Effects of Drillstring-Eccentricity, -Rotation, and -Buckling Configurations on Annular Frictional...

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The Effects of Drillstring-Eccentricity,-Rotation, and -Buckling Configurations

on Annular Frictional Pressure Losses WhileCirculating Yield-Power-Law Fluids

Oney Erge, Evren M. Ozbayoglu, Stefan Z. Miska, Mengjiao Yu, and Nicholas Takach, University of Tulsa;

Arild Saasen, Det Norske Oljeselskap ASA and University of Stavanger, and Roland May, Baker Hughes

Summary

An experimental study followed by comprehensive flow modelingis presented. The experiments were conducted on a horizontalannulus with drillstring under compression, considering the influ-ence of rotation on frictional pressure losses of yield-power-law(YPL) fluids. Flow through various buckling configurations withand without drillstring rotation was investigated. Correlations of critical Reynolds numbers are presented that predict the onset andoffset of transition from laminar- to turbulent-flow regions in con-

centric and eccentric annuli. A broad model of flow of YPL fluidsis proposed for concentric, eccentric, and buckled configurations.The model includes the effects of rotation in laminar, transitional,and turbulent flow.

A 91-ft inner pipe was rotated while applying axial compres-sion during flow. At the no-compression case, eccentricity of theinner pipe is varied as the drillstring rotated. The aim for sucha design was to simulate actual drilling operations. The testmatrix involves flow through sinusoidal, transitional, and helicallybuckled drillstring. The effect of pitch length is investigated.Helical modes with two different pitch lengths were tested. Eightdistinct YPL fluids were used to examine the dependence of pres-sure losses on fluid parameters. In the theoretical part, a stabilitycriterion is modified to determine the onset of transitional flowof YPL fluids and a correlation is proposed for practical purposes.

In addition, pressure-loss-prediction models are presented for theflow of YPL fluids through concentric, eccentric, free, andbuckled configurations of the drillstring, with and without rota-tion. The proposed models are compared with data from the litera-ture and the experiments.

It has been observed that increasing eccentricity and rotaryspeed causes an earlier transition from laminar to turbulent flow.Results suggest reduced pressure losses with an eccentric pipe. Inaddition, buckled configurations showed a further decrease of fric-tional pressure losses as the compression increases. In the helicalmode, two pitch lengths are compared, and decreasing the pitchlength resulted in a decrease in pressure losses. Rotation tests areconducted with free and buckled configurations. Rotation in thefree-drillstring mode showed an increase in pressure losses as therotary speed of the drillstring increases.

Amplified vigorous motion of the drillstring is visuallyobserved as the drillstring is buckled while rotating. Rotating thedrillstring while buckled showed a further increase in pressurelosses compared to rotating in free mode. This additional increasein pressure losses is attributed to the more-dynamic motion of thedrillstring. Distinct differences of pressure losses in the effects of buckling and rotation are observed in laminar, transitional, andturbulent flow. Significant differences are measured in the transi-tion region.

Introduction

Keeping the drilling-fluid equivalent circulating density (ECD) inthe operating window between the pore and fracture pressure is achallenge, particularly when the gap between these two is narrow,such as in high-temperature and high-pressure offshore applica-tions. To overcome this challenge, accurate estimation of fric-tional pressure loss in the annulus is essential, especially for multilateral, extended-reach, and slimhole drilling applicationsusually encountered in shale-gas and/or oil drilling.

First, flow state should be determined to predict the pressurelosses. The flow in the annulus can be laminar, transitional, or tur-bulent, depending on the parameters such as drillstring rotaryspeed, eccentricity, diameter ratio, fluid properties, and flow rate.At each flow state, prediction of frictional pressure losses will behandled with a corresponding model. Most transition-criterionmodels in literature do not consider the effect of eccentricity or the diameter ratio (the ratio of the diameter of the inner pipe tothat of the outer pipe). Therefore, a more-precise model isrequired to estimate the onset and the offset of the transitionregion for the flow in annuli, which consequently will lead tomore-accurate estimation of the frictional pressure losses.

Rotating the drillstring causes additional shear to the fluid, andadditional shear affects both the shear-thinning property and iner-tial effects. Some investigations reported an increase and some

reported a decrease in frictional pressure losses as the drillstring isbeing rotated, whereas most field measurements showed an in-crease. In Fig. 1, pressure-while-drilling measurements collectedfrom an offshore drilling operation are presented. ECD shows adirect response to rotary-speed change of the drillstring. As thedrillstring-rotation speed decreases, a decrease in pressure loss isobserved. Investigating the reasons for the difference between theexperimental measurements in literature and the field measure-ments while the drillstring is being rotated is one of the objectivesof this study. To accomplish this objective, an experimental setupsimulating drilling conditions is introduced. Free-drillstring rota-tion without prefixed eccentricity is in the scope of this study.

Rotating the drillstring in laminar flow of yield-power-law(YPL) fluids through concentric annuli can decrease the pressure

losses because of the shear-thinning ability of YPL fluids, espe-cially in laboratory experiments. When eccentricity is introduced,rotation will increase the magnitude of inertial effects. Rotatingthe inner pipe will trigger both shear-thinning ability and inertialeffects, and they try to counteract each other. As a result, anincrease or a decrease in frictional pressure losses can be observed.

Literature Review

Flow in the annuli is complex because of many variables inaction, such as drillstring eccentricity, rotation, lateral motion,time, cuttings, and fluid parameters (Cartalos and Dupuis 1993;Ahmed et al. 2010; Saasen 2013). Drillstring eccentricity isreported to cause a decrease in pressure losses up to 60% (Uner et al. 1988; Haciislamoglu 1989; Haciislamoglu and Langlinais

1990; Nouri et al. 1993; Azouz 1994; Haciislamoglu and Cartalos1994; Ozgen and Tosun 1997; Fang 1998; Sestak et al. 2001;

CopyrightVC 2015 Society of Petroleum Engineers

This paper (SPE 167950) was accepted for presentation at the IADC/SPE DrillingConference and Exhibition, Fort Worth, Texas, USA, 4–6 March 2014, and revised forpublication. Original manuscript received for review 7 March 2014. Revised manuscriptreceived for review 26 March 2015. Paper peer approved22 July 2015.

September 2015 SPE Drilling & Completion 257



Escudier et al. 2002; Hashemian 2005; Kelessidis et al. 2011;May et al. 2013).

In laminar flow, rotation tends to decrease pressure lossesbecause of added shear to the apparent viscosity, whereas ampli-fied inertial effects increase pressure losses. Rotation of the drill-string in concentric annuli in laminar flow of yield-power-law(YPL) fluids can result in decreasing the pressure losses because

of the shear-thinning ability of YPL fluids (Erge 2013). As theflow stability decreases with increasing axial-flow rate and rota-tion speed, pressure losses increases in transitional and turbulentflow (Escudier and Gouldson 1995).

Rotation in eccentric annuli introduces additional shear and in-ertial effects. For non-Newtonian fluids, few experimental meas-urements of both rotation and eccentricity effects are presented(Nouri and Whitelaw 1997; Escudier et al. 2002, Ozbayoglu andSorgun 2010). Nouri and Whitelaw (1997) conducted experimentsfor the dimensionless eccentricity of 0.5, diameter ratio of 0.5,and 300-rev/min rotary speed. Their experiments show anincreased flow resistance of more than 30% with rotation of theinner pipe for both Newtonian and non-Newtonian fluids. Escud-ier et al. (2002) numerically and experimentally investigated theeffects of pipe rotation on frictional pressure loss in concentric

and eccentric annular geometries for Newtonian and non-Newto-nian fluids. Inner-cylinder rotation is found to increase the magni-tude of the axial-pressure gradient for the flow of a Newtonianfluid in eccentric annuli because of elevated inertial effects. Theyshow that shear thinning effects can be dominant in a slightlyeccentric annulus and even surpass the inertial effects, whereas ina highly eccentric annulus, inertial effects can be dominant for non-Newtonian fluids. Ozbayoglu and Sorgun (2010) conductedexperiments for estimation of frictional pressure loss in a fullyeccentric annular geometry. The experiments show that increasingrotation speed in a fully eccentric annulus significantly increasesfrictional pressure losses at lower flow rates, whereas the effect of rotation vanishes at high axial flows.

