Statistical Analysis of Supercooling in Fuel Gas Hydrate Systems

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Statistical Analysis of Supercooling in Fuel Gas Hydrate Systems Nobuo Maeda,*,† Darrell Wells,† Patrick G. Hartley,† and Karen A. Kozielski‡ †

CSIRO Materials Science & Engineering and ‡CSIRO Earth Science & Resource Engineering, Ian Wark Laboratory, Bayview Avenue, Clayton, VIC 3168, Australia S Supporting Information *

ABSTRACT: The recently developed high pressure automated lag time apparatus (HP-ALTA) was applied to the study of the formation and growth of interfacial gas hydrate films for three gases; methane (C1), 90% methane/10% propane gas mix (C1/ C3), and synthetic gas mix (SGM). The effects of gas pressure and cooling rate were studied for each gas. Some degree of supercooling was observed in all cases. The probability distributions of formation temperature (Tf) were often found to be bimodal, due to the formation of either gas hydrate or ice in sequential experimental runs. The width of the Tf distribution of ice was about 3−4 K. In contrast, the width of the Tf distribution of gas hydrates was about 20 K which reflects the importance of mass transfer (gas diffusion) processes in nucleation. Differences in hydrate and ice nucleation probability distributions were observed for different gases, reflecting differences in both thermodynamic equilibrium phase behavior and hydrate formation mechanisms. For all gases studied, Tf generally increased with increasing gas pressure. A minimum threshold pressure for hydrate formation was observed, with magnitude decreasing in the order C1 > SGM > C1/C3. The effect of cooling rate on gas hydrate nucleation probability was also studied. The median of the distribution of Tf (Tf50) was found to decrease with an increased cooling rate, consistent with the increases in effective induction time as samples were cooled more slowly. Our results clearly highlight the value in collecting large data sets which can be used to assemble probability distributions when studying intrinsically stochastic processes such as gas hydrate nucleation and growth.

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INTRODUCTION Gas hydrate nucleation is a stochastic event and hence statistical analyses are required for understanding and predicting of gas hydrate formation. Such information is vital for flow assurance in oil and gas pipelines. The focus of gas hydrate related flow assurance has shifted in recent years from prevention/control through the use of hydrate inhibitors or anti-agglomerates (AA), to a risk management approach based on understanding the likelihood of gas hydrate formation.1,2 One approach to hydrate risk management is to quantitatively assess the pressure−temperature (P−T) conditions where the formation of gas hydrates is thermodynamically favored.1 This approach requires reliable quantification of nucleation probability which in turn requires the collection of a large amount of hydrate nucleation and growth data.3,4 Some progress was made for collection of a large number of statistical hydrate nucleation and growth data at elevated pressures. Zeng et al.5 studied the formation of propane hydrates and methane hydrates with the use of high pressure NMR. In this study, the number of the samples was not given but appears to be about 10. The recent advent of high pressure differential scanning calorimetry (HP-DSC)6 offered a novel approach to this otherwise time-consuming process. HP-DSC was applied for the collection of a large number of statistical hydrate nucleation and growth data with the use of dilute water-in-oil (W/O) emulsions7−10 or silica gels11 in which individual water droplets can be regarded as being independent from each other. For this application, heat flow profiles during the induction time at a constant supercooling were interpreted to reflect the nucleation and growth probability distribution profiles of gas hydrates. The heat flow profiles were symmetric and narrow at high driving forces (supercooling >25 K) but © 2012 American Chemical Society

became broad and asymmetric (not Gaussian) at low driving forces (supercooling 15 MPa for methane and ≈5 MPa for the other two gases) was boosted using a pressure booster pump (Model AG-62, Haskel Australasia Pty Ltd., Queensland, Australia) to reach desired experimental pressures of up to 15 MPa. Operation Procedure. The details of the operating procedure for the “interfacial transmittance configuration” of the HP-ALTA are given elsewhere.12 Briefly, approximately 130 μL of milli-Q water (18.2 MΩ resistivity) was placed in a custom-made glass “boat” and mounted horizontally in the pressure chamber of HP-ALTA. The Milli-Q water purification unit (Millipore Cooperation, USA) purifies water with the use of up to date technologies including ion-exchange (Quantum EX cartridge), synthetic activated carbon, ultraviolet oxidation, and 1822

