Accurate Vaporizing GasDrive Minimum Miscibility Pressure Prediction

SPE
SPE 15677
Accurate Vaporizing Gas-Drive Minimum Miscibility
Pressure Prediction
by E.-H. Benmekki and G,A. Mansoori,
●SPE

●

U. of ///inois

Member

Copyright 19S6, Sociely of Petroleum Engineers
This paper was prepared tor presentation at the 61st Annual Technical Conference and Exhibition of the Societ y of Pelroleum Engineers held in New
Orfeans, LA October 5-8, 1986.
This paper waa selected for presentation by an SPE Program Commiltee following review of information contained in an abatract aubmittad by the
author(a). Conlents ot the paper, aa presented, have not been reviewad by the Society of Petrolebm Engineers and are subject to correction by the
aulhor(s). The material, as presented, doas not necessarily reflect any position of the Society of Petroleum Engineers, ita officers, or members, Papera
presented at SPE meetings are subject to publication review by Editorial Committees of the S@ety of Petroleum Engineers. Permission to copy is
reetrtcted to an abstract of not more than 3W words. Illustration may not be copied, The abstract ahoutd contain conspicuous acknowledgment of
where and by whom the paper ia praeenlad. Write Publications Manager, SPF, PO. Sox 83383S, Richardson, TX 75083.3836. Telex, 7S098S SPEDAL.

ABSTRACT
Prediction
of
The 14inimum
Miscibility
Pressure
(IMP)
of the Vaporizing

Gas Drive
(VGD)
process
is
of state
with
different
modeled
using
an ●quation
mixing
rUle8
joined
with
●
newly
formulated
expression
for
the unlike-three-body
interactions
●nd the resarvolr
fluid.
between
tha injection
gas
The comparison
of the
numericel
results

with
the
evailable
experimental
data
indicates
that
an
●quation
of
state
alone
overestimate
the
MMP.
However,
when tha equation
of stata
is Joined
with
the
unlike-threa-body
interaction
term,
the
MP
will
be
predicted
●ccurately.

proposed
The
technique
is used to develop
a simple
and reliable
correlation
for
the accurate
vaporizing
gas drive
MMP prediction
INTRODUCTION
The Ternary
or pseudoternary
diagram
is a useful
way
development
to
visualize
the
of
miscible
displacement
in enhanced
oi I
recovery.
The phasa
behavior

of a
reservoir
fluid
for which
the axact
composition
is
never
known
can
be
represented
approximately
on
a triangular
diagram
by grouping
the components
of the
reservoir
fluid
into
three
pseudocomponents,
Such
diagram
is
called
pseudoternary
diagram.

Tha scopa
of this
paper
involvas
the use of the
Peng-Robinson
equation
of
atate
couplad
with
coherent
mixing
and combining
rules
derivad
from
statistical
mechanical
consideration,
●nd the
●ffects
in
Implementation
of the
three
body
the
●valuation
of the phaae

behavior
of ternary
systems
and
tha
prediction
of
the
minimum
mlacibility
TO support
prassura
of simulatad
reservoir
fluids,
the application
of the model,
it was preferable
to
obtain
phase
bahavior
data
for
true
tarnary
systems

Refareneas

●nd illuatratione

at

●nd of

papar

such
●s
carbon
dioxide-n-butana-n-deeane
●nd
methane-n-butana-n-decane,
with
are
rigorously
described
by
ternary
diagrams.
Moreover,
●xperimental
vapor-liquid
data
for
the
above
●t pressures
●nd tamperature$

aystama
are ●vailable
which
fall
within
the range
of tha majority
of oil
reservoirs.
eqUatlOn
The utility
of the
Peng-Robinson
(PR)
of state
has baen
tested’*2
with
Iimitad
aucce$
in
predicting
the
phasa
●nd minimtm
behavior
miscibility
reservoir
pressures
of

simulated
fluids.
By using
the
PR
equation
of
stste
●n
overprediction
of the !4RP of the methane-n-butane●nd it
was balaived
n-decana
system
was observed
that
this
was
due to
the limitations
of the
PR
●ccurately
predict
equation
which
doss
not
the
phasa

behavior
of
the
mathana-n-butane-n-decane
system
In addition
tha
in the
critical
region.
prediction
of
the vapor-liquid
coexi$tenca
curves
of the carbon
dioxida-n-butana-n-decane
sytems
was
not
satisfactory
in
all
ranges
of presaurea
and
compositions.

