optimal determination herschel-bulkley rheol paramaters-kelessidis

ngineering 53 (2006) 203–224www.elsevier.com/locate/petrol

Journal of Petroleum Science and E

Optimal determination of rheological parameters forHerschel–Bulkley drilling fluids and impact on pressure drop,

velocity profiles and penetration rates during drilling

V.C. Kelessidis a,⁎, R. Maglione b, C. Tsamantaki a, Y. Aspirtakis a

a Mineral Resources Engineering Department, Technical University of Crete, Chania, Greeceb Vercelli, Italy

Received 1 June 2005; accepted 16 June 2006

Abstract

Drilling fluids containing bentonite and bentonite–lignite as additives exhibit non-Newtonian rheological behavior which canbe described well by the three parameter Herschel–Bulkley rheological model. It is shown that determination of these parametersusing standard techniques can sometimes provide non-optimal and even unrealistic solutions which could be detrimental to theestimation of hydraulic parameters during drilling. An optimal procedure is proposed whereby the best value of the yield stress isestimated using the Golden Section search methodology while the fluid consistency and fluid behavior indices are determined withlinear regression on the transformed rheometric data. The technique yields in many cases results which are as accurate as theseobtained by non-linear regression but also gives positive yield stress in cases where numerical schemes give negative yield stressvalues. It is shown that the impact of the values of the model parameters can be significant for pressure drop estimation but lesssignificant for velocity profile estimation for flow of these fluids in drill pipes and concentric annuli. It is demonstrated that verysmall differences among the values of the model parameters determined by different techniques can lead to substantial differencesin most operational hydraulic parameters in oil-well drilling, particularly pressure drop and apparent viscosity of the fluid at thedrilling bit affecting penetration rates, signifying thus the importance of making the best simulation of the rheological behavior ofdrilling fluids.© 2006 Elsevier B.V. All rights reserved.

Keywords: Drilling fluids; Rheology; Herschel–Bulkley; Pressure drop; Velocity profile; Penetration rates

1. Introduction

In oil-well drilling, bentonite is added in drillingfluids for viscosity control, to aid the transfer of cuttingsfrom the bottom of the well to the surface, and forfiltration control to prevent filtration of drilling fluidsinto the pores of productive formations. It is long known

⁎ Corresponding author.E-mail address: [email protected] (V.C. Kelessidis).

0920-4105/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2006.06.004

that above around 120 °C and in conditions of highsalinity, bentonite slurries begin to thicken catastrophi-cally (Gray and Darley, 1980; Bleler, 1990; Elward-Berry and Darby, 1992). Attempts to describe and topredict the gelling tendencies of bentonite suspensionshave not yet resulted in a concise method which couldpredict rheological and filtration properties, given theamount of added bentonite and its physical character-istics. The flocculation of bentonite suspensions at hightemperatures could be resolved with the addition of

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http://dx.doi.org/10.1016/j.petrol.2006.06.004

204 V.C. Kelessidis et al. / Journal of Petroleum Science and Engineering 53 (2006) 203–224

thinners to reduce the rheology of the mixture but manythinners degrade over the same temperature range. Athinner with high thermal stability is lignite (Clark,1994; Briscoe et al., 1994; Miano and Rabaioli, 1994)and recent evidence (Mihalakis et al., 2004; Kelessidiset al., 2005) demonstrated the stabilizing effect of Greeklignite in terms of rheological and filtration control ofbentonite slurries. Their measurements also showed thatthe three parameter Herschel–Bulkley model describeswell the rheology of these bentonite–lignite watersuspensions.

Various rheological models have been proposed todescribe the rheological behavior of bentonite mixtures,particularly for drilling applications. The two parameterBingham plastic model (Bingham, 1922) or the powerlaw model (Govier and Aziz, 1972; Bourgoyne et al.,1991) are used most often because of their simplicityand the fair agreement of predictions with therheograms. The power law model, although useful as afirst correction to Newtonian behavior, it may lead tosubstantial errors if the fluid exhibits yield stress. Othertwo parameter models like the Casson model (Casson,1959; Hanks, 1989) or the Prandl–Eyring model(Govier and Aziz, 1972) have not found wideacceptance. Three constant parameter models havebeen proposed by Herschel and Bulkley (1926), byGraves and Collins (1978), by Gucuyener (1983) and byRobertson and Stiff (1976). More complex fourparameters models (Shulman, 1968; Mnatsakanov etal., 1991) or even five parameter models (Maglione etal., 1996) have also been proposed. Detailed descriptionof the various rheological models proposed andderivation of the appropriate flow equations have beengiven by Bird et al. (1982) and by Maglione andRomagnoli (1999).

The more complex rheological models are deemedmore accurate in predicting the behavior of drilling fluidsthan the two parameter models that are widely acceptedat present. However, there is not wide acceptance andwide application of the more complex models because ofthe difficulty in finding analytical solutions for thedifferential equations of motion and because of thecomplexity of the calculations for the derivation of theappropriate hydraulic parameters such as Reynoldsnumber, flow velocity profiles, circular and annularpressure drops and criteria for transition from laminar toturbulent flow. Simulation of rotational viscometer dataof non-Newtonian fluids appears to be better when alarger number of rheological parameters is used but inthis case, the hydraulic parameters can be obtained onlyby numerical methods for most of the more complexrheological models. As of today, a compromise between

the accuracy in the calculations and the simplicity of theuse is required and the best way to achieve this is with theuse of the Herschel–Bulkley rheological model. Thethree parameter Herschel–Bulkley model has not beenused widely until very recently, although it was not onlyproposed almost at the same time as the Bingham plasticmodel but it also describes most drilling fluid rheologicaldata much better (Fordham et al., 1991; Hemphil et al.,1993; Maglione and Ferrario, 1996; Kelessidis et al.,2005). The reason for the nonfrequent use is thatderivation of the model's three parameters is complex(Nguyen and Boger, 1987; Hemphil et al., 1993).Furthermore, analytical solutions for laminar flow inpipe and annuli are not possible, requiring eithergraphical or trial-and-error solutions (Hanks, 1979;Govier and Aziz, 1972; Fordham et al., 1991). Theadvent of personal computers and their online use in thefield, however, made trial-and-error solutions trivialtasks, hence, more and more investigators opt to useHerschel–Bulkley rheological models in fluid me-chanics computations of drilling fluids (Maglione etal., 1999a; Maglione et al., 2000; Becker et al., 2003). Asearch in the Society of Petroleum Engineers electroniclibrary of scientific articles, covering the period of 1975–2003, resulted in 319 articles having as keywords ‘powerlaw’, 131 articles with keywords ‘Bingham’, 51 articleswith keywords ‘Herschel–Bulkley’, and 16 articles withkeywords ‘Casson’.