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slightly lower pressure of 10 MPa and using a different sample cell (boat). A strikingly different behavior was observed. Here, the Tf distribution in part (a) apparently consists of two distinct groups, one with a range of 274−289 K, and the second with a range of 261−270 K. The S-curve in part (b) shows a clear gap in the distribution of Tf values, clearly highlighting this bimodal behavior. As described earlier, this bimodal behavior likely resulted from formation of either ice or gas hydrate in different runs and led us to devise a new data presentation method, as described in the Materials and Methods section. Pressure Dependence. Figure 4a shows the fraction of the number of runs at 0.025 K/s cooling rate for which Tf < 273 K under various pressures of C1/C3 gas. Figure 4b shows summary data for runs in which Tf > 273 K. The calculated thermodynamic phase boundaries for hydrate (CSM Gem software1) and ice (Clausius−Clapeyron17) formation as a function of temperature and pressure are also shown for comparison. Figure 4a shows that below 4 MPa, 100% of the samples resulted in Tf values below 273 K. Detailed analysis of these data sets indicates Tf values in the range 255−270 K. This is comparable to the values measured for ice at ambient conditions (see the Supporting Information), which also showed some variation depending on the precise boat (sample cell) used. Therefore we conclude that ice nucleation was a major factor at this P−T condition. At P > 4 MPa, however, a significant fraction of samples showed nucleation at temperatures above 273 K. This must be due to C1/C3 hydrate formation. In fact, there were several experiments in which 100% of the data points fell in this category. We conclude that this reflects the increased hydrate nucleation probability at these elevated gas pressures. When C1/C3 hydrates formed at P > 4 MPa, the median Tf50 was 8−14 K below Teq (i.e., ΔT50 of 11 ± 3 K). Broadly speaking, it appears that Tf50 values run parallel to the thermodynamic equilibrium phase boundary for hydrate formation with this offset. However, the “error bars” show that individual values of Tf can be offset from Teq by as little as −4 K on rare occasions. Figure 5 shows similar data recorded for various pressures of methane (C1) gas, at a cooling rate of 0.025 K/s. Figure 5a demonstrates that methane hydrate did not form above 273 K for values of P < 10 MPa (i.e., the fraction of the number of runs for which Tf < 273K is 1). For methane pressures of 10−14 MPa, nucleation rarely occurred at temperatures above 273 K. On the occasions which it did, the median Tf50 was 8−14 K below Teq (ΔT50 of 11 ± 3 K). However, the “error bars” show that individual values of Tf can be offset from Teq by as little as −3 K on rare occasions. Once again, the distribution of Tf values below 273 K resembled that of ice at ambient conditions. Figures 4 and 5 show that methane (C1) gas hydrate is much less likely to form under comparable supercooling P and T conditions relative to C1/C3 gas hydrates (the fraction of the number of runs for which Tf < 273 K was much higher for methane gas). However, the offset of Tf50 values from Teq was similar for both gas compositions (i.e., ΔT50 ≡ Teq − Tf50 of 11 ± 3 K). Figure 6 shows similar data recorded for various pressures of synthetic gas mix (SGM), at a cooling rate of 0.025 K/s. Figure 6a demonstrates that SGM hydrate formed frequently above 273 K at values of P > 6 MPa. When the SGM hydrate formed at P > 6 MPa, the median Tf50 was about 10 K below Teq (ΔT50 of about 10 K).

