A New Diminishing Interface Method for Determining the Minimum

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A NEW DIMINISHING INTERFACE METHOD FOR DETERMINING THE MINIMUM MISCIBILITY PRESSURES OF LIGHT OILCO2 SYSTEMS IN BULK PHASE AND NANOPORES Kaiqiang Zhang, Na Jia, Fanhua Zeng, and Peng Luo Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b02439 • Publication Date (Web): 16 Oct 2017 Downloaded from http://pubs.acs.org on October 17, 2017

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Measured IFT Predicted IFT Interfacial thickness Slope of interfacial thickness

15

2

1.0

1

0.5

0

5

0.0

-1

-0.5

-2

-1.0

MMP = 22.1 MPa 0 0

5

10

15

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25

P (MPa)

ACS Paragon Plus Environment

30

d /dP

10

(nm)

2

(mJ/m )

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A NEW DIMINISHING INTERFACE METHOD FOR DETERMINING THE MINIMUM MISCIBILITY PRESSURES OF LIGHT OIL−CO2 SYSTEMS IN BULK PHASE AND NANOPORES

by Kaiqiang Zhang1, Na Jia1*, Fanhua Zeng1, and Peng Luo2

1

Petroleum Technology Research Centre (PTRC) Petroleum Systems Engineering Faculty of Engineering and Applied Science University of Regina Regina, Saskatchewan S4S 0A2 Canada

2

Saskatchewan Research Council (SRC) Regina, Saskatchewan S4S 7J7 Canada

*Corresponding author. Tel: 1-306-337-3287; Fax: 1-306-585-4855. E-mail: [email protected] (N. Jia)


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ABSTRACT In this paper, a new interfacial thickness-based method, namely, the diminishing interface method (DIM), is developed to determine the minimum miscibility pressures (MMPs) of light oil−CO2 systems in bulk phase and nanopores. First, a Peng−Robinson equation of state (PR-EOS) is modified to calculate the vapour−liquid equilibrium in nanopores by considering the effects of capillary pressure and shifts of critical temperature and pressure. Second, the parachor model is coupled with the modified PREOS to predict the interfacial tensions (IFTs) in bulk phase and nanopores. Third, a formula of the interfacial thickness between two mutually soluble phases is derived, based on which the novel DIM is developed by considering two-way mass transfer across the interface. The MMP is determined by extrapolating the derivative of the interfacial thickness with respect to the pressure (∂δ / ∂P) T to zero. It is found that the modified PREOS coupled with the parachor model is accurate for predicting the phase behaviour and IFTs in bulk phase and nanopores. More specifically, in nanopores, the lighter components prefer to be in vapour phase by increasing the temperature or decreasing the pressure and the IFTs are decreased with the pore radius, especially at low pressures. The determined MMPs of 12.4, 15.0, and 22.1 MPa from the DIM agree well with the laboratory measured results for the three Pembina light oil−CO2 systems in bulk phase at Tres = 53.0°C. Moreover, the MMPs of the Pembina and Bakken live oil−pure CO2 systems in the nanopores of 100, 20, 4 nm are determined from the DIM, which tend to be decreased at a smaller pore level. Physically, the interface between the light oil and CO2 diminishes and the two-phase compositional change reaches its maximum at the determined MMP from the DIM.

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Keywords: Interfacial thickness; MMP determination; Nanopores; Equation of state; Parachor model; Light oil−CO2 systems. 1. Introduction In the petroleum industry, carbon dioxide (CO2) has long been used as an efficient solvent to displace and extract the remaining oil in an oil reservoir.1 In many oilfield applications worldwide, the miscibility development between the residual oil and injected gas is desired for a gas injection project.2 The minimum miscibility pressure (MMP) is defined as the lowest operating pressure at which the oil and gas phases can become miscible in any portions through a dynamic multi-contact miscibility (MCM) process at the reservoir temperature.3,4 Thus an accurate determination of the MMP for a given oil−CO2 system is required to ensure a miscible CO2 enhanced oil recovery (CO2-EOR) project in an oilfield. A number of theoretical models,5,6 numerical simulations,7 and experimental methods8,9 have been developed to determine the MMPs of various oil−CO2 systems in bulk phase. The experimental methods are considered to be the most accurate and reliable.10‒13 Therein, the slim-tube tests14 and the rising-bubble apparatus15 are the most commonly used experimental methods for measuring the MMPs. More recently, the vanishing interfacial tension (VIT) technique has been developed as an alternative experimental method for measuring the MMPs of various oil−CO2 systems.16 The VIT technique was first proposed for the MMP determination by Rao in 199716 and applied subsequently by several researchers.17,18 In general, the interfacial tensions (IFTs) between the oil and CO2 phases can be measured at different pressures and the reservoir temperature by applying, for example, the axisymmetric drop shape analysis (ADSA) technique for the pendant drop case.19 Alternatively, they can be predicted by using, for instance, the parachor model.20 Then, the VIT technique is applied to determine

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the MMP of the oil−CO2 system by linearly extrapolating the measured/predicted IFT vs. test pressure data to zero IFT. The theoretical basis of the VIT technique is that the miscibility between the oil and CO2 phases is achieved at the zero-IFT condition.21 In practice, some higher IFTs measured/predicted at lower test pressures were always used and extrapolated to zero IFT for the MMP determinations by convention, which was justified through its relatively small difference with the determined MMP from the slimtube tests.22 In addition to the IFT, some other interfacial properties (e.g., interfacial thickness) are also very important for the oil−gas miscibility study. Gibbs established the relations between the IFT and several surface excess quantities by treating the interface as a twodimensional mathematical surface, which is known as the Gibbs dividing surface model (DSM).23 In general, an interface between two bulk phases (e.g., the oil and CO2 phase) is a nonhomogeneous region, whose thickness is called the interfacial thickness. The interfacial thickness is obtained from the molecular theories because it cannot be measured directly.24 In terms of the Gibbs DSM, the partial derivative of the IFT with respective to the pressure between the two bulk phases involved, (∂γ ∂P) T , is related to an increase in the interfacial area in the two-dimensional domain.25 In 1966, (∂γ ∂P) T was proven to represent the interfacial thickness for the first time as it has the dimension of a length.26 Based on the Gibbs DSM, the geometrical relationship between two Gibbs dividing surfaces was examined and (∂γ ∂P) T was equal to the distance between the two zero-mass Gibbs dividing surfaces in the interfacial region.27 At a later time, the interface was considered to be a three-dimensional region in the Hansen’s approach,28 which was a closer physical representation of a real interface. In this approach, an alternative

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convention was developed that two chemical potential terms were eliminated by introducing two Lagrange multipliers. Therefore, (∂γ ∂P) T was also considered as the interfacial thickness and called the Hansen thickness. In surface physics and chemistry, the interfacial thickness was subsequently studied by several researchers,29,30 however, most of them were limited between two immiscible or not highly mutually soluble phases. The presence of nanopores in tight formation and its effect on liquid phase behaviour have been introduced in some studies.31 Cubic equation of state (EOS) is treated as an available and appropriate approach to calculate the vapour−liquid equilibrium (VLE) properties in nanopores.32 However, most existing MMP determination methods are targeted for bulk phase and unsuitable for nanopores. An initial trial for predicting the MMP in nanopores was undertaken by combining the Peng−Robinson EOS (PR-EOS)33 with multiple-mixing cell (MMC) algorithm.34 It is found that the reduction of the MMP is almost 0.90 MPa (130 psi) for a light oil−pure CO2 system and up to 3.45 MPa (500 psi) for a light oil−CO2+CH4 system for a pore radius of 4 nm compared with unconfined pores. Moreover, the confinement effect on the MMP is considered to be marginal if the pore radius is 20 nm or higher.35 Later, another study for MMP prediction in nanopores was conducted by applying the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) EOS and VIT technique.36 The PC-SAFT EOS was associated with the parachor model to predict the IFTs in nanopores, based on which the MMP was estimated by extrapolating to zero IFT. The confinement effect on the MMP is found to be significant when pore radius is smaller than 10 nm. A reduction of 23.5% in MMP is obtained if the pore radius is reduced from infinite to 3 nm. In general, the MMP is decreased under the confinement effect, especially at some extremely small pore levels.

