Dear Author, Please, note that changes made to the HTML content will be added to the article before publication, but are not reflected in this PDF. Note also that this file should not be used for submitting corrections.

Our reference: JNNFM 3626 P-authorquery-v11

AUTHOR QUERY FORM

Journal: JNNFM

Article Number: 3626

Please e-mail or fax your responses and any corrections to:

E-mail: corrections.esch@elsevier.sps.co.in

Fax: +31 2048 52799

Dear Author,

Please check your proof carefully and mark all corrections at the appropriate place in the proof (e.g., by using on-screen annotation in the PDFfile) or compile them in a separate list. Note: if you opt to annotate the file with software other than Adobe Reader then please also highlightthe appropriate place in the PDF file. To ensure fast publication of your paper please return your corrections within 48 hours.

For correction or revision of any artwork, please consult http://www.elsevier.com/artworkinstructions.

Any queries or remarks that have arisen during the processing of your manuscript are listed below and highlighted by flags in the proof. Clickon the ‘Q’ link to go to the location in the proof.

Location inarticle

Query / Remark: click on the Q link to goPlease insert your reply or correction at the corresponding line in the proof

Q1 Your article is registered as belonging to the Special Issue/Collection entitled ‘‘VPF2013’’. If this is NOTcorrect and your article is a regular item or belongs to a different Special Issue please contacts.selvi@elsevier.com immediately prior to returning your corrections.

Q2 Please confirm that given name(s) and surname(s) have been identified correctly.

Q3 Please check whether the designated corresponding author is correct, and amend if necessary.

Q4 Please check the edit made in Eqs. (23)-(26), and correct if necessary.

Q5 Please check the hierarchy of the section headings.

Q6 Please check the year in Ref. [20].

Thank you for your assistance.

Please check this box if you have nocorrections to make to the PDF file

Highlights

� The flow start-up depend on the material time response and compressibility. � Two time scales are important on gel break: shearing andmaterial response time. � The required time for the flow start depends on the inlet pressure. � The maximum pressure reached in the restartis a function of the imposed flow rate.

JNNFM 3626 No. of Pages 1, Model 5G

10 January 2015

1

1

3 The effect of compressibility on flow start-up of waxy crude oils

4

5

6 Gabriel Merhy de Oliveira, Cezar O.R. Negrão ⇑7 Research Center for Non-Newtonian Fluids (CERNN), Post-graduate Program in Mechanical and Materials Engineering (PPGEM), Federal University of Technology – Paraná8 (UTFPR), Av. Sete de Setembro, 3165, CEP 80.230-901, Curitiba, PR, Brazil

910

1 2a r t i c l e i n f o

13 Article history:14 Received 2 June 201415 Received in revised form 20 October 201416 Accepted 23 December 201417 Available online xxxx

18 Keywords:19 Waxy crude oil20 Structured material21 Compressible flow22 Transient simulation23

2 4a b s t r a c t

25The current work presents a mathematical model to simulate the flow start-up of gelled oils in pipelines.26The model comprises the conservation equations of mass and momentum and an elasto-viscoplastic con-27stitutive equation to account for thixotropy of the material. The flow is considered one-dimensional, lam-28inar and compressible and the cross-section shear stress is admitted to vary linearly so that the29constitutive equation can be integrated along the pipe radius. The balance equations are solved by the30method of characteristics, the constitutive equation by the finite difference method and finally they31are altogether integrated iteratively. Two kinds of boundary conditions are investigated: a constant inlet32pressure and a constant inlet flow rate. Two time-scales are identified in the current study: the pressure33wave propagation time and the material response time. It can be anticipated that the ratio of this two34times are quite important on the restart time in case of a constant pressure boundary condition and35on the magnitude of the maximum pressure for a constant flow rate boundary condition.36� 2015 Elsevier B.V. All rights reserved.37

38

39

40 1. Introduction

41 Oil fields such as those recently found along the Brazilian coast42 line can be as far as 300 km away from the seashore so as to convey43 oil from platforms to the land is a huge challenge. An alternative44 for the transport is to pump oil from the offshore production plat-45 form to the coast by using long pipelines. As the oil found in these46 oil fields is highly paraffinic, wax precipitation can take place at47 high temperatures in order that the oil can gel at the ambient tem-48 perature (�20 �C). Therefore, the sea bottom temperature (�4 �C),49 where the pipelines are placed, provide favorable conditions for oil50 gelation. Not only wax precipitation increases the oil viscosity but51 also impairs flow start-ups. High pump pressures may be required52 at the flow start-up in order to break-up the gel structure. Such53 high pressures lead to overestimation of pipeline resistance and/54 or pipe dimension, making the project unfeasible.55 A reliable prediction of gelled fluid start-ups demands appropri-56 ate robust models for the transient phenomenon. Several works57 have been dedicated to model flow start-ups in pipelines, being58 most of them concerned specifically with waxy crude oils [1–12].59 Despite the fluid being compressed at inlet in order to displace60 the material throughout the whole pipe, some works have consid-61 ered the problem as incompressible [1,2,4,12], as the liquid com-

62pressibility is usually very small. Whereas the incompressible63fluid motion is usually considered quasi-steady so that the tran-64sient phenomenon depends only on the changes of fluid properties65(thixotropy), the compressible flow analysis includes both the iner-66tia and the transient terms.67The low temperature at the seabed shrinks the oil [13] as an68outcome of density reduction and of wax crystallization. The69shrinkage may create spaces within the gelled material providing70compressibility to the material. Therefore, fluid compressibility71may play a role in flow start-ups of oils as a pressure variation,72resulting from fluid compression at the pipeline inlet, takes some73time to be sensed at the outlet. For instance, a pressure wave with74a typical speed of 1000 m/s takes about 5 min to reach the other75pipe end of a 300 km long pipeline.76Flow start-up problems can be viewed from two different per-77spectives: (i) the determination of the minimum inlet pressure that78overcomes the material yield stress and consequently, starts up the79flow or; (ii) the determination of the maximum pressure that80results from a imposed inlet flow rate. Inasmuch as the first can81be treated as quasi-steady incompressible, the second, which is82the most common case in real applications, can only be dealt by83using a transient compressible flow model as only one fluid veloc-84ity is admitted throughout the pipe in quasi-steady approaches.85Gel breaking is another important issue that must be86addressed in flow start-ups. As reported by review papers of87Mewis [14], Barnes [15] and Mewis and Wagner [16], time

http://dx.doi.org/10.1016/j.jnnfm.2014.12.0100377-0257/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +55 41 3310 4658; fax: +55 41 3310 4852.E-mail address: negrao@utfpr.edu.br (C.O.R. Negrão).