Viscometric data reduction procedures applicable tovarious rheological models have been proposed bymany investigators, addressing also some of the inherentproblems associated with data reduction (Krieger, 1968;Darby, 1985; Borgia and Spera, 1990; Yeow et al.,2000). The standard procedure for the estimation of thethree rheological parameters for Herschel–Bulkleyliquids, with rheological equation,

s ¼ sy þ Kγn ð1Þ

where τ, τy are the shear stress and the yield stressrespectively, K, n are the fluid consistency and fluidbehavior indices respectively and γ is the shear rate, isthrough non-linear regression of the viscometric datafrom concentric cylinder geometry. This is normallydone using a numerical package, minimizing the sum oferror squares and judging the goodness of fit through thevalue of the correlation coefficient Rc

2 from thelinearized form of Eq. (1), as in Eq. (2),

lnðs−syÞ ¼ lnK þ nlnðγÞ ð2ÞHowever, non-linear fit to various data in this

laboratory with a numerical package sometimes has

205V.C. Kelessidis et al. / Journal of Petroleum Science and Engineering 53 (2006) 203–224

given the best fit (highest correlation coefficient Rc2) with

negative values for the yield stress, which is mean-ingless. In these situations, the condition τy>0 must beimposed to get meaningful results affecting thus theoptimum determination of all three parameters, whilethere is also a possibility of non-unique solutions.Moreover, there is concern over the use of the correlationcoefficient as an indicator of the goodness of fit becauseit should be used only for linear functions (Helland,1988; Ohen and Blick, 1990). In addition, considerationshould be given to differences of rheological parametersand their effect on subsequent computations, whenchoosing data with similar correlation coefficients (Rc

2),like for example data with Rc

2 =0.95 versus data withRc2 =0.96, or between Rc

2 =0.98 and Rc2 =0.99.

Many times, especially when curve fitting non-linearfunctions, the sum of square errors (Q2) is utilized forthe goodness of fit, which is defined as,

Q2 ¼Xi

ðyi− yiÞ2 ð3Þ

where yi, ŷi are the measured and the predictedquantities. Another method to determine the goodnessof fit of a particular rheological model to rheometric datais the method based on the best index value (BIV). It isdefined as the ratio between the sum of the squares ofthe deviation of the predicted value from the averagevalue y and the sum of the squares of the deviation ofthe measured value from the average value (Maglioneand Romagnoli, 1999).

BIV ¼

Xi

ð yi− yÞ2Xi

ðyi− yÞ2 ð4Þ

The closer the value of BIV to one, the better is thecapacity of the analytical equation (or the rheologicalmodel) to approximate viscometer data. Values of BIVlarger than one indicate tendency for over-predictionwith the model while values lower than one indicatetendency for under-prediction.

The final choice of the particular parameters canalways be questioned when the choice could be, forexample, between data fits with Rc

2 =0.995 or Rc2=0.998,

between Q2=0.9 or Q2=1.1, and between BIV=0.99 orBIV=1.01, as it will be demonstrated later. Most of thetime, two or even more rheological parameter sets canapproximate well and in a similar way rotationalviscometer data, but in turn they could have large effectson the end results. For example, parameters like Reynoldsnumber, velocity profile, pressure drop for the circular and

the annular sections in a drilling circuit can all be largelyinfluenced by this choice and can have desirable orundesirable carry-over effects. For instance, a variationin the velocity profile and the Reynolds number canaffect the cuttings carrying capacity of the drilling fluid.So, the same drilling fluid can be estimated to performbetter if a rheological parameter set is preferred toanother, even though both may have very similar valuesof Rc

2, Q2 or BIV.As most flows during drilling oil and gas wells are

laminar, for which analytical solutions exist for the flowof Herschel–Bulkley fluids in pipes and annuli, althoughnot explicit, effort should be made to determine theimpact of the use of different sets of rheological para-meters, derived with different methodologies, on themain parameters of interest, pressure drop and velocityprofiles, thus providing a more robust indicator for theproper choice of the appropriate rheological parameters.Prior work (Maglione and Romagnoli, 1998; Maglione,1999) has suggested that the variation of the flowbehavior index of the Herschel–Bulkley model could bethe most important factor because it can influence allhydraulic parameters of the drilling hydraulic circuit,from flow regimes, to velocity profiles, to pressure dropand to rates of penetration. It was shown that the lowerthe flow behavior index is, the lower the pressure drop is,in the circular and the annular sections. As the maximumpressure available is normally constant, the gainedpressure drop (due to a lower n) can be used to increasethe flow rate thus providing more hydraulic power at thedrilling bit or allowing the drilling of longer sections.The authors concluded, however, that more studies wererequired to better define and resolve this problem.

The scope of the present work is to propose a differentand optimal methodology to determine the threeHerschel–Bulkley rheological parameters of drillingfluids from concentric cylinder viscometric data, avoid-ing the observed pitfalls of current non-linear regressiontechniques, which may give meaningless negative yieldstress values. Furthermore, the paper aims to demon-strate that the particular choice of the rheologicalparameters, among similarly performing data fit curves,can greatly affect the determination of pressure drop andvelocity profiles of drilling fluids flowing in drillinghydraulic circuits (pipe and annuli) and to present fieldcases of these implications. An evaluation is alsoattempted about the conditions for preferring a particularrheological parameter set, whether it is the best fit of aparticular rheological model to rheometric data or alsothe impact of the specific choice of rheological parametervalues on pressure drop and velocity profile estimationfor flow of drilling fluids during drilling.

Fig. 1. Rheograms of 6.42% w/v bentonite suspensions in water,hydrated (sample S1) and thermally aged at 177 °C for 16 h (sampleS2). The rheological measurements were taken at 25 °C.


