Generalized Cubic EOS

7/27/2019 Generalized Cubic EOS

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Generalized Cubic Equations of State

Ivar Aavatsmark, Sarah Gasda

Uni CIPRUniversity of Bergen

Research Summit, Foundation CMG

Calgary, 7-8 October 2013
http://find/http://-/?-http://-/?-


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Outline

Motivation

Equations Test cases

Conclusions


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Simulation of CO2 storage with different EOS

Furthest up-dip point of CO2 plume
http://find/


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Isobars of CO2

0 200 400 600 800 1000 1200 1400220

240

260

280

300

320

340

360

380

Density (kg/m3)

Temperature(K)

(spacing 2 MPa)
http://find/http://find/


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Isobars of CO2 near critical point (468 kg/m3, 304.1 K)

300 350 400 450 500 550 600 650300

301

302

303

304

305

306

307

308

309

310

Density (kg/m3)

Temperature(K)

(spacing 0.1 MPa)
http://-/?-http://find/http://find/http://-/?-http://-/?-


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Motivation

Most oil recovery processes are dominated by

pressure-driven flow (viscous flow).

Most CO2 storage processes are dominated by

buoyancy-driven flow.

Moreover, CO2 storage sites typically lie in the

pressure-temperature domain just above the critical point

of CO2 (Tc = 304.1 K, pc = 7.38 MPa).

Hence, density accuracy is more important in CO2 storage

than in oil recovery simulations.


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Generalized cubic equation of state

p=RT

v b

a(T)

(v 1b)(v 2b)

where

1 < 2 < 1

Examples:

Soave-Redlich-Kwong: 1 =

1 and 2 = 0

Peng Robinson: 1 = (

2 + 1) and 2 =

2 1
http://find/


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Fugacity coefficients, pure substance

=exp(Z 1)

ZB

Z 2BZ

1B

A(21)B

where

A =p a(T)

R2T2, B=

pb

RT
http://find/


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Fugacity coefficients, mixtures

i =exp

Bi(Z 1)

Z B

Z 2BZ 1B

A(Ai

B

i

)

(21)B

where

a(T) =

ni=1

nj=1

wiwjai(T)aj(T)(1 dij), b=

ni=1

wibi

A =p a(T)

R2T2, B=

pb

RT

Ai =2

a(T)

nj=1

wj

ai(T)aj(T)(1 dij), Bi =

bib
http://find/http://goback/


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Determination of parameters in cubic EOS

Expressions for a(T) and b:

a(T) = aR2T2cpc

1 +

1

T/Tc

2

, b= bRTc

pc

Compressibility factor

Z = pvRT

Cubic polynomial (expressed in critical point)

P(Z) = Z3 [1 + b(1 + 1 + 2)]Z2+ [a+ (b+

2b)(1 + 2) +

2b12]Z

[ab+ (2b+ 3b)12]
http://find/


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Determination of parameters in cubic EOS

In the critical point

p

v

T

= 0,

2p

v2

T

= 0

Hence,

P(Zc) = PZ(Zc) = PZZ(Zc) = 0

Three equations with three unknowns (a, b and Zc):

3Zc = 1 + b(1 + 1 + 2)3Z2c = a+ (b+

2b)(1 + 2) +

2b12

Z3c = ab+ (2b+

3b)12
http://find/


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12 diagram

8 6 4 2 03

2

1

0

1

1

2

PR

SRKvdWZc

=0.2

6

Z c=0

.2

7

Zc(C

O2)

Zc=

0.28

Zc=

0.29

Zc=

0.3074

Zc=1/3

Zc = 0.375
http://find/


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Test case 1: Johansen formation

Supercritical fluid

Temperature range: 345375 K

Pressure range: 2030 MPa. Critical point of CO2:

Tc = 304.1 K,pc = 7.38 MPa
http://find/


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Johansen case

8 6 4 2 03

2

1

0

1

1

2

PR

SRKvdWZc

=0.2

6

Z c=0

.2

7

Zc(C

O2)

Zc=

0.28

Zc=

0.29

Zc=

0.3074

Zc=1/3

Zc = 0.375
http://find/


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Johansen case: Isochores of CO2

345 350 355 360 365 370 37520

22

24

26

28

30

Temperature (K)

