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Wax Deposition and Aging in Flowlines from Irreversible

Thermodynamics

Hussein Hoteit, Reza Banki,, and Abbas Firoozabadi*,,

Imperial College London, London SW7 2AZ, U.K., ReserVoir Engineering Research Institute (RERI),385 Sherman AVenue Suite 5, Palo Alto, California 94306, and Department of Chemical Engineering, Yale

UniVersity, 9 Hillhouse AVenue ML103, New HaVen, Connecticut 06511

ReceiVed NoVember 30, 2007. ReVised Manuscript ReceiVed April 4, 2008

The development of waxy crude oil and some gas condensate fields can lead to serious operational problemsbecause of solidification of the paraffin components of the fluid in flowlines. Many numerical models in theliterature predict the thickness of the wax deposit. However, most of these models assume that the wax-oil(gel) deposit has a constant wax content. In this work, we analyze wax deposition in laminar flow regime topredict the thickness and the composition of the gel layer as a function of position and time. The wax-oil gelregion is considered as a porous medium. The velocity field and the pressure drop are calculated from theNavier-Stokes equation in the liquid region and from a combined Darcy-type equation and the Navier-Stokesequation in the gel region. The wax amount is estimated as a result of a decrease in fluid temperature belowthe wax appearance temperature (WAT), counterdiffusion processes from thermal and molecular diffusions,

and radial convection which occurs because of nonuniform gel layer thickness. We compare predicted resultsfrom our model with several experimental data from the literature. The results which are in agreement withdata cannot be predicted by formulations in which chain rule is used to replace concentration gradient withtemperature gradient in the molecular diffusion expression.

1. Introduction

Wax consists mainly of heavy paraffins and some naphthenes.

At temperatures below the wax appearance temperature (WAT),

wax components drop out of the solution and crystallize. Wax

deposition in production tubing and pipelines is a common

problem in cases where the fluid temperature is less than the

WAT. Wax deposition along the inner walls of the pipelineincreases the pressure drop, decreases the flow rate, and causes

operational problems. To prevent blockage of pipelines, wax

deposits should be removed periodically. Different mechanical,

thermal, and chemical techniques can be used for wax removal.15

There is considerable interest in predicting the deposition rate,

wax thickness, and wax content as this helps to its prevention

and use of various methods for its removal.

A large number of studies have been conducted on various

aspects of wax deposition in pipelines. Mechanisms considered

in these studies include the following: (1) molecular diffusion,

(2) shear dispersion, (3) Brownian diffusion, and (4) gravity

settling.611 Burger et al.7 reported that molecular diffusion of

species is a dominant process at high temperatures and high

heat flux conditions whereas shear dispersion is the dominant

flux at low temperatures and low heat fluxes. Also, the

contribution of the Brownian motion is small compared with

other mechanisms. Hamouda and Ravnoy9 concluded that

molecular diffusion is the main mechanism. Brown et al.12

concluded that increasing shear rates (flow rates) decreases the

rate of deposition. Hamouda and Davidsen13 investigated the

effect of flow rate on wax deposition. Their studies reveal a

significant change of the wax deposition rate when the flow

* Corresponding author. E-mail: [email protected]. Imperial College London. RERI. Yale University.(1) Shock, D.; Sudburg, J. D.; Crockett, J. J. Studies of the mechanism

of paraffin deposition and its control. J. Pet. Technol. 1955, 7 (9), 2330.(2) Jorda, R. M. Paraffin deposition and prevention in oil wells. J. Pet.

Technol. Trans. AIME. 1966, 237, 16051612.(3) Narvaez, C.; Ferrer, A. A.; Corpoven, S. A. Prevention of paraffin

well plugging by Plunger-Lift use, SPE 21640. Presented at The SPEProduction Operation Symposium, Oklahoma City, OK, April 7-9, 1991.

(4) Svetgoff, J. Paraffin problems can be resolved with chemicals. OilGas J. 1984, 82 (9), 7982.

(5) Eastund, B. J.; Schmitt, K. J.; Meek, D. L.; Anderson, D. C.;Grisham, G. New system stops paraffin buildup. Pet. Eng. Int. 1989, 61(1), 4651.

(6) Bern, P. A.; Withers, V. R.; Cairns, J. R. Wax Deposition in CrudeOil Pipelines. Proceedings of the European Offshore Petroleum Conference,London, England, Oct 21-24, 1980; exhibit paper EUR 206.

(7) Burger, E. D.; Perkins, T. K.; Striegler, J. H. Studies of WaxDeposition in the Trans Alaska Pipeline. J. Pet. Technol. 1981, 33 (6), 10751086.

(8) Majeed, A.; Bringedal, B.; Overa, S. Model Calculates WaxDeposition for N. Sea Oils. Oil Gas J. 1990, 88 (25), 6369.

(9) Hamouda, A. A.; Ravnoy, J. M. Prediction of Wax Deposition inPipelines and Field Experience on the Influence of Wax on Drag-ReducerPerformance. 24th Annual Offshore Technical Conference, OTC 7060,Houston, TX, May 4-7, 1992.

(10) Rygg, O. B.; Rydahl, A. K.; Ronningsen, H. P. Wax deposition inOffshore Pipeline Systems. 1st North American Conference, June 1998.

(11) Weingarten, J. S.; Euchner, J. A. Methods for Predicting Waxprecipitation and Deposition, SPE 15654-PA. SPE Prod. Eng. 1988, 3 (1),121126.