Q3

Q1

Q2

Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journal homepage: ht tp : / /www.elsevier .com/locate / jnnfm

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

88 dependent properties of gels have been studied for a long time.89 Although the material elasticity has been recognized as impor-90 tant in gel properties [17–21], many thixotropy models used91 to approach flow start-ups [1–10] have considered the gels as92 viscoplastic [22–26]. Few works such as those after Negrão93 et al. [11] and Souza Mendes et al. [12] have already dealt gel94 breaking as an elasto-viscoplastic phenomenon. The first95 employed the thixotropy model after Dullaert and Mewis [18]96 whereas the second used his own model [19].97 All thixotropic models are based on a structure parameter fol-98 lowing [23] to represent the structuring level of the material. The99 structure parameter that usually varies from zero (unstructured)

100 to one (fully structured) is governed by a kinetic equation com-101 posed of a build-up and break-down term. The first represents102 aggregation forces, such as the Brownian motion, and the latter103 is either shear rate [18] or shear stress [19–21] induced.104 Whether the material flows or not and how long it takes to flow105 depends on the material structure strength and on the stress level106 the structure is submitted. The material time response is usually107 represented in the evolution equation of the structure parameter108 by a characteristic time, e.g., the equilibrium time, teq, defined by109 [19–21]. Considering the fluid motion as compressible, two charac-110 teristic times are important in flow start-up: the material time111 response and the time for the pressure wave to travel throughout112 the whole pipeline. Since the fluid is stressed as the pressure wave113 propagates, the material structure can only be broken during or114 after the pressure propagation. Therefore, the relative magnitude115 of these two characteristic times rules the time required for the116 flow start-up.117 The current work proposes a different approach for the flow118 start-up of gelled oils in pipelines by considering that the time119 for the flow restart and the magnitude of pressures obtained120 within the pipe depend on both the material time response and121 on the fluid compressibility. The model is based on the conserva-122 tion equation of mass and momentum assuming the fluid motion123 as weakly compressible and on a thixotropy model.124 Despite the elasto-viscoplastic thixotropy model after Dullart125 and Mewis [18] being used in a previous work [11] to deal with126 compressible flow start-ups in pipelines, the model does not127 clearly establish a material structure characteristic time that is128 required in the current analysis. In addition to that, the kinetic129 equation for the structure parameter and elastic deformation goes130 to infinity as time tends to zero which does not allow changes of131 the structure parameter and elastic deformation at time zero and132 makes the numerical solution quite sensitive to the time-step at133 the beginning of the simulation. Therefore, the thixotropy model134 after Souza Mendes and Thompson [21] is chosen instead of the135 Dullart and Mewis’s [18] model.136 Two kinds of boundary conditions are considered: a constant137 inlet pressure and a constant inlet flow rate. In the first, the138 required time for starting-up the flow for a given pressure is deter-139 mined whereas in the latter, the maximum pressure reached in the140 restart for an imposed flow rate is obtained. Considering the mate-141 rial as elasto-viscoplastic, the effect of the material elasticity is also142 investigated.

1432. Mathematical model

144The problem was devised as a long horizontal pipeline in which145the fluid is at rest (under no stress) at time zero, as shown in Fig. 1.146At the outlet, the pressure is considered to be zero and at the inlet,147two types of boundary conditions are admitted: (i) a constant inlet148pressure and (ii) a constant flow rate.

1492.1. Governing equations

150By considering the flow as laminar, one-dimensional, the con-151servation equations of mass and momentum are thus written,152respectively, as,153

@q@tþ q

@V@zþ V

@q@z¼ 0 ð1Þ 155155

156and,157

q@V@tþ qV

@V@zþ @p@z¼ �4

dsw ð2Þ 159159

160where q is the fluid density, V is the axial velocity, p is the pressure161and sw shear stress at the pipe wall, d is the pipe diameter, t and z162are the time and axial coordinates, respectively. All dependent vari-163ables are averaged throughout the pipe cross sectional area. By164substituting the definition of isothermal compressibility,

165a ¼ 1q@q@p

���T¼ 1

qc2, into Eq. (1), the mass balance equation is reduced to,166

@p@tþ 1

a@V@zþ V

@ p@z¼ 0 ð3Þ 168168

169where c is the pressure-wave speed and N is the temperature.

1702.2. Constitutive equation

171The viscoelastic thixotropy model proposed by Souza Mendes172and Thompson [21] is used to account for gel-breaking and recov-173ery. The model comprises a constitutive equation based on the Jef-174freys model:175

_cþ h2ðkÞ€c ¼h2ðkÞg1

sh1ðkÞ

þ _s� �

ð4Þ177177

178where _c, €c, s and _s are, respectively, the shear rate, the rate of179change of the shear rate, the shear stress and rate of change of180the shear stress. h1 and h2 are the time-dependent relaxation and181retardation times, respectively, which depend on a structure182parameter (k):183

h1ðkÞ ¼ 1� g1gvðkÞ

� �gvðkÞGsðkÞ

ð5Þ185185

186

h2ðkÞ ¼ 1� g1gvðkÞ

� �g1

GsðkÞð6Þ

188188

189where g1 is the viscosity at infinite shear rate and gv and Gs are the190structure-dependent material viscosity and elastic modulus:191

GsðkÞ ¼ G0emð1k�1k0Þ ð7Þ 193193

z r

d

l

Fig. 1. Illustration of the pipeline geometry.

2 G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

194

gvðkÞ ¼ g1ek ð8Þ196196

197 where k0 and G0 are the structure parameter and the elastic modu-198 lus for the fully structured material, respectively, and m is a con-199 stant value. This structure parameter is computed by using the200 following kinetic equation,201

dkdt¼ 1

teq

1k� 1

k0

� �a

� kkeq

� �b 1keq� 1

k0

� �a" #

ð9Þ203203

204 where teq is characteristic time for the structure recover or break-205 down for a given level of stress and keq is the equilibrium structure206 parameter calculated as:207

keqðsÞ ¼ lngeqðsÞg1

� �ð10Þ

209209

210 and geq is the equilibrium viscosity computed according to:211

geqð _ceqÞ ¼ ð1� e�g0 _ceq=sy Þ sy � syd

_ceqe� _ceq= _cyd þ syd

_ceqþ K _cn�1

eq

� �þ g1 ð11Þ213213

214 where K and n are, respectively, the consistency index and power-215 law index, sy and syd are, respectively, the static and dynamic yield216 stresses and _cyd, the shear rate that marks the transition in stress217 from sy to syd. As the pipeline aspect ratio (d/l) is quite small, the218 pressure is admitted to be constant across the pipeline section,219 resulting in a linear shear stress distribution along the cross section:220

s ¼ sw2rd

ð12Þ222222

223 where d is the inner pipe diameter, r is the radial position and sw is224 the shear stress on the pipe wall which depends on time and axial225 position, z. By substituting Eq. (12) into Eq. (4) and defining �ov/or226 as the shear rate, the following equation is found:227

� @v@rþ h2ðkÞ

@

@t� @v@r

� �¼ 2r

dh2ðkÞg1

sw

h1ðkÞþ _sw

� �ð13Þ

229229

230 where v is the axial velocity. Assuming sw as a known time-depen-231 dent value, Eq. (13) can be integrated to obtain the radial velocity232 profile along the time.