Pressure(MP

a)

Span-Wagner

345 350 355 360 365 370 37520

22

24

26

28

30

200

300

400

500

600

700

800

Temperature (K)

Peng-Robinson

(spacing 50 kg/m3)

J h D i i f S W CO
http://find/


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Johansen case: Deviation from Span-Wagner CO2

345 350 355 360 365 370 37520

22

24

26

28

30

Temperature (K)

Pressure(MPa)

Peng-Robinson

RMS: 22.8 kg/m3

345 350 355 360 365 370 37520

22

24

26

28

30

Temperature (K)

Volume-translated

Peng-Robinson

RMS: 8.8 kg/m3

345 350 355 360 365 370 37520

22

24

26

28

30

40

30

20

10

0

10

20

Temperature (K)

Optimized cubic

EOS

RMS: 3.2 kg/m

3

(spacing 10 kg/m3)

J h I h f CO
http://find/


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345 350 355 360 365 370 37520

22

24

26

28

30

Temperature (K)

Pressure(MP

a)

Span-Wagner

345 350 355 360 365 370 37520

22

24

26

28

30

200

300

400

500

600

700

800

Temperature (K)

Optimized cubic EOS

(spacing 50 kg/m3)

T t 2 Sl i i j ti i th Ut i f ti
http://find/


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Test case 2: Sleiper injection in the Utsira formation

Liquid and supercritical fluid


Pressure range: 810 MPa. Critical point of CO2:

Tc = 304.1 K,pc = 7.38 MPa

Sl i
http://find/


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Sleipner case

8 6 4 2 03

2

1

0

1

1

2

PR

SRKvdWZc

=0.2

6

Z c=0

.2

7

Zc(C

O2)

Zc=

0.28

Zc=

0.29

Zc=

0.3074

Zc=1/3

Zc = 0.375

Sleipner case Isochores of CO
http://find/


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Sleipner case: Isochores of CO2

300 305 310 315 3208

8.5

9

9.5

10

Temperature (K)

Pressure(MP

a)

Span-Wagner

300 305 310 315 3208

8.5

9

9.5

10

200

300

400

500

600

700

800

Temperature (K)

Peng-Robinson

(spacing 50 kg/m3)

Sleipner case: Deviation from Span Wagner CO
http://find/


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Sleipner case: Deviation from Span-Wagner CO2

300 305 310 315 3208

8.5

9

9.5

10

Temperature (K)

Pressure(MPa)

Peng-Robinson

RMS: 51.6 kg/m3

300 305 310 315 3208

8.5

9

9.5

10

Temperature (K)

Volume-translated

Peng-Robinson

RMS: 18.4 kg/m3

300 305 310 315 3208

8.5

9

9.5

10

80

60

40

20

0

20

40

Temperature (K)

Optimized cubic

EOS

RMS: 17.6 kg/m3

(spacing 10 kg/m3)

Sleipner case: Isochores of CO
http://find/


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300 305 310 315 3208

8.5

9

9.5

10

Temperature (K)

Pressure(MP

a)

Span-Wagner

300 305 310 315 3208

8.5

9

9.5

10

200

300

400

500

600

700

800

Temperature (K)

Optimized cubic EOS

(spacing 50 kg/m3)

Test case 3: Modified Sleiper to include critical point


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Test case 3: Modified Sleiper to include critical point

Liquid and supercritical fluid


Pressure range: 78 MPa.

Critical point of CO2:

Tc = 304.1 K,pc = 7.38 MPa,c = 468 kg/m

3

Sleipner with critical point
http://find/


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8 6 4 2 03

2

1

0

1

1

2

PR

SRKvdWZc

=0.2

6

Z c=0

.

