SPE 148040

Three-phase Hydrocarbon Thermodynamic Liquid-Liquid-Vapour Equilibrium in CO2 Process Lins Jr., Abel G., SPE, Petrobras; Nghiem, Long X., SPE, CMG and Harding, Thomas G., SPE, UofC

Copyright 2011, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Characterisation and Simulation Conference and Exhibition held in Abu Dhabi, UAE, 9–11 October 2011. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Hydrocarbon phase behaviour must be rigorously represented when there is a need to properly account for mass transfer between phases in a porous medium. The overly simplified black-oil formulation, although appropriate for primary depletion and waterflooding, provides inadequate representation of miscible displacement processes. As a result, compositional simulation has evolved to provide thermodynamically consistent means to accurately describe the phases and compositions present within the porous reservoir rocks.

Compositional simulators have become essential modelling tools for CO2 processes in the Petroleum Industry. Advances in computational power have encouraged the development of meaningful improvements and refinements that were not possible until very recently.

CO2 injection into an oil reservoir at low temperatures causes the appearance of a three-phase hydrocarbon thermodynamic Liquid-Liquid-Vapour (LLV) equilibrium. The traditional use of a two-phase flash calculation in this three-phase region may lead to instability problems. Besides, commercial compositional simulators normally do not consider two-phase hydrocarbon Liquid-Liquid (LL) thermodynamic equilibrium that appears in oil reservoirs at low temperatures in the presence of CO2. Instead, it is treated as a Liquid-Vapour (LV) thermodynamic equilibrium and the fluid flux behaviour is not well represented. A compositional simulator must be able to represent adequately the LL hydrocarbon thermodynamic equilibrium when it is present in order to rigorously model the reservoir phase behaviour in the presence of CO2.

A novel procedure has been developed to overcome instabilities which may arise in calculation of multiphase liquid-liquid-vapour (LLV) hydrocarbon phase equilibrium. In addition, a new procedure has been developed for representing the thermodynamic liquid-liquid hydrocarbon equilibrium in a compositional simulator. This new procedure represents the real behaviour of the fluid flux. It is more rigorous than the traditional approach of lumping of the two liquid phases into a pseudo single liquid phase or as a liquid-vapour (LV) thermodynamic equilibrium. The results of this implementation are presented and analyzed in detail.

1. Introduction

The computational cost of compositional simulation is considerably higher than that of black-oil modelling. Compositional simulation involves a large number of unknowns and complex flash calculations to represent thermodynamic equilibrium in each gridblock of the spatially discretized reservoir. Not only are the number of equations and the effort in updating their coefficients greater, but also the equations are more non-linear as the fluids become more volatile, thus increasing the computational effort for achieving convergence for each computing timestep.

When seeking to identify possible causes of instabilities when simulating inside a three-phase region LLV (oil, CO2-rich oil and gas phase) with a two-phase flash algorithm, the use of software such as WinProp is essential. Developed by the Computer Modelling Group, WinProp is an equation of state modelling tool used for multiphase equilibrium and properties determination (WinProp, 2010).

With WinProp, it is possible to make a P-X phase diagram for multiple contact miscibility between the CO2 and oil and so to find the LLV locus region where instabilities may arise.

This compositional simulator got the ability to accurately model fluid systems where two liquid hydrocarbon phases are in equilibrium. The implementation in GEM is presented in detail.

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2. Multiphase Thermodynamic LLV Equilibrium

The thermodynamic equilibrium equations can be written in terms of K-values, or equilibrium ratios (Li and Nghiem, 1986), assuming that all the phases – oil, gas and water – are in thermodynamic equilibrium.

ioigiggi K ϕϕψ lnlnln, −+= = 0; i = 1, 2, …, nc (1)

( )∑=

=++

−=

cn

j jwwjggo

jjggnw KFKFF

zK

1, 0

1ψ

(2)

ioiwiwwi K ϕϕψ lnlnln, −+= = 0 ; i = 1, 2, …, nc (3)

( )∑=

=++

−=

cn

j jwwjggo

jjwwnw KFKFF

zK

1, 0

1ψ

(4)

Equations (1) through (4) are necessary to compute fluid phase equilibrium. The solution by Newton’s method is fast,

but only if it starts with a reasonable initial guess. If the guess is not close enough to the solution, then Newton’s method does not converge. Nghiem and Li (1984) propose to start the iterative process by using the Quasi Newton Successive Substitution (QNSS) method, which converges even for poor initial guesses, and then switch to Newton’s method.

