)
 Liquid Phase (when two phases exist)
 Single phase when only one phase exists
Equations of State (EOS)
 Single Component Systems
Equations of State (EOS) are mathematical
relations between pressure (P) temperature (T),
and molar volume (V).
 Multicomponent Systems
For multicomponent mixtures in addition to (P, T
& V) , the overall molar composition and a set of
mixing rules are needed.
Uses of Equations of State (EOS)
 Evaluation of gas injection processes
(miscible and immiscible)
 Evaluation of properties of a reservoir oil
(liquid) coexisting with a gas cap (gas)
 Simulation of volatile and gas condensate
production through constant volume
depletion evaluations
 Recombination tests using separator oil
and gas streams
 Many more…
Equations of State (EOS)
 One of the most used EOS’ is the PengRobinson EOS (1975). This is a threeparameter corresponding states model.
P
P
RT
V b V (V
Prep
Pattr
a
b) b(V
b)
Equations of State (EOS)
 Peng-Robinson EOS is a three-parameter
corresponding states model.
 Critical Temperature Tc
 Critical Pressure Pc
 Acentric factor
PV Phase Behavior
Pressurevolume
behavior
indicating
isotherms for
a pure
component
system
Tc
Pressure
CP
T2
v
P1
T1
L
2 - Phases
V
L
V
Molar Volume
Equations of State (EOS)
 The critical point conditions are used
to determine the EOS parameters
P
V
0
Tc
2
P
2
V
0
Tc
Equations of State (EOS)
 Solving these two equations
simultaneously for the Peng-Robinson
EOS provides
2
a
a
2
c
RT
Pc
and
b
b
RTc
Pc
Equations of State (EOS)
Where
and
with
a
0.45724
b
0.07780
1 m1
Tr
m 0.37464 1.54226
2
0.2699
2
EOS for a Pure Component
Pres sur e
CP
T2
4
v
P
1
3
1
L
A1
2
10
0
5
A2
P
~
V
0
V
T
2 - P has
hases
L
7
1
2
V
Mo la r V o lum e
T1
6
EOS for a Pure Component




Maxwell equal area rule
(Van der Waals loops)
For a fixed Temperature
lower than Tc the vapor
pressure is found when A1
= A2
Equations of State cannot
be quadratic polynomials
Lowest root is liquid molar
volume, largest root is gas
molar volume
Middle root has no
physical significance
CP
Pressure

T2
v
3
P1 1
L A1 1
10
0
2
4
A2
P
V~
5
0
V
T
2 - Phas
es
L
7
1
2
V
Molar Volum
Volume
e
T1
6
Equations of State (EOS)
 Phase equilibrium for a single
component at a given temperature
can be graphically determined by
selecting the saturation pressure such
that the areas above and below the
loop are equal, these are known as
the van der Waals loops.
Equations of State (EOS)
 PR equation can be expressed as a
cubic polynomial in V, density, or Z.
Z
3
( B 1) Z
( A 3B
2
2
2 B) Z
( AB B
2
3
B) 0
A
with B
a P
2
RT
bP
RT
Equations of State (EOS)
 When working with mixtures (a ) and
(b) are evaluated using a set of
mixing rules
 The most common mixing rules are:
 Quadratic for a
 Linear for b
Quadratic MR for a
0.5
Nc Nc
a
xi x j ai a j
m
i
j
1 ki j
i 1 j 1
 where kij’s are the binary interaction
parameters and by definition
kij k ji
kii 0
Linear MR for b
Nc
bm
xibi
i 1
Example
 For a three-component mixture (Nc =
3) the attraction (a) and the repulsion
constant (b) are given by
a
m
2 x1 x2 a1a2
0.5
1
2 x1 x3 a1a3
x32 a3
bm
x1b1
2
0.5
1 3
3
x2b2
(1 k12 ) 2 x2 x3 a2a3
x3b3
(1 k13 )
x12 a1
1
0.5
2
3
x22 a2
(1 k23 )
2
Equations of State (EOS)
 The constants a and b are evaluated
using
 Overall compositions zi with i = 1, 2…Nc
 Liquid compositions xi with i = 1, 2…Nc
 Vapor compositions yi with i = 1, 2…Nc
Equations of State (EOS)
 The cubic expression for a mixture is then
evaluated using
Am
a
P
m
RT
2
Bm
bm P
RT
Analytical Solution of Cubic
Equations
 The cubic EOS can be arranged into a
polynomial and be solved analytically
as follows.