(12) Brown, T. S.; Niesen, V. G.; Erickson, D. D. Measurement andPrediction of the Kinetics of Paraffin Deposition. SPE Annual TechnicalConference and Exhibition, Houston, TX, Oct 3-6, 1993; SPE 26548-MS.

(13) Hamouda, A. A.; Davidsen, S. An Approach for Simulation ofParaffin Deposition in Pipelines as a Function of Flow Characteristics withReference to Tesside Oil Pipeline. Proceedings of the SPE InternationalSymposium on Oilfield Chemistry, San Antonio, TX, Feb 14-17, 1995;SPE 28966-MS.

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shifts from laminar to turbulent flow. Creek et al.14 concludedthat the deposition rate decreases with increasing flow rate ratherthan increasing as suggested by a number of authors. They also

concluded that the liquid fraction of the gel layer in turbulentflow is significantly lower than that in laminar flow. Severalauthors have argued that shear dispersion, Brownian diffusion,and gravity settling can be neglected in the deposition ofwax.1216 Molecular diffusion and heat transfer are oftensuggested to be the main cause of deposition.12,13,15

The numerical simulation of wax deposition in pipelines is adifficult task. The numerical model should take into consider-ation the thermodynamic nature of the problem and the movingboundary between the gel layer and the liquid bulk. Thethermodynamic modeling for solid/liquid equilibrium has beensuccessfully studied by many authors.17,18 However, modelingfluid dynamics, thermal conduction, and multicomponent dif-

fusions owing to molecular and thermal gradients in the wax-oil gel layer and the liquid region are a challenge. Most of thenumerical models in the literature are based on some assump-tions which may not be valid. Sevendsen19 is perhaps the firstauthor who introduced a mathematical model for quantitativeprediction of wax components in the deposit for a horizontalpipe. The model uses simplified fluid dynamic correlations topredict the velocity field and the temperature profile. Thevelocity is assumed essentially axial which is calculated froma power-law correlation that depends on the radius of the wax-deposit layer. This correlation may not be accurate if there is asignificant variation in the thickness of the deposit along thepipeline. Sevendsen ignored the horizontal temperature gradient

which can only be justified for long enough pipes. The modelis based on the assumption that the wax content is constant inthe gel layer. However, experiments have shown that wax

content may vary significantly in the axial and radial directionsin the gel layer.16 Greenhill et al.20 used a semiempirical waxmodel which includes the shear removal mechanisms. Themodel predicts the wax volume but disregards the wax content.Singh et al.15,16 developed a mathematical model to describethe wax formation and aging in a laboratory flow-loop apparatus.Based on experimental observations, their model takes intoconsideration the radial variation of wax content in a thick wax-deposit and assume constant content in a thin wax-deposit. Thevelocity field and temperature profile are calculated based onassumptions similar to Sevendsen.19 Ramirez-Jaramillo et al.21

presented a model for wax deposition in oil pipelines thatincorporates molecular diffusion, shear removal, and aging ofthe gel layer. The authors used the multisolid model from Lira-Galeana et al.18 to describe the solid/liquid phase behavior. Theyassumed a quasi-steady state for mass, momentum, and energybalances and assumed incompressible fluids. Hernandez et al.22

introduced a kinetic resistance factor in a wax model to allowfor the departure from the bulk phase equilibrium at theinterface. They also incorporated the effect of shear strippingin their work. Assumptions in the model include the following:(1) constant wax content in the deposit layer and (2) a single

(14) Creek, J. L.; Matzain, B.; Apte, M.; Volk, M.; Delle Case, E.; Lund,H Mechanisms for Wax Deposition. AIChE Spring National Meeting,Houston, TX, March 1999.

(15) Singh, P.; Fogler, H. S.; Venkatsean, R.; Nagarajan, N. Formationand aging of Incipient Thin film Wax-oil Gels. AIChE J. 2000, 46(5), 10591074.

(16) Singh, P.; Fogler, H. S.; Venkatesan, R. Morphological evolutionof thick wax deposits during aging. AIChE J. 2001, 47 (1), 618.

(17) Pedersen, K. S. Prediction of cloud point temperature and amountof wax precipitation. SPE Prod. Facil. 1995, 10 (1), 4649.

(18) Lira-Galeana, C.; Firoozabadi, A.; Prausnitz, J. M. Thermodynamicsof Wax Precipitation in Petroleum Mixtures. AIChE J. 1996, 42 (1), 239

248.(19) Svendsen, J. A. Mathematical Modeling of Wax Deposition in oilPipeline Systems. AIChE J. 1993, 39 (8), 13771388.

(20) Elphingston, G. M.; Greenhill, K. L.; Hsu, J. J. C. Modeling ofMultiphase Wax Deposition. J. Energy Res. Technol. 1999, 121 (2), 8185.

(21) Ramirez-Jaramillo, E.; Lira-Galeana, C.; Manero, O. Modeling WaxDeposition in Pipelines. Pet. Sci. Technol. 2004, 22 (7-8), 821861.

(22) Hernandez, O. C.; Hensley, H.; Sarica, C.; Brill, J. P.; Volk, S. P. E.;Delle-Case, E. Improvements in Single-Phase Paraffin Deposition Modeling,SPE 84502-PA. SPE Prod. Facil. 2004, 19 (4), 237244.

