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Abstract:Production in reservoirs located in deep and ultra-deep water that contain waxy crude oils faces a huge obsta-cle imposed by the low temperatures of the environment. When the waxy crude oil is subjected to a tempera-ture below the Gelation Temperature, as in the case investigated in the present work, it exhibits a variety of non-Newtonian features: elasticity, plasticity, viscous effects, and time-dependency, which renders to this materiala highly complex behavior. A crucial feature that is frequently ignored when the determination of the yield stressis being carried out, is the time-dependency nature of these materials. We demonstrate how significantly dif-ferent values of yield stress can be obtained if this character of the material is neglected. We use the asphaltenesproperties as inhibitors of wax formation and propose a protocol to capture yield-stress parameters. One impor-tant conclusion is that waxy crude oils can be classified as apparent-yield-stress fluids, and not (true-) yield-stress materials, with the presence of a dynamic and a static yield-stresses.

Zusammenfassung:Die Produktion in Reservoiren, die im Bereich von tiefem und ultratiefem Wasser liegen und wachsartige Rohölebeinhalten, wird durch ein großes Hindernis aufgrund der niedrigen Umgebungstemperaturen erschwert. Wennwachsartige Rohöle Temperaturen, die niedriger als die Gelierungstemperatur sind, ausgesetzt werden, wie esin dieser Arbeit untersucht wird, treten eine Vielzahl nicht-Newtonscher Erscheinungen auf: Elastizität, Plastizi-tät, viskose Effekte und zeitabhängige Phänomene, die hochkomplex sind. Eine bedeutende Erscheinung, die häu-fig bei der Bestimmung der Fließspannung ignoriert wird, ist das zeitabhängige Verhalten dieser Materialien. Wirzeigen, wie bedeutend unterschiedliche Werte der Fließspannung erhalten werden können, wenn diese Materi-aleigenschaft vernachlässigt wird. Wir verwenden asphaltartige Eigenschaften als Inhibitoren der Wachsbildungund schlagen ein Protokoll vor, um die Fließspannungsparameter zu bestimmen. Eine bedeutende Schlussfolge-rung ist, dass wachsartige Rohöle klassifiziert werden können als scheinbare Fließspannungsfluide und „falsche“Fließspannungsmaterialien mit dynamischen und statischen Fließspannnungswerten.

Résumé:La production de pétrole brut cireux dans des réservoirs localisés dans les eaux profondes et ultra profondes faitface à un énorme obstacle : les basses températures imposées par l’environnement. Lorsque le pétrole brut cireuxest assujetti à une température inférieure à la température de gélification, comme dans le cas de la présenteétude, il présente une variété de propriétés non Newtoniennes : élasticité, plasticité, effets visqueux et dépen-dance temporelle, qui donnent à ce matériau un comportement très complexe. Un aspect crucial qui est fré-quemment ignoré lors de la détermination de la contrainte seuil, est la nature transitoire de ces matériaux. Nousdémontrons que des valeurs de contrainte seuil significativement différentes peuvent être obtenues si cet aspectdu matériau est négligé. Nous utilisons les propriétés inhibitrices des asphaltènes pour la formation de la cire,et nous proposons un protocole pour mesurer la contrainte seuil. Une conclusion importante est que les brutsde pétrole cireux peuvent être classifiés comme des fluides à contraintes seuils apparentes, et non comme desmatériaux possédant de vraies contraintes seuils, avec la présence de contraintes seuils dynamiques et statiques.

Key words: waxy crude oils; re-start problem; yield stress measurement; time-dependent material; wax for-mation inhibitors

Measuring the Yielding of Waxy Crude Oils Considering itsTime-dependency and Apparent-yield-stress Nature

Edson J. Soares1*, Roney L. Thompson2, and Andre Machado3

1 LabReo, Department of Mechanical Engineering, Universidade Federal do Espirito Santo,Avenida Fernando Ferrari, 514, 29075-910, Vitoria, ES, Brazil

2 LMTA, Department of Mechanical Engineering (PGMEC), Universidade Federal Fluminense,Rua Passo da Patria 156, 24210-240, Niteroi, RJ, Brazil

