Toward accurate density and interfacial tension modeling for carbon dioxide/water mixtures

Zixuan Cui; Huazhou Li

doi:10.1007/s12182-020-00526-x

Toward accurate density and interfacial tension modeling for carbon dioxide/water mixtures

Cui, Zixuan; Li, Huazhou 2020-11-19 00:00:00 Phase behavior of carbon dioxide/water binary mixtures plays an important role in various CO -based industry processes. This work aims to screen a thermodynamic model out of a number of promising candidate models to capture the vapor–liq- uid equilibria, liquid–liquid equilibria, and phase densities of C O /H O mixtures. A comprehensive analysis reveals that 2 2 Peng–Robinson equation of state (PR EOS) (Peng and Robinson 1976), Twu α function (Twu et al. 1991), Huron–Vidal mixing rule (Huron and Vidal 1979), and Abudour et al. (2013) volume translation model (Abudour et al. 2013) is the best model among the ones examined; it yields average absolute percentage errors of 5.49% and 2.90% in reproducing the experimental phase composition data and density data collected in the literature. After achieving the reliable modeling of phase compositions and densities, a new IFT correlation based on the aforementioned PR EOS model is proposed through a nonlinear regression of the measured IFT data collected from the literature over 278.15–477.59 K and 1.00–1200.96 bar. Although the newly proposed IFT correlation only slightly improves the prediction accuracy yielded by the refitted Chen and Yang (2019)’s correlation (Chen and Yang 2019), the proposed correlation avoids the inconsistent predictions present in Chen and Yang (2019)’s correlation and yields smooth IFT predictions. Keywords Phase behavior · CO /H O binary mixtures · Volume translation · Interfacial tension 2 2 1 Introduction Due to their simplicity and good reliability, CEOSs such as SRK EOS (Soave 1972) and PR EOS (Peng and Robin- The interaction of CO with H O is frequently seen in sev- son 1976) are the most widely used thermodynamic models 2 2 eral subterranean processes (such as CO -based enhanced for the phase behavior modeling of C O /H O binary mix- 2 2 2 recovery and CO storage). Phase behavior of CO /H O tures (Aasen et al. 2017; Michelsen and Mollerup 2007). 2 2 2 mixtures under subterranean conditions plays a great role in Numerous articles have addressed phase-composition mod- affecting the overall efficiency of these processes. Thus how eling of CO /H O mixtures. Two types of methods, ϕ–ϕ 2 2 to accurately model the phase behavior of C O /H O mix- (fugacity–fugacity) approach and γ–ϕ (activity–fugacity) 2 2 tures becomes drastically important. Overall, an appropriate approach (Trusler 2017; Zhao and Lvov 2016), are often combination of cubic equation of state (CEOS), mixing rule applied in such modeling processes. Because γ-ϕ approach in CEOS, α function, volume translation, and interfacial ten- has a discontinuity issue in the phase diagram near the sion (IFT) model should be determined to well capture the critical region (Zhao and Lvov 2016), this work focuses vapor–liquid equilibrium (VLE), liquid–liquid equilibrium on ϕ–ϕ based methods. Valtz et al. (2004) found that the (LLE), phase density, and IFT of CO /H O mixtures. most accurate model is PR EOS (Peng and Robinson 1976), 2 2 Mathias–Copeman α function (Mathias and Copeman 1983), and Wong–Sandler mixing rule (Wong and Sandler Edited by Yan-Hua Sun 1992) with average absolute percentage deviation (AAD) of 5.4% in reproducing the measured phase composition * Huazhou Li data for C O /H O mixtures. However, the temperature 2 2 huazhou@ualberta.ca and pressure ranges used by Valtz et al. (2004) were nar- Department of Civil and Environmental Engineering, School row (278.2–318.2 K and 4.64–79.63 bar, respectively). In of Mining and Petroleum Engineering, University of Alberta, addition, the parameters in the Wong–Sandler mixing rule Edmonton, AB T6G 1H9, Canada Vol.:(0123456789) 1 3 510 Petroleum Science (2021) 18:509–529 (Wong and Sandler 1992) are given as discrete values at of CO /H O mixtures and achieved a significant improve- 2 2 different isotherms. Zhao and Lvov (2016) applied PRSV ment in density-prediction accuracies. However, a more EOS (Stryjek and Vera 1986) and the Wong–Sandler mix- accurate volume translation function, the one proposed by ing rule to calculate phase compositions, obtaining an AAD Abudour et al. (2012b, 2013), was not applied in Aasen et al. of 7.12% in reproducing the measured phase composition (2017)’s study; furthermore, it should be noted that Aasen data of CO /H O mixtures over a wide range of tempera- et al. (2017) used GERG-2008 (Kunz and Wagner 2012) 2 2 tures and pressures. Similar to Valtz et al. (2004)’s study, in and EOS-CG (Gernert and Span 2016) calculated densities the study by Zhao and Lvov (2016), the parameters in the as reference densities instead of experimental data. In this Wong–Sandler mixing rule are provided as discrete values at study, we apply the volume translation method by Abudour different isotherms, instead of generalized correlations; their et al. (2012b, 2013) to see if the use of this model can further model is inconvenient to use since one has to make extrapo- improve phase-density predictions for CO /H O mixtures; 2 2 lations based on the provided values when making predic- these predictions are compared to the measured density data tions at conditions different from those given by Zhao and documented in the literature. Lvov (2016). Abudour et al. (2012a) applied van der Waals Parachor model (Sugden 1930) is one of the most widely (1873) one-fluid (vdW) mixing rule with several tempera - applied models in predicting mixtures’ IFT (Schechter and ture-dependent BIP correlations in PR EOS in determining Guo 1998). However, its accuracy heavily relies on the den- phase compositions of CO /H O mixtures. With the tuned sity difference between the two coexisting phases in a VLE 2 2 BIPs, their model yielded good accuracy (i.e., AAD of 5.0%) or an LLE. Our experience in using Parachor model to cal- in aqueous phase-composition predictions but lower accu- culate IFT of CO /H O mixtures shows that Parachor model 2 2 racy (i.e., AAD of 13.0%) in C O -rich phase-composition is generally appropriate for the IFT estimation for VLE of predictions, respectively. CO /H O systems, but less suitable for the IFT estimation 2 2 A recent comprehensive study by Aasen et al. (2017) for LLE of C O /H O systems. This is primarily because an 2 2 revealed that the most accurate thermodynamic model LLE of a CO /H O mixture has a smaller density difference 2 2 (among the ones examined by them) in phase-composition than a VLE. Several empirical IFT correlations for CO / and phase-density predictions for CO /H O mixtures is PR H O mixtures have been proposed in the literature. However, 2 2 2 EOS, Twu α function (Twu et al. 1991), Huron-Vidal mix- most of these correlations are only applicable to a limited ing rule, and constant volume translation. This model only temperature and pressure range (Zhang et al. 2016). Hebach yields an AAD of 4.5% in phase-composition calculations et al. (2002) proposed a new correlation which correlated and an AAD of 2.8% in phase-density calculations for C O / IFT with phase densities. Hebach et al. (2002)’s model is H O mixtures. Aasen et al. (2017), Valtz et al. (2004), and suitable over a wide range of temperature and pressure con- Zhao and Lvov (2016) also pointed out that more advanced ditions, although the prediction accuracy decreases with an models [e.g., the Cubic-Plus-Association (CPA) EOS (Kon- increase in temperature or pressure. Chen and Yang (2019) togeorgis et al. 1996)] do not guarantee an improvement in proposed a new empirical IFT correlation for C O /CH /H O 2 4 2 the phase-composition predictions for CO /H O mixtures. ternary systems based on mutual solubility, and this model 2 2 With regards to phase-density calculations, CEOS based performs well for C O /H O binary mixtures. However, 2 2 methods tend to overestimate liquid-phase molar volumes. A our experience in applying Chen and Yang (2019)’s model detailed discussion of this issue can be found in the studies shows that some breaking points can be observed in the by Matheis et al. (2016) and Young et al. (2017). In order to predicted IFT curves under some conditions, hampering its address this problem, Martin (1979) introduced the volume ability in providing consistent and smooth IFT predictions. translation concept in CEOS to improve liquid-phase volu- In addition, using two sets of BIPs (as applied in Chen and metric predictions. Peneloux et al. (1982) developed vol- Yang (2019)’s study) in the aqueous phase and non-aqueous ume translation schemes in SRK EOS for pure substances. phase can lead to thermodynamic inconsistency issue near Jhaveri and Youngren (1988) applied volume translation the critical region as demonstrated by Li and Li (2019). into PR EOS, leading to the improvement of liquid phase- The discussion above reveals that the previous studies of density predictions. A thorough comparison of different phase behavior modeling of the CO /H O mixtures tend to 2 2 types of volume translation methods can be found in Young primarily focus on phase-composition modeling and pay less et al. (2017)’s work. According to the study by Young et al. attention to phase-density calculations (especially for the (2017), the temperature-dependent volume translation CO -rich phase). Whereas, phase-density is one important method developed by Abudour et al. (2012b, 2013) pro- property in VLE and LLE since IFT calculations and flow vides the most accurate estimates on liquid-phase densities simulations can heavily rely on such property. As for the IFT without thermodynamic inconsistencies (e.g., crossover of modeling, we are currently lacking a reliable IFT correla- pressure–volume isotherms). Aasen et al. (2017) applied tion that not only pays due tribute to the phase composition constant volume translation to phase-density calculations and density of CO /H O mixtures but also gives smooth and 2 2 1 3 Petroleum Science (2021) 18:509–529 511 consistent IFT predictions over a wider range of tempera- 2.2 Mixing rules ture/pressure conditions. In this study, we first conduct a thorough literature review Mixing rules have a great impact on phase equilibrium cal- to select the most promising thermodynamic models that culations. Huron and Vidal (1979) proposed a new expres- can well capture the VLE and LLE of CO /H O mixtures. sion by considering the excess Gibbs energy for CEOS, 2 2 Then, we conduct phase-composition calculations by using which made more accurate the phase-composition predic- PR EOS, Twu α function, and Huron-Vidal mixing rule [as tions for mixtures containing polar substances. Furthermore, suggested by Aasen et al. (2017)], and validate the accu- according to the comprehensive study by Aasen et al. (2017), racy of this model by comparing the calculated VLE and the most accurate thermodynamic model among the ones LLE phase compositions to the measured ones. Then, we examined by them is PR EOS coupled with Twu α function introduce Abudour et al. (2012b, 2013) volume translation and Huron-Vidal mixing rule, which provides an AAD of model in phase-density calculations to check if applying this 4.5% in reproducing the phase-composition data measured model can further improve the density-prediction accuracies. for CO /H O mixtures. Hence, in the first part of this study, 2 2 A new empirical IFT correlation for CO /H O mixtures is we collect more phase equilibria data for C O /H O mixtures 2 2 2 2 then proposed based on the reliable thermodynamic model to verify the performance of the model suggested by Aasen that incorporates the Huron-Vidal mixing rule and the Abu- et al. (2017). These additional experimental data are not dour et al. (2013) volume translation model. included in the study by Aasen et al. (2017). The vdW mixing rule is one of the most commonly used mixing rules in petroleum industry (Pedersen et al. 2014). 2 Methodology Although vdW mixing rule is originally developed for non- polar systems, the vdW mixing rule coupled with the tuned 2.1 PR EOS and α functions BIPs can be reliably used for describing the phase behav- ior of mixtures containing polar components (e.g., water). In this study, PR EOS (Peng and Robinson 1976) is imple- Besides, based on the study by Abudour et al. (2012a), mented because of its simplicity and more accurate liquid- Gasem α function with vdW mixing rule and their tem- density predictions compared with SRK EOS (Aasen et al. perature-dependent volume translation function provided 2017). The expression of PR EOS is detailed in “Appendix a promising means to well reproduce the measured liquid- A”. phase densities for CO /H O mixtures. Therefore, in this 2 2 With regards to α functions in PR EOS, Twu α func- study, we also employ the model suggested by Abudour tion and Gasem α function are used in this study. Compared et al. (2012a) to test if it outperforms the model suggested with the other types of α functions, the Twu α function can by Aasen et al. (2017). The expressions of vdW mixing rule describe the thermodynamic properties of polar compounds and Huron–Vidal mixing rule and their BIPs are detailed in more accurately and perform better in the supercritical “Appendices B and C”. region (Young et al. 2016). In addition, according to the study by Aasen et al. (2017), Twu α function coupled with 2.3 Volume translation models PR EOS and Huron-Vidal mixing rule yields the most accu- rate phase-composition estimations on CO /H O mixtures Volume translation is used to overcome the inherent defi- 2 2 among the models evaluated by them. Therefore, we select ciency of CEOS in liquid-phase-density predictions. In order Twu α function as one of the evaluated α functions in this to improve liquid-phase-density calculations, Peneloux et al. study. (1982) developed a constant volume translation model in The Gasem α function provides more accurate represen- SRK EOS, while Jhaveri and Youngren (1988) developed tation of supercritical phase behavior (Gasem et al. 2001). a constant volume translation model in PR EOS. Abudour Besides, based on the study by Abudour et al. (2013), Gasem et al. (2012b, 2013) revised the temperature-dependent α function coupled with PR EOS, vdW mixing rule and volume translation function to improve both saturated and Abudour volume translation yields the most accurate liq- single-phase liquid density calculations. Furthermore, unlike uid-phase-density predictions for the chemical compounds other temperature-dependent volume translation models, examined by their study. Therefore, we select the Twu α the volume translation model developed by Abudour et al. function and the Gasem α function in VLE/LLE and phase- (2012b, 2013) does not yield thermodynamic inconsist- density calculations. “Appendix A” shows the expressions ency issues unless at extremely high pressures. Therefore, of Twu α function and Gasem α function. we select the constant and Abudour et al. volume transla- tion models in this study for phase-density predictions. The expressions of these two volume translation models are detailed in “Appendix D”. 1 3 512 Petroleum Science (2021) 18:509–529 where σ in the interfacial tension; α is the introduced com- 2.4 IFT correlations for CO /H O mixtures 2 2 ponent-dependent correction term; x and y are the mole i i fractions of component i in liquid and vapor phases, respec- In this study, we select Parachor model (Sugden 1930), Hebach et al. (2002)’s correlation, and Chen and Yang tively; P is the Parachor value of component i ( P = 52 , i H O P = 78 ) (Liu et al. 2016); is the molar density of (2019)’s correlation to predict IFT of CO /H O mixtures. 