Z
3
( B 1) Z
( A 3B
2
2
2 B) Z
( AB B
2
3
B) 0
Analytical Solution of Cubic
Equations
 Let’s write the polynomial in the
following way
x
3
a1 x
2
a2 x a3
0
Note: “x” could be either the molar volume, or
the density, or the z-factor
Analytical Solution of Cubic
Equations
 When the equation is expressed in
terms of the z factor, the coefficients
a1 to a3 are:
a1
a2
a3
( B 1)
( A 3B
2
( AB B
2 B)
2
3
B )
Procedure to Evaluate the Roots of
a Cubic Equation Analytically
 Let
Q
R
3a2
9
9a1a2
S
3
T
3
R
R
2
1
a
3
1
27 a3
54
Q
3
Q
3
2a
R
2
R
2
Procedure to Evaluate the Roots of
a Cubic Equation Analytically
 The solutions are,
x1
x2
x3
S T
1
a1
3
1
S T
2
1
S T
2
1
1
a1
i 3 S T
3
2
1
1
a1
i 3 S T
3
2
Procedure to Evaluate the Roots of
a Cubic Equation Analytically
 If a1, a2 and a3 are real (always here)
The discriminant is
D = Q 3 + R2
Then
 One root is real and two complex
conjugate if D > 0;
 All roots are real and at least two are
equal if D = 0;
 All roots are real and unequal if D < 0.
Procedure to Evaluate the Roots of
a Cubic Equation Analytically
x1
If D 0
x2
where
cos
R
Q3
x3
2
1
Q cos
3
2
1
Q cos
3
2
1
Q cos
3
1
a1
3
120
1
a1
3
240
1
a1
3
Procedure to Evaluate the Roots of
a Cubic Equation Analytically
x1
x2
x1 x2
x1 x2 x3
x3
a1
x2 x3
x3 x1
a2
a3
where x1, x2 and x3 are the three roots.
Procedure to Evaluate the Roots of
a Cubic Equation Analytically
 The range of solutions useful for
engineers are those for positive
volumes and pressures, we are not
concerned about imaginary numbers.
Solutions of a Cubic Polynomial
We are only
interested in
the first
quadrant.
Solutions of a Cubic Polynomial
 http://van-der-waals.pc.unikoeln.de/quartic/quartic.html
contains Fortran codes to solve the roots of
polynomials up to fifth degree.
Web site to download Fortran source codes to
solve polynomials up to fifth degree
EOS for a Pure Component
Pres sur e
CP
T2
4
v
P
1
3
1
L
A1
2
10
0
5
A2
P
~
V
0
V
T
2 - P has
hases
L
7
1
2
V
Mo la r V o lum e
T1
6
Parameters needed to solve
EOS
 Tc, Pc, (acentric factor for some
equations i.e. Peng Robinson)
 Compositions (when dealing with
mixtures)
 For a single component
 Specify P and T  determine Vm
 Specify P and Vm  determine T
 Specify T and Vm  determine P
Tartaglia: the solver of cubic
equations
http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/tartalia.html
Cubic Equation Solver
http://www.1728.com/cubic.htm
WWW Cubic Equation Solver
 Only to check your results
 You will not be able to use it in the
exam if needed
 Special bonus HW will be invalid if
using this code, you MUST provide
evidence of work
 Write your own code (Excel is OK)
Two-phase VLE
 The phase equilibria equations are
expressed in terms of the equilibrium
ratios, the “K-values”.
Ki
yi
xi
ˆl
i
ˆv
i
Dew Point Calculations
 Equilibrium is always stated as:
l
ˆ
xi i P
v
ˆ
yi i P
(i = 1, 2, 3 ,…Nc)
 with the following material balance
constraints
Nc
Nc
xi 1,
i 1
Nc
yi 1,
i 1
zi 1
i 1
Dew Point Calculations
 At the dew-point
l
ˆ
xi i
xi Ki
v
ˆ
zi i
zi
(i = 1, 2, 3 ,…Nc)
Dew Point Calculations
 Rearranging, we obtain the Dew-Point
objective function
Nc
i 1
zi
Ki
1 0
Bubble Point Equilibrium
Calculations
 For a Bubble-point
Nc
zi Ki 1 0
i 1
Flash Equilibrium Calculations
 Flash calculations are the work-horse
of any compositional reservoir
simulation package.