3 UO-ES, ENGP, EES, PETROBRAS, Avenida Nossa Senhora da Penha 1688,29057-550, Vitoria, ES, Brazil

* Corresponding author: edson@ct.ufes.brFax: x55.27.40092151

Received: 18.3.2013, Final version: 11.6.2013

© Appl. Rheol. 23 (2013) 62798 DOI: 10.3933/ApplRheol-23-62798

subjected to a cooling process that acceleratesthe deposition of paran at the pipeline wall. Theproduction losses and the necessary investmentsto remediate such kind of problem are responsi-ble for significant costs in the petroleum indus-try and the understanding of this complexprocess is a challenging task which is worthy tofollow [4]. An important correlated problem isthe so-called re-start problem. It occurs whenthere is a shutdown of the pumping process, thatis caused for operational reasons. In this case,when there is no flow, the oil is cooled and itstemperature decreases below the GT and so thegel structure is inevitably formed. These differ-ent crystallization processes, in quiescent andunder shearing conditions were analyzed byKane et al. [5] on trying to establish a correlationbetween structure and rheology. After hydrateprecipitates, wax precipitates are the most com-mon cause for blockage in oil pipes [6]. In orderto re-start production, the pumping pressure atthe pipe inlet must be high enough so that it canrupture the internal gel structure and sustain asteady-state flow. In other words, the pumpingpressure must overcome the resistance imposedby the yield stress of the material. This re-startpressure can be significantly different from thepressure necessary to operate in normal condi-tions, i.e. when the steady-state is achieved. Thepump that can handle both situations has tooperate in a wider pressure range. In this sense,a good estimative of the yield stress of the oilwhen it is in the complete gelled state would beuseful for the hydraulic plant project.

From the discussion above, we can concludethat gel formation and gel breakage processesare extremely relevant for the logistic of trans-portation of waxy crude oils in ultra deep water[7]. These two processes are highly influenced byshear history, temperature history and the com-position of the waxy crude oils. Hence, the previ-ous shear flow conditions before shutdown, theinitial temperatures of the oil and the environ-ment, the thermal conductivities of fluids andsolid materials involved, and the temperaturedependence of the parameters that control thekinetics of gelation are important factors todetermine the final state of the gel formation.The gelation process is responsible for a struc-turing in the material that can be quantified bythe mechanical property yield stress. Under-standing the yielding of the gel structures is cru-

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1 INTRODUCTIONAfter a long tradition of oil production on shoreand a period of off-shore exploration, reservoirslocated in deep and ultra-deep water are anattractive source for the increasing demands ofoil. Typically, waxy crude oils are the type of oilfound in these deep water reservoirs. Waxy crudeoils are paranic systems which are composed bya complex mixture of hydrocarbons containingsaturates, aromatics, naphthenes, asphaltenes,and resins. When these materials are cooled andachieve the so-called Wax Appearance Tempera-ture (WAT), also called Cloud Point (CP), there is aformation of wax crystallites. Theoretically, theWAT is defined in a purely thermodynamic basis,being independent of operational conditions [1].Below the WAT, another important characteristictemperature of the problem is the so-called Gela-tion Temperature (GT), where the oil presents agel character. At these low temperatures, theseparan waxes precipitate out of the crude oil mix-ture and form a gel-like structure consisting of asolid paran crystal network and entrapped liquidoil [1]. The GT is dependent on the flow conditions.The structuring evolution of the oil takes place bya growth of these crystallites and the formationof crystals of larger sizes that continue to inter-act to each other in this flocculating process. At aeven lower temperature, the so-called Pour Point(PP) is achieved. The PP is defined as the lowesttemperature at which the oil flow freely undercondition specified by the American Society forTesting and Materials (ASTM). Below the GT theoil looses its Newtonian character and assumes aviscoplastic character, with an associated yieldstress. Besides that, the yield stress of the gelledoil is strongly dependent on temperature, beinga decreasing function of this variable [2]. Theextent of increase in the yield stress values withthe decreasing temperature is greater when thepercentage of wax is higher [3].

Typically, the oil leaves the reservoirs andenters the pipeline at a temperature ranging 60to 100 °C. The temperature in very deep waterenvironment can achieve 4 °C. This temperaturedifference generates a heat flux from the oil tothe environment. During normal operationalconditions, the main problem faced by the oilindustry is related to the paran precipitation anddeposition in the pipe. In contact with this lowtemperature environment, the wax crude oil is

cial for the strategies of gel breakage and re-start. Singh et al. [8] reported that slower coolingrates lead to greater reduction on the yield stress.However, determining the re-start pressure froma simple measure of the yield stress is not a sim-ple task. In fact, there is a significant discrepan-cy between what is measured for a waxy crudeoil in the rheology laboratory and what is ob -tained in pipeline tests [9]. There are strong evi-dences that laboratory rheological measure-ments overestimates the upstream pressureneeded for re-starting the flow.

A first challenge to face is that the mea-surement of the yield stress of a material doesnot have a standard protocol consensually accep -ted. In fact, this issue is a matter of profound dis-cussion in the literature, where a fundamentalpoint concerns questioning the very existence ofthe yield stress [10 – 14]. Together with the termyield stress there are other expressions to desig-nate correlated entities such as elastic yieldstress, static yield stress, dynamic yield stress [15],apparent yield stress [11], a terminology that willbe discussed later. Besides that, an additionalcomplication comes from the fact that gelledmaterial exhibits time dependent properties dueto changes in microstructure in a time scale thatneeds to be accounted for [16 – 19]. The hindranceis that is not easy to evaluate if the steady statewas achieved, and the value of the yield stressbecomes a function of the experimental proto-col, the accuracy of the rheo meter at low shearrates and the patience of the experimentalist.These time dependent materials are sensitive tothe time scale over which the gel is subjected toa given shear stress. Therefore, another source ofdifferent behavior can be identified if laboratoryand pipeline are submitted to different timescales of imposed shear stress [20].

