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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
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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
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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)
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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)
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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
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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
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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
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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]
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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
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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
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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
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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
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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
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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
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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)
Optimized cubic EOS
(spacing 50 kg/m3)
T t 2 Sl i i j ti i th Ut i f ti
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Test case 2: Sleiper injection in the Utsira formation
Liquid and supercritical fluid
Temperature range: 300320 K
Pressure range: 810 MPa. Critical point of CO2:
Tc = 304.1 K,pc = 7.38 MPa
Sl i
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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
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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
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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
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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)
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
Temperature range: 300310 K
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
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Sleipner with critical point
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
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Sleipner with CP: Isochores of CO2
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)
Sleipner with CP: Isochores of CO2
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Sleipner with CP: Isochores of CO2
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
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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)
Johansen case: Isochores of CO2
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Johansen case: Isochores of CO2
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)
Sleipner case: Isochores of CO2
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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)
Sleipner with CP: Isochores of CO2
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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)
Sleipner with critical point
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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
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