Table 1. Relevant Data for Example 1

parameter value

L (m) 0.254Rin (mm) 1.841Rout (mm) 3.175znC8 0.67zcycloC6C19 0.33Tin (C) 0Ta (C) 25Q (gpm) 0.0158 (10-6 m3/s)

Re 526knC8 (W/m K) 0.13kcycloC6C19 (W/m K) 0.19kglass (W/m K) (pipe wall) 17.4FcycloC6C19 (kg/m

3) at 25 C 827FnC8 (kg/m

3) at 25 C 699Fglass (kg/m3) at 25 C 2230

species Tc (K) Pc (bar) acentric factor MW (g/mol)

nC8 568.7 24.92 0.39 114.0cycloC6C19 840.0 10.86 0.94 350.0

Figure 1. Computational domain and definition of the two main regions.

Figure 2. Deposit thickness ratio vs time; comparison with repeatedmeasured data of Cordoba and Schall.27 Example 1.

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diffusion coefficient for multicomponent condensate and crudesystem. Other aspects of the work by Hernandez et al. are similarto the work of Singh et al.15

In all existing models, the convection flow in the gel layer isneglected and Ficks law is used to describe the moleculardiffusion. The convection flow can be important during the earlyformation of the gel layer. It is reported that as little as 2%weight fraction of precipitated wax is sufficient to form the gellayer.16,23,24 The gel layer, with significant amount of oil trappedin it, behaves as a pseudoporous medium with nonzero convec-tion flow.7 To the best of our knowledge, none of the available

studies in the literature have accounted for the radial convectionflux in the wax formation. There are two concerns about theuse of Ficks law in describing the molecular and thermaldiffusions. The description of mass diffusion by Ficks law isonly valid for isothermal process. Ficks law may not directlyapply to describe diffusion processes in the gel layer or in theliquid region because of the radial temperature variation. Thesecond concern is that the available correlations for predictingdiffusion coefficients are only appropriate for binary mixtures.25

Most of the available models that support multicomponent flow

use one average diffusion coefficient by treating the oil solventcomponents and the wax components as a binary mixture.7,15,16,19

Singh et al.15,16 suggested that the aging process is because ofa counter-diffusion process where the heavy components diffuseinto the gel layer and the light components diffuse in thecounterdirection. The carbon number at which the oil and waxmolecules segregate and counterdiffuse is called as the criticalcarbon number by Singh et al.15,16 The critical carbon numbercould be different for different compositions, and is also afunction of the wax precipitation conditions, like the pipe walltemperature.15,16 Models that simplify the diffusion processes

by using an average molecular diffusion coefficient may havea drawback because they cannot predict the critical carbonnumber. Furthermore, the critical carbon number cannot bepredetermined. The appropriate diffusion flux in flowlines isthe result of both molecular and thermal diffusion processes.14,25

In a recent work,26 we have developed a mathematical modelfor wax deposition in pipelines that has two basic differences fromthe existing models. First, in this model, the diffusion processesare described by the molecular diffusion and the thermal diffusion.The molecular and thermal diffusions are driven by compositionaland thermal gradients, respectively. The thermal and moleculardiffusion coefficients of components vary with position and time.(23) Holder, G. A.; Winkler, J. Wax Crystallization from Distillate Fuels:

I. Cloud and pour Phenomena Exhibited by Solutions of Binary n-paraffinMixtures. J. Inst. Pet. 1965, 51 (499), 228235.

(24) Holder, G. A.; Winkler, J. Wax Crystallization from Distillate Fuels,Part II: Mechanisms of Pour Depression. J. Inst. Pet. 1965, 51 (499), 235243.

(25) Ghorayeb, K.; Firoozabadi, A. Molecular, Pressure, and ThermalDiffusion in Nonideal Multicomponent Mixtures. AIChE J. 2000, 46 (5),883891.

Figure 3. Predicted solid saturation (fraction) and temperature at different times: Re ) 526, example 1.

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position and time. It can capture the effect of flow rate and pipewall temperature on wax formation.

This paper is organized as follows. In the next section, themodel for wax deposition is briefly reviewed. We then showthe results for a two-component mixture where the effect offlow rate and diffusion fluxes are discussed. Our model is thenused to predict and analyze the results from different experi-ments. The paper is concluded with some remarks.

2. Wax Deposition ModelThe gel layer forms when the fluid temperature falls below

the WAT. When the gel layer forms and the fluid temperature

stops decreasing, wax continues to grow within the gel layermainly as a result of diffusion mechanisms. Our mathematicalmodel for wax deposition in the laminar regime is based on theassumptions that the shear removal of the deposited waxparticles, Brownian motion, and gravity settling are negligible.There is a belief by many authors that these mechanisms arenot significant in wax deposition9,12,14,15,28 and the moleculardiffusion is the dominating process. Neglecting the gravityeffects, the 3D physical problem is simplified to a 2D

computational problem, where computations are preformed ina cross section with dimensions equal to the length and the innerradius of the pipe. In this work, we consider the gel layer as apseudoporous medium, where the precipitated solid phase isimmobile. Observations have shown that as little as 2% weightfraction of precipitated wax is enough to form the gel layer.15,23,24

The species in the liquid phase continue to flow into the gellayer because of convection and diffusion processes. Duringthe early formation of the gel layer, the convection process couldbe dominant.