233 2.3. Dimensionless equations and scale analysis

234 In contrast to incompressible flows in which the pressure wave235 speed is infinite so that any pressure variation at one pipe end is236 immediately sensed at the other end, a pressure change in com-237 pressible flows travel at a finite wave speed. Therefore, the pres-238 sure wave takes a finite time to reach the pipe outlet after the239 fluid being compressed at the inlet. As a consequence, the material240 is only shear stressed at a certain pipe position when the pressure241 wave hits that position. However, the material structure may not242 change immediately after being stressed once the time for chang-243 ing the structure depends on the initial strength of the material244 and on the level of stress. As defined by Souza Mendes and Thomp-245 son [21], teq in Eq. (9) is a characteristic time for changing the246 structure parameter, k, which can also be understood as a time247 scale for material break-down at a certain level of stress. The248 break-down time is in the order of magnitude of teq if the shearing249 time is smaller than the equilibrium time and the material breaks250 down as it is sheared when the shearing time is larger than teq.251 Since the material is stressed in flow start-ups as pressure propa-252 gates along the pipe, the relative order of magnitude between253 the start-up shearing time scale, l/c, and teq may play a significant254 hole in the material break-down and consequently, in the start-up255 time. The analysis here presented is based on the relative order of

256magnitude of these time scales as they not only affect the start-up257time but also the magnitude of pressures throughout the pipeline.258The problem parameters are now scaled following the ideas of259Souza Mendes and Thompson [21]. By choosing the characteristic260shear rate to be c/d, the characteristic shear stress to be the261dynamic yield stress, syd, and the characteristic time to be the time262for pressure propagation along the pipe length (l/c), the following263dimensionless parameters for the constitutive model are obtained:264s�y ¼ sy=syd, t�eq ¼ teqðc=lÞ, G�0 ¼ G0l=ðg1cÞ, _c�yd ¼ _cyd=ðc=dÞ,265g�1 ¼ g1ðc=dÞ=syd, g�0 ¼ g0ðc=dÞ=syd and K� ¼ Ksn�1

yd =gn1. The pres-

266sure is scaled by the force that balances the dynamic yield stress267in the pipe and the velocities by the wave speed so that the dimen-268sionless counterparts are given by p⁄ = p/(4sydl/d), v⁄ = v/c and269V⁄ = V/c, respectively. The dimensionless forms of time, axial and270radial positions are defined, respectively, as t⁄ = t(c/l), z⁄ = z/l and271r⁄ = r/d. By scaling the governing equations using these dimension-272less parameters, the following equations are obtained: 273

@p�

@t�þ g�1/

@V�

@z�þ V�

@p�

@z�¼ 0 ð14Þ 275275

276

g�1/q�@V�

@t�þ g�1/q�V�

@V�

@z�þ @ p�

@z�þ s�w ¼ 0 ð15Þ 278278

279where s�w ¼ sw=syd and / is equal to 1=4(qcd/g1)d/l, which can be can280be understood as the relationship between the inertia of the pres-281sure wave and the viscous dissipation, as discussed elsewhere282[27]. / can also be expressed as a function of the Reynolds number283(Re = qVd/g1), Mach number (M = V/c) and the pipe aspect ratio284(d = d/l), resulting in / = 1=4d Re/M. The product g�1 by / can be285understood as the inverse of a dimensionless compressibility:286

g�1/ ¼ qc2dsydl

¼ 1a�

ð16Þ288288

289where a� ¼ sydl=ðqc2dÞ ¼ asydl=d. Notably, the pressure wave only290propagates at the speed of sound, c, when the problem is thermody-291namic reversible whereas in a real situation, the pressure wave292speed depends on the magnitude of the viscous dissipation. There-293fore, the higher is a⁄ the more dissipating is the flow.294The dimensionless form of Eq. (13) is thus given by,295

� @v�

@r�þ h�2ðkÞ

@

@t�� @v

�

@r�

� �¼ 2

g�1r�h�2ðkÞ

s�wh�1ðkÞ

þ _s�W� �

ð17Þ297297

298where _s�w ¼ _swc=sydl, h�1ðkÞ ¼h1ðkÞc

l ¼ 1G�0ð1� e�kÞe

k�m 1k�

1k0

� �and

299h�2ðkÞ ¼h2ðkÞc

l ¼ 1G�0ð1� e�kÞe

�m 1k�

1k0

� �.

300A scale analysis of the terms of Eqs. (14) and (15), based on the301methodology described by [28], is now carried out in order to eval-302uate their relative order of magnitude during the flow start-up. For303the weakly compressible flows considered here, the order of mag-304nitude of the Mach number is assumed to be smaller than 0.1. In305other words, the dimensionless velocity is in the order of the mag-306nitude of the Mach number.307For the current analysis, it is now assumed that the pressure at308t = 0 changes from zero to a maximum value within a time interval309l/c and also that the pressure varies from a maximum value at the310inlet to p = 0 at the outlet. Additionally, the dimensionless velocity311is considered to change from zero to the maximum Mach number312within the same time and length scales. Finally, the dimensionless313density and shear stress at the wall take the order of magnitude of3141.0 and of the maximum pressure, respectively. The following315dimensionless scales are thus identified for the changes,316

p� � p�in; V� � M; q� � 1; s�w � p�in; t� � 1; z� � 1 ð18Þ 318318

319By assuming p�in ¼ 10 and the maximum Mach number equal to 0.1320for a weakly compressible flow, the order of magnitude of the terms321of Eqs. (14) and (15) will be,

G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx 3

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

32210; 0:1g�1/; 1 ð19Þ324324

325

0:1g�1/; 0:12 g�1/; 10; 10 ð20Þ327327

328 Notably, if g�1/ P 102, the third term of continuity equation and the329 second term of the momentum equation is at least one order of330 magnitude smaller than the other terms. In the current analysis,331 the product of / by g�1 is considered to be at least equal to 100332 so that the second and the third terms of Eqs. (14) and (15) can333 be disregarded. Additionally, the q� in the momentum balance is334 considered to be constant and equal to 1 once the changes of den-335 sity are computed by changes of pressure in both equations – mass336 and momentum. Therefore, the equations to be solved take the337 form:338

@p�

@t�þ g�1/

@V�

@z�¼ 0 ð21Þ340340

341

g�1/@V�

@t�þ @ p�

@z�þ s�w ¼ 0 ð22Þ343343

344 3. Solution algorithm

345 The governing equations are solved by the method of character-346 istics (MOC) that is typically used for solving hyperbolic partial dif-347 ferential equations [29]. The method consists in simplifying partial348 differential equations to a family of ordinary differential equations,349 along which the solution can be integrated from an initial condi-350 tion. In the current case, Eqs. (21) and (22) are reduced to two total351 differential equations, which are valid over the characteristic lines352 dz⁄/dt⁄ = ± 1:353

dp�

dt�þ g�1/

dV�

dt�þ s�w ¼ 0 Cþ ð23Þ355355

356dz�

dt�¼ þ1 Cþ ð24Þ358358

359

� dp�

dt�þ g�1/

dV�

dt�þ s�w ¼ 0 C� ð25Þ361361

362dz�

dt�¼ �1 C� ð26Þ364364

365 Notably, plots of Eqs. (24) and (26) provide straight lines on the z–t366 plane, as shown in Fig. 2, which are named characteristic lines C+

367 and C�, respectively. It is worth noting that Eqs. (23) and (25) are368 valid only over the C+ and C� characteristic line, respectively.