 The objective is to find the fv in a VL
mixture at a specified T and P such
that
Nc
i
zi ( K i 1)
f v ( K i 1)
11
0
Evaluation of Fugacity Coefficients
and K-values from an EOS
 The general expression to evaluate
the fugacity coefficient for component
“i” is
RT ln ˆ
P
v
i
Vi
0
RT
dP
P
T fixed
Evaluation of Fugacity Coefficients
and K-values from an EOS
 The final expression to evaluate the
fugacity coefficient of component ‘i’ in the
vapor phase using an EOS is.
RT ln ˆiv
Vtv
P
v
ni
T , n vj i
RT
v
dV
t
v
Vt
RT ln Z v
 A similar expression replacing v by l is used
for the liquid
Equations of State are not perfect…
 EOS provide self consistent fluid
properties
 Density (o & g) trends are correctly
predicted with pressure, temperature,
and compositions (and all derived
properties…)
 Same phase equilibrium model for gas
and liquid phases (material balance
consistency)
Equations of State are not perfect…
 However… predicted fluid property values
may differ substantially from data
 EOS are routinely “calibrated” to selected &
limited experimental data
 After “calibration” EOS predictions beyond
range of data can be used with confidence
 EOS are extensively used in reservoir
simulation
What is EOS calibration?
 Minimization of squared differences between
experimental and predicted fluid properties
Ndata
gi
predicted
gi
exp erimental
2
min
i 1
 These Properties (gi) include:
 Densities, saturation pressures
 Relative amounts of gas and liquid phases
 Compositions, etc.
What is EOS calibration?
 Accomplished by changing within certain limits
selected EOS parameters
 Minor adjustments (1 to 2%) of binary interaction
parameters (kij) can change saturation pressures by
20 to 30%
 Different properties of the C7+ fraction affect liquid
dropout and densities. These properties include
 Molecular weight (uncertainty is +/- 10%)
 Specific gravity
 Critical properties and acentric factors which are
highly dependent on correlations – Cannot be easily
measured and not usually done.
Pre and post calibration predictions
from an EOS
Pre and post calibration predictions
from an EOS
Pre and post calibration predictions
from an EOS
Pre and post calibration predictions
from an EOS
Problems to Think About…
 Determine the equilibrium ratio of C1
from multiple flash calculations using
SOPE. Select a mixture and a suitable
pressure temperature range
 Discuss the trends, how does kC1change
with T at a fixed P?
 Discuss the trends, how does kC1change
with P at a fixed T?
 Provide well documented graphs
Problems to Think About…
 Compare the equilibrium ratio of C1 at 4000
psia and at 200 oF with that of the
convergence pressure chart using.




A mixture of C1 and C2
A mixture of C1 and C4
A mixture of C1 and C8
Discuss the results obtained and provide
overlapped plots
 Calibrate one of EOS’s in SOPE to the
bubble point data reported by Standings in
the following table
Problems to Think About…
Problems to Think About…
 Mole fraction of C1
 Dew point pressure
 Bubblepoint
pressure
 Z-factors of mixture
(gas and liquid)
 Molar volumes of
mixture gas & liquid
 All at T = 160oF
(not shown here)
Problems to Think About…
 Select one EOS
 Select the best kij
(Vdw, RK, SRK, PR,
that matches the
or Cubic-4G)
bubble point
pressure
 Select one bubble
point pressure for
 Compare the
one composition of
values of
methane
experimental vs.
predicted molar
 Plot pb predicted vs
volumes
binary interaction
parameter selected
You should be obtaining a plot like
this one…
Bubble Point Pressure C 1-C 4 Mixture (10% C1) at T =
o
160 F
2500.0
Experimental
pb is 339 psia
Pressure, psia
2000.0
1500.0
You CANNOT use
this same
composition in
Your homework
1000.0
500.0
0.0
-0.5
-0.3
Cubic-4G
-0.1
P-R
kij
0.1
S-R-K
0.3
R-K
0.5
VDW
This is the end, we survived!!!