The ultimate
objective

of this
paper
it to show
the impact
of the mixing
and combining
rules
on the
prediction
of
and
the
phase
envelops
tha
contribution
of the
three
body-effects
on
phaae
behavior
predictions
naar
the critical
ragion.
THE

VAN

OER

WAALS

MIXING

From
conformal
tha
statistical
machanica
it
intarmolecular
potantial
molecules
of
a mixture
potential
energy
function
the following
axpresslon:

RULES
solution
theory
of
can ba
ahown that
palranergy

function
of any two
can
ba
ralated
to
the
of a raferenca
fluid
by


ACCURATEVAPORIZING
GAS DRIVEMINIMUN

RT

1/3
.. (r)
‘IJ

= f;juo(r,thli

“

(1)

)

hx

- ~

~

n
Z

=;

(4)
V (v+b)

+b (v-b)

where

a(T)

= a(Tc){l

+ K(l-T~/2)}2

(5)

a(Tc)

= 0.4572fI

R2T~iPc

(6)

b u 0.07180

nn

‘Xhx

a (T)

p.—.
v-b

potential
energy
In the above
equation
U.
is the
function
of the reference
pure
fluid,
fi.
is the
conformal
molecular
energy
parameter

and AiJ1/3
is
the
conformal
molecular
length
parameter
of
interactions
betwean
molecules
i and
j
of
the
mixture.
By
using
Eq.1
in
tha
statistical
mechanical
virial
or energy
equations
of state
and
application
of the conformal

solution
approximation
to the radial
distribution
functions
of components
of a mixturas
it can be shown
that

SPE 15677

MISCIBILITY
PRBSSURRPREDICTION

RTc/Pc

(7)

(2)

‘ixjfijhij

the
characteristic
following
relationship:
xixjhij

constant

u

is

gi~;en

by

tha

(3)

i]

K - o.37k6k
where
hx ●nd
fx
●re
the conformal
solution
parameters
of
● hypothetical
pure
fluid
which
represent
the mixture

●nd xi,
x ●re the mole
fractione.
This
means
that
for]the
extension
applicability
of ● pure
fluid
equation
of state
mixtures
one
has to
replace
moiecular
●nergy
of state
with
length
parameters
of tha ●quation
above
mixing
rules.

of
to

●nd
the

In ordar
to apply
the van der Waais
mixing
rules
in
different
equations
of
state,
one
has
to
consider
the fol iowing
guidelines
of the conformal
solution
theory
of statistical
machanics:
(i)
The
vander
Waals
mixing
rules

are for
constants
of an equation
of stata.
(ii)
Equation
2
ia
a mixing
rule
for
parameters
that
are proportional
to (molecuiar
3
length)
3.(moiecuiar
energy)
and
Equation
is
a mlxlng
rula
for
parameters
that
are
proportional
to (molecular

Iangth)j

In

the

the Peng-Robinson5
received
a
wide
calculations
is
parform
vapor-liquid

Peng-Robinson

aquatlon

of

equation
of
acceptance
in
chosen
in this
equilibrium

state

- 0.26992Q2

(6)

can

identical
with
the mixing
Equations
2 and 3 ●re
rules
which
were
originally
proposed
by van
der
Waals4
for
the van der Waals
equation
of atate
as
it was applied
to simple
mixturas.

Aa an

exampie
state
which
has
process
engineering
investigation
to
calculations.

+ 1.5ii226w

It Is
customary,
for
parameters
e
and b with
which
●re known ●a thair

nn
a = Z 2

the mixture,
the
following
mixing
rules

to calculate
expressions

(9)

XiXjaij

ij

b = !
i

aij

=

(lo)

xibi

(1-4ij)(aiiajj)

This
set
of mixing
with
the
guidelines
solution
theory

of
In order
correctly
we
must
constants
write
the
foliowlng

’/2

rules
is
dictated
statistical

(11)

however
inconsistent
by
tha
conformal
mechanics.

to apply
the van der W&ale mixing
in
tha Pang-Robinson

equation
of
tharmodynamlc
variables
separate
of
etate.
Thus ,
of tha ●quation
Peng-Robineon
equation
of stata
form:

rules
atata,
from
wa can
in tha


v------

-.

.

v

c/RT

+ d -

.. . -—-—-

- - ---

-----------

THEORY

2 4 (cd/RT)

z.—.

OF THE

THREE

BOOY

FORCES

(12)
(v+b)

v-b

where

-- -—.-

c = a(Tc)

+

(1+[)2

(b/v)

and

In
a
potential
be wri~.ten

(v-b)

nn
c = ~>x.x.c..
IJIJ
ij

(13)

b = ~~

(lb)

xixjbij

i.i

nn
(15)

d = ;;xixjdij

l/3
b ii
bij=

(1-l

ij)3

[

for
that
be

the unlika
●re corwiatent

interaction
with
the

U=!u(ij)
i
~ijk(l
u(ijk)

bJl

,3

(16)

[

where
i,
triangle
interior
evaluation
is possible

[cii.

+

...

(19)

+ 3cos?icosTjcos~k)

(20)

=

cjj/bii.

j and k are the three
mclecules
with
sides
‘ij’
a~~kY;~d;&
●ngles yi,
?j
of the triple-dipole
constant
to show9 that

forming

a

%
UiJk

it

1/3
+

‘j] _]

3

(17)
3h
‘ijk-

(l-kij)

u(ijk)

i
In
the
above
equation
u(ij)
is
the
pair
intermolecular
potential
energy
between
molecules
i
and j,

and u(ijk)
is the
triplet
intermolecular
potentia
●nergy
between
molecules
i,
j and k.
It
is shown i
that
the
contribution
of
tha triplet
intermolecular
to
the
total
interaction
energy
intermolecular
potential
energy
is
of the order
af
5 to 10%.