369The pipeline is divided into N equal reaches, Dz⁄, as shown in370Fig. 2. In order to satisfy both Eqs. (24) and (26), the time-step is371computed according to Dt⁄ = Dz⁄. If p⁄ and V⁄ are both known at372position i � 1 and known values are assumed for s�w at past and373present times, Eq. (23) can be integrated over the characteristic374line C+ and therefore, be written in terms of unknown variables375p⁄ and V⁄ at point i. Integration of Eq. (25) along line C�, with p⁄,376V⁄ and s�w known, leads to a second equation in terms of the same377two unknown variables at i. By solving the two resulting algebraic378equations, p⁄ and V⁄ can be obtained at point i as a function of379known previous time-step values at points i � 1 and i + 1:380

ðp�Þkþ1i ¼ Fþ þ F�

2ð27Þ 382382

383

ðV�Þkþ1i ¼ Fþ � F� � ðs�wÞ

kþ1i Dz�

2g�1/ð28Þ

385385

386where387

Fþ ¼ ðp�Þki�1 þ g�1/ðV�Þki�1 �12ðs�wÞ

ki�1Dz� ð29Þ 389389

390

F� ¼ ðp�Þkiþ1 � g�1/ðV�Þkiþ1 þ12ðs�wÞ

kiþ1Dz� ð30Þ 392392

393By dividing the pipe radius in M points, the constitutive Eq. (17) can394be discretized by using the implicit finite difference method and the395following algebraic equation is found:396

ð1þ eÞðv�Þkþ1j;i ¼ ð1þ eÞðv�Þkþ1

jþ1;i þ eððv�Þk�1j;i � ðv�Þ

k�1jþ1;iÞ

þððr�Þ2jþ1 � ðr�Þ

2j Þ

g�1½ðeþ f Þðs�wÞ

kþ1i � eðs�wÞ

k�1i � ð31Þ

398398

399where e ¼ ðh�2Þ

kþ1j;i

Dt� , f ¼ ðh�2Þ

kþ1j;i

ðh�1Þkþ1j;i

. Positions j and j + 1 mean two adjacent

400radial positions; an outer and an innermost, respectively, and k,401k � 1 and k + 1 stand for the current, the past and the future time.402By knowing all variables and parameters at the previous time-step,403the wall shear stress at past and future times and the velocity at the

404outermost position at present time, the velocity ðv�Þkþ1j;i can be

405determined. As h�1 and h�2 are functions of k, the discrete values of406k are obtained from the explicit discretization of Eq. (9):407

kkþ1j;i ¼ kk

j;i þDt�

t�eq

1kk

j;i

� 1k0

!a

�kk

j;i

kkþ1eq;j;i

!b1

kkþ1eq;j;i

� 1k0

!a24

35 ð32Þ

409409

410where keq,j,i is obtained from Eq. (10) as a function of the local value411of shear stress (sw,i r/d).412The integration of the radial velocity profile (Eq. (31)) along the413pipe cross section results in the average velocity:414

ðV�Þkþ1i ¼ 2

XM

j¼1

ððv�Þkþ1jþ1;i þ ðv�Þ

kþ1j;i Þððr�Þ

2jþ1 � ðr�Þ

2j Þ ð33Þ

416416

417As wall shear stress is not a known value at time k + 1, an iterative418solution must be carried out between Eqs. (27), (28), (31) and (33)419to obtain s�w. The problem solution begins with the fluid standing420still unstressed at time zero, so that p⁄, V⁄ and s�w are all zero at each421computing point i. The solution consists in finding p⁄ and V⁄ for each422grid point along t⁄ = Dt⁄ by using Eqs. (27) and (28) and attributing a

423wall shear stress at k + 1, ðs�wÞkþ1i . Eq. (31) along with Eq. (32) is

424solved to provide the radial velocity profile by using the same

425attributed ðs�wÞkþ1i value. An average velocity can then be obtained

426from Eq. (33) and compared with the average counterpart given427by Eq. (28). Inasmuch as the average velocity values are different,

428ðs�wÞkþ1i is corrected and ðV�Þkþ1

i and ðV�Þkþ1i are computed again.

C-

Δz

z =0t 0=

t

i - 1

i

i + 1

C+

z l=

Δt

k

k+1

Fig. 2. z–t Grid for solving the governing equations.

Q4

4 G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

429 ðs�wÞkþ1i is corrected using an iterative procedure based on the root

430 finding secant method until ðV�Þkþ1i and ðV�Þkþ1

i are very close. As431 soon as convergence has been reached, the solution evolves to the432 next time-step and then proceeds until the simulation time has433 been covered.

434 4. Results

435 As an initial condition, the fluid velocity and pressure are set to436 zero throughout the domain, V�ðz�; t� ¼ 0Þ ¼ 0 and p�ðz�; t�437 ¼ 0Þ ¼ 0. The pressure is assumed to be zero at the outlet through-438 out the whole simulation, p�ðz� ¼ 1; t�Þ ¼ 0, whereas two types of439 inlet boundary conditions are considered: p�ðz� ¼ 0; t�Þ ¼ p�0 and440 V�ðz� ¼ 0; t�Þ ¼ V�0, where p�0 and V�0 are constant values.

441 4.1. Model validation

442 In order to validate the model, the numerical solution is com-443 pared with the analytical solution presented by Oliveira et al.444 [10]. As the analytical solution applies to a Newtonian fluid, the445 elastic terms of the constitutive equation were disregarded and446 the viscosity was set to a constant value. Only the constant pressure447 case was considered and p�0 was fixed at 10. A grid size of 20 points448 was established in the radial direction and two grid sizes of 50 and449 100 points were considered, respectively, in the axial direction. The450 numerical results, as displayed in Fig. 3, agree quite well with the451 analytical solution for both axial grid sizes. Assuming that the452 20 � 100 (Nr � Nz) mesh size provides accurate enough results, this453 mesh was used for most simulations shown in the next section.454 However, some highly oscillating results required a more refined455 mesh and in these cases, a mesh size of 20 � 400 was used instead.

456 4.2. Case studies

457 The purpose of the current section is mainly to evaluate how458 compressibility, by means of changing /, and the equilibrium time,459 t�eq, affect the flow start-up in the pipeline. In addition to / and t�eq,460 only the effect G�0 is also investigated in the sensitivity analysis461 because of the large number of model parameters. The following462 parameters are then chosen: s�y ¼ 1, _c�yd ¼ 10�8, g�1 ¼ 102,463 g�0 ¼ 109 and K� ¼ 10, n = 0.5, m = 1.0, a = 1.0, b = 1.0.