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The MMP reduction is considered to be caused by the effects of capillary pressure and shifts of the critical temperature and pressure.37 Overall, few studies have been found to specifically study the MMP prediction in nanopores. In this study, first, a PR-EOS is modified for the VLE calculations in nanopores by considering the effects of capillary pressure and shifts of the critical temperature and pressure, which is also coupled with the parachor model to predict the IFTs. Second, the interfacial thickness between two mutually soluble phases (e.g., light oil and CO2 phases) is determined by considering the two-way mass transfer, i.e., CO2 dissolution into the oil phase and light hydrocarbons (HCs)-extraction from the oil phase by CO2. Based on the determined interfacial thicknesses, a new technical method, namely, the diminishing interface method (DIM), is proposed and applied to determine the MMPs. Finally, the phase behaviour and MMPs of three liquid−vapour systems, a pure HC system (i.e., iC4−nC4−C8)34 and two live light oil−pure CO2 systems (i.e., Pembina and Bakken live light oil−pure CO2 systems)9,35, are predicted by using the modified PR-EOS and new DIM in bulk phase and nanopores, which are subsequently compared with and validated by the literature results. 2. Experimental 2.1 Materials In Table 1, the detailed compositions of a pure HC system and two live light oil−CO2 systems used in this study are listed. More specifically, a ternary mixture of 4.53 mol.% n-C4H10 + 15.47 mol.% i-C4H10 + 80.00 mol.% C8H1838 is used to be the pure HC system and a Pembina9 and a Bakken35 live light oil together with pure CO2 are the two live light oil−CO2 systems in this study. The properties of the pure HC system and the Bakken live

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oil−CO2 system were introduced in the literature.35,38 Moreover, a Pembina dead light oil sample was collected from the Pembina oilfield, Cardium formation in Alberta (Canada). The gas chromatography (GC) compositional analysis of the cleaned Pembina dead oil was performed and the detailed results can be found elsewhere.9,39 The Pembina live oil with the gas−oil ratio (GOR) of 15:1 sm3/sm3 was reconstituted by saturating the Pembina dead oil sample with the produced HC gas. The actual composition of the produced gas was equal to 66.50 mol.% CH4 + 11.41 mol.% C2H6 + 11.39 mol.% C3H8 + 10.70 mol.% n-C4H10. In addition, a Pembina dead light oil−pure CO2 system and a Pembina dead light oil−impure CO2 system are used for MMP determinations in bulk phase. The detailed experimental setups and procedures for preparing the Pembina live oil sample and the impure CO2 sample were described elsewhere.18,39 2.2 PVT tests A mercury-free pressure−volume−temperature (PVT) system (PVT-0150-100-200316-155, DBR, Canada) was used to measure the PVT data of the Pembina dead light oil−CO2 system with four different CO2 concentrations at Tres = 53.0°C.9 The measured PVT data are summarized in Table 2. The experimental setup and procedure of the PVT tests were described previously.9,39 It is found that the experimentally measured saturation pressure, oil density, and oil-swelling factor (SF) increase with CO2 concentration due to the CO2 dissolution. In this work, the measured Pembina oil PVT data were used to tune the modified Peng−Robinson EOS (PR-EOS) for bulk phase calculations. Moreover, the PVT data for the iC4−nC4−C8 system from the literature38 are summarized and listed in Table 3. 2.3 IFT tests

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The IFTs between the Pembina light oil and CO2 are measured by applying the ADSA technique for the pendant drop case, which was described in detail elsewhere.40 The high-pressure IFT cell (IFT-10, Temco, USA) is rated to 69.0 MPa and 177.0°C. The IFT-cell volume is 49.5 cm3. The ADSA program requires the density difference between the oil drop and the CO2 phase at the test conditions and the local gravitational acceleration as the input data. The dead/live oil sample densities at different test conditions were measured experimentally by using a densitometer (DMA512P, Anton Paar, USA).39 The CO2 density was predicted by using the CMG WinProp module (Version 2016.10, Computer Modelling Group Limited, Canada) under the same test conditions. Three respective series of the IFT tests for the Pembina dead light oil−pure CO2 system, live light oil−pure CO2 system, and dead light oil−impure CO2 system were conducted at Tres = 53.0°C.9,40 The detailed experimental data of these three series of the IFT tests are listed in Table 4. 3. Theory 3.1 Modified equation of state In this study, a modified PR-EOS is proposed to calculate the VLE properties in nanopores.41 More specifically, first, the PR-EOS29 is shown as follows:

P=

RT a − v − b v (v + b ) + b (v − b )

where 0.45724 R 2Tc2 a= α (T ) Pc b=

0.0778RTc Pc

8


(1)

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α (T ) = [1 + m (1 − Tr )] 2 m = 0.37464 + 1.54226ω − 0.26992ω 2

where P is the system pressure; R is the universal gas constant; T is the temperature; a and b are EOS constants; v is the molar volume; Tc is the critical temperature in bulk phase; Pc is the critical pressure in bulk phase; Tr is the reduced temperature; and ω is the acentric factor. The shifts of critical properties (i.e., critical temperature and pressure) of the confined fluids are considered to occur in nanopores, which is specifically studied in the previous study42 and related with the ratio of the Lennard-Jones size diameter (σ LJ ) and the pore radius as follows,

Tcp = Tc − Tc [0.9409

Pcp = Pc − Pc [0.9409

where σ LJ= 0.2443

σ LJ rp

σ LJ rp

− 0.2415(

− 0.2415(

σ LJ rp

σ LJ rp

)2 ]

(2)

)2 ]

(3)

Tc ; Tcp is the critical temperature in nanopores; Pcp is the critical Pc

pressure in nanopores; r p is the pore radius. The initial K-value of each component can be estimated from Wilson’s equation,43

Ki =

Pci T exp[5.37(1 + ωi )(1 − ci )] P T

(4)

where Pci is the critical pressure of component i; Tci is the critical temperature of component i; ωi is the acentric factor of component i. Then the Rachford−Rice equation is applied to calculate x i and yi ,

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N

z i ( K i − 1) =0 i − 1) β

∑ 1 + (K i =1

(5)

where β is the vapour fraction. The compressibility of the liquid or vapour phase can be determined,

Z L3 − (1 − BL ) Z L2 + ( AL − 3BL2 − 2 BL ) Z L − ( AL BL − BL2 − BL3 ) = 0

(6a)

Z V3 − (1 − BV ) Z V2 + ( AV − 3BV2 − 2 BV ) Z V − ( AV BV − BV2 − BV3 ) = 0

(6b)

where Z L and ZV are the respective compressibility factors of the liquid and vapour phases; AL =

aP bP aPL bP , BL = L , AV = 2 V 2 , BV = V . Constants of a and b are 2 2 RT RT RT RT

obtained by applying the van der Waals mixing rule,

a = ∑∑ xi x j aij

(7a)

b = ∑ xi bi

(7b)

i

j

i

where a ij is the binary interaction of component i and component j , aij = (1 − k ij ) ai a j ; k ij is the binary interaction coefficient of component i and component j ; k ij = k ji and k ii = k jj = 0 . Minimum Gibbs free energy is applied to select roots of the compressibility

factors for the liquid and vapour phases.44 The liquid and vapour phases are assumed to be the wetting phase and non-wetting phase, respectively.45 Thus the capillary pressure ( Pcap ) is, Pcap = PV − PL

(8)

where PV is the pressure of the vapour phase and PL is the pressure of the liquid phase. On the other hand, the capillary pressure can be expressed by Young−Laplace equation,

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Pcap =

2γ cosθ rp

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(9)

where γ is the interfacial tension and θ is the contact angle of the vapour−liquid interface with respect to the pore surface, which is assumed to be 30° according to the experimental results in the literature.38 Therein, the IFT is estimated by means of the Macleod−Sugden equation, which will be specifically introduced in 3.2. The fugacity coefficient of a mixture is, b a b AL Z + (1 + 2 ) B L ln φ iL = ( i ) L ( Z L − 1) − ln( Z L − B L ) − [ 2( i ) 0L.5 − ( i ) L ] ln( L ) b a b 2 2 BL Z L + (1 − 2 ) B L

(10a) b AV a b Z + (1 + 2 ) BV ln φiV = ( i ) V ( Z V − 1) − ln( Z V − BV ) − [ 2( i )V0.5 − ( i ) V ] ln( V ) b a b 2 2 BV Z V + (1 − 2 ) BV

(10b) The VLE calculations based on the modified PR-EOS require a series of iterative computation through, for example, the Newton−Raphson method. Fig. 1 shows the flowchart of the VLE calculation process. 3.2 Parachor model The parachor model is most commonly used by the petroleum industry to predict the IFT of a liquid−vapour (e.g., oil−CO2) system.46 Macleod20 first related the surface tension between the liquid and vapour phases of a pure component to its parachor and the molar density difference between the two phases,

σ = [ p(ρ L − ρ V )]4 where σ is the surface tension in mJ/m2; p is the parachor.

