Simulating Wax Deposition in Pipelines for Flow Assurance

Energy & Fuels 2008, 22, 729–741

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Simulating Wax Deposition in Pipelines for Flow Assurance† Beryl Edmonds, Tony Moorwood,* Richard Szczepanski, and Xiaohong Zhang Infochem Computer SerVices Ltd., 13 Swan Court, 9 Tanner Street, London SE1 3LE, United Kingdom ReceiVed July 24, 2007. ReVised Manuscript ReceiVed September 18, 2007

A method is presented for simulating wax deposition in pipelines in which the wax phase is represented as a continuous distribution of n-paraffin components. The thermodynamic properties of the wax are predicted using a previously developed thermodynamic model. We have adopted a mass transfer model to predict the likely rate of wax deposition. Predictions from the model are compared with observations of flow-loop experiments, and predictions for full-scale pipelines are presented. The impact of the removal of deposited wax by shearing is also investigated. We demonstrate the effect of assuming molecular diffusion to be the dominant mechanism of wax deposition, and we also show that not representing the wax with its full n-paraffin distribution leads to serious distortions of the pipeline simulations. Practical software development issues in the implementation of the model are outlined.

Introduction In a previous paper,1 we looked at the thermodynamics of wax precipitation from oils. We adapted the Coutinho model for the wax phase so that it could be used in conjunction with conventional cubic equations of state for the oil and gas phases. A large number of results were presented to illustrate the reliability of the method. Wax thermodynamics is important in any deposition model as the thermodynamic driving force to equilibrium is a fundamental factor affecting the deposition rate. The thermodynamics of wax formation can be defined and measured more precisely than wax deposition rates. However, the deposition problem is the major engineering issue because oil pipelines have to be constructed in environments, both subsea and onshore, where the temperature of the surroundings is below the oil’s wax appearance temperature (WAT), i.e. wax will tend to build up on the pipeline walls. Sometimes deposition can be avoided by heating or thermal insulation, but usually deposition is inevitable, and remedial measures such as periodic pigging have to be incorporated in the design. In a shut-in, the temperature may fall below the wax pour point causing the entire contents of a pipeline to gel; in extreme cases, restart may be impossible leading the pipeline to be abandoned. Wax remediation therefore has enormous cost implications for the oil industry worldwide. In this paper we look at one aspect of this, the simulation of wax deposition. In 1993, Brown, Niesen, and Erickson2 proposed a practical method to calculate the rate of wax deposition on the walls of pipelines based on molecular diffusion. Citing earlier investigations, they suggested that other mechanisms such as shear dispersion, Brownian diffusion, and gravitational settling can be neglected. Although they performed thermodynamic equilibrium calculations assuming that wax is formed from a mixture † Presented at the 8th International Conference on Petroleum Phase Behavior and Fouling. * Corresponding author. Fax: 0044 207 407 3927. E-mail: [email protected]. (1) Coutinho, J. A. P.; Edmonds, B.; Moorwood, T.; Szczepanski, R.; Zhang, X. Energy Fuels 2006, 20, 1081. (2) Brown, T. S.; Niesen V. G.; Erickson, D. D. Measurement and Prediction of the Kinetics of Paraffin Deposition. SPE 26548, Society of Petroleum Engineers: Richardson, TX, 1993.

of n-paraffins, they treated the wax as a single component for deposition calculations. Rygg et al.3 proposed a simulation model based on a similar approach except they included a term for shear dispersion, and they treated the wax phase as consisting of a limited number of wax-forming pseudocomponents. Lindeloff and Krejbjerg4 presented a similar model. Noncompositional molecular diffusion has also been used by various researchers at the University of Tulsa to model their experimental results.5–8 Fogler and co-workers9,10 considered the internal diffusion mechanisms of waxy gel deposits. Besides considering the inward diffusion of wax-forming components towards the cold pipe wall, they also considered the counterdiffusion of oil molecules out of the wax deposits which they used to explain the phenomenon of wax aging. (However, aging may be caused by a combination of processes: Ostwald ripening is another possible mechanism.11) Fogler’s model explains many features observed in wax deposition, but it is computationally demanding because it requires the solution of coupled differential equations. Hernandez12 proposed a much simplified version of such a model where deposition was described by a mass transfer model together with an empirical term to describe the transfer of oil molecules out of the gel. (3) Rygg, O. B.; Rydahl, A. K.; Rønningsen, H. P. Wax Deposition in Offshore Pipeline Systems. Proceedings of the 1st North America Conference on Multiphase Technology, Banff, AB, Canada, 1998; pp 10–11. (4) Lindeloff, N.; Krejbjerg, K. Energy Fuels 2002, 16, 887. (5) Matzain, A. Single Phase Liquid Paraffin Deposition Modeling. M.S. thesis, University of Tulsa: Tulsa, OK, 1996. (6) Lund, H.-J. Investigation of Paraffin Deposition during Single-Phase Liquid Flow in Pipelines. M.S. thesis, University of Tulsa: Tulsa, OK, 1998. (7) Matzain, A. Multiphase Flow Paraffin Deposition Modeling. Ph.D. thesis, University of Tulsa: Tulsa, OK, 1999. (8) Apte, M. S. Investigation of Paraffin Deposition during Multiphase Flow in Pipelines and Wellbores. M.S. thesis, University of Tulsa: Tulsa, OK, 1999. (9) Singh, P.; Venkatesan, R.; Fogler, H. S.; Nagarajan, N. AIChE J. 2000, 46, 1059. (10) Singh, P.; Venkatesan, R.; Fogler, H. S.; Nagarajan, N. R. AIChE J. 2001, 47, 6. (11) Coutinho, J. A. P.; Lopes da Silva, J. A.; Ferreira, A.; Soares, M. S.; Daridon, J.-L. Pet. Sci. Technol. 2003, 21, 381. (12) Hernandez Perez, O. C. Investigation of Single-Phase Paraffin Deposition Characteristics. M.S. thesis, University of Tulsa: Tulsa, OK, 2002.