464 4.2.1. Inlet pressure case465 In this case, the dimensionless inlet pressure at time zero is set466 to 10.0 meaning that the inlet pressure is ten times larger than the467 pressure required to balance the dynamic yield stress, syd. Fig. 4(a)468 shows the time evolution of pressure for different axial positions,

469whereas Fig. 4(b) depicts the fluid average velocity at z⁄ = 0.0,470z⁄ = 0.5, and at z⁄ = 1.0, for / = 1.0, t�eq ¼ 1:0 and G�0 ¼ 102. The high471frequency pressure and velocity oscillations at early simulation472times are the result of pressure wave propagation and reflection473within the fully structured portion of the material that is highly474elastic. The pressure fluctuations, however, are fast dissipated475because the initially fully structured material is also highly viscous.476After the high frequency oscillation being dissipated, the material477remains stressed near the inlet by a high pressure gradient which478gradually breaks down the gel from the inlet to the outlet propa-479gating the pressure in a low frequency. A comparison of Fig. 4(a)480and (b) shows that this low frequency pressure wave propagates481at the same speed the fluid starts moving throughout the pipe,482indicating the continuous material break-down. After about 50483units of time, the fluid starts exiting the pipe and the outlet pres-484sure reaches its maximum value. From this time on, the pressure485within the whole pipe and the outlet velocity are reduced to reach486the equilibrium and the flow in fact starts up.487Fig. 5 presents the time change of the structure parameter for488different axial and radial positions. As noted, the variation of the489structure parameter is not only faster but also higher at the pipe490wall in comparison with the other radial positions. Additionally,491the closer to the inlet the faster is the variation of the structure492parameter. The speed in which the structure parameter changes493in the axial direction is in agreement with low frequency wave494speed described in the previous paragraph. In other words, the495flow only starts up when the pressure wave reaches the pipe outlet496changing the material structure throughout the whole pipe. As497expected, the steady-state structure parameter depends on the498radius but not on the axial position. Although not shown in Fig. 5499the structure parameter at pipe centerline remains unchanged at500its initial value because the material is not sheared at that position.501Fig. 6(a) and (b) shows the pressure at the middle of the pipe,502z⁄ = 0.5, as a function of time for / = 1.0 and / = 100, respectively,503and different values of t�eq. As shown, the higher the / the less dis-504sipative is the system so that the steady-state is reached faster and505the result is more oscillating in comparison with the lower / case.506Fig. 6 also shows that the time for the system to reach the steady-507state increases with t�eq. For the less dissipative case and t�eq ¼ 1:0508(Fig. 6(b)), the material breaks down as the pressure wave propa-509gates, and the flow stabilizes as soon as the pressure oscillation510dissipates. As t�eq increases to 10, the pressure also oscillates but511the flow does not start up because the material does not break512down immediately. It means that the material is shaken within513the material elastic regime, as a result of pressure propagation,514but does not change its initial structure instantly. Similarly to the515less dissipative case, the material also breaks down as soon as516the pressure wave propagates throughout the pipe for the more

t* [-]0 2 4 6 8 10

0

5

10

Analytical SolutionN=50N=100

p* [-

]

z* = 0.9

z* = 0.1

(a)

z* = 0.5

t* [-]0 2 4 6 8 10

0

0.0025

0.005

0.0075

0.01

0.0125

Analytical SolutionN=50N=100

v* [-

]

(b)z* = 0.5

Fig. 3. Comparison between the numerical result and the analytical solution for a Newtonian fluid. (a) Time evolution of pressure at different axial positions and (b) timeevolution of the velocity at z⁄ = 0.5.

Q5

G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx 5

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

517 viscous case (/ = 1.0 in Fig. 6(a)) and t�eq ¼ 0:1, but the pressure518 does not oscillate because of the higher dissipation. Nevertheless,519 the pressure propagation along the pipe is delayed with the aug-520 mentation of the equilibrium time so that the material begins to521 break down only when the pressure starts changing at a certain522 axial position. It is worth noting that the required time for the523 pressure to start changing is in the order of 10 times t�eq. Excepting524 the high frequency oscillations, the maximum pressure in both525 high and low dissipative cases takes place when the material

526breaks down and consequently, starts flowing throughout the527whole pipe.528Despite the abnormal aspect of the oscillations in Fig. 6(b) that529reach negative pressures and also values as high as the inlet pres-530sure, the oscillations are caused by pressure propagation within531the fully structured material that is significantly elastic and not532by numerical instability. Fig. 7 displays the axial pressure distribu-533tion for different times in order to explain that. As exhibited in534Fig. 7(a), the pressure propagates at sound speed similarly to an535inviscid inelastic case but differently from a low viscous Newto-536nian fluid case that presents an almost linear pressure distribution537upwind the wave front. Additionally, the axial distribution pre-538sents an adverse gradient with pressures lower than the wave front539values. This nonlinear pressure distribution results from the mate-540rial elastic stretch. Differently from an inviscid model that reduces541the pressure to zero after being reflected at the pipe outlet, the cur-542rent case pressure is reduced to negative values after the wave543reflection, as shown in Fig. 7(b). The reason for that are the lower544pressure values found upwind the wave crest before the wave545reflection.546Although the inlet pressure has being imposed as a step func-547tion this is improbable to take place in a real situation because of548pumping system inertia. This unlikely fast pressure change at the549pipe inlet may be, therefore, the cause of the exaggerated pressure550oscillations observed at early simulation times. In order to evaluate551the effect of the step change, the inlet pressure was gradually552imposed by using a ramp function combined with a step function:553

p�ð0; tÞ ¼p�0

t�t�r; 0 6 t� < t�r

p�0; t� P t�r

(ð34Þ

555555

t* [-]0 100 200 300 400

-10

-5

0

5

10

15

teq* = 0.1

teq* = 1.0

teq* = 10.0

φ = 1.0 G0* = 102(a)

p*[-

]

t* [-]0 50 100 150 200

-10

-5

0

5

10

15

teq* = 1.0

teq* = 10.0

φ = 100.0 G0* = 102(b)

p*[-

]

Fig. 6. Time variation of pressure at z⁄ = 0.5 for different equilibrium times. (a) / = 1.0 and (b) / = 100.0. Inlet pressure case with G�0 ¼ 104.

t* [-]0 20 40 60 80 100

0

2

4

6

8

10

12φ = 1.0 teq

* = 1.0 G0* = 102(a)

z* = 0.9

p* [-

]

0.80.70.60.50.40.30.20.10.0

t* [-]0 20 40 60 80 100

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03(b)

z* = 0.0

z* = 0.5

z* = 1.0

v* [-

]

Fig. 4. Time variation of (a) pressure and (b) average velocity at different axial positions. Inlet pressure case with / = 1.0, t�eq ¼ 1:0 and G�0 ¼ 104.

t* [-]0 20 40 60 80 100

0

5

10

15

20

25

r* = 0.5r* = 0.3r* = 0.1

z* = 0.0

z* = 0.5z* = 1.0

λ [-]

φ = 1.0 teq* = 1.0 G0

* = 102

Fig. 5. Time variation of the structure parameter for different radial and axialpositions. Inlet pressure case with / = 1.0, t�eq ¼ 1:0 and G�0 ¼ 104.