However,
higher
order
terms
(four
body
interactions
and
higher)
in
Equation
19
are
negligible.
Noreovar,
whan a
third
order
quantum
mechanical
perturbation
interaction
is
carried
eut~~,
~!eca~r~r~~~~f
that
the leading
term
in the three-body

interaction
●nergy
is
the dipole-dipole-dipole
term which
is
known
as
the
Axilrod-Teller
triple-dipole
dispersion
energy.
Tha Axilrod-Teller
potentitl
is
given
by the following
expression:

2

Cij=

N
Z

+

1/3

+

1/3
d ii
(1-miJ)3

may

(rijrjkrik)3

2

dijw

intermolecular
molecules

d = a(Tc)~2/RTc

This
new
form
indicates
that
this
equation
of
state
has tl.ree
independent

constants
which
ara b,
c,
Parameters
b
and d are proportional
to
and d.
while
length)3
or (b-h
and
d-h),
(molecular
(molecular
is
proportional
to
parameter
c
langth)
3. (molecular
energy)
or
(c-fh).
Thus,
the
mixing
rules

for c, b, and d will
be

The combining
rules
c ●nd d
parameters
b,
●bove mixing
rules
will

fluid
system
the
total
energy
of tha
interacting
in the following
form:

bjJ]l/2

bij

~-~i(i~)
m(hnfo)z

a~(iw)

~k(i~)

d~

(21)

o

(18)

kij,
Iij
and
in
Equations
16-18
parameters
that
mij
are
interaction
parameters
the binary
can
be adjustad
to provide
the bast
fit
to

the
experimental
data,
In the
next
section
we will
discuss
tha
shortcoming
of using
mixing
rulas
for
multicomponent
mixtures
(three
components
and more)
●nd wa will
propose
the
concept
of
unlike
threebody
interactions
to correct
this
problem.

whera
ai(iu)
is
the
dipole
polarizability
of
h is the
molecule
i at the imaginary
frequency
iu,
constant
is
the
Planck
●nd
Several
ap~oximate
express~%’?or
permittivity$
Uijk
have
the
been
tripie-dipole
constant
the expression
which

is very
proposed:
however,
●ssociated
with
the Axilrod-Teller
potentiai
often
function
has the following
form:


PRESSURZ
ACCURATEVAPORIZING
GAS DRIVEN2NINUI MISCIBILITY

(Ii+lj+lk)

3
v jk’

liljlk~iaj~k
(22)

~
(ii+lj)

whe[ e Ii
~i

is the

is the
static

(I]+lk)

(Ii+lk)

first
ionization
polarizabilty

potential
molecule

of

Nv

f 1 (?)

#

f? (n)

(23)

where

fl(v)

fz(n)

v =

= 9.87749q2

=

(T/6)

1 -

+

1.12789?+

11.76739?3

-

4.20030#

0.73166w2

(26)

in
in which

N
is the number
of molecules
and d
is a
hard
core
molecular
diameter.
relation
holds
result
the
following
Helmholtz
free
energy
of pure
fluids:

where
A2b
is the
Helmholtz
free
pair
intermolecular
interactions,
Helmholtz
free

energy
due to triplet
interactions.
Basic
statistical
which
incorporate
in
of pair
intermolecular
derive
the expression

due to pair
interactions
A2b.
Then
if We replace
A2b in
Equation
27
will
have
an
the resulting
expression
which
can
be
used

for
realistic
fluids’”.
However,
for
such equations
of state
the intermolecular
potential
energy
parameters
are
not availabie
for highly
asymmetric
compounds.
In
extending
Equation
27 to
mixtures
one
can
either
use an
exact
mixture
theory
or
a set

of
mixing
rules.
In a
multicomponent
mixture,
in
addition
to
binary
interaction
parameters
there
to
wiil
be numerous
three-body
parameters
PiJk’s
be dealt
with.
For example,
in a binary
mixture,
in
addition
to
binary
parameters,
wa will

have four
ternary
parameters
(U1lI,
“112*
V122
and
which
are all
different
from
each other.
“222)
In a
ternary
mixture
we will
have
the
following
tarnary
parameters
and
V123 . . . . .
(“111 . . . . .
Which
add up to nine
ternary
parameters.
y33)

T e excessive
number
of
interaction
parameters
and
the lack of experimental
data
for
these
parameters
demonstrate
the difficulty
which
presently
exist
in
the
practical
utilization
of
such
statistical
mechanical
equation
of
state.
As a result
in the
present

rePort
*J
are
using
the
Peng-Robinson
equation
of
state
which
is
● fairly
accurate
for
thermodynamic
empirical
equation
property
calculations
of hydrocarbon
mixtures.
However,
when
●n empirical
equation
of
atate
is
used for
pure

fluid
it would
be
rather
difficult
to
separsts
contributions
of
and
three-body
the
two-body
interactions
into
the equation
of state.