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Hsu and co-workers13–15 also started from the concept of mass diffusion, but they emphasized on the basis of their observations that shearing of deposited wax, or prevention of deposition, must be considered in cases of turbulent flow. Most pipelines operate in the turbulent region in normal production. Hsu’s approach was to use semiempirical scale-up rules that take into account the effect of turbulence to reduce or prevent wax deposition. Matzain7 tried to take account of wax shearing by empirically modifying Fick’s Law; in effect, his model uses an effective wax diffusion coefficient, its value being reduced to account for the effect of shearing. Hovden et al.16 described a simulation model that combined the Matzain expression for shearing with the compositional version of Fick’s Law proposed by Rygg at al. Nazar et al.17 also used Fick’s Law to simulate deposition of wax as a single component combined with a wax shearing term. Venkatesan18 has investigated experimentally and theoretically the effect of shearing under turbulent flow. In this work, we investigate the methodology of simulating wax deposition within the framework of producing a practical engineering tool that can simulate deposition in a real pipeline with a reasonable computation time. We have therefore focused on quasi-steady-state one-dimensional methods that can be solved by finite difference. Deposition Rate We initially considered the flow of single-phase waxy oil through a pipe using standard chemical engineering methods19 to calculate the heat transfer to the surroundings and the pressure drops. We assume within the bulk oil that local thermodynamic equilibrium pertains, so wax may precipitate if the temperature is below the wax appearance temperature (WAT) and is entrained by the oil. The oil–wax equilibrium is calculated with our implementation of the Coutinho model.1 On the basis of this simple but well-tried fluid flow model, we have investigated methods for simulating the rate of wax deposition on pipeline walls. When calculating deposition rates, the most frequently used assumption is that the deposition rate on the pipeline wall is principally controlled by molecular diffusion. In simulating multiphase fluid flow, it is commonly assumed that the phase equilibrium can be calculated on the basis of local thermodynamic equilibrium, as experience suggests that this is a good approximation to the behavior of fluid phases. As discussed by Venkatesan and Fogler,20 the molecular diffusion approach is based on the assumption that the wax phase as well as the fluid phases are everywhere in local thermodynamic equilibrium, including in the region of wax deposition. For turbulent flow, the molecular-diffusion method can be regarded as linking the heat and mass transfer to the pipeline wall using film theory. Taking the simplest analysis, (13) Hsu, J. J. C.; Brubaker, J. P. Wax Deposition Measurement and Scale-Up Modeling for Waxy Live Crudes under Turbulent Flow Conditions. SPE 29976, Society of Petroleum Engineers: Richardson, TX, 1995. (14) Hsu, J. J. C.; Lian, S. J.; Liu, M.; Bi, H. X.; Guo, C. Z. Validation of Wax Deposition Model by a Field Test. SPE 48867, Society of Petroleum Engineers: Richardson, TX, 1998. (15) Elphingstone, G. M.; Greenhill, K. L.; Hsu, J. J. C. J. Energy Resour. Technol. 1999, 121, 81. (16) Hovden, L.; Xu, Z. G.; Rønningsen, H. P.; Labes-Carrier, C.; Rydahl, A. In Multiphase Technology, BHR Group: Cranfield, UK, 2004. (17) Nazar, A. R. S.; Dabir, B.; Islam, M. R. Energy Sources 2005, 27, 185. (18) Venkatesan, R. The Deposition and Rheology of Organic Gels. Ph.D. thesis, University of Michigan: Ann Arbor, MI, 2004. (19) Coulson, J. M.; Richardson, J. F. Chemical Engineering: Fluid Flow, Heat Transfer and Mass Transfer, 6th revised ed.; ButterworthHeinemann: Oxford, 1999; Vol. 1. (20) Venkatesan, R.; Fogler, H. S. AIChE J. 2004, 50, 1623.

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consider the case of a waxy oil flowing through a pipeline: the heat flux Q of heat lost by the oil to the wall can be written as Q ) λ(Tb - Tw)/x

(1)

where λ is the thermal conductivity, x is the thickness of the laminar film adjacent to the wall, Tb is the bulk temperature of the flowing oil, and Tw is the temperature of the oil in contact with the wall, or the waxy gel deposited on the wall. For mass transfer, the transfer rate Mi for wax-forming component i per unit area is given by the analogous expression Mi ) Di(Cib - Ciw)/x

(2)

where Di is the diffusion coefficient of component i in the oil, Cib is the concentration of component i in the bulk oil, and Ciw is the concentration of component i in the oil at the interface with the wall or the deposited gel. If the temperature at the wall Tw lies below the WAT and if local thermodynamic equilibrium applies, the concentrations Ciw can be calculated from a thermodynamic model for wax solubility in the oil. (If the diffusion rates are expressed in molar terms, the concentrations can be represented as mole fractions.) Because the wax-forming components are individually present in very small amounts, it is reasonable to treat the mass transfer rates given by eq 2 as mutually independent. Many investigators treat the wax as a single component which reduces eq 2 to a scalar expression in which the mass transfer rates and concentrations are total values for the wax, and the diffusion coefficient is an average value. The film thickness in eqs 1 and 2 is assumed to be a singlevalued unknown which can be eliminated by calculating the mass transfer rates from the heat transfer rate. Equation 1 can be rewritten as Q ) Nuλ(Tb - Tw)/d

(3)

where d is the pipeline diameter and Nu is the Nusselt number which can be estimated from many standard correlations for fluid flow in pipes. Likewise, eq 2 can be expressed as Mi ) ShiDi(Cib - Ciw)/d

(4)

where Shi is the Sherwood number for component i. Comparison of eqs 1–4 shows that the molecular diffusion approach is equivalent to assuming that all the Sherwood numbers are equal to the Nusselt number. (Venkatesan and Fogler20 give a more precise treatment of the relation between the Sherwood and Nusselt numbers.) There are conceptual difficulties in assuming local thermodynamic equilibrium in the deposition region. Consider the situation in the laminar film illustrated schematically in Figure 1. The temperature decreases towards the wall reaching the WAT at plane AA′. Plane BB′ is the location of the interface between the oil and the wall or the deposited waxy gel. To the right of plane AA′, no precipitation will occur as the oil is above the WAT; however, mass balance requires that there must be a diffusion of wax components towards plane AA′ which implies that there must also be a concentration gradient to the right of plane AA′ created not by the wax solubilities but by the deposition process. To the left of plane AA′, if local equilibrium applies, the concentration of wax in oil below the WAT must decrease with decreasing temperature causing wax to crystallize out immediately. These wax crystals will not be attached to the gel surface, so it is reasonable to assume they will be entrained by the oil and carried with the flow, unless the whole mixture gels. The concentration gradient predicted from the wax solubility will control the rate of diffusion towards the wall, but changes in the diffusion rate at different distances from the

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Figure 2. Thermodynamic models compared with experimental data for a wax precipitation curve.