2. Experimental data

The rheometric data used in this work were takenfrom an on-going research project on the use of Greeklignite as thinning agent in bentonite suspensions,particularly when exposed to high temperatures (Miha-lakis et al., 2004; Kelessidis et al., 2005), samples S1through S12, as well as from published field data fromdrilling operations reported by Merlo et al. (1995) anddrilling fluid data of Blick (1992), as reported by Al-Zahrani (1997).

The experimental data of Mihalakis et al. (2004) andKelessidis et al. (2005) were taken with water–bentonitesuspensions at 6.42% w/v, either hydrated for 24 h atroom temperature or aged statically in an aging cell for16 h at 177 °C. Various types of lignites from differentplaces in Greece were added at 0.5% w/v and 3% w/v inthe suspension in a similar fashion (hydrated or agedthermally). The bentonite used was a commercialproduct used in oil-well drilling (Zenith©) provided byS&B Industrial Minerals S.A. All samples were preparedfollowing American Petroleum Institute procedures(2000). The samples were agitated vigorously for5 min before testing and measurements were madewith a continuously varying rotational speed rotatingviscometer (Grace, M3500) at two sample temperatures,25 °C and 65 °C and 12 speeds: 3, 6, 10, 20, 30, 60, 100,200, 300, 400, 500 and 600 rpm. The data is reproducedin Appendix B (Table B1).

The data of Merlo et al. (1995), reproduced in TableB2, were derived with drilling fluid from fieldoperations during drilling circulation tests at varioussections of the well, taking fluid samples from the outletof the drilling circuit. The rheometric data were derivedwith a Huxley–Bertran high pressure high temperaturerotational viscometer for samples S13, S14, S15, S16and S17, while data for sample S18 were derived with aFann VG 35 six speed rotational viscometer (Merlo etal., 1995). Al-Zahrani (1997) reports that the rheologicaldata of Blick (1992), samples S19 to S21 reproduced inTable B3, were taken with a rotary viscometer fordifferent suspensions prepared by adding variousquantities of Wyoming bentonite in water.

3. Rheological parameter estimation forHerschel–Bulkley drilling fluids

3.1. Current methodology

Some of the rheometric data, given in Table B1, arepresented in Figs. 1 and 2. The three rheologicalHerschel–Bulkley parameters were derived according to

standard methodology, using non-linear regression withan appropriate numerical package. These values aregiven in Table 1 together with the correlation coeffi-cient, the sum of errors squared and the best index value,as defined above. From the rheograms presented in Figs.1 and 2 above, as well as from the values of therheological parameters listed in Table 1, it is evident thatthe rheological behavior of these suspensions is typicalof yield-pseudoplastic fluids and modeling their rheo-logical behavior as Herschel–Bulkley fluids is morethan appropriate.

There exist, however, situations where application ofnon-linear regression to rheometric data gives as optimalset of the three rheological parameters, a set withnegative values of the yield stress, which of course ismeaningless. Reported data showing this behavior ispresented in Table 2 for data of Kelessidis et al. (2005),of Merlo et al. (1995) and of Blick (1992). There is thepossibility of imposing the condition τy>0 whenderiving the rheological parameters with non-linearregression which, however, leads to non-optimal solu-tions, as will be demonstrated later.

It is evident that the use of a numerical package toapply non-linear regression to rheometric data in orderto derive Herschel–Bulkley parameters is not alwaysthe optimal procedure since the method can lead toinappropriate negative values for the yield stress. Foryield-pseudoplastic fluids, as most bentonite suspen-sions are, a yield value exists either as an engineeringreality or as an inherent fluid property, although thereis considerable dispute about this issue (Barnes andWalters, 1985; Chen, 1986; Hartnett and Hu, 1989;Evans, 1992; Schurz, 1992; De Kee and Chan ManFong, 1993; Barnes, 1999). The yield value of thefluid should be derived first using an appropriatetechnique, followed by the estimation of the other twoparameters with non-linear regression applied to the

Fig. 2. Rheograms of bentonite–lignite–water suspensions (samplesS7 and S12) both thermally aged at 177 °C for 16 h. The rheologicalmeasurements were taken at 25 °C.


linearized form of Herschel–Bulkley rheologicalequation (Eq. (2)).

This could be accomplished through a trial-and-errorprocedure, where various values of τy are assumed andthe original (τ−γ) rheological data is transformed to [ln(τ−τy)− ln γ] data in order to establish the best linearcurve, determined from the correlation coefficient andthe sum of error squares, giving as y-intercept, (ln K),while the slope of the curve will be the flow behaviorindex, (n) (Hemphil et al., 1993; Turian et al., 1997).With this approach, however, there is no guarantee thatthe optimum values have been derived since thedetermination of (K,n) depends on the correct estimationof τy. Turian et al. (1997) found the procedure verytedious and the results very sensitive to the assumedvalue of the yield stress. They further stated that a majorweakness of the Herschel–Bulkley model has been theinability to unambiguously establish the model para-meters since different sets of these values can provideequivalent fits of the experimental data, exactly the kindof behavior described in this work. Furthermore, a set ofHerschel–Bulkley rheological parameters derived witha correlation coefficient of 0.998 could be different than

Table 1Herschel–Bulkley rheological parameters of bentonite–water and bentonite–

Sample τy (Pa) K (Pa sn) n

S1 8.4748 3.4010 0.255S2 11.3025 5.9115 0.264S3 0.0788 2.3861 0.340S4 0.0000 0.2050 0.493S5 0.6751 0.0732 0.700S6 6.3938 0.4498 0.500S7 2.4095 0.1251 0.701S8 1.1843 0.1265 0.643S9 3.4701 0.0313 0.804S10 0.3793 0.0567 0.619

a set of parameters derived with a correlation coefficientof 0.996, and the final choice, which will have an impacton the end result, pressure drop and velocity profiles,can be a dilemma.

The procedure could be improved if a proper initialvalue of τy is assumed, either taken from the originalrheogram or from a semi-log plot, with τy the y-intercept, as shown in Fig. 3. In this semi-log plot of[τ− log(γ)], the last points at low shear rates should lienormally on a straight line, which when extended tovery low values of the shear rate (τ→0) it should crossthe τ-axis at the value of the yield stress, a procedurefollowed by some investigators (Benna et al., 1999;Barnes, 1999). This method, however, is not deemedvery accurate because measurements at very low shearrates may suffer from wall slip effects (Guillot, 1990;Barnes, 1995).