(24)

(25)

(nd3/V)

SPE 15677

and
i.

will

be
a
positive
In
Equation
20
Uijk
provided
the
repulsion,
indicating
quantity,
and it
wili
be
.zute
triangle,
molecules
form an
negative,
indicating
attraction,
when the moiecules
triangle.
Cor,tribution
of threeform an obtuse
body effects
to the Helmholtz
free
energy

of a pure
mechanical
the
statistical
i Iuid
using
for
the
molecu:ar
approximation
superposition
describe
by a Pad&
radial
distribution
function
is
approximantb

A3b=

PREDICTION

volume
V
As a
for
the

energ

and A~bd!~
t;:
intermolecular

mechanical
equations
of state
their
formulations
the concept
can be used
to
interactions
for
the Helmholtz
free
energy

An emp!rical
equationof
state
ia usually
Joined
with
● aet
of mixing
and combining
rules
when \te
●pplication

is
extended
to
mixtures.
By
●
comparison
of ● mixture
empirical
equation
of state
with
● statistical
mechanical
equation
of state
we
can
conclude
chat,
for
pure
fluid
and
binary
mixtures,
an
empirical
equation
of

state
can
rapresant
mixture
properties
correctly
since
the
interaction
which
energy
of
is related
to P112
●mpirical
and to
accountad
for
by the
“122 is
interaction
●auation
of
state
through
the binary
parameters
which
are used
in the

combining
rules.
when we
use a mixture
equation
However,
of state
which
is based
on the above
concapt
of mixing
rules
for
multicomponent
mixtures
(ternary
and
higher
systems)
,
thsre
wili
be a deficiency
in
the
mixture
property
representation.
This

deficiency
is due to
the lack of consideration
of any unlike
interaction
term
in
such
empirical
three-body
This
equations
of
stata.
deficiency
can
be
corrected
by adding
the
contribution
of the unlike
three-body
term,
resulting
from the Axilrod-Teliar
equation
of
potential,
to tha

empirical
state.
Consequently
for
the
Heimholtz
free
ene~gy
of
●
muiticomponent
mixture
we can write

A~-A;

(a,b)

+~~:xixjxk

‘~k

,i#J#k

(28)


arL

Ant

I

n,

U*

9-.-—.

---

--

~-=nf,uldsyst.
,.,.,.O

where

~

Am -

and

~=

while
would

example,

a ternary

for
be

●

Am= As(a,b)

binary

a

ha’s
f’:re
is
:+?
t~~

mixtu~t

w

+xlx2x3A~3

i=~tion

four

mm 3

A~~3+

IrIXZX4

A~4+

X2X3X4

A~4

pressure.
can
elso
which
the

Equation

Formulation

Thhova#ae:qu:j~&l~f
.
a
following
determinant

28

minimum
achieved

to

The
be
defined
critical

of

the

MNP

the
●re

critical
given

state

by

of
the

equationa:

A~~~

ii&8x:
+ X1X3X4

the
be

●s
the
minimum
es
the
tie
line
(tangent
to
the
binodal
curve
at
the
critical
POint)
passes
through
the point
representing
the

oil
composition
(Figure
1). Dynamic
miscibility
can
be achieved
when the reservoir
fluid
lies
to the
right
of the limiting
tie
line.
miscibility
pressure
pressure
at

Mathematical

(31)

compenent mixture

+ X1X2X3

28

~t~?

.

●nd
temperature,
●t
which
miscibility
can
multiple
contects
is refarred

In evaluating
a petroleum
reservoir
field
for
possible
C02 or natural
gas flooding
certain
data
are
required
which
can
be
measured

in
the
laboratory,
in
the absence
of
measurements
such
information
can be estimated
from
fundamentals
and
theoretical
considerate
ions.
The
required
information
include
the MMP,
PVT data,
asphaltene
precipitation,
viscosity
reduction,
the swelling
of
crude
oil.

It
is
obvious
that
accurata
predictions
of PVT
data
and
MMP have
important
consequences
for
the
design
of
●
miscible
displacement
process.
In tha
following
section
●
mathematical
model
is presented
for
the evaluation
of the minimum

miscibility
prassure.

(30)

mixtura

●gent)

“~ture

m=2

,

A~(a,b)

mixture
average
Ame(a,b)
and
evaluated
wit
state.

for

A~(a,b)

for

~

f2 (d)

is
the
d
diameter,
free
energy
equation
of

in
which
molecular
Helmhol’:z
empirical
For
becomes

f , (d)

NUijk
——
d9

A~;k.

(:?’