To transform the equation to a mass transfer expression, the Nusselt number is replaced by the Sherwood number, and the Prandtl number, by the Schmidt number: Shi ) f(Re, Sci)

(6)

In its strictest form, the analogy assumes that the functions in eqs 5 and 6 are identical. There are various different forms proposed in the literature. For turbulent flow, we have adopted forms of the Colburn and Dittus–Boelter equations,22 Nu ) 0.023Re0.8Pr1/3

(7)

Shi ) 0.023Re0.8Sc0.4 i

(8)

and

Figure 1. Schematic diagram of a laminar film of oil in contact with the gel deposit or pipeline wall.

wall will control the amount of wax that must precipitate to preserve mass balance. The diffusion rates cannot therefore be solely controlled by the wax solubilities; in fact, the concentration gradient is also generated by the deposition process, just as it is to the right of plane AA′. Experience of measuring thermodynamic equilibria suggests that solid phases take considerably longer to equilibrate than fluid phases. Standard nucleation theory of crystal formation21 is based on the premise that the dissolved components may become supersaturated, and the rate of crystal nucleation is related to the degree of supersaturation. The points above suggest that it may not be correct to assume that the wax deposition process is governed by local thermodynamic equilibrium. An alternative approach, which does not assume thermodynamic equilibrium, is the well-known technique in chemical engineering of calculating mass transfer using the heat and mass transfer analogy. Essentially, the approach disregards the detail of the mass transfer process, which may be too complex to understand, and instead relates the heat and mass transfer rates to empirical correlations based on dimensionless groups. (See the Appendix for the standard definitions.) Starting with heat transfer, which is described by the Nusselt number, it is common to estimate the Nusselt number from the Reynolds and Prandtl numbers, Nu ) f(Re, Pr)

(5)

(21) Jones, H. G. Crystallization Process Systems, Butterworth-Heinemann, Oxford, 2002.

The difference between the molecular diffusion and mass transfer approaches can be regarded therefore as the difference between setting the Sherwood number equal to the Nusselt number, or else using expressions like eqs 7 or 8. Thermodynamic Relationships Whatever approach to deposition is adopted, it is necessary to use a thermodynamic model for the solubility of wax components in the oil as a function of temperature in order to calculate the concentrations in eq 4. As an indication of the importance of the thermodynamics, we have taken oil 6 reported by Erickson et al.23 as an example. This is one of the few cases in the public domain where the n-paraffin distribution of the oil is reported. Figure 2 shows the experimental precipitation curve together with the calculations using three different thermodynamics models: these are the Coutinho model,1 the model of Rønningsen et al.,24 and the model of Erickson, Niesen, and Brown.23 As Erickson et al. did not propose details of how to treat the oil phase, we have used our own equation-of-state formulation, but the wax phase properties are as described by Erickson et al.23 The figure clearly shows that the selected thermodynamic models give quite different results. The figure also shows that the differences between the models are least marked around the WAT; here, the spread between the three models is about 8 °C. (The measured WAT appears somewhat (22) Incropera, F. P.; De Witt D. P. Fundamentals of Heat and Mass Transfer, 5th ed.; Wiley: New York, 2002. (23) Erickson, D. D.; Niesen, V. G.; Brown, T. S. Thermodynamic Measurement and Prediction of Paraffin Precipitation in Crude Oil. SPE 26604, Society of Petroleum Engineers: Richardson, TX, 1993. (24) Rønningsen, H. P.; Sømme, B. F.; Pedersen, K. F. An Improved Thermodynamic Model for Wax Precipitation: Experimental Foundation and Application. In Proceedings of Multiphase ′97, BHR Group: Cranfield, UK, 1997.

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Figure 3. Flow-loop measurements for Garden Banks fluid; simulated results with no wax shearing term.

low.) However, for simulating deposition, it is necessary to know the wax solubilities at temperatures below the WAT, i.e. the shape of the wax precipitation curve as a whole needs to be accurate. The figure illustrates the good results obtained from the Coutinho model. Wax Porosity The wax deposited in pipelines is found to be a gel of wax crystals and occluded oil. The fraction of occluded oil, or porosity, is usually high and can vary between typically 60–90 mass %. Currently, there is no method that reliably predicts wax porosity, so we follow the normal practice of treating wax porosity as an input quantity that must be specified. Its value has to be determined experimentally or estimated for each oil. The porosity has a major impact on the simulation as the total volume of the deposited gel is sensitive to its assumed value.

Edmonds et al.

deposits corresponding to the highest flowrates. The same trend of reducing thickness of the deposit with increasing rates of flow has been observed by other experimenters, e.g. Venkatesan.18 In contrast, the simulation suggests that the deposit thickness is not very sensitive to changes in flowrate. The model actually predicts that mass transfer rates should increase slightly with increasing flowrates, but this trend is masked by the insulating effect of the deposited gel which tends to reduce the deposition rates. We considered a number of explanations for the discrepancy above. One possibility is that we have overestimated the thermal conductivity of the waxy gel and that the temperature of the bulk oil rises with time above the WAT causing deposition to cease. This explanation is unconvincing as it would suggest that the final thickness of the deposit would be about the same for all flowrates, whereas it is actually quite sensitive to the flowrate. It would also imply that the gel thermal conductivity is very much lower than we assume to be the case based on the known properties of paraffins. Another explanation could be that we have neglected the effect of wax aging, i.e. the tendency of oil to diffuse out of the waxy gel as a result of the oil concentration gradient that is created in the waxy gel by the temperature gradient across it. An estimation of aging time can be made by simple dimensional analysis combined with Fick’s Law. Assume the oil concentration on the outer surface of the gel in contact with the pipe wall exceeds the concentration on the inner surface in contact with the oil by ∆C. From Fick’s Law, the rate of diffusion is given by R ) DA∆C/x where D is the diffusion coefficient, A is the area covered by the deposit, and x is the deposit thickness. The diffusion rate can also be equated to R ) Ax∆C/τ where Ax∆C is the amount of oil that has to diffuse out and τ is the aging time. Combining these two expressions, and cancelling A and ∆C gives

Flow-Loop Simulations

τ ) x2/D

has presented results for wax deposition in flow loops. Hernandez’s work has a number of advantages: her results show a good mutual consistency, sufficient details are reported to allow the experiments to be simulated, she used real waxy petroleum fluids for which detailed analyses are available, and she used large-scale flow loops that are easier to relate to fullscale pipelines than bench-scale tests. Figure 3 shows measured and calculated wax thicknesses as a function of time for different flowrates of Garden Banks fluid. The wax thickness was estimated from the measured pressure drop and was also directly measured in sections of the test pipe at the end of the test. In the figures, pressure drop estimations are represented as points, and other direct measurements are shown as vertical error bars. The calculations use the masstransfer model, eq 8, combined with the Coutinho model for wax thermodynamics.1 The wax content of the gel was set to the values reported by Hernandez: 35.4, 39.1, and 39.6 mass % for flow rates 1800, 1500, and 1000 barrels per day (bpd) or 3.31 × 10-3, 2.76 × 10-3, and 1.84 × 10-3 m3 s-1, respectively. The temperature difference between the oil and coolant entering the test section was 30 °F (16.67 °C). The wax is represented as a continuous distribution of n-paraffins up to a carbon number of 82 (nC82) as given in the reported analysis. No parameters were adjusted to match the experimental data. The initial rate of wax deposition is reasonably well-represented by the model given the number of different effects that are being simulated. However, with increasing time, the measurements show increasing differences between the different flowrates with the thinnest

