Compositional simulation of reservoir performance by a reduced thermodynamic model

Compurers them. Engng, Vol. 18, No. 2. pp. 75-81, Printed in Great Britain. All rights reserved

0098-I 354/94 56.00 + 0.00 Copyright 0 1994 Pergamon Press L:d

1994

COMPOSITIONAL SIMULATION PERFORMANCE BY A THERMODYNAMIC P. WANG Engineering

Research

Center,

(Received

4 December

and

E.

H.

OF RESERVOIR REDUCED MODEL

STENsYt

IVC-SEP. Department of Chemical Engineering. of Denmark, Building 229, DK-2800, Lyngby

1992, final revision received

The Technical

I7 June 1993: received for publication

University

I July

1993)

Abstract-In compositional reservoir simulations, it is usually assumed that a local thermodynamic equilibrium between all phases exists throughout the reservoir within each time step. Phase behavior calculations are conducted based on an equation of state (EOS) in each block. This computation normally takes a very significant percentage of the total CPU time. In this work, an extension of the reduced thermodynamic model (Wang and Stenby, Compurers &em. Engng 16, 5449-5456, 1992) to the processes vary is demonstrated. This modified model has been tested in which both pressure and feed composition against the experimental data of gas condensates and the PVT data computed from the EOS. The results indicate that the new model can well represent the influences of changes in the feed composition and the pressure on the K-values. The new model has been implemented into a compositional reservoir simulator, UTCOMP (Chang Ph.D. Thesis, Univ. of Texas, Austin, 1990). Simulations of natural depletion for three well-defined mixtures and of a dry gas recycle process for a gas condensate in the North Sea are performed. The results simulated with the new method are comparable to those computed by the PR EOS. But the CPU time is reduced by approximately 50% by means of the new method compared to the one required by the PR EOS. In addition, this work shows that the CPU time can be further decreased by using the newly suggested procedure (Leibovici and Neoschil, Fluid Phase E9uil. 74, 303-308, 1992) for solving the Rachford-Rice equation for phase split.

assumption Two

types

of

reservoir

reservoirs

are

industry;

black

simulators.

being oil

The

components

the

equilibria

pseudo-oil

dependent

of the reservoir

solubility

mixture sitions,

gas

constant

no volatility

solubility

of

a

simple

in the oil of

function

pressure

processes

dependent

techniques

where

and

This

the fluid

on composition,

properties

however,

treated

the reser-

as

a

separate

in the water phase,

between

water

and

and no

oil or gas

ture. The

best predictive of state (EOS) behavior

pressure.

Once

are

For

the number 10’.

to whom all correspondence should be addressed.

by

methods

simulator

behavior

of

hydrocarbon may

of hydrocarbon

the mix-

be based

on

are usually description

mixtures

is employed,

at high the com-

very slow since the fluid properties iteration.

For

a

usual

in which the compositional of isothermal

It is generaily

has strongly 75

PVT

since cubic EOS

this approach

becomes

obtained

simulation

oil

of

to be able to give an accurate

of phase putation

as a

in the compositional

prediction

equations

simulation tAuthor

occurs

and the thermodynthroughout

fluid which is a complex

considered

are also

the black

efficient

reservoir

oil model

pressure.

only

The major difficulty

in the reservoirs

saturation

transfer

is an

and zero

can be expressed

normally

present

to describe

phases.

and

Based on these

it may be said that the black

voir. mass

pressure-

phase.

is required

In both models it is assumed

is isothermal

is

gas

model

the reservoir fluid with more than

state prevails

Water

immiscible

compositional

amic equilibrium component

oil and gas phase compo-

the fluid properties of

the

A

components

that the reservoir

the hydrocarbon

of oil in the gas phase,

is used to study recovery for which

two

fluid/rock

by a pseudo-gas

of gas and oil in the water.

assumptions,

some

with

representation

presumes

how

composition,

is invalid.

these recovery processes.