In the proposed wax deposition model, we incorporatemomentum, energy, and species balance equations and athermodynamic model for liquid/solid phase equilibria. We

couple the Darcy law and the Navier-Stokes equation toapproximate the pressure and the velocity field in the gel layerand in the liquid region. We solve the transient Navier-Stokes

Table 3. Critical Properties, Acentric Factor, and MolecularWeight of the Oil and Wax Components, Example 2

species Tc (K) Pc (bar) acentric factor MW (g/mol)

C11 638.8 19.48 0.53 156.0C12 658.4 18.10 0.57 170.0C13 675.9 16.79 0.61 184.0C14 692.3 15.73 0.65 198.3C15 707.8 14.79 0.69 212.3Blandol and kerosene 670.8 6.23 1.80 500.0C22-C26 808.8 8.79 1.18 353.9

C27-C30 832.6 7.72 1.21 403.6C31-C34 852.8 6.77 1.35 454.8C35+ 868.6 5.99 1.37 502.5

Figure 6. Wax deposition as a function of time for different outside wall temperatures: (a) wax thickness ratio; (b-d) amount of wax species. Q) 1 gpm, initial Re ) 540, initial wax content ) 0.67 wt %, example 2.


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equation for compressible fluid in the liquid region. In the gellayer, a source term of Darcy-type is added to the momentumequation. This source term aims to slow down the velocity asthe amount of solid wax increases in the gel layer. The volume

fraction of the liquid phase (liquid saturation) can be viewedas the porosity of the porous medium. The permeability of the

porous medium is written as a function of porosity by usingthe Carman-Koseny equation.29 The energy balance equationis written in terms of the enthalpies of liquid and solid phases.

The Peng-Robinson equation of state (PR-EOS) is used to relatethe enthalpy of the liquid phase to the temperature. The enthalpy

Figure 7. Predicted solid saturation and composition vs radial distanceat the pipe midsection for three different wall temperatures: (a) solidsaturation, (b) composition of Blandol, and (c) overall C35+ composition.Initial wax content ) 0.67 wt %, time ) 5 days, example 2.

Figure 8. Predicted temperature and velocities vs radial distance atthe pipe midsection for three different wall temperatures: (a) temperatureand (b) axial and (c) redial velocities. Initial wax content ) 0.67 wt%, time ) 5 days, example 2.


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of the solid phase is calculated in terms of the molar latent heat,heat capacity, and enthalpy of pure components.30 In the speciesbalance equations, the convection and diffusion processes aretaken into consideration in the gel layer and in the liquid region.The diffusion flux is written as a combination of the molecularand thermal diffusions. The model of Ghorayeb and Firooza-badi25 is used to compute molecular and thermal diffusioncoefficients. For a given temperature, pressure and composition,the multisolid wax model by Lira-Galeana et al.18 is used forsolid/liquid phase split calculation. This model is also used byRamirez-Jaramillo21 for a similar purpose. From our experience,this model is reliable. The successive substitution iteration (SSI)technique introduced in Banki et al.26 is numerically veryefficient. Unlike most wax deposition models, we do not assumea quasi-steady state for the momentum and energy balanceequations. All equations are approximated in the transient state.This allows the use of our model for shut down conditions.

A finite-difference based method is used for the spatialdiscretization of the 2D computational domain. The algorithmstarts by approximating the pressure and the velocity field. TheSIMPLER method of Patankar31 is used to approximate themomentum equation that is coupled with the overall mass

balance equation. After calculating the pressure and the velocityfield, the temperature and the composition are predicted by

solving the energy and species balance equations. For a giventemperature, pressure and composition, the phase-equilibriamodel is applied to calculate solid and liquid amounts and thedensities. We refer to Banki et al.26 for a complete descriptionof the mathematical model and the algorithm.

3. Results and Discussions

We use our model to predict deposition in the laboratory

setups from Cordoba and Schall27,32 and the comprehensive workof Singh et al.15,16 Our predictions are compared with themeasurements by these authors. We also report on the effect ofmolecular and thermal diffusion on the deposition. Then wepresent calculated results for a long pipe.

(26) Banki, R.; Hoteit, H.; Firoozabadi, A. Int. J. Heat Mass Transfer,published online Feb 7, http://dx.doi.org/10.1016/j.ijheatmasstrnasfer.2007.11.012.

(27) Cordoba, A. J.; Schall, C. A. Solvent migration in a paraffin deposit.Fuel 2001, 80, 12791284.

(28) Chen, X. T.; Butler, T.; Volk, M.; Brill, J. P. Techniques formeasuring wax thickness during single and multiphase flow. SPE AnnualConference and Exhibition, San Antonio, TX, Oct 5-8, 1997; SPE 38773-MS.

(29) Carman, P. C. Trans. Inst. Chem. Eng. 1937, 15a, 150166.(30) Firoozabadi, A. Thermodynamics of hydrocarbon ReserVoirs;McGraw-Hill: New York, 1999.

Figure 9. Driving forces for molecular and thermal diffusions vs radialdistance at the pipe midsection: (a) composition gradient of C35+ inliquid phase; (b) temperature gradient. Initial wax content ) 0.67 wt%, time ) 5 days, Ta ) 4.4 C, example 2.

Figure 10. (a) Wax thickness ratio as a function of time for differentwall temperatures. (b) Gel layer profiles in the computational domainfor different wall temperatures: Q ) 1 gpm, initial Re ) 260, time )4 days, initial wax content ) 3 wt %, example 2.