For the Garden Banks experiments, the maximum gel thickness was about 0.35 mm for the 1800 bpd flow rate. To estimate the diffusion coefficient, we approximate the fluid by a component of carbon number 13 (C13) which has a similar molecular weight, and we approximate the wax by nC40 which has a similar molecular weight to the predicted wax precipitate at 55 °F, the temperature of the flow loop’s coolant. Using the Hayduk–Minhas method,25 the diffusion coefficient is found to be ∼1.7 × 10-10 m2 s-1, giving an aging time of ∼12 min. However, the diffusion of oil through the gel is restricted by the precipitated wax crystals, so a more realistic estimate can be obtained using an effective diffusion coefficient estimated from an equation proposed by Cussler et al.26 as suggested by Singh et al.:10

Hernandez12

Deff )

D 1 + R Fw2/(1 - Fw) 2

(9)

(10)

where Fw is the mass fraction of wax crystals in the gel and R is the mean aspect ratio of the wax crystals. For Garden Banks fluid at 1800 bpd, Fw is reported to be 35.4 mass %. Singh et al. suggest that R initially starts at unity in a wax deposit and then increases with increasing wax content. Taking R ) 1 gives Deff/D ) 0.84, which makes little difference to the aging time. (25) Hayduk, W.; Minhas, B. S. Can. J. Chem. Eng. 1982, 60, 295. (26) Cussler, E. L.; Hughes, S. E.; Ward, W. J.; Aris, R. J. Membr. Sci. 1988, 38, 161.

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Figure 4. Empirical regression of flow-loop measurements for Garden Banks fluid at different temperature differences. The flowrate is 1800 bpd.

Figure 5. Empirical regression of flow-loop measurements under turbulent flow by Venkatesan.18

However, Paso et al.27 report observing mean aspect ratios as high as 61 for polydisperse paraffin crystals; using R ) 61 gives Deff/D ) 1.4 × 10-3 resulting in an aging time of τ ) 145 h. It is consequently difficult to assess the likely rate of aging as it depends so strongly on the assumed aspect ratio of the wax crystals. We have not attempted to incorporate aging into the simulation methods used in this study. Lastly, we considered the possibility that the observed deposit thicknesses are controlled by shearing of the waxy gel by the forces exerted by the flowing oil. The experimental evidence suggests that there is a tendency of wax deposits to “plateau” with time, i.e. for the existence of a limiting maximum thickness at which the overall deposition rate becomes zero. We attempted to fit the observed growth of deposits with time using empirical correlations and found that many cases can be fitted well with an expression of the type x ) A - B exp(-Ct)

(11)

where x is the deposit thickness and t is the elapsed time. Figure 4 shows two fits to runs with Garden Banks fluid, and Figure 5 shows results for a flow loop with a model fluid investigated by Venkatesan.18 Equation 11 supports the idea that the deposit thickness approaches a limiting thickness as x f A as t f ∞. The exponential decay is compatible with the assumption that the rate of wax removal by shearing (from what would otherwise be deposited) is proportional to the deposit thickness. Approximating the deposition rate R as a constant and writing the removal rate as kx, the increase in thickness dx is related to the time increment dt by (R - kx)

dt ) F dx

(12)

(27) Paso, K.; Senra, M.; Yi, Y.; Sastry, A. M.; Fogler, H. S. Ind. Eng. Chem. Res. 2005, 44, 7242.

Figure 6. Flow-loop measurements for Garden Banks fluid. The flow rate is 1800 bpd. Simulated results are shown with and without the wax shearing term.

where F is the density of the gel (an effective value to allow for porosity). Integration of eq 12 yields eq 11. Nazar et al. made the assumption that the shearing rate is proportional to deposit thickness, citing the work of Kern and Seaton,28 but other investigators have not made this assumption. Without this assumption, the predicted increase of deposit thickness with time will follow quite a different functional form, and modeling the tendency to plateau is more difficult. The assumption can be supported by comparison with theories of maximum particle size in waxy gels under imposed fluid flow; maximum particle diameter can be related to maximum sustainable torque, i.e. to the product of the applied force and the particle diameter.29 By analogy, we assume that the average wax shearing rate is proportional to the product of the average stress exerted on the deposit by the fluid and the thickness of the deposit, even though the actual mechanism of wax shearing by the turbulence of the fluid will be somewhat random. We also tentatively assume that the shearing rate is inversely proportional to the strength of the gel which, following the work of Venkatesan et al.,30 we set to Fw2.3. Hence, S ) Kσx ⁄ Fw2.3

(13)

where S is the wax shearing rate, K is an empirical constant, and σ is the stress exerted by the fluid on the deposit, which is given by the usual expression19 1

σ ) ⁄2 fFu2

(14)

where f is the Fanning friction factor, F is the mass density of the fluid, and u is the mean velocity of the fluid. Figures 6–8 show the results for Garden Banks fluid with the wax shearing term eq 13 added to the simulation. K was adjusted until the simulation for 1800 bpd, Figure 6, gave a reasonable result. The tendency to plateau is now clearly predicted. Using the same value of K, Figure 7 gives the simulation for 1500 bpd. Here, the simulated result is low relative to the thickness estimated by the pressure drop, but they are compatible with the direct measurements. Figure 8 gives the comparison for the 1000 bpd flowrate. Two experimental runs were carried out at 1000 bpd under identical conditions for the first 24 h. Unfortunately, the thickness of the wax deposit estimated from the pressure drop is rather different for the two runs suggesting that the measurements can be subject to significant error. Again, the simulated thickness is low relative (28) Kern, D. Q.; Seaton, R. E. Brit. Chem. Eng. 1959, 4, 258. (29) Lorge, O.; Djabourov, M.; Brucy, F. ReV. Inst. Fr. Pét. 1997, 52, 235. (30) Venkatesan, R.; Nagarajan, N. R.; Yi, Y.-B.; Sastry, A. M.; Fogler, H. S. Chem. Eng. Sci. 2005, 60, 3587.

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Figure 7. Flow-loop measurements for Garden Banks fluid. The flow rate is 1500 bpd. Simulated results are shown with and without the wax shearing term.

Figure 8. Flow-loop measurements for Garden Banks fluid. The flowrate is 1000 bpd. Simulated results are shown with and without the wax shearing term.

Edmonds et al.

Figure 10. Flow-loop measurements for South Pelto fluid. The flow rate is 1800 bpd. Simulated results are shown with and without the wax shearing term.