The black oil model charac-

component

two-component

and

of constant

phases

two hydrocarbon

of appropriately

process,

terizes the reservoir fluid simply a

these

oil

which characterizes

compositional

between

as a means

system are characterized.

isothermal

arises from how many

displacement

thermodynamic

for

in the petroleum

and

difference

essentially

are chosen

describing

used

simulators

main

types of simulators

simulators

widely

and

flashes is often

true that

time is spent

over

50%

reservoir

model

is used,

as high as of the total

on these calculations.

restricted the practical

application

This of the

P.

16

WANG

and E. H.

EOS compositional simulator. In order to reduce the computation time, many attempts have been made in literature to improve the existing flash calculation algorithm or develop new flash routines. The objective of this work is to develop a reduced model to determine the distribution of components between phases of hydrocarbon mixtures in order to save computation time in the reservoir simulations. The phase fraction is then obtained by simply solving the Rachford-Rice equation. The parameters of the reduced mode1 are fluid-dependent and easily estimated by fitting the experimental K-values or those computed from the EOS.

STENBY

The K-value changes caused by the variations of the feed composition, which is the second term of equation (6) can be presented by means of the following way:

By a material balancing and elimination of phase fraction variation, we may reach: InKy=

1

2 j=18(Kj-11)+

REDUCED

THERMODYNAMIC

One of the most fundamental relationships dealing with the thermodynamic properties of a substance in a two-phase region is the Clapeyron equation, which may be expressed as: dP=’ dT=-’

1

MODEL

AH T-AV

Assuming that gas phase is ideal gas and AH constant at low pressure, we may have:

(1)

is

x

aIn&:

(I--@--

NC

zi(Ki -

SF, (K, -

p”’ K!d,_L_ P’

where

C~=ln(P.)(j&--f) , ---1 1 . Tai

(5)

Tci

For the real component K-values of a mixture under reservoir conditions, the following equation can be written if the temperature is assumed to be constant: lnKi=lnK~+InK~.

(6)

The first term of the right-hand side of the above equation accounts for the contribution of the pressure variation to the K-values. It is found in this work that this term can be well described by: lnK:=(a+bP-lnP)+(c+dP)C,.

(7)

(9)

1)

(10)

1)e + 1 = ‘*

(11)

(3)

(4)

dn,.

The composition of the liquid and vapor phases is then computed by means of the equations below:

y; = K;xi.

By application of equation (2), we may reach the equation below (P, = 1 atm): In Kid I = C I - In P 3

1

This term was neglected in our earlier work (Wang and Stenby, 1992) since the pressure term is dominating in a natural depletion process of a reservoir where the feed compositions vary slightly during the displacement compared to a gas injection process. Once the K-values of each component are known, the fraction of vapor phase can readily be obtained by solving the Rachford-Rice equation:

In PsaL= In P, + where P, is the vapor pressure. The term In P, goes to zero if the boiling temperature is evaluated at 1 atm. For a mixture, the ideal K-value for component i is defined as:

8Kaln4: J any

an)

[

VALIDATION I.

Experimental

OF

THE

K-values

(12) REDUCED

from

a CVD

MODEL

process

The K-values at each pressure stage of a constantvolume depletion [(CVD), Moses and Donohoe, 19891 process for the gas condensates can be obtained by a mass-balance calculation based on the measured PVT data. These K-values may be considered as the experimental data. In Fig. 1 the typical result of the K-values correlated by equation (6) is shown for a gas condensate (Kenyon and Behie, 1987). The estimations of the critical properties and acentric factors of C,, fractions required by equation (6) are given in the next section. 2. K-values calculated from

an EOS

It is generally accepted that the cubic EOS can be used for high-pressure PVT calculations of hydrocarbon fluids. Therefore, investigation of the Kvalues computed from an EOS is also the way of

77

Simulation of reservoir performance .

pressure at which the above equation is satisfied can be found. The K-value at the dew point can also be obtained from the above equation. The liquid dropout at each pressure stage is evaluated from:

Expl.