●iscible
Pressure
through
minimum
miscibility
minimwn

,

m.4

&(4x16x2)

lJ-

(32)

=0

(33)

-0

(34)

6;/6x;

6;/(6xlJx2)

It
is
a
proven
fact
that
unlika
three-body
interaction
terms
●re the major
part
of the threebody potential
in multicomponent
mixtures”.
As
three-body
correction
terms
in
tha
result
the
a
above
equations
would
make a substantial
improvement
specially

in the region
of equimolar
mixture.
&bx;
VAPORIZING

GAS

DRIVE

ANO

MINIRUMHISCIBILiTY

PRESSUR

E

,

6;/(3x,6x2)

v.
13u/6x2

NJ/axl
The vaporizing
gas drive
mechanism
is a process

used
in
enhanced
oil
recovery
to
achieve
dynamic
miscible
displacement
or
multiple
contact
miscibla
displacement.
Miscible
displacement
processes
rely
on multiple
contact
of injected
gas
and reservoir
a0
in-situ
vaporization
of
oil
to

develop
intermediate
moiecular
weight
hydrocarbons
from
the
into
the injected
gas and
create
a
reservoir
oil
miscible
transition
zone’2
The miscible
agents
which
●re used
in such
●
process
may
include
natural
gas,
inert
gases

and
Dynamic
miscibility
with
C02 has
carbon
dioxide.
it can
be achieved
at a
advantage
since
a major
inert
lower
pressure
than
with
natural
gas
or
gases.

of the
molar
Gibbs
where
the
partial
derivatives

free
energy
g(P,T,xi)
are obtained
at constant
P,
When
the above
determinant
equations
T and x .
the
are
so ? ved
for
the
critical
compositions,
tangent
to the binodal
curve
at the critical
point
will
be obtained
as the following:

x:

-

c
X2 -

dPn

xl

at

=—
X2

dx2

critical

point

(35)


ACCURATEVAPORIZINGGAS DRIVZMZNIMUMMZSCIBILI~PRESSUREPREDICTION

i

where
Xlc
●nd X2C
are

the
critical
intermediate
compositions
of
the
light
●nd
components,
respectively.
Pn is the interpolating
poiynomi&l
of
the
binodal
curve
and
the
first
derivative
of
the
interpolating
polynomial
at the
is
approximated
by
a
central

eriticai
point
it should
be pointed
out that
difference
formula.
a good estimate
of the
critical
point
of a mixture
can
be obtained
from
the
coexisting
curves
and
35
to generate
the critical
combined
with
Equation
tie
line.
VAPOR-LIOUID

EOUILiBRIUtl

where
A = cp/(RT)2
B = bp/RT
Cm~+RTd-

2d(cdRT)

●quation
of
state,
The
original
Peng-Robinson
was used
in the derivation
of Equation
Equation
h,
However,
with
the implementation
of the three
39.
body effects
the mixture
equation
of state
will
be

P =(6A/6V)T

in
the
equilibrium
atate,
intensive
the
properties
pressure
and
chemical
- temperature,
potentials
of ●ach component
are constant
in the
overall
system.
Since
the
chemicai
potentials
are
functions
of
temperature,
pressure
and

compositions,
the equilik-ium
condition

where
equation

Pe

*n=

is
of

Pe+x 123
x x

the
state

expression
and

=PiL(T.p,

i=l,2,

{X})

.,.,n

be

expressed

6]”=

Xi

for

the

(ko)

empirics!

+35.30217v2-16.80120?3

(42)

(36)
1.46332?

(k3)

by
The
the

yi

j

f2

(41)

f;=(3f2/6v)=-l.12789+
can

- flf;

s = (8/27)TNokU123

f~=(3f}/6q)=lg.754g8q
~iv(T.P.{y])

f;fz

M
—(
b3v

CALCULATIONS

When
applying
a
aingie

equation
of state
to
describe
both
liquid
and vapor
phases,
the success
of
the vapor-liquid
equilibrium
predictions
will
depend
on the accuracy
of the •quatl~~
of state
and
on the mixing
rules
which
are used.

SPB 1%77

+iL

i=l,2,..

.,n

co-volume
fol iowing

parameter,
reiation:

b,

is

related

to

q with

(37)

q = (b/4v)
The ●xpression
depends
on the
is the same ?or

for
the
equation
the vapor

fugacity
coefficient
of st~te
that
is
and iiquid
phases

m
RT

in

dim

f

[(3P/bni)T,V,n

v

-(RT/V)]dV-RTlnZ

(38)

Now if we derive
the
fugacity
coefficient

from
38,
integral
form,
Equation
and using
Equation
we obtain
for
the
PR equation
of state
with
originai
mixing
rules
the fo! iowing
expression:

of the correct
version
of the van
rules,
the foi lowing
expression
coefficient
wiil
be derivad:

der

for

di

=

(hi/b)
-(a(2

(Pev

+2RT2xjdij
/~(cd)
((Z+

(l+J2)B)

)/C
/(Z+

‘2d(RT)(c
-

(2 Xxjbij
(1-d2) B)))

-

RT)/RT

2xjai~/a

the

-

-ln(P(v-b)/RT)
bi/b)/(2v’2bRT)

ln((v+(l+42)b)/(v+(l-42)b))
+d(xix2x3A~~3

b)/b)(Z-1)-ln(Z-B)-(A/(2d2

(In

the

40,

J
In

With
the use
Waals
mixing
the fugacity

(44)

&i
used and

)/dni

(45)

B))
~Xjdij

- b)/b)

(39)

while
the following
expression
PR
equation
of
state
is
version
of the van der Waals

is derived
used with
the
mixing

ruiea!

when the
correct


rL

L20{6

n,

Ih.