2 2 CO 2 L 3 M The Parachor model (Sugden 1930) is one of the most liquid phase in mol/cm ; is the molar density of vapor phase in mol/cm . N is the number of component; n is the widely used methods in determining mixtures’ IFT. It cor- relates IFT with phase compositions and molar densities of exponent. First, the component-dependent correction term α is set each phase. Parachor is a component-dependent constant. The expression of the Parachor model is shown in “Appen- as a constant for each component, and the exponential term n can be expressed by the equilibrium ratios of C O -rich dix E”. Hebach et al. (2002)’s correlation correlates IFT with temperature, pressure, and phase densities. Phase composi- phase and aqueous phase: tions are not included in their correlation. To make fair com- n = C ln K + C ln K + C 1 CO 2 H O 3 (2) 2 2 parison, we also refit coefficients in their correlation based on the IFT database employed in this study. Values of the where C , C , and C are empirical coefficients; K and 1 2 3 CO original and refitted coefficients as well as the Hebach et al. K are the equilibrium ratios (as known as K-values) of H O (2002)’s correlation are shown in “Appendix E”. CO and H O: 2 2 Chen and Yang (2019)’s correlation correlates IFT with K = y ∕x phase equilibrium ratios (K-values) and the reduced pres- (3) i i i sure of CO . Unlike the Parachor model and the Hebach Since using one coefficient set for both α and n on the et al. (2002)’s correlation, the density of the two equilibrat- whole CO -rich-phse density range cannot converge after ing phases is not one input in the Chen and Yang (2019)’s reaching the maximum iterations, we use two coefficient correlation. Chen and Yang (2019)’s correlation, they pro- sets based on C O -rich-phase densities. Table 1 listed the posed four groups of coefficient sets, i.e., one coefficient values of these coefficients and α determined by fitting the set (using one coefficient set on the whole pressure range) proposed correlation (abbreviated as Scenario #1) to the IFT with or without the reduced pressure term, and two coef- training dataset. ficient sets (dedicated to the pressure ranges of p ≤ 73.8 and Since using constants to represent α leads to a larger p > 73.8 bar) with or without the reduced pressure term. AAD compared with the refitted Chen and Yang (2019)’s Since using the reduced pressure term can improve pre- correlation (i.e., 8.8746% vs. 7.8520%, respectively), we diction accuracy (Chen and Yang 2019), we introduce the correlate equilibrium ratios to α to see if it can improve reduced pressure term in this study. Similarly, we refit these the IFT predictions. The expression of n in this scenario coefficients based on the IFT databased employed in this (abbreviated as Scenario #2) is the same as that in Scenario study to make fair comparison. Values of the original and #1. The expression for α is given as: refitted coefficients as well as the Chen and Yang (2019)’s correlation are detailed in “Appendix E”. = C ln K + C ln K + C i 1 CO 2 H O 3 (4) 2 2 2.5 IFT correlation proposed in this study Specifically, when the CO -rich-phase density is greater than 0.2 g/cm , can be simplified as: H O Before we finalize our IFT correlation, we tried several sce- = C lnK + C 1 CO 3 (5) H O 2 narios to find the optimal one to correlate the IFT of CO / 2 H O mixtures. Since the Parachor model is one of the most widely used models in mixtures’ IFT predictions, we revise the original Parachor model by introducing a component- Table 1 Values of the correlation coefficients and α in Scenario #1 dependent correction term α ; furthermore, we replace the i 3a 3 constant exponential term in the original Parachor model Coefficients 𝜌 < 0.2 g/cm ≥ 0.2 g/cm CO -rich H O-rich 2 2 by correlating it with several physical properties (e.g., equi- 0.7957 0.1520 CO librium ratios). The new IFT correlation can be expressed 0.8855 0.9509 H O as follows: 2 C −0.0727 0.1026 C 0.1044 0.0736 M M = P (x − y ) (1) i i i i C 5.5730 3.9154 L V i=1 is the density of CO -rich phase CO -rich 2 1 3 Petroleum Science (2021) 18:509–529 513 Table 2 listed the values of these coefficients determined Table 3 Coefficients in the α term for H O and CO i 2 2 by fitting the proposed correlation to the IFT training data- Component C C C C C 1 2 3 4 5 set. Since using one coefficient set for α and n on the whole CO -rich-phse density range in Scenario #2 cannot converge H O 1.1325 −0.0085 −0.0083 0.0134 0.0089 after reaching the maximum iterations, we use two coeffi- CO −0.4193 −0.0057 −0.0320 0.0209 −0.1430 cients based on C O -rich-phase densities (the same as Sce- nario #1). We find that using correlations to represent α can slightly 3 Results and discussion improve the IFT predictions (i.e., AAD of 8.31% in Sce- nario #2 vs. AAD of 8.87% in Scenario #1). Besides, we The values of critical pressure (p ), critical temperature (T ), c c find that the value of n is around 4 over a wide range of tem- acentric factor (ω), molecular weight (M), critical compress- perature/pressure conditions in all scenarios (i.e., its value ibility factor (Z ) used in this study are retrieved from the only slightly changes with the change of equilibrium ratios); NIST database (Lemmon et al. 2011). therefore, we set the value of n as 4 for simplicity. However, our experience in applying Scenarios #1 and 3.1 Performance comparison of thermodynamic #2 shows that some breaking points can be observed in models in phase equilibrium calculations the correlated IFT curves due to the fact that two differ - ent sets of coefficients are adopted under the conditions of Table 4 summarizes the measured phase equilibrium data of 3 3 𝜌 < 0.2 g/cm and ≥ 0.2 g/cm , respectively. CO /H O mixtures over 278–378.15 K and 0.92–709.3 bar CO -rich H O -rich 2 2 2 2 In addition, based on the study by Chen and Yang (2019), reported in the literature. Note that these experimental data introducing the reduced pressure of CO can improve IFT were not included in the study by Aasen et al. (2017). Com- predictions. Thus, we introduce the reduced pressure of C O parison between the measured and calculated phase-compo- in the expressions of α and use one coefficient set to see if sition results is evaluated by the average absolute percentage these settings can further improve the prediction accuracies deviation (AAD) defined as: without yielding inconsistent IFT predictions. Based on the x − x calculation results, the following IFT correlation yields the 1 CAL EXP AAD = × 100% (8) lowest AAD among the ones examined in this study: N x EXP where AAD is the average absolute percentage deviation; N M M = P x − y (6) i i i i is the number of data points; x and x are the calculated L V CAL EXP i=1 and measured mole fraction of C O or H O in the aqueous 2 2 phase (or the CO -rich phase), respectively. where the α term in the new correlation can be expressed as: Table 5 details the settings of the four thermodynamic = C +(C p + C ) ln K +(C p + C ) ln K models examined in this work. Table 6 summarizes the 1 2 r 3 CO 4 r 5 H O (7) 2 2 performance of different thermodynamic models in phase- where p is the reduced pressure of CO . r 2 composition predictions. Table 3 lists the values of these coefficients determined by As shown in Table 6, although AAD for x of Case CO fitting the proposed correlation to the IFT training dataset. 3 (Twu + HV) is slightly higher than that of Case 4 (Gasem + HV), i.e., 4.73% of Case 3 vs. 3.64% of Case 4, Case 3 (Twu + HV) significantly outperforms the other models in y predictions, i.e., AAD of 10.37% of Case H O 3 vs. > 16% of other cases. Thus, given the overall perfor- mance, Case 3 (Twu + HV) is found to be the best model Table 2 Values of the correlation coefficients and α in Scenario #2 3 3 Coefficients 𝜌 < 0.2 g/cm ≥ 0.2 g/cm CO -rich H O-rich 2 2 C C C C C C 1 2 3 1 2 3 –0.4685 –0.2177 1.7944 0.4583 0.0107 –1.3451 CO –0.1033 0.0311 1.8397 0.5259 – –0.3583 H O n 0.3599 –0.0855 1.3153 –0.2685 0.0124 3.5123 1 3 514 Petroleum Science (2021) 18:509–529 Table 4 Phase equilibrium data of C O /H O mixtures employed in this study 2 2 a b c T, K p, bar x , % y , % N References AAD, % CO H O 2 2 323.15–373.15 25.3–709.3 0.429–3.002 – 29 Weibe and Gaddy (1939) 2.09 285.15–313.15 25.3–506.6 0.925–3.196 – 42 Weibe and Gaddy (1940) 5.50 323.15–373.15 200–500 2–2.8 1–3 4 Tödheide and Frank (1962) 1.90/34.69 288.71–366.45 6.9–202.7 0.0973–2.63 0.0819–12.03 24 Gillepsie and Wilson (1982) 6.68/5.93 323.15 68.2–176.8 1.651–2.262 0.339–0.643 8 Briones et al. (1987) 3.17/5.26 285.15–304.21 6.9–103.4 – 0.0603–0.33739 9 Song and Kobayashi (1987) 6.40 323.15–378.15 101.33–152 1.56–2.1 0.55–0.9 4 D’Souza et al. (1988) 4.62/16.81 348.15 103.4–209.4 1.91–1.92 0.63–0.84 2/3 Sako et al. (1991) 7.97/25.38 323.15 101–301 2.075–2.514 0.547–0.782 3 Dohrn et al. (1993) 1.85/15.53 278–293 64.4–294.9 2.5–3.49 – 24 Teng et al. (1997) 7.36 288–323 0.92–4.73 0.038–0.365 – 49 Dalmolin et al. (2006) 2.89 313.2–343.2 43.3–183.4 1.13–2.40 – 28 Han et al. (2009) 3.37 273.15–573.15 100–1200 0.89–14.96 – 130 Guo et al. (2014) 5.43 323.15–423.15 150 1.77–2.19 – 3 Zhao et al. (2015) 4.18 Solubility of CO in the aqueous phase Solubility of H O in CO -rich phase 2 2 AAD yielded by the Case 3 model (PR EOS, Twu α function, and Huron–Vidal mixing rule) in x and/or y predictions. If two numbers are CO H O 2 2 shown in table, the former indicates AAD in x prediction and the latter indicates AAD in y prediction. CO H O 2 2 d,e,f,g,h,i These data are already summarized by Spycher et al. (2003). We directly use these data mentioned in their paper for convenience N is 2 for x and 3 for y , respectively CO H O 2 2 Only experimental data for CO /pure water are selected in the study by Zhao et al. (2015) Table 5 Settings of four thermodynamic models examined in this work Case No. α function Mixing rule BIPs Case 1 Gasem et al. (2001) vdW (1873) kc = 0.27; kd = −0.21 (Abudour et al. 2012a) ij ij Case 2 Gasem et al. (2001) vdW (1873) kc (T); kd (T) (Abudour et al. 2012a) ij ij Case 3 Twu et al. (1991) Huron and Vidal (1979) Aasen et al. (2017) Case 4 Gasem et al. (2001) Huron and Vidal (1979) Aasen et al. (2017) The expressions of kc (T) and kd (T) are listed in “Appendix B” ij ij in phase-composition predictions. Figure 1 compares 3.2 Evaluation of thermodynamic models in density the performance of different models at T = 323.15 K and calculations T = 348.15 K. As can be seen from these two figures, the thermodynamic model Case 3 (Twu + HV) can well capture Table 7 summarizes the experimental aqueous-phase the trend exhibited by the measured solubility data over a and CO -rich-phase densities of CO /H O mixtures over 2 2 2 wide pressure range. 278–478.35 K and 2.5–1291.1 bar documented in the lit- erature. The pressure–temperature coverage of the phase density data collected from the literature are shown in Table 6 AAD of calculated mole fraction of CO in the aqueous phase “Appendix F”. ( x ) and mole fraction of H O in the CO -rich phase ( y ) by dif- CO 2 2 H O 2 2 ferent thermodynamic models Since Case 3 (Twu + HV) outperforms other thermody- namic models in phase-composition predictions for C O / Case No. AAD for x , % AAD for y , % CO H O 2 2 H O mixtures, we only focus on the performance of Case Case 1 57.81 19.26 3 coupled with volume translation in phase-density pre- Case 2 8.85 16.90 dictions. Table 8 summarizes the performance of differ - Case 3 4.73 10.37 ent volume translation models in both aqueous-phase and Case 4 3.64 16.59 CO -rich-phase density calculations. 1 3 Petroleum Science (2021) 18:509–529 515 10 10 Experimental data Experimental data Case 1 (Gasem + vdW + constant BIPs) Case 1 (Gasem + vdW + constant BIPs) Case 2 (Gasem + vdW + temperature-dependent BIPs) Case 2 (Gasem + vdW + temperature-dependent BIPs) Case 3 (Twu + HV) Case 3 (Twu + HV) Case 4 (Gasem + HV) Case 4 (Gasem + HV) 01020850 900950 1000 01020 700 800 900 1000 1000x , 1000y 1000x , 1000y CO2 H2O CO2 H2O (a)(b) Fig. 1 Measured and calculated pressure-composition data for CO /H O mixtures at T = 323.15 K (a) and T = 348.15 K (b). Solid circles are the 2 2 experimental data from the studies by Briones et al. (1987) and Gillepsie and Wilson (1982) Table 7 Aqueous-phase ( ) and CO -rich-phase ( ) density data of CO /H O mixtures employed in this study H O 2 CO -rich 2 2 2 2 3 3 a T, K p, bar , kg/m , kg/m N References AAD, % H O CO -rich 2 2 352.85–471.25 21.1–102.1 840–963 – 32 Nighswander et al. (1989) 3.01 288.15–298.15 60.8–202.7 1015–1027 – 27 King et al. (1992) 2.17 278–293 64.4–294.9 1013.68–1025.33 – 24 Teng et al. (1997) 1.35 304.1 10–70 999.4–1011.8 18.8–254.2 8 Yaginuma et al. (2000) 2.67/5.12 332.15 33.4–285.9 990.5–1010.3 – 29 Li et al. (2004) 3.80 283.8–333.19 10.8–306.6 983.7–1031.77 – 203 Hebach et al. (2004) 2.47 307.4–384.2 50–450 950.6–1026.1 80.8–987.5 43 Chiquet et al. (2007) 3.61/2.01 322.8–322.9 11–224.5 988.52–1009.13 18.8484–812.725 11 Kvamme et al. (2007) 3.37/1.94 382.41–478.35 34.82–1291.9 871.535–994.984 36.943–944.965 32/40 Tabasinejad et al. (2010) 4.85/4.43 298.15–333.15 14.8–207.9 984.6–1022 24.6–907.1 36 Bikkina et al. (2011) 3.08/3.15 292.7–449.6 2.5–638.9 905.9–1034.9 4.6–1023.4 145/128 Efika et al. (2016) 3.56/2.01 AAD yielded by Case 3–1 model (PR EOS, Twu α function, Huron–Vidal mixing rule, and Abudour volume translation model) for and/or H O predictions. If two numbers are shown in table, the former indicates AAD for prediction and the latter indicates AAD for CO -rich H O CO -rich 2 2 2 prediction N is 30 for and N is 40 for H O CO -rich 2 2 N is 144 for and N is 128 for H O CO -rich 2 2 As shown in Table 8, incorporation of VT into the ther- It can be seen from Fig. 2 that, regarding aqueous- modynamic framework can generally improve the phase- phase density predictions, the performance of Case 3–2 density prediction accuracy. Case 3–1 (Twu + HV + Abudour (Twu + HV + Constant VT) improves dramatically as tem- VT) provides the most accurate estimates of both aqueous- perature rises. As shown in Fig. 2e, f, at high temperature phase and C O -rich-phase density, yielding AAD of 2.90% conditions, Cases 3–2 yields the most accurate aqueous- in reproducing the measured phase-density data. Figure 2 phase density predictions; however, it fails to accurately further visualizes some of the calculation results by these predict CO -rich-phase densities. As a lighter phase, three different models at different pressure/temperature CO -rich-phase density can be accurately predicted without conditions. the use of volume translation functions. Applying Abudour VT method is able to only slightly improve the prediction 1 3 Pressure, bar Pressure, bar 516 Petroleum Science (2021) 18:509–529 Table 8 AAD of the calculated aqueous-phase density ( ) and CO -rich-phase density ( ) by different thermodynamic models H O 2 CO -rich 2 2 Model AAD for , % AAD for ,% Average AAD, % H O CO -rich 2 2 Case 3–1 (Twu + HV + Abudour VT ) 3.04 2.62 2.90 Case 3–2 (Twu + HV + Constant VT) 4.49 7.86 5.51 Case 3 (Base case) (Twu + HV) 15.08 3.38 11.44 VT: volume translation accuracy (i.e., AAD of 2.62%). In contrast, applying con- provides reliable phase-composition and phase-density pre- stant VT in CO -rich-phase density predictions can lead to dictions that can be fed into the proposed IFT correlation. larger errors than the case without the use of VT. Mean absolute errors (MAE), AAD, and coefficient of Figure 3 compares the performance of different mod- determination (R ) are used as performance measures. The els in terms of their accuracy in phase-density predictions expressions of MAE and R are as follows: over 382.14–478.35 K and 35.3–1291.9 bar. Note that the results of CPA EOS model from the work by Tabasinejad MAE = − (9) EXP,i CAL,i et al. (2010) focuses on the same pressure and temperature i=1 ranges. As can be seen from Fig. 3, although the CPA EOS model can accurately predict the aqueous-phase density, it � � EXP,i CAL,i i=1 tends to be less accurate in determining the C O -rich-phase 2 R = 1 − (10) � � density. Overall, the thermodynamic model Cases 3–1 EXP,i EXP i=1 (Twu + HV + Abudour VT) give an accuracy comparable to where σ is the measured IFT data in mN/m; σ is the the more complex CPA EOS model. EXP CAL In addition, according to the study by Aasen et al. (2017), calculated IFT in mN/m by different correlations; is the EXP average of the measured IFTs in mN/m. CPA EOS model yields higher percentage errors (AAD) in reproducing phase-composition data for C O /H O mixtures 2 2 3.3.1 Performance of different IFT correlations compared with Case 3 (PR EOS + Twu + HV), i.e., 9.5% vs. 4.5% (Aasen et al. 2017). Therefore, overall, Case 3–1 Table 10 shows the details of the different IFT models exam- (Twu + HV + Abudour VT) is a more accurate model in both phase-composition and phase-density predictions for CO / ined in this study. Table 11 summarize the performance of different correlations in IFT estimations. As can be seen, the H O mixtures. most accurate IFT model is Model 3 proposed in this study, although it only shows a marginal edge over Model 2. 