11


(11)

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At a later time, Eq. (11) was extended to a multi-component mixture, which is the socalled Macleod−Sugden equation,47 r

r

i =1

i =1

γ = (ρ L ∑ x i p i − ρ V ∑ y i p i ) 4

(12)

where γ is the IFT between the vapour and liquid bulk phases of the multi-component mixture; xi and yi are the respective mole percentages of the ith component in the liquid and vapour bulk phases, i = 1, 2, …, r; r is the component number in the mixture; and pi is the parachor of the ith component. Since ρ L =

P MWV PL MWL , and ρ V = V , Eq. (12) is Z L RT Z V RT

rewritten as,

γ =(

PL MWL Z L RT

r

∑x p i

i =1

i

−

PV MWV Z V RT

r

∑y p ) i

4

i

(13)

i =1

where MWL is the molecular weight of the liquid phase and MWV is the molecular weight of the vapour phase. The parachor model in this study can be used to calculate the IFTs of light oil‒CO2 systems in nanopores because it is coupled with the modified PREOS, some variables of which have considered the effects of capillary pressure and shifts of critical temperature and pressure. 3.3 Derivation of interfacial thickness In this study, a formula for determining the interfacial thickness between two mutually soluble phases (e.g., oil and CO2 phases) is derived by taking account of the two-way mass transfer. Suppose that a closed system, as shown in Fig. 2, consists of two mutually soluble phases ( α and β ), each of which has two components (1 and 2), the Gibbs free energy as an interfacial excess quantity is given by,48

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G( p, T ) = γA + µ1 N1 + µ 2 N 2 = U + PV − TS

(14)

where A is the surface area of the interface, µ is the chemical potential, N j is the mole number of the j th component, U is the internal energy, V is the volume, S is the entropy. Two-way mass transfer occurs so that there is not only internal energy (U) change but also external potential energy (Y) change,

(U α + Y α ) + (U β + Y β ) = TS − PV + γA + ( µ1α N1α + µ1β N1β ) + ( µ 2β N 2β + µ 2α N 2α ) (15) Legendre transforms of the internal energy plus external potential energy gives,

d [(U α + Y α ) + (U β + Y β )] = TdS − PdV + γdA + ( µ1α dN1α + µ1β dN1β ) + ( µ 2β dN 2β + µ 2α dN 2α ) (16) α β Given the fact that ( N 1α + N 1β ) and ( N 2 + N 2 ) are constant because of mass

conservation, Eq. (16) is rewritten as,

d[(U α + Y α ) + (U β + Y β )] = TdS − PdV + γdA + (µ1α − µ1β )dN1α + (µ 2β − µ 2α )dN2β

(17)

In Eq. (17), [( µ1α − µ1β ) dN 1α + ( µ 2β − µ 2α ) dN 2β ] is referred to as the chemical potential changes due to the interfacial mass transfer. Physically, [(µ1α − µ1β )dN1α ] or [( µ 2β − µ 2α ) dN 2β ] represents the change of the internal energy and external potential

energy for each component. Then, subtracting differentiation of (γA + µ1 N1 + µ 2 N 2 ) from the full differentiation of [(U + Y ) + PV − TS ] yields,

− SdT + VdP − N1 dµ1 − N 2 dµ 2 = Adγ

(18)

The Gibbs-Duhem equation for each phase can be written,30 − S α dT + V α dP α + N1α dµ1 + N 2α dµ 2 = 0

13


Phase α

(19a)

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− S β dT + V β dP β + N 2β dµ 2 + N1β dµ1 = 0

Phase β

(19b)

Two undetermined Lagrange multipliers, λα and λβ , are introduced and applied into Eqs. (19a) and (19b). Afterwards, Eq. (18) is used to subtract them, which is then divided by interfacial area A, dγ =

( N − λα N1α − λβ N 1β ) − ( S − λα S α − λβ S β ) (V − λα V α − λβ V β ) dT + dP − 1 d µ1 A A A ( N 2 − λα N 2α − λβ N 2β ) − dµ 2 (20) A

Let, s = ( S − λα S α − λ β S β ) / A

(21a)

δ = (V − λα V α − λ β V β ) / A

(21b)

Γ1 = ( N1 − λα N 1α − λβ N 1β ) / A

(21c)

Γ2 = ( N 2 − λα N 2α − λβ N 2β ) / A

(21d)

Then, Eq. (20) can be represented as,

dγ = − s dT + δdP − Γ1 dµ1 − Γ2 dµ 2

(22)

Theoretically, two Lagrange multipliers λα and λβ could be determined by setting any two of the interfacial excess quantities (i.e., s , δ , Γ1 , Γ2 ) to be zero, the choices of which are purely conventional.49 At a given P and T, the chemical potentials cannot be determined and should be eliminated as independent variables so that Γ1 and Γ2 are set to be zero, dγ = − s dT + δdP

T remains constant in the study, i.e., dT = 0 , so,

14


(23)

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δ =(

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∂γ )T ∂P

(24)

In Eq. (24), δ is the distance between two miscible phases, which is also denoted as the interfacial thickness. In addition to the above derivations, there is another series of derivations for the interfacial thickness based on the Gibbs convention is proposed. First, dividing Eqs. (19a) and (19b) by volume V α and V β at the constant temperature, − dP α = n1α dµ1 + n 2α dµ 2

Phase α

(25a)

− dP β = n2β dµ2 + n1β dµ1

Phase β

(25b)

where n1α and n1β are the molar concentrations of the first component in α and β phases, respectively; n2α and n2β are the molar concentrations of the second component in α and

β phases, respectively. Gibbs convention at a constant temperature states,23 −(

i ∂µ ∂γ ) T = ∑ Γi ( i ) T ∂P ∂P i =1

(26)

Assuming the location of Γ1 = 0 to be the reference surface,

N1 = z1n1β + ( H − z1 )n1α

(27)

where z1 is the surface location and H is the total height. For Component 1, with respect to the surface at z 2 and a random location b,

N1 = z 2 n1β + ( H − z 2 )n1α + Γ1( z2 )

= bn1β + ( H − b)n1α + Γ1(b ) Similar equations can be obtained for substance 2.

15


(28)

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Eqs.

(27)

and

(28)

are

subtracted

to

obtain

Γ1(b ) = δ (n1α − n1β ). Similarly,

Γ2(b ) = δ (n2α − n2β ). Given the assumption of Pα = P β = P, Eqs. (25a) and (25b) are rearranged to be, ∂µ1 n2α − n2β ( )= α β ∂P n1 n2 − n2α n1β

(29a)

∂µ 2 n β − n1α ) = α 1β ∂P n1 n2 − n2α n1β

(29b)

(

Then, Eqs. (29a) and (29b) are combined into Eq. (26),

Since δ =

(

n α − n2β ∂γ ) = −Γ1( z2 ) ( α 2β ) ∂P n1 n2 − n2α n1β

(30a)

(

n β − n1α ∂γ ) = −Γ2( z1 ) ( α 1β ) ∂P n1 n2 − n2α n1β

(30b)

Γib − Γid , b or d means any location in the system. Thus the interfacial ∆ ni

thickness is,

δ = −(

∂γ  n1α n2β − n2α n1β  )T   ∂P  (n1β − n1α )(n2α − n2β ) 

(31)

In this study, phase α is the vapour phase, phase β is the liquid phase, substance 1 is V L L  nCO  noil − nCO nV 2 oil ≈ −1 . CO2, and substance 2 is oil. For a light oil−CO2 system,  L 2 V V L   (nCO2 − nCO2 )(noil − noil ) 

Thus Eq. (31) is simplified to be δ = (

∂γ )T , which represents the interfacial thickness ∂P

between two mutually soluble phases as defined in Eq. (24).