2

7

Zc(C

O2)

Zc=

0.28

Zc=

0.29

Zc=

0.3074

Zc=1/3

Zc = 0.375

Sleipner with CP: Isochores of CO2
http://find/


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300 302 304 306 308 3107

7.2

7.4

7.6

7.8

8

Temperature (K)

Pressure(MP

a)

Span-Wagner

300 302 304 306 308 3107

7.2

7.4

7.6

7.8

8

200

300

400

500

600

700

800

Temperature (K)

Peng-Robinson

(spacing 50 kg/m3)

Sleipner with CP: Deviation from Span-Wagner CO2


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Sleipner with CP: Deviation from Span-Wagner CO2

300 302 304 306 308 3107

7.2

7.4

7.6

7.8

8

Temperature (K)

Pressure(M

Pa)

Peng-Robinson

RMS: 56.7 kg/m3

300 302 304 306 308 3107

7.2

7.4

7.6

7.8

8

Temperature (K)

Volume-translated

Peng-Robinson

RMS: 34.5 kg/m3

300 302 304 306 308 310

7

7.2

7.4

7.6

7.8

8

100

80

60

40

20

0

20

40

60

80

Temperature (K)

Optimized cubic

EOS

RMS: 27.4 kg/m3

(spacing 10 kg/m3)



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300 302 304 306 308 3107

7.2

7.4

7.6

7.8

8

Temperature (K)

Pressure(MP

a)

Span-Wagner

300 302 304 306 308 3107

7.2

7.4

7.6

7.8

8

200

300

400

500

600

700

800

Temperature (K)

Optimized cubic EOS

(spacing 50 kg/m3)

Summary of optimal cubic EOS parameters 1 and 2


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Summary of optimal cubic EOS parameters, 1 and 2

8 6 4 2 03

2

1

0

1

1

2

PR

SRKvdWZc

=0.2

6

Z c=0

.27

Zc(C

O2)

Zc=

0.28

Zc=

0.29

Zc=

0.3074

Zc=1/3

Zc = 0.375

Conclusions
http://goforward/http://find/http://goback/http://-/?-http://-/?-


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Conclusions

The generalized cubic EOS is precisely as simple andcomputationally effective as PR or SRK.

Through appropriate choices of the parameters 1, 2 and, increased density accuracy may be obtained in

predefined pressure-temperature domains. Increased density accuracy is important in CO2 storage

and could be decisive for optimizing the composition of the

injection fluid.

An implementation of the generalized cubic EOS in GEM

would be appreciated.

Isotherms of CO2


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Isotherms of CO2

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

16

18

20

Density (kg/m3)

Pressure(MPa)

(spacing 5 K)

http://find/


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345 350 355 360 365 370 375

20

22

24

26

28

30

Temperature (K)

Pressure(M

Pa)

Peng-Robinson

RMS: 22.8 kg/m3

345 350 355 360 365 370 375

20

22

24

26

28

30

Temperature (K)

Volume-translated

Peng-Robinson

RMS: 8.8 kg/m3

345 350 355 360 365 370 375

20

22

24

26

28

30

200

300

400

500

600

700

800

Temperature (K)

Optimized cubic

EOS

RMS: 3.2 kg/m3

(spacing 50 kg/m3)

http://goback/http://find/http://goback/


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p 2

300 305 310 315 320

8

8.5

9

9.5

10

Temperature (K)

Pressure(M

Pa)

Peng-Robinson

RMS: 51.6 kg/m3

300 305 310 315 320

8

8.5

9

9.5

10

Temperature (K)

Volume-translated

Peng-Robinson

RMS: 18.4 kg/m3

300 305 310 315 320

8

8.5

9

9.5

10

200

300

400

500

600

700

800

Temperature (K)

Optimized cubic

EOS

RMS: 17.6 kg/m3

(spacing 50 kg/m3)

http://find/


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p 2

300 302 304 306 308 310

7

7.2

7.4

7.6

7.8

8

Temperature (K)

Pressure(M

Pa)

Peng-Robinson

RMS: 56.7 kg/m3

300 302 304 306 308 310

7

7.2

7.4

7.6

7.8

8

Temperature (K)

Volume-translated

Peng-Robinson

RMS: 34.5 kg/m3

300 302 304 306 308 310

7

7.2

7.4

7.6

7.8

8

200

300

400

500

600

700

800

Temperature (K)

Optimized cubic

EOS

RMS: 27.4 kg/m3

(spacing 50 kg/m3)



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p p

8 6 4 2 03

2

1

0

1

1

2

PR

SRKvdWZc

=0.2

6

Z c=0

.

27

Zc(C

O2)

Zc=

0.28

Zc=

0.29

Zc=

0.3074

Zc=1/3

Zc = 0.375

Generalized Cubic EOS

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