3.3 Evaluation of the newly proposed IFT correlation Figure 4 visually compares the measured IFTs vs. pres- sure and the calculated ones by different IFT models at Table 9 summarizes the experimental IFT data of C O /H O selected temperatures. As shown in these plots, in general, 2 2 Model 3 (this study) outperforms other empirical correla- mixtures over 278.15–477.59 K and 1–1200.96 bar docu- mented in the literature. Ideally, phase densities should be tions over a wide range of temperatures and pressures. It can be also observed from these plots that breaking points appear directly measured; however, only Chiquet et al. (2007), Kvamme et al. (2007), Bikkina et al. (2011), Bachu and in the predicted IFT curves at p = 73.8 bar by Model 2 (Refit- ted Chen and Yang (2019)’s correlation with two coefficient Bennion (2009), and Shariat et al. (2012) applied measured phase densities in IFT calculations. In order to expand our sets). Such discontinuous IFT prediction can be attributed to the fact that two different sets of coefficients are adopted IFT database, IFT data with precisely determined phase densities are also included in our IFT database. The col- under the conditions of p ≤ 73.8 and p > 73.8 bar, respec- tively, in Chen and Yang (2019)’s correlation. Although lected IFT data are randomly placed into two bins: a training dataset (including 589 data points) and a validation dataset using one coefficient in Chen and Yang (2019)’s correlation (e.g., Model 5) can avoid such discontinuous IFT predic- (including 189 data points). Results in Sects. 3.1 and 3.2 reveal that the thermody- tions, it yields larger percentage errors. Therefore, Model 3 (this study) is the best model in IFT predictions for C O /H O namic model using PR EOS, Twu α function, Huron-Vidal 2 2 mixing rule, and Abudour et al. (2013) VT yields the most mixtures over a wide range of temperatures and pressures. Figure 5 illustrates how the IFTs predicted by Model 3 accurate estimates on both phase compositions and densi- ties. Therefore, the aforementioned thermodynamic model (this study) vary with pressure at different temperatures. It can be observed from Fig. 5 that the new IFT correlation 1 3 Petroleum Science (2021) 18:509–529 517 500 500 Aqueous-phase density Aqueous-phase density CO2-rich-phase density CO2-rich-phase density 400 400 300 300 200 200 100 100 0 0 00.2 0.40.6 0.81.0 1.2 00.2 0.40.6 0.81.0 1.2 3 3 Aqueous-phase/CO -rich-phase density, g/cm Aqueous-phase/CO -rich-phase density, g/cm 2 2 (a) T = 297.8 K (b) T =322.8 K 500 500 Aqueous-phase density Aqueous-phase density CO2-rich-phase density CO2-rich-phase density 400 400 300 300 200 200 100 100 0 0 00.2 0.40.6 0.81.0 1.2 00.2 0.40.6 0.81.0 1.2 3 3 Aqueous-phase/CO -rich-phase density, g/cm Aqueous-phase/CO -rich-phase density, g/cm 2 2 (c) T = 342.8 K (d) T =373.0 K 500 500 Aqueous-phase density Aqueous-phase density CO -rich-phase density CO -rich-phase density 2 2 400 400 300 300 200 200 100 100 0 0 00.2 0.40.6 0.81.0 1.2 00.2 0.40.6 0.81.0 3 3 Aqueous-phase/CO -rich-phase density, g/cm Aqueous-phase/CO -rich-phase density, g/cm 2 2 (e) T = 398.4 K (f) T =448.5 K Fig. 2 Predictions of aqueous-phase and C O -rich-phase density by Case 3–1 (Twu + HV + Abudour VT, dashed line), Case 3–2 (Twu + HV + constant VT, dotted line) and Case 3 (Base case, solid line) at different temperature conditions. The circles are the measured phase- density data from the study by Efika et al. (2016) 1 3 Pressure, bar Pressure, bar Pressure, bar Pressure, bar Pressure, bar Pressure, bar 518 Petroleum Science (2021) 18:509–529 lower temperature conditions (i.e., T < 378 K). It is interest- Aqueous-phase density 16.15 ing to observe from Fig. 5a that when the pressure is less CO -rich-phase density than around 15 bar and the temperature is between 278.15 and 368.15 K, an increase in temperature leads to a decrease in the predicted IFT under the same pressure. In comparison, 11.11 when the pressure is larger than around 15 bar, an increase in temperature leads to an increase in the predicted IFT. At higher temperatures of 378.15–478.15 K, an increase in temperature always results in a decline in the predicted 5.60 4.57 IFT under the same pressure, as seen in Fig. 5b. Most of 4.30 4 3.37 the measured IFTs documented in the literature follow this 1.77 2 trend (Akutsu et al. 2007; Chalbaud et al. 2009; Chiquet 0.25 et al. 2007; Chun and Wilkinson 1995; Da Rocha et al. 1999; Abudour VT Constant VT No VT appliedCPA EOS Georgiadis et al. 2010; Hebach et al. 2002; Heuer 1957; Hough et al. 1959; Kvamme et al. 2007; Khosharay and Fig. 3 Bar chart plots comparing the AAD in aqueous-phase (black) Varaminian 2014; Liu et al. 2016; Park et al. 2005; Pereira and CO -rich-phase (gray) density predictions by different models et al. 2016; Shariat et al. 2012), except for the studies by over 382.14–478.35 K and 35.3–1291.9 bar. Calculation results by Bachu and Bennion (2009) and Bikkina et al. (2011), i.e., the CPA EOS method are from the study by Tabasinejad et al. (2010) an increase in temperature leads to an increase in IFT at a temperature range of 373.15–398.15 K in the study by Bachu provides smooth and consistent IFT predictions at differ - and Bennion (2009), and an increase in temperature leads ent pressures and temperatures. Overall, Model 3 proposed to an increase in IFT over 298.15–333.15 K in the study in this study yields accurate and consistent IFT predictions by Bikkina et al. (2011). Again, the sharp drops in the IFT over the wide range of temperatures and pressures, although curves at lower temperatures (where C O remains subcriti- it yields relatively higher percentage errors at higher tem- cal) are due to the transformation of VLE to LLE. perature conditions (e.g., T = 478 K) compared with that at Table 9 Measured IFT data for CO /H O mixtures used in this study 2 2 T, K p, bar IFT, mN/m N References AAD, % 311–411 1–689.48 17.40–58.40 58 Heuer (1957) 9.13 311.15–344.15 1–197.8 17.63–69.20 28 Hough et al. (1959) 16.73 278.15–344.15 1–186.1 18.27–74.27 114 Chun and Wilkinson (1995) 6.18 311.15–344.15 1.6–310.7 19.38–56.86 20 Da Rocha et al. (1999) 13.26 278.4–333.3 1–200.3 12.4–74 85 Hebach et al. (2002) 3.76 293.15–344.15 1–173.2 20.55–78.01 26 Park et al. (2005) 7.19 318.15 11.6–165.6 25.4–70.5 14 Akutsu et al. (2007) 8.38 322.8–322.9 11–224.5 29.1–63.7 11 Kvamme et al. (2007) 4.80 307.4–384.2 50–450 45.8–22.8 43 Chiquet et al. (2007) 8.57 293.15–398.15 20–270 18.9–68.1 87 Bachu and Bennion (2009) 8.89 344.15 28.57–245.24 25.49–45.01 11 Chalbaud et al. (2009) 10.86 297.8–374.3 10–600.6 21.23–65.73 80 Georgiadis et al. (2010) 3.13 298.15–333.15 14.8–207.9 22.16–59.66 36 Bikkina et al. (2011) 11.29 323.15–477.59 77.78–1200.96 10.37–35.38 21 Shariat et al. (2012) 15.89 284.15–312.15 10–60 29.02–66.98 30 Khosharay and Varaminian (2014) 3.37 298.4–469.4 3.4–691.4 12.65–68.52 78 Pereira et al. (2016) 6.46 299.8–398.15 7.86–344.12 28.04–68.23 36 Liu et al. (2016) 9.89 AAD of the new IFT correlation proposed in this study (abbreviated as Model 3) b,c,e These data are already summarized by Park et al. (2005) and Shariat et al. (2012). We directly use these data mentioned in their papers for convenience d,f,g Some experimental data appear to be outliers and hence excluded for further analysis due to the significant deviation from other experimental data at similar temperature and pressure conditions (see Appendix G) 1 3 Average absolute percentage deviation, % Petroleum Science (2021) 18:509–529 519 (this study) is 0.0069, P <𝛼 ( = 0.05 ); therefore, it is rea- Table 10 Technical Characteristics of different IFT models examined in this study sonable to say that Model 3 statistically outperforms Model 2. In addition, the new model does not give discontinuous IFT model No. Characteristics IFT predictions, while Chen and Yang (2019)’s IFT model Model 1 Original Parachor model bears such issue. Model 2 Refitted Chen and Yang (2019)’s correla- tion with two coefficient sets Model 3 Newly proposed correlation (this study) 4 Conclusions Model 4 Refitted Hebach et al. (2002)’s correlation Model 5 Refitted Chen and Yang (2019)’s correla- The objective of this study is to screen and develop reliable tion with one coefficient set models for describing the VLE, LLE, phase density, and IFT of CO /H O mixtures. Based on the comparison between 2 2 the experimental data and the calculated ones from different 3.3.2 Statistical significance tests of IFT correlations models, we can reach the following conclusions: As shown in Table 11, the AADs yielded by Model 2 (refit- 1. The most accurate method to represent C O /H O VLE 2 2 ted Chen and Yang (2019)’s correlation) and Model 3 (this and LLE is PR EOS, Twu α function, and Huron-Vidal study) are on the same scale. Therefore, it is necessary to mixing rule, which only yields AAD of 5.49% and 2.90% conduct statistical significance tests to check if the marginal in reproducing measured C O /H O phase-composition 2 2 edge of Model 3 over Model 2 is statistically significant. data and phase-density data over a temperature range of Figure 1 shows the frequency distribution of the differences 278–378.15 and 278–478.35 K and over a pressure range between the measured IFT data (i.e., whole dataset includ- of 6.9–709.3 and 2.5–1291.1 bar, respectively. ing 778 data points) and calculated ones by Model 2, while 2. Applying either constant or Abudour et al. (2013) VT Fig. 7 shows the same information for Model 3. As can be method can significantly improve aqueous-phase density seen from Figs. 6 and 7, the distribution of the deviations calculations. In addition, when the temperature is higher generated by the two models can be considered to follow than 373 K, constant VT method can yield lower error in Gaussian distributions. As such, paired one-tailed t-tests are reproducing measured phase-density data than Abudour applied as the statistical significance test method (Japkowicz et al. (2013) VT method; and Shah 2011). 3. Constant VT method cannot improve the prediction P-value is used to check if one model is better than accuracy of C O -rich-phase density. Abudour et al. another one. Typically, the significance threshold α is 0.05; (2013) VT method can slightly improve CO -rich-phase when P >𝛼 , two models have the same performance. In density predictions, but such improvement is more obvi- contrast, when P ≤ , it is reasonable to say that one model ous at low to moderate temperature conditions. is significantly better than another one (Japkowicz and Shah 4. The new IFT correlation based on the aforementioned 2011). PR EOS model outperforms other empirical correlations P-value of Model 2 (refitted original Chen and Yang with an overall AAD of 7.77% in reproducing measured (2019)’s correlation with two coefficient sets) and Model 3 IFT data of C O /H O mixtures. The new IFT correla- 2 2 Table 11 Summary of the performance of different correlations in IFT estimations Evaluation metricsModel 1 Model 2 Model 3 Model 4 Model 5 Training dataset AAD, % – 7.5218 6.6893 10.6901 11.4002 MAE - 2.4232 2.1311 3.2532 3.8349 R – 0.9416 0.9547 0.9008 0.8586 Validation dataset AAD, % – 8.8812 8.8684 11.6494 13.3408 MAE – 2.6446 2.6064 3.4174 4.1864 R – 0.9116 0.9325 0.9044 0.8402 AAD, % 47.0902 7.8520 7.7683 10.9231 11.8716 Overall MAE 13.7870 2.4770 2.3586 3.2931 3.9203 R −0.7053 0.9372 0.9420 0.9017 0.8541 No refitted coefficients are applied in Parachor model. Instead, we directly apply Parachor model in IFT calculations. Thus, it is not necessary to distinguish between training and validation datasets 1 3 520 Petroleum Science (2021) 18:509–529 80 80 Experimental data Experimental data Model 1 (Parachor model) Model 1 (Parachor model) 70 70 Model 2 (Refitted Chen and Yang (2019)'s Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) correlation with two coefficient sets) Model 3 (This study) Model 3 (This study) 60 60 Model 4 (Refitted Hebach et al. (2002)'s Model 4 (Refitted Hebach et al. (2002)'s correlation) correlation) 50 50 Model 5 (Refitted Chen and Yang (2019)'s Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) correlation with one coefficient set) 40 40 30 30 20 20 VLE region LLE region 10 10 0 0 020406080 100 120 140 160 180 200 050100 150200 250 Pressure, bar Pressure, bar (a) T = 297.9 K (b) T =322.8 K 80 70 Experimental data Experimental data Model 1 (Parachor model) Model 1 (Parachor model) Model 2 (Refitted Chen and Yang (2019)'s Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) correlation with two coefficient sets) Model 3 (This study) Model 3 (This study) Model 4 (Refitted Hebach et al. (2002)'s Model 4 (Refitted Hebach et al. (2002)'s correlation) correlation) Model 5 (Refitted Chen and Yang (2019)'s Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) 40 correlation with one coefficient set) 0 0 050 100 150 200 250 300 350 400 0100 200300 400500 600 Pressure, bar Pressure, bar (c) T = 343.3 K (d) T =374.3 K 60 50 Experimental data Model 1 (Parachor model) Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) Model 3 (This study) Model 4 (Refitted Hebach et al. (2002)'s correlation) Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) Experimental data Model 1 (Parachor model) Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) Model 3 (This study) Model 4 (Refitted Hebach et al. (2002)'s correlation) Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) 0 0 050 100 150 200 250 300 0200 400600 8001000 Pressure, bar Pressure, bar (e) T = 398.15 K (f) T =422.04 K Fig. 4 IFT predictions at different temperature conditions by different models. At T = 297.9 K (a), VLE is transformed to LLE at p = 64 bar. Model 1 (Parachor model) shows a more deteriorating performance when the vapor CO -rich phase changes to a liquid phase. Experimental data are from the studies by Kvamme et al. (2007), Georgiadis et al. (2010), Liu et al. (2016), and Shariat et al. (2012) 1 3 IFT, mN/m IFT, mN/m IFT, mN/m IFT, mN/m IFT, mN/m IFT, mN/m Petroleum Science (2021) 18:509–529 521 Correlated IFT Correlated IFT 278 K 378 K 70 298 K 398 K 308 K 478 K 318 K 348 K 368 K 378 K 278 K 478 K 10 10 0 200 400 600 800 1000 0200 400600 8001000 Pressure, bar Pressure, bar (a) (b) Fig. 5 Plots of predicted IFTs vs. pressure by the newly proposed IFT correlation Model 5 at the temperature ranges of 278–368 K (a) and 378–478 K (b). The curves are plotted with an interval of 10 K. Experimental data are taken from previous studies by Heuer (1957), Chun and Wilkinson (1995), Park et