The flash calculation implies the solution of Equations (1) through (4) for specific pressure, temperature, and compositions. In order to solve them, it is necessary to know a priori the number and nature of the phases present. These equations provide no information as to whether a variation in pressure, temperature, or composition increases the number of phases in the system. Therefore, it is essential to execute a stability test to verify that there is no other possible state with a lower Gibbs free energy. The system is in equilibrium whenever it is at its lowest Gibbs free energy.

The stability test can be stated in terms of the tangent plane criterion: a mixture in a certain composition, pressure and temperature is stable if and only if the tangent plane to the Gibbs free energy surface in this composition is everywhere below the Gibbs free energy surface (Nghiem and Li, 1988).

In this section, the oil from Wasson field, with properties described by Orr and Jensen (1984), is used. Wasson field is located in south-western Yoakum and northwestern Gaines counties on the Llano Estacado of West Texas, five miles east of the New Mexico border. The field covers 62,500 acres. The 33oAPI oil used is a recombination of dead oil and gas from Wasson field with a solution gas-oil ratio of 312 scf/bbl. Nghiem and Li (1986) report a full description of the oil and gas composition, as well as the match of the phase diagram with experimental data.

With WinProp, it is possible to make a P-X phase diagram for multiple contact miscibility between the CO2 and the Wasson oil. Figure 1 shows a P-X diagram for a CO2/Wasson oil mixture at 32.2oC. This picture, produced by CMG-WinProp, presents the LLV locus, observed inside the green curve.

Oil + CO2 at 32.2 C : P-X Diagram

2.50E+3

5.00E+3

7.50E+3

1.00E+4

1.25E+4

1.50E+4

1.75E+4

2.00E+4

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

CO2 Composition (mole fraction)

Pres

sure

(kPa

)

2-Phase boundary

3-Phase boundary

LLV Critical Point

L LL

LV

LLV

Figure 1 - Phase Diagram P-X for CO2/Wasson Oil Mixture at 32.2oC

Considering that the mixture CO2/Wasson oil may be multi-contact miscible, taking a sample with 80% CO2 content

after the first contact and mixing again in a proportion of 80% concentration of CO2, too many iterations are required to reach the equilibrium solution.

SPE 148040 3

oil rich #2 : P-X Diagram

6000

6200

6400

6600

6800

7000

7200

7400

7600

7800

8000

0.80 0.85 0.90 0.95 1.00

CO2 Composition (mole fraction)

Pres

sure

(kPa

)

3-Phase boundaryCriticalTest Interval

LV

LL

LLV

Figure 2 - Phase Diagram for CO2/Rich Oil with the Pressure Test Interval

Figure 2 shows a phase diagram with the interval of pressure used in the test. A two-phase flash routine from WinProp

is used for pressure varying from 7375 kPa to 7415 kPa with increments of 2 kPa in each flash calculation, with a total of 21 flash calculations, considering a total concentration of 96% of CO2 and temperature of 32.2 oC.

Table 1a presents the number of iterations required to calculate the equilibrium when using two-phase flash calculation in each pressure. Note the high number of iterations going from LV equilibrium to LL equilibrium at 7409 kPa.

Table 1 - Number of Iterations on Two-Phase Flash Calculation a) Without Correction b) With Correction

Pressure (KPa) No of iterations Equilibrium7375 13 L-V7377 4 L-V7379 4 L-V7381 4 L-V7383 4 L-V7385 5 L-V7387 5 L-V7389 5 L-V7391 5 L-V7393 5 L-V7395 5 L-V7397 5 L-V7399 7 L-V7401 9 L-V7403 9 L-V7405 10 L-V7407 12 L-V7409 97 L-L7411 3 L-L7413 3 L-L7415 3 L-L

Pressure (KPa) No of iterations Equilibrium7375 16 L-L7377 3 L-L7379 3 L-L7381 3 L-L7383 3 L-L7385 3 L-L7387 3 L-L7389 3 L-L7391 3 L-L7393 3 L-L7395 3 L-L7397 3 L-L7399 3 L-L7401 3 L-L7403 3 L-L7405 3 L-L7407 3 L-L7409 3 L-L7411 3 L-L7413 3 L-L7415 3 L-L

Note that for the first pressure, 7375 kPa, WinProp starts considering a single-phase test for stability. As the system is

not in the single-phase region, the stability test indicates that the single phase is unstable. Then it splits into two-phases and by using the guess from the stability test, the two-phase flash reaches the LV equilibrium after about 13 iterations. For the second pressure, 7377 kPa, the two-phase flash procedure starts from the last equilibrium solution, using that equilibrium value as the initial guess.