A second point concerns the gelation pro -cess. The temperature difference provokes a spa-tial temperature gradient, which acts togetherwith the diffusion of wax forming molecules ren-dering a spatial distribution of wax crystals in thegel formation. These different layers of structur-ing often present different yield stresses associ-ated, turning the gel in the oil pipeline into a in -homogeneous material with respect to itsme chanical properties such as viscosity and yieldstress. This intricate behavior makes difficult theprediction of the re-start pressure by models thatnot take this fact into account [6, 21]. Another dif-

ference in gel formation is the possible presenceof voids in the pipe due to shrinkage during thegelation process [22]. Therefore, different proto-cols followed in the laboratory render differentshear and temperature histories to the material,what leads to different final bulk properties.

In order to minimize transport problems,wax inhibiting strategies are often adopted. Oneimportant example is to add chemical sub-stances that act as wax crystals modifiers or assurface active agents [23]. Some substances fre-quently present in waxy crude oil compositioncan act as natural inhibitors of paraffin deposi-tion and wax crystallization. Polar compoundssuch as asphaltenes and resins are known to havethis property. Hence, even being part of the oildistillation residues and not being desirable com-ponents of oil production from the refine per-spective, the presence of asphaltenes and resinscan be extremely helpful in the case of the neces-sity of stopping the production, when the oil issubjected to a gelation process. Asphaltenes aregenerically defined as compounds of petroleumthat are insoluble in alkanes. They are not aunique substance, but a mixture of poly-disperseand chemically heterogeneous species [24]. Theasphaltene fraction is formed by many series ofrelatively large molecules containing aromaticrings, several heteroaromatic and napthenic ringplus relatively short paraffinic branches. Theyalso constitute the most polar and heaviestspecies of oil distillation residues compound.They usually form aggregates or flocculates andthis character is used to explain their effect asinhibitors of wax crystals formation in a oil pipe -line. These aggregates form obstacles for waxycrystals to grow and to merge with other crystals.In the flow of oil in a pipeline, the pressure dropin a pipe is the main cause of the solvent degra-dation with respect to the solute asphaltene. Inother words, a pressure decrease to the bubblepoint is responsible for the precipitation of theasphaltenes in the solution and hence they couldloose their capacity of inhibiting the formationof the gel structure. Hence, the final effect due topressure drop is quite similar to that encounteredafter a flocculant such as n-pentane or n-hepa -tane is added after the mixture is cooled.

The great majority of the models intendedto represent the flow curve of waxy crude oilshave a yield stress as a parameter. However, dueto the difficulties on defining and measuring the

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yield stress of the material and the time depen-dent character of the gelled oil, a discussion onthe measurement of the yield stress and on theavailable viscoplastic models is pertinent in thepresent context. The intrinsic problem that is inthe very concept of the yield stress was clearlystated by Evans [12]. He observe that the defini-tion of yield stress excludes the possibility ofexperimental proof of its existence. In the con-text of time dependent materials, this observa-tion is even more critical. If we apply a certainstress, and apparently there is no flow associat-ed, how can we guarantee that this state ofaffairs will remain? This fact was observed byMoller at al. [17], who proposed an alternativedefinition that the experimentalists need toemploy: "the highest stress at which no flow isdetectable within the duration of the experi-ment." However, the strain rate can be so low inthis case that, although no detectable flow isoccurring, one cannot prove that after a longtime, the material will stay in the same unde-formed state. In fact, Barnes and Walters [10]show that some materials that were thought tobe yield-stress materials do flow when theapplied stress is below the value of the previous-ly measured yield stress. Even though, the yieldstress concept is useful for these materials, andthe expression apparent yield stress exemplifiesthis point. Although there is flow for stressesbelow a certain value, for these materials, therate of strain associated is very small. When theapparent yield stress is achieved, a major collapseof microstructure takes place with a pronounceddecrease of the viscosity.