6 G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

556 and p�0 is the final value of the rump function and t�r is the dimen-557 sionless time (=tr c/l) for p�0 to be established. In this case, p�0 was558 made equal to 10.0 and the results are shown in Fig. 8(a). In com-559 parison with Fig. 6(b), the magnitude of the oscillation has reduced560 significantly without showing negative values.561 To investigate the effect of the material elasticity on the pres-562 sure propagation for / = 100 case, the elastic effects were removed563 from the constitutive equation. Fig. 8(b) shows the results for this564 exclusively viscous model. A comparison of Fig. 8(b) with Figs. 8(a)565 and 6(b) show that the pressure oscillations have completely dis-566 appeared and also that the pressure propagation has significantly567 being delayed. The elimination of the elastic effect made the prob-568 lem highly dissipative because of the high viscosity of the fully569 structured material. It can be concluded that the elastic effect is570 the reason for the pressure propagation in this low / value case.571 Notably, the removal of the elastic effect of the / = 100 case made572 the pressure response quite similar to the quite dissipative / = 1573 case of Fig. 6(a) for t�eq ¼ 10. Nevertheless, the exclusively viscous574 problem responds faster than the visco-elastic model.575 Fig. 9 shows the axial distributions of pressure and of the struc-576 ture parameter at the wall for three / values and t�eq ¼ 100, after a577 period of t⁄ = 50. Whereas the pressure wave has reached the pipe578 outlet for the less dissipative case (/ = 100) establishing a pressure579 gradient throughout the whole pipe, the pressure has only propa-580 gated near the pipe inlet for / = 1. In other words, the material is581 stressed differently along the axial position depending on the /582 value so that the high / value case is firstly affected throughout

583the whole pipe in comparison to the others. According to584Fig. 9(b), only the less dissipative flow has started after 50 time585unit, once the structure parameter at the wall has changed586throughout the whole pipe.587Fig. 10 depicts the time variation of pressure in different axial588positions, for two different / values and t�eq ¼ 100. In the less dis-589sipative case found in Fig. 10(b), the pressure wave propagates fast,590oscillates in high frequency throughout the whole pipe and the591oscillation is also rapidly dissipated. Despite the fluid being592stressed in all axial positions, the material structure has not been593fully broken after the dissipation of the high frequency oscillation.594The material structure starts breaking down throughout the whole595pipe almost simultaneously as the pressure in all axial positions596increases (t⁄ � 250) and drops (t⁄ � 1000) almost at the same time.597On the contrary, the pressure does not oscillate and does not prop-598agate immediately throughout the whole pipe in the higher dissi-599pative case, as shown in Fig. 10(a). In fact, the material structure600starts changing from inlet to the outlet delaying the complete601break-down. Whereas in the less dissipative case of Fig. 10(b) the602flow takes about 1000 time units to start up, in the higher dissipa-603tive case of Fig. 10(a) the flow starts after 2000 time units. It is604worth mentioning that in both cases the flow only starts up when605the pressure begins to drop throughout the whole pipe to reach the606steady-state.607The effect of the material elasticity is now investigated by608changing the values of G�0. Notably, the lower the G�0 the more609elastic is the material and the higher the G�0 the more rigid is

t* [-]0 50 100 150 200

-10

-5

0

5

10

15(b)

p*[-

]

φ = 100.0 teq* = 10.0 G0

* = 102

t* [-]0 50 100 150 200

-10

-5

0

5

10

15φ = 100.0 teq

* = 10.0 G0* = 102(a)

p*[-

]

Fig. 8. Time variation of pressure at z⁄ = 0.5 for / ¼ 100, t�eq ¼ 102 and G�0 ¼ 102. (a) inlet pressure applied with ramp duration of t�r ¼ 1 and (b) inelastic model. Inlet pressurecase.

z* [-]0 0.2 0.4 0.6 0.8 1

-10

-5

0

5

10

15

p*[-]

φ = 100.0 teq* = 10.0 G0

* = 102(a)

t* = 0.2 0.4 0.6 0.8 1.0

z* [-]0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10

15

p*[-]

φ = 100.0 teq* = 10.0 G0

* = 102(b)

t* = 1.9

1.71.5 1.3 1.1

Fig. 7. Axial pressure distribution for different simulation times. (a) t⁄ = 0.2, 0.4, 0.6, 0.8 and 1.0 and (b) t⁄ = 1.1, 1.3, 1.5, 1.7 and 1.9. Inlet pressure case with / ¼ 100, t�eq ¼ 102

and G�0 ¼ 102.

G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx 7

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

610 the fluid. Fig. 11 shows the effect of elasticity on the pressure611 variation at the middle of the pipe. In spite of the high value612 of t�eq in the more dissipative case (/ = 1.0), the reduction of613 G�0 enlarges the portion of the fully structured material that is614 stressed within the pipe, reducing the pressure peaks at the615 flow start-up. For the less dissipative case (/ = 100), on the616 other hand, the reduction of G�0 not only increases the pressure617 oscillation because of the higher elasticity but also reduces the618 level of the material structure at the start-up so that the mate-619 rial breaks down as the pressure propagates.

6204.2.2. Inlet flow rate case621The inlet flow rates set in this section are chosen to provide the622same steady-state pressure obtained in the constant inlet pressure623cases. Fig. 12 presents the time variation of pressure and average624velocity for different axial positions, / = 10.0, t�eq ¼ 1:0 and625G�0 ¼ 102. As noted, the pressure increases up to the time the fluid626starts flowing at the pipe outlet and then is relieved. The fast pres-627sure increase at the inlet takes place to overcome the elastic yield628of the structured material and the first relief of the inlet pressure at629t⁄ = 2.9 is caused by the partial material break-down near the pipe

t* [-]0 1000 2000 3000

-10

-5

0

5

10

15

G0* = 10-1

G0* = 102

G0* = 105

φ =1.0 teq* = 100.0(a)

p*[-

]

t* [-]0 1000 2000 3000

-10

-5

0

5

10

15

G0* = 10-1

G0* = 102

G0* = 105

φ = 100.0 teq* = 100.0(b)

p*[-

]

Fig. 11. Time variation of pressure at z⁄ = 0.5 for different three G�0 values and two / values: (a) / = 1 and (b) / = 100.0. Inlet pressure case with t�eq ¼ 100.

z* [-]0 0.2 0.4 0.6 0.8 1

-2

0

2

4

6

8

10

12

φ = 1.0φ = 10.0φ = 100.0

p*[-

]

t* = 50.0 teq* = 100.0 G0

* = 102(a)

z* [-]0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

φ = 1.0φ = 10.0φ = 100.0

λ w[-

]

t* = 50.0 teq* = 100.0 G0

* = 102(b)

Fig. 9. Axial distribution of (a) pressure and of the (b) structure parameter at the wall after a period of t⁄ = 50 for different values of /. Inlet pressure case with t�eq ¼ 100 andG�0 ¼ 102.

t* [-]0 1000 2000 3000

-2

0

2

4

6

8

10

12φ = 1.0 teq

* = 100.0 G0* = 102(a)

p*[-

]

z* = 0.9

0.70.8

0.60.50.40.30.20.10.0

t* [-]0 1000 2000 3000

-2

0

2

4

6

8

10

12φ = 100.0 teq

* = 100.0 G0* = 102(b)

p*[-

]

z* = 0.9

0.70.8

0.60.50.40.30.20.10.0

Fig. 10. Time variation of pressure at different axial positions for t�eq ¼ 100 and two / values: (a) / = 1 and (b) / = 100.0. Inlet pressure case with G�0 ¼ 102.