Inside the three-phase region using a two-phase flash calculation, it is possible to obtain an LL or an LV equilibrium solution. Both are solutions; the equilibrium that is obtained, whether LL or LV, depends on the initial guess used.

The same test is executed dropping the pressure instead of raising the pressure. This test results in no instability and equilibrium LL for all of the range 7415-7375 kPa. Similar tests are exhaustively executed by dropping the pressure in several ranges and using several options of delta P, with no occurrence of instability.

The problem arises when just crossing the border from LLV to LL equilibrium, and only in specific conditions. In the problem demonstrated above, the pressure 7409 kPa is inside the LL equilibrium locus, but too close to the LLV equilibrium. As the initial guess is almost a solution, the QNSS-Newton procedure to find the equilibrium cannot easily pass from the solution close to the initial guess to the real solution.

Suggestions to avoid instability that can lead to non-convergence in difficult cases are as follows:

4 SPE 148040

1. Find the region where the conditions in terms of pressure, temperature and CO2 concentration can yield LLV equilibrium.

2. Once in this region, whenever the QNSS procedure uses more than a certain number of iterations to be close enough to the solution, start a flash calculation using a higher pressure, above the LLV equilibrium, and for the next iteration use the regular pressure.

Using a procedure similar to the above, it is possible to solve the same problem without the instability observed - see Table 1b. All the points show LL equilibrium and use a low number of iterations per equilibrium, except the first one. In this case, the first one uses three more iterations than is the case without correction, because it is starting from a higher-pressure value. The advantage is that the solution is consistent and does not yield a non-convergence condition.

3. Identification of Liquid-Liquid-Vapour Region

Orr Jr. and Jensen (1984) show that Equation (5), with vapour pressure P in kPa and temperature T, in Kelvin, may be used to estimate the vapour pressure of CO2. This is a valid estimate for the Minimum Miscibility Pressure (MMP) of CO2 and the hydrocarbon mixture, which is where the LLV equilibrium region appears, if it exists.

⎟⎠⎞

⎜⎝⎛ +−

=91.10015,2

3.101 TeP (5)

The estimate is quite accurate for dead oil, but an adequate safety margin, 200 to 300 psi (1379 to 2068 KPa), should be added if the oil contains dissolved gas (Orr Jr. and Jensen, 1984).

Figure 3 shows the LLV behaviour for the Wasson Field oil at different temperatures. Figure 3 and Table 2 confirm that the pressure from Equation (2), here used as a minimum miscibility pressure (MMP) estimate, has a good correlation with the critical point. The LLV behaviour for temperatures below 25 oC is represented not as a region, but as a line, and the critical point, although not identified, can be associated with the more external point of the line.

Orr Jr. and Jensen (1984) explain how a correlation that does not take into account variations in oil composition can fit MMPs reported in the literature. They state that the density of CO2 increases so sharply with increasing pressure in the neighbourhood of the vapour pressure that, even if oils are sufficiently different in composition to require substantially different CO2 densities for efficient extraction, the pressure required to produce those densities can be similar.

Table 2 - Comparative Table between Critical Point and MMP CO2 LLV CO2 MMP Estimative

Temperature Critical Point estimative Relative ErroroC KPa KPa %

10 4493.18 4482.54 -0.24

20 5725.71 5715.59 -0.18

25 6500.00 6414.66 -1.31

32.2 7501.65 7524.04 0.30

35 7781.49 7989.46 2.67

4000

4500

5000

5500

6000

6500

7000

7500

8000

8500

9000

0.80 0.85 0.90 0.95 1.00Composition (mole fraction)

Pres

sure

(kPa

)

LLV EquilibriumCritical PointMMP estimativeT 20 'C

10 oC

20 oC

25 oC

32.2 oC

35 oC

Figure 3 - LLV Thermodynamic Equilibrium for Wasson Oil in Different Temperatures.