In #.l= ((2Z

xjbiJ-b)/b)

(c/(242

RTb))

(Pev/RT-l)

(c2xjdij+d2xjci
(ln

((v+

j)/d(cd))

(l+J2)b)

+d(/l~2x3A~~3

Mu

-ln(P(v-b)/RT)

~xjclj+

((2

maWA

2RT~xjdlj-2~(RT’)
/C-(2

/(v+

(1-~2)

~xjbij-b)

/b)

b)))

u,

A.

-,7wn*

f

this
figure
that
tha devletion
of the
PR equstion
the
critical
point
is
around
substantially
corrected.
Figure
~ shows
the phase
behevior
of
the
methene-n-butane-n-decane
system17.
S i nce
the value
of the triple-dipole
constant

“123 is
obtained
from
an
approximate
expression,
en
adjustable
parameter,
~,
is introduced
in Equation
23 as the following:

(46)

)/dni

3
With
the aid of a computational
37
through
46
are used
to
ternary
and
curves
of binary

discussed
below.
RESULTS

AND

algorithm
Equations
generate
the
binodal
systems
as
it
is

DISCUSSION

calculations
experimental
binary
in the present
data
are
used
in
the
vapor-liquid
equilibrium
the binary

interaction
parameters
evaluation
of
which
minimize
the following
objective
function:

P (exp)
OF=!

-

P(cal)

[
ie~

1;

(47)

P (exp)

experimental
data
where
H

is
the
number
of
con$!dered,
P(exp)
and P(cal)
are the experimental
bubb I e
pressures,
and
calculated
point
A three
parameter
search
routina
is
respectively.
used
to evaluate
the
binary
interaction
parameters
of the correct
version
of
tha van der Waals mixing
the

binary
interaction
rules.
The
values
of
parameters
of all
the systems
studied
in this
paper
are reported
in Table
1.
In
Figure
2 the
experimental
and
calculated
fcr
the
methane-n-decane
results
●re
compared
influence
on the prediction
system

which
has a big
of
the methane-n-butane-n-decane
ternary
system.
In
this
case
both
mixing
rules
provide
a
good
correlation
of the experimental
results:
however,
a
!s observed
for
bigger
deviation,
overprediction,
the original
mixing
rules
in
the vicinity

of the
critical
point.
The carbon
dioxide-n-decane
system
3 whera we can sea that
the
is illustrated
in Figure
PR equation
with
the classical
mixing
rules
fails
the VLE data
in
all
ranges
to properly
correlate
of prassures
and compositions
while
an excellent
is obtained
with
the correct
mixing

correlation
rules.
mixtures
it has been
shown
that
For asymmetric
PR equation
of state
could
not represent
the sharp
near
the mixture
critical
region.
changes
of slopes
●lso observed
in ● simple
This
same problem
can be
ternary
mixture
of methane-ethane-propane16
as it
in
Figure
4.

is
demonstrated
However,
by
incorporating
the three-body
effects
it is shown
in

“ijk=

(li+lJ+lk)

liljlk’li~jak
(48)

c ~
(Ii+lj)

(Ij+lk)

(Ii+lk)

In this
equation,
c is adjusted
to provide
the best
correlation

possible
of
the
ternary
system.
In
Figure
5
the value
of
c is
found
to be
equal
to
0.5.
carbon
dioxide-n-butane-n-decane
shown
in
Figure
6
where
the
phase
system~8
Tle
behavior
prediction
with

the correct
verston
of the
an der Waals
mixing
rule
is clearly
superior
than
with
the classical
mixing
rules.
The chain-dotted
line
is for
the PR equation
with
the correct
mixing
rules
and including
the
three-body
effacts
with
6=1.
Figure
7 illustrates
the contribution

of tho
three-body
effects
on ths phase
behavior
prediction
of
the
ternary
system
in
the
vicinity
of
the
critical
region
which
ia very
important
for
the
with
tho
prediction
of
the MP.
The
PR ●quatian
●nd including

the
three
Classictl
mixing
ru18s
c=O.5 is rapresentad
by the sulid
body-effects
with
lines
while
the PR equation
with
the
$ameatlxlng
rules
Wi thout
●ffect.
but
three-body
the
dashed
lines.
It should
be
overpredicts
the FU4P,
pointed
out
that

in this
case the critical
point
is
curves
and
the
avaluated
from
the
coexisting
with
● quadratic
is ●pproximated
binodal
curve
8
In Figure
polynomial
around
the critical
point.
tha critical
point
is
obtained
from
Equations
Z9
and