Figure 11. Flow-loop measurements for Garden Banks fluid. The flow rate is 1800 bpd. Simulated results are shown with no wax shearing term to compare mass transfer and Fick’s Law assumptions.

thickness tends to increase too rapidly, but if we add the wax shearing effect, using the same coefficient value as for Garden Banks oil, the results give a better trend. The results with shearing are still a little high relative to the pressure-drop measurements; however, the other measurement techniques give higher values for the wax deposit suggesting a smaller discrepancy between simulation and observation. The overall agreement between the model and the flow-loop measurements is as good as one could reasonably expect given that the mass transfer coefficients are estimated in all cases from a very simple general correlation, eq 8. Figure 9. Flow-loop measurements for South Pelto fluid. The flow rate is 2900 bpd. Simulated results are shown with and without the wax shearing term.

to the measurements indicating that perhaps that the mass transfer coefficients are being underestimated for the lower flowrates although the calculated results exceed one of the direct measurements of wax thickness. The discrepancy certainly cannot be due to the wax shearing effect as it is quite small for the 1000 bpd flowrate. The same simulation model has also been tested against Hernandez’s result for South Pelto fluid. The fluid was represented by a continuous distribution of n-paraffin components up to nC80 as in the reported analysis. These experiments were performed in a different flow loop from the Garden Banks study. Figures 9 and 10 show the results for a 45 °F (25 °C) temperature difference at flowrates of 2900 and 1800 bpd. The thickness of the wax deposit was estimated from the pressure drop, by direct measurement of a removed sample, and by liquid displacement from a section of the pipe. The simulated wax

Model Sensitivities for Flow-Loop Simulations The effects of different model assumptions are shown using the run for Garden Banks oil at 1800 bpd as an example. The base case in each example is the simulation without wax shearing. Figure 11 shows the effect of assuming that molecular diffusion is the mechanism of wax deposition, which is achieved by setting the Sherwood number for each wax-forming component equal to the Nusselt number. In this example, molecular diffusion underestimates the initial observed rate of deposition by a factor of about 8 before thermal insulation by the waxy gel becomes significant. Other investigators have also found that the molecular diffusion method can underpredict the observed rate of wax deposition.2,6 To this point, the calculations have been performed using all the individual n-paraffins in the fluids. It is tempting to reduce the complexity of the simulations by grouping the n-paraffins together as a smaller number of pseudocomponents, as is normally done for equation-of-state calculations for petroleum fluids. To test this, the n-paraffins in Garden Banks oil were

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Figure 12. Wax precipitation curve for Garden Banks fluid using a full n-paraffin distribution compared with using seven and three wax pseudocomponents.

Figure 14. Simulation of the development of the wax thickness profile for a pipeline reported by Brown et al.2 No wax shearing term is included.

Figure 13. Flow-loop measurements for Garden Banks fluid. The flow rate is 1800 bpd. Simulated results are shown with no wax shearing term to compare the use of a full n-paraffin distribution with using seven and three wax pseudocomponents.

Figure 15. Simulation of the development of the temperature profile for a pipeline reported by Brown et al.2 No wax shearing term is included.

grouped into seven and three wax pseudocomponents. Figure 12 shows a comparison of the wax precipitation curves calculated with the Coutinho model. Reducing the number of components reduces the predicted WAT and raises the predicted amount of wax precipitated at lower temperatures. Seven pseudocomponents gives an acceptable approximation to the precipitation curve, but three pseudocomponents causes the predictions to be unreasonable; steps start to appear corresponding to the freeze-out of each individual pseudocomponent. The corresponding simulations for the flow loop are presented in Figure 13, which shows that reducing the wax to seven pseudocomponents gives some increase in the predicted wax thickness with time but using only three pseudocomponents causes the predictions to depart markedly from the simulation with the full n-paraffin distribution. It will be shown below that grouping the n-paraffins into pseudocomponents has a much more noticeable effect on the simulation of full-scale pipelines. Simulation of Pipelines The purpose of simulating flow-loop experiments is to show that the assumptions made in the model are reasonable by comparison with the measurements, but the object is to provide a tool for simulating wax deposition in full-scale pipelines. Unfortunately, there is limited information available to provide direct verification of the performance of the model for pipelines. As a first example, the case reported by Brown et al.2 for a 6-mile (9.65-km) section of a subsea pipeline of internal diameter 12 inch (305 mm) carrying 20 000 bpd (0.0368 m3 s-1) of oil was investigated. Figure 14 shows the calculated profile of wax deposited at a number of different times assuming mass transfer but no wax shearing. The wax content of the gel

Figure 16. Simulation of the development of the wax molecular weight profile for a pipeline reported by Brown et al.2 No wax shearing term is included.

was set at 51 mass % from the reported pigging data. Figure 15 shows the corresponding prediction of the temperature profile. Initially, the temperature falls to that of the surrounding sea water within the first 10 km of pipeline with most wax being deposited in the same section. As the wax deposit increases, the resulting insulation of the flowing oil causes the temperature to take longer to reach ambient. Consequently, the wax deposit profile is stretched farther along the pipe because deposition is driven by the temperature difference between the bulk and the oil–gel interface. Figure 16 shows the predicted molecular weight of the wax crystals deposited which emphasizes that the simulation is fully compositional. Near the inlet of the pipe, the temperature is high, so only the heaviest wax components will precipitate; however, as the temperature falls along the pipeline, more components will tend to precipitate, so the average molecular weight decreases. In contrast, the simulation

736 Energy & Fuels, Vol. 22, No. 2, 2008

Figure 17. Simulation of the development of the wax thickness profile for a pipeline reported by Brown et al.2 The wax shearing term is included.

method of Brown et al. treats the wax as a single component, so changes in wax properties cannot be modeled. The total volume of waxy gel deposited after 1000 days is predicted to be about 950 bbl (150 m3). The simulations by Brown et al. gave a similar prediction with a total volume of 1070 bbl (170 m3) of waxy gel deposited after 1000 days. In fact, a pig became stuck in the pipeline after 1000 days of operation and had to be removed by back-flowing the pipeline. During the course of the operation, the total volume of deposit recovered was reported as “more than 50 bbl”, which suggests that the simulated deposit thickness may be too high. The same simulation was repeated but with the wax shearing term, eq 13, added. In all the pipeline simulations, the same value of the shear parameter K was used as in the flow-loop simulations reported above. Figure 17 shows the corresponding predictions of the wax thickness with time. Although initially the wax deposit builds up as before, the simulation suggests that the shear forces halt the deposition process after a while and little then changes. This resembles the simulation results for flow loops for which no further increase in deposition occurs after a while (plateauing). The difference is the timescale: for a flow loop, it is a few days, whereas for a full-scale pipeline, it is measured in months. After 1000 days, the total volume of the gel deposited is predicted to be 114 bbl (18 m3) which is reasonable in comparison with the data (considering that not all the wax deposit would have been recovered). The simulation suggests that the initial pressure drop across the pipeline is 13 psi (0.9 bar) before any wax is deposited which is in agreement with the reported value for the actual pipeline which was said to be “less than 20 psi”. However, the simulation predicts that the pressure drop after 1000 days is only slightly greater at 20 psi (1.4 bar) whereas the reported value is 150 psi (10.3 bar). Calculations of friction factors using Moody diagrams or similar methods suggest that for the observed thickness of wax deposited, the pressure drop should be less than twice that for a bare pipe, a result in line with the simulation, so the reported increase in pressure drop remains unexplained. Another example of a full-scale subsea pipeline is given by Lindeloff and Krejbjerg.4 The pipeline is 12.5 km long with an internal diameter of 190.5 mm. In their first case, the pipeline is lightly insulated and positioned on the sea bed, and the pressure is sufficiently high to maintain the oil as a single liquid phase. We characterized the oil using the somewhat limited reported analysis; fortunately, wax calculations are not particularly sensitive to the treatment of the non-normal paraffin components in the equation-of-state model. The n-paraffin content of the fluid was estimated using the method of Coutinho

Edmonds et al.