Wardhaugh and Boger [25] identified threeyield stress quantities in order to fully character-ize the yielding of a gelled waxy crude oil. The firstone is the elastic yield stress. It is the value of stressbelow which the material is in the elastic (recov-erable) regime. The static yield stress is more asso-ciated to the classical definition of yield stress, andis defined as the stress value where the gel mate-rial starts to fracture and the breaking process. Thedynamic yield stress can be viewed as the value ofstress towards to the material is attracted, whenthe shear rate is decreased departing from theunstructured state. As discussed in [25] this vari-ety of "yield stresses" are features of solid-likebehavior that are uncommon in general vis-coplastic materials, and are signals of a very com-plicated yielding process. It is worth noticing that

this complicated behavior is present in other sys-tems besides waxy crude oils [26, 27]. The identi-fication of an elastic yield stress in waxy crude oilsin the gelled state adds another degree of com-plexity to the problem. This puts this kind of mate-rial into the category of elasto-viscoplastic timedependent materials. In general, there are notmany examples of models that can handle thiskind of material. A very important subcategory ofmaterials with such behavior is when the timedependency has a thixotropic character, i.e. whenthe flow curve of the material is unique and not afunction of the previous history undergone by thematerial.

As discussed by de Souza Mendes andThomp son [19] there are two classes of modelsthat were conceived to model elasto-visco -plastic-thixotropic models. The first type, calledType I, takes a Bingham-like viscoplastic consti-tutive equation and includes elasticity andthixotropy. The second kind, called Type II, takesa Maxwell-like constitutive equation and addsviscoplastic and thixotropic features. The line ofarguments they developed resulted in the con-clusion that there are some advantages in adopt-ing Type II models. Among these advantages are:� the capacity of the model to tend smoothly to

particular cases such as viscoplastic-thixo tro -pic systems and elasto-viscoplastic systems

� the constitutive equation does not need anextra-evolution equation for the elastic strain

� there is no necessity to relax the definition ofthe yield stress, by making this quantity afunction of the structuring level. Additionalfeatures that would be interesting for bothtypes were are also stressed, such as the inclu-sion of the flow curve as an input of the mod-el and the attribution to the stress level (andnot the strain rate level) the responsibility forthe breakage of the material structure.

In the present work we perform yield stressmeasurements of a Brazilian crude oil with differ-ent concentrations of asphaltenes and resins. Inthis connection, we use the yielding character ofthe material to quantify the ability of the as -phaltenes and resins as natural inhibitors of waxformation. We show that not taking into accountthe time-dependent nature of waxy crude oils,turns the yield stress measurement dependent onthe protocol. However, a more carefully approachresults into an independent flow curve.

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2 VISCOPLASTIC MODELSA non-Newtonian viscosity proposed by Peder-sen and Ronningsen [28] is a function of the vol-ume fraction of wax particles in the combined oilplus wax phase Fwax. It is given by

(1)

where g· is the shear rate and D, E, and F are para-meters used to fit the viscosity function. Thisfunction has a yielding-like character, since vis-cosity diverges in the limit g· ® 0. Pedersen andRonningsen [29] tested a number of waxy crudeoils obtaining a good fitting using Equation 1. Thework of [6] and [30] explores the viscosity depen-dency on temperature. Both works highlight thenon-Arrhenius character of this dependency,below the Gelation Temperature. An Arrheniusdependency is characterized by an equation ofthe following type

(2)

where href is the viscosity at a reference temper-ature qref and a is a quantity with dimension oftemperature which rules the intensity of thetemperature dependency. As shown by March-esini et al. [30], when the oil is being cooled froma liquid unstructured state, the material beginswith an Arrhenius-like dependency, but, at a cer-tain point it starts a different kind of behavior.This point can be a good estimate for the GT. Dim-itriou et al. [6] argues that, since the pure waxystate is well capture by a VFT (Vogel-Fulcher-Tammann) model, given by h = hvexp [b(q - q

∞)-1],

the combined behavior must be something inbetween these two extremes. In order to fit theconstant temperature data of a model waxycrude oil [6] uses the Cross model

(3)

where h0 and h∞

are limiting values at low andhigh shear rates, respectively, g· * a characteristicshear rate, and m a parameter that controls the

shear thinning intensity. They also used the Her-schel-Bulkley model. They found that, while theCross model was successful on fitting the viscos-ity data in the whole range of shear rates, the Her-schel-Bulkley model was capable to provide agood fitting only in the high limit range. Thisresult, together with others present in the liter-ature give evidence to arm that waxy crude oilsare apparent-yield-stress fluids and not yield-stress materials [31]. In order to consider the timedependent nature of waxy crude oils, Zhao et al.[7, 32] use an equation for the evolution of a struc-ture parameter proposed in [33] given by

(4)

and developed a model that is based on a timedependent yield stress of the form

(5)

where tA and tB are the residual and original yieldstresses, t – ty the local duration of shearing, andl·c the gel degradation rate. As discussed previ-

ously a different perspective of understandingyielding can be employed from the framework ofType II models [19]. In this case, another philoso-phy is adopted where the time dependencynature of the material is handled without mak-ing the yield stress a function of time (or equiva-lently, structure). The yield stress is interpretedas the stress value for starting the major break-down process. In this case the flow curve, whichis determined experimentally is considered thelocus of the attractor points which are in a non-equilibrium state, i.e. when the microstructure ischanging. A quite general function for the vis-cosity at the equilibrium state that is conceivedto model apparent-yield-stress fluids was pro-posed in [34, 35] and is given by