8 G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

630 inlet. After that, the pressure wave progresses breaking down the631 material structure throughout the pipe up to t⁄ = 20 when the pipe632 is finally relieved and the flow starts up.633 Fig. 13 presents the time variation of the inlet pressure for three634 different equilibrium times and two / values. As noted, the higher635 the equilibrium time the higher is the pressure magnitude and the636 time for breaking down the material structure. Additionally, the637 pressure peaks increase and the breaking down time diminishes638 with the reduction of dissipation (increase of /). The results for639 Newtonian fluid with the same viscous dissipation in steady-state640 are also shown in Fig. 13. Whereas the pressure the Newtonian641 fluid does not fluctuate for the high dissipative case (/ = 1), the642 pressure for / = 10 oscillates at c/l frequency for a short period of643 time so as to double the inlet pressure at t⁄ = 2.0. The Newtonian644 fluid starts flowing much faster than the structured fluids, regard-645 less the / value or the equilibrium time. The maximum inlet pres-646 sure for the Newtonian fluid can be larger or smaller than those for647 the structured fluid depending on the magnitude of the equilib-648 rium time.649 In order to check if the magnitudes of the start-up pressures are650 affected by the speed in which the material is stressed, the inlet651 flow rate is imposed from zero to its steady-state value based on652 a ramp function combined with a step function:653

Q �ð0; tÞ ¼Q �0

t�t�r; 0 6 t� < t�r

Q �0; t� P t�r

(ð35Þ

655655

656 where Q �0 is the dimensionless flow rate that provides the same657 steady-state result as the constant inlet pressure case. Fig. 14 pre-658 sents the time variation of the inlet pressure for different values

659of t�r , t�eq ¼ 10 and / = 10.0. In spite of the pressure peak being660reduced and the pressure time response being increased with t�r ,661the pressure magnitude is only slightly affected for t�r ¼ t�eq. The662pressure peaks are diminished because the increase of t�r renders663time for the material to break down so that the flow resistance is664reduced as the stress is increased. Notably, the time for the material665break-down depends is larger, equal and smaller than t�r for t�r smal-666ler, equal and larger than 100, respectively.

t* [-]0 10 20 30 40 50

0

5

10

15

20

z* = 0.0z* = 0.5z* = 0.9

(a)

p* [-

]

φ = 10.0 teq* = 1.0 G0

* =102

t* [-]0 10 20 30 40 50

0

0.0005

0.001

0.0015

0.002

z* = 0.1z* = 0.5z* = 1.0

(b)

V*

[-]

Fig. 12. Time variation of (a) pressure and of (b) average velocity for different axial positions. Inlet flow rate case with / = 10.0, t�eq ¼ 1:0 and G�0 ¼ 102.

t* [-]0 100 200 300 400

0

10

20

30

40

50

Newt.t*eq = 0.1t*eq = 1.0t*eq = 10.0

φ = 1.0 G0* = 102(a)

p*in

[-]

t* [-]0 50 100 150

0

10

20

30

40

50

Newt.t*eq = 0.1t*eq = 1.0t*eq = 10.0

φ = 10.0 G0* = 102(b)

p*in

[-]

Fig. 13. Time variation of the inlet pressure for different t�eq and two / values: (a) / = 1.0 and (b) / = 10.0. Inlet flow rate case with G�0 ¼ 102.

t* [-]0 100 200 300 400 500

0

10

20

30

40

50

tr* = 0.0tr* = 10.0tr* = 100.0tr* = 200.0tr* = 400.0

φ = 10.0 teq* = 10.0 G0

* = 102

p*in

[-]

Fig. 14. Time variation of the inlet pressure for different changes of the inlet flowrate. Inlet flow rate case with / = 10.0, t�eq ¼ 10:0 and G�0 ¼ 102.

G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx 9

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

667 The effect of the material elasticity on the start-up pressure is668 now investigated for a constant inlet flow rate. As shown in669 Fig. 15, the less elastic (high G�0) is the material the higher is the670 pressure needed for the material break-down and the faster is671 the material response. In spite of the pressure change being inde-672 pendent on t�eq up to material initial break-down, the smaller the673 equilibrium time the faster is the material break-down and the674 smaller the pressure required for break-down. Both Fig. 15(a)675 and (b) shows an amplification of the region where the high fre-676 quency pressure oscillations take place. For the more elastic case677 (G�0 ¼ 10�1), the frequency of oscillation coincides with the pres-678 sure wave frequency (c/l) and by enlarging G�0 to 105, the oscilla-679 tions take place because of material break-down at the inlet.

680 5. Conclusions

681 The current work puts forward a mathematical model to predict682 pressure changes during the flow start-up of structured fluids in683 pipelines. The flow is considered weakly compressible whereas684 the structured fluid is modeled by using the Souza Mendes and685 Thompson’s [21] constitutive equations. Two time scales were686 identified as important parameters for gel breaking: the shearing687 time and the time for material break-down. The first was defined688 as the pressure wave propagation time and the second as the equi-689 librium time defined in the structure parameter model of Souza690 Mendes and Thompson [21]. Two types of inlet boundary condi-691 tions were considered: constant pressure and constant flow rate.692 Whereas in the first, the required time for the flow start-up for a693 given pressure is determined, in the latter, the maximum pressure694 reached in the restart for an imposed flow rate is obtained.695 The conclusions can thus be summarized as follow:

696 (i) If the equilibrium time is smaller than the pressure propaga-697 tion time, the flow starts up as soon as the pressure wave698 reaches the pipe outlet;699 (ii) An avalanche effect is observed for equilibrium time higher700 than the pressure propagation time;701 (iii) The lower the viscous dissipation in comparison to the fluid702 compressibility the faster is the flow start-up;703 (iv) The reduction of the elastic modulus reduces the pressure704 peaks for high dissipative material and reduces the start-705 up time for less dissipative materials;706 (v) For a constant inlet flow rate boundary condition, the higher707 the equilibrium time or the higher the dissipation the larger708 is the pressure peak necessary for the flow start-up;709 (vi) By slowing down the variation of the inlet flow rate, the inlet710 pressure peaks are reduced.