Figure 4. (a) Fraction of number of runs for which HP-ALTA reached the equilibrium melting point of water without formation of C1/C3 hydrate is shown as a function of gas pressure. The vertical dashed line shows the equilibrium gas pressure of C1/C3 hydrate at 273.2 K. The cooling rate was 0.025 K/s. (b) Distribution in Tf for the subset of the data shown in part a where C1/C3 gas hydrate formed above 273.2 K. The open symbols represent the median of Tf for each subset. The “error bars” shown here are not the traditional standard deviation, but represent 100% of the data scatter. The solid curve represents the equilibrium hydrate dissociation temperature, Teq. The dashed line represents the equilibrium melting temperature of ice. Differently shaped symbols refer to the results obtained using different glass “boats”. ultrafiltration (0.22 μm Millipak-20 Express membrane filter). Light was passed vertically through the gas−water interface. Once pressurized at the desired gas pressure, automated cooling was commenced and the temperature at which light intensity dropped due to solid (ice or hydrate) formed (Tf) was recorded. Subsequently the sample was warmed to dissociate hydrates. This cooling/detection/ warming cycle was repeated for >100 runs for each sample in order to collect statistically significant data sets.

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RESULTS Figure 2 shows a typical “Manhattan” histogram data of Tf vs the chronological measurement number (a) and the corresponding S-curve (b) for C1/C3 gas at 11.5 MPa. In this instance, it can be seen that approximately 200 runs were collected over >5 days and the resulting Tf values ranged from 259 to 285 K. The stochastic nature of nucleation is highlighted by the apparently random distribution of Tf with time. This range of Tf is larger than that observed previously in the ice and the THF hydrate systems.3,4,16 This wide range is reflected in the broad width of the corresponding S-curve (Figure 2b). Tf50 of this distribution is 276 K, which is 21 K below the Teq of the system at this pressure.1 Figure 3 shows another “Manhattan” histogram (a) and the corresponding S-curve (b) for this gas mixture recorded at a 1823

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that individual values of Tf can be offset from Teq by as little as −6 K on rare occasions. Compared with methane gas, the formation of SGM gas hydrate is much more likely under comparable P and T conditions (the fraction of the number of runs for which Tf < 273 K is much lower). Both the fraction of the number of runs for which Tf < 273 K and the range of median Tf50 for which the formation of gas hydrates occurred were similar to those found for C1/C3 gas. Given the composition of SGM (1.04 ± 0.02 mol % n-butane, 3.89 ± 0.08% propane, 9.99 ± 0.2% ethane, the balance is methane), it is perhaps not surprising that its behavior is closer to that of C1/C3 than to methane (C1). It appears that the limitation in mass transfer (gas diffusion) is still a dominant factor in our results, which will be discussed later in the Discussion section. Cooling Rate Dependence. Figure 7 shows data recorded for methane at a pressure of 13 MPa at different cooling rates. Figure 5. (a) Fraction of number of runs for which HP-ALTA reached the equilibrium melting point of water without formation of methane hydrate is shown as a function of gas pressure. The vertical dashed line shows the equilibrium gas pressure of methane hydrate at 273.2 K. The cooling rate was 0.025 K/s. (b) Distribution in Tf for the subset of the data shown in part a where methane gas hydrate formed above 273.2 K. The open symbols represent the median of Tf for each subset. The “error bars” shown here are not the traditional standard deviation, but represent 100% of the data scatter. The solid curve represents equilibrium hydrate dissociation temperature, Teq. The dashed line represents the equilibrium melting temperature of ice. Differently shaped symbols refer to the results obtained using different glass boats.

Figure 7. (a) Fraction of number of runs for which the cooling ramp of HP-ALTA reached 273 K without detection of methane hydrate is shown as a function of the cooling rate. The methane gas pressure was 13 MPa. (b) Range of Tf for the runs in which Tf > 273 K. The “error bars” shown here are not the traditional standard deviation, but represent 100% of the data scatter. The horizontal solid line represents the equilibrium hydrate dissociation temperature, Teq. The horizontal dashed line represents the equilibrium melting temperature of ice.

Figure 7a demonstrates that nucleation frequently occurs at temperatures Tf < 273 K, as was observed and discussed above. A clear trend was observed in that the formation of methane hydrate becomes more likely (i.e., the fraction of the number of runs for which Tf > 273 K becomes higher) with decreased cooling rate. At a cooling rate of 0.01 K/s, the slowest cooling rate that can be used in the HP-ALTA, the fraction for which Tf < 273 K dropped to about 35%. Figure 7b shows that the median Tf50 climbed to about 281 K (about 8 K below Teq) at this slowest cooling rate studied. Figure 8 shows data recorded for C1/C3 gas at a pressure of 12 MPa at different cooling rates. Figure 8a demonstrates that nucleation occurs frequently (often >80% of runs) at temperatures Tf > 273 K for all cooling rates at or below 0.05 K/s. At the fastest cooling rate studied (0.075 K/s); however, the fraction of the number of runs for which Tf < 273 K became considerably higher, suggesting increased likelihood of ice formation.