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It should be noted that the sign of δ is determined by the characteristics of the two bulk phases. More specifically, if the two phases are barely mutually soluble and repulsive intermolecular interaction dominates in the interfacial region, δ > 0 . If the two phases are mutually soluble and two-way mass transfer occurs across the interface,

δ < 0. In this study, the interfacial tension of the light oil−CO2 system is decreased with the pressure so that δ is negative. Although the sign of δ can be positive, zero, or negative, the physical interfacial thickness has to be positive.49

4. Results and discussion 4.1 Phase behaviour in bulk phase and nanopores The bulk phase PVT tests with four different CO2 concentrations of 0.00, 35.90, 42.70, and 51.70 mol.% in the Pembina light oil were conducted at the reservoir temperature of Tres = 53.0°C39 and are summarized in Table 2. It is found that the measured saturation pressure, oil density, and oil-swelling factor (SF) increase with CO2 concentration. The modified PR-EOS was tuned by using a set of major tuning parameters.50,51 More specifically, the binary interaction coefficient (BIC) between CO2 and C30+, critical pressure and temperature, and acentric factor of C30+ were adjusted to match the measured PVT data. The final predicted saturation pressures, oil densities, and oil-swelling factors are compared with the measured PVT data in Table 2. The predicted data are found to agree well with the measured PVT data since their relative errors are rather small. It should be noted that the above-mentioned tuning parameters are adjusted for one time and applied to predict the phase behaviour of the three liquid‒vapour systems in this study. It is well known that some important interfacial properties (e.g., IFT) between the equilibrated oil and gas phases are as a result of the two-way mass transfer.52 On the 17


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other hand, the results of the two-way mass transfer are always represented by means of the VLE calculations. Hence, it is necessary to study the phase behaviour of a given light oil−CO2 system in order to better analyze and understand the two-way mass transfer and interfacial properties. Figs. 3a−c show x CO , yHCs , and x CO + y HCs vs. pressure data of 2

2

the three Pembina light oil−CO2 systems at Tres = 53.0°C. Here, x CO stands for 2

pure/impure CO2 mole fraction in the liquid (oil) phase due to solvent dissolution and y HCs denotes the extracted HCs mole fraction in the gas phase. Physically, x CO + y HCs 2

represents the overall or total compositional change that is attributed to the two-way mass transfer. In this study, x CO + y HCs is referred to as the two-way mass transfer index 2

(MTI) for brevity, which changes in the range of 0 to 1. The MTI is equal to zero if the two phases are completely insoluble. It is equal to unity if the two phases are miscible or if one phase (gas) is completely dissolved into the other phase (liquid), i.e., the complete dissolution. It is found from Figs. 3a−c that x CO is quickly increased with the pressure up to a 2

certain pressure threshold, above which x CO is increased gradually and finally reaches its 2

maximum. The gaseous CO2 is much more easily dissolved into the oil phase than the liquid CO2.53 On the other hand, y HCs is slightly decreased at a lower pressure, which is followed by an obvious increase starting from the threshold pressure, at which a strong HCs-extraction occurs because the liquid CO2 has a stronger extraction ability than the gaseous CO2. At an even higher pressure, y HCs increases much more gradually because most light to intermediate HCs in the oil phase have already been extracted.

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In addition, the two-way MTI ( x CO + y HCs ) has a similar trend to x CO because x CO 2

2

2

represents over 90% of the total compositional change and becomes a dominant component in the liquid phase at a high pressure. Hence, three pressure ranges can be defined and determined in terms of the two-way MTI by using two dividing or threshold A B pressures, P and P , which are marked in Figs. 3a−c. It is found from Eq. (13) that the

predicted IFT from the parachor model largely depends on the molecular weight ( MWV ) and compressibility ( Z V ) of the vapour phase. Fig. 4 shows the forward finite difference approximation of the partial derivative, ∂ ( MW V / Z V ) ∂P , for each Pembina light A oil−CO2 system. Three respective threshold pressures of P = 10.8, 11.3, and 13.5 MPa

are determined for the three Pembina light oil−CO2 systems and marked in Fig. 4, where A each partial derivative reaches its maximum. Since y HCs is sufficiently small at P ≤ P ,

x CO 2 + y HCs also reaches its maximum at P

A

because a quick CO2 dissolution is

completed. On the other hand, the incremental two-way MTI x CO + y HCs change per 2

incremental pressure is reduced to and remains under 1 mol.%/MPa at an even higher B threshold pressure ( P = P ), above which the two-way MTI remains almost the same

and the two-way mass transfer is almost completed. In this study, three respective B threshold pressures of P = 13, 15, and 23 MPa are determined by using the 1

mol.%/MPa criterion of the two-way MTI change for the three light oil−CO2 systems and marked in Figs. 3a−c, respectively. In micro- and nano-channels, the vapour and liquid compositions before and after flash calculations for the iC4−nC4−C8 system at two different conditions (i.e., constant 19


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pressure and constant temperature) are obtained from literature34 and summarized in Tables 3a and b. Some detailed analyses can be found in the previous study.38 Most importantly, it is found that the lighter components (i.e., iC4 and nC4) prefer to be in the vapour phase by increasing the temperature or decreasing the pressure. The predicted compositions and fractions of the liquid and vapour phases as well as IFTs are listed in Tables 3a and 3b, which agree well with the literature results. The predicted capillary pressure either in micro-channel (10 µm) or nano-channel (100 nm) is slightly lower than the literature data for both cases. In a similar manner with the literature, the predicted capillary pressure in the nano-channel is almost two orders of magnitude higher than that in the micro-channel. Besides, the modified PR-EOS is also applied to calculate the bubble-point and dew-point pressures of the Bakken live light oil−pure CO2 system at pore radius of 10 and 3 nm, which are also in a good agreement with the literature figure.35 The comparison figure cannot be presented because no precise data but a figure was given in the literature. Overall, the modified PR-EOS in this study is capable of predicting the phase behaviour of the pure HC system and/or light oil−CO2 systems in bulk phase and/or nanopores, whose results agree well with the literature data. 4.2 In bulk phase 4.2.1 IFT predictions In this study, the measured18,40 and predicted IFTs are plotted in Figs. 5a−c for the Pembina dead light oil−pure CO2 system, live light oil−pure CO2 system, and dead light oil−impure CO2 system at different pressures and Tres = 53.0°C, respectively. The measured and predicted IFTs match well especially at lower pressures, both of which are quickly reduced with the pressure. When the pressure is higher than 10 MPa, the predicted IFT is lower than the measured IFT. In the IFT tests, the pendant oil drop that 20


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was formed eventually is mainly consisted of relatively heavy paraffinic (i.e., HCs), aromatic, or asphaltic components of the original dead light oil after the initial quick and subsequent slow HCs-extractions by CO2 at a higher pressure.22 Thus the measured IFT at a higher pressure is between the remaining oil phase and the CO2 phase with some extracted light to intermediate HCs. In the EOS modeling, however, the light oil is characterized and represented by using a series of alkanes (i.e., C3−C30+) rather than a combination of the paraffinic, aromatic, or asphaltic molecules. The predicted IFT is between the intermediate to heavy alkanes of the light oil and the CO2 phase with some extracted light to intermediate alkanes at a higher pressure. This is why the predicted IFT is slightly lower than the measured IFT at a higher pressure. Overall, the modified PREOS coupled with the parachor model is proven to be able to accurately predict the IFTs in bulk phase. 4.2.2 MMP determinations In theoretical section, the interfacial thickness (δ ) is defined as the partial derivative of the IFT (γ ) with respect to the pressure (P) at a constant temperature, i.e.,

δ = (∂γ ∂P) T . In this study, the interfacial thickness is obtained by using the forward finite difference approximation (FDA) of the partial derivative of the IFT (γ ) with respect to the pressure (P) at a constant temperature, i.e., δ = (∆γ ∆P) T . The IFTs and interfacial thicknesses between the bulk oil and CO2 phases as well as the FDA of the partial derivative of the interfacial thickness (second derivative of the IFT) with respect to the pressure at a constant temperature, i.e., (∆δ / ∆P) T , for the three Pembina light oil−CO2 systems are plotted in Figs. 5a−c. It is found that the IFT of the each light oil−CO2 system is reduced with the pressure since the dead/live light oil and pure/impure

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CO2 phase are mutually soluble. Thus the sign of δ is negative in this study. However, it should be noted the physical interfacial thickness is always positive even if the sign of δ could be positive, zero, or negative.49 It is seen from Figs. 5a−c that the interfacial thicknesses of the three Pembina light oil−CO2 systems are different, but overall, they are decreased with the pressure. More specifically, the interfacial thickness of the Pembina dead light oil−pure CO2 system is slightly increased initially, then it is quickly decreased and finally tends to be stabilized. It is slightly different in the Pembina live light oil−pure CO2 system that the interfacial thickness is level off at the initial stage. However, the interfacial thickness of the dead light oil−impure CO2 system is continuously decreased with the pressure. It is worthwhile to mention that among the three systems, the interfacial thickness of the dead light oil−pure CO2 system is quickly reduced to be the minimum at a high pressure while that of the dead light oil−impure CO2 system is decreased slowly and tends to be the largest at a high pressure. Furthermore, the inflection point of the derivative of the interfacial thickness with respect to the pressure (∆δ / ∆P) T vs. pressure curve is found to be in a A good agreement with P for each Pembina light oil−CO2 system.

In terms of the DIM, the MMP is determined by linearly regressing and extrapolating the derivative of the interfacial thickness with respect to the pressure (∆δ / ∆P) T vs. pressure data to zero. Physically, (∆δ / ∆P) T = 0 means the interfacial thickness between the oil and CO2 phases becomes constant and does not change with the pressure. Thus it is inferred that a stable interfacial thickness between the oil and CO2 phases rather than a conventional zero-IFT condition is obtained when the miscibility is achieved.