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The numerical model requires, as input data, the pipediameter, fluid composition and critical properties, thermalconductivities of pure components in the liquid state, and

thermal conductivities of the solid-wax components. Themelting-point properties, heat capacity, and enthalpy of fusionfor paraffins, naphthenes, and aromatics can be calculated byusing existing correlations.25 Heat capacities and molar latentheat of pure components can be found in Chickos et al. 33 andHimrad et al.34

3.1. Example 1: Comparison with Data from Cordoba

and Schall. In this example, we compare our predictions withexperimental data by Cordoba and Schall;27 these authors

measured the wax deposition thickness in a flow loop system.The fluid mixture consists of a light oil species (n-octane) anda heavy cycloalkane (nonadecylcyclohexane; cycloC6C19). Waxformation is investigated as a function of time in a 25.4-cmtesting tube. The fluid is recycled in the flow loop at a flowrate of 63 mL/min, which corresponds to a clean-tube Reynoldsnumber of 526. The wall thickness of the testing tube is about1.3 mm, and the inner tubing radius is 1.8 mm. Therefore, thepipe wall resistance to heat transfer cannot be neglected. Wecalculated the transient solution of the energy balance equationin the computational domain which includes the pipe wall. Theflow conditions and other relevant data are presented in Table1. Cordoba and Schall32 evaluated the amount of the depositionby using two methods: (1) the heat transfer method and (2) thegravimetric method. They compared the two methods andconcluded that the former is more precise than the latter,especially with thick deposition. They explained that the errorin the gravimetric method is due to the difficulty in drainingthe trapped oil from the collected gel sample. Our predictionsare thus compared with the results reported based on the heattransfer method. Cordoba and Schall32 reported that 0.08 molfraction of cycloC6C19 is soluble in nC8 at 17 C. In our work,we use the measured solubility data to adjust the criticalproperties of the heavy component. The parameters used in thephase-split calculations are listed in Table 1. The inlet fluidtemperature was fixed in the 25-28 C range and outsidewall temperature was held at 0 C. As a result, there is a high

temperature gradient in the system. We first used one of theexperiments to calculate the parameter that represents themorphology of the porous media comprised of wax crystallites(see ref 26). The value of C ) 106 1/m2 describes waxmorphology. In all the other experiments in this example andin the rest of the paper, this value will be used. The computa-tional domain is discretized into a 35 29 grid. The time stepis adaptively controlled by using a simple algorithm that allowsfor limited variations of the pressure and composition duringeach time step.

Figure 2 presents the results for the deposition thickness ratio/Rin ( is the average deposit thickness and Rin is the pipe innerradius) at a flow rate of 63 mL/min. Our predictions and

measured data show good agreement. The figure shows that ourmodel can capture (1) the rapid formation of the gel regionwhich occurs during the first few minutes of the experimentand (2) the increase in wax thickness, where the depositthickness reaches an asymptotic value after about 60 min. Thesolid saturation in the computational domain is presented inFigure 3a-d at t) 1, 10, 40, and 120 min. Note the evolution

(31) Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Hemi-sphere Publishing Corp.: Washington, 1980.

(32) Cordoba, A. J.; Schall, C. A. Application of a heat transfer methodto determine wax deposition in a hydrocarbon binary mixture. Fuel 2001,80 (9), 12851291.

(33) Chickos, S.; Hosseini, S.; Hesse, D. G.; Liebman, J. F. HeatCapacity Corrections to a Standard State: A Comparison of New and SomeLiterature Methods For Organic Liquids and Solids. Struct. Chem. 1993, 4(4), 271278.

(34) Himrad, S.; Sunwono, A.; Mansoori, G. A. Characterization ofAlkanes and Paraffin Waxes for Application as Phase Change EnergyStorage Medium. Energy Sources 1994, 16 (1), 117128.

Figure 11. Wax deposition as a function of time for different flowrates: (a) wax thickness ratio; (b and c) amount of wax species. Initialwax content ) 0.67 wt %, example 2.


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of the saturation profiles and the hardening (aging) of the gellayer. Figure 3e and 3f shows the temperature profiles at t) 1and 120 min. Figure 4a depicts the radial variation in thetemperature at the middle of the pipe at t ) 1, 10, and 120min. The temperature difference is nearly 6 C across the innerradius of the tube at t ) 120 min. Figure 4b shows thetemperature averaged over radius along the tube length. Thepredicted temperature difference between the inlet and the outletafter 2 h is about 3 C, which is in agreement with themeasurements reported by Cordoba and Schall.27 Figure 4c and4d depicts the evolution of the solid saturation and the overall

composition of n-octane as a function of time at positions Aand B (see Figure 1). It can be noticed that solid saturation(that is, wax amount) increases very quickly close to the wall,which is mainly because of a large thermal flux. The lightcomponent (n-octane) diffuses out of the gel region and the waxcomponent (cycloC6C19) moves toward the pipe wall (Figure4d).

3.2. Example 2: Comparison with Data from Singh et

al.15,16. Singh et al.15,16 have conducted extensive measurementson deposition from several experiments. They investigated theinfluence of the operating conditions on the formation and theaging of the gel region. They also presented a numerical modelto predict the measured data. Depending on the thickness ofthe gel layer, they distinguished between two types of wax

deposition models. In one case, where the gel layer thicknessis up to 20% of the inner pipe radius (thin gel), they assume a

constant wax content in the gel region in the radial direction.15

The other case corresponds to a thick wax deposit, where the

thickness of the gel layer is up to 50% of the inner radius. 16

Their model, in this case, accounts for the variation of wax

content along the radial distance. Singh et al.15,16 adjusted the

aspect ratio of the wax crystals to match the results from their

experiments and their model. They assumed a linear variation

of average aspect ratio with wax content of the deposit resulting

in changes from 1 to 24 in the aspect ratio in some of their

experiments.