Figure 6. (a) Fraction of number of runs for which HP-ALTA reached the equilibrium melting point of water without formation of SGM hydrates is shown as a function of gas pressure. The vertical dashed line shows the equilibrium gas pressure of SGM hydrate at 273.2 K. The cooling rate was 0.025 K/s. (b) Distribution in Tf for the subset of the data shown in part a where SGM gas hydrate formed above 273.2 K. The open symbols represent the median of Tf for each subset. The “error bars” shown here are not the traditional standard deviation, but represent 100% of the data scatter. The solid curve represents the equilibrium hydrate dissociation temperature, Teq. The dashed line represents the equilibrium melting temperature of ice. Differently shaped symbols refer to results obtained using different glass boats.

Once again it appears that Tf50 values run parallel to the thermodynamic equilibrium phase boundary for hydrate formation with a 10 K offset. However, the “error bars” show 1824

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DISCUSSION

General Characteristics of Gas Hydrate Nucleation Data. As noted earlier, HP-ALTA cannot distinguish formation of ice from formation of gas hydrate and therefore detection of a phase change in the sample at a temperature below 273 K requires careful analysis. We found that the formation of ice in a sample cell using the “interfacial transmittance configuration” (i.e., a boat) at ambient conditions (0.1 MPa in air) was in the range 261−270 K. This is significantly higher than was found in the “bulk transmittance configuration” (i.e., a glass tube12) where the observed range was 255−262 K. The width of the S-curve for a specific sample cell was about 3 K in each case. We hypothesize that the observed differences in Tf between sample cells arises from their unique surface defects or scratches. Regardless of the precise mechanisms behind the variations in Tf, values below 273 K at elevated pressures could be due to gas hydrate and/or ice formation. Therefore, unambiguous detection of formation of gas hydrate can only be made above 273 K; hence, measurements above and below 273 K were treated differently as described earlier. We found that the width of the S-curves at low gas pressures (for which all Tf values were 273 K, an increased width of S-curves of ≈20 K was observed. Given that the latter must have resulted from the formation of gas hydrate, we postulate that a narrow width for an S-curve may be a signature of the formation of ice. We note that, in a study of nucleation and growth of THF hydrate at ambient conditions, Wilson et al. reported an S-curve width which was similar to that of ice (2−3 K).16 A key difference between gas hydrate formation and THF hydrate or ice formation is that the latter are bulk phase transitions whereas the formation of gas hydrate occurs at the gas−water interface. This is due to the low solubility of nonpolar gases in water and associated gas mass transfer limitation. We postulate that this intrinsic difference in the mechanisms of nucleation and growth is reflected in the differences in the width of the S-curves. We consider that the very broad S-curves for gas hydrate observed here have arisen from the mass transfer of gas into water, not from the “intrinsic” nucleation probability of crystalline solids when all the constituting components are present, as would be the case for ice or THF hydrates. We may then hypothesize that the intrinsic range of distribution of nucleating a crystalline solid, for which all the necessary components are already in place, is 3−4 K. Where mass transfer of gas into the aqueous phase is required as an additional step in the nucleation and growth process, this additional kinetic consideration causes the range of the probability distribution to be larger. It may be surprising that the mass transfer limitation becomes such a dominant issue even for the formation of hydrate films at the gas−water interface where the mass transfer limitation is expected to be the smallest in the system. It is interesting that Mathews et al. observed that the minimum supercooling of fine mists in a large flow loop was in the range of 3−4 K,18 which is similar to our findings in static (quiescent) samples . Effect of Gas Pressure on Tf. Data presented in Figures 4−6 show some common features. In all cases, a large degree of supercooling was observed, as demonstrated by the occurrence of Tf at temperatures below Teq. The degree of supercooling was in the range 3−35 K across all samples and

Figure 8. (a) Fraction of number of runs for which the cooling ramp of the HP-ALTA reached 273 K without detection of the formation of C1/C3 hydrates is shown as a function of cooling rate. The C1/C3 gas pressure was 12 MPa. (b) Range of Tf for runs in which Tf > 273 K. The horizontal solid line represents the equilibrium hydrate dissociation temperature, Teq. The horizontal dashed line represents the equilibrium melting temperature of ice.