In the literature, many experimental and numerical studiesconducted on the effect of rotation showed that annular pressurelosses can either increase or decrease with increasing drillstring-rotation speed (Marken et al. 1992; Fang 1998; Escudier et al.2002; Ahmed and Miska 2008; Ahmed et al. 2010; May et al.2013). Field measurements explicitly show that increasing drill-string-rotation speed increases the frictional pressure losses(Charlez et al. 1998; Isambourg et al. 1998; Ward and Andreassen1998; Green et al. 1999; Hemphill et al. 2007).

Complex configurations of drillstring may be present in thewellbore. As the drillstring is rotated, lateral movement or wob-bling of the drillstring and drillstring-motion patterns such assnaking and whirling will affect the pressure losses. Gao andMiska (2008) presented examples of drillstring-motion patternsexpected in a wellbore. In the literature, very-limited informationexists on deflected and rotating inner pipes. Haciislamoglu and

Cartalos (1994) considered for the first time the degree to which adrillstring is skewed. They conducted experiments for the flow of

Newtonian and power-law fluids through skewed annuli. Asafaand Shah (2012) conducted experiments for the flow of power-law guar fluids through buckled drillstrings. They suggested thatthe deflection of the drillstring will mostly reduce the pressurelosses compared with the concentric geometry.

Annular-pressure losses can be accurately estimated consider-ing the pseudosteady conditions, as Saasen (2013) discussed.Pseudosteady conditions include the drillstring-motion effects. Inthis study, pseudosteady conditions are considered and drillstringmodes are analyzed.

Most current drilling fluids show highly non-Newtonian flowbehavior and exhibit high shear thinning performance. These flu-ids can be characterized as YPL. The YPL-fluid model better describes the fluid at low and high shear rates than the Bingham-plastic or power-law models (Hemphill et al. 1993; Friedheim andConn 1996; Bern et al. 2007). Yield stress of the drilling fluid sus-pends the cuttings and shear thinning ability enables lower pres-sure losses at the high flow rates. This behavior is practicallywell-explained by the YPL model.

Herschel and Bulkley (1926) presented the Herschel-Bulkleyfluid model (YPL), which is given as

s ¼ s y þ K _cm: ð1Þ

Several approaches have been reported in the literature to evalu-ate the behavior of YPL fluids through concentric and eccentric

annuli with and without inner-pipe rotation. There are approxima-tions that show high accuracy for a range of diameter ratios (Kozickiet al. 1966; Uner et al. 1988; Luo and Peden 1990; Marken et al.1992; Reed and Pilehvari 1993; Ozgen and Tosun 1997; Sestaket al. 2001; Ahmed and Miska 2009; Naesgaard 2012). Narrow-slotapproximation of the concentric annuli is widely used for practicalreasons. The annular geometry is represented as a slot that has thesame area and the same height with the annulus. The approximationreduces the complexity of the flow equations and shows reasonableaccuracy for the annuli with diameter ratio greater than 0.3. In addi-tion, empirical and semiempirical correlations are available by useof experiments, field measurements, and computational-fluid-dy-namics studies (Haciislamoglu and Langlinais 1990; Haciislamogluand Cartalos 1994; Hemphill et al. 2008; Ahmed et al. 2010;Ozbayoglu and Sorgun 2010; Naesgaard 2012). In a few studies, ec-

centricity and rotation have been handled with conformal-mappingtechniques and numerical solutions to the system (Fang 1998;Escudier et al. 2002; Sorgun et al. 2010; May et al. 2013). Thesestudies are conducted for a prefixed eccentricity and consider onlythe rotation about the inner pipe’s own axis. Free-drillstring rotationand deflected configurations presented with this study more closelysimulate actual drilling conditions.

Scope ofWork

In this study, laminar flow, transitional flow, and turbulent flowof yield-power-law (YPL) fluids through annuli, including theeffects of drillstring eccentricity, rotation, and buckling, are inves-tigated. The focus of this paper is on two topics:

• Transition from the laminar to turbulent flow of YPL fluidsin concentric and eccentric annuli

• Annular frictional pressure losses of YPL fluids, includingthe effects of the drillstring eccentricity, rotation, andbuckling

An experimental approach and a theoretical approach are fol-lowed for both topics. Experiments are conducted for free, sinu-soidal, transitional, and helically buckled drillstrings with andwithout rotation. In helical configuration, the effect of pitch-length decrease is investigated; eight YPL fluids are tested, andconsistent results of pressure-loss change with inner-pipe rotationare observed. Laminar, transition, and turbulent regions are inves-tigated separately.

A modification for a stability parameter is proposed, extendingits applicability to eccentric annuli for YPL fluids. A practical

correlation is presented to determine the onset and offset of thetransition region for both concentric and eccentric geometries.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0:00:00

1.345

1.34

1.335

1.33

1.3250:02:53 0:05:46

Time

E C D

( s p e c i f i c g r a v i t y )

0:08:38 0:11:31

160

ECD

Rev/min120

80

40

0

R e v / m i n

Fig. 1—ECD at various drillstring-rotation speeds vs. time forWell-A (Erge et al. 2014). PWD ¼ pressure while drilling.

258 September 2015 SPE Drilling & Completion



Then, a comprehensive but practical model is presented thatincorporates nearly all drillstring configurations while circulatingYPL fluids. A method to separate the wellbore into sections isused to increase the accuracy in pressure-loss prediction.

Experimental Setup and TestMatrix

The purpose of the experimental work is to analyze the pressurelosses in various drillstring configurations with and without the

rotation of the drillstring while circulating yield-power-law (YPL)fluids. Free-drillstring rotation is experimented to simulate thehorizontal-drilling applications, where the most-significant pres-sure losses occur. Analyzing free-drillstring rotation better ap-proximates the actual drilling conditions, and therefore thedifference between the field measurements and the experimentalmeasurements in literature is aimed to be better understood bymeans of the results from this experimental setup.

Saasen (2013) discussed the importance of the dimensionless-variables consistency between the laboratory measurements andfield applications. At the design phase, dimensionless variables of the experimental facility and the fluids in the test matrix are com-pared with the actual sample drilling cases. Reynolds numbers, di-ameter ratio, Taylor numbers, dimensionless yield stresses, andflow-behavior indices are covering actual drilling parameters.

A dynamic test facility was used to rotate and compress theinner pipe. Observations on the effect of drillstring rotation andbuckling were made optically through the long and transparent testsection. At the initial position the inner pipe is lying at the bottombecause of its extensive length, suggesting a fully eccentric annu-lar geometry. When the inner pipe is rotated, whirling, snaking,

and irregular motions were observed. This state is considered to befree-drillstring configuration because there is no prefixed eccen-tricity imposed on the drillstring. Various drillstring-motion pat-terns can be observed when rotated at free-drillstring mode. Thereason for this design is to simulate the actual drilling operations,especially the highly inclined and horizontal-drilling operations.

Properties of the experimental facility are shown in Table 1. Adetailed schematic of the test facility is given in Fig. 2.

Pressure is measured by use of point pressure transducers

(PTs) and a differential-pressure transducer installed to the annu-lar section. The readings of the pressure difference between PT5and PT3, which is the longest section, are presented in this study.

A centrifugal pump is used for fluid circulation. In the testphase, additives are mixed through the hopper and fluid goesthrough the pump to the supply line. At the inlet of the supplyline, a flowmeter is present. It measures both density and flowrate. The fluid goes through the supply line and connects to theannular-test section. Inner pipe is rotated by means of a variable-speed motor at desired rev/min. A variable-frequency drive is in-stalled and a proportional-integral-derivative-controller system isused to control the flow rate by use of the data-acquisition com-puter, where all the sensor readings are visualized and recorded.

Water runs are conducted before experimentation, and the cali-bration of the pressure transducers on the flow loop is performed.The equations to calculate pressure losses for the flow of a Newto-nian fluid through annuli can be found in Ahmed and Miska(2009). The results obtained from the water tests show a goodagreement with the calculated pressure losses (Erge 2013).

YPL-fluid flow through various buckling configurations istested for the effects of rotation and nonrotation cases. Before

Table 1 — Specifications of the experimental facility.