-

Calc.

(14)

c = 0.824559 d = -0.002827 0.01

0

I

I

I

I

I

50

100

150

200

250

Pressure (atm) Fig. 1. Comparison between experimentalK-values and computed ones for a gas condensate(Kenyon and Behie, 1987). testing the validity of equation (6). Many hydrocarbon mixtures with defined components (Yarborough, 1972) have been used for this test. A typical comparison of the K-values computed from the PR (Peng and Robinson, 1976) EOS and the reduced model is shown in Fig. 2. It appears from Figs 1 and 2 that the K-values measured and computed from the EOS can be well represented by the reduced model with a single set of parameters

in both

high-

REGENERATION

and low-pressure

regions.

[$,$=

A l-._.-.-•

nC5

*r-*-,/r

(15)

T, = 0.556 exp(4.2009i?~“6’SSG0.046’4),

(16)

e

1.

I

(1 - T,sIT,)CT

p =exp

(13)

By adjusting the pressure in equation (6) at the given composition zi and the temperature, the dew point

fz: c3

T, = 169.43822MW”~45534SGo~~2*‘,

OF THE PVT DATA

The above well-defined mixtures and gas condensates are also used to test the capability of the reduced model to reproduce the PVT data. Phase volumes are estimated from the PR EOS. The dew point pressure is estimated from:

10

are calculated from The phase compositions equations (11) and (12). The 0 -value is obtained from equation (10) at the given pressure. Since C,+ fractions are involved in the gas condensates, the critical properties (T, and P,) and acentric factor (w) of the plus fractions have to be known in advance, namely plus fraction characterization. In this work, the following procedure is used: (a) by extrapolation of equation (6) the Ci values of the C,+ fractions can be found as the K-values of the C,+ fractions at each pressure stage are known from the mass-balance calculations; (b) the Ta value in equation (5) is found from the Whitson-relation (Whitson, 1983). The critical temperature T, is obtained from the Winn-relation (Winn, 1957). The PC is then estimated from equation (5) as follows:

T -

TB

;

(17)

1

(c) by consideration of each of the C,+ fractions as a pure component, the acentric factor (w) may be found by fitting the T, values estimated by means of the EOS to those obtained from equation (15). Figures 3 and 4 demonstrate the comparisons of the liquid dropout curves computed from the reduced model and the PR EOS for the mixtures examined. The evaluation of the K-values is EOS-independent in this work. The EOS is only used to estimate the phase volumes. This may slightly affect the feed mole number in the next stage. It appears from these two figures that the reduced model can reproduce the

i--_-a ,-:

2 _,

10-I

r/

;i M

lc7 10-Z

nCl0

--

.‘/ ,. a = -0.1247893

/*

b = 0.0210786

/-

E = 0.8817808

l

d = -0.0032097 10-j

. 0

I 50

l

-

I

I

I

I

100

150

200

250

Pressure

(atm)

Fig. 2. Comparison between experimental K-values and computed ones for a well-defined mixture (Yarborough, 1972).

f,l

Expl. Reduced

model \

0

I

I

I

50

100

150

I\,, 200

250

Pressure (atm) Fig. 3. Liquid dropout curve of the gas condensate.

78

and E. H.

P. WANG

STENBY

reservoir gas condensate, constituent cycling

of

of

gas.

in order

variations element

during

the

A 3-D Pressure (atm)

reservoir.

produced

mixture.

An

dropout

Besides,

with

satisfactory

accuracy.

the dew point pressure is obtained

extrapolation dew

curves of equation

point

balance

pressure

cannot

calculations.

lated dew points

from

(6) since the K-values be

obtained

by

Table

the

well

the upper

from

with the measured

the reduced

model

given

from

the

PR EOS. COMPOSlTlONAL SIMULATION REDUCED MODEL

reduced model proposed

The

implemented reservoir

in

a newly

simulator,

simulations

of

have

been

flow

rate well model For

to estimate

with

reduced

model.