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Mathematically, the linearity of such a linear regression can be represented by the socalled linear correlation coefficient (LCC) or R 2 . More specifically, the LCC of the linear regression of the data points from the highest (∆δ / ∆P) T point at the lowest pressure to any point at an arbitrarily higher pressure is obtained for the MMP determination. In the previous study,18 Rc2 = 0.990 is considered to be a critical value of the LCC criterion. Hence, in this study, the MMPs of the Pembina dead light oil−pure CO2 system, live light oil−pure CO2 system, and dead light oil−impure CO2 system are determined to be 12.4, 15.0, and 22.1 MPa by using the LCC criterion from the DIM at Tres = 53.0°C and shown in Figs. 5a−c. It is seen from Table 5 that the determined MMPs from the DIM for the Pembina dead and live light oil−pure CO2 systems agree well with 12.4−12.9 MPa from the coreflood tests and 15.2−15.4 MPa from the slim-tube tests.9 In the Pembina dead light oil−impure CO2 system, the determined MMP from the DIM is slightly lower than that from the RBA tests but similar to that from the VIT technique. It is inferred that the determined MMP may be overestimated by means of the RBA tests.54 As mentioned B above, P is determined to be 13, 15, and 23 MPa by using the 1 mol.%/MPa criterion,

at which the two-phase compositional change is considered to reach its maximum. It is found that the determined MMPs of the three light oil−CO2 systems from the DIM in B Figs. 5a−c are in good agreement with P in Figs. 3a−c. Thus the DIM is concluded to

be physically meaningful for determining the MMP since the two-phase compositional change is found to reach the maximum at the determined MMP from the DIM. In

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summary, the DIM is proven to be accurate and in a physically meaningful way for determining the MMPs in bulk phase. 4.3 In nanopores 4.3.1 IFT predictions The modified PR-EOS coupled with the parachor model is applied to predict the IFTs of the Bakken live light oil−pure CO2 system at different pressures and Tres = 116.1°C in nanopores, which are summarized, plotted and compared with the predicted data from the literature35 in Fig. 6a. The predicted IFTs in this study agree well with the recorded IFTs from the literature, both of which always decrease with pressure increases. At a constant pressure, it is found that the confinement effect is negligible and the predicted IFTs remain almost unchanged when the pore radius is larger than 100 nm. Once the pore radius is smaller than 100 nm, the predicted IFT is decreased with a reduction of pore radius. This is because the pressure of the vapour phase is increased with the addition of the capillary pressure (note that the vapour phase is assumed to be the non-wetting phase), which leads the density of the vapour phase (i.e., second term of the Macleod−Sugden equation) to increase. That is why the predicted IFTs from the parachor model become smaller with a reduction of pore radius/an increase of confinement effect. The predicted IFTs at a lower pressure are also found to decrease more significantly by reducing the pore radius when compared with those at a higher pressure. In a similar manner with the Bakken live light oil−pure CO2 system, the predicted IFTs of the Pembina live light oil−pure CO2 system in the pore radius range of 2 to 1,000,000 nm at three different pressures and Tres = 53.0°C are plotted in Fig. 6b. It should be noted that an obvious IFT drop occurs starting from 1,000 nm at P = 4.0 MPa while starting from 100 nm at P = 7.5

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and 9.5 MPa. Thus it is inferred that the IFT is more sensitive to the variation of the pore radius at a lower pressure. In other word, the confinement effect on the IFT tends to be weaker at a higher pressure. 4.3.2 MMP determinations In Figs. 7a−c, the MMPs of the Pembina live light oil−pure CO2 system in the nanopores with pore radius of 100, 20, and 4 nm are determined to be 15.4, 13.7, and 13.4 MPa by using the LCC criterion from the DIM at Tres = 53.0°C, which are listed and compared with the measured MMPs from the slim-tube tests9 for bulk phase in Table 6. It is found that the MMPs of the Pembina live light oil−pure CO2 system in bulk phase and in nanopore with pore radius of 100 nm are almost same. The MMPs are found to be decreased with a decrease of pore level when the pore radius is smaller than 100 nm. Furthermore, the MMPs of the Bakken live light oil−pure CO2 system in the nanopores with the three same pore radius are determined to be 24.1, 21.4, and 20.6 MPa at Tres = 116.1°C and shown in Figs. 8a−c. On the other hand, the MMP of the Bakken live light oil−pure CO2 system in bulk phase is estimated to be 24.7 MPa by using the multiplemixing cell method in CMG WinProp module and listed in Table 6. Thus it is concluded that the decrease of pore radius (i.e., increase of the confinement effect) lowers the MMPs significantly for the both Pembina and Bakken live light oil−pure CO2 systems. In comparison with the literature results, the determined MMPs of the Bakken live light oil−pure CO2 system from the DIM in this study are higher either in bulk phase or in nanopores. It is inferred that the literature results were underestimated and the determined MMPs from the DIM are more accurate, which is concluded based on the following two reasons: first, the determined MMPs of the Pembina dead/live light oil−pure/impure CO2

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systems from the DIM in bulk phase agree well with the measured MMPs from some laboratory tests, which means the DIM together with the modified PR-EOS is accurate to determine MMPs; second, the calculated MMP of the Bakken live light oil−pure CO2 system in bulk phase is 24.7 MPa, which is comparable to the determined MMPs from the DIM but much higher than the literature data. In addition, it should be noted that the MMPs of the Bakken live light oil−pure CO2 system are much higher than those of the Pembina live light oil−pure CO2 system either in bulk phase or in nanopores. This is because the feed oil and solvent ratio of the former system is 0.50:0.50 by mole while that of the latter system is 0.01:0.99 by mole. It means the amount of CH4 and heavy components from the oil in the former system is much higher than that of the latter system. Besides, the reservoir temperature of the Bakken live light oil−pure CO2 system is 116.1°C, which is much higher than 53.0°C for the Pembina live light oil−pure CO2 system. Last but not least, the determined MMPs from the DIM are found to be more accurate in comparison with the calculated ones from some existing theoretical methods, whose technical details are presented in the Supporting Information.

6. Conclusions The following four major conclusions can be drawn from this work: First, the modified Peng−Robinson equation of state (PR-EOS) is found to be accurate for vapour−liquid equilibrium (VLE) calculations of the Pembina light oil−CO2 system in bulk phase and the iC4−nC4−C8 system and Bakken live light oil−pure CO2 system in nanopores: •

In bulk phase: CO2 dissolution is found to be a dominant mass-transfer process, which accounts for 90% of the total compositional change. Moreover, three

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A B pressure ranges, which are divided by P and P , are identified and explained in

the two-phase compositions vs. pressure curves. •

In nanopores: the lighter components are found to prefer to be in the vapour phase by increasing the temperature or decreasing the pressure. Furthermore, the predicted capillary pressure in the nano-channel (100 nm) is almost two orders of magnitude higher than that in the micro-channel (10 µm).

Second, the parachor model coupled with the modified PR-EOS is proven to be accurate for predicting the interfacial tensions (IFTs) in bulk phase and nanopores: •

In bulk phase: the predicted IFTs of the three Pembina light oil−CO2 systems are found to agree well with the measured IFTs at Tres = 53.0°C. The predicted IFT is slightly lower at a relatively higher pressure, which is because the light oil is not completely and accurately characterized in the EOS modeling. For example, no heavy aromatic or asphaltic components are considered.

•

In nanopores: the predicted IFTs of the Bakken live light oil−pure CO2 system agree well with the recorded IFTs from the literature in the pore radius range of 4−1,000 nm at Tres = 116.1°C. It is found that the IFT remains constant but decreases with the pore radius from 100 nm for Bakken oil case and from 1,000 nm for Pembina oil case. Moreover, the IFT is more sensitive to the variation of the pore radius at a lower pressure for both two systems.