Other investigators have proposed to measureseparately τy by other means, for example utilizing thevane method (Nguyen and Boger, 1987; Alderman etal., 1991; Barnes and Nguyen, 2001) which, although itleads to better determination for the yield stress, it istime consuming and not applicable for drillingoperations.

3.2. New methodology

A new methodology is proposed, amenable tocomputer implementation, which determines the threerheological Herschel–Bulkley parameters avoidingpotential errors of deriving negative yield stress values.The methodology is based on an initial optimaldetermination of τy, using a near optimum form of theGolden Section search, sometimes called the Fibonaccisearch, followed by linear regression of the linearizedform of the Herschel–Bulkley rheological equation. Thetechnique has been proposed by Ohen and Blick (1990)for Robertson and Stiff fluids (Robertson and Stiff, 1976)

lignite–water suspensions (data of Kelessidis et al., 2005)

Rc2 Q2 (Pa2) BIV

6 0.9876 3.8520 0.99805 0.9885 12.6820 0.98837 0.9340 44.8843 0.93430 0.9450 4.1802 0.63991 0.9937 0.7718 0.99271 0.9951 1.2885 0.99452 0.9975 0.9188 0.99806 0.9965 0.5671 0.99615 0.9666 3.3209 0.96876 0.9983 0.0395 0.9982

Table 2Herschel–Bulkley rheological parameters of bentonite–water and bentonite–lignite–water suspensions with negative τy, derived from nonlinearregression

Sample τy (Pa) K (Pa sn) n Rc2 Q2 (Pa2) BIV Source

S11 −0.2685 0.2486 0.5312 0.9957 0.5402 0.9848 Kelessidis et al. (2005)S12 −0.2880 0.2210 0.5841 0.9963 0.7890 0.9960 Kelessidis et al. (2005)S14 −0.0932 1.9580 0.3488 0.9990 0.2595 0.9991 Merlo et al. (1995)S15 −1.1650 2.3990 0.3158 0.9988 0.2850 0.9981 Merlo et al. (1995)S17 −0.6213 1.1400 0.3704 0.9994 0.0739 0.9995 Merlo et al. (1995)S19 −0.0906 1.1371 0.4393 0.9964 1.3320 0.9958 Blick (1992)S20 −3.7262 5.4440 0.3222 0.9982 2.4130 0.9985 Blick (1992)S21 −6.0980 9.2885 0.3208 0.9988 4.7160 0.9996 Blick (1992)


which follow rheological Eq. (5) below, with A, B, C theRobertson–Stiff rheological parameters, and it has beenextended to Herschel–Bulkley fluids in this work.

s ¼ Aðγþ CÞB ð5Þ

The aim is to find the value of the yield stress, τy,which minimizes the error of the difference between thepredicted and the measured shear stress values, in otherwords, to bracket a minimum for a given function f (x)in the interval (a, c). A minimum is known to bebracketed only when there is a triplet of points a<b<c(or c<b<a) such that f (b) is less than both f (a) and f(c) (Press et al., 1992). The function is then evaluated atan intermediate point, x, which is chosen, either betweena and b or between b and c. After evaluating f (x), if f(b)< f (x), the new bracketing triplet becomes (a,b, x),otherwise, if f (b)> f (x), the triplet becomes (b, x, c).The process continues until the bracketing interval istolerably small. The strategy for choosing point b (or x),given (a, c) leads to the Golden Section search, whichstates that the optimal bracketing interval a, b, c has its

Fig. 3. Graphical determination of yield stress, τy, taken as the y-intercept from a semi-log plot of the original rheogram.

middle point b a fractional distance from one end, forexample a, of a value equal to the golden ratio of0.61803 and a fractional distance from the other end of0.38197 (Fig. 4). At each stage of the search of thebracketing interval of the minimum, the next point to betried is the point x which is a fractional distance 0.61803from one end and a distance of 0.38197 from the otherend (Press et al., 1992).

To estimate the three rheological parameters τy, K, n,a combination of iteration, to determine the yield stressusing the Golden Section search method, and a leastsquares fitting of the linearized Herschel–Bulkley equa-tion, is used (Ohen and Blick, 1990). This is accomplishedby first picking an initial interval of search (L, U), with Lthe lower limit and U the upper limit, defined as,

U ¼ sy0 þ tol ð6Þ

L ¼ sy0−tol ð7Þ

where tol is the half width of the search interval and τy0is an initial estimate of the yield stress. The lower limit istaken very close to zero while the upper limit can betaken as 2·τy0. Although the value of τy0 can be anynon-zero value and the procedure will converge, τy0 canbe estimated following the graphical procedure proposedby Robertson and Stiff (1976) as modified by Ohen andBlick (1990). Given a rheological data set of τ−γ, threedata sets are defined, one at the lowest shear stress, τmin

−γmin, one at the highest shear stress, τmax− γmax, whilethe third set is based on the geometric mean shear stress

Fig. 4. Schematic explaining the Golden Section ratio.


from the first two points, that is, (τ)2 =(τmin*τmax), withγ derived by interpolation. A system of three equationswith three unknowns is thus derived as,

smin ¼ sy0 þ K0γn0min ð8Þ

smax ¼ sy0 þ K0γn0max ð9Þs ¼ sy0 þ K0 γn0 ð10Þ

This system can be solved to get the initial value ofτy0. Combination of Eqs. (8)–(10) gives,

smax−smin

smax− s¼ γn0max−γ

n0min

γn0max− γn0ð11Þ

with 0<n0<1. Eq. (11) can be solved easily numericallyto get n0 and it is then used to derive the initial value foryield stress, τy0,

sy0 ¼ sð γ2n0−γn0minγn0maxÞ

γn0ðγn0min þ γn0max−2 γn0Þ þ γ2n0−γn0minγn0max

ð12Þ

The triplet of points (L, τy0, U) is thus establishedfollowing the above procedure. The new points, in thesearch of the minimum, are evaluated using the goldenratio of 0.61803 as,

sy1 ¼ Lþ 0:61803ðU−LÞ ð13Þ

for the point between (L, τy0), while for the pointbetween (τy0, U) the new value becomes,

sy2 ¼ U−0:61803ðU−LÞ ð14Þ

The functional relationship for optimization isprovided by the correlation coefficient Rc

2 of Eq. (2)for the chosen value of τy. Schematically, the relation-ship Rc

2− τy is shown in Fig. 5 for successiveapproximations of τy. It is a unimodal function implying

Fig. 5. Variation of correlation coefficient with assumed values of yieldstress (the case represents real data).

quick convergence. The search interval varies accordingto the conditions,

R2c1 < R2

c2YU ¼ sy1 ð15Þ

R2c1 > R2

c2YL ¼ sy2 ð16Þ

R2c1 ¼ R2

c2YU ¼ sy1; L ¼ sy2 ð17ÞThe whole procedure can be performed with a

numerical package.