Simulation

developed

by

the

1990).

of

The

three

well-

UTCOMP

with

the

results

of the fluids in order to

from

the

runs

with

the

The same runs as made in our earlier work (Wang 1992) are performed

new reduced the fluid work.

apparent

made

observed. K-values be

to those difference

in this

This

in this work with the

The reservoir

are identical

No

lations

model.

work

implies

used

system and

in the earlier

between

and

that

block

the

the

previous

simu-

work

influence

on

of small change in the feed composition

neglected

in

the

natural

depletion

of

the

is the can gas

condensates. Simulation Part

of

of gas cychg

process

the produced

gas

reservoir to maintain minimize

the injected

gas.

into

the

the reservoir pressure. This can

the amount

sation and vaporize

is re-injected

of retrograde

liquid

the in situ condensed

Dry

gas

is usually

conden-

liquid into

miscible

For rate

with

the

to 5%

proposed

to characterize

initial

from

the PR

the dew point volume Reservoir

well,

a total is

per annum.

by Pedersen

et ai. (1988)

the reservoir fluid for the curve estimated

is compared

EOS

as shown

to the one in Fig.

liquid volumes

’

Layer I 2 3

5. It

relative to

by both the reduced

rock propertics. well for the 3-D run

Permeability

gas

specified,

data

NX=lS, NY=5, NZ=3 DX(ft) = 52.72, 13 x 105.45, 52.72: DY(ft) = 5 x 32.08 Datum (subsurface, ft) Capillary pressure Initial reservoir pressure (atm) Experimental dew point pressure (atm) Calculated dew point pressure (atm, PR EOS) Calculated dew point pressure (am, reduced model) Formation properties Temperature (“F) Compressibility, Psi Injection well Radius (ft) Permeability (md) Constanr bottomhole pressure (atm) Production well Radius (ft) Permeability (md) Constant production rate (lb-mol day-‘) Thickness

pressure

of the injected fluid

depletion

calculated

grid data, conditions

bottomhole

reservoir

The liquid dropout model

well

for the production

908.6 lb-mol/day

appears that the retrograde

I.

The

is also

pressure

well the flowing

the production of

the simplified

computed

Table

of natural depletion

and Stenby,

is employed

in

well, while a constant

atm), and the composition

The procedure

from

fluid are listed.

is applied

equals

EOS calculations.

the

The PR EOS is also used in these runs behavior

that

production

is

block

of the initial

bottomhole

the injection

which is equivalent

compositional

(Chang,

and gas cycling of a gas condensate

the PVT

compare

(289.6

corner,

for the EOS calculation

flowing

well.

are specified.

depletions

conducted

reduced model.

THE

in this work has been

UTCOMP

natural

defined mixtures

FROM

the

gas

initial water saturation

is used for the injection

pressure

in

wet

2.

constant

model

the

of each layer are presented

fluid and the injected

in Table

and

2 the compositions

mass-

agree well

ones or those computed

1. In Table

is located

right hand

porosity,

necessary information A

to large

displacement,

(1,1,3)

at the

It can be seen that the extrapo-

cycling

block

Thickness,

reservoir

Due

in each small volume

(15 x 5 x 3) is used to describe

and rock permeability liquid

recovery.

injection

comer, from

(15,5,1).

be a

of such a process.

grid model

bottom-left

Fig. 4. Liquid dropout curve of the well-defined

gas

may

of hydrocarbon

term, lnKF, has to be taken into account

in the simulation the

displacement

to improve

the dry gas

reservoir

in feed compositions

composition

is the primary

Therefore,

the gas condensate

special case of miscible fluids

and methane

the dry

Porosity

(fi)

(md)

(%)

47.5 78.5 33.0

3.17 I .48 6.39

41.47 33.52 40.86

and

initial

6500.0 0.0 289.6 289.6 291.6 2X9.6 160.0 I+.-6 16 lCUKl.0 289.6 16

1000.0 908.6

water saturation 30.0 30.0 30.0

(%)

Simulation of 2.