711

712

713

714Acknowledgements

715The authors acknowledge the financial support of PETROBRAS716S/A, ANP (Brazilian National Oil Agency) and CNPq (The Brazilian717Council for Scientific and Technological Development).

718References

719[1] R.A. Ritter, J.P. Batycky, Numerical prediction of the pipeline flow720characteristics of thixotropic liquids, SPE J. 7 (1967) 369–376.721[2] J. Sestak, M.G. Cawkwell, M.E. Charles, M. Houska, Start-up of gelled crude oil722pipelines, J. Pipeline. 6 (1987) 15–24.723[3] M.G. Cawkwell, M.E. Charles, An improved model for start-up of pipelines724containing gelled crude oil, J. Pipeline. 7 (1987) 41–52.725[4] C. Chang, Q.D. Nguyen, H.P. Rønningsten, Isothermal start-up of pipeline726transporting waxy crude oil, J. Non-Newtonian Fluid Mech. 87 (1999) 127–727154.728[5] M.R. Davidson, Q.D. Nguyen, C. Chang, H.P. Rønningsten, A model for restart of729a pipeline with compressible gelled waxy crude oil, J. Non-Newtonian Fluid730Mech. 123 (2004) 269–280.731[6] G. Vinay, A. Wachs, J.F. Agassant, Numerical simulation of weakly compressible732Bingham flows: the restart of pipeline flows of waxy crude oils, J. Non-733Newtonian Fluid Mech. 136 (2006) 93–105.734[7] I.A. Frigaard, G. Vinay, A. Wachs, Compressible displacement of waxy crude oils735in long pipelines startup flows, J. Non-Newton. Fluid Mech. 147 (2007) 45–64.736[8] G. Vinay, A. Wachs, I. Frigaard, Start-up transients and efficient computation of737isothermal waxy crude oil flows, J. Non-Newtonian Fluid Mech. 143 (2007)738141–156.739[9] A. Wachs, G. Vinay, I. Frigaard, A 1.5D numerical model for the start up of740weakly compressible flow of a viscoplastic and thixotropic fluid in pipelines, J.741Non-Newtonian Fluid Mech. 159 (2009) 81–94.742[10] G.M. Oliveira, L.L.V. Rocha, A.T. Franco, C.O.R. Negrão, Numerical simulation of743the start-up of Bingham fluid flows in pipelines, J. Non-Newtonian F. Mech.744165 (2010) 1114–1128.745[11] C.O.R. Negrão, A.T. Franco, L.L.V. Rocha, A weakly compressible flow model for746the restart of thixotropic drilling fluids, J. of Non-Newtonian F. Mech. 16747(2011) 1369–1381.748[12] P.R. Souza Mendes, F.S.M.A. Soares, C.M. Ziglio, M. Gonçalves, Startup flow of749gelled crudes in pipelines, J. Non-Newtonian F. Mech. 179–180 (2012) 23–31.750[13] D.A. Phillips, I.N. Forsdyke, I.R. McCraken, P.D. Ravenscroft, Novel approaches751to waxy crude restart: Part 1: thermal shrinkage of waxy crude oil and the752impact for pipeline restart, J. Petroleum Sci. Eng. 77 (2011) 237–253.753[14] J. Mewis, Thixotropy – a general review, J. Non-Newtonian Fluid Mech. 6754(1979) 1–20.755[15] H. Barnes, Thixotropy – a review, J. Non-Newtonian Fluid Mech. 70 (1997) 1–75633.757[16] J. Mewis, N.J. Wagner, Thixotropy, Adv. Colloid Interface Sci. 147–148 (2009)758214–227.759[17] A. Mujumdar, A.N. Beris, A.B. Metzner, Transient phenomena in thixotropic760systems, J. Non-Newtonian Fluid Mech. 102 (2002) 157–178.761[18] K. Dullaert, J. Mewis, A structural kinetics model for thixotropy, J. Non-762Newtonian Fluid Mech. 139 (2006) 21–30.763[19] P.R. de Souza, Mendes, modeling the thixotropic behavior of structured fluids,764J. Non-Newtonian Fluid Mech. 164 (2009) 66–75.765[20] P.R. Souza Mendes, Thixotropic elasto-viscoplastic model for structured fluid,766Soft Matter. 7 (2017) 2471-2483.767[21] P.R. Souza Mendes, R.L. Thompson, A unified approach to model elasto-768viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids,769Rheol. Acta 52 (7) (2013) 673–694.

t* [-]0 100 200 300 400 500

0

10

20

30

40

50

t*eq = 1.0t*eq = 10.0

φ = 10.0 G0* = 10-1(a)

p*in

[-]

t* [-]0 100 200 300 400 500

0

10

20

30

40

50

t*eq = 1.0t*eq = 10.0

φ = 10.0 G0* = 105(b)

p*in

[-]

0 2 4 6 830

35

40

0 2 4 6 80

0.5

1

1.5

Fig. 15. Time variation of the inlet pressure for different values of equilibrium time and two elasticity values. (a) G�0 ¼ 10�1 and (b) G�0 ¼ 105. Inlet flow rate case with / = 10.0.

Q6

10 G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010

770 [22] D.C.H. Cheng, F. Evans, Phenomenological characterization of the rheological771 behavior of inelastic reversible thixotropic an antithixotropic fluids, Brit. J.772 Appl. Phys. 16 (1965) 1599–1617.773 [23] M. Houska, Inzenyrske aspekty reologie tixotropnich kapalin, PhD thesis,774 Techn. Univ. Prague, 1980.775 [24] W. Worrall, S. Tulianni, Viscosity changes during the aging of clay-water776 suspensions, Trans. Br. Ceram. Soc. 63 (1964) 167–185.777 [25] E.A. Toorman, Modeling the thixotropic behaviour of dense cohesive sediment778 suspensions, Rheol. Acta. 36 (1997) 56–65.

779[26] Q. Nguyen, D. Boger, Thixotropic behaviour of concentrated bauxite residue780suspensions, Rheol. Acta. 24 (1985) 427–437.781[27] G.M. Oliveira, C.O.R. Negrão, A.T. Franco, Pressure transmission in Bingham782fluids compressed within a closed pipe, J. Non-Newtonian F. Mech. 169–170783(2012) 121–125.784[28] A. Bejan, Convection Heat Transfer, 3rd ed., John Wiley & Sons, 2004.785[29] E.B. Wylie, V.L. Streeter, L. Suo, Fluid Transients in Systems, Prentice Hall, N.786Jersey, 1993.

787

G.M. de Oliveira, C.O.R. Negrão / Journal of Non-Newtonian Fluid Mechanics xxx (2015) xxx–xxx 11

JNNFM 3626 No. of Pages 12, Model 5G

10 January 2015

Please cite this article in press as: G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow start-up of waxy crude oils, J. Non-Newtonian FluidMech. (2015), http://dx.doi.org/10.1016/j.jnnfm.2014.12.010