Figure 18. Simulation of the development of the wax thickness profile for a pipeline reported by Lindeloff and Krejbjerg.4 This is the singlephase unburied case. No wax shearing term is included.

Figure 19. Simulation of the development of the temperature profile for a pipeline reported by Lindeloff and Krejbjerg.4 This is the singlephase unburied case. No wax shearing term is included.

Figure 20. Simulation of the development of the pressure profile for a pipeline reported by Lindeloff and Krejbjerg.4 This is the single-phase unburied case. No wax shearing term is included.

and Daridon,32 and the wax content of the fluid was adjusted to a value of 3.38 mass % which gave a calculated WAT of 38.4 °C. The reported WAT for the oil was 34 °C (see ref 3). The n-paraffin distribution in the model runs continuously up to nC71 at which point it is truncated as the concentrations were judged to be negligible at that point. The wax content of the gel was set to 40 mass % following the work of Rygg et al.3 The oil flow rate was 15.55 kg s-1. Figures 18–21 show the simulated deposition, temperature, pressure, and wax molecular weight profiles with no wax shearing. It is noticeable that our calculations produce profiles that have a smoother shape than Lindeloff and Krejbjerg’s, and with increasing time the deposition is shifted farther along the (31) Brill, J. P.; Mukherjee, H. Multiphase Flow in Wells; SPE Monograph; Society of Petroleum Engineers: Richardson, TX, 1999; Vol. 17.

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Figure 21. Simulation of the development of the wax molecular weight profile for a pipeline reported by Lindeloff and Krejbjerg.4 This is the single-phase unburied case. No wax shearing term is included.

Figure 23. Simulation of the development of the wax thickness profile for a pipeline reported by Lindeloff and Krejbjerg.4 This is the multiphase buried case. No wax shearing term is included.

Figure 22. Simulation of the development of the wax thickness profile for a pipeline reported by Lindeloff and Krejbjerg.4 This is the singlephase unburied case. The wax shearing term is included.

Figure 24. Simulation of the development of the wax thickness profile for a pipeline reported by Lindeloff and Krejbjerg.4 This is the multiphase buried case. The wax shearing term is included.

pipe. The pressure profile is similar to that predicted by Lindeloff and Krejbjerg, but our calculations give a larger pressure drop after 50 days. However, the pressure drop is very sensitive to the assumed roughness of the wax deposit, and consequently, it is largely adjustable. The large predicted variation in wax molecular weight reflects the wide range of molecular weight of the continuous n-paraffin distribution assumed for this oil. We then repeated the same simulation adding the wax shearing term. The resulting wax thickness profile is shown in Figure 22. By comparison with Figure 18, it can be seen that at 5 days there is little difference, but as the wax thickness increases, the wax shearing term predicts an increasing rate of erosion of the deposit; by 50 days, the predicted deposit thickness is significantly reduced from a maximum of about 12 to 5 mm. The second case of Lindeloff and Krejbjerg is for the same subsea pipeline, but this time, the pipe is buried to a depth of 750 mm in the sea bed and operated at a lower pressure to give multiphase flow. The case corresponds to an actual pipeline for which more details are given by Rygg et al.3 We extended our model to cover multiphase flow by using the Beggs and Brill method as described by Brill and Mukherjee.31 The flow model was again solved using a simple forward-difference technique. The fraction of the pipeline wall in contact with the oil was estimated from the flow model, and the rate of wax deposition was reduced accordingly. Lindeloff and Krejbjerg adopted a similar procedure. Figure 23 shows the results with no wax shearing; as in the previous case of the unburied pipeline, the wax thickness is smoother than that predicted by Lindeloff and Krejbjerg and (32) Coutinho, J. A. P.; Daridon, J.-L. Energy Fuels 2001, 15, 730.

the predicted deposition starts nearer to the pipe inlet. The predictions for temperature and pressure profiles show much less variation with time than in the first case because deposition only becomes significant toward the end of the pipeline. The predicted wax molecular weights are much higher than Lindeloff and Krejbjerg’s because our simulation suggests that the initial wax deposits are caused by the heaviest n-paraffins which we have included in our model, which is also why we obtain a smooth deposition curve. As for the example of Brown et al., there exists a serious discrepancy regarding the predictions for this pipeline when wax shearing is not considered. Even after 200 days, the predicted wax deposit is still thickening and the calculations of Rygg et al. and Lindeloff and Krejbjerg appear to show the same trend with no obvious decrease in deposition rate with time. However, operational experience3 for the pipeline is that after 6 months no increase in pressure drop was detected, which was taken to mean that no further wax deposition was occurring. Also, the pipeline does not need to be pigged. When the wax shearing term is added, the corresponding profile for wax thickness is shown in Figure 24. After about 60 days, the deposition rate has become almost insignificant, a result that is consistent with the operational experience for the pipeline. Labes-Carrier et al.33 reported two cases of pipelines with a waxy oil and waxy condensate which they simulated with the deposition model of Rygg et al.3 as implemented in the Olga pipeline simulator combined with the thermodynamic wax model of Rønningsen et al.24 To simulate their oil pipeline, we set up (33) Labes-Carrier, C.; Rønningsen, H. P; Kolnes, J.; Leporcher, E. Wax Deposition in North Sea Gas Condensate and Oil Systems: Comparison between Operational Experience and Model Prediction. SPE 77573, Society of Petroleum Engineers: Richardson, TX, 2002.

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Figure 25. Simulation of the development of the wax thickness profile for an oil pipeline reported by Labes-Carrier et al.33 No wax shearing term is included.