Table

model

and

imental

C,

C,

EOS

agree well with

by using the reduced of liquid

reduction. the

supports

would

not

in the estimation permeability,

the liquid

it is

vertical

horizontal

the gas-relative

phase

on

one.

permeability

The

is esti-

(18)

is set to be equal

capillary

pressures

in the simulation

is assumed

simulations

of

model

voir pressure average

reservoir

between

are not accounted

a period

MW

0.016 0.098 0. I52 0.176 0.193 0.274 0.462 0.503 0.597 0.792

16.44 30.07 44.09 58.12 58.12 80.04 94.00 108.00 139.57 211.34

production

since only the

the earlier production is mainly

caused

in Fig. 6 as a function

progress,

significantly

of

during

period because the production

by the expansion

fluid within this period.

of the reservoir

As the dry gas injection

more and more condensed

condensation

is in

oil is evaporated.

mated are

by both

almost

profiles

of

those blocks

Figure

esti-

and the PR EOS

7 shows

the

pressure

layer in X-direction

pre-

It appears that the pressure is nearly linear, except

close to the production

profile

The

average

reduced model As

indicated

dicted

estimated

from

well. Again,

both

models

in the

agree

oil

the initial

model,

saturations

computed

in Fig.

5, the dew point

EOS

reservoir

is around

the

pressure.

of 0.001

pre-

higher

than

Therefore,

the initial

state with an oil

for the simulation

while,

pressure

2 atm

fluid is in the two-phase EOS,

from

and the PR EOS are shown in Fig. 8.

by the PR

the PR

at the

model

region.

behaviors

well

saturation

the pressure

identical.

drop in most of the blocks

been

and

the reduced

dicted by the two models.

ther-

runs. The

that the pressure

of the production

15 yr have

An initial reser-

pressure,

if the fluid is in the retrograde

it appears

reservoir

atm is used in both

pressure

well are plotted

time. Both pressures are reduced

with the reduced

and the PR EOS.

of 289.4

to the

to be mobile.

out by the UTCOMP

modynamic

0

46.31 48.20 41.90 36.00 37.50 30.90 29.55 27.10 23.88 21.36

pressure

for. But it does not pose any problem gas phase The

the

way as:

permeability

phases appearing

carried

only

Due to the

k,, = s, The

of

of the gas phase and no relative permeability in the simplest

(2) 194.3 305.4 369.8 408.1 425.2 490.6 526.2 552.8 605.3 696.8

as seen in the graph, which can also reduce the liquid

impact

mated

gas

This results in the increase of the reservoir

relative

of

dry

PC Wm)

Besides,

of the presence

injected

also

made

of

the

pro-

that the water and oil are not mobile,

available,

and

the

during

to be mobile.

data

in

fluid

This observation

be large

the gas phase is considered mobility

the

permeability. calculation

assumed

small

and

that the liquid condensation

the assumption

In the

can be

Besides,

are found

with a dry gas cycling.

gas-relative

the exper-

curve is not steep during the pressure

reservoir

duction

model.

condensation

It induces

reservoir

4.86 2.08 0.39 0.70 0.00 0.82 0.00 0.00 0.00

It seems that the better match

achieved

initial

91.14

5.133 2.275 0.502 1.014 I .929 0.659 0.379 0.541 0.439

amounts

liquid dropout

the

87.130

the PR

values.

of

79

performance

Injection fluid mole (%)

Reservoir fluid mole (%)

Compound N,, CO,. C, C, LC, nC, ic,, ncs. C, C, C, 12 C13+

Compositions

reservoir

in the study

conducted

with

with

the reduced

the reservoir fluid is initially at the dew point.

This leads to the oil saturations

estimated

run with the reduced model being around

T 5

o .

PR EOS

-

Reduced

from

0.001

the

lower

model

-~._,_.-.-.--r--r--r-. ;

:

Ave.

res.

P.