Singh et al.15,16 report correctly that the aging process is due

to diffusion but only molecular diffusion is singled out as thedominant process. Their work is based on the definition of a

critical carbon number for the wax-forming molecules. Hydro-

carbons that have carbon numbers more than the critical carbon

number diffuse into the gel layer and those with carbon numbers

less than the critical carbon number diffuse out of the gel layer

toward the pipe center. The critical carbon number is allowed

to bea functionof compositionsand thepipewalltemperature;15,16

that is obtained from adjusting their model to the measurements.

Our model for diffusion flux is free of a critical carbon number.

Singh et al.15,16 use the effective molecular diffusion coefficient

in a binary mixture from the critical carbon number; this

coefficient is assumed to be related to the square of the aspect

ratio.15,16 A correlation from the literature is used for binariesby these authors.

Figure 12. Predicted temperature, solid saturation, and wax composition radial distance at the pipe midsection for two flow rates: (a) temperature,(b) solid saturation, (c) composition (overall) of Blandol, and (d) C 35+. Initial wax content ) 0.67 wt %, time ) 5 days, example 2.


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In this example, we used our model without parameteradjustment in the deposition and without the introduction ofthe critical number. The model is applied without adjustmentsfor thin and thick wax depositions. As we will see later, one ofthe experiments that was judged by previous authors to be athin wax deposition in the past, may have features of a thickwax deposition.

In the experiments for a thin gel layer, Singh et al.15 used aliquid solvent mixture (Blandol and kerosene) with 0.67%

weight of wax-forming species (food grade wax). In the work

of Singh et al.,16 the same solvent mixture and wax-formingspecies are used but the amount of wax-forming species wasincreased to 3 wt % in order to conduct experiments for thick

deposits. Singh et al.15,16 studied the effects of the flow rate,the pipe wall temperature on the thickness of the gel layer, and

the increase of the wax content in the deposition. The relevantdata on the testing tube and fluids and solids are presented in

Table 2. The liquid solvent (that is, oil solvent) used in theexperiments is a 3:1 mixture (volume basis) of mineral oil

(Blandol) and kerosene (see also, ref 35). The density andviscosity of the oil solvent mixture are 838.5 kg/m3 and 8.7

mPa s, respectively. The molecular weight distribution of thefood grade wax (Mobil M140) is given by Singh et al.,15,35 and

the carbon number varies from 23 to 38. The solubility of the

(35) Singh, P.; Fogler, H. S.; Nagarajan, N. Prediction of the WaxContent of the Incipient Wax-Oil Gel in a Flowloop: An application of theControlled-Stress Rheometer. J. Rheol. 1999, 43 (6), 14371459.

Figure 13. (a) Wax thickness ratio as a function of time for differentflow rates. (b and c) Effect of molecular and thermal diffusions ondeposition thickness and wax aging. Initial wax content ) 3 wt %, Q) 2 gpm, example 2.

Figure 14. Radial flux of species by convection flow, moleculardiffusion, and thermal diffusion. Initial wax content ) 3 wt %, Q ) 2gpm, example 2.


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wax-forming species in the oil solvent is provided as a functionof temperature:35,36

Wwax) (T(F)

63.187)5977

(1)

where Wwax is the weight percent of wax-forming species inthe mixture (solubility) and T is the temperature in degreesFarenheit.

Figure 5 presents the measured solubility of wax-formingspecies in the solvent as a function of temperature. We usedthe solubility data in Figure 5 to make minor adjustments incritical properties of the heavy species of the solid/liquid phase-equilibria model. The critical properties of the species areprovided in Table 3 (after minor adjustments from ref 30). Notethat the adjustment of the heavy species critical properties isnot related to the flow model which is the central theme of thiswork.

3.2.1. Effect of Pipe Wall Temperature. Singh et al.15

investigated the effect of the difference between the inlet fluidtemperature and the pipe wall temperature. They presented aseries of flow loop experiments where the inlet temperature is

kept at 22.2 C with constant pipe wall temperatures of 4.4,7.2, and 8.3 C. A constant flow rate of 1 gpm (correspondingto Re ) 540 for a clean tube) was used to perform the threeexperiments. Measured data and our predictions for the waxthickness ratio as a function of time are presented in Figure 6a;the agreement is good. For all the three wall temperatures, oneobserves the rapid formation of the gel layer to an asymptoticvalue after which the gel thickness stops growing. As expected,the gel thickness is smaller at the higher wall temperature. The

average wax content of the gel layer for the three walltemperatures predicted from our model and the measured valuesare shown in Figure 6b-d. The amount of wax continues toincrease even after the gel layer thickness stops growing. It canbe observed that wax concentration increase in the gel layer ishigher for the higher wall temperatures. To analyze thisbehavior, we show, in Figures 7, the wax saturation and thecomposition of Blandol (oil solvent component) and C35+ (ofwax) vs the radial distance at pipe midsection. In Figure 7a,the wax saturation near the pipe wall is higher for the higherwall temperature, which is consistent with Figure 6. A similarbehavior can be seen in Figure 7c for the composition of C35+.However, the composition of Blandol has an opposite trend,

that is, lower Blandol amount for higher wall temperature. Thisbehavior, which is also consistent with Figures 6 and 7a, isbecause of the counterdiffusion fluxes of different species thatsegregate the wax components toward the pipe wall and thesolvent components away from the wall. We still need to explainwhy the deposition rate is higher for thinner gel layers(corresponding to higher wall temperature, Figure 6). Figure8a shows that the temperature gradient close to the wall is nearlythe same for the three wall temperatures. However, in the thingel layer, wax components travel a shorter distance from thebulk liquid before they accumulate at the pipe wall.