A thermodynamic three-phase LLV equilibrium occurs in a CO2-hydrocarbon mixture only when a high percentage of CO2 is present (Fong et al., 1992; Larson et al., 1989; Orr Jr. et al., 1981; Orr Jr. and Jensen, 1984). Fong et al. (1982), working with two kinds of oil, reach the three-phase LLV for CO2 concentration in the range 70-100 %; Orr Jr. and Jansen

SPE 148040 5

(1984) conclude that the three-phase could appear only for CO2 concentrations higher than 60%. Based on the information above, a new procedure is developed to avoid possible instabilities in working with any

hydrocarbon mixture and CO2 and using the numerical compositional simulator – GEM. 1. Find the region where the conditions in terms of pressure, temperature, and CO2 concentration can yield LLV

equilibrium. The bounds of this region are: - Temperature lowers than 50o C, - Vapour pressure, as defined by Equation (2.22), in the range (P-1, P+3) with P in MPa, - CO2 concentration higher than 60%, - The two-phase flash in the preceding time-step indicates L-V equilibrium.

2. Once in this region, whenever the QNSS procedure uses more than 30 iterations to be close enough to the solution, begin a flash calculation using a higher pressure. Pressure = P+3 (P in MPa); this pressure is above the LLV equilibrium, and from the next iteration on, use the regular pressure.

4. Simulation of a Liquid-Liquid Thermodynamic Equilibrium Region

Once a region is identified as being like liquid-liquid equilibrium, there are two options: (i) The first and easier option is to consider a lumping of the liquid phases. The system works as if it were in the single

hydrocarbon liquid phase. The problem is that these two liquid phases are moving with different velocities. The liquid phases have different densities and viscosities, and the resulting simulation may not represent satisfactorily the real behaviour of the reservoir.

(ii) The second option is to consider the two liquid phases as individual phases flowing through the reservoir. In this case, component flux equation must be restated as indicated by Equation (6). Note that instead of the gas term, another oil term appears. The difference between these two-oil phases is the subscript h and l, referring to the higher and lower root solution from the cubic equation of state.

( ) ( )+Δ−ΔΔ+Δ−ΔΔ= dgpyTdgpyT oliololohiohohi γγψ ( )dgPpyT wcwoiww Δ−Δ+ΔΔ γ

( )ni

nii NN

tVq −Δ

−+ +1 = 0 ; i = 1,…,nc,nw (6)

The equilibrium constants are necessary to determine the molar fraction for each liquid phase. The densities, molar densities, and viscosities are computed in a similar way to the oil-gas equilibrium, because the composition of each phase is known after the flash calculation. The difficulty remains for computing the relative permeability of each phase that is necessary for calculating the transmissibilities.

Although immiscible, the wettability of these two-liquid phases has greater similarity than the wettability of oil-gas phases. Hence, it is not necessary to compute the capillary pressure, and the relative permeability is equal to the oil phase for the two-liquid phases.

The gas saturation is zero and instead of one oil saturation So, there are two oil saturations, one for each phase, Soh and Sol. The total oil saturation is the sum of the two oil phase saturation.

oloho SSS += (7)

The oil viscosity for each oil phase is computed based in its individual composition of the oil phase. The relative permeability curve for each oil phase is assumed, in a simplified form, as proportional to its oil phase saturation, Equation (8).

o

ohroroh S

SKK = ;

o

olrorol S

SKK = (8)

The 5th SPE comparative project (Killough and Kossack, 1987) uses light oil, and it is a good example for observing the behaviour of the compositional simulator in the liquid-liquid equilibrium region. The 5th SPE comparative project is used, but the Water Alternating Gas Injection (WAG) is modified to include CO2 injection and the original oil composition is adjusted to that of Wasson oil. The project is a 7x7x3 grid with one producer, producing with a constant BHP of 1000 psi, and one injector well, with a constant gas rate of 1.2x107 scf/d, for a period of 8 years.

5. Multiphase Thermodynamic LLV Equilibrium Results and Analyses

Figure 4 presents the gas saturation distribution after 8 years of injection for a compositional simulator without LL equilibrium detection. Every time that a cell reaches CO2 saturation above 60%, the composition of the oil, the pressure, and the temperature lead to a liquid-liquid equilibrium condition. When the system in Figure 4 is in a two-phase region, it uses multiphase thermodynamic LV equilibrium; hence whenever LL equilibrium is present, in this system, the lighter liquid phase works like a vapour phase.

6 SPE 148040

Figure 4 - Gas Saturation after 8 Years of Simulation with Thermodynamic LV Equilibrium

Figure 5 shows the oil rate production for three different conditions when the system reaches a thermodynamic liquid-

liquid equilibrium. ‘LV’ means that the system considers the lighter liquid phase as being like a vapour phase; ‘LL’ means that the system takes the two-liquid phase into account; and ‘L’ means that the system makes a lumping of the two-liquid phases, and they flow like a single liquid phase.