3.3. Computational results and comparison withexperimental data

This methodology has been applied to several datasets, given in Tables B1–B3 in Appendix B, and hasproved to work extremely well. In Table 3, therheological parameters that have been calculated usingboth procedures, non-linear regression with a numericalpackage and with the Golden Section technique arepresented for the cases where non-linear regressionprovides meaningful results, in other words, positiveyield stress values. Rheograms of some of the samplesare shown in Figs. 6 and 7 together with the curvesderived from non-linear regression (NL) and for GoldenSection search (GS).

The results show minor differences among the twomethods both in terms of all three rheologicalparameters and the two of the three indices of correlation(Rc

2, Q2) while the third index (BIV) does not reallyreflect the similarities between the estimated parameters.This close agreement demonstrates the ability of the newproposed scheme to properly determine the rheologicalparameters with similar success as the application ofnonlinear regression using numerical packages, for thecases that the latter predict positive yield stress values.

The results show also that for all fluid samples,except sample S8, the flow behavior index, n,determined with the GS method, is smaller whencompared to the value determined with the NL method,while the yield point may be smaller or larger. Inpractical terms, and while keeping all other rheologicalparameters constant, a smaller n-value results inflattening of the velocity profile improving the carryingcapacity of the drilling fluid, extending laminar flowregime and decreasing pressure drop (Maglione andRobotti, 1996; Maglione et al., 1999a,b).

The close agreement between the rheological para-meters obtained by the two methodologies raises thepoint about the optimal data set to be used when the

Table 3Comparison of rheological parameters derived from non-linear regression (NL) and by Golden Section (GS)

Sample and method τy (Pa) K (Pa sn) n Rc2 Q2 BIV Source

S7NL 2.4095 0.1251 0.7012 0.9975 0.9188 0.9980 Kelessidis et al. (2005)GS 2.4141 0.1369 0.6842 0.9910 1.3124 0.9401




S13NL 1.7020 1.2063 0.4352 0.9971 1.0847 0.9959 Merlo et al. (1995)GS 0.0000 1.9940 0.3704 0.9952 1.8105 1.0246

S16NL 0.1747 0.9448 0.4097 0.9990 0.1563 0.9989 Merlo et al. (1995)GS 0.0379 1.0200 0.3993 0.9995 0.1636 0.9956

S18NL 2.6750 0.2492 0.6607 0.9982 0.7375 0.9977 Merlo et al. (1995)GS 1.6813 0.6496 0.5173 0.9950 5.1244 0.8035

Positive yield stress values.


choice is among values giving very similar data fit.Furthermore it raises questions about the implicationsand the real impact on the operational parameters ofinterest, pressure drop and velocity profiles.

The new methodology has been applied also torheometric data which give negative yield stress valueswhen applying non-linear regression, with the resultsshown in Table 4. The listed rheological parameters arederived using three different techniques: application ofnon-linear regression (NL, numerical-no penalty),application of non-linear regression with imposition ofτy>0 (NLP, numerical with penalty) and application ofthe Golden Section search (GS). Imposing τy>0 may

Fig. 6. Comparison of rheograms of original data with results fromrheological models for bentonite–lignite–water suspensions.

sometimes result in non-optimum solutions givinglower correlation coefficients, much higher sum ofsquared errors and values of best index value signifi-cantly different than one, compared to cases of non-linear regression but without this imposition and tocases of Golden Section search methodology (samplesS11, S12, S15). For some of the cases though, NLPmethod may result in as good predictions as the GoldenSection search (samples S14, S17, S19, S20 and S21).The new proposed scheme provides good solutions, asevidenced by the sum of squared errors, the correlation

Fig. 7. Comparison of rheograms of original data with results fromrheological models for drilling mud.

Table 4Comparison of rheological parameters derived from numerical package and by proposed scheme, non-optimal solutions

Sample and method τy (Pa) K (Pa sn) n Rc2 Q2 BIV Source

S11NL −0.2685 0.2486 0.5312 0.9957 0.5402 0.9948 Kelessidis et al. (2005)NLP 0.0000 0.2326 0.5166 0.9590 5.1540 0.9149GS 0.0718 0.1462 0.6068 0.9942 0.7865 1.0263

S12NL −0.2880 0.2210 0.5841 0.9963 0.7890 0.9960 Kelessidis et al. (2005)NLP 0.0001 1.0493 0.3218 0.9160 34.15 0.4435GS 0.3976 0.0940 0.7036 0.9942 1.5646 0.9881

S14NL −0.09323 1.9580 0.3488 0.9990 0.2595 0.9991 Merlo et al. (1995)NLP 0.0000 1.8900 0.3530 0.9990 0.2627 0.9938GS 0.0000 1.9050 0.3523 0.9990 0.2608 0.9987



S19NL −0.0906 1.1371 0.4393 0.9964 1.3320 0.9958 Blick (1992)NLP 0.0000 1.1170 0.4414 0.9964 1.3380 0.9930GS 1.4701 0.6234 0.5203 0.9927 2.2244 1.0398

S20NL −3.7262 5.4440 0.3222 0.9982 2.4130 0.9985 Blick (1992)NLP 0.0000 3.6872 0.3683 0.9974 3.3610 0.9890GS 0.0000 3.5776 0.3739 0.9975 3.7578 1.0204

S21NL −6.0980 9.2885 0.3208 0.9988 4.7160 0.9996 Blick (1992)NLP 0.0003 6.3864 0.3650 0.9980 7.2000 0.9900GS 0.0000 6.1803 0.3712 0.9987 8.4888 1.0262

(NL) is the application of non-linear regression, (NLP) is the application of non-linear regression with the imposition of τy>0 and (GS) is theapplication of the new technique.


coefficients and the best index values. The suitability ofthe proposed scheme is further demonstrated in Figs. 8–12, where the rheograms for some of the samples of

Fig. 8. Comparison of rheograms of original data with results fromrheological models for bentonite–lignite–water suspensions (sampleS11).