Figure 7a also shows the interesting behavior of the higherwax saturation at the single phase liquid-gel interface than at

the pipe wall for the wall temperature of 4.4

C, which resultsin the thick gel layer. This unusual behavior has been reportedby Singh et al.16 from the measurements for the thick gel layer.They measured a higher concentration of wax deposition at thesingle phase liquid-gel interface than the pipe wall. We canonly explain such a composition profile from the balancebetween thermal diffusion and the molecular diffusion and themulticomponent nature of deposition. The thermal diffusion fluxdepends on the temperature gradient, which is a maximum atthe pipe wall (Figure 9b). However, the molecular diffusionflux depends on the composition (mole fraction) gradient, whichfor some of the precipitated components may be a maximumclose to the interface and for some others in the gel layer awayfrom the interface (Figure 9a). The effect of the moleculardiffusion can also be seen in Figure 7c where C35+ mainlyaccumulates close to the interface (detailed discussion will comelater). Another process that may also contribute to the waxgrowth rate is the radial convection flow. Further discussionon molecular, thermal and convection fluxes and a more detaileddiscussion will be presented later. Figure 8b and 8c shows theaxial and radial velocity at the pipe midsection for the threewall temperatures. A thicker gel layer leads to higher velocity,as expected.

Singh et al.16 also presented a series of flow loop experimentsusing the mixture with wax-forming species of 3 wt % in theliquid mixture (Table 2). The inlet temperature was 29.5 Cand the pipe wall temperatures were 4.5, 9.5, and 14.5 C. The

experiments were performed with constant flow rate of 1 gpm(corresponding to Re ) 260). Figure 10a shows good agreement

(36) Rnningsen, H. P.; Bjrndal, B.; Hansen, A. B.; Pedersen, W. B.

Wax precipitation from north sea crude oils I. Crystallization and dissolutiontemperatures, and Newtonian and non-Newtonian flow properties. EnergyFuels 1991, 5 (6), 895908.

Figure 15. Solid saturation and composition at the pipe midsectionfor different flow rates. (a) Solid saturation. (b) Overall compositionof C35+. Initial wax content ) 3 wt %, time ) 4 days, example 2.


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between our predictions and measured data for the gel layerthickness, which is presented as a function of time. The profileof the gel layer in the computational domain is presented inFigure 10b.

3.2.2. Effect of Flow Rate. Experiments with different flowrates were also conducted by Singh et al.15,16 to study the effectof the flow rate on deposition. In the first set of experiments,where the solvent oil-wax mixture with 0.67 wt % of wax isused, the injected fluid temperature and the pipe wall temper-

ature are held at 7.2 and 8.3 C. Three flow rates were used: 1,2.5, and 4 gpm (corresponding to Re ) 540, 1340, and 2140 in

a clean tube). Since a turbulence flow could occur with the flowrate of 4 gpm (Re ) 2140), we compared our predictions forthe wax thickness ratio and the wax content in the gel regionwith the measured data at flow rates of 1 and 2.5 gpm. Figure11 shows that the measured data and predictions are in goodagreement. Again, one notices that a higher flow rate results ina thinner gel region with higher wax content. The same behavior(thinner gel having higher wax content) was also observed byvarying the wall temperature. In addition to the explanation

given previously, a higher flow rate leads to a higher temperaturegradient close to the pipe wall (see Figure 12a) resulting in

Figure 16. Predicted composition in liquid phase of wax components vs radial distance at the pipe midsection: Q ) 1 gpm, initial wax content )0.67 wt %, time ) 5 days, example 2.


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higher thermal flux. In Figure 12b-d, we show the waxsaturation and composition of Blandol and C35+ at t) 5 days.In another set of experiments by Singh et al.,16 where 3 wt %wax (Table 2) was dissolved in the solvent mixture, the inletfluid temperature was 29.5 C and the wall temperature was4.5 C. The deposit thickness was monitored as a function oftime at flow rates of 1, 2, and 4 gpm (corresponding to initialReynolds numbers of 260, 520, and 1040). Comparison betweenpredictions and measured data of the deposit thickness ratio ispresented in Figure 13a. The predictions show a slight under-estimation of the deposit thickness as the flow rate increases.

3.2.3. Effect of Diffusion Flux and Radial ConVection

Flux. We investigated the effect of molecular and thermaldiffusions on wax composition using our numerical model forthe following cases: (1) without diffusion, (2) without thermaldiffusion (only molecular diffusion), and (3) with thermal andmolecular diffusions. Figure 13b and 13c show that molecularand thermal diffusions have an appreciable effect on thethickness (Figure 13b), but the effect on the aging of the gellayer is very pronounced especially from thermal diffusion(Figure 13c). Thermal diffusion effect has a more pronouncedeffect on aging than molecular diffusion.