(6)

where h0 and h∞

are the viscosities values asso-ciated to g· ® 0 and g· ® ∞, respectively, ty thestatic yield stress, tyd the dynamic yield stress,

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and K and n are the power-law parameters (con-sistency and exponent), respectively. Further, g· ydis the characteristic dynamic shear rate, whichmarks the transition between static and dynam-ic yield stresses. In order to complete the thixo -tropic model there is a evolution equation for thestructure parameter [14]

(7)

where l0 is the maximum value of the structureparameter l and the subscript eq is associated toequilibrium quantities. The parameters a and bare for fitting the structure evolution of thematerial. The evolution equation for the relationstress-strain rate given by

(8)

where the general parameters q1 and q2 are func-tions of the structure level l. There are severalmethods for determining the yield stress [31]. Themethods depend on being indirect or direct meth-ods (indirect would correspond to extrapolationor determination with respect to a curve fittingof a model), on the geometrical device (cone-plate, plate-plate, vane, couette), on imposingcontrolled stress or controlled strain. A challengethat the characterization of waxy crude oil impos-es is associated to its time dependency character.The yield stress can only be achieved if the wait-ing time for reading the stress value is longenough so that the stress value is not changinganymore. Concerning oscillatory tests we canobserve two criteria for determining the yieldstress: one is the point where there is a more pro-nounced drop on the value of G’ representing thatthe material is losing its elastic character andanother is defined when the G’ and G’’ have thesame value representing the point after whichviscous effects are dominant over elastic ones.

3 EXPERIMENTAL PROCEDUREThe measurements were carried out with a com-mercial rheometer, model HAAKE MARS II, man-ufactured by Thermo Scientific, Germany, byusing a standard cone-plate geometry, in which

a sample of 0.2 ml is required for each test. Oursamples of waxy crude oil were supplied by theBrazilian Oil Company (Petrobras). The oil, lightand free of salt and water, contains, at least, 3 %of asphaltenes and resins. In order to study theeffect of such kind of hydrocarbons as naturalparan inhibitors, we used two different floccu-lants, n-heptane and n-pentane, to remove as -phal tenes and resins from the original oil.

As the rheological parameters of waxy crudeoil are highly dependent on temperature history,all the samples were previously submitted to arigorous step-by-step procedure. First of all, eachsample was heated up to 60 °C. This level of tem-perature was kept for 20 minutes. After that, bya precise thermal control equipment, the oil tem-perature was decreased at a constant rate of0.62 °C/min until the test temperature of 4 °Cwas achieved. The sample rested for 1 hour and20 minutes at this temperature before the begin-ning of the test. At this level of temperature theparan crystals formation was quite intense.

4 RESULTS AND DISCUSSIONA very important conclusion, from the discussionconducted in Section 1, is that the complex time-dependent character of waxy crude oils at lowtemperatures, requires yield stress measure-ments to follow a protocol that takes this time-dependency into consideration. We show thisfact by quantifying the effect of the asphaltenesand resins present in the waxy crude oil compo-sition as natural inhibitors of waxy formation.Figure 1 shows the relative amount of asphalteneplus resin removed from the original oil as a func-tion of solvent volume used. It was observed dif-ferent levels of precipitated hydrocarbons byusing n-heptane (black symbols) or n-pentane(red symbols). With 0.5 ml of the former solvent,2.3 % of hydrocarbons were removed from the oil.Using 2 ml of n-pentane, 3 % of hydrocarbonswere removed from the oil. The amount ofasphaltenes in oil is usually defined operational-ly by using the n-heptane. If we assume that noresin is precipitated with aid of such a solvent, wecould infer that the difference on the amount ofprecipitated hydrocarbons obtained with thetwo solvents is because the n-pentane causesalso the precipitation of resin. Hence, we couldassume that, from the total amount of precipi-tated hydrocarbons occurred by the action of then-pentane, around 2.3 % is asphaltene and 0.7 %

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is resin. However, this line of arguments is anapproximation and we are taking into accountthe combined effect, actually. For both floccu-lants, n-heptane and n-pentane, we can assumethat the amount of solvent used to remove themaximum quantity of asphaltenes and resinsdoes not change significantly the rheologicalbehavior of the original oil. This hypothesis issupported by the data shown in Figure 2, in which h(g· ) is displayed for three different values of sol-vent volume. The difference between the origi-nal oil viscosity and the one with 0.5 ml of n-hep-tane (Sample 1) and 2 ml of n-pentane (Sample 2)diluted is hardly perceived. Thus, we assume thatany other rheological parameter is unchanged atall. The crystals start to precipitate at a very hightemperature, as can be inferred from Figure 3,where the shear-viscosity is displayed as a func-tion of the shear-rate for a range of tempera-tures. We can see that the shear-viscosity ex -hibits Newtonian-like behavior at temperaturevalues higher than T = 20 °C. For temperaturebelow this value, the non-Newtonian characterof the viscosity is clearly noticeable. Our testswere performed at T = 4 °C, in which the waxycrude oil is highly structured.