Figure 26. Simulation of the development of the wax thickness profile for an oil pipeline reported by Labes-Carrier et al.33 The wax shearing term is included.

our thermodynamic model for wax using the reported total wax content of 3.2 mass %. The calculated WAT was found to be 51 °C which is somewhat higher than the reported WAT of 34.2 °C measured by cross-polar microscopy (CPM). However, the model predicts that the wax precipitation curve has a quite small but extended “tail”; at 34.2 °C, the predicted amount of wax precipitated is ∼0.055 mass % of the oil. In our experience, the result is entirely consistent with the CPM measurement which requires a small but finite amount of wax to form before it can be detected. The wax model was therefore used without any matching. Figures 25 and 26 show the simulated wax deposit profile in the pipeline for a flow rate of 8000 standard m3 (Sm3) of oil per day with and without wax shearing. Again, our calculation predicts a smoother profile with the wax deposit starting closer to the inlet of the pipeline compared with results reported for the Olga simulation. Like Labes-Carrier et al., we have assumed a wax porosity of 60%. Also, we predict that the rate of wax deposition is higher; under normal operations when the flow rate is 12 000 Sm3/day, we calculate that the pigging intervals are 7 and 11 days without or with wax shearing on the basis of a 2 mm pigging criterion. The result is comparable with the normal operational practice of pigging the pipeline once a week; the Olga simulation suggests a pigging interval of once a month. For the case of the condensate pipeline reported by LabesCarrier et al., we set up a wax model based on the reported wax content of 5 mass % of the condensate. The calculated WAT was 56.6 °C compared with a measured value of 26 °C by differential scanning calorimetry (DSC). At the reported WAT, the model predicts that the amount of wax precipitated is 0.46 mass % of the condensate. The value is higher than for the oil WAT above which was determined by CPM, which reflects the fact that DSC is generally a less sensitive method of detecting WAT. However, the value obtained is still small, so we again chose not to match the wax model to the data.

Edmonds et al.

Figure 27. Simulation of the development of the wax thickness profile for a condensate pipeline reported by Labes-Carrier et al.33 60% wax porosity is assumed. No wax shearing term is included.

Figure 28. Simulation of the development of the wax thickness profile for a condensate pipeline reported by Labes-Carrier et al.33 60% wax porosity is assumed. The wax shearing term is included.

Figure 29. Simulation of the development of the wax thickness profile for a condensate pipeline reported by Labes-Carrier et al.33 80% wax porosity is assumed. The wax shearing term is included.

Figures 27 and 28 show the calculated wax deposition profiles for the condensate pipeline for a total flow rate of 106 Sm3/day with and without wax shearing. Here, 60% wax porosity was assumed, although its actual value was not reported. The wax shearing term makes a very marked difference because of the highly turbulent flow regime in the pipeline. Without wax shearing, we predict that wax buildup would be rapid with pigging required every 4 days for a 2 mm criterion. With wax shearing, the pigging interval goes up slightly to 4.5 days, but from Figure 28, it can be seen that no further growth in the wax deposit occurs after about 30 days. Figure 29 shows that if a wax porosity of 80% is assumed, the wax deposit will only just reach the 2 mm pigging criterion and wax growth effectively stops after about 1 week. The condensate pipeline showed no evidence of wax deposition during its time of operation and was never pigged. Our results suggest that the pipeline is on the borderline of requiring pigging when wax shearing is considered, which appears to be compatible with the field experience. Our simulations predict that the wax profile is

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Figure 30. Simulation of the development of the wax thickness profile using Fick’s Law for a pipeline reported by Lindeloff and Krejbjerg.4 This is the single-phase unburied case. No wax shearing term is included.

Figure 32. Simulation of the wax thickness profile after a fixed time interval for a pipeline reported by Lindeloff and Krejbjerg.4 This is the single-phase unburied case. No wax shearing term is included. The figure compares the effect on the profile of using of a full n-paraffin distribution with using seven and three wax pseudocomponents.

Figure 31. Simulation of the wax thickness profile after a fixed time interval for a pipeline reported by Lindeloff and Krejbjerg.4 This is the single-phase unburied case. No wax shearing term is included. The figure shows the effect on the profile of truncating the n-paraffin distribution.

Figure 33. Simulation of the development of the wax thickness profile using seven wax pseudocomponents for a pipeline reported by Lindeloff and Krejbjerg.4 This is the multiphase buried case. No wax shearing term is included.

smoothly varying in thickness and molecular weight and extends over a significant proportion of the pipeline. The Olga simulation is very different with sharp spikes appearing which then develop into steplike changes in the wax thickness. The Olga results seem incompatible with the observation that the pipeline did not need pigging. It is interesting that shear forces may limit the buildup of wax in real pipelines thereby reducing the need for remediation measures, which raises the question of whether some pipelines are being pigged unnecessarily on the basis of mass transfer predictions. However, it is important to remember that, even if the deposit thickness is not increasing, the wax may continue to age making it harder and therefore more difficult to remove. Model Sensitivities for Pipeline Simulations In the absence of much quantitative information about wax deposition in full-scale pipelines, it is useful to explore the consequences of changing aspects of the model to assess how physically reasonable the various assumptions are. Taking Lindeloff and Krejbjerg’s unburied pipeline case, Figure 30 shows the result with no wax shearing if molecular diffusion (Fick’s Law) is used instead of the mass transfer model, cf. Figure 18. The predicted thickness of the wax deposit is reduced by a factor of about 7 which is a major difference. Our interpretation of the flow-loop data suggests that the mass transfer assumption should be preferred. Figure 31 illustrates the differences caused by truncating the n-paraffin distribution. The n-paraffins of the full model run to

nC71, but in the truncated model, the n-paraffin distribution is terminated at nC49. The truncated model shows a much more abrupt start of the deposition zone which we believe is caused by the absence of the heaviest wax components; the full model predicts a smoother distribution. Much more than a flow loop, a full-scale pipeline tends to separate out the different deposited wax components into different zones along the pipe because of their different melting points. The full model therefore gives a more physically realistic prediction of the deposition profile along the pipe. Figure 32 shows the impact of grouping the n-paraffin components together into a smaller number of pseudocomponents in order to reduce the number of components in the wax model. The effect of using seven and three pseudocomponents is shown for the unburied pipeline with spikes occurring at the points along the pipeline where the temperature falls below the value needed to cause a particular pseudocomponent to precipitate. Figures 33 and 34 show how in the case of multiphase flow in the buried pipeline the deposition profiles for seven and three pseudocomponents develop with time. Figure 34 resembles Lindeloff and Krejbjerg’s results which were also obtained by representing the wax with pseudocomponents. We interpret the discontinuous start of the deposition zone as an artifact of the model caused by the onset of deposition of the heaviest pseudocomponent. We believe the result in Figure 23 is more physically realistic because it was obtained with the full n-paraffin distribution. We feel the Olga simulations reported by Labes-Carrier et al. also show features that are artificial and

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Edmonds et al.

version took about 20 s for the physical property correlations to be regressed, followed by the actual simulation of the pipeline which took about 1 min. Software Structure

Figure 34. Simulation of the development of the wax thickness profile using three wax pseudocomponents for a pipeline reported by Lindeloff and Krejbjerg.4 This is the multiphase buried case. No wax shearing term is included.