al. (2005), Akutsu et al. (2007), Bachu and Bennion (2009), Bikkina et al. (2011), Shariat et al. (2012), and Liu et al. (2016) 120 0.12 140 0.14 120 0.12 100 0.10 100 0.10 80 0.08 80 0.08 60 0.06 60 0.06 40 0.04 40 0.04 20 0.02 20 0.02 0 0 0 0 -15 -10 -50 51015 -20 -15 -10 -50 51015 Differences between measured and correlated IFT, mN/m Differences between measured and correlated IFT, mN/m Fig. 7 Frequency distribution of the difference between the measured Fig. 6 Frequency distribution of the differences between the meas- IFT data (i.e., the whole dataset including 778 data points) and calcu- ured IFT data (i.e., the whole dataset including 778 data points) and lated ones by Model 3 (this study). Blue columns are instances, and calculated ones by Model 2 (refitted Chen and Yang (2019)’s correla- the red curve is probability density function which follows Gaussian tion with two coefficient sets). Blue columns are instances, and the distribution with μ = -0.2051 and σ = 3.2781 red curve is probability density function which follows Gaussian dis- tribution with μ = 0.0941 and σ = 3.3457 as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes tion is only slightly more accurate than the refitted Chen were made. The images or other third party material in this article are and Yang (2019)’s correlation with two coefficient sets. included in the article’s Creative Commons licence, unless indicated But the new correlation yields smooth IFT predictions, otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not avoiding the issue of discontinuous IFT predictions permitted by statutory regulation or exceeds the permitted use, you will yielded by Chen and Yang (2019)’s correlation. need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long 1 3 Instances Correlated/measured IFT, mN/m Probability density function Correlated/measured IFT, mN/m Instances Probability density function 522 Petroleum Science (2021) 18:509–529 Appendix A: PR EOS and α functions Table 12 BIPs correlations in the van der Waals mixing rule as obtained by Abudour et al. (2012a, b) The PR EOS (Peng and Robinson 1976) can be expressed as: Case No. kc = AT + B kd = AT + B ij ij RT a A B A B p = − (11) v − b v(v + b) + b(v − b) Case 2 0.00058 0.08149 0.00029 −0.31262 where p is the pressure in bar; v stands for the molar volume in cm /mol; T is the temperature in K; a is the attraction 6 2 kc and kd are the BIPs that need to be fitted. In this study, ij ij parameter with unit of bar cm /mol , and b is the repulsion 3 the linear temperature-dependent BIP correlations from the parameter with unit of cm /mol; a and b can be determined study by Abudour et al. (2012a, b) are applied for CO /H O 2 2 by Eqs. (12) and (13): mixtures. Table 12 lists the BIP correlations obtained by 2 2 R T Abudour et al. (2012a, b). a = 0.457535 (12) When vdW mixing rule is used in PR EOS, the fugacity coefficient can be written as: � � RT c ⎛ ⎞ � � Z + 1 + 2 B b = 0.077796 (13) bb 2aa bb ⎜ ⎟ i A i i p ln = (Z − 1) − ln(Z − B) − − ln � � i ⎜ √ ⎟ b a b m m m 2 2B ⎜ ⎟ Z − 1 + 2 B ⎝ ⎠ where R is the universal gas constant in J/(mol K); T is the (18) critical temperature in K; p is the critical pressure in bar; where and α is the so-called alpha function. The expression of Twu α function can be written as (Twu b + b i j bb = 2 z 1 + kd − b (19) et al. 1991): i j ij m j=1 N(M−1) MN T = T exp L 1 − T (14) r r � � where T is the reduced temperature; L, M and N are com- aa = z a a 1 − kc r (20) i j i j ij j=1 pound-specific parameters. The values of these parameters regressed by Martinez et al. (2018) are used in this study. where Z is the compressibility factor. For PR EOS, Z can be Gasem α function can be expressed by (Gasem et al. calculated by Eq. (21). 2001): 3 2 2 2 3 Z − (1 − B)Z + A − 3B − 2B Z − AB − B − B = 0 C+D+E (T) = exp A + BT 1 − T (15) (21) where where the values of correlation parameters A through E are a p 2.0, 0.836, 0.134, 0.508 and −0.0467, respectively. A = (22) 2 2 R T Appendix B: Summary of van der Waals b p B = . (23) one‑fluid mixing rule and its BIPs RT The van der Waals one-fluid mixing rule can be expressed as (van der Waals 1873): Appendix C: Summary of Huron–Vidal n n �� mixing rule and its BIPs a = z z a a 1 − kc (16) m i j i j ij i=1 j=1 In the Huron–Vidal mixing rule, the following equations are applied to calculate a and b (Huron and Vidal 1979): m m n n b + b i j b = z z 1 + kd n n (17) m i j ij b + b 2 i j i=1 j=1 b = z z (24) m i j i=1 j=1 where z is the molar fraction of the ith component in the mixture; a and b can be calculated by Eqs. (12) and (13); i i 1 3 Petroleum Science (2021) 18:509–529 523 E The derivation of the expression of the activity coefficient i ∞ a = b z − (25) in Huron-Vidal mixing rule is detailed in “Appendix H”. m m i i=1 where G is the excess Gibbs energy at infinite pressure, and Appendix D: Summary of volume translation Λ is an EOS-dependent parameter. For PR EOS, Λ = 0.62323 models applied in this study (Huron and Vidal 1979). The excess Gibbs energy corresponding to the non-ran- The constant volume translation can be expressed as (Penel- dom two-liquid (NRTL) (Zhao and Lvov 2016; Wong and oux et al. 1982; Jhaveri and Youngren 1988): Sandler 1992) model can be expressed by (Aasen et al. 2017; Huron and Vidal 1979): v = v − z c � � ∑ (34) corr EOS i i b z exp − i=1 ji j j ji ji j=1 G = RT z � � (26) ∞ n b z exp − i=1 k k ki ki where v is corrected molar volume in cm /mol; v k=1 corr EOS stands for PR-EOS-calculated molar volume in cm /mol; c where is the component-dependent volume shift parameter which can be determined by Eq. (35) (Young et al. 2017). Δg ji (27) ji RT c = s × b (35) i i i a The values of s used by Liu et al. (2016) are applied in g =− ii (28) this study ( s = 0.23170 and s =−0.15400). H O CO i 2 2 Abudour volume translation model can be expressed as (Abudour et al. 2012a, b): b b � � i j √ g =−2 g g 1 − k (29) ij ii jj ij 0.35 b + b i j v = v + c − (36) corr EOS c 0.35 + d The generalized BIP correlations for τ obtained by Aasen ij where is volume correction at the critical temperature in et al. (2017) are given below: cm /mol; d is the dimensionless distance function given by (Mathias et al. 1989; Abudour et al. 2012b): = 5.831 − 2.559 (30) RT T PR 0 0 d = (37) RT =−3.311 + 0.03770 (31) 3 where ρ is the molar density in mol/cm . The volume trans- RT T 0 0 lation function proposed by Abudour et al. (2013) was where T = 1000 K is the reference temperature. extended to mixtures by the following equations (Abudour When the Huron–Vidal mixing rule is used in PR EOS, et al. 2013): the fugacity coefficient can be calculated by (Zhao and Lvov 2016): � � ⎛ ⎞ � � Z + 1 + 2 B b a ln ⎜ ⎟ i 1 i i ln = (Z − 1) − ln(Z − B) − + ln √ � � ⎜ √ ⎟ (32) b b RT m i 2 2 ⎜ ⎟ Z − 1 + 2 B ⎝ ⎠ where lnγ is the activity coefficient of component i and can be expressed as (Zhao and Lvov 2016): � � ∑ � � � � � �� z b exp − � b z exp − z b exp − ji j j ji ji j=1 i j ij ij lj l l lj lj l=1 ln = � � + � � ⋅ − � � ∑ ∑ ∑ (33) i ij n n n z b exp − z b exp − z b exp − k k ki ki j=1 k k kj kj k k kj kj k=1 k=1 k=1 1 3 524 Petroleum Science (2021) 18:509–529 0.35 Table 13 Coefficients in Hebach et al. (2002)’s correlation v = v + c − (38) corr EOS m cm 0.35 + d Coefficients Original value Refitted value where (Abudour et al. 2013): b , g/(cm K) 0.00022 0.00022 b −1.9085 −1.9085 RT cm −2d k , mN/m 27.514 25.6836 c = c − 0.004 + c e (39) 0 m 1m 1m 6 2 cm k, cm /g −35.25 −218.4717 12 4 k, cm /g 31.916 9.3192 18 6 k, cm /g −91.016 −0.9621 c = z c (40) 3 1m i 1i k, cm /g 103.233 33.4068 i=1 k , mN/m 4.513 14.4970 2 6 k, g /cm 351.903 10.9290 PR 1 1 v1 d = − (41) RT RT a cm T cm 11 where ω is the acentric factor of the mixture (Abudour where T , p and δ are the critical temperature, critical cm cm cm et al. 2013): pressure and volume correction of the mixture at the criti- cal point, respectively. c is the specie-specific parameter 1i = z (48) m i i of component i and has a linear relationship with critical i=1 compressibility ( Z ) (Abudour et al. 2012b): where ω is the acentric factor of component i. c = 0.4266Z − 0.1101 (42) 1 c The term d can be derived using the original PR EOS Appendix E: Summary of existing IFT (Matheis et al. 2016): correlations for CO /H O mixtures 2 2 2a(v + b) v RT d = − (43) m Parachor model (Sugden 1930) can be expressed as below 2 2 RT 2 2 (v − b) cm −b + 2bv + v (Schechter and Guo 1998): The volume correction of the given mixture at the criti- M M cal point, δ , can be determined by (Abudour et al. 2013): cm = P x − y (49) i i i L V i=1 RT cm = 0.3074 − v (44) cm i ci where x and y are the mole fractions of component i in liq- i i cm i=1 uid and vapor phases, respectively; P is the Parachor value where v is the critical volume of component i; θ is the of component i ( P = 52 , P = 78 ) (Liu et al. 2016); ci i H O CO 2 2 M 3 M surface fraction of component i defined by (Abudour et al. is the molar density of liquid phase in mol/cm ; is the L V 2013): molar density of vapor phase in mol/cm . Hebach et al. (2002)’s correlation can be expressed as: 2∕3 z v � � �� ci √ (45) 2 i ∑ n 2∕3 = k 1 − exp k dd + k ⋅ dd + k ⋅ dd 0 1 2 3 z v i=1 ci (50) � � + k ⋅ dd + k exp k (dd − 0.9958) 4 5 6 The critical temperature of the mixture can be calculated via the following mixing rule (Abudour et al. 2013). where (Hebach et al. 2002): dd = − (51) T = T H O corr (46) cm i ci 2 i=1 𝜌 +b (304−T)(10×p) CO 0 2 3 3 The critical pressure of the mixture can be determined by 0.025 g∕cm <𝜌 < 0.25 g∕cm CO 𝜌 = 1000 corr the correlation proposed by Aalto et al. (1996): 𝜌 in other cases CO � � (52) 0.2905 − 0.085 RT m cm p = ∑ (47) cm i ci i=1 1 3 Petroleum Science (2021) 18:509–529 525 where is the CO -rich-phase density in g/cm ; is CO 2 H O 2 2 Aqueous-phase density data CO -rich-phase density data the aqueous-phase density in g/cm ; k to k and b to b are 2 0 6 0 1 Interfacial tension data empirical coefficients. The units of T , p, and dd are K, bar, Pure-component saturation curves 2 6 and g /cm , respectively. Table 13 lists the values of original and refitted coefficients. Chen and Yang (2019)’s correlation is given as: = C + C p + C lnK + C p + C lnK (53) 1 2 r 3 CO 4 r 5 H O 2 2 where σ is IFT in mN/m; p is the reduced pressure of CO ; r 2 C to C are empirical coefficients. To make fair compari- 1 5 sons, we refit these coefficients based on the IFT database employed in this study. Table 14 summarizes the values of 250300 350400 450500 these refitted coefficients. Temperature, K Appendix F: Pressure–temperature coverage Fig. 8 Pressure–temperature coverage of phase-density and IFT data collected from the literature. The solid curves stand for pure-CO of phase‑density and IFT data collected 2 (left) and pure-H O (right) saturation curves, respectively from the literature. Figure 8 depicts the pressure–temperature coverage of IFT data (i.e., around 28–31 mN/m) obtained by other stud- phase-density and IFT data collected from the literature over ies under similar conditions. 278–478.35 K and 2.5–1291.1 bar, and 278.15–477.59 K Figure 10 indicates that the measured data by Bachu and and 1–1200.96 bar, respectively. Bennion (2009) fall into the range of 16–19 mN/m over 307.15–314.15 K and 120–270 bar, which are significantly lower than the measured IFT values (i.e., around 30 mN/m) Appendix G: Experimental data selection obtained by other studies under similar conditions. No out- in IFT database employed in this study lier exists at other temperature and pressure conditions. We have removed these outliers in the IFT regression analysis. The collected IFT data are further screened to remove any obvious outliers. Figure 9 shows the identification of the outliers from the collected data over 40–60 bar and Appendix H: Derivation of activity 278.15–298.15 K, while Fig. 10 shows the identification coefficient in the fugacity expression of outliers from the collected data over 100–270 bar and when Huron‑Vidal mixing rule is used. 307.15–314.15 K. As seen in Fig. 9a, the measured IFT data by Chun and Similar to the approach used by Wong and Sandler (1992), Wilkinson (1995) and Park et al. (2005) fall into the range of the activity coefficient of component i can be expressed by 5–8 mN/m over 278.15–288.15 K and 40–60 bar, which are the following formula: significantly lower than the measured IFT data (i.e., around 22–28 mN/m) obtained by other studies under similar con- ln = (54) ditions. Figure 9b indicates that the measured IFT data i RT z by Chun and Wilkinson (1995) and Park et al. (2005) fall into the range of 10–14 mN/m over 293.15–298.15 K and where the excess Gibbs free energy can be expressed as 50–70 bar, which are significantly lower than the measured (Huron and Vidal 1979): Table 14 Refitted coefficients in Chen and Yang’s correlation Coefficient set Pressure range C C C C C 1 2 3 4 5 1 Full −64.7356 2.3405 16.3306 2.0919 −7.1593 2 p ≤ 73.8 bar −34.3182 6.5500 10.8716 7.9611 −7.9076 Else −49.7215 0.2460 18.0648 0.1813 −1.9879 1 3 Pressure, bar 526 Petroleum Science (2021) 18:509–529 80 80 T = 278.15 K, Chun and Wilkinson (1995) T = 293.15 K, Bachu and Bennion (2009) T = 278.5 K, Hebach et al. (2002) T = 293.15 K, Park et al. (2005) 70 70 T = 283.15 K, Chun and Wilkinson (1995) T = 298.15 K, Chun and Wilkinson (1995) T = 287 K, Hebach et al. (2002) T = 298.15 K, Bachu and Bennion (2009) 60 60 T = 288.15 K, Chun and Wilkinson (1995) T = 298.15 K, Park et al. (2005) 50 50 40 40 30 30 20 20 Outliers 10 Outliers 10 0 0 04 20 06080 100 120 140 160 180 200 050 100 150 200 250 300 Pressure, bar Pressure, bar (a) (b) Fig. 9 Identification of the outliers at T = 278.15–288.15 K (a) and T = 293.15–298.15 K (b). Outliers are from the studies by Chun and Wilkin- son (1995) and Park et al. (2005) � � b z exp − ji j j ji ji j=1 G = RT z � � (55) ∞ n b z exp − i=1 k k ki ki k=1 T = 307.15-314.15 K, Bachu and Bennion (2009) T = 307.8-309.6 K, Chiquet et al. (2007) T = 307.9-308.2 K, Hebach et al. (2002) To make the derivation process more intuitive, we can T = 312.8 K, Georgiadis et al. (2010) set i = 1 (the first component) and n = 2 (two compounds T = 313.3 K, Pereira et al. (2016) in the system). Then the excess Gibbs free energy can be expressed as: � � b z exp − j1 j j j1 j1 j=1 40 ∞ = z ⋅ � � RT b z exp − k k k1 k1 k=1 30 � � ∑ (56) b z exp − j2 j j j2 j2 j=1 Outliers + z ⋅ � � b z exp − k k k2 k2 k=1 Taking the partial derivative of the first term in the right 0 100 200 300400 500 hand side of Eq. (56) yields: Pressure, bar Fig. 10 Identification of the outliers at moderate temperature (307.15–314.15 K) conditions. Outliers are from the study by Bachu and Bennion (2009) � � �� b z exp − j1 j j j1 j1 j=1 z ⋅ � � b z exp − k k k1 k1 k=1 � � b z exp − j1 j j j1 j1 j=1 = � � + z (57) b z exp − k k k1 k1 k=1 � � � � � � ⎛ � � � � �� n ⎞ b exp − ⋅ b z exp − b exp − ⋅ b z exp − 1 11 11 j1 j j j1 j1 j=1 ⎜ ⎟ 11 1 11 11 k k k1 k1 k=1 ⋅ − ⎜ � � �� ⎟ ∑ ∑ 2 2 n n b z exp − b z exp − ⎜ ⎟ k k k1 k1 k k k1 k1 k=1 k=1 ⎝ ⎠ 1 3 Measured IFT, mN/m Measured IFT, mN/m Measured IFT, mN/m Petroleum Science (2021) 18:509–529 527 Taking the partial derivative of the second term in the right hand side of Eq. (56) yields: � � �� b z exp − j2 j j j2 j2 j=1 z ⋅ � � b z exp − k k k2 k2 k=1 � � � � � � ∑ (58) ⎛ � � � � �� n ⎞ b exp − ⋅ b z exp − b exp − ⋅ b z exp − 1 12 12 j2 j j j2 j2 j=1 ⎜ ⎟ 12 1 12 12 k k k2 k2 k=1 = z ⋅ − 2 ⎜ � � �� ⎟ ∑ ∑ 2 2 n n b z exp − b z exp − ⎜ ⎟ k k k2 k2 k k k2 k2 k=1 k=1 ⎝ ⎠ single-phase liquid densities. Fluid Phase Equilib. 2012;335:74– As such, Eq. 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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2020
ISSN: 1672-5107
eISSN: 1995-8226
DOI: 10.1007/s12182-020-00526-x
Publisher site: See Article on Publisher Site

Abstract