Table 4 are given for the various approaches. Similargraphs produced for the rest of the samples have furtherdemonstrated the suitability of the proposed technique.

Fig. 9. Comparison of rheograms of original data with results fromrheological models for bentonite–lignite–water suspensions (sampleS12).

Fig. 10. Comparison of rheograms of original data with results fromrheological models for drilling mud (sample S15).



From the data reported in Table 4 it is evident thatthe values of the flow behavior index, determinedwith the three different methods, do not differsignificantly except for sample S12. The maindifference is the yield point which can affect bothcuttings carrying capacity of drilling fluids andpressure drop. The results of Tables 3 and 4 showthat when the fluid shows a positive yield point,when determined by non-linear regression, the GSmethod determines quite different rheological para-meters with the larger difference observed on thevalue of the flow behavior index. When the fluidshows negative value of the yield point, when derivedby non-linear regression, the only difference derivedwhen using the GS method is on the yield pointitself.

3.4. Discussion

The results presented above have indicated thatoptimal determination of the rheological parameters ofdrilling fluids described by the Herschel–Bulkley


rheological model is not an easy task. There will becases where application of non-linear regression withavailable numerical packages can give estimates of thethree rheological parameters with high degree ofconfidence. Caution, however, should be exercisedwhen these techniques give meaningless negativeyield stress values. Imposition of the condition thatyield stress is positive may lead to significant differ-ences of the estimated rheological parameters and thechoice of an inappropriate rheological model, other thanthe Herschel–Bulkley model which might have beenderived if determination of the rheological parameters isperformed following the Golden Section search meth-odology proposed here.

It has been demonstrated that the proposedtechnique, which relies on the proper choice of theyield stress using the Golden Section search methodol-ogy and on the minimization of the sum of squarederrors on the linearized form of the Herschel–Bulkleyrheological equation, can lead to meaningful andappropriate values of all three rheological parameterswith high degree of confidence. The techniqueeliminates the ambiguity and tediousness in findingthe correct parameter set, evident by previous usedmethodologies.

Data fitting with the appropriate model and theapplication of the most applicable technique alwaysposes the question about a relevant index of “correct-ness”. There are concerns over real differences betweenparameters derived with curve fitting giving similarcorrelation coefficients, as for example, differencesbetween Rc

2 =0.97 and Rc2 =0.98. The values in Tables 3

and 4 indicate that the differences in Herschel–Bulkleyparameters can sometimes be very subtle and sometimesvery large, but the impact on the main parameters ofinterest, pressure drop and velocity profiles is not reallyknown nor it has been investigated thoroughly in thepast.

Fig. 13. Pressure drop-flow rate graph for the three fluids, withrheological parameters determined by Golden Section (GS) and bynon-linear regression with τy>0 (NLP) in a 0.311 m by 0.127 mconcentric annulus. Laminar flow computations.


4. Impact on pressure drop and velocity profile ofrheological parameter estimation by differenttechniques for drilling operations

4.1. Theoretical considerations

The rheological parameters determined by the threetechniques described above, non-linear regression, non-linear regression with imposition of penalty for negativeyield stress and the new methodology utilizing theGolden Section search methodology, have been utilizedto determine pressure drop and velocity profiles intypical oil-well drilling situations. Two pipe geometriesare used, a pipe with internal diameter of 0.0883 m and apipe with internal diameter of 0.1264 m. In addition,three annulus geometries are used, an annulus with aninternal diameter of the outer pipe of 0.4445 m and anexternal diameter of the inner pipe of 0.127 m, anannulus with an internal diameter of the outer pipe of0.311 m and an external diameter of the inner pipe of0.127 m and an annulus with an internal diameter of theouter pipe of 0.216 m and an external diameter of theinner pipe of 0.089 m. Results have been derived for arange of flow rates encountered in oil-well drillingsituations, but keeping the flow laminar, which is thenormal condition encountered in drilling applications,particularly for flow in annulus. The appropriate flowequations for Herschel–Bulkley fluids are summarizedin Appendix A, both for pipe flow and for concentricannulus, utilizing slot flow solutions for the latter, acommon approach in drilling hydraulics, particularly forannulus diameter ratios greater than 0.3 (Bourgoyne etal., 1991; Maglione and Romagnoli, 1999).

Computations of pressure drop-flow rate data setshave been performed for all samples with rheologicaldata presented in Tables B1–B3. Results, however, arepresented here for samples S12, S17 and S19, for whichnegative yield values have been estimated as well as forsome of the samples giving positive yield stress values(samples S7, S9, S10 and S18). The rheologicalparameters were determined with the new methodologyapplying Golden Section search (GS) and with non-linear regression with penalty for τy>0 (NLP) and arelisted in Tables 3 and 4.

In Fig. 13, data is presented for the pressure dropvariation with the flow rate, for flow in the 0.311 mby 0.127 m concentric annulus. For sample S12, thereis a 2 to 3 times variation between the computedpressure drop using rheological parameters derivedwith the GS method when compared with the valuescomputed when the rheological parameters wereobtained according to method NLP. For this sample,

the correlation coefficient for NLP is a low 0.9160(Table 4) while for method GS is 0.9942, hence thevariation observed in Fig. 13 may be partly explainedby the bad fit of the rheogram when using methodNLP. For sample S17, the variation in the predictedpressure drop values with the rheological parametersderived by methods GS and NLP is between 1.5 to 2.5times. For this case, however, the correlation coeffi-cient for the estimation of the rheological parametersis a high 0.9987 for both methods (Table 4). Forsample S19, for which the rheological parameterswere determined with GS have a correlation coeffi-cient of 0.9927 and with NLP have a correlationcoefficient of 0.9964 (Table 4), the difference in thepressure drop predictions is smaller than the previouscases but it is still significant ranging from 10% athigh flow rates to 50% at low flow rates.