CJ l-

-

” 50

Fig. 5. CME

Reduced

Prod.

model

I

I

I

I

I

100

150

200

250

300

Pressure

(atm)

liquid dropout curve of the North Sea gas condensate.

I

270 0

1000

well

I

I

I

2000

3000

4000

Time

P. I 5000

I 6000

(day)

Fig. 6. Production well and average reservoir pressures a function of time.

as

P. WANG

80

and

E. H.

STENBY

r

. *r

.

l

_

.I

-Reducedmodel

.

.

..

.

PR (EOS)

Reduced model

. 270

0

I

I

I

I

I

I

I

I

2

4

6

8

10

12

14

16

0

I 1000

I 2000

Block number in X-direction Fig. 7. Pressure profiles in the block different times.

(1 :I 5,5,1)

I 5000

I 4000

I 6000

Time (day) at three

than those from the PR EOS. However, almost no difference of the pressure profiles simulated by the two models is observed. It may be concluded that the pressure estimation is not sensitive to the saturation in the cases where the S,, is very small. Figure 9 shows the ratio of cumulative gas injection to cumulative oil production simulated from the two models. In the later stages, the increase of this ratio means that the injected dry gas has probably reached the production well (breakthrough). The oil recovery estimated from the reduced model and the PR EOS is, as shown in Fig. 10, also agreed well. The purpose of introducing the reduced model is to see how much computation time can be saved. For the simulations performed in this study, 55% of the CPU time is saved by using the reduced model compared to the one required by the simulation with the PR EOS. TRANSFORMATION

I 3000

Fig. 9. Ratio of cumulative gas injected to oil produced.

inal equation was transformed into the following form:

WJ- %.)(cra -

8),$* ,;y;, ‘1 t 1

0,

(19)

where OLL = --_1, K -1 cfR= --.

1

(20)

1 r&,-l

(21)

The bounds of the phase split 8 in equation (19) are defined as below instead of (0,l):

(22) for Ki> 1, ea=min(-$+),

OF THE RACHFORD-RICE EQUATION

(23)

for K.<

A newly developed technique for solving the phase split suggested by Leibovici and Neoschil (1992) has been employed in our work as a tool of saving computational expense. In this procedure, the orig-

=

1.

80 O I60 -

PR EOS Reduced model

0.5

E .Ig z ;: 3 .s 0

0.4

-0. .

._

l

.

0.3 0.2 .

0.1

,

-

PR EOS Reduced model

I 0

I

0

I 1000

I 2000

I 3000

I 4000

I 5000

I

0

2000

6000

Time (day) Fig. 8. Average oil saturationas a functionof time.

4000

I 6000

Time (day) Fig. 10. Oil recovery computed by the runs with the PR EOS

and the reduced model.

Simulation

It is found (22)

that this technique

is helpful.

and (23) can often provide

aries within

which

this range,

(19)

solution

like

can thus generally

iterations, ations

in most

are

in

this

Rachford-Rice 2,069,533

For

and Neoschil,

a linear

function.

be reached

within

the for

work,

equation

cycling

example,

[equation

the three

only two iter-

gas

process

the

(lo)]

times flash calculations.

form,

equation

by only 610,004 iterations equation.

(19),

original

is solved

This

by

the

results

by

to be solved of the

Rachford-Rice

in a further

reduction

of the

time spent on the phase behavior

is analogous

to IMPES

tation

procedure

each

time

an

fully implicit

around

of the CPU

compu-

and noniterative

additional

is only

larger reduction

calcu-

of the UTCOMP

type and the overall

is sequential

step,

(19)

scheme

to

the trans-

is just one-third

original

Since the solution

equation

While

is required

times, which

needed

computation lations.

Within

times for the phase split corresponding

the 290,612 formed

is searched.

of the calculations

required.

simulated

bound-

by Leibovici

behaves

saving

2.5%.

over

by

using

A considerably

time can be expected

for

simulators. CONCLUSION

The results in this work may lead to the following conclusions: model

calculations

of carrying

out the P-T

for the hydrocarbon

been developed.