Phase behavior of carbon dioxide/water binary mixtures plays an important role in various CO -based industry processes. This work aims to screen a thermodynamic model out of a number of promising candidate models to capture the vapor–liq- uid equilibria, liquid–liquid equilibria, and phase densities of C O /H O mixtures. A comprehensive analysis reveals that 2 2 Peng–Robinson equation of state (PR EOS) (Peng and Robinson 1976), Twu α function (Twu et al. 1991), Huron–Vidal mixing rule (Huron and Vidal 1979), and Abudour et al. (2013) volume translation model (Abudour et al. 2013) is the best model among the ones examined; it yields average absolute percentage errors of 5.49% and 2.90% in reproducing the experimental phase composition data and density data collected in the literature. After achieving the reliable modeling of phase compositions and densities, a new IFT correlation based on the aforementioned PR EOS model is proposed through a nonlinear regression of the measured IFT data collected from the literature over 278.15–477.59 K and 1.00–1200.96 bar. Although the newly proposed IFT correlation only slightly improves the prediction accuracy yielded by the refitted Chen and Yang (2019)’s correlation (Chen and Yang 2019), the proposed correlation avoids the inconsistent predictions present in Chen and Yang (2019)’s correlation and yields smooth IFT predictions. Keywords Phase behavior · CO /H O binary mixtures · Volume translation · Interfacial tension 2 2 1 Introduction Due to their simplicity and good reliability, CEOSs such as SRK EOS (Soave 1972) and PR EOS (Peng and Robin- The interaction of CO with H O is frequently seen in sev- son 1976) are the most widely used thermodynamic models 2 2 eral subterranean processes (such as CO -based enhanced for the phase behavior modeling of C O /H O binary mix- 2 2 2 recovery and CO storage). Phase behavior of CO /H O tures (Aasen et al. 2017; Michelsen and Mollerup 2007). 2 2 2 mixtures under subterranean conditions plays a great role in Numerous articles have addressed phase-composition mod- affecting the overall efficiency of these processes. Thus how eling of CO /H O mixtures. Two types of methods, ϕ–ϕ 2 2 to accurately model the phase behavior of C O /H O mix- (fugacity–fugacity) approach and γ–ϕ (activity–fugacity) 2 2 tures becomes drastically important. Overall, an appropriate approach (Trusler 2017; Zhao and Lvov 2016), are often combination of cubic equation of state (CEOS), mixing rule applied in such modeling processes. Because γ-ϕ approach in CEOS, α function, volume translation, and interfacial ten- has a discontinuity issue in the phase diagram near the sion (IFT) model should be determined to well capture the critical region (Zhao and Lvov 2016), this work focuses vapor–liquid equilibrium (VLE), liquid–liquid equilibrium on ϕ–ϕ based methods. Valtz et al. (2004) found that the (LLE), phase density, and IFT of CO /H O mixtures. most accurate model is PR EOS (Peng and Robinson 1976), 2 2 Mathias–Copeman α function (Mathias and Copeman 1983), and Wong–Sandler mixing rule (Wong and Sandler Edited by Yan-Hua Sun 1992) with average absolute percentage deviation (AAD) of 5.4% in reproducing the measured phase composition * Huazhou Li data for C O /H O mixtures. However, the temperature 2 2 huazhou@ualberta.ca and pressure ranges used by Valtz et al. (2004) were nar- Department of Civil and Environmental Engineering, School row (278.2–318.2 K and 4.64–79.63 bar, respectively). In of Mining and Petroleum Engineering, University of Alberta, addition, the parameters in the Wong–Sandler mixing rule Edmonton, AB T6G 1H9, Canada Vol.:(0123456789) 1 3 510 Petroleum Science (2021) 18:509–529 (Wong and Sandler 1992) are given as discrete values at of CO /H O mixtures and achieved a significant improve- 2 2 different isotherms. Zhao and Lvov (2016) applied PRSV ment in density-prediction accuracies. However, a more EOS (Stryjek and Vera 1986) and the Wong–Sandler mix- accurate volume translation function, the one proposed by ing rule to calculate phase compositions, obtaining an AAD Abudour et al. (2012b, 2013), was not applied in Aasen et al. of 7.12% in reproducing the measured phase composition (2017)’s study; furthermore, it should be noted that Aasen data of CO /H O mixtures over a wide range of tempera- et al. (2017) used GERG-2008 (Kunz and Wagner 2012) 2 2 tures and pressures. Similar to Valtz et al. (2004)’s study, in and EOS-CG (Gernert and Span 2016) calculated densities the study by Zhao and Lvov (2016), the parameters in the as reference densities instead of experimental data. In this Wong–Sandler mixing rule are provided as discrete values at study, we apply the volume translation method by Abudour different isotherms, instead of generalized correlations; their et al. (2012b, 2013) to see if the use of this model can further model is inconvenient to use since one has to make extrapo- improve phase-density predictions for CO /H O mixtures; 2 2 lations based on the provided values when making predic- these predictions are compared to the measured density data tions at conditions different from those given by Zhao and documented in the literature. Lvov (2016). Abudour et al. (2012a) applied van der Waals Parachor model (Sugden 1930) is one of the most widely (1873) one-fluid (vdW) mixing rule with several tempera - applied models in predicting mixtures’ IFT (Schechter and ture-dependent BIP correlations in PR EOS in determining Guo 1998). However, its accuracy heavily relies on the den- phase compositions of CO /H O mixtures. With the tuned sity difference between the two coexisting phases in a VLE 2 2 BIPs, their model yielded good accuracy (i.e., AAD of 5.0%) or an LLE. Our experience in using Parachor model to cal- in aqueous phase-composition predictions but lower accu- culate IFT of CO /H O mixtures shows that Parachor model 2 2 racy (i.e., AAD of 13.0%) in C O -rich phase-composition is generally appropriate for the IFT estimation for VLE of predictions, respectively. CO /H O systems, but less suitable for the IFT estimation 2 2 A recent comprehensive study by Aasen et al. (2017) for LLE of C O /H O systems. This is primarily because an 2 2 revealed that the most accurate thermodynamic model LLE of a CO /H O mixture has a smaller density difference 2 2 (among the ones examined by them) in phase-composition than a VLE. Several empirical IFT correlations for CO / and phase-density predictions for CO /H O mixtures is PR H O mixtures have been proposed in the literature. However, 2 2 2 EOS, Twu α function (Twu et al. 1991), Huron-Vidal mix- most of these correlations are only applicable to a limited ing rule, and constant volume translation. This model only temperature and pressure range (Zhang et al. 2016). Hebach yields an AAD of 4.5% in phase-composition calculations et al. (2002) proposed a new correlation which correlated and an AAD of 2.8% in phase-density calculations for C O / IFT with phase densities. Hebach et al. (2002)’s model is H O mixtures. Aasen et al. (2017), Valtz et al. (2004), and suitable over a wide range of temperature and pressure con- Zhao and Lvov (2016) also pointed out that more advanced ditions, although the prediction accuracy decreases with an models [e.g., the Cubic-Plus-Association (CPA) EOS (Kon- increase in temperature or pressure. Chen and Yang (2019) togeorgis et al. 1996)] do not guarantee an improvement in proposed a new empirical IFT correlation for C O /CH /H O 2 4 2 the phase-composition predictions for CO /H O mixtures. ternary systems based on mutual solubility, and this model 2 2 With regards to phase-density calculations, CEOS based performs well for C O /H O binary mixtures. However, 2 2 methods tend to overestimate liquid-phase molar volumes. A our experience in applying Chen and Yang (2019)’s model detailed discussion of this issue can be found in the studies shows that some breaking points can be observed in the by Matheis et al. (2016) and Young et al. (2017). In order to predicted IFT curves under some conditions, hampering its address this problem, Martin (1979) introduced the volume ability in providing consistent and smooth IFT predictions. translation concept in CEOS to improve liquid-phase volu- In addition, using two sets of BIPs (as applied in Chen and metric predictions. Peneloux et al. (1982) developed vol- Yang (2019)’s study) in the aqueous phase and non-aqueous ume translation schemes in SRK EOS for pure substances. phase can lead to thermodynamic inconsistency issue near Jhaveri and Youngren (1988) applied volume translation the critical region as demonstrated by Li and Li (2019). into PR EOS, leading to the improvement of liquid phase- The discussion above reveals that the previous studies of density predictions. A thorough comparison of different phase behavior modeling of the CO /H O mixtures tend to 2 2 types of volume translation methods can be found in Young primarily focus on phase-composition modeling and pay less et al. (2017)’s work. According to the study by Young et al. attention to phase-density calculations (especially for the (2017), the temperature-dependent volume translation CO -rich phase). Whereas, phase-density is one important method developed by Abudour et al. (2012b, 2013) pro- property in VLE and LLE since IFT calculations and flow vides the most accurate estimates on liquid-phase densities simulations can heavily rely on such property. As for the IFT without thermodynamic inconsistencies (e.g., crossover of modeling, we are currently lacking a reliable IFT correla- pressure–volume isotherms). Aasen et al. (2017) applied tion that not only pays due tribute to the phase composition constant volume translation to phase-density calculations and density of CO /H O mixtures but also gives smooth and 2 2 1 3 Petroleum Science (2021) 18:509–529 511 consistent IFT predictions over a wider range of tempera- 2.2 Mixing rules ture/pressure conditions. In this study, we first conduct a thorough literature review Mixing rules have a great impact on phase equilibrium cal- to select the most promising thermodynamic models that culations. Huron and Vidal (1979) proposed a new expres- can well capture the VLE and LLE of CO /H O mixtures. sion by considering the excess Gibbs energy for CEOS, 2 2 Then, we conduct phase-composition calculations by using which made more accurate the phase-composition predic- PR EOS, Twu α function, and Huron-Vidal mixing rule [as tions for mixtures containing polar substances. Furthermore, suggested by Aasen et al. (2017)], and validate the accu- according to the comprehensive study by Aasen et al. (2017), racy of this model by comparing the calculated VLE and the most accurate thermodynamic model among the ones LLE phase compositions to the measured ones. Then, we examined by them is PR EOS coupled with Twu α function introduce Abudour et al. (2012b, 2013) volume translation and Huron-Vidal mixing rule, which provides an AAD of model in phase-density calculations to check if applying this 4.5% in reproducing the phase-composition data measured model can further improve the density-prediction accuracies. for CO /H O mixtures. Hence, in the first part of this study, 2 2 A new empirical IFT correlation for CO /H O mixtures is we collect more phase equilibria data for C O /H O mixtures 2 2 2 2 then proposed based on the reliable thermodynamic model to verify the performance of the model suggested by Aasen that incorporates the Huron-Vidal mixing rule and the Abu- et al. (2017). These additional experimental data are not dour et al. (2013) volume translation model. included in the study by Aasen et al. (2017). The vdW mixing rule is one of the most commonly used mixing rules in petroleum industry (Pedersen et al. 2014). 2 Methodology Although vdW mixing rule is originally developed for non- polar systems, the vdW mixing rule coupled with the tuned 2.1 PR EOS and α functions BIPs can be reliably used for describing the phase behav- ior of mixtures containing polar components (e.g., water). In this study, PR EOS (Peng and Robinson 1976) is imple- Besides, based on the study by Abudour et al. (2012a), mented because of its simplicity and more accurate liquid- Gasem α function with vdW mixing rule and their tem- density predictions compared with SRK EOS (Aasen et al. perature-dependent volume translation function provided 2017). The expression of PR EOS is detailed in “Appendix a promising means to well reproduce the measured liquid- A”. phase densities for CO /H O mixtures. Therefore, in this 2 2 With regards to α functions in PR EOS, Twu α func- study, we also employ the model suggested by Abudour tion and Gasem α function are used in this study. Compared et al. (2012a) to test if it outperforms the model suggested with the other types of α functions, the Twu α function can by Aasen et al. (2017). The expressions of vdW mixing rule describe the thermodynamic properties of polar compounds and Huron–Vidal mixing rule and their BIPs are detailed in more accurately and perform better in the supercritical “Appendices B and C”. region (Young et al. 2016). In addition, according to the study by Aasen et al. (2017), Twu α function coupled with 2.3 Volume translation models PR EOS and Huron-Vidal mixing rule yields the most accu- rate phase-composition estimations on CO /H O mixtures Volume translation is used to overcome the inherent defi- 2 2 among the models evaluated by them. Therefore, we select ciency of CEOS in liquid-phase-density predictions. In order Twu α function as one of the evaluated α functions in this to improve liquid-phase-density calculations, Peneloux et al. study. (1982) developed a constant volume translation model in The Gasem α function provides more accurate represen- SRK EOS, while Jhaveri and Youngren (1988) developed tation of supercritical phase behavior (Gasem et al. 2001). a constant volume translation model in PR EOS. Abudour Besides, based on the study by Abudour et al. (2013), Gasem et al. (2012b, 2013) revised the temperature-dependent α function coupled with PR EOS, vdW mixing rule and volume translation function to improve both saturated and Abudour volume translation yields the most accurate liq- single-phase liquid density calculations. Furthermore, unlike uid-phase-density predictions for the chemical compounds other temperature-dependent volume translation models, examined by their study. Therefore, we select the Twu α the volume translation model developed by Abudour et al. function and the Gasem α function in VLE/LLE and phase- (2012b, 2013) does not yield thermodynamic inconsist- density calculations. “Appendix A” shows the expressions ency issues unless at extremely high pressures. Therefore, of Twu α function and Gasem α function. we select the constant and Abudour et al. volume transla- tion models in this study for phase-density predictions. The expressions of these two volume translation models are detailed in “Appendix D”. 1 3 512 Petroleum Science (2021) 18:509–529 where σ in the interfacial tension; α is the introduced com- 2.4 IFT correlations for CO /H O mixtures 2 2 ponent-dependent correction term; x and y are the mole i i fractions of component i in liquid and vapor phases, respec- In this study, we select Parachor model (Sugden 1930), Hebach et al. (2002)’s correlation, and Chen and Yang tively; P is the Parachor value of component i ( P = 52 , i H O P = 78 ) (Liu et al. 2016); is the molar density of (2019)’s correlation to predict IFT of CO /H O mixtures. 