Centrifugal pumpHopper

12

12

511

6

9

7.92 m 14.63 m 3.05 m

3.73 m 7.85 m

19.51 m

1.52 m 1.52 m 7.86 m 5.49 m

PT5 6PT4

7DP1

6

38

9

10

PT2

6PT3

6PT1

832

4

1

13

Storage

tank

1 Displacement transducer

2 Moving “Top” end

3 Load cell

4 Torque meter

5 50.8 mm hose

6 Point-pressure transducer (PT)

7 Differential-pressure transducer (DP)

8 50.8 mm × 25.4 mm annulus

9 53.3 mm lD supply line

10 Bottom end

11 Flowmeter

12 Three-way valve

13 Variable frequency drive

Fig. 2—Schematic of the experimental facility.




each test where the rotation or the configuration is changed, a

loading run is conducted. The objective of the loading run is todifferentiate the loads required for sinusoidal, transition, and heli-cal configurations so that the tests can be performed at the desireddrillstring configuration. In Fig. 3a, a loading run is shown, and inFig. 3b, pitch lengths for various compression rates are presented.

In Fig. 3a, the following points can be noted:• At Point 1, the inner pipe is touching the top load cell and is

free at the bottom end. The displacement motor in the topend moves the inner pipe to the bottom load cell. When theinner pipe touches the bottom load cell, compression of thepipe starts.

• At Point 2, pipe is no longer straight and lateral displacements

are observed. This is the starting point for the sinusoidalmode. As the pipe compressed further, lateral displacementsincrease, and at Point 4 the first helixes form.

• Point 3 represents the transition between the sinusoidal- andhelical-buckling configurations.

• At Point 4, the inner pipe helically buckles. As the compres-sion increases, new helixes form and the pitch lengthdecreases. Point 5 is selected as another test point to under-stand the effect of decreased pitch length on the frictionalpressure losses. Point 5 represents a near-lockup state.

Test fluids are prepared by use of Laponite RD and XCD andPAC R additives. Laponite RD is a colorless clay (sodium magne-sium silicate) that is used as a rheology modifier in different indus-tries. XCD is a long-chain water-based polymer that results frommicrobial action on a high-molecular-weight polysaccharide. Lapon-ite RD and XCD give a fluid thixotropic behavior. PAC R is polya-nionic cellulose, which controls fluid loss in freshwater, seawater,and saltwater drilling-fluid systems. PAC R is used as a viscosifier.

Eight YPL fluids with various yield stresses, consistencies,and flow-behavior indices are tested to collect comprehensive ex-perimental data to be used during model validation and correla-tion development. The YPL model is related to the test fluids’behavior, which showed both a yield and a shear-thinning prop-erty. Compositions of the test fluids and fluid parameters are pre-sented in Table 2. Fluid characterization is conducted by use of both Chan 35 rotational viscometer and Anton Paar MCR 301 rhe-ometer with duo-gap geometry. The fluid sample is taken directlyfrom the flow loop when stable data acquisition occurred. Therheometer allowed the obtaining of precise shear-rate and shear-

stress measurements at different temperatures. The rheologymeasurements of the tested fluid are presented in Fig. 4.

400

Top load

Bottom load (5)

(4)

(3)

(2)

(1)

400

350

350

300

300

250

250

200

200

150

150

L o a d ( k g )

P i t c h L e n g t h ( m )

Load (kg)

100

100

50

500

00 0.002 0.004 0.006 0.008 0.01 0.012 0.0146.5

7

7.5

8

8.5

9

Displacement (m)

Fig. 3—(a) Loading of the inner pipe, no rotation, in water, displacement speed of 25.4 mm/s. (b) Observed pitch length for every45.36-kg increase in load.

TestFluid

sr etemar aPdiulFnoitisopmoC

LaponiteXCD (wt%)

PAC R (wt%) Laponite RD (wt%)Temperature

(°F)τ y (Pa) K (Pa·s

m) m R

2

YPL1 0.03% – 0.28% 70 0.29 0.07 0.55 0.999

YPL2 – 0.03% 1.96% 65 3.15 1.44 0.37 0.997

YPL3 – 0.03% 2.51% 75 4.09 2.44 0.33 0.997

YPL4 – – 2.51% 80 5.93 1.69 0.35 0.997

YPL5 – 0.06% 1.50% 80 1.59 0.39 0.51 0.996

YPL6 – 0.03% 1.82% 80 4.04 0.89 0.39 0.997

YPL7 – 0.03% 2.11% 90 7.10 1.09 0.40 0.960

YPL8 – 0.08% 2.23% 90 9.65 3.33 0.31 0.983

Table 2 — Composition of the test fluids and rheological properties of the test fluids measured by use of Anton Paar MCR 301 rheometer

(reference pressure5atmospheric).

40

35

30

25

20

S h e a r S t r e s s ( P a )

15

10

5

0

0 200 400 600

Shear Rate (1/s)

800 1,000 1,200

YPL1

YPL2

YPL3

YPL4

YPL5

YPL6

YPL7

YPL8

Fig. 4—Rheology measurements of the tested fluids.




The pressure readings for one fluid at 11–20 flow rates androtation speeds of 0–120 rev/min with an increment of 30 rev/minare recorded. The configuration is set before each test, and nochange in configuration is observed during testing. Each test isrecorded for approximately more than 4 minutes of stable datawhere no significant transient behavior is observed. Severalsequences of testing are conducted, such as from 0 to 120 rev/minand from 120 to 0 rev/min. Several tests are repeated and agood reproducibility and no significant deviation is observed.Pressure vs. velocity plots are drawn, and the onset and offset of the transition are interpreted from the pressure loss vs. velocitycurves by analyzing slope changes. The first data point thatshows a deflection from the laminar-flow curve is selected asthe onset of the transition. The first data point that is deflectedfrom the turbulent-flow curve is marked as the offset of thetransition. This information is also verified with the Fanning-fric-tion-factor and generalized Reynolds-number curves, and theonset and offset of transitional flow is marked accordingly (Ergeet al. 2015a).

Stability Criterion

To predict the frictional pressure losses while drilling, flow stateshould be determined as laminar, transitional, or turbulent flow.For each, the prediction of frictional pressures losses will behandled with a corresponding model.

Flow Stability in Concentric Annuli. In concentric annuli,because of the degree of curvature difference between the inner and outer pipes, a distinct transition behavior is expected in theinner- and outer-flow regions. The inner-flow region is the regionfrom the inner pipe to the location where the shear stress becomeszero. The outer region is from the outer pipe to the location whenthe shear stress becomes zero. Unlike in pipe flow, the laminar-velocity profile is asymmetric in concentric and eccentric annular geometries. This effect is even more pronounced at smaller diam-eter ratios. Most theoretical studies (Hanks 1963; Gucuyener andMehmetoglu 1996; Dou et al. 2010) suggest an earlier transitionfor non-Newtonian fluids in the inner shear region, where the ve-locity gradient is mostly greater than in the outer shear region.The proposed theoretical approaches by the authors suggest a

higher instability at the location of the higher velocity gradientalong the velocity profile. Asymmetrical-velocity profile of annuliresults in a higher-velocity gradient close to the inner pipe. Hence,the proposed theoretical studies show an earlier transition in theinner shear region. But experimental studies show an early transi-tion near the outer pipe (Hanks and Peterson 1982; Japper-Jaafar et al. 2010).

Hanks and Peterson (1982) measured the turbulence intensitiesat several locations by use of a hot film anemometer. Their resultssuggest two distinct critical Reynolds numbers for flow in concen-tric annuli. Their results show that for Newtonian fluids, the firstdisturbance in laminar flow occurs in the outer region of concen-tric annuli. Japper-Jaafar et al. 2010 measured the axial fluctua-tions in a concentric annulus by use of laser doppler anemometry.At the experiments conducted for the yield-power-law (YPL) flu-ids, more turbulent fluctuations are found near to the outer wall,suggesting earlier transition occurred at the outer region.

An earlier transition in the outer region is calculated for mostof the cases with the approach described in this study. Also, amethod to predict the onset and offset of the transition in eccentricannuli is proposed.