To appreciate the importance of the radial convection flux,we compare the radial fluxes driven by thermal, molecular, andconvection processes across a cross section in the gel layer of

length L/2 and radius 0.85Rin (the location of the cross section

is shown in Figure 1). Figure 14 shows that, during the early

time of the gel layer formation, the radial convection flux is

important. However, as the solid saturation increases, the

molecular and thermal diffusions become more dominant. Figure

14a reveals that the flux of Blandol is mainly due to thermal

diffusion (see the inset); however, the flux of for the C35+ is

mostly from molecular diffusion (from Figure 14b). Note that

the plot in Figure 14a shows a reversal in molecular diffusion

flux due to sign change in composition gradient in agreement

with results in Figure 9a.

As a final presentation for the extensive tests by Singh et

al.,16 we show the computed wax and saturation and composition

of C35+ for the mixture with the high wax-forming species.

Figure 15 shows that as the flow rate increases the wax content

of the gel layer close to the interface increases substantially

above the concentration at the wall. We attribute this behavior,

which is in line with the measurements, to the multicomponent

nature of molecular diffusion and thermal diffusions. Compari-

son of the results in Figure 7a for a wall temperature of 4.4 C,

and all the results in Figure 15 shows that one may not arbitrarily

divide wax deposition as thick or thin. The same figure alsoshows a nonmonotonic wax saturation and C35+ concentration

in the radial direction. The wax saturation is close to wax

fraction in weight fraction based on the density of various

species. The results in Figure 15 are close to measured data by

Singh et al.16 However, our predicted results have different

trends from the calculated values by Singh et al.16 which show

a monotonic behavior. Figure 16 presents the predicted com-

position profile of the groupings of the wax-forming species

and may explain the nonmonotonic behavior of the composition

profiles. The figure contains the results which provide the

essence of our argument that (1) there may not exist a sound

basis to adopt a critical carbon number for deposition, (2) the

maximum concentration for different species in the gel regionmay not be always at the interface, and (3) thermal diffusion

may contribute significantly to the species profiles.

Table 4. Relevant Data for Example 3

parameter value

L (m) 1000Rin (cm) 15znC8 0.67zcycloC6C19 0.33Tin (C) 39Ta (C) 24Q (gpm) 2

Re 500

knC8 (W/m K) 0.13kcycloC6C19 (W/m K) 0.19FcycloC6C19 (kg/m

3) at 25 C 827FnC8 (kg/m

3) at 25 C 699

Figure 17. Predicted solid saturation and composition at different times. (a and b) Solid saturation (fraction). (c and d) Composition of cycloC 6C19(mole fraction). Re ) 500, example 3.


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3.3. Example 3: Large-Scale System. The purpose of thisexample is to show the wax deposition behavior in a longpipeline. We study the deposition of the binary mixture of nC8/cycloC6C19 from example 1. The pipe length is 1 km, and theinner radius is 15 cm. The flow conditions and other relevantdata are listed in Table 4. The solid saturation in the compu-tational domain is illustrated in Figure 17a and 17b at t ) 5and 12 months. It can be noticed that the profile of the waxdeposit in a long pipeline is different from that in laboratory-

scale tubes as shown in Figure 17a and 17b. A maximumdeposition thickness may occur far from the outlet where thefluid temperature is the minimum. Such a behavior has beenreported by many authors.8,1921 Figure 17c and 17d showscomposition profiles for cycloC6C19 at t ) 5 and 12 months.The segregation of the heavy and light components is similarto a laboratory-scale system.

4. Concluding Remarks

There are two main features that characterize our model:We study the effect of molecular diffusion and thermal

diffusion in wax deposition predictions. Both affect the flux ofspecies toward the pipe wall by diffusion. The driving force

for molecular diffusion is the concentration gradient. The drivingforce for thermal diffusion is temperature gradient. These twogradients may not be proportional in the gel layer. Thetemperature gradient is expected to be high close to the pipewall. The concentration gradient, because of phase change, isexpected to be high at the point where a species crystallizesfirst in the gel layer. Furthermore, the temperature gradient closeto the wall may reach a pseudo steady state quickly and staylarge close to the wall. This may not be true for the concentration

gradients. All these aspects may lead to incorrect results by thecommonly accepted use of the chain rule to replace concentra-tion gradient by temperature gradient in the Ficks law.

Our numerical model for wax deposition in petroleumflowlines has been verified using various measurements in theliterature. The proposed model predicts the wax formation andaging as a result of a decrease in the temperature below thewax appearance temperature (WAT) and the continuous growthof wax content due to thermal and molecular fluxes. Published

experimental data are used to investigate the reliability of themodel and to show that it can capture various observations inthe experiments. The predicted results show that the thicknessof the gel layer increases rapidly in the early stages, then thedeposit thickness virtually stops increasing. The wax contentof the gel layer grows as a result of the combined molecularand thermal diffusion fluxes. It is also shown that higher flowrates (similarly, higher wall temperatures) result in a thinnergel layer with higher wax content. The prediction is consistentwith the measured data. We have also investigated the effectof molecular, thermal diffusion fluxes, and the radial convectionflux. Both molecular and thermal diffusions affect the waxdeposition; the effect of thermal diffusion can be even more

pronounced than molecular diffusion. All these predictions aremade without adjustments of the parameters unlike the past workwhen thermal diffusion is neglected.

Acknowledgment. The funding for this work was provided bythe member companies of the Reservoir Engineering ResearchInstitute (RERI). We thank Daniel Rosner and Alana Leahy-Diosof Yale University for reading the work and their comments.

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