2 2 CO 2 L 3 M The Parachor model (Sugden 1930) is one of the most liquid phase in mol/cm ; is the molar density of vapor phase in mol/cm . N is the number of component; n is the widely used methods in determining mixtures’ IFT. It cor- relates IFT with phase compositions and molar densities of exponent. First, the component-dependent correction term α is set each phase. Parachor is a component-dependent constant. The expression of the Parachor model is shown in “Appen- as a constant for each component, and the exponential term n can be expressed by the equilibrium ratios of C O -rich dix E”. Hebach et al. (2002)’s correlation correlates IFT with temperature, pressure, and phase densities. Phase composi- phase and aqueous phase: tions are not included in their correlation. To make fair com- n = C ln K + C ln K + C 1 CO 2 H O 3 (2) 2 2 parison, we also refit coefficients in their correlation based on the IFT database employed in this study. Values of the where C , C , and C are empirical coefficients; K and 1 2 3 CO original and refitted coefficients as well as the Hebach et al. K are the equilibrium ratios (as known as K-values) of H O (2002)’s correlation are shown in “Appendix E”. CO and H O: 2 2 Chen and Yang (2019)’s correlation correlates IFT with K = y ∕x phase equilibrium ratios (K-values) and the reduced pres- (3) i i i sure of CO . Unlike the Parachor model and the Hebach Since using one coefficient set for both α and n on the et al. (2002)’s correlation, the density of the two equilibrat- whole CO -rich-phse density range cannot converge after ing phases is not one input in the Chen and Yang (2019)’s reaching the maximum iterations, we use two coefficient correlation. Chen and Yang (2019)’s correlation, they pro- sets based on C O -rich-phase densities. Table 1 listed the posed four groups of coefficient sets, i.e., one coefficient values of these coefficients and α determined by fitting the set (using one coefficient set on the whole pressure range) proposed correlation (abbreviated as Scenario #1) to the IFT with or without the reduced pressure term, and two coef- training dataset. ficient sets (dedicated to the pressure ranges of p ≤ 73.8 and Since using constants to represent α leads to a larger p > 73.8 bar) with or without the reduced pressure term. AAD compared with the refitted Chen and Yang (2019)’s Since using the reduced pressure term can improve pre- correlation (i.e., 8.8746% vs. 7.8520%, respectively), we diction accuracy (Chen and Yang 2019), we introduce the correlate equilibrium ratios to α to see if it can improve reduced pressure term in this study. Similarly, we refit these the IFT predictions. The expression of n in this scenario coefficients based on the IFT databased employed in this (abbreviated as Scenario #2) is the same as that in Scenario study to make fair comparison. Values of the original and #1. The expression for α is given as: refitted coefficients as well as the Chen and Yang (2019)’s correlation are detailed in “Appendix E”. = C ln K + C ln K + C i 1 CO 2 H O 3 (4) 2 2 2.5 IFT correlation proposed in this study Specifically, when the CO -rich-phase density is greater than 0.2 g/cm , can be simplified as: H O Before we finalize our IFT correlation, we tried several sce- = C lnK + C 1 CO 3 (5) H O 2 narios to find the optimal one to correlate the IFT of CO / 2 H O mixtures. Since the Parachor model is one of the most widely used models in mixtures’ IFT predictions, we revise the original Parachor model by introducing a component- Table 1 Values of the correlation coefficients and α in Scenario #1 dependent correction term α ; furthermore, we replace the i 3a 3 constant exponential term in the original Parachor model Coefficients 𝜌 < 0.2 g/cm ≥ 0.2 g/cm CO -rich H O-rich 2 2 by correlating it with several physical properties (e.g., equi- 0.7957 0.1520 CO librium ratios). The new IFT correlation can be expressed 0.8855 0.9509 H O as follows: 2 C −0.0727 0.1026 C 0.1044 0.0736 M M = P (x − y ) (1) i i i i C 5.5730 3.9154 L V i=1 is the density of CO -rich phase CO -rich 2 1 3 Petroleum Science (2021) 18:509–529 513 Table 2 listed the values of these coefficients determined Table 3 Coefficients in the α term for H O and CO i 2 2 by fitting the proposed correlation to the IFT training data- Component C C C C C 1 2 3 4 5 set. Since using one coefficient set for α and n on the whole CO -rich-phse density range in Scenario #2 cannot converge H O 1.1325 −0.0085 −0.0083 0.0134 0.0089 after reaching the maximum iterations, we use two coeffi- CO −0.4193 −0.0057 −0.0320 0.0209 −0.1430 cients based on C O -rich-phase densities (the same as Sce- nario #1). We find that using correlations to represent α can slightly 3 Results and discussion improve the IFT predictions (i.e., AAD of 8.31% in Sce- nario #2 vs. AAD of 8.87% in Scenario #1). Besides, we The values of critical pressure (p ), critical temperature (T ), c c find that the value of n is around 4 over a wide range of tem- acentric factor (ω), molecular weight (M), critical compress- perature/pressure conditions in all scenarios (i.e., its value ibility factor (Z ) used in this study are retrieved from the only slightly changes with the change of equilibrium ratios); NIST database (Lemmon et al. 2011). therefore, we set the value of n as 4 for simplicity. However, our experience in applying Scenarios #1 and 3.1 Performance comparison of thermodynamic #2 shows that some breaking points can be observed in models in phase equilibrium calculations the correlated IFT curves due to the fact that two differ - ent sets of coefficients are adopted under the conditions of Table 4 summarizes the measured phase equilibrium data of 3 3 𝜌 < 0.2 g/cm and ≥ 0.2 g/cm , respectively. CO /H O mixtures over 278–378.15 K and 0.92–709.3 bar CO -rich H O -rich 2 2 2 2 In addition, based on the study by Chen and Yang (2019), reported in the literature. Note that these experimental data introducing the reduced pressure of CO can improve IFT were not included in the study by Aasen et al. (2017). Com- predictions. Thus, we introduce the reduced pressure of C O parison between the measured and calculated phase-compo- in the expressions of α and use one coefficient set to see if sition results is evaluated by the average absolute percentage these settings can further improve the prediction accuracies deviation (AAD) defined as: without yielding inconsistent IFT predictions. Based on the x − x calculation results, the following IFT correlation yields the 1 CAL EXP AAD = × 100% (8) lowest AAD among the ones examined in this study: N x EXP where AAD is the average absolute percentage deviation; N M M = P x − y (6) i i i i is the number of data points; x and x are the calculated L V CAL EXP i=1 and measured mole fraction of C O or H O in the aqueous 2 2 phase (or the CO -rich phase), respectively. where the α term in the new correlation can be expressed as: Table 5 details the settings of the four thermodynamic = C +(C p + C ) ln K +(C p + C ) ln K models examined in this work. Table 6 summarizes the 1 2 r 3 CO 4 r 5 H O (7) 2 2 performance of different thermodynamic models in phase- where p is the reduced pressure of CO . r 2 composition predictions. Table 3 lists the values of these coefficients determined by As shown in Table 6, although AAD for x of Case CO fitting the proposed correlation to the IFT training dataset. 3 (Twu + HV) is slightly higher than that of Case 4 (Gasem + HV), i.e., 4.73% of Case 3 vs. 3.64% of Case 4, Case 3 (Twu + HV) significantly outperforms the other models in y predictions, i.e., AAD of 10.37% of Case H O 3 vs. > 16% of other cases. Thus, given the overall perfor- mance, Case 3 (Twu + HV) is found to be the best model Table 2 Values of the correlation coefficients and α in Scenario #2 3 3 Coefficients 𝜌 < 0.2 g/cm ≥ 0.2 g/cm CO -rich H O-rich 2 2 C C C C C C 1 2 3 1 2 3 –0.4685 –0.2177 1.7944 0.4583 0.0107 –1.3451 CO –0.1033 0.0311 1.8397 0.5259 – –0.3583 H O n 0.3599 –0.0855 1.3153 –0.2685 0.0124 3.5123 1 3 514 Petroleum Science (2021) 18:509–529 Table 4 Phase equilibrium data of C O /H O mixtures employed in this study 2 2 a b c T, K p, bar x , % y , % N References AAD, % CO H O 2 2 323.15–373.15 25.3–709.3 0.429–3.002 – 29 Weibe and Gaddy (1939) 2.09 285.15–313.15 25.3–506.6 0.925–3.196 – 42 Weibe and Gaddy (1940) 5.50 323.15–373.15 200–500 2–2.8 1–3 4 Tödheide and Frank (1962) 1.90/34.69 288.71–366.45 6.9–202.7 0.0973–2.63 0.0819–12.03 24 Gillepsie and Wilson (1982) 6.68/5.93 323.15 68.2–176.8 1.651–2.262 0.339–0.643 8 Briones et al. (1987) 3.17/5.26 285.15–304.21 6.9–103.4 – 0.0603–0.33739 9 Song and Kobayashi (1987) 6.40 323.15–378.15 101.33–152 1.56–2.1 0.55–0.9 4 D’Souza et al. (1988) 4.62/16.81 348.15 103.4–209.4 1.91–1.92 0.63–0.84 2/3 Sako et al. (1991) 7.97/25.38 323.15 101–301 2.075–2.514 0.547–0.782 3 Dohrn et al. (1993) 1.85/15.53 278–293 64.4–294.9 2.5–3.49 – 24 Teng et al. (1997) 7.36 288–323 0.92–4.73 0.038–0.365 – 49 Dalmolin et al. (2006) 2.89 313.2–343.2 43.3–183.4 1.13–2.40 – 28 Han et al. (2009) 3.37 273.15–573.15 100–1200 0.89–14.96 – 130 Guo et al. (2014) 5.43 323.15–423.15 150 1.77–2.19 – 3 Zhao et al. (2015) 4.18 Solubility of CO in the aqueous phase Solubility of H O in CO -rich phase 2 2 AAD yielded by the Case 3 model (PR EOS, Twu α function, and Huron–Vidal mixing rule) in x and/or y predictions. If two numbers are CO H O 2 2 shown in table, the former indicates AAD in x prediction and the latter indicates AAD in y prediction. CO H O 2 2 d,e,f,g,h,i These data are already summarized by Spycher et al. (2003). We directly use these data mentioned in their paper for convenience N is 2 for x and 3 for y , respectively CO H O 2 2 Only experimental data for CO /pure water are selected in the study by Zhao et al. (2015) Table 5 Settings of four thermodynamic models examined in this work Case No. α function Mixing rule BIPs Case 1 Gasem et al. (2001) vdW (1873) kc = 0.27; kd = −0.21 (Abudour et al. 2012a) ij ij Case 2 Gasem et al. (2001) vdW (1873) kc (T); kd (T) (Abudour et al. 2012a) ij ij Case 3 Twu et al. (1991) Huron and Vidal (1979) Aasen et al. (2017) Case 4 Gasem et al. (2001) Huron and Vidal (1979) Aasen et al. (2017) The expressions of kc (T) and kd (T) are listed in “Appendix B” ij ij in phase-composition predictions. Figure 1 compares 3.2 Evaluation of thermodynamic models in density the performance of different models at T = 323.15 K and calculations T = 348.15 K. As can be seen from these two figures, the thermodynamic model Case 3 (Twu + HV) can well capture Table 7 summarizes the experimental aqueous-phase the trend exhibited by the measured solubility data over a and CO -rich-phase densities of CO /H O mixtures over 2 2 2 wide pressure range. 278–478.35 K and 2.5–1291.1 bar documented in the lit- erature. The pressure–temperature coverage of the phase density data collected from the literature are shown in Table 6 AAD of calculated mole fraction of CO in the aqueous phase “Appendix F”. ( x ) and mole fraction of H O in the CO -rich phase ( y ) by dif- CO 2 2 H O 2 2 ferent thermodynamic models Since Case 3 (Twu + HV) outperforms other thermody- namic models in phase-composition predictions for C O / Case No. AAD for x , % AAD for y , % CO H O 2 2 H O mixtures, we only focus on the performance of Case Case 1 57.81 19.26 3 coupled with volume translation in phase-density pre- Case 2 8.85 16.90 dictions. Table 8 summarizes the performance of differ - Case 3 4.73 10.37 ent volume translation models in both aqueous-phase and Case 4 3.64 16.59 CO -rich-phase density calculations. 1 3 Petroleum Science (2021) 18:509–529 515 10 10 Experimental data Experimental data Case 1 (Gasem + vdW + constant BIPs) Case 1 (Gasem + vdW + constant BIPs) Case 2 (Gasem + vdW + temperature-dependent BIPs) Case 2 (Gasem + vdW + temperature-dependent BIPs) Case 3 (Twu + HV) Case 3 (Twu + HV) Case 4 (Gasem + HV) Case 4 (Gasem + HV) 01020850 900950 1000 01020 700 800 900 1000 1000x , 1000y 1000x , 1000y CO2 H2O CO2 H2O (a)(b) Fig. 1 Measured and calculated pressure-composition data for CO /H O mixtures at T = 323.15 K (a) and T = 348.15 K (b). Solid circles are the 2 2 experimental data from the studies by Briones et al. (1987) and Gillepsie and Wilson (1982) Table 7 Aqueous-phase ( ) and CO -rich-phase ( ) density data of CO /H O mixtures employed in this study H O 2 CO -rich 2 2 2 2 3 3 a T, K p, bar , kg/m , kg/m N References AAD, % H O CO -rich 2 2 352.85–471.25 21.1–102.1 840–963 – 32 Nighswander et al. (1989) 3.01 288.15–298.15 60.8–202.7 1015–1027 – 27 King et al. (1992) 2.17 278–293 64.4–294.9 1013.68–1025.33 – 24 Teng et al. (1997) 1.35 304.1 10–70 999.4–1011.8 18.8–254.2 8 Yaginuma et al. (2000) 2.67/5.12 332.15 33.4–285.9 990.5–1010.3 – 29 Li et al. (2004) 3.80 283.8–333.19 10.8–306.6 983.7–1031.77 – 203 Hebach et al. (2004) 2.47 307.4–384.2 50–450 950.6–1026.1 80.8–987.5 43 Chiquet et al. (2007) 3.61/2.01 322.8–322.9 11–224.5 988.52–1009.13 18.8484–812.725 11 Kvamme et al. (2007) 3.37/1.94 382.41–478.35 34.82–1291.9 871.535–994.984 36.943–944.965 32/40 Tabasinejad et al. (2010) 4.85/4.43 298.15–333.15 14.8–207.9 984.6–1022 24.6–907.1 36 Bikkina et al. (2011) 3.08/3.15 292.7–449.6 2.5–638.9 905.9–1034.9 4.6–1023.4 145/128 Efika et al. (2016) 3.56/2.01 AAD yielded by Case 3–1 model (PR EOS, Twu α function, Huron–Vidal mixing rule, and Abudour volume translation model) for and/or H O predictions. If two numbers are shown in table, the former indicates AAD for prediction and the latter indicates AAD for CO -rich H O CO -rich 2 2 2 prediction N is 30 for and N is 40 for H O CO -rich 2 2 N is 144 for and N is 128 for H O CO -rich 2 2 As shown in Table 8, incorporation of VT into the ther- It can be seen from Fig. 2 that, regarding aqueous- modynamic framework can generally improve the phase- phase density predictions, the performance of Case 3–2 density prediction accuracy. Case 3–1 (Twu + HV + Abudour (Twu + HV + Constant VT) improves dramatically as tem- VT) provides the most accurate estimates of both aqueous- perature rises. As shown in Fig. 2e, f, at high temperature phase and C O -rich-phase density, yielding AAD of 2.90% conditions, Cases 3–2 yields the most accurate aqueous- in reproducing the measured phase-density data. Figure 2 phase density predictions; however, it fails to accurately further visualizes some of the calculation results by these predict CO -rich-phase densities. As a lighter phase, three different models at different pressure/temperature CO -rich-phase density can be accurately predicted without conditions. the use of volume translation functions. Applying Abudour VT method is able to only slightly improve the prediction 1 3 Pressure, bar Pressure, bar 516 Petroleum Science (2021) 18:509–529 Table 8 AAD of the calculated aqueous-phase density ( ) and CO -rich-phase density ( ) by different thermodynamic models H O 2 CO -rich 2 2 Model AAD for , % AAD for ,% Average AAD, % H O CO -rich 2 2 Case 3–1 (Twu + HV + Abudour VT ) 3.04 2.62 2.90 Case 3–2 (Twu + HV + Constant VT) 4.49 7.86 5.51 Case 3 (Base case) (Twu + HV) 15.08 3.38 11.44 VT: volume translation accuracy (i.e., AAD of 2.62%). In contrast, applying con- provides reliable phase-composition and phase-density pre- stant VT in CO -rich-phase density predictions can lead to dictions that can be fed into the proposed IFT correlation. larger errors than the case without the use of VT. Mean absolute errors (MAE), AAD, and coefficient of Figure 3 compares the performance of different mod- determination (R ) are used as performance measures. The els in terms of their accuracy in phase-density predictions expressions of MAE and R are as follows: over 382.14–478.35 K and 35.3–1291.9 bar. Note that the results of CPA EOS model from the work by Tabasinejad MAE = − (9) EXP,i CAL,i et al. (2010) focuses on the same pressure and temperature i=1 ranges. As can be seen from Fig. 3, although the CPA EOS model can accurately predict the aqueous-phase density, it � � EXP,i CAL,i i=1 tends to be less accurate in determining the C O -rich-phase 2 R = 1 − (10) � � density. Overall, the thermodynamic model Cases 3–1 EXP,i EXP i=1 (Twu + HV + Abudour VT) give an accuracy comparable to where σ is the measured IFT data in mN/m; σ is the the more complex CPA EOS model. EXP CAL In addition, according to the study by Aasen et al. (2017), calculated IFT in mN/m by different correlations; is the EXP average of the measured IFTs in mN/m. CPA EOS model yields higher percentage errors (AAD) in reproducing phase-composition data for C O /H O mixtures 2 2 3.3.1 Performance of different IFT correlations compared with Case 3 (PR EOS + Twu + HV), i.e., 9.5% vs. 4.5% (Aasen et al. 2017). Therefore, overall, Case 3–1 Table 10 shows the details of the different IFT models exam- (Twu + HV + Abudour VT) is a more accurate model in both phase-composition and phase-density predictions for CO / ined in this study. Table 11 summarize the performance of different correlations in IFT estimations. As can be seen, the H O mixtures. most accurate IFT model is Model 3 proposed in this study, although it only shows a marginal edge over Model 2. 