Third, a formula for the interfacial thickness between two mutually soluble phases (e.g., oil and CO2), i.e., δ = (

∂γ )T , is developed by considering the two-way mass ∂P

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transfer, i.e., CO2 dissolution into the oil through the convective dispersion and molecular diffusion and hydrocarbons (HCs)-extraction from the oil phase by CO2. Fourth, a new interfacial thickness-based method, the diminishing interface method (DIM), is developed and applied to determine the minimum miscibility pressures (MMPs) of different light oil−CO2 systems in bulk phase and nanopores by extrapolating the derivative of the interfacial thickness with respect to the pressure (∆δ / ∆P) T to zero: •

In bulk phase: the MMPs of the Pembina dead light oil−pure CO2 system, live light oil−pure CO2 system, and dead light oil−impure CO2 system at Tres = 53.0°C are determined to be 12.4, 15.0, and 22.1 MPa by using the linear correlation coefficient (LCC) criterion from the DIM, which agree well with 12.4−12.9 MPa from the coreflood tests, 15.2−15.4 MPa from the slim-tube tests, and 23.4−23.5 MPa from the rising-bubble apparatus (RBA) tests, respectively. The determined B MMP from the DIM is found to be in good agreement with P . Thus the

determined MMP from the DIM is proven to be physically meaningful, at which not only the interfacial thickness between the light oil and CO2 phases tends to be minimum and stabilized with the pressure but the two-phase compositional change also reaches its maximum. •

In nanopores: the MMPs of the Pembina live light oil−pure CO2 system in the nanopores with pore radius of 100, 20, and 4 nm are determined to be 15.4, 13.7, and 13.4 MPa by using the LCC criterion from the DIM at Tres = 53.0°C. In addition, the MMP of the Bakken live light oil−pure CO2 system in bulk phase Tres = 116.1°C is estimated to be 24.7 MPa from multiple-mixing cell method and the MMPs in the nanopores with pore radius of 100, 20, and 4 nm are determined 28


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to be 24.1, 21.4, and 20.6 MPa, respectively. In comparison with the measured MMPs in bulk phase, the MMPs are found to be decreased with a decrease of pore level when the pore radius is smaller than 100 nm.

Acknowledgements The authors would like to acknowledge the Petroleum Systems Engineering at the Faculty of Engineering and Applied Science of the University of Regina. They also want to thank some technical discussions with Dr. Yongan Gu from the University of Regina.

References [1] Jarrell, P. M.; Fox, C. E.; Stein, M. H.; Webb, S. L. Practical Aspects of CO2 Flooding. SPE Monograph Series, Vol. 22. Richardson, TX, SPE, 2002. [2] Trivedi, J. J.; Babadagli, T. Oil Recovery and Sequestration Potential of Naturally Fractured Reservoirs during CO2 Injection. Energy Fuels 2009, 23 (8), 4025‒4036. [3] Elsharkawy, A. M.; Poettmann, F. H.; Christiansen, R. L. Measuring CO2 Minimum Miscibility Pressures: Slim-Tube or Rising-Bubble Apparatus? Energy Fuels

1996, 10 (2), 443‒449. [4] Ahmadi, M. A.; Zahedzadeh, M.; Shadizadeh, S. R.; Abbassi, R. Connectionist Model for Predicting Minimum Gas Miscibility Pressure: Application to Gas Injection Process. Fuel 2015, 148 (5), 202−211. [5] Haghtalab, A.; Moghaddam, A. K. Prediction of Minimum Miscibility Pressure Using the UNIFAC Group Contribution Activity Coefficient Model and the LCVM Mixing Rule. Ind. Eng. Chem. Res. 2016, 55 (10), 2840−2851.

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[6] Shokrollahi, A.; Arabloo, M.; Gharagheizi, F.; Mohammadi, A. H. Intelligent Model for Prediction of CO2‒Reservoir Oil Minimum Miscibility Pressure. Fuel 2013, 112 (10), 375‒384. [7] Høier, L.; Whitson, C. H. Miscibility Variation in Compositionally Grading Reservoirs. SPE Res. Eval. Eng. 2001, 4 (1), 36−43. [8] Hawthorne, S. B.; Miller, D. J.; Jin, L.; Gorecki, C. D. Rapid and Simple CapillaryRise/Vanishing Interfacial Tension Method to Determine Crude Oil Minimum Miscibility Pressure: Pure and Mixed CO2, Methane, and Ethane. Energy Fuels

2016, 30 (8), 6365‒6372. [9] Zhang, K.; Gu, Y. Two Different Technical Criteria for Determining the Minimum Miscibility Pressures (MMPs) from the Slim-Tube and Coreflood Tests. Fuel 2015, 161 (12), 146‒156. [10] Czarnota, R.; Janiga, D.; Stopa, J.; Wojnarowski, P. Determination of Minimum Miscibility Pressure for CO2 and Oil System Using Acoustically Monitored Separator. J. CO2 Util. 2017, 17 (1), 32‒36. [11] Czarnota, R.; Janiga, D.; Stopa, J.; Wojnarowski, P.; Kosowski, P. Minimum Miscibility Pressure Measurement for CO2 and Oil Using Pressure Increase Method, J. CO2 Util. 2017, 21 (10), 156‒161. [12] Liu, Y.; Jiang, L.; Tang, L.; Song, Y.; Zhao, J.; Zhang, Y.; Wang, D.; Yang, M. Minimum Miscibility Pressure Estimation for a CO2/n-Decane System in Porous Media by X-Ray CT. Exp. Fluids 2015, 56 (7), 154‒162. [13] Liu, Y.; Jiang, L.; Song, L.; Zhao, Y.; Zhang, Y.; Wang, D. Estimation of Minimum Miscibility Pressure (MMP) of CO2 and Liquid n-Alkane Systems Using an Improved MRI Technique. Magn. Reson. Imaging. 2016, 34 (2), 97‒104.

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[14] Yellig, W. F.; Metcalfe, R. S. Determination and Prediction of CO2 Minimum Miscibility Pressures. J. Pet. Technol. 1980, 32 (1), 160−168. [15] Christiansen, R. L.; Haines, H. K. Rapid Measurement of Minimum Miscibility Pressure with the Rising-Bubble Apparatus. SPE Res. Eng. 1987, 2 (4), 523‒527. [16] Rao, D. N. A New Technique of Vanishing Interfacial Tension for Miscibility Determination. Fluid Phase Equilibr. 1997, 139 (1−2), 311−324. [17] Naseri, A.; GhareSheikhloo, A. A.; Kamari, A.; Hemmati‒Sarapardeh, A.; Mohammadi, A. H. Experimental Measurement of Equilibrium Interfacial Tension of Enriched Miscible Gas‒Crude Oil Systems. J Mol. Liq. 2015, 211 (11), 63‒70. [18] Zhang, K.; Gu, Y. Two New Quantitative Technical Criteria for Determining the Minimum Miscibility Pressures (MMPs) from the Vanishing Interfacial Tension (VIT) Technique. Fuel 2016, 184 (11), 136−144. [19] Rio, O. I.; Neumann, A. W. Axisymmetric Drop Shape Analysis: Computational Methods for the Measurement of Interfacial Properties from the Shape and Dimensions of Pendant and Sessile Drops. J. Colloid Interface Sci. 1997, 196 (2), 136‒147. [20] Macleod, D. B. On a Relation between Surface Tension and Density. Trans Faraday Soc 1923, 19 (2), 38−42. [21] Rao, D. N.; Lee, J. I. Application of the New Vanishing Interfacial Tension Technique to Evaluate Miscibility Conditions for the Terra Nova Offshore Project. J. Pet. Sci. Eng. 2002, 35 (1), 247−262. [22] Escrochi, M.; Mehranbod, N.; Ayatollahi, S. The Gas−Oil Interfacial Behaviour during Gas Injection into an Asphaltenic Oil Recovery. J. Chem. Eng. Data 2013, 58 (9), 2513−2526. 31


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[23] Gibbs, J. W. The scientific papers of J Willard Gibbs: Vol. 1: Thermodynamics. New York, Dover Publications, 1961. [24] Motomura, K.; Iyota, H.; Aratono, M.; Yamanaka, M.; Matuura, R. Thermodynamic Consideration of the Pressure Dependence of Interfacial Tension. J. Colloid Interface Sci. 1983, 93 (1), 264−269. [25] Motomura, K.; Aratono, M. Geometric Formalism of the Thermodynamics of Adsorption at Interfaces between Two Fluid Phases. Langmuir 1987, 3 (2), 304−306. [26] Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption. New York, John Wiley & Sons. Inc., 1966. [27] Good, R. J. Thermodynamic of Adsorption and Gibbsian Distance Parameters: ІІ. The Pressure Coefficient of Interfacial Tension in Ternary Two- and Three-Phase Systems. J. Colloid Interface Sci. 1982, 85 (1), 128−140. [28] Hansen, R. S. Thermodynamic of Interfaces between Condensed Phases. J. Phys. Chem. 1962, 66 (3), 410−415. [29] Turkevich, L. A.; Mann, J. A. Pressure Dependence of the Interfacial Tension between Fluid Phases. 2. Application to Liquid−Vapour Interfaces and to Interfaces of Amphiphilic Solutions. Langmuir 1990, 6 (2), 457−470. [30] Lyklema, J. Fundamental of Interface and Colloid Science. Volume І: Fundamentals. San Diego, Academic Press Inc., 1991. [31] Dong, X.; Liu, H.; Hou, J.; Wu, K.; Chen, Z. Phase Equilibria of Confined Fluids in Nanopores of Tight and Shale Rocks Considering the Effect of Capillary Pressure and Adsorption Film. Ind. Eng. Chem. Res. 2016, 55 (3), 798−811.