The principal purpose of this work was to investigate wax deposition in pipelines, but in order to do this, we needed to set up a fluid flow model. There are many, widely available engineering models for fluid flow in pipelines or pipe networks that use far more advanced techniques than the ones we have implemented, and it is not a constructive use of time and effort to attempt to reproduce them. Instead, we have structured our deposition model to be as compatible as possible with existing fluid flow models. The procedures for calculating the rate of wax deposition with or without wax shearing are all implemented in one code module. The procedures to calculate physical properties, including phase equilibria, using the approximate methods mentioned above are all contained in another module. On a PC, the modules are packaged as dynamic link libraries (DLLs) which are callable from code written in a wide range of languages including C++, Java, Visual Basic, and Fortran. Both modules are called from our own relatively simple treatment of fluid flow, but they could equally well be called from another fluid flow simulation program. Conclusions

Figure 35. Comparison between the results from the rigorous and fast approximate versions of the deposition model.

likewise are a consequence of grouping the n-paraffins into a small number of pseudocomponents. Computational Speed In order to perform a pipeline simulation in a reasonable time, it is desirable to use only a small number of components. However, our investigations suggest that to obtain physically convincing results it is important not to lump the n-paraffins into pseudocomponents, but to represent the components of the wax as a continuous distribution. Consequently, the simulations can easily have a number of components approaching 100. As the phase equilibria of both the flowing fluids and the deposited gel must be repeatedly calculated along with the physical properties of the phases present, the simulations are computationally intensive taking several hours for a typical real pipeline. To make the model into a practical engineering tool, the calculations have to be performed much more quickly. Our method to achieve this is to convert the phase equilibrium and physical property data into empirical expressions that are fitted to the rigorous model. The technique is a much more sophisticated version of the familiar approach of reducing the phase equilibria and physical properties to a table, except our method preserves the compositional nature of the phase equilibrium and preserves the smoothness of the physical property data. Figure 35gives a comparison between the rigorous and approximate forms of the model. The visual impression is that there is little appreciable difference in the results. The differences in the predictions are normally within about 1%, but the impact on run time is enormous. Taking the example in Figure 35 which uses 74 components, the rigorous version took several hours to run, whereas, using the same computer, the approximate

A new model for simulating wax deposition has been developed that uses a fully compositional treatment of the wax phase. The treatment of the fluid flow is simple but general; the model can be applied both to single-phase and multiphase flow, and it can handle deposition from waxy condensates as well as waxy oils. The model is necessarily an idealization, but we have attempted to make its underlying assumptions as physically reasonable as possible so that the predicted trends are reliable. The model has been validated by comparison with laboratory flow-loop experiments. Our investigations suggest that using a compositional treatment gives predictions of the wax deposition profiles in pipelines that appear more physically realistic than those given by simulations that are noncompositional or use only limited numbers of n-paraffin pseudocomponents. The model takes account of the removal of wax by shearing using a new expression that has been devised to be compatible with observations from flow-loop studies. Wax shearing may be an important part of the deposition mechanism with implications for real pipelines. A difficulty in interpreting flow-loop experiments is the number of physical processes that are involved. It is not possible to isolate each effect such as heat transfer coefficients, mass transfer coefficients, wax aging rates, individual physical properties of the wax and the oil, etc. As a result, the overall predictions of a model can be compared with experimental results, but it is difficult to analyze the accuracy of individual assumptions within the simulation. It is therefore not easy to assess how well models will scale-up to predict real pipeline behavior, especially as there is a lack of information about waxing in real pipelines that can be used to test the simulation models. More information for full-scale pipelines that could be used for validation would be of considerable commercial benefit to the oil industry. We have not investigated the effect of wax aging in this study. There are some practical difficulties in incorporating aging into a simple model; uncertainty of the effective diffusivity of oil in a waxy gel is probably the most serious. In common with many others, we have to assume in our simulations a value for

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the oil porosity of the deposited waxy gel. The inability to predict porosity is perhaps an even bigger source of uncertainty than the inability to predict aging, as porosity has a large impact on predictions of required pigging rates, for example. Understanding porosity is another area, like scale-up, where greater knowledge would bring practical benefits for the oil industry. Appendix The heat transfer rate can be conventionally expressed as Q ) h(Tb - Tw)

(A1)

where h is the heat transfer coefficient; the mass transfer rate of component i can similarly be expressed as Mi ) mi(Cib - Ciw)

(A2)

where mi is the mass transfer coefficient. The coefficients are obtained from the definition of the Nusselt number Nu ) hd/λ

Shi ) mid/Di

(A4)

where d is a characteristic dimension of the process, here the internal diameter of the pipe. The Nusselt and Sherwood numbers are found using expressions like eqs 7 and 8 from the Reynolds, Prandtl, and Schmidt numbers, Re, Pr, and Sc, respectively. The conventional definitions of these quantities are Re ) Fud/η

(A5)

(A6)

Sci ) η/FDi

(A7)

where F, u, η, λ, and Cp are the density, mean velocity, viscosity, thermal conductivity, and specific heat capacity at constant pressure of the fluid. Di is the diffusion coefficient of component i. For an oil phase, the physical properties must be estimated using suitable methods. For the fluid-phase behavior and thermodynamic properties we use the SRK equation of state coupled with our own methodology to represent oils with a number of pseudocomponents,34 although the Peng–Robinson equation could also be used. The viscosity is obtained using the method of Pedersen and Fredenslund,35 and the thermal conductivity is found using the Chung–Lee–Starling method.36,37 Diffusion coefficients are obtained from the Hayduk–Minhas method.25 If the oil is below the WAT, wax crystals are entrained in the oil thereby raising the oil’s effective viscosity ηeff. The effective viscosity is estimated using the relation

(A3)

and the Sherwood number (sometimes referred to as the mass transfer Nusselt number)

Pr ) Cpη/λ

ηeff ) ηexp(AFv)

(A8)

where η is the viscosity of the pure oil phase, Fv is the volume fraction of the precipitated wax, and A is a constant that is adjusted to match the available data for the effective viscosity. EF700434H (34) Infochem Computer Services Ltd. User Guide for Models and Physical Properties, version 3.5; 2006.. (35) Pedersen, K. S.; Fredenslund, Aa Chem. Eng. Sci. 1987, 42, 182. (36) Chung, T.-H.; Lee, L. L.; Starling, K. E. Ind. Eng. Chem. Fundam. 1984, 23, 8. (37) Chung, T.-H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Ind. Eng. Chem. Res. 1988, 27, 671.