3.3 Evaluation of the newly proposed IFT correlation Figure 4 visually compares the measured IFTs vs. pres- sure and the calculated ones by different IFT models at Table 9 summarizes the experimental IFT data of C O /H O selected temperatures. As shown in these plots, in general, 2 2 Model 3 (this study) outperforms other empirical correla- mixtures over 278.15–477.59 K and 1–1200.96 bar docu- mented in the literature. Ideally, phase densities should be tions over a wide range of temperatures and pressures. It can be also observed from these plots that breaking points appear directly measured; however, only Chiquet et al. (2007), Kvamme et al. (2007), Bikkina et al. (2011), Bachu and in the predicted IFT curves at p = 73.8 bar by Model 2 (Refit- ted Chen and Yang (2019)’s correlation with two coefficient Bennion (2009), and Shariat et al. (2012) applied measured phase densities in IFT calculations. In order to expand our sets). Such discontinuous IFT prediction can be attributed to the fact that two different sets of coefficients are adopted IFT database, IFT data with precisely determined phase densities are also included in our IFT database. The col- under the conditions of p ≤ 73.8 and p > 73.8 bar, respec- tively, in Chen and Yang (2019)’s correlation. Although lected IFT data are randomly placed into two bins: a training dataset (including 589 data points) and a validation dataset using one coefficient in Chen and Yang (2019)’s correlation (e.g., Model 5) can avoid such discontinuous IFT predic- (including 189 data points). Results in Sects. 3.1 and 3.2 reveal that the thermody- tions, it yields larger percentage errors. Therefore, Model 3 (this study) is the best model in IFT predictions for C O /H O namic model using PR EOS, Twu α function, Huron-Vidal 2 2 mixing rule, and Abudour et al. (2013) VT yields the most mixtures over a wide range of temperatures and pressures. Figure 5 illustrates how the IFTs predicted by Model 3 accurate estimates on both phase compositions and densi- ties. Therefore, the aforementioned thermodynamic model (this study) vary with pressure at different temperatures. It can be observed from Fig. 5 that the new IFT correlation 1 3 Petroleum Science (2021) 18:509–529 517 500 500 Aqueous-phase density Aqueous-phase density CO2-rich-phase density CO2-rich-phase density 400 400 300 300 200 200 100 100 0 0 00.2 0.40.6 0.81.0 1.2 00.2 0.40.6 0.81.0 1.2 3 3 Aqueous-phase/CO -rich-phase density, g/cm Aqueous-phase/CO -rich-phase density, g/cm 2 2 (a) T = 297.8 K (b) T =322.8 K 500 500 Aqueous-phase density Aqueous-phase density CO2-rich-phase density CO2-rich-phase density 400 400 300 300 200 200 100 100 0 0 00.2 0.40.6 0.81.0 1.2 00.2 0.40.6 0.81.0 1.2 3 3 Aqueous-phase/CO -rich-phase density, g/cm Aqueous-phase/CO -rich-phase density, g/cm 2 2 (c) T = 342.8 K (d) T =373.0 K 500 500 Aqueous-phase density Aqueous-phase density CO -rich-phase density CO -rich-phase density 2 2 400 400 300 300 200 200 100 100 0 0 00.2 0.40.6 0.81.0 1.2 00.2 0.40.6 0.81.0 3 3 Aqueous-phase/CO -rich-phase density, g/cm Aqueous-phase/CO -rich-phase density, g/cm 2 2 (e) T = 398.4 K (f) T =448.5 K Fig. 2 Predictions of aqueous-phase and C O -rich-phase density by Case 3–1 (Twu + HV + Abudour VT, dashed line), Case 3–2 (Twu + HV + constant VT, dotted line) and Case 3 (Base case, solid line) at different temperature conditions. The circles are the measured phase- density data from the study by Efika et al. (2016) 1 3 Pressure, bar Pressure, bar Pressure, bar Pressure, bar Pressure, bar Pressure, bar 518 Petroleum Science (2021) 18:509–529 lower temperature conditions (i.e., T < 378 K). It is interest- Aqueous-phase density 16.15 ing to observe from Fig. 5a that when the pressure is less CO -rich-phase density than around 15 bar and the temperature is between 278.15 and 368.15 K, an increase in temperature leads to a decrease in the predicted IFT under the same pressure. In comparison, 11.11 when the pressure is larger than around 15 bar, an increase in temperature leads to an increase in the predicted IFT. At higher temperatures of 378.15–478.15 K, an increase in temperature always results in a decline in the predicted 5.60 4.57 IFT under the same pressure, as seen in Fig. 5b. Most of 4.30 4 3.37 the measured IFTs documented in the literature follow this 1.77 2 trend (Akutsu et al. 2007; Chalbaud et al. 2009; Chiquet 0.25 et al. 2007; Chun and Wilkinson 1995; Da Rocha et al. 1999; Abudour VT Constant VT No VT appliedCPA EOS Georgiadis et al. 2010; Hebach et al. 2002; Heuer 1957; Hough et al. 1959; Kvamme et al. 2007; Khosharay and Fig. 3 Bar chart plots comparing the AAD in aqueous-phase (black) Varaminian 2014; Liu et al. 2016; Park et al. 2005; Pereira and CO -rich-phase (gray) density predictions by different models et al. 2016; Shariat et al. 2012), except for the studies by over 382.14–478.35 K and 35.3–1291.9 bar. Calculation results by Bachu and Bennion (2009) and Bikkina et al. (2011), i.e., the CPA EOS method are from the study by Tabasinejad et al. (2010) an increase in temperature leads to an increase in IFT at a temperature range of 373.15–398.15 K in the study by Bachu provides smooth and consistent IFT predictions at differ - and Bennion (2009), and an increase in temperature leads ent pressures and temperatures. Overall, Model 3 proposed to an increase in IFT over 298.15–333.15 K in the study in this study yields accurate and consistent IFT predictions by Bikkina et al. (2011). Again, the sharp drops in the IFT over the wide range of temperatures and pressures, although curves at lower temperatures (where C O remains subcriti- it yields relatively higher percentage errors at higher tem- cal) are due to the transformation of VLE to LLE. perature conditions (e.g., T = 478 K) compared with that at Table 9 Measured IFT data for CO /H O mixtures used in this study 2 2 T, K p, bar IFT, mN/m N References AAD, % 311–411 1–689.48 17.40–58.40 58 Heuer (1957) 9.13 311.15–344.15 1–197.8 17.63–69.20 28 Hough et al. (1959) 16.73 278.15–344.15 1–186.1 18.27–74.27 114 Chun and Wilkinson (1995) 6.18 311.15–344.15 1.6–310.7 19.38–56.86 20 Da Rocha et al. (1999) 13.26 278.4–333.3 1–200.3 12.4–74 85 Hebach et al. (2002) 3.76 293.15–344.15 1–173.2 20.55–78.01 26 Park et al. (2005) 7.19 318.15 11.6–165.6 25.4–70.5 14 Akutsu et al. (2007) 8.38 322.8–322.9 11–224.5 29.1–63.7 11 Kvamme et al. (2007) 4.80 307.4–384.2 50–450 45.8–22.8 43 Chiquet et al. (2007) 8.57 293.15–398.15 20–270 18.9–68.1 87 Bachu and Bennion (2009) 8.89 344.15 28.57–245.24 25.49–45.01 11 Chalbaud et al. (2009) 10.86 297.8–374.3 10–600.6 21.23–65.73 80 Georgiadis et al. (2010) 3.13 298.15–333.15 14.8–207.9 22.16–59.66 36 Bikkina et al. (2011) 11.29 323.15–477.59 77.78–1200.96 10.37–35.38 21 Shariat et al. (2012) 15.89 284.15–312.15 10–60 29.02–66.98 30 Khosharay and Varaminian (2014) 3.37 298.4–469.4 3.4–691.4 12.65–68.52 78 Pereira et al. (2016) 6.46 299.8–398.15 7.86–344.12 28.04–68.23 36 Liu et al. (2016) 9.89 AAD of the new IFT correlation proposed in this study (abbreviated as Model 3) b,c,e These data are already summarized by Park et al. (2005) and Shariat et al. (2012). We directly use these data mentioned in their papers for convenience d,f,g Some experimental data appear to be outliers and hence excluded for further analysis due to the significant deviation from other experimental data at similar temperature and pressure conditions (see Appendix G) 1 3 Average absolute percentage deviation, % Petroleum Science (2021) 18:509–529 519 (this study) is 0.0069, P <𝛼 ( = 0.05 ); therefore, it is rea- Table 10 Technical Characteristics of different IFT models examined in this study sonable to say that Model 3 statistically outperforms Model 2. In addition, the new model does not give discontinuous IFT model No. Characteristics IFT predictions, while Chen and Yang (2019)’s IFT model Model 1 Original Parachor model bears such issue. Model 2 Refitted Chen and Yang (2019)’s correla- tion with two coefficient sets Model 3 Newly proposed correlation (this study) 4 Conclusions Model 4 Refitted Hebach et al. (2002)’s correlation Model 5 Refitted Chen and Yang (2019)’s correla- The objective of this study is to screen and develop reliable tion with one coefficient set models for describing the VLE, LLE, phase density, and IFT of CO /H O mixtures. Based on the comparison between 2 2 the experimental data and the calculated ones from different 3.3.2 Statistical significance tests of IFT correlations models, we can reach the following conclusions: As shown in Table 11, the AADs yielded by Model 2 (refit- 1. The most accurate method to represent C O /H O VLE 2 2 ted Chen and Yang (2019)’s correlation) and Model 3 (this and LLE is PR EOS, Twu α function, and Huron-Vidal study) are on the same scale. Therefore, it is necessary to mixing rule, which only yields AAD of 5.49% and 2.90% conduct statistical significance tests to check if the marginal in reproducing measured C O /H O phase-composition 2 2 edge of Model 3 over Model 2 is statistically significant. data and phase-density data over a temperature range of Figure 1 shows the frequency distribution of the differences 278–378.15 and 278–478.35 K and over a pressure range between the measured IFT data (i.e., whole dataset includ- of 6.9–709.3 and 2.5–1291.1 bar, respectively. ing 778 data points) and calculated ones by Model 2, while 2. Applying either constant or Abudour et al. (2013) VT Fig. 7 shows the same information for Model 3. As can be method can significantly improve aqueous-phase density seen from Figs. 6 and 7, the distribution of the deviations calculations. In addition, when the temperature is higher generated by the two models can be considered to follow than 373 K, constant VT method can yield lower error in Gaussian distributions. As such, paired one-tailed t-tests are reproducing measured phase-density data than Abudour applied as the statistical significance test method (Japkowicz et al. (2013) VT method; and Shah 2011). 3. Constant VT method cannot improve the prediction P-value is used to check if one model is better than accuracy of C O -rich-phase density. Abudour et al. another one. Typically, the significance threshold α is 0.05; (2013) VT method can slightly improve CO -rich-phase when P >𝛼 , two models have the same performance. In density predictions, but such improvement is more obvi- contrast, when P ≤ , it is reasonable to say that one model ous at low to moderate temperature conditions. is significantly better than another one (Japkowicz and Shah 4. The new IFT correlation based on the aforementioned 2011). PR EOS model outperforms other empirical correlations P-value of Model 2 (refitted original Chen and Yang with an overall AAD of 7.77% in reproducing measured (2019)’s correlation with two coefficient sets) and Model 3 IFT data of C O /H O mixtures. The new IFT correla- 2 2 Table 11 Summary of the performance of different correlations in IFT estimations Evaluation metricsModel 1 Model 2 Model 3 Model 4 Model 5 Training dataset AAD, % – 7.5218 6.6893 10.6901 11.4002 MAE - 2.4232 2.1311 3.2532 3.8349 R – 0.9416 0.9547 0.9008 0.8586 Validation dataset AAD, % – 8.8812 8.8684 11.6494 13.3408 MAE – 2.6446 2.6064 3.4174 4.1864 R – 0.9116 0.9325 0.9044 0.8402 AAD, % 47.0902 7.8520 7.7683 10.9231 11.8716 Overall MAE 13.7870 2.4770 2.3586 3.2931 3.9203 R −0.7053 0.9372 0.9420 0.9017 0.8541 No refitted coefficients are applied in Parachor model. Instead, we directly apply Parachor model in IFT calculations. Thus, it is not necessary to distinguish between training and validation datasets 1 3 520 Petroleum Science (2021) 18:509–529 80 80 Experimental data Experimental data Model 1 (Parachor model) Model 1 (Parachor model) 70 70 Model 2 (Refitted Chen and Yang (2019)'s Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) correlation with two coefficient sets) Model 3 (This study) Model 3 (This study) 60 60 Model 4 (Refitted Hebach et al. (2002)'s Model 4 (Refitted Hebach et al. (2002)'s correlation) correlation) 50 50 Model 5 (Refitted Chen and Yang (2019)'s Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) correlation with one coefficient set) 40 40 30 30 20 20 VLE region LLE region 10 10 0 0 020406080 100 120 140 160 180 200 050100 150200 250 Pressure, bar Pressure, bar (a) T = 297.9 K (b) T =322.8 K 80 70 Experimental data Experimental data Model 1 (Parachor model) Model 1 (Parachor model) Model 2 (Refitted Chen and Yang (2019)'s Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) correlation with two coefficient sets) Model 3 (This study) Model 3 (This study) Model 4 (Refitted Hebach et al. (2002)'s Model 4 (Refitted Hebach et al. (2002)'s correlation) correlation) Model 5 (Refitted Chen and Yang (2019)'s Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) 40 correlation with one coefficient set) 0 0 050 100 150 200 250 300 350 400 0100 200300 400500 600 Pressure, bar Pressure, bar (c) T = 343.3 K (d) T =374.3 K 60 50 Experimental data Model 1 (Parachor model) Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) Model 3 (This study) Model 4 (Refitted Hebach et al. (2002)'s correlation) Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) Experimental data Model 1 (Parachor model) Model 2 (Refitted Chen and Yang (2019)'s correlation with two coefficient sets) Model 3 (This study) Model 4 (Refitted Hebach et al. (2002)'s correlation) Model 5 (Refitted Chen and Yang (2019)'s correlation with one coefficient set) 0 0 050 100 150 200 250 300 0200 400600 8001000 Pressure, bar Pressure, bar (e) T = 398.15 K (f) T =422.04 K Fig. 4 IFT predictions at different temperature conditions by different models. At T = 297.9 K (a), VLE is transformed to LLE at p = 64 bar. Model 1 (Parachor model) shows a more deteriorating performance when the vapor CO -rich phase changes to a liquid phase. Experimental data are from the studies by Kvamme et al. (2007), Georgiadis et al. (2010), Liu et al. (2016), and Shariat et al. (2012) 1 3 IFT, mN/m IFT, mN/m IFT, mN/m IFT, mN/m IFT, mN/m IFT, mN/m Petroleum Science (2021) 18:509–529 521 Correlated IFT Correlated IFT 278 K 378 K 70 298 K 398 K 308 K 478 K 318 K 348 K 368 K 378 K 278 K 478 K 10 10 0 200 400 600 800 1000 0200 400600 8001000 Pressure, bar Pressure, bar (a) (b) Fig. 5 Plots of predicted IFTs vs. pressure by the newly proposed IFT correlation Model 5 at the temperature ranges of 278–368 K (a) and 378–478 K (b). The curves are plotted with an interval of 10 K. Experimental data are taken from previous studies by Heuer (1957), Chun and Wilkinson (1995), Park et al. (2005), Akutsu et al. (2007), Bachu and Bennion (2009), Bikkina et al. (2011), Shariat et al. (2012), and Liu et al. (2016) 120 0.12 140 0.14 120 0.12 100 0.10 100 0.10 80 0.08 80 0.08 60 0.06 60 0.06 40 0.04 40 0.04 20 0.02 20 0.02 0 0 0 0 -15 -10 -50 51015 -20 -15 -10 -50 51015 Differences between measured and correlated IFT, mN/m Differences between measured and correlated IFT, mN/m Fig. 7 Frequency distribution of the difference between the measured Fig. 6 Frequency distribution of the differences between the meas- IFT data (i.e., the whole dataset including 778 data points) and calcu- ured IFT data (i.e., the whole dataset including 778 data points) and lated ones by Model 3 (this study). Blue columns are instances, and calculated ones by Model 2 (refitted Chen and Yang (2019)’s correla- the red curve is probability density function which follows Gaussian tion with two coefficient sets). Blue columns are instances, and the distribution with μ = -0.2051 and σ = 3.2781 red curve is probability density function which follows Gaussian dis- tribution with μ = 0.0941 and σ = 3.3457 as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes tion is only slightly more accurate than the refitted Chen were made. The images or other third party material in this article are and Yang (2019)’s correlation with two coefficient sets. included in the article’s Creative Commons licence, unless indicated But the new correlation yields smooth IFT predictions, otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not avoiding the issue of discontinuous IFT predictions permitted by statutory regulation or exceeds the permitted use, you will yielded by Chen and Yang (2019)’s correlation. need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long 1 3 Instances Correlated/measured IFT, mN/m Probability density function Correlated/measured IFT, mN/m Instances Probability density function 522 Petroleum Science (2021) 18:509–529 Appendix A: PR EOS and α functions Table 12 BIPs correlations in the van der Waals mixing rule as obtained by Abudour et al. (2012a, b) The PR EOS (Peng and Robinson 1976) can be expressed as: Case No. kc = AT + B kd = AT + B ij ij RT a A B A B p = − (11) v − b v(v + b) + b(v − b) Case 2 0.00058 0.08149 0.00029 −0.31262 where p is the pressure in bar; v stands for the molar volume in cm /mol; T is the temperature in K; a is the attraction 6 2 kc and kd are the BIPs that need to be fitted. In this study, ij ij parameter with unit of bar cm /mol , and b is the repulsion 3 the linear temperature-dependent BIP correlations from the parameter with unit of cm /mol; a and b can be determined study by Abudour et al. (2012a, b) are applied for CO /H O 2 2 by Eqs. (12) and (13): mixtures. Table 12 lists the BIP correlations obtained by 2 2 R T Abudour et al. (2012a, b). a = 0.457535 (12) When vdW mixing rule is used in PR EOS, the fugacity coefficient can be written as: � � RT c ⎛ ⎞ � � Z + 1 + 2 B b = 0.077796 (13) bb 2aa bb ⎜ ⎟ i A i i p ln = (Z − 1) − ln(Z − B) − − ln � � i ⎜ √ ⎟ b a b m m m 2 2B ⎜ ⎟ Z − 1 + 2 B ⎝ ⎠ where R is the universal gas constant in J/(mol K); T is the (18) critical temperature in K; p is the critical pressure in bar; where and α is the so-called alpha function. The expression of Twu α function can be written as (Twu b + b i j bb = 2 z 1 + kd − b (19) et al. 1991): i j ij m j=1 N(M−1) MN T = T exp L 1 − T (14) r r � � where T is the reduced temperature; L, M and N are com- aa = z a a 1 − kc r (20) i j i j ij j=1 pound-specific parameters. The values of these parameters regressed by Martinez et al. (2018) are used in this study. where Z is the compressibility factor. For PR EOS, Z can be Gasem α function can be expressed by (Gasem et al. calculated by Eq. (21). 2001): 3 2 2 2 3 Z − (1 − B)Z + A − 3B − 2B Z − AB − B − B = 0 C+D+E (T) = exp A + BT 1 − T (15) (21) where where the values of correlation parameters A through E are a p 2.0, 0.836, 0.134, 0.508 and −0.0467, respectively. A = (22) 2 2 R T Appendix B: Summary of van der Waals b p B = . (23) one‑fluid mixing rule and its BIPs RT The van der Waals one-fluid mixing rule can be expressed as (van der Waals 1873): Appendix C: Summary of Huron–Vidal n n �� mixing rule and its BIPs a = z z a a 1 − kc (16) m i j i j ij i=1 j=1 In the Huron–Vidal mixing rule, the following equations are applied to calculate a and b (Huron and Vidal 1979): m m n n b + b i j b = z z 1 + kd n n (17) m i j ij b + b 2 i j i=1 j=1 b = z z (24) m i j i=1 j=1 where z is the molar fraction of the ith component in the mixture; a and b can be calculated by Eqs. (12) and (13); i i 1 3 Petroleum Science (2021) 18:509–529 523 E The derivation of the expression of the activity coefficient i ∞ a = b z − (25) in Huron-Vidal mixing rule is detailed in “Appendix H”. m m i i=1 where G is the excess Gibbs energy at infinite pressure, and Appendix D: Summary of volume translation Λ is an EOS-dependent parameter. For PR EOS, Λ = 0.62323 models applied in this study (Huron and Vidal 1979). The excess Gibbs energy corresponding to the non-ran- The constant volume translation can be expressed as (Penel- dom two-liquid (NRTL) (Zhao and Lvov 2016; Wong and oux et al. 1982; Jhaveri and Youngren 1988): Sandler 1992) model can be expressed by (Aasen et al. 2017; Huron and Vidal 1979): v = v − z c � � ∑ (34) corr EOS i i b z exp − i=1 ji j j ji ji j=1 G = RT z � � (26) ∞ n b z exp − i=1 k k ki ki where v is corrected molar volume in cm /mol; v k=1 corr EOS stands for PR-EOS-calculated molar volume in cm /mol; c where is the component-dependent volume shift parameter which can be determined by Eq. (35) (Young et al. 2017). Δg ji (27) ji RT c = s × b (35) i i i a The values of s used by Liu et al. (2016) are applied in g =− ii (28) this study ( s = 0.23170 and s =−0.15400). H O CO i 2 2 Abudour volume translation model can be expressed as (Abudour et al. 2012a, b): b b � � i j √ g =−2 g g 1 − k (29) ij ii jj ij 0.35 b + b i j v = v + c − (36) corr EOS c 0.35 + d The generalized BIP correlations for τ obtained by Aasen ij where is volume correction at the critical temperature in et al. (2017) are given below: cm /mol; d is the dimensionless distance function given by (Mathias et al. 1989; Abudour et al. 2012b): = 5.831 − 2.559 (30) RT T PR 0 0 d = (37) RT =−3.311 + 0.03770 (31) 3 where ρ is the molar density in mol/cm . The volume trans- RT T 0 0 lation function proposed by Abudour et al. (2013) was where T = 1000 K is the reference temperature. extended to mixtures by the following equations (Abudour When the Huron–Vidal mixing rule is used in PR EOS, et al. 2013): the fugacity coefficient can be calculated by (Zhao and Lvov 2016): � � ⎛ ⎞ � � Z + 1 + 2 B b a ln ⎜ ⎟ i 1 i i ln = (Z − 1) − ln(Z − B) − + ln √ � � ⎜ √ ⎟ (32) b b RT m i 2 2 ⎜ ⎟ Z − 1 + 2 B ⎝ ⎠ where lnγ is the activity coefficient of component i and can be expressed as (Zhao and Lvov 2016): � � ∑ � � � � � �� z b exp − � b z exp − z b exp − ji j j ji ji j=1 i j ij ij lj l l lj lj l=1 ln = � � + � � ⋅ − � � ∑ ∑ ∑ (33) i ij n n n z b exp − z b exp − z b exp − k k ki ki j=1 k k kj kj k k kj kj k=1 k=1 k=1 1 3 524 Petroleum Science (2021) 18:509–529 0.35 Table 13 Coefficients in Hebach et al. (2002)’s correlation v = v + c − (38) corr EOS m cm 0.35 + d Coefficients Original value Refitted value where (Abudour et al. 2013): b , g/(cm K) 0.00022 0.00022 b −1.9085 −1.9085 RT cm −2d k , mN/m 27.514 25.6836 c = c − 0.004 + c e (39) 0 m 1m 1m 6 2 cm k, cm /g −35.25 −218.4717 12 4 k, cm /g 31.916 9.3192 18 6 k, cm /g −91.016 −0.9621 c = z c (40) 3 1m i 1i k, cm /g 103.233 33.4068 i=1 k , mN/m 4.513 14.4970 2 6 k, g /cm 351.903 10.9290 PR 1 1 v1 d = − (41) RT RT a cm T cm 11 where ω is the acentric factor of the mixture (Abudour where T , p and δ are the critical temperature, critical cm cm cm et al. 2013): pressure and volume correction of the mixture at the criti- cal point, respectively. c is the specie-specific parameter 1i = z (48) m i i of component i and has a linear relationship with critical i=1 compressibility ( Z ) (Abudour et al. 2012b): where ω is the acentric factor of component i. c = 0.4266Z − 0.1101 (42) 1 c The term d can be derived using the original PR EOS Appendix E: Summary of existing IFT (Matheis et al. 2016): correlations for CO /H O mixtures 2 2 2a(v + b) v RT d = − (43) m Parachor model (Sugden 1930) can be expressed as below 2 2 RT 2 2 (v − b) cm −b + 2bv + v (Schechter and Guo 1998): The volume correction of the given mixture at the criti- M M cal point, δ , can be determined by (Abudour et al. 2013): cm = P x − y (49) i i i L V i=1 RT cm = 0.3074 − v (44) cm i ci where x and y are the mole fractions of component i in liq- i i cm i=1 uid and vapor phases, respectively; P is the Parachor value where v is the critical volume of component i; θ is the of component i ( P = 52 , P = 78 ) (Liu et al. 2016); ci i H O CO 2 2 M 3 M surface fraction of component i defined by (Abudour et al. is the molar density of liquid phase in mol/cm ; is the L V 2013): molar density of vapor phase in mol/cm . Hebach et al. (2002)’s correlation can be expressed as: 2∕3 z v � � �� ci √ (45) 2 i ∑ n 2∕3 = k 1 − exp k dd + k ⋅ dd + k ⋅ dd 0 1 2 3 z v i=1 ci (50) � � + k ⋅ dd + k exp k (dd − 0.9958) 4 5 6 The critical temperature of the mixture can be calculated via the following mixing rule (Abudour et al. 2013). where (Hebach et al. 2002): dd = − (51) T = T H O corr (46) cm i ci 2 i=1 𝜌 +b (304−T)(10×p) CO 0 2 3 3 The critical pressure of the mixture can be determined by 0.025 g∕cm <𝜌 < 0.25 g∕cm CO 𝜌 = 1000 corr the correlation proposed by Aalto et al. (1996): 𝜌 in other cases CO � � (52) 0.2905 − 0.085 RT m cm p = ∑ (47) cm i ci i=1 1 3 Petroleum Science (2021) 18:509–529 525 where is the CO -rich-phase density in g/cm ; is CO 2 H O 2 2 Aqueous-phase density data CO -rich-phase density data the aqueous-phase density in g/cm ; k to k and b to b are 2 0 6 0 1 Interfacial tension data empirical coefficients. The units of T , p, and dd are K, bar, Pure-component saturation curves 2 6 and g /cm , respectively. Table 13 lists the values of original and refitted coefficients. Chen and Yang (2019)’s correlation is given as: = C + C p + C lnK + C p + C lnK (53) 1 2 r 3 CO 4 r 5 H O 2 2 where σ is IFT in mN/m; p is the reduced pressure of CO ; r 2 C to C are empirical coefficients. To make fair compari- 1 5 sons, we refit these coefficients based on the IFT database employed in this study. Table 14 summarizes the values of 250300 350400 450500 these refitted coefficients. Temperature, K Appendix F: Pressure–temperature coverage Fig. 8 Pressure–temperature coverage of phase-density and IFT data collected from the literature. The solid curves stand for pure-CO of phase‑density and IFT data collected 2 (left) and pure-H O (right) saturation curves, respectively from the literature. Figure 8 depicts the pressure–temperature coverage of IFT data (i.e., around 28–31 mN/m) obtained by other stud- phase-density and IFT data collected from the literature over ies under similar conditions. 278–478.35 K and 2.5–1291.1 bar, and 278.15–477.59 K Figure 10 indicates that the measured data by Bachu and and 1–1200.96 bar, respectively. Bennion (2009) fall into the range of 16–19 mN/m over 307.15–314.15 K and 120–270 bar, which are significantly lower than the measured IFT values (i.e., around 30 mN/m) Appendix G: Experimental data selection obtained by other studies under similar conditions. No out- in IFT database employed in this study lier exists at other temperature and pressure conditions. We have removed these outliers in the IFT regression analysis. The collected IFT data are further screened to remove any obvious outliers. Figure 9 shows the identification of the outliers from the collected data over 40–60 bar and Appendix H: Derivation of activity 278.15–298.15 K, while Fig. 10 shows the identification coefficient in the fugacity expression of outliers from the collected data over 100–270 bar and when Huron‑Vidal mixing rule is used. 307.15–314.15 K. As seen in Fig. 9a, the measured IFT data by Chun and Similar to the approach used by Wong and Sandler (1992), Wilkinson (1995) and Park et al. (2005) fall into the range of the activity coefficient of component i can be expressed by 5–8 mN/m over 278.15–288.15 K and 40–60 bar, which are the following formula: significantly lower than the measured IFT data (i.e., around 22–28 mN/m) obtained by other studies under similar con- ln = (54) ditions. Figure 9b indicates that the measured IFT data i RT z by Chun and Wilkinson (1995) and Park et al. (2005) fall into the range of 10–14 mN/m over 293.15–298.15 K and where the excess Gibbs free energy can be expressed as 50–70 bar, which are significantly lower than the measured (Huron and Vidal 1979): Table 14 Refitted coefficients in Chen and Yang’s correlation Coefficient set Pressure range C C C C C 1 2 3 4 5 1 Full −64.7356 2.3405 16.3306 2.0919 −7.1593 2 p ≤ 73.8 bar −34.3182 6.5500 10.8716 7.9611 −7.9076 Else −49.7215 0.2460 18.0648 0.1813 −1.9879 1 3 Pressure, bar 526 Petroleum Science (2021) 18:509–529 80 80 T = 278.15 K, Chun and Wilkinson (1995) T = 293.15 K, Bachu and Bennion (2009) T = 278.5 K, Hebach et al. (2002) T = 293.15 K, Park et al. (2005) 70 70 T = 283.15 K, Chun and Wilkinson (1995) T = 298.15 K, Chun and Wilkinson (1995) T = 287 K, Hebach et al. (2002) T = 298.15 K, Bachu and Bennion (2009) 60 60 T = 288.15 K, Chun and Wilkinson (1995) T = 298.15 K, Park et al. (2005) 50 50 40 40 30 30 20 20 Outliers 10 Outliers 10 0 0 04 20 06080 100 120 140 160 180 200 050 100 150 200 250 300 Pressure, bar Pressure, bar (a) (b) Fig. 9 Identification of the outliers at T = 278.15–288.15 K (a) and T = 293.15–298.15 K (b). Outliers are from the studies by Chun and Wilkin- son (1995) and Park et al. (2005) � � b z exp − ji j j ji ji j=1 G = RT z � � (55) ∞ n b z exp − i=1 k k ki ki k=1 T = 307.15-314.15 K, Bachu and Bennion (2009) T = 307.8-309.6 K, Chiquet et al. (2007) T = 307.9-308.2 K, Hebach et al. (2002) To make the derivation process more intuitive, we can T = 312.8 K, Georgiadis et al. (2010) set i = 1 (the first component) and n = 2 (two compounds T = 313.3 K, Pereira et al. (2016) in the system). Then the excess Gibbs free energy can be expressed as: � � b z exp − j1 j j j1 j1 j=1 40 ∞ = z ⋅ � � RT b z exp − k k k1 k1 k=1 30 � � ∑ (56) b z exp − j2 j j j2 j2 j=1 Outliers + z ⋅ � � b z exp − k k k2 k2 k=1 Taking the partial derivative of the first term in the right 0 100 200 300400 500 hand side of Eq. (56) yields: Pressure, bar Fig. 10 Identification of the outliers at moderate temperature (307.15–314.15 K) conditions. Outliers are from the study by Bachu and Bennion (2009) � � �� b z exp − j1 j j j1 j1 j=1 z ⋅ � � b z exp − k k k1 k1 k=1 � � b z exp − j1 j j j1 j1 j=1 = � � + z (57) b z exp − k k k1 k1 k=1 � � � � � � ⎛ � � � � �� n ⎞ b exp − ⋅ b z exp − b exp − ⋅ b z exp − 1 11 11 j1 j j j1 j1 j=1 ⎜ ⎟ 11 1 11 11 k k k1 k1 k=1 ⋅ − ⎜ � � �� ⎟ ∑ ∑ 2 2 n n b z exp − b z exp − ⎜ ⎟ k k k1 k1 k k k1 k1 k=1 k=1 ⎝ ⎠ 1 3 Measured IFT, mN/m Measured IFT, mN/m Measured IFT, mN/m Petroleum Science (2021) 18:509–529 527 Taking the partial derivative of the second term in the right hand side of Eq. (56) yields: � � �� b z exp − j2 j j j2 j2 j=1 z ⋅ � � b z exp − k k k2 k2 k=1 � � � � � � ∑ (58) ⎛ � � � � �� n ⎞ b exp − ⋅ b z exp − b exp − ⋅ b z exp − 1 12 12 j2 j j j2 j2 j=1 ⎜ ⎟ 12 1 12 12 k k k2 k2 k=1 = z ⋅ − 2 ⎜ � � �� ⎟ ∑ ∑ 2 2 n n b z exp − b z exp − ⎜ ⎟ k k k2 k2 k k k2 k2 k=1 k=1 ⎝ ⎠ single-phase liquid densities. Fluid Phase Equilib. 2012;335:74– As such, Eq. 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Toward accurate density and interfacial tension modeling for carbon dioxide/water mixtures

Toward accurate density and interfacial tension modeling for carbon dioxide/water mixtures

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Toward accurate density and interfacial tension modeling for carbon dioxide/water mixtures

Toward accurate density and interfacial tension modeling for carbon dioxide/water mixtures

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