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[32] Ma, Y.; Jin, L.; Jamili, A. Modifying van der Waals Equation of State to Consider Influence of Confinement on Phase Behaviour. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, September 30−October 2, 2013; Paper SPE 166476. [33] Peng, D.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15 (1), 58−64. [34] Ahmadi, K.; Johns, R. T. Multiple-Mixing-Cell for MMP Calculations. SPE J. 2011, 16 (4), 733−742. [35] Teklu, T. W.; Alharthy, N.; Kazemi, H.; Yin, X.; Graves, R. M.; AlSumaiti, A. M. Phase Behavior and Minimum Miscibility Pressure in Nanopores. SPE Res. Eval. Eng. 2014, 17 (3), 396–403. [36] Wang, S.; Ma, M.; Chen, S. Application of PC-SAFT Equation of State for CO2 Minimum Miscibility Pressure Prediction in Nanopores. Presented at the SPE Improved Oil Recovery Conference, Tulsa, OK, April 11−13, 2016; Paper SPE 179535. [37] Travalloni, L.; Castier, M.; Tavares, F. W.; Sandler, S. I. Thermodynamic Modeling of Confined Fluids Using an Extension of the Generalized Van Der Waals Theory. Chem. Eng. Sci. 2010, 65 (10), 3088−3099. [38] Wang, L.; Parsa, E.; Gao, Y.; Ok, J. T.; Neeves, K.; Yin, X.; Ozkan, E. Experimental Study and Modeling of the Effect of Nanoconfinement on Hydrocarbon Phase Behaviour in Unconventional Reservoirs. Presented at the SPE Western North American and Rocky Mountain Joint Regional Meeting, Denver, CO, April 16−18, 2014; Paper SPE 169581.

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[39] Zhang, K. Qualitative and Quantitative Technical Criteria for Determining the Minimum Miscibility Pressures from Four Experimental Methods. MASc. Thesis, University of Regina, Regina, Saskatchewan, Canada, 2016. [40] Gu, Y.; Hou, P.; Luo, W. Effects of Four Important Factors on the Measured Minimum Miscibility Pressure and First-Contact Miscibility Pressure. J. Chem. Eng. Data 2013, 58 (5), 1361−1370. [41] Brusllovsky, A. I. Mathematical Simulation of Phase Behaviour of Natural Multicomponent Systems at High Pressures with an Equation of State. SPE Res. Eng.

1992, 7 (1), 117−122. [42] Zarragoicoechea, G. J.; Kuz, V. A. Critical Shift of a Confined Fluid in a Nanopore. Fluid Phase Equilib. 2004, 220 (1), 7‒9. [43] Wilson, G. M. Vapour−Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86 (2), 127−130. [44] Whitson, C. H.; Brule, M. R. Phase Behaviour. SPE Monograph Series, Vol. 20, Richardson, TX, SPE, 2000. [45] Nojabaei, B.; Johns, R. T.; Chu, L. Effect of Capillary Pressure on Phase Behavior in Tight Rocks and Shales. SPE Res. Eval. Eng. 2013, 16 (3), 281−289. [46] Nobakht, M.; Moghadam, S.; Gu, Y. Determination of CO2 Minimum Miscibility Pressure from Measured and Predicted Equilibrium Interfacial Tensions. Ind. Eng. Chem. Res. 2008, 47 (22), 8918−8925. [47] Sugden, S. The Variation of Surface Tension with Temperature and Some Related Functions. J. Chem. Soc. Trans. 1924, 125, 32−41. [48] Guggenheim, E. A. Thermodynamics−An Advanced Treatment for Chemists and Physicists. North-Holland, Amsterdam, 1985. 34


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[49] Yang, C.; Li, D. A Method of Determining the Thickness of Liquid−Liquid Interfaces. Colloid Surf. A: Physicochem. Eng. Aspects 1996, 113 (1−2), 51−59. [50] Agarwal, R. K.; Li, Y. K.; Nghiem, L. A Regression Technique with Dynamic Parameter Selection for Phase Behaviour Matching. SPE Res. Eng. 1990, 5 (1), 115−120. [51] Jindrova, T.; Mikyska, J.; Firoozabadi, A. Phase Behaviour Modeling of Bitumen and Light Normal Alkanes and CO2 by PR-EOS and CPA-EOS. Energy Fuels

2016, 30 (1), 515‒525. [52] Ayirala, S. C.; Rao, D. N. Solubility, Miscibility and Their Relation to Interfacial Tension in Ternary Liquid Systems. Fluid Phase Equilibr. 2006, 249 (1−2), 82−91. [53] Hu, R.; Trusler, J. P. M.; Crawshaw, J. P. Effect of CO2 Dissolution on the Rheology of a Heavy Oil/Water Emulsion. Energy Fuels 2017, 31 (4), 3399‒3408. [54] Zhang, K.; Gu, Y. New Qualitative and Quantitative Technical Criteria for Determining the Minimum Miscibility Pressures (MMPs) with the Rising-Bubble Apparatus (RBA). Fuel 2016, 175 (7), 172−181.

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Table 1 Compositions of liquid and vapour phases for a pure hydrocarbon system (i.e., nC4−iC4−C8 system)38 and two light oil−pure CO2 systems (i.e., Pembina live light oil−pure CO2 system9 and Bakken live light oil−pure CO2 system35) used in this study.

Pure HCs Pure HC Composition Component38 (mol.%) nC4

4.53

iC4

15.47

C8

80.00

Oil Pembina oil Component9 C1 C2 C3 C4 C5−6 C7−12 C13−29 C30+ Feed oil−solvent ratio (by mole)

Composition (mol.%) 62.35 10.70 10.69 10.10 0.54 2.70 2.30 0.62 0.01 : 0.99

36

Bakken oil Component35 C1 C2 C3 C4 C5−6 C7−12 C13−21 C22−80 Feed oil−solvent ratio ( by mole )


Composition (mol.%) 36.74 14.89 9.33 5.75 6.41 15.85 7.33 3.70 0.50 : 0.50

Solvent Pure CO2 (mol.%)

100.00

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Table 2 Measured39 and calculated saturation pressures, oil densities, and oil-swelling factors (SFs) of the Pembina light oil–pure CO2 systems at the reservoir temperature of Tres = 53.0°C.

x CO 2

Test no.

wt.% 0.00 10.40 13.40 18.20

1 2 3 4

Notes:

mol.% 0.00 35.90 42.70 51.70

m Psat

c Psat

(MPa) − 6.50 7.80 9.60

(MPa) − 6.52 7.77 9.63

ε p (%) − 0.31 0.38 0.31

m ρ oil (g/cm3) 0.8300 0.8432 0.8440 0.8485

ρ coil (g/cm3) 0.8311 0.8439 0.8446 0.8488

ερ

SF m

SF c

ε SF

(%) 0.13 0.08 0.07 0.04

at Psat

at Psat

(%)

− 1.16 1.20 1.28

− 1.16 1.19 1.30

− 0.00 0.80 1.56

x CO 2 :

weight or mole percentage of CO2 dissolved into in the dead light oil

m Psat :

measured saturation pressure

c Psat :

calculated saturation pressure

m : ρ oil

measured oil density

ρ coil :

calculated oil density

SF m :

measured oil-swelling factor

SF c :

calculated oil-swelling factor

ε:

relative error between the calculated and measured data

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Table 3 Measured38 and calculated pressure−volume−temperature data for iC4−nC4−C8 system in the micro-channel of 10 µm and nanochannel of 100 nm at (a) constant pressure and (b) constant temperature. Parameters

Before flash calculation38

After flash calculation (this study)

After flash calculation38

(a) constant pressure case Temperature (°C ) Pressure (Pa) Liquid (iC4−nC4−C8, mol.%) Vapour (iC4−nC4−C8, mol.%) Liquid fraction (mol.%) Vapour fraction (mol.%) IFT (mJ/m2) Pcap in micro-channel (kPa) Pcap in nano-channel (kPa) Temperature (°C) Pressure (Pa) Liquid (iC4−nC4−C8, mol.%) Vapour (iC4−nC4−C8, mol.%) Liquid fraction (mol.%) Vapour fraction (mol.%) IFT (mJ/m2) Pcap in micro-channel (kPa) Pcap in nano-channel (kPa)

24.9 15.47 0

61.89 0

71.9

4.53 0 100.00 0.00 − − −

80.00 0

4.88 64.35

85,260 1.87 16.82 82.20 17.80 16.24 3.38 286.91

(b) constant temperature case 71.9 839,925 18.11 20.00 28.59 11.15 0 0 75.82 21.01 100.00 29.50 0.00 70.50 13.33 − 2.77 − 235.54 −

38


93.25 18.83

5.09 62.84

60.26 3.16

426,300 18.98 76.66

1.68 17.56 82.03 17.97 15.68 1.92 185.63

93.23 19.60

6.44 19.35 30.38 69.62 12.89 1.88 182.49

74.58 3.99

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Page 40 of 50

Table 4 Measured39 and calculated twelve interfacial tensions (IFTs) at twelve different pressures and the reservoir temperature of Tres = 53.0°C for the Pembina dead light oil−pure CO2 system, live light oil−pure CO2 system, and dead light oil−impure CO2 system, respectively.