The impacts

feed composition

variations

model.

the parameters

which

are

flash

mixtures

has

of the pressure and are included

in the

of the reduced

fluid-dependent,

are

model,

easily

deter-

mined; 2. The K-values EOS

can

model.

measured

be

well

When

reduced

involved

can be simulated

results

are

by the simulation

However,

around

schil could

the

natural of

the

by the reduced

to

those

with the PR EOS.

half of the computation technique

suggested

time

of the Rachford-

by Leibovici

time for fully-implicit

5. The pressures

estimated

simulator

saturation

in

processes

and Neo-

be efficient for the reduction

computation sitional

the

be saved;

The transformation Rice equation

the

reproduce

comparable

computed

can generally

reduced EOS,

data;

behavior

The

from the

the

the PR

and the gas cycling

gas condensates model.

with

by

can satisfactorily

PVT

phase

depletions

and computed

correlated

coupled

model

experimental

4.

81 NOMENCLATURE

a,b,c,d

=

Parameters of equation (3)

C, = Characteristic constant of component i defined in equation (4) dni = Variation of the feed mole number for component F = AH = K, = = K KII = L., = MW = N, = nF = rz! = Pe = P = P, = SG = T = T, = T, = Vo = V, = X, = y, = z, = Greek

i Feed mole number Latent heat Equilibrium constant

of component

i

Maximum one of the K, values Maximum one of the K, values Liquid volume % of dew point volume Molecular weight Number of components Mole number of component i in liquid phase Mole number of component i in vapor phase Vapor pressure Pressure, atm Critical pressure, atm Specific gravity Temperature, K True boiling point, K Critical temperature, K Dew point volume Liquid volume Liquid composition of component i Vapor composition of component i Feed composition of component i

letters

aL = 0~~= 8 = 4 F= 4:’ =

Left bound of 0 Right bound of 0 Mole fraction of vapor phase to the feed Fugacity coefficient of component i in liquid phase Fugacity coefficient of component i in vapor phase REFERENCES

1. A reduced

3. The

performance

Equations

very narrow

the solution

as mentioned

equation

of reservoir

by the IMPES

are not sensitive

of the order

of magnitude

of the

simulators; compo-

to the low of 10e3.

Chang Y. B. Development of an equation of state compositional simulator. Ph.D Thesis. The University of Texas at Austin (1990). Kenyon D. E. and G. A. Behie, Third SPE comparative solution project: gas cycling of retrograde condensate reservoirs. J. Petrol. Technol. 981-997 (1987). Leibovici C. F. and J. Neoschil, A new look at the Rachford-Rice equation. Ffuid Phase Equil. 74, 303-308 (1992). Moses P. L. and C. W. Donohoe, Gas-condensate rescrvoirs, Chap. 39. Petroleum Engineering Handbook (H. B. Bradley, Ed.) SPE (1989). Pedersen K. S., P. Thomassen and A. Fredenslund, Characterization of gas condensate mixtures. paper presented at the 1988 AIChE Spring National Meeting, New Orleans (1988). Peng D.-Y. and D. B. Robinson, A new two-constant equation of state. ind Engng Chem. Fundam. f5, 5964 (1976). Turek E. A., R. S. Metcalfe, L. Yarborough and R. L. Robinson, Phase equilibria in CO,-multicomponent hydrocarbon systems: experimental data and improved prediction technique. Sot. Petrol. Engrs J. Jun 308423 (1984). Yarborough L., Vapor-liquid equilibrium data for multicomponent mixtures containing hydrocarbon and nonhydrocarbon components. f. Chem. Engng. Data 17, 129-133 (1972). Wang P. and E. Stenby, Phase equilibrium calculation in compositional reservoir simulation. Computers them. Engng 16, s449-S456 (1992). Whitson C. H., Characterizing hydrocarbon plus fractions. Sot. Petrol Engrs. J. 683494 (1983). Winn F. W., Petrol. Refiner. 36, 157-162 (1957).