We adopt the classical concept of yield-stress i.e. that its measurement must be per-formed in a steady state regime. Since the waxycrude oils are highly thixotropic, we must, first ofall, certify ourselves that our measurements are

time independent. With a virgin oil, i.e. withoutany flocculant addition that could act to provokeasphaltene precipitation, we conducted the testdisplayed in Figure 4, where the shear-stress isshown as function of the shear rate. The differ-ent flow curves reflect different time intervals forthe software of the rheometer to provide the out-put values. When we input in the rheometer ashear rate sweep, we must define the range ofimposed shear rates and the total time the exper-iment will take. Hence, when the shear rate ischanged to the next one, the rheometer takes atime interval to consider that the new steadystate is achieved and to compute the stress val-ue associated to the new shear rate value. We arecalling this interval elapsed measurement time.For a strain-controlled test, we define the elapsedmeasurement time as the time the rheometerwaits from the instant that the new shear ratevalue is imposed until the new stress value iscomputed. Figure 4 shows that very differentflow curves can be obtained from different choic-es of the elapsed measurement time. The curvet(g· ) shift down when the time of getting a pointis increased from 10 to 30 seconds, which indi-cates a great stress time dependence. A furtherincrease on time from 30 to 40 seconds shows nodifference on t(g· ). Thus, we assume that 30 sec-onds is a long enough time to ensure that thesteady state measure is achieved. From what wasdiscussed, we conclude that the 30 seconds

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Figure 1 (left above):Fraction of Asphaltene andResin removed from theoriginal oil as a function ofthe solvent volume intro-duced.

Figure 2 (right above):Shear-viscosity h of the orig-inal oil as a function ofshear-rate g· at different sol-vent concentrations fortemperature fixed at 40 °C.

Figure 3 (left below):Shear-viscosity h of the orig-inal oil as a function ofshear-rate g· for a range oftemperature.

Figure 4 (right below):Shear-stress as a function ofshear-rate. The test wasconducted in a attempt todetermine the minimalmeasuring point time tohave a steady state t(g· )flow-curve (black symbols).

elapsed measurement time corresponds to theflow curve. It is worth noting here that a value ofyield-stress computed before the steady state beachieved can be dramatically different andinduces to misunderstandings. In order to checkand eliminate any doubt if the flow curve wasachieved or not, we choose samples of shear-ratevalues to displace over the time. The stress startswith a high value, decreases, and achieves thesteady state values of the shear-stress corre-sponding to ones obtained by the 30 secondselapsed time experiment. This result gives suffi-cient evidence that our procedure to obtain thesteady state stress curve is good enough. How-ever, an important difference from the previousexperiment was the longer time interval neces-sary for the achievement of the steady state val-ues, showing that, an optimized elapsed mea-surement time depends on the other settings ofthe experiment.

Figure 5 shows the flow curve for the origi-nal waxy crude oil, a sample that contains, atleast, 3 % of asphaltenes and resins. The shear-stress is measured for a wide range of shear-ratefrom a very small value, g· = 5 · 10-5 s-1 to g· = 300 s- 1.Because we were able to achieve low shear ratevalues, we could identify that the yield process ofthis material is classified in the category of appar-ent-yield-stress fluids. We can see that, for shearrate values below 10-3 s-1 the flow curve falls downsome decades in the shear stress measurement.Hence, we identify a creep flow below the “yieldstress”. Although it could be argued that thisbehavior could be caused by slip, this result is inaccordance to the findings of [6] for anotherwaxy crude oil, where the Hershel-Buckley mod-el could not fit the low shear-rate regime andtherefore a Casson model was used. In that work,a cone-plate geometry with roughened surfaceswas used to avoid slip. The disadvantage of usingthe Casson model is that there is no yield stressparameter associated. As discussed in [6], theyield-stress value is important even for apparent-yield-stress fluids, because this stress valuemarks a major break down of microstructure.