Figure 8b shows a slight trend toward increased supercooling with increasing cooling rates. Figure 9 shows data recorded for SGM at a pressure of 11 MPa at different cooling rates. Figure 9a demonstrates that

Figure 9. (a) Fraction of number of runs for which the cooling ramp of HP-ALTA reached 273 K without detection of SGM hydrate is shown as a function of cooling rate. The SGM gas pressure was 11 MPa. (b) Range of Tf for the runs in which Tf > 273 K. The horizontal solid line represents the equilibrium hydrate dissociation temperature, Teq. The horizontal dashed line represents the equilibrium melting temperature of ice.

nucleation occurs very frequently (often 100% of runs) at temperatures Tf > 273 K for all cooling rates studied (0.01− 0.075 K/s). Figure 9b shows that a slight trend toward increased supercooling with increasing cooling rates may be discernible. 1825

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Effect of Cooling Rate on Tf. It was generally observed that the fraction of samples nucleating below 273 K and the extent of supercooling slightly diminished with decreases in the cooling rate. This trend may be explained because the sample is cooled below Teq for longer as the cooling rate is decreased. It is generally accepted that nucleation probability increases with duration of the supercooled state (i.e., the “induction time” is effectively lengthened). In addition, the longer times to reach a given supercooled temperature at slow cooling rates allows more time for gas diffusion into the sample. Other than this rather subtle trend, the effect of cooling rate appears of secondary importance, at least in the range that can be studied using an HP-ALTA.

gas pressures/compositions studied. This range of supercooling is large compared with that of ice (≈13 K)4 and THF hydrates (≈17 K).16 A minimum threshold pressure for gas hydrate formation above 273 K was observed for all gases. This pressure was found to be much higher for methane (≈10 MPa) than for the other two gases (≈3 MPa). Above the threshold gas pressure, hydrate nucleation probability (above 273 K) generally increased with the gas pressure. This difference in the threshold pressure between the different gases might be explained by the different thermodynamic phase boundaries in the P−T diagrams of the different gases. Teq values of methane are typically 10 K lower (i.e., Teq closer to 273 K) than those of the other two gases at similar pressures (see Figures 4−6), and hence, a smaller degree of supercooling is required to reach 273 K. The observed trend further correlates well with the enthalpy of dissociation of hydrocarbon gas hydrates which increases with the molecular weight of the gas (≈54 kJ/mol for C1, ≈71 kJ/mol for C2, ≈126 kJ/mol for C3, ≈130 kJ/mol for C4).1 In other words, lower enthalpies of dissociation indicate less stable hydrates and, hence, requirement for greater incentive (driving force) for formation. A second possible reason for the difference is that methane is known to form structure I (sI) hydrate while C1/C3 and SGM are structure II (sII) forming compositions. A rather unexpected finding was that ΔT50 for Tf > 273 K was similar at different pressures (i.e., the profile of Tf followed the profile of Teq with a relatively constant offset) and was similar among the three different gases studied. ΔTf50 was 11 ± 3 K (minimum 3 K) for methane gas, 11 ± 3 K (minimum 4 K) for C1/C3 gas, and about 10 K (minimum 6 K) for SGM (The uncertainty bound here is referring to the range of variation in ΔT50 of each gas. Each Tf data set has a large distribution, and the median of each distribution is Tf50. Tf50 varies with the gas pressure in parallel to the thermodynamic equilibrium phase boundary. Consequently, the median of the supercooling, ΔT50 ≡ Teq − Tf50, becomes relatively constant with the change of gas pressure.). This was despite differences in the ease of hydrate formation above 273 K between the different gases (Figures 4a, 5a, 6a; methane requires a much larger threshold overpressure than the other two gases). It is possible that the similar ΔT50 values measured for the three systems reflect a requirement for a similar driving force (i.e., chemical potential difference)19 for hydrate nucleation in these supercooled systems. These results suggest that the system can tolerate 3−6 K of supercooling with a low probability of formation of hydrates for all three gases, which has important implications with respect to the management of gas hydrate formation risk in natural gas pipelines during shut-in periods. Davies et al. measured the supercooling of ice and methane hydrate using the cooling ramp mode of a HP-DSC.10 They reported that methane hydrate nucleated before ice during the cooling ramp when the gas pressure was higher than 15 MPa whereas ice nucleated before methane hydrate when the pressure was lower than 15 MPa. While our results on methane gas are in broad agreement with this observation, we have also shown that methane hydrate can be detected below 15 MPa (and above the thermodynamic ice melting point of 273 K) for a small fraction of runs. We attribute this difference to the HP-ALTA being capable of collecting hundreds of hydrate nucleation events in an automated manner and hence rare nucleation events can be detected.