30 and
the
quadratic
polynomial.
araund
the
critical
point
is obtained
with
two
additional
points
from
the blnodal
curve.
In this
case we also
observe
an overprediction
of the MMP from
the PR
equation
and the classical
mixing
rules.
CONCLUSION
As

a conclusion

the

following

may

be

pointed

out:
prediction
of
phase
(i)
For
a
successful
behavior
of
ternary
and
multi component
systems,
we must
first
be abble
to correlate
binary

data
of species
constituting
the mixture
correctly.
In
the
present
report
this
has
been
achievad
by
the van
der
utilizing
the correct
version
of
Waals
mixing
rules
for
the
PR aquation
of state.
the binary
VLE data ●ra correlated
As a resuit,

which
was
not
achievad
with
an
accuracy
previously
with
the PR equation.
(ii)

To

Improve

prediction

of

tha

phase

behavior


SPB 1s677

ACCURATEVAPORIZING

GAS DRIVEMINIMUMMISCIBILITY
PRESSURSPREDICTION

]

of ternery
●nd multi component
mixtures
araund
the
critical
ragion
it it
necessary
to
incorporate
the three-body
effects
in the equation
of :tata
calculation.
The contribution
of the three-body
effects
around
the
critical
point
must
not

be
Wi th
phenomena”
the
“critical
confused
effect’9.
Deviations
of
the
PR equation
of
state
from
experimental
data
of
ternary
systems
the
critical
point
are
generaliy
much
around
what
the “non-classical”
ef~~ct
due

bigger
than
to “critical
phenomena”
can cause.
The authors
have demonstrated
that
a
large
portion
of this
deviation
can
be corrected
by
incorporating
the
in the
phase
behavior
unlike-three-body
effects
studied
the noncalculations.
In most
instances
classical
contribution
is so small

that
for
the
scale
of the
graphs
and
the accuracy
of
the
available
experimental
data
it is insignificant.
(iii)
statistical
mixtures
~improving
capabilities
engineering

The

utilization
of
the
concept
of
thermodynamics
of

multicomponent
has
provided
us with
a strong
tool
of
the
correlation
and
predictive
of
the
empirical
existing
thermodynamic
models.

(iv)
The
ternary
mixture
computation
I
presented
technique
here
based
on
the

incorporation
of the corracted
version
of the van
der Waals
mixing
rulo$
●nd the three-body
effects
can
be
readiiy
extended
to
multicomponen
t
have developed
● ntssber
calculation.
The ●uthora
packagea
with
of computer
●re
capablo
of
performing
such caiculationa.
In the forthcoming
calculations

for
publication
reeulta
of ●uch
multicomponent
syatame will
be alss
reported.

I = c~ponent
L = Liquid

identification
state

m = Mixture

prOPertY

r = Reduced

property

V = vapor

state

ACKNOWLEDGEMENTS
The
authora

are
indebted
to
Dr.
Abbas
Firoozabadi
of
Stanford
University
for
his advice
during
the preparation
of this
work.
This
research
is supported
by the
Division
of Chemical
Sciences,
Sciences
of
the
U.S.
Office
of
Basic
Energy

Oepartmsnt
of energy
Grant
DE-FGL2-84ER13229.
REFERENCES
1.

Kuan,
D.Y.,
Kilpatrick,
P.R.,
Sahimi,
M.,
Striven,
L.E.
●nd Davis,
H. T.:
“Multicomponent
Carbon
Dioxide/
Water/
Hydrocarbon
Phase Behavior
Modeling:
A Comparative
Study,”
●t the 58th Annual
paper
SPE 11961 presented
●nd Exhibition,

San
Technical
Conference
1983.
Franciaco,
CA, Oct. s-8,

2.

Firoozabadi,
Correlation
Miscibility

3,

~nt~ri,

K.t
“Analyaia
A. ●nd Aziz,
of Nitrogen
and Lean Gas
SPE 13669
Pressure,”
paper

G. A.: Wixing

Rules

Equatione
of State,”
papar
1985 ACS National
Meeting,
28-itsY 3.
April

for

●nd

Cubic

presented
●t the
tliami,
Florida,

NOMENCLATURE
k,
$ = Binary

inter~ction

v = Reduced

dar

Waa}e,

J.

l).:

l’Over

de

Vioeistofloestand,”
Leiden
(1873).

COntlflUltOlt

van

Doctoral

density

M = Chemical

potential

~ _ Acantric

factor

5.

Pang,
D. Y. and Robineon,
O. B. : “A New TwoConstant
Equation
of State,”
ind.
Eng.
Chem.
Fund.
(1976)
volume
15, 59-64.

6.

Barker,
I}Three

J. A., Henderson,
D. and Smith,
W. R.:
Body Forces
in Dense
Syetaln$t”
phy$icai

Raview

Letters

P = Pressure
R = Universai

gas

intermolecular

(1968)

voiume

21,

134-136.

constant
7,

r =

Van

den Gas-en
Dissertation,

parameter

distance

Axi irod,
S,
the van der
Chem. Phys.

R. and Teller,
E.I
Waals
Type Between
(1943)
voluma
11,

Axi irod,

B,

M.:

Th~ory,U

J,

Chem.