The computed velocity profiles for the 0.311 m by0.127 m concentric annulus at various flow rates forsample S19, the sample that showed the smallestvariation in pressure drop values, are shown in Fig.14. Small variations on velocity profiles are observedwith larger variations in plug velocities, particularly athigh flow rates. Larger variations are expected and havebeen determined for the other fluid samples for whichlarger pressure drop variations have been observedwhen using rheological parameters derived with the twotechniques.

Fig. 15 shows the results of the pressure dropcomputations for the smaller annulus and similar trendsare observed as in Fig. 13. For sample S12, the variationbetween the pressure drop values computed with

Fig. 14. Velocity profiles for fluid S19 with rheological parametersdetermined by Golden Section (GS) and by non-linear regression withpenalty (NLP) in the 0.311 m by 0.127 m concentric annulus. Laminarflow computations for five flow rates (379 l/min, 1136 l/min, 1893 l/min, 3028 l/min and 3785 l/min).

Fig. 16. Velocity profiles for fluid S12 with rheological parametersdetermined by Golden Section (GS) and non-linear regression withpenalty (NLP), for the 0.216 m by 0.089 m concentric annulus, for fiveflow rates.


rheological parameters obtained by methods GS andNLP, is two to three times. For sample S17, thisvariation is between 1.5 to 2 times, while for sample S19there are no significant differences for flow rates higherthan 600 l/min. The velocity profiles for sample S12, thesample that showed the largest variation in the estimatedpressure drop values, are shown in Fig. 16, anddifferences are observed both in velocity profiles butalso in maximum (plug) velocities for the computedvalues using the rheological parameters from the twocases.

Fig. 15. Pressure drop-flow rate graph for three fluids, with rheologicalparameters determined by Golden Section (GS) and by non-linearregression with penalty (NLP), for the 0.216 m by 0.089 m concentricannulus. Laminar flow computations.

For pipes, the results for pressure drop estimation areshown in Fig. 17 for the pipe size of 0.1264 m and Fig.18 for the pipe size of 0.0883 m for samples S12, S17and S19, for laminar flow. The variation observed issimilar to the presented cases for the concentric annulus,with significant variation for samples S12 and S17 forboth pipe sizes, while the variation is very small forsample S19 for both pipe sizes.

Similar computations, for the samples for which theyield stress was found positive and with no significantdifferences among the model parameters listed in Table3, have revealed that the differences in pressure drop

Fig. 17. Pressure drop-flow rate graph for three fluids, with rheologicalparameters determined by Golden Section (GS) and by non-linearregression with penalty (NLP), for the 0.1264 m pipe. Laminar flowcomputations.

Fig. 18. Pressure drop-flow rate graph for three fluids, with rheologicalparameters determined by Golden Section (GS) and by non-linearregression with penalty (NLP), for the 0.0883 m pipe. Laminar flowcomputations.


computation are much smaller in this case while thevelocity profiles for the cases of these samples do notvary much. Some of the results for the pressure dropestimation are shown in Fig. 19 for samples S7, S9, S10and S18.

The presented results demonstrate that differences incomputed pressure drop values and velocity profilesexist when computations are performed using either therheological parameters obtained by GS (the mostcorrect one) or by NLP. The variations can be ratherlarge for typical drilling situations not only on pressuredrop but also on velocity profiles. This is particularly

Fig. 19. Pressure drop-flow rate graph for fluids S7, S9, S10 and S18,with rheological parameters determined by Golden Section (GS) andby non-linear regression (NL), for the 0.311 m by 0.127 m concentricannulus. Laminar flow computations.

important for pressure drop estimation because of thesmall margin for frictional pressure drop normallyallowed for flow in the annulus. It is equally importantfor the estimation of velocity profiles which affectcuttings transport and intermixing of fluids in theannulus, where laminar flow conditions normallyprevail. The analysis further demonstrates that selec-tion of rheological parameters based on the goodnessof fit of the rheogram, the correlation coefficient or thebest index value may not always be sufficient.Estimation of the impact of the choice of therheological parameters by an appropriate methodologyshould be made by computing pressure drop andvelocity profiles for typical drilling situations. Thisapproach can be further extended and refined in orderto derive an ‘index of appropriateness’ for estimatingrheological parameters.

4.2. Implications on drilling operations

The implications on drilling from use of differentrheological parameters but from same rheological dataset have been assessed with field data of Merlo et al.(1995). An analysis is presented for the circulationtest performed in well A at the depths of 555 m and2008 m with the flow geometry shown in Fig. 20.The circulation test at 555 m depth is deemed as themost reliable in terms of data analysis because thedrill pipe was located inside the casing with anominal diameter of 0.508 m (inside diameter of0.4826 m) with a well-known geometry and becausethe circulating drilling fluid is not much affected by

Fig. 20. Geometry of well A for the reported drilling circulation test at3200 l/min. Relevant geometrical data are given by Merlo et al.(1995).

Fig. 22. Velocity profiles of NL and GS fluids for flow of the S13drilling fluid in a 0.483 m by 0.127 m concentric annulus, for a flowrate of 3202 lpm.


the temperature of the well. For purposes ofsimulation, the S13 sample was considered, forwhich the rheological parameters have been deter-mined by using both non-linear regression (NL) andGolden Section method (GS), which exhibit smalldifferences in the rheological parameters but verysimilar correlation coefficients, sum of square errorsand best index values (Table 3).

The calculated value of pressure drop in the drillinghydraulic circuit (excluding the drilling bit), using themethodology of Merlo et al. (1995), is estimated as44.000 Pa for the NL fluid and about 37.800 Pa forthe GS fluid. Measured field data gave a value of29.100 Pa (Merlo et al., 1995). From the above data itcan be seen that the calculated pressure drop in thedrilling circuit is 29.9% and 51.2% more than the fieldmeasured value when using the GS fluid or the NLfluid respectively. Furthermore, the pressure dropcalculated by using the GS fluid is less than thepressure drop calculated with the NL fluid by anamount equal to 16.4% of the GS fluid value,indicating that the GS fluid performs better than theNL fluid, in terms of pressure drop. This holds eventhough both fluids have the same formulation andcharacteristics and the viscometer data are very welldescribed by both methods (Rc

2 = 0.9971 andRc2 =0.9952 for the NL fluid and the GS fluid

respectively). In Fig. 21, the variation of the calculatedfrictional pressure loss in the drilling hydraulic circuitis plotted against the flow rate for the NL fluid andGS fluid, together with measured data.