0

1000

2000

3000

4000

5000

6000

7000

1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995

Qo

(bbl

/d)

LVLLL

Figure 5 - Oil Rate Considering L, LV or LL Equilibrium

Figure 6 shows the Gas Oil Ratio (GOR) behaviour for the three different conditions when the system reaches thermodynamic liquid-liquid equilibrium. When the lighter hydrocarbon liquid is considered like a vapour phase, LV case, it has a higher relative permeability than the liquid phase and so travels faster than the liquid. The gas reaches the production well in a shorter time, increasing the GOR.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995

GO

R (s

cf/b

bl)

LVLLL

Figure 6 - Gas Oil Ratio Considering Multiphase Thermodynamic L, LV, and LL Equilibrium

SPE 148040 7

When both liquid hydrocarbons are considered like a single phase, L case, the lighter hydrocarbon is a gas when produced, but while flowing into the reservoir it has a low velocity, because it is travelling as a single phase. The gas takes longer to reach the production well, decreasing the GOR. Figure 7 reveals the total oil production (Np) and Figure 8 the gas production rate (Qg) for the three different conditions when the system reaches the thermodynamic liquid-liquid equilibrium.

0

2

4

6

8

10

12

14

16

18

1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995

NP

(106 b

bl)

LVLLL

Figure 7 - Cumulative Oil Production Considering Multiphase Thermodynamic L, LV, and LL Equilibrium

0

1

2

3

4

5

6

1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995

Qg

(106 s

cf/d

)

LVLLL

Figure 8 - Gas Rate Production Considering Multiphase Thermodynamic L, LV, and LL Equilibrium

6. Conclusions

- A novel procedure has been proposed to overcome instabilities which may arise in calculation of multiphase liquid-liquid-vapour (LLV) hydrocarbon phase equilibrium; - A new procedure has been developed for representing the thermodynamic liquid-liquid hydrocarbon equilibrium in a compositional simulator that represents the real behaviour of the fluid flux as a more rigorous alternative to the lumping of the two liquid phases to be represented as an average single liquid phase when applicable;

References

Fong, W.S., Sheffield, J.M., Ehrlich, R., and Emanuel, A.S. 1992: Phase Behavior Modeling Techniques for Low-Temperature CO2 Applied to McElroy and North Ward Estes Projects. Paper SPE 24194, Proceedings of SPE/DOE Eight Symposium on Enhanced Oil Recovery held in Tulsa, Oklahoma, Apr. 22-24. Killough, J.E. and Kossack, C.A. 1987: Fifth Comparative Solution Project: Evaluation of Miscible Flood Simulators. Paper SPE 16000, proceedings to ninth SPE Symposium on Reservoir Simulation held in San Antonio, Texas, February 1-4. Larson, L.L., Silva, M.K., Taylor, M.A., and Orr Jr., F.M. 1989: Temperature Dependence of L1/L2/V Behavior in CO2/Hydrocarbon Systems. Paper SPE15399. SPE Reservoir Engineering, February. 105:114. Nghiem, L.X. and Li, Y.K. 1984: Computation of Multiphase Equilibrium Phenomena with an Equation of State. Fluid Phase Equilibria, Amsterdam. 17: 77-95.

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Nghiem, L.X. and Li, Y.K. 1986: Effect of Phase Behavior on CO2 Displacement Efficiency at Low Temperatures: Model Studies With an Equation of State. Paper SPE 13116, SPERE, July. 414-422. Nghiem, L.X. and Li, Y.K. 1988: Phase-Equilibrium Calculations for Reservoir Engineering and Compositional Simulation. Proceedings for First International Forum on Reservoir Simulation held in Alpbach, Austria, Sept. 12-16. Orr Jr., F.M. and Jensen, C.M. 1984: Interpretation of Pressure-Composition Phase Diagrams for CO2/Crude Oil Systems. Paper SPE11125, SPE Journal, Oct. 485-497. Orr Jr., F.M., Yu. A.D., and Lien, C.L. 1981: Phase Behaviour of CO2 and Crude Oil in Low-Temperature Reservoirs. Paper SPE08813, SPE Journal, May. 480-492. WinProp User's Guide - Version 2010, CMG-Computer Modeling Group Calgary-AB, Canada.