Ryan and Johnson (1959) defined a dimensionless local-stabil-ity parameter. By use of Reynolds decomposition, the kinetic-energy equation is rewritten in a form that includes the velocityfluctuations. This form of the kinetic-energy equation is sub-tracted from the laminar-kinetic-energy equation and an equationof turbulence fluctuations is obtained. The stability parameter isthe ratio of the turbulent-energy production and the rate of workperformed by the viscous stresses. A good agreement is reported

with their parameter and the experiments conducted for the New-tonian and non-Newtonian fluids. Their stability criterion is modi-

fied for the flow of YPL fluids in concentric and eccentric annulito predict the flow state locally in annuli. The Ryan and Johnson(1959) stability criterion is given as

Z r ð Þ ¼ r wqt r ð Þsw

@ t r ð Þ@ r

: ð2Þ

First, instead of averaging shear stress at the wall, inner andouter shear stresses at the wall are evaluated separately. Equationsto calculate the inner and outer shear stresses at the wall are givenin Eqs. 3 and 4, respectively (Ahmed and Miska 2009):

sw;i ¼ s ya

Ri

d P

dl

a2 R2i

2 Ri

; ð3Þ

sw;o ¼ sw;i Ri

Ro

d P

dl

R2o R2

i

2 Ro

; ð4Þ

The distance from the center to the start of the plug region isdefined as a and the distance from the center to the end of theplug region is defined as b. Explicit equations for a and b are

a

¼

s y ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi

s2 y þ

d P

dl

2

R2ok

2

" #v uut

d Pdl

;

ð5

Þ

b ¼s y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffis2

y þ d P

dl

2

R2ok

2

" #v uut d P

dl

: ð6Þ

Instead of using the concept of the hydraulic radius, the inner shear region is evaluated as if the velocity profile near the inner region is symmetrical. A representation of the approach andanalysis of the flow of YPL-fluid-velocity profile is presented inFig. 5a. The same approach is followed for outer shear region.The Ryan and Johnson (1959) stability criterion is initiallydefined for the symmetrical-velocity profiles. Two (one for theinner and one for the outer region) hydraulic radii are defined tofollow a similar approach with the original parameter and evalu-ate the shear regions separately because a distinct behavior isobserved in the literature for each shear region. They are givenin Eqs. 7 and 8:

r H ;i ¼ b þ a 2 Rið Þ; ð7Þ

r H ;o ¼ 2 Ro a bð Þ: ð8Þ

Combining the information so far, the modified Ryan andJohnson (1959) stabiliy criterion for the flow of YPL fluids inannuli is proposed as

Z inner r ð Þ ¼ r H ;iqV r ð Þsw;i

@ V r ð Þ@ r

; ð9Þ

Z outer r ð Þ ¼ r H ;oqV r ð Þsw;o

@ V r ð Þ@ r

: ð10Þ

To evaluate Eqs. 9 and 10, the numerical solution for the lami-nar flow of YPL fluids in concentric annuli can be used (Erge2013). No analytical solution exists for laminar or turbulent flow of YPL fluids in concentric annuli. To obtain a solution for laminar flow, four equations with four unknowns must be solved numeri-cally to obtain the pressure gradient at a given flow velocity. An an-alytical integration is not possible because of the yield-stress term.

A numerical integration and an iterative procedure are necessary toobtain a solution for YPL-fluid flow in a concentric annulus.

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. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .




The velocity profile for the flow of YPL fluids in concentricannuli is given as

z;ð I Þ ¼ð r

Ri

r

2 K

d P

dl R2

ok2

2rK

d P

dl s y

K

1

m

dr ; ð11Þ

z;ð II Þ ¼ ð Ro

r

r

2 K

d P

dl þ R2

ok2

2rK

d P

dl s y

K 1

m

dr ; ð12Þ

where z;ð I Þ defines the velocity profile in the z axis of the inner region, which is from inner pipe to the plug region; and z;ð II Þ isthe velocity profile in the z-axis of the second shear region, whichis from the plug to the outer wall. Derivation of Eqs. 11 and 12can be found in a previous work (Erge 2013); a similar solution isgiven by Ahmed and Miska (2009). By use of the proposed Eqs. 9and 10, an early transition occurs at the outer region for mostcases. This phenomenon shows good agreement with the experi-mental measurements on the stability of the flow of Newtonian

and YPL fluids that are present in the literature. Detailed informa-tion on the modification derivation, comparisons with the experi-ments and the models in the literature (Metzner and Reed 1955;Hanks 1963; Maglione 1995; Desouky and Al-Awad 1998; Douet al. 2010), and applicability are available in the literature (Erge2013). A sample schematic for the stability parameter, shear stressdistributions, and velocity profile for the flow of YPL fluidsthrough concentric annuli is presented in Fig. 5b.

Flow Stability in Eccentric Annuli. The proposed stability crite-rion in Eqs. 9 and 10 is extended to eccentric annuli by use of anumerical solution of laminar flow of YPL fluids in eccentricannuli (Luo and Peden 1990; Hashemian 2005). Sample 3D ve-locity profiles are obtained by use of a numerical solution andLuo and Peden’s (1990) approximation, and are given in Figs. 6aand 6b, respectively. After obtaining the 3D velocity profile, the3D stability-parameter profile can be obtained and laminar/turbu-lent transition can be predicted.

The narrow-slot approximation may be used for practical pur-poses to obtain a Z -profile. However, it will neglect the effect of

. . . . . . . . . . .

. . . . . . . . .

Drillpipe

Drillpipe

1.5

1

0.5

0

0 0.2 0.4

(b –a )/2+a –Ri (b –a )/2+Ro –b

X = (r –Ri )/(Ro –Ri )

V / V b

(a –Ri ) (Ro –b )

(b –a )

0.6 0.8

Ri

a

λ Ro

b

1

Velocity profile

Borehole wall

Borehole wall

Stability-parameterdistribution Z (r )

Shear stress

distributionτ rz (r )

Velocity profileυ z (r )

Z ml Z ml

Z mo

Z mo

Ro

r

z

+τ y

–τ y

Fig. 5—(a) Analysis of laminar velocity profile of YPL fluids in concentric annuli. (b) Laminar flow of YPL fluids in concentricannuli, velocity profile, shear-stress distribution, and stability-parameter distribution.

1.4

1.2

1

0.8

0.6

0.4

0.2

00.04

0.030.02

0.010–0.01

X (m) Y (m)X (m)

Y (m)

V ( m / s )

1.4

1.2

1

0.8

0.6

0.4

0.2

00.04

V ( m / s )

–0.02

0.030.02

0.010–0.01

–0.02–0.03

–0.03

–0.02

–0.02

–0.01

–0.010 0

0.01

0.01

0.02

0.02

0.03

0.030

0.2

0.4

0.6

0.8

1

1.2

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 6—3D velocity profile with R o ¼ 0.0254 m, R i ¼ 0.0127 m, sy ¼ 5 Pa, K ¼ 1 Pa sm

, m ¼ 0.5, V ¼ 0.35m/s, and q¼ 1000 kg/m3

obtained by use of (a) conformal-mapping technique (Hashemian 2005) and (b) Luo and Peden (1990) approach.




degree of curvature and only a rough approximation can beobtained. The Z -parameter and gradient obtained by use of thenarrow-slot approximation is given as

Z yð Þ

¼qh2m

2sw y

h s y

1

m 2sw y

h s y

m þ

1

m sw s y

m þ 1

m

24 352 K

2

ms2w m þ 1ð Þ

;

ð13Þ@ Z yð Þ@ y

¼ 0

¼qh

2sw y

h s y

2

mm þ 2ð Þ 2sw y

h s y

1 m

msw s y

m þ 1

m

24

35

K

2

msw m þ 1ð Þ:

ð14ÞTo find the maximum of the stability-parameter’s profile, Eq.

14 should be solved for y. Because the equation is nonlinear, anexplicit solution for y cannot be obtained, and therefore a numeri-cal evaluation is necessary. Once the value of y where Z becomesmaximum is obtained, it is inserted into Eq. 13. This way, themaximum Z -value is obtained for a given fluid’s parameters, aswell as the outer and inner diameters of the wellbore.

Previously, an approximation of the Reynolds numbers ( Re)for the onset and offset of transition was presented by Guillot andDenis (1988) and Guillot (1990). It is widely used and also recom-mended in API Spec. 10A (2010) and API RP 13D. The general-

ized flow-behavior index is implemented by Founargiotakis et al.(2008) into equations, which are given as

Re1 ¼ 3;250 1;150 N ; ð15Þ

Re2 ¼ 4;150 1;150 N : ð16Þ

Correlations are developed for the onset and offset of transi-tion, including the effects of eccentricity and diameter ratio. Eqs.13 and 14 are used to develop the correlation for predicting theonset of the transitional flow. Luo and Peden’s (1990) approach isselected for eccentricity. It should be noted that the use of a nu-merical solution for concentric or eccentric annuli can result in alarge computational time depending on the convergence and thenode count. It can take up to several hours with a common com-

mercial hardware. However, the readers are encouraged to use thenumerical solution together with the proposed approach for more-

accurate results. The correlation to predict the offset of transi-tional flow is developed by use of the experimental data collectedduring this study.