Pembina dead light oil−pure CO2 system

P (MPa) γ eq m (mJ/m2)

2.0

3.0

4.0

5.5

6.5

7.5

8.5

9.5

10.0

12.0

15.0

20.0

16.89

15.50

14.02

11.57

10.02

7.39

5.01

3.74

3.00

2.05

1.45

1.10

γ eq c (mJ/m2)

16.58

14.92

13.36

10.77

9.01

7.00

5.13

3.31

2.43

1.49

1.15

0.89

Pembina live light oil−pure CO2 system


1.8

3.0

3.8

4.8

5.6

7.0

8.6

10.1

11.4

15.0

16.6

18.1

17.77

15.24

13.79

12.67

10.81

8.75

5.89

4.34

4.04

3.28

2.56

1.75

γ eq c (mJ/m2)

17.17

15.21

13.96

12.42

11.07

8.76

6.22

4.15

2.97

1.94

1.35

0.99

Pembina dead light oil−impure CO2 system


2.0

3.0

4.5

6.0

7.0

8.5

10.0

12.5

15.0

19.0

21.0

24.0

17.56

15.99

13.93

11.58

9.69

7.82

5.95

4.11

3.01

2.05

1.88

1.34

γ eq c (mJ/m2)

16.79

15.41

13.11

10.88

9.46

7.50

5.76

3.22

2.33

1.02

0.88

0.57

Notes:

m: c:

Measured IFTs calculated IFTs

39


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Table 5 Determined minimum miscibility pressures (MMPs) of the Pembina dead light oil−pure CO2 system, live light oil−pure CO2 system, and dead light oil−impure CO2 system in bulk phase from the vanishing interfacial tension (VIT) technique, coreflood tests, slim-tube tests, rising-bubble apparatus (RBA) tests, and diminishing interface method (DIM) at the reservoir temperature of Tres = 53.0°C.

Test system Pembina dead light oil−pure CO2 Pembina live light oil−pure CO2 Pembina dead light oil−impure CO2 Reference

VIT-MMP (MPa) Traditional Improved 10.6 12.9 12.5 13.2 21.4 21.8 (40) (18)

40

Coreflood tests MMP (MPa)

Slim-tube tests MMP (MPa)

RBA-MMP (MPa)

12.4−12.9 − − (9)

− 15.2−15.4 − (9)

− − 23.4−23.5 (54)


DIM-MMP (MPa) 12.4 15.0 22.1 This study

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Table 6 Determined minimum miscibility pressures (MMPs) from the diminishing interface method (DIM) in the nanopores with different pore radius and measured/predicted bulk-phase MMPs from the slim-tube tests and multiple-mixing cell method for the Pembina live light oil−pure CO2 system at 53.0°C and Bakken live light oil−CO2 system at 116.1°C.

Notes:

Oil

Gas

Temperature (°C)

Pembina live light oil9

Pure CO2

53.0

Bakken live light oil35

Pure CO2

116.1

a: b:

Pore radius (nm) Inf (bulk phase) 100 20 4 Inf (bulk phase) 100 20 4

determined from the slim-tube tests calculated from the multiple-mixing cell method

41


MMP (MPa) 15.2−15.4a 15.4 13.7 13.4 24.7b 24.1 21.4 20.6

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Parachor model

Fig. 1 Flowchart of the modified Peng‒Robinson equation of state for phase property predictions and parachor model for interfacial tension calculations in nanopores.

42


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H

Page 44 of 50

H

Bulk

n1α

Bulk

n2α

phase

α

n2α

n1α

phase α

Γ2 = 0, Γ1 > 0

z2

Border

Interfacial

Region

Region

(Absorption)

δ 0

phase

Bulk

n2β

n1β

β

phase

β ni

(a)

z =b ni

(b)

Fig. 2 Schematic diagram of the interfacial structure between two miscible phases: (a) real case and; (b) ideal case.

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Page 45 of 50

1.0

0.10

x CO + yHCs 2

x CO

2

0.08

yHCs

0.6

0.06

0.4

0.04

yHCs

x CO 2 or x CO 2+ yHCs

0.8

0.02

0.2

P

PA

0.0 0

5

B 0.00

10

15

20

P (MPa)

(a) 1.0

0.10

0.08

x CO2 + yHCs x CO 0.6

2

0.06

yHCs

yHCs

x CO 2 or x CO 2+ yHCs

0.8

0.4

0.04

0.2

0.02

P

A

P

B

0.0

0.00 0

5

10

15

20

P (MPa)

(b) 1.0

0.10

x CO + yHCs 2

x CO

0.8

0.08

2

yHCs 0.6

0.06

0.4

0.04

y HCs

x CO 2 or x CO 2+ yHC s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

0.02

0.2

P

A

P

B

0.0

0.00 0

5

10

15

20

25

30

P (MPa)

(c)

Fig. 3 Predicted x CO , y HCs , and x CO + y HCs of (a) the Pembina dead light oil−pure 2

2

CO2 system; (b) the Pembina live light oil−pure CO2 system; and (c) the Pembina dead light oil−impure CO2 system at Tres = 53.0°C.

44


Energy & Fuels

30

Pembina dead light oil−pure CO2 system Pembina live light oil−pure CO2 system Pembina dead light oil−impure CO2 system

20

∂d((MW /Zg))/∂ ∂P MWgg/Z g / dP eq

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 50

10

0

PA -10 0

5

10

15

20

25

30

P (MPa)

Fig. 4 Calculated forward finite difference approximation of the partial derivative ∂ ( MWg Z g ) A for the Pembina dead light oil−pure CO2 system with P = 10.8 MPa, the ∂P A Pembina live light oil−pure CO2 system with P = 11.3 MPa, and the Pembina dead A light oil−impure CO2 system with P = 13.5 MPa at Tres = 53.0°C.

45


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Energy & Fuels

Fig. 5 Determined minimum miscibility pressures of (a) the Pembina dead light oil−pure CO2 system; (b) the Pembina live light oil−pure CO2 system; and (c) the Pembina dead light oil−impure CO2 system from the diminishing interface method (DIM) at Tres = 53.0°C. 46


Energy & Fuels

25 6.89 MPa (literature) 6.89 MPa (this study) 10.34 MPa (literature) 10.34 MPa (this study) 13.79 MPa (literature) 13.79 MPa (this study) 17.24 MPa (literature) 17.24 MPa (this study)

2

γ (mJ/m )

20

15

10

5

0 1

10

100

1000

rp (nm)

(a) 20

4.0 MPa 7.5 MPa 9.5 MPa 15

2

γ (mJ/m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 48 of 50

10

5

0 1e+0

1e+1

1e+2

1e+3

1e+4

1e+5

1e+6

rp (nm)

(b) Fig. 6 Confinement effect on predicted interfacial tensions of (a) the Bakken live light oil−pure CO2 system from the literature31 and the model in this study in the pore radius range of 4−1,000 nm at four different pressures and Tres = 116.1°C and (b) the Pembina live light oil−pure CO2 system from the model in this study in the pore radius range of 2−1,000,000 nm at three different pressures and Tres = 53.0°C.

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Energy & Fuels

Fig. 7 Determined minimum miscibility pressures of Pembina live light oil−pure CO2 system in the nanopores of (a) 100 nm; (b) 20 nm; and (c) 4 nm at Tres = 53.0°C.

48


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Fig. 8 Determined minimum miscibility pressures of the Bakken live light oil−pure CO2 system in the nanopores of (a) 100 nm; (b) 20 nm; and (c) 4 nm at Tres = 116.1°C.

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A New Diminishing Interface Method for Determining the Minimum

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