When the flow velocity distribution in the annulus(0.483 m by 0.127 m) is computed for the two fluids at a

Fig. 21. Pressure drop-flow rate curves for drilling circulation test forWell A, using NL and GS derived rheological parameters for drillingfluid sample S13. The measured data at the one flow rate (3202 lpm) isalso shown.

flow rate of 3202 lpm, the results show similar velocityprofile in the annular section (Fig. 22) meaning that bothfluids can have the same cuttings carrying capacity.Cuttings transport efficiencies, calculated as per Walkerand Mayes (1975) but extended for Herschel–Bulkleyfluids by Gallino and Maglione (1996), are estimated at78.9% and 77.5% for the NL fluid and GS fluidrespectively.

Finally, considering the apparent viscosity of thedrilling fluid at the nozzles of the drilling bit, μa,which affects the rate of penetration among otherparameters, differences in the behavior of the NL andGS fluids are expected which can be evaluated. Theapparent viscosity at the drilling bit can becomputed, following standard drilling hydraulicsprocedures (Bourgoyne et al., 1991; Maglione andRomagnoli, 1999), by estimating the wall shearstress, τw, and the wall shear rate, γw, of the drillingfluid at the bit nozzles, and computing the apparentviscosity, μa, as,

la ¼swγw

ð18Þ

The apparent viscosity is estimated at a value ofabout 1.707 cP for the NL fluid and at a value of about1.261 cP for the GS fluid for the flow rate of 3202 lpm.Fig. 23 shows the behavior of apparent viscosity at thedrilling bit nozzles for the NL and GS fluids, forvarious flow rates. The results show that the apparentviscosity of the GS fluid has always lower values thanthe values of the NL fluid with the differences varyingbetween 18 and 27%, for the range of the consideredflow rates.

The rate of penetration during drilling (ROP) isdirectly affected by fluid density (ρ), fluid velocity in the

Fig. 23. Variation of apparent viscosity, at the drilling bit, with the flowrate, for the NL and GS fluids, for sample S13 and Well A (concentricannulus of 0.4826 m by 0.127 m).


drilling bit nozzles (Vn), bit nozzle diameter (dn), whileit is inversely proportional to drilling fluid apparentviscosity at the drilling bit (μa,bit) (Eckel, 1967; Beck etal., 1995; Gallino et al., 1996). This variation isdescribed well by Eq. (19),

ROP~qVndnla;bit

!c

ð19Þ

where c is a constant, which depends on the drilledformation. From Eq. (19) one can notice that, keepingall other parameters constant, the rate of penetration isinversely proportional to the apparent viscosity at thebit. Considering for instance the case for well Areported above (Merlo et al., 1995), if a value for therate of penetration of 10 m/h is assumed for the NLfluid, the corresponding rate of penetration for the GSfluid would be predicted around 12.6 m/h, an increaseof 26%. This particular example reveals that just byconsidering and studying better the best fit curves ofviscometer data of drilling fluids, a surprising resultwith respect to the rate of penetration can be derived.This significant increase has been computed usingrheological parameters that were very close to eachother (Table 3, sample S13). The results could bemore dramatic, if larger differences occur amongrheological parameters derived using the two differenttechniques.

All the above results could be considered as thedirect effect of the decrease of the flow behaviorindex of the drilling fluid when determined using theGS method and its consequences on hydraulicparameters. It seems that the GS fluid performs betterthan the NL fluid, even though they are the same

fluid, with the only difference in the approachfollowed to determine the best fit rheological para-meters from viscometer data. The variation of therheological parameters affects mostly frictional pres-sure drop and apparent viscosity at the bit. Hence, atthe design phase and during oil-well drilling opera-tions, caution should be exercised when using therheological parameters from the best fit curves of theviscometer data, because, as it has been shown, verysmall differences at the design phase can leadsometimes to substantial differences in most ofoperational parameters.

5. Conclusions

In this study it was shown that the Herschel–Bulkleyrheological model properly described all rheologicaldata of drilling fluids obtained with rotational visc-ometer data. It has been demonstrated that the threerheological Herschel–Bulkley parameters can bederived with a numerical package using non-linearregression but the procedure may not always lead tooptimal solutions, because sometimes, meaninglessnegative yield stress values are determined. Impositionof the condition for positive values of the yield stressgives non-optimal solutions. A methodology has beenproposed to alleviate this problem. It estimates the bestvalue for the yield stress using the Golden Sectionsearch methodology, and then applies linear regressionto the transformed data, yielding as accurate results asthe numerical schemes in normal cases but also givingpositive values for the yield stress in situations wherenumerical schemes determine negative values. Therecommended approach leads to unique solutions andcan be easily implemented.

It was further shown that pressure drop andvelocity profiles for laminar flow in pipes andconcentric annulus can be significantly affected byproper choice of rheological parameters. The mostappropriate set should be determined utilizing not onlythe statistically best fit indices but also the impact onpressure drop and velocity profiles. Use of therelevant flow equations for laminar flow in typicaloil-well drilling situations in concentric annulus andpipe can aid significantly in determining the impact ofthe particular choice of rheological parameters onpressure drop, on velocity profiles and on rate ofpenetration, aiding in choosing the most appropriaterheological parameters. The computed results havedemonstrated that it is very important to make the bestsimulation of rheological behavior of drilling fluidsbefore computing hydraulic parameters. Once the best


rheological parameters are obtained, utilization ofappropriate calculation tools for estimation of pressuredrop and velocity profiles can aid in avoiding manyproblems during drilling operations and can allowbetter exploitation of the properties of existing drillingfluids.

NomenclatureA, B, C Robertson–Stiff rheological parametersBIV Best index valuedp/dL Pressure drop per unit length (M/L2T2)dn Bit nozzle diameter (L)h Gap of annulus, distance between plates in the

slot (L)K Fluid consistency index (M/LT2−n)L Lower limit for the estimation of yield stressn Fluid behavior indexp Pressure (M/LT2)r Radius (L)R pipe radius (L)R1 radius of outer pipe of annulus (L)R2 radius of inner pipe of annulus (L)Rc2 Correlation coefficient

ROP Rate of penetration