Besides this apparent-yield-stress nature of thewaxy crude oil tested, we can observe a non-monotonic behavior which can be addressed byidentifying a static and a dynamic yield stresswhere the former is greater than the latter. Thestatic and dynamic yield-stress can be directlyinferred from the data, as Figure 5 indicates. Thestatic yield stress ty is the maximum stress val-ue, starting from a very low shear rate andincreasing this parameter, in this case ty = 27.8 Pa.The dynamic yield stress tyd can be defined hereas the first minimum stress value, starting froma high shear rate and decreasing this parameter,in this case tyd = 11.4 Pa. The shear-rate is reallysmall before the static yield-stress is achievedwhen the visco-plastic structure starts to breakand a further increase in g· is followed by adecreasing t. The shear-stress falls until thedynamic yield-stress is achieved. This level ofstress is characterized by a jump on the shear-rate. In fact, g· jumps from around 0.02 s-1 to 10 s- 1

while the shear-tress changes just very slightly.This is caused by a major breakdown on the mate-rial structure when ty is achieved. For the start-up flow problem, where the material is initiallyat rest, we assume that a stress at least equal thestatic one is required. Otherwise, as an apparent-yield-stress fluid, the material will undergo acreep flow. Thus, from a practical point of view,the static yield stress ty is the value of stress thatneeds to be confronted with the re-start pressureto guarantee a steady state flow. It is worthy not-ing that the material behavior is quite well pre-dicted by Equation 6. At shear-rates larger thang· = 0.001 s-1 the data is quite well fitted. Belowsuch shear-rate the material is shearthickeningand, as the referred model consider a constantlevel of viscosity, the points are not well fitted inthe range of g· £ 0.001 s-1.

As mentioned before, there are other waysto measure the yield-stress. Oscillatory tests thattake into account the storage and loss modulusas a function of shear-stress are widely used bythe industry. Such kind of test is displayed in Fig-ure 6. The modulus G’ and G’’ is computed by

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Figure 5 (left):Flow curve of the original oil.The symbols are measure-ments and the line is a curvefitting provided by a modi-fied SMD model. The rheo-logical curve fitting parame-ters are: ty = 27.8 Pa, tyd =11.4 Pa, n = 0.35, n0 = 1.8,K = 2 Pasn+1, K0 = 90000000Pasn+1, h

∞= 0.2 Pas, and

t0d = 0.01 s-1.

Figure 6:The storage and Loss Modu-lus, G’ and G’’, respectively,as a function of the Stressintensity, t, at three differ-ent test time. The test isconducted by increasing theamplitude of deformationat a fixed frequency w.

increasing the deformation amplitude with afixed frequency. As mentioned previously, thedynamic yield-stress can be assumed to be thevalue of stress where the two curves intercept toeach other, i.e. when the loss modulus becomesmore important than the storage modulus. As therheological properties of our oil are time depen-dent, the same care, when the flow curve wasdetermined, must be taken, i.e. tests must be car-ried out over a long enough time to certify peri-odic steady state was achieved. Keeping in mindsuch time dependency, we measured G’ and G’’over a range of stress, 0 £ tod £ 1000 Pa, for dif-ferent periods of time. The value of stress at theintercepting points falls dramatically when theexperimental time is increased. In fact, this criti-cal value fell from 55.0 Pa, at a test of 1200 s, to10.5 Pa, at a test of 4800 s. Such time dependen-cy is very pronounced and a difference of 423 %between the two extreme values is worth ofattention. The dynamic yield-stress inferred bythe test of 4800 s (tyd = 10.5 Pa) is quite close thatone obtained by the flow curve, Figure 5, in whichtyd = 11.5 Pa. Hence, we can assume that such kindof test could, eventually, be a good alternative toaccomplish the value of the dy namic yield-stressclose enough to the standard one, obtained by theflow curve. One possible way to infer the staticyield stress that is sometimes used is to assign tothis entity the stress value where the G’ deviatesfrom the low-shear-stress plateau. However, thisprocedure is not unequivocally correlated to theconcept the static yield-stress ty. If we assumethat ty is the main quantity that needs to be cal-culated for the re-start problem, the analysis giv-en by the flow curve seems to be more appropri-ate for practical applications and this is the kindof test we conducted here to investigate the roleof as phalt enes and resins on the paraffin crystalformation. By adding 0.5 ml of n-heptane, anamount of 2.3 % of asphaltenes and resins wasremoved from the original oil. A new flow curvewas obtained and displayed in Figure 7. The val-ues of the static and dynamic yield-stress in -creased dramatically. The former has risen 76.3 %

while the latter, 163.2 %. With addition of 2 ml ofn-pentane a larger amount of asphaltenes andresins was taken away from the oil and, again, theflow curve was measured and shown in Figure 8.With respect to the previous results, the dynam-ic yield stress slightly changed, but the static onein creased around 18 %. Related to the originalwaxy crude oil, both ty and tyd increased with 175.4and 111.5 %, respectively.