“interaction
Threa
Atoms,”
299-300.

of
J.

T = Temperature
8.
v -

Noiar

volume

llTripia-Oipole
Phyat
(1951)

interaction.

1.

vO1uma “g);

719-729,
x = Nole

fraction

9.
Z = Compraeaibiiity
SuperScrip
c = Criticai

Q

~

factor

●ubscriDts

state

Maitiand,
Wakeham,
Clarandon

G. C.,
W, A.t
Prees,

Rigby,
N.,
Smith,
E. B.
intormolacular
Force%,
Oxford
(1981)
Chapter
2.

●nd


,,

F..U. BENMEKKIAND G. A. MANSOORI

sm 1!%77 .
------

10.

Alem,
A. H. and Mznsoorl,
G, A.:
“ The VIM
Theory
of Molecular
Thermodynamics:
Analytic
Equation
of State
of Nonpolar
Fluids,”
AIChE
Journal
(1984)
volume
30, 468-474.

11.

Bell,
Waals
Phys.

12.

13.

R. J, and Kingston,
Interaction
of two
(1966)
volume
Sot.

Stalkup,
Monograph
(1984) .

f’, l.:
Volume

Immiscible
8,

Henry

A, E.:

or Three

88,

“The van
Atoms,”

der
Proc.

901-907.

DisplacementO”
L,

?aherty

Series

Perg,
D. Y, and Robinson,
L), B.:
11 A Rigorous
Method
for
Predicting
the Critical
Properties
of Hulticomponent
Systems

from an Equation
of
AlchE Journal
(1977)
volume 23,
$.tate.l!

137-141i.
lk.

W. N.:
Some Properties
Sege, B. H. and Lacey,
of the Liahter
Hydrocarbons,
Hydrogen
Sulfide
and Carbon
Dioxide,
American
Petroleum
(1955).
Institute

15.

Raamer,

16.

Price,
A. R. @nd Kob@yashi,
R.:
lILow
Temperature
Vapor-Liquid
Equilibrium
In Light
Hydrocarbon
Mixtures:
14ethane-Ethano-Propane
System,”
J. Chers. Eng.
Data
(1959)
volume
4,
40-52.

17.

Reamer,
$lphase

H. H. and Sage,
B, H.:
“Phase
Equilibrium
In Hydrocarbon
Systems.

Volumetric
and Phaae
Behavior
of the n-Decsne-Carbon
Dioxide
System,”
J. Chem. ‘Eng.
Data (1963)
Volume 8, 508-s13.

Behavior
System,l*

H. H.,
Equilibria

Fiskin,
in

J. !!. ●nd
Hydrocarbon

Sage,
B.
SYstam$:

in the Methane-n-Butano-n-Oecane
Ind.
Eng.
Chem.

(1949)
VolUMe

H.:
phase

41?

287}-2875.
18.

19.

Hetcalfe,
R. S. and
of Phase
Equilibria
Rechanism,[l
Society

(1979)

242-252.

Hahne,

F. J.
Berlin

Verlag

Yarborough,
on the C02
of Petroleum

N.:
Critical
Heidelberg

L,:
‘The
Displacement
Engineers

Phenomena,

(1983).

Effect
J.

Springer-

9


rtible

1.

V81WS

of

binary

interaction

parametcra

for

the

different

mtxlng

Binary

-Pact

rules

Interactions

Mixing

Pz?ameters

Rules

Origins)

Mixing
Rules

Reference

system

Temperature
(Kelvln)

Pressure
Range
(bera)

k~a

04s2

1s>

mv?

[,>

-0.0848

-0.0178

0.0139

-0.6300

‘0.0624

0.0440

0.0195

-0.0261

0.0100

methane
n-8utene

-

[14]

344.26

10-103

*thene
n-Oecane

-

[ 14]

344.26

4-120

0.06s1

[14]

344.26

i -9

0.W46

[14]

344.26

11-66

o.ooa7 -0.0857

0.0006

0.1351

[ 15]

344.26

13-120

0.2054 -0.0650

-0.0792

0.1075

n-8utane
n-oecane
Carbon
n-tlutane

-

O~oxlde-

carbon
oloxlden-Oecane

-o.

v
Q

(i-

41s,6
4

8000.

1

09s.0

COMPONENT 1

,.*””

T . 344,2@K

,’
,’
*’

100,0
110.0

OWC

I

m}%o.
a

‘it

:

●;‘1
&

,’

LEGEND

*16 0.

i

,’

c Exporlmontal Dcta
PR (Ea.4) + Ea..l3=lE
,.., Pfi.(~s4). .+. . E@Qll
.. .. ... .

abwo

,’

{

/
f

/
(Oe,tj

1/
I

14P.*.
11s0.
or o

00!0
400

I@od ! --+--

0,0 0.$

COMPONENT 3

COMPONENT 2

0.:

0,s

0,4

0.8

0.O

0$7

0.0

0,0

x(1) , Y(1)

Wm.

100.0-

s
;

00.0-

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