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CONCLUSIONS The recently developed HP-ALTA was applied to the study of formation and growth of interfacial gas hydrate films for three gases; methane, C1/C3 gas mix, and synthetic gas mix. All nucleation and growth events occurred in the temperature range of 255−290 K. It was found that the distribution of formation temperatures (Tf) was sometimes bimodal, due to the formation of both gas hydrate and/or ice, where temperatures were below 273 K. A new way of analyzing data which decouples the ice and gas hydrate contributions to the probability distribution was developed. The width of the Tf distribution of ice was about 3−4 K. In contrast, the width of the Tf distribution of gas hydrates was about 20 K and reflects the requirement for gas mass transfer into the aqueous phase in the latter case. For all gases studied, Tf generally increased with increasing gas pressure. It was further found that a minimum threshold pressure exists for each gas below which hydrate could not form, which decreased in the order C1 > SGM > C1/ C3. This trend may result from the different enthalpies of dissociation and/or different hydrate structure of methane (sI) relative to the other gas compositions (sII). For all gases studied, both the median Tf (Tf50) and the highest Tf (Tfmax) tracked the equilibrium dissociation temperature (Teq) with fixed offsets of 10−11 K (ΔT50 ≡ Teq − Tf50) and 3−6 K (ΔTmin ≡ Teq − Tfmax), respectively. Therefore, despite the large differences in gas composition, the probability of nucleation at a specified value of supercooling was remarkably similar. Our results suggest that 3−6 K of supercooling can be tolerated without significant gas hydrate nucleation, an observation which could have significance with respect to the management of gas hydrate formation risk in natural gas pipelines. The effect of cooling rate on gas hydrate nucleation probability was also studied. For all gases and pressures studied, Tf slightly decreased with increasing cooling rate, consistent with the increased effective induction time as samples were cooled more slowly. Our results clearly highlight the value in collecting large data sets which can be used to assemble probability distributions when studying intrinsically stochastic processes such as gas hydrate nucleation and growth.

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ASSOCIATED CONTENT

S Supporting Information *

Statistical data of formation of ice in the sample cells (boat) of the HP-ALTA of the interfacial transmittance configuration. The ice formation data are provided for both at 0.1 MPa in air and at low methane pressures for which only ice can form (i.e., outside the thermodynamic phase boundary of methane hydrate). This material is available free of charge via the Internet at http://pubs.acs.org/. 1826

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by CSIRO’s Petroleum and Geothermal Research Portfolio and NM’s Australian Research Council Future Fellowship (FT0991892). We thank Prof. Tony Haymet and Dr. Peter Wilson for assistance with the original HP-ALTA design project, as described in ref 12. We further acknowledge Dr. Ramesh Kini (Chevron Energy Technology Company) and Dr. Richard Chapman (BP) for valuable discussions regarding data analysis and interpretation during preliminary experimental work.

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REFERENCES

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