By use of all this information, practical correlations are con-structed by use of a nonlinear regression analysis. The followingcorrelations are proposed:

Re1 ¼ 2;100 ½ N 0:331 1þ 1:402j 0:977j2 0:019e N 0:868j;

ð17Þ Re2 ¼ 2;900 ½ N 0:039 Re0:307

1 ; ð18Þ

where

j ¼ Di

Do

; ð19Þ

e ¼ 2 E

Do Di

: ð20Þ

Generalized flow-behavior index ( N ) can be estimated as pre-sented in Appendix A. Eq. 17 reduces to Re1 ¼ 2; 100 for e ¼ 0,

N ¼ 1, and j ¼ 0, which is the critical Reynolds number for theflow of Newtonian fluids through pipes. The suggested interval touse Eq. 17 is 0.1< N < 1. In the drilling industry the 0.1 > N caseis not practiced. Eq. 17 is developed by use of the analytical workon the Ryan and Johnson (1959)’s criterion proposed in this study,

therefore it is expected to show fair agreement with the shear thin-ning fluids such as Bingham plastic and power law. Because notheoretical study exists on the offset of transition, instead, experi-mental and field measurements are mostly used to construct thecorrelations. Eq. 18 is developed by use of experimental datafrom the eight YPL fluids tested in this study. The recommendedinterval to use Eq. 18 is 0.15 < N < 0.4, and it should be notedthat only the tests with diameter ratio of 0.5 is used while con-structing Eq. 18.

Comparison of the correlation with the model is shown inFig. 7. Critical Reynolds numbers obtained with the correlation,given in Eq. 17, are compared with the critical Reynolds numbersestimated by the model using Eqs. 13 and 14. Approximately5,000 data points are generated by use of Eqs. 13 and 14. TheReynolds numbers are calculated by use of the set of equationsgiven in Appendix A, Eqs. A-1 through A-6. These points arecompared with the correlation’s predictions. The results showgood agreement between the correlation and the model when thecritical Reynolds number is greater than approximately 1,500,which is usually encountered in actual drilling operations.

Flow ThroughAnnuliWith Deflected Drillstrings

Effect of Rotation Speed. Free-drillstring-rotation tests wereconducted in the no-compression state. Because there was no pre-fixed eccentricity imposed on the drillstring, most of it was fullyeccentric, lying on the lower side of the outer pipe. Detailed infor-mation on the experimental results can be found in the literature(Erge et al. 2014; 2015a, 2015b); rotation tests are conducted sim-ilarly for the yield-power-law (YPL) fluid numbers 1, 2, 5, and 8,and similar results are observed.

The pressure loss vs. velocity curves for YPL3, YPL4, andYPL8 are plotted in Figs. 8a, 8b, and 9, respectively. For YPL3,the flow-rate range corresponds to Reynolds numbers of 200– 5,000. YPL4 covers Reynolds numbers of 60–4,900 and YPL8 isin the range of 30–5,200. These figures show that for the laminar region, increased pressure loss is mostly observed as rotationspeed increases. In the transition region, rotation of the drillstringcauses an increase in pressure losses, and when the flow becomesturbulent, the effect of rotation diminishes. In Fig. 9b, Fanningfriction factor vs. Reynolds number plot of YPL8 is presented.Also, from these figures it can be observed that rotation of theinner pipe causes an earlier transition to turbulent flow comparedwith the no-rotation case. The slope changes in Figs. 9a and 9b

show the critical Reynolds number for the no-rotation case is1,174, whereas the critical Reynolds number is 788 with rotary

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3,000

2,500

2,000

1,500

1,000

500

00 500 1,000

Critical Reynolds Number Estimated by the Model

C r

i t i c a l R e y n o l d s N u m b e r E s t i m a t e d

b y t h e C o r r e l a t i o n

1,500 2,000 2,500 3,000

–12.5%

+12.5%

Fig. 7—Comparison of the proposed correlation with the Ryanand Johnson (1959) stability criterion with narrow-slot and theLuo and Peden (1990) approximation.




speed of 120 rev/min. The Reynolds number can be calculated byuse of the set of equations given in Appendix A, specifically Eqs.A-1 through A-6. The friction factors can be calculated as

f ¼ r H

qV 2d P

dl ; ð21Þ

in which r H

is

r H ¼ Ro Ri: ð22Þ

The most pronounced effect of rotation is observed in the tran-sition region, which is also the region mostly observed in drillingapplications.

In Fig. 8a at approximately the velocity of 2.05 m/s and in Fig.8b at 1.7 m/s, the flow is transitional and the rotation of the inner pipe showed a significant increase in pressure losses as the pipe isrotated. An approximately 23% increase in pressure losses ismeasured when the pipe is rotated with 120 rev/min with YPL4.The rotation caused an earlier transition to the turbulent regionbecause of added shear with the rotation of the inner pipe. YPL4is more shear thinning compared with YPL3, and YPL4’s results

showed a more-significant effect of the pipe rotation. Similarly,YPL8 shows up to a 30% increase in pressure losses when theinner pipe is rotated. This effect is observed with the rest of thefluids as well. Fluids with high yield stress, consistency index,and low-flow-behavior index showed significant rotation effectson pressure losses.

The dashed lines in Figs. 8 and 9 show the onset and offset of the transition that is determined from the pressure-loss curves bydetermining the deviations and slope changes of the experimentaldata. The lines are drawn dependent on the pressure-loss curve of the nonrotating pipe.

Laminar and transition regions of YPL6, which corresponds toReynolds numbers of 40–3,100, and YPL7, which covers theReynolds numbers of 20–2,800, are shown in Figs. 10a and 10b,respectively. Rotating the drillstring causes a decrease in pressurelosses at lower Reynolds numbers for YPL6 (40–160) and YPL7(20–300). Again, it should be noted that there are several forcesthat counteract each other, such as shear thinning and inertial

effects. A shift is observed at approximately 0.2m/s for YPL6(Reynolds number of approximately 80) and 0.4m/s for YPL7(Reynolds number of approximately 250), and the shift is attrib-uted to the observed increase in inertial effects surpassing theshear thinning ability. After these critical velocities, gradualincreases in pressure losses are observed as the rotation speedincreases. For this case, increase in pressure loss is attributed tothe additional flow disturbance caused by the observed wobblingand lateral motion of the drillstring. Fig. 10 verifies observationsin the literature, suggesting annular frictional pressure losses candecrease or increase with increasing rotary speed.

Effect of Buckling Configurations. The pressure loss vs. veloc-ity plots for YPL2 and YPL4 are shown in Figs. 11a and 11b,

respectively. The flow-rate range corresponds to Reynolds num-bers of 1,800–5,500 for YPL2 and 60–4,900 for YPL4. The plotssuggest a significant decrease in pressure loss when compressionis started. In Fig. 11b, a 27% decrease in pressure loss is recordedat approximately 1.7 m/s, when the pipe is compressed to the heli-cal mode from the free mode. The separation in the pressure-losslines increases as the axial velocity increases, comparing the non-compressed and the compressed drillstrings. From Fig. 11, it canbe concluded that drillstring buckling can cause a significantreduction in frictional pressure losses, and as the bucklingincreases, a further reduction occurs. Similar results are reported

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

00

2000

4000

6000

8000

10 000

Onset of transition Offset of transition Onset of transition Offset of transition

1 2 3

V (m/s)

d P / d l ( P a / m )

d P / d l ( P a / m )

0

2000

4000

6000

8000

10 000

4 5 0 1 2 3

V (m/s)

4 5

Fig. 8—Measured pressure loss vs. velocity rotation tests for (a) YPL3 with Re 1 ¼ 1,042 and Re 2 ¼ 2,092; (b) YPL4 with Re 1 ¼ 1,167and Re 2 ¼ 2,361. Onset and offset of transition are given for nonrotating inner pipe.

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

0

0

2000

4000

6000

8000

10 000Onset of transition Offset of transition Onset of transition Offset of transition

1 2 3

V (m/s)

d P / d l ( P a / m )

F a n n i n g F r i c t i o n F a c t o r

4 50.001

0.01

0.1

110 100 1000 10 000

Reynolds Number

Fig. 9—(a) Measured pressure loss vs. velocity rotation tests for YPL8. (b) Fanning friction factor vs. Reynolds number rotationtests for YPL8 with Re 1 ¼ 1,174 and Re 2 ¼ 2,692. Onset and offset of transition are given for nonrotating inner pipe.




in the literature (Haciislamoglu and Cartalos 1994; Haciislamogluand Cartalos 1996; Asafa and Shah 2012; Taghipour et al. 2013).