Figures 5, 7, and 8 were fitted by Equation 6as represented by the blue curve. We found agood agreement between this curve and theexperimental data, specially in the shear rateregion above the static yield stress. However, forvery low values of the shear rate this curvedetaches from the experimental one. It isobserved a shear-thickening behavior below thestatic yield stress. A different expression is pro-posed here that can account for this behavior. Itis a slight modification of the Equation 6 and isgiven by

(9)

Figures 5, 7, and 8 show that Equation 9 has a bet-ter agreement with the experimental results (thered curves) allowing a good description at lowshear rates values. A decrease in the consistencyindex Ko and the thickening power-law exponentno is obtained for oil with less quantities ofasphaltenes and resins. The viscosity in the min-imum structure level differs from the maximumone by a factor of 105 to 106. Hence, when thestress value is above the dynamic yield stress, thewaxy crude oil achieves a more mobile state. Bycomparing these results, we notice that with lessasphaltenes and resins in the oil, changes in vis-cosity occur at a lower range of stress intensity,assuming a more vertical line in the graph.Another important result is that not only theyield stress level increases with the removal ofasphaltenes and resins, but the maximum vis-cosity level also increases.

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Figure 7 (left):Flow curve with 2.3 % ofAsphaltene and Resinremoved from the originaloil. The symbols are mea-surements and the line is acurve fitting provided by amodified SMD model. Therheological curve fittingparameters are: ty = 49 Pa,t0d = 30 Pa, n0 = 1.62,n = 0.8, K = 0.5 Pasn+1,K0 = 80000000 Pa.sn+1,h

∞= 0.1 Pas, and

g·0d = 0.01 s-1.

Figure 8:Flow curve with 3 % ofAsphaltene and Resinremoved from the originaloil. The symbols are mea-surements and the line is acurve fitting provided by amodified SMD model. Therheological curve fittingparameters are: ty = 58.8 Pa,tyd = 31.4 Pa, n = 0.45,n0 = 1.6, K = 2 Pasn+1,K0 = 70000000 Pasn+1, h

∞= 0.28 Pas, and

g· 0d = 0.01 s-1.

As the literature also reports, our resultsstrongly suggest that asphaltenes and resinsplay a significant role on the paraffin crystal for-mation. However, the real effect of asphaltenesand resins on the yield stress needs to be mea-sured considering the time dependency natureof the waxy crude oil. A protocol that does nottake this time dependency into account can leadto very different values for the yield stress and,consequently the measured quantity will notrepresent the real influence of the asphaltenesand resins as wax inhibitors from the rheologicalperspective. Since the yield stress is a crucialparameter on the determination of the re-startpressure and, consequently, on the pumping pro-ject, the accuracy of its measurement is some-thing that the oil industry must pursue.

5 CONCLUSIONWe analyzed the rheology of a waxy crude oil atthe temperature of 4 °C in a process of constantcooling rate that started from 60 °C, and nullshear rate, simulating the shut down conditionsobtained in ultra-deep oil reservoirs, where a re-start of the production is necessary. The mainconclusion of the present analysis concerns thenecessity of taking into account the time-depen-dency nature of the material at these low tem-peratures. In particular we found that for theBrazilian oil tested, the differences in the dynam-ic yield stress can achieve more than 300 %. Thesedifferences can make an enormous impact on theprediction of the re-start pressure and can be areason for the known fact that laboratory char-acterization overestimates the yield stress andconsequently the re-start pressure.

We used the known property of the asphal -tenes as inhibitors of wax formation to quantifyits effects from the rheology perspective. To thisend we added different amounts of two types offlocculants: n-pentane and n-heptane in order toprecipitate asphaltenes and resins. When the vir-gin oil was characterized, its corresponding sta-tic yield stress was ty = 11 Pa. For a precipitationof ≈ 2.3 % of asphaltenes and resins and a furtherrheological test made this value rise to ty = 49 Pa;while the associated static yield stress value fora precipitation of 3 % of asphaltenes and resinswas ty = 58.8 Pa.

We also found that the Brazilian waxy crudeoil tested is an apparent-yield stress fluid and,therefore it flows at very low shear rates below

the static yield stress. This feature combined withthe presence of a dynamic and a static yield stressrenders to the flow curve a non-monotonicbehavior with a very particular character. In fact,a new flow curve fitting is proposed, adaptedfrom a previous one proposed by [35]. This newcurve fitting reveals a shear-thickening behaviorfor very low stress values, in the pre-yieldingregime. The whole non-monotonic flow curvecan only be obtained in a constant shear rateinput, because a constant shear stress inputleads to more than one value as a correspondingshear rate. Because of this, a part of the flowcurve is considered not stable [36] and leads tothe possibility of shear banding effects [37].

ACKNOWLEDGEMENTSThis research was partially funded by grants fromCNPq (Conselho Nacional de Pesquisa e Desen-volvimento), ANP (Agência Nacional de Petróleo)and PETROBRAS.

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