Effect of Rotation Speed With a Buckled Drillstring. The rota-tion test for YPL3 in the sinusoidal mode is shown in Fig. 12a.The most pronounced effect is observed at the transition region.Similar results are obtained for other buckling modes, and therotation test in helical-buckling configuration is presented in Fig.12b. In Fig. 12, the measured pressure-loss lines show greater sep-aration in the transition region and the separation increases as theamount of compression increases. With this information, com-pression is found to magnify the effect of rotation on frictionalpressure losses. This behavior depends on more-dynamic motionof the inner pipe as compression increases. Buckling and rotation

tests are conducted for most of the fluids in the test matrix(Erge 2013).

Model Development. Narrow-slot calculation procedure is modi-fied to include the transitional flow and drillstring configurations,such as fully eccentric and buckled geometries. A step-by-stepcalculation procedure is given in Appendix A and a simple exam-ple is presented in Appendix B. To implement the eccentricityinto the approximation, the Haciislamoglu and Langlinais (1990)and Haciislamoglu and Cartalos (1994) correlations are selected.Onset and offset of transition region is determined by the pro-posed correlation. A transitional-flow-friction factor is estimatedby a linear interpolation between laminar- and turbulent-frictionfactors (Founargiotakis et al. 2008; Kelessidis et al. 2011). The

0

2000

4000

6000

8000

Helical, pitch length = 6.9 m

10 000

d P / d

l ( P a / m )

0

2000

4000

6000

8000

10 000

d P / d

l ( P a / m )

0 1 2 3

V (m/s)

4 5 0 1 2 3

V (m/s)

4 5

Onset of transitionOnset of transition

Offset of transitionOffset of transitionNo compression


Sinusoidal

Transitional


No compression


Sinusoidal

Transitional

Fig. 11—Measured pressure loss vs. velocity without rotation for various buckling configurations for (a) YPL2 with Re 1 ¼ 1,351 andRe 2 ¼ 2,674; (b) YPL4 with Re 1¼ 1,167 and Re 2 ¼ 2,361.

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

500

1000

1500

2000

2500Onset of transition Onset of transition

V (m/s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

V (m/s)

d P / d l ( P a / m )

0

500

1000

1500

2000

2500

d P / d l ( P a / m )

Fig. 10—Measured pressure loss vs. velocity, rotation tests of laminar and transition region for (a) YPL6 with Re 1 ¼ 1,741; (b) YPL7with Re 1 ¼ 1,212. Onset and offset of transition are given for nonrotating inner pipe.

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

0

0

2000

4000

6000

8000

10 000

Onset of transitionOffset of transition

1 2 3

V (m/s)

d P / d l ( P a / m )

4 5

0 rev/min

30 rev/min

60 rev/min

90 rev/min

120 rev/min

0

0

2000

4000

6000

8000

10 000

Onset of transition Offset of transition

1 2 3

V (m/s)

d P / d l ( P a / m )

4 5

Fig. 12—Measured pressure loss vs. velocity rotation tests for YPL3 with Re 1 ¼ 995 and Re 2 ¼ 1,998 for (a) sinusoidal buckled innerpipe; (b) helically buckled inner pipe with Re 1 ¼ 1,003 and Re 2 ¼ 2,021.




annulus is divided into many cross sections in the z-direction.Each is evaluated separately and averaged to obtain an accuratepressure-loss prediction. The approach is presented in Fig. 13a.Optimized friction-factor concept is proposed to evaluate the fric-tional pressure losses in various annular geometries:

f O ¼ cð Þ f Mod: NS:; ð23Þ

where c constants are presented to link the friction factors withmodified narrow-slot calculation to optimized friction factors for the annular frictional pressure losses of YPL fluids through free or buckled drillstrings, including the rotation effects.

By use of the experimental data, the following correlations areproposed for practical estimation of the annular frictional pressure

losses with free and buckled drillstrings, including the rotation of the inner pipe.

The correlations are developed by use of the tables given in thestudy of Erge (2013). The tables contain extensive information of theexperiments conducted. The results do not necessarily follow a mo-notonous trend because of counteracting inertial effects (caused byrotation, lateral vibration, and irregular motion of the drillstring) andshear thinning ability of the fluid. A nonlinear regression analysis isconducted and the best-matching equation and constants are deter-mined. The coefficient of determination obtained with the analysisresulted in lowest R2 ¼ 0:65, highest R2 ¼ 0:99, and the average of all the points used to develop the correlations resulted in R2 ¼ 0:83.The correlations are developed for practical purposes, and for in-creased accuracy one can refer to the original tables in Erge (2013).In this study, Eqs. 24, 25, and 26 are developed by use of the experi-

mental data, and where the diameter ratio of 0.5 is investigated, thesuggested interval to use in the equations is 0.15< N < 0.4.

For laminar flow,

c ¼ 0:2287 N 0:0580 Fd þ 0:1237 xd þ 0:4289:

ð24ÞFor transitional flow,

c ¼ 1:0267 N 0:0096 Fd þ 0:0390 xd þ 1:2422:

ð25ÞFor turbulent flow,

c

¼ 1:7821

N

0:0132

Fd

þ 0:1388

xd

þ 1:7983;

ð26Þ

where the dimensionless force is defined as the ratio of the effectiveforce to the force required to buckle the drillstring sinusoidally:

Fd ¼ F

Fs

; ð27Þ

where the dimensionless rotation rate is

xd ¼ rev=min

500 c=min: ð28Þ

The force required to sinusoidally buckle the drillstring can beestimated by use of the equation (Ahmed and Miska 2009)

Fs ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi EIw psina

r cr

: ð29Þ

Comparisons With Literature. The proposed models and the cal-culation procedure are compared with the experimental data fromliterature. Fairly agreeing results are obtained with the Fordhamet al. (1991), Ahmed (2005), Pilehvari and Serth (2009), and Keles-sidis et al. (2011) experimental data for both concentric and eccen-tric annuli. Pilehvari and Serth (2009) rearranged the experimentaldata by Subramanian and Azar (2000) to show laminar, transitional,and turbulent flow. The results are given in Figs. 13b, 14, and 15.

The results showed a good agreement with the experimentaldata of flow of YPL fluids in various geometries such as concen-tric, partially eccentric, and fully eccentric annuli. The lesser agreement in Fig. 15a may be explained by the lesser shear-thin-

ning ability of the test fluid, especially when the flow is turbulent.The observations on the lesser-shear-thinning fluids and the New-tonian fluids such as water suggested lesser effect of eccentricityon pressure losses, which is also observed in the results of experi-ments conducted by Ozbayoglu (2002).

Conclusions

After analyzing the experimental results and the model predic-tions, the following conclusions are made. The pressure-losscurves obtained from the experiments are analyzed on the stabilityof the flow of yield-power-law (YPL) fluids. Pressure-loss curvesshowed an earlier deflection from the laminar region in the eccen-tric annuli, which suggested an earlier transition with increasingeccentricity compared with concentric annuli; this is also reported

in Erge et al. (2015a). Rotation tests are conducted to investigatethe stability of the flow under dynamic conditions. An earlier

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

Proposed model, modified narrow-slot

Kelessidis et al. (2011) data

V (m/s)

0

0

500

1000

1500

2000

2500

0.5 1 1.5 2 2.5

d P / d l ( P a / m )

z

L

di

Ro

Ro

Ri

y

z ε y

ε x

ω

Ri

x y

ε = ε x 2 +

ε y 2

√

(a) (b)

Fig. 13—(a) Wellbore grids. (b) Comparison of the proposed model (modified narrow-slot approximation) with the experimentaldata from literature, concentric annulus, and nonrotating inner pipe for sy ¼ 1.073 Pa, K ¼ 0.0088 Pa sm , and m ¼ 0.8798.




transition is observed when the inner pipe is rotated, and this ear-lier transition is attributed to the added shear and observed lateralvibrations of the inner pipe. As the rotation speed of the inner pipeincreases, the critical velocity to start transitional flow decreaseswith the fluids tested in this study. No significant influence of drill-string buckling is observed on onset or offset of transition.

Experiments on the flow of YPL fluids through annuli sug-gested that annular frictional pressure losses can increase or decrease with increasing drillstring rotary speed. The reason mostfield measurements show an increase is because of dominant iner-tial effects during drilling operations. Experiments showed a sig-nificant increase in pressure losses as the pipe is rotated whilecompressed or free. The examples given in the paper, (e.g., Fig. 8b,a 23% increase, and Fig. 9a, up to a 30% increase in pressure losseswhen the pipe is rotated) suggest that the effect of rotation is an im-portant factor that should be taken into account during the designphase and during the operation to avoid fracturing the formation.Significant differences in frictional pressure losses are observedcomparing concentric, fully eccentric, and buckled configurations.Buckled modes showed potential to significantly reduce frictionalpressure losses. Inner pipe is rotated when it is buckled and com-

pression is found to affect rotation by making it more dynamic,which results in additional disturbances in the flow. The rotationcauses a more-significant increase in pressure losses for the com-pressed and rotating inner pipes compared with the noncompressedand rotating inner pipes. Two pitch lengths are tested, and decreas-ing pitch length resulted in a decrease in pressure losses.

Eight fluids with various yield stresses, consistencies, andflow-behavior indices are tested and laminar, transitional, and tur-bulent regions are investigated. Fluids with higher yield stress andhigher shear-thinning ability showed a more-significant increasein the pressure losses while rotating the inner pipe. The effect of

rotation is more pronounced in the laminar and transition regionwhere the rotation increases the pertubations in flow and causesan earlier transition to the turbulent region. In the turbulentregion, the effect of rotation is insignificant, especially for low-viscosity, low-yield fluids. After a certain fluid velocity, axialflow suppresses the effect of rotation.

Discussion

To our knowledge, this is the first theoretical study that predicts thetransition in an eccentric geometry. The effect of fluid properties,diameter ratio, and the eccentricity of the inner pipe is includedinto the stability parameter, which can predict the flow state whiledrilling horizontally where the pipe is usually eccentric.

A modified narrow-slot procedure is proposed in Appendix A,and a simple example is given in Appendix B on how to use theoptimized friction factors. Slot approximation is widely used for geometries of diameter ratio greater than 0.3, at which the error may be considered acceptable depending on the application. Theapproximation is given for practical purposes, and readers arestrongly encouraged to use the numerical solution given in Eqs. 9

through 12 to obtain more-accurate results.The correlations in Eqs. 18, 22, 23, and 24 are developed byuse of the experimental results, and therefore more experimentswith various diameter ratios, degrees of eccentricity, and wideranges of flow-behavior indices and yield stress are recommendedto obtain more-generalized correlations.

Nomenclature

a, b ¼ geometric parameters, mc ¼ dimensionless correlation coefficient

D ¼ diameter, m

Proposed model, modified narrow-slot Proposed model, modified narrow-slot

Ahmed (2005) dataKelessidis et al. (2011) data

V (m/s)

d P / d l ( P a / m )

d P / d l ( P a / m )

1600

1400

1200

1000

800

600

400

200

00 0.5 1 1.5 2 2.5

V (m/s)

0 0.5 1 1.5 2 2.5 3

70 000

60 000

50 000

40 000

30 000

20 000

10 000

0

Fig. 14—Comparison of the proposed model (modified narrow-slot approximation) with the experimental data from literature: (a)fully eccentric annulus, nonrotating inner pipe for sy ¼ 0.886 Pa, K ¼ 0.013 Pa sm , and m ¼ 0.8343; (b) 80% eccentric annulus, nonrotating inner pipe for sy ¼ 3.5 Pa, K ¼ 3.27 Pa sm , and m ¼ 0.39.

Proposed model, modified narrow-slot Proposed model, modified narrow-slot

Pilehvari and Serth (2009) data Fordham et al. (1991) data

d P / d l ( P a / m )

d P / d l ( P a / m )

1600

1800

1400

1200

1000

800

600

400

200

0

4000

3500

3000

2500

2000

1500

1000

500

0

0 1 2 3 4

V (m/s)

0 0.2 0.4 0.6 0.8 1

V (m/s)

Fig. 15—Comparison of the proposed model (modified narrow-slot approximation) with the experimental data from literature: (a)

fully eccentric annulus, nonrotating inner pipe for sy ¼ 1:05 Pa, K ¼ 0:42Pa sm and m ¼ 0.63; (b) concentric annulus, nonrotatinginner pipe for sy ¼ 1:59 Pa, K ¼ 0:143Pa sm and m ¼ 0.54.




d P ¼ frictional pressure loss, Pad P /dl ¼ frictional pressure loss gradient, Pa/m

E ¼ offset distance, m EI ¼ bending stiffness of a pipe, N m2

f ¼ friction factor F ¼ axial compression force, N

Fs ¼ axial-compression force to start pipe buckling, Nh ¼ height of the slot, m

K ¼ consistency index, Pa sm

m ¼ flow-behavior index N ¼ generalized flow-behavior indexQ

¼ flow rate, m

3 /s

r, R ¼ radius, mr c ¼ radial clearance between a borehole and a pipe, m

Re ¼ Reynolds number V ¼ mean axial-fluid velocity, m/sw ¼ width of the slot, m

w p ¼ unit weight of a pipe in fluid, N/m x ¼ dimensionless yield stressc ¼ shear rate, 1/se ¼ dimensionless eccentricity

l, g ¼ viscosity, Pa sj ¼ diameter ratiok ¼ geometrical constant, mq ¼ density, kg/m3

s

¼ shear stress, Pa

t ¼ velocity, m/sx ¼ angular speed, rad/s

Subscripts

b ¼ bulkbuc ¼ buckledcon ¼ concentric

d ¼ dimensionlessecc ¼ eccentric

h, H ¼ hydraulici ¼ inner

lam ¼ laminar Mod.N.S.

¼ modified narrow slot

p ¼ pipeopt ¼ optimized

o ¼ outer tra ¼ transitionaltur ¼ turbulent

w ¼ wall y ¼ yield

YPL ¼ yield power law

Acknowledgments

The authors of this paper wish to express their appreciation to theUniversity of Tulsa Drilling Research Projects, Det norske oljesel-

skap ASA, and Baker Hughes for their valued support.

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Appendix A—Procedure for Annular-Frictional-

Pressure-Loss CalculationWhileCirculating

Yield-Power-Law (YPL) Fluids

Step 1: Solve equation for sw for a given Q,

Q ¼ wh2

2 K 1

ms2w

m

1 þ 2m

sw s y

1 þ m

m sw þ m

1 þ ms y

;

ðA-1Þwhere

h ¼ Do Di

2 ; ðA-2Þ

w ¼ p

2 Do þ Dið Þ: ðA-3Þ

Step 2: Calculate x from

x ¼ s y

sw: ðA-4Þ

Step 3: Calculate generalized flow-behavior index from(Ahmed and Miska 2009)

3 N 1 þ 2 N

¼ 3m1 þ 2m

1 11 þ m

x m

1 þ m

x2

:

ðA-5ÞStep 4: Calculate ReYPL from

ReYPL ¼ 12qV 2

sw

: ðA-6Þ

Step 5: To estimate whether flow is in laminar, transition, or turbulent region, use the proposed correlations:

Re1 ¼ 2;100 ½ N 0:331 1 þ 1:402j 0:977j2 0:019e N 0:868j;

ðA-7Þ Re2 ¼ 2;900 ½ N 0:039 Re0:307

1 : ðA-8Þ

Step 6: If Re1 > ReYPLð Þ, for laminar flow, then continue bychoosing from Steps 7 through 9.

Step 7: If the drillstring is concentric, calculate the frictionfactor as

f lam:;con: ¼ 24

ReYPL

: ðA-9Þ

Step 8: If the drillstring is eccentric, calculate f lam:;con: by useof Eq. A-9 and multiply it with ½ Rlaminar :

½ Rlaminar ¼ 1 0:072 e

N j0:8454 1:5e2

ffiffiffiffi N

p j0:1852

hþ 0:96e3 ffiffiffiffi

N p j0:2527i; ðA-10Þ

f lam:;ecc: ¼ ½ Rlaminar f lam:;con:: ðA-11Þ

Step 9: If the drillstring is buckled, calculate f lam:;con: by use of Eq. A-9 and multiply it with the corresponding proposed constant,which is given in Eq. 24:

f lam:;buc: ¼ cf lam:;con:: ðA-12Þ

Step 10: Calculate pressure loss for laminar flow:

d P

dl ¼ 2

f qV 2

Do

Di

: ðA-13Þ

Step 11: If ReYPL > Re2ð Þ, for turbulent flow, continue bychoosing from Steps 12 through 14.

Step 12: If the drillstring is concentric, calculate the frictionfactor as

1 ffiffiffiffiffiffiffiffiffiffiffiffiffi f tur :;con:

p ¼ 4

N 0:75 log10 ReYPL f tur :;con:

1 N =2ð Þh i 0:4

N 1:2 :

ðA-14ÞStep 13: If the drillstring is eccentric, calculate f tur :;con: by use

of Eq. A-14 and multiply it by ½ Rturbulent:

½ R

turbulent

¼ 1

0:048

e

N

j0:8454

2

3

e2 ffiffiffiffi N p

j0:1852þ 0:258e3

ffiffiffiffi N

p j0:2527

i; ðA-15Þ

f tur :; ecc: ¼ ½ Rturbulent f tur :; con:: ðA-16Þ

Step 14: If the drillstring is buckled, calculate f tur :;con: by use of Eq. A-14 and multiply it by the corresponding proposed constant,which is given in Eq. 26:

f tur :;buc ¼ cf tur :;con:: ðA-17Þ

Step 15: Calculate pressure loss by use of Eq. A-13.Step 16: If Rec2 > ReYPL > Rec1ð Þ, for transitional flow in a

concentric or eccentric annuli,

f tra: ¼ f lam: þ ReYPL Re1ð Þ f tur : f lam:ð Þ Re2 Re1

: ðA-18Þ

Step 17: If the drillstring is buckled, calculate f tra: by use of Eq. A-18 and multiply it by the corresponding proposed constant,which can be obtained by use of Eq. 25:

f tra:;buc ¼ cf tra:;con: ðA-19Þ

Step 18: Calculate pressure loss by use of Eq. A-13.Here, f lam:;con:, f tur :;con:, and f tra:;con: are referred to as f Mod: NS:,

f lam:;buc:, and f tur :;buc:, and f tra:;buc: is referred to as f o. In the generalform, it is

f opt ¼ cf Mod:N:S:: ðA-20ÞAppendixB—Example Calculation

A simple example is presented here that shows how to use the pro-posed c constants. Considering an imaginary well, annular-fric-tional-pressure-loss gradient will be calculated while circulating ayield-power-law (YPL) fluid for a horizontal section, at which thedrillstring is helically buckled. There are only drillpipes helicallybuckled at the section of interest and they are inside the casing.The inputs are: casing¼ 24.45 cm (inner diameter ¼ 22.66cm);drillpipe ¼ 11.43 9.718 cm; bending stiffness ( EI ) of the drillpipe ¼95 865 011N/m

2; flow rate ¼ 3028.3L/min; mud density ¼ 1200 kg/

m2; sy ¼ 6 Pa; K ¼ 1.5 Pa sm; and m ¼ 0.4.Solving Eq. A-1 for sw gives

sw ¼ 21:5 Pa:

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .




Following Appendix A, the dimensionless yield stress in Eq.A-4 is calculated as

x ¼ 0:28:

Eq. A-5 is solved for N . Generalized flow-behavior indexbecomes

N ¼ 0:26:

ReYPL is calculated according to Eq. A-6:

ReYPL

¼ 1887:

As an approximation, the correlations given in Eqs. A-7 andA-8 can be used to determine the onset and the offset of transition:

Re1 ¼ 1;909;

Re2 ¼ 4;913:

Flow is determined as laminar for the case of Re1 > ReYPL.Friction factor for the laminar flow in concentric annuli that isgiven in in Eq. A-9 is calculated as

f lam:;con: ¼ 0:1272:

Dimensionless rotation rate and dimensionless force are calcu-

lated by use of Eqs. 27 through 29:

Fs ¼ 12186 kg;

Fd ¼ 2:978;

xd ¼ 0:3:

To obtain the optimized friction factors, the c constant needsto be calculated. For the case of laminar flow, Eq. 24 is used:

c ¼ 0:3538:

By use of the proposed c constant, the flow in concentricannuli is linked to the flow in helical-buckled geometry:

f lam:;buc: ¼ cf lam:;con: ¼ 0:004:

Finally, the pressure-loss gradient is calculated by use of Eq. A-13:

d P

dl ¼ 243:04 Pa=m:

Oney Erge works as a research engineer at Schlumberger. Hiscurrent research interests include wellbore hydraulics andmanaged-pressure drilling. Erge has authored or coauthoredmore than eight technical papers and holds five patents. He isa member of SPE and holds a master’s degree in petroleumengineering from the University of Tulsa.

Evren M. Ozbayoglu is the Chapman Endowed Wellspring As-sociate Professor in the Petroleum Engineering Department atthe University of Tulsa and is one of the associate directors ofthe University of Tulsa Drilling Research Projects. Previously, hewas a full-time faculty member with the Petroleum & NaturalGas Engineering Department at Middle East Technical Univer-sity. Ozbayoglu’s research interests include non-Newtonian-fluid flow, multiphase-fluid flow, underbalanced drilling, holecleaning, tubular mechanics, horizontal and directional dril-ling, artificial-neural-network applications, and computational

fluid dynamics. He has authored or coauthored more than 80technical papers and more than 40 journal manuscripts.Ozbayoglu is a member of SPE. He holds bachelor’s and mas-ter’s degrees in petroleum and natural gas engineering fromMiddle East Technical University and a PhD degree in petro-leum engineering from the University of Tulsa.

Stefan Z. Miska is currently the Jonathan Detwiler EndowedChair Professor of Petroleum Engineering and Director of TulsaUniversity Drilling Research Projects at the University of Tulsa.Miska’s current research interests are focused on wellborehydraulics, mechanics of tubulars, directional drilling, and rockmechanics. He has taught at the University of Mining and Met-

allurgy, Poland; the Norwegian Institute of Technology; andNew Mexico Tech. In 1992, Miska joined the University of Tulsaas chair of its Petroleum Engineering Department. He wasinvolved in the successful design and development of adownhole, turbine-type motor for air drilling and has beeninstrumental in development of research facilities for wellborehydraulics at simulated downhole conditions. Miska has alsomade contributions to the development of new buckling con-cepts and the axial force transfer in extended-reach drilling.He has published more than 190 technical papers and con-tributed to several books. Miska is involved with SPE in manyways: He has served as technical editor for SPE Drilling and Completion and is a member of the SPE Drilling and Comple-tion Advisory Committee. Miska has been the recipient of twoSPE international awards: the 2000 Distinguished Petroleum En-gineering Faculty Award, recognizing his many years in aca-demia, and the 2004 Drilling Engineering Award. He is also anSPE Distinguished Member. Recently, Miska was awarded thetitle of Professor of Technical Sciences by the President ofPoland. He earned his master’s and PhD degrees from the Uni-versity of Mining and Metallurgy.

Mengjiao Yu is associate professor at the University of Tulsa. Hisresearch interests include drilling engineering, cuttings trans-port, shale stability, wellbore hydraulics, and downhole mea-surement. Yu has authored or coauthored more than 80technical papers. He is a member of SPE and holds a PhDdegree in petroleum engineering from the University of Texasat Austin.

Nicholas E. Takach is a professor of chemistry at the Universityof Tulsa. Takach joined Tulsa University Drilling Research Proj-ects in 1996 and became senior associate director in January2002. His main research interests include the properties of dril-

ling and completion fluids for oil and gas wells. Takach haspublished in both chemistry and petroleum-related journals,and has given presentations in both areas at national andinternational conferences. He is a member of SPE and holds abachelor’s degree in chemistry from California State Polytech-nic University and a PhD degree in inorganic chemistry fromthe University of Nevada, Reno.

Arild Saasen works as technology adviser in Det norske oljesel-skap ASA in Norway. He also holds a position as adjunct pro-fessor in drilling and well fluids at the Department of PetroleumEngineering at the University of Stavanger. Saasen has previ-ously worked as a specialist in fluid technology in Statoil andas a researcher in Rogaland Research (now IRIS). His researchinterests cover a wide range within technology, ranging fromanchor handling of semisubmersible rigs to downhole tools

and drilling and well fluids. Saasen holds a master’s degree influid mechanics from the University of Oslo and a PhD degreein rheology from the Technical University of Denmark.

Roland May is a research engineer at Baker Hughes CelleTechnology Center in Celle, Germany. His research interestsinclude various aspects of fluid dynamics during well construc-tion. May has authored or coauthored multiple technicalpapers and holds several patents related to drilling technol-ogy. He holds a Diplom Ingenieur degree in mechanical engi-neering from the Technical University of Karlsruhe, Germany.

Effects of Drillstring-Eccentricity, -Rotation, and -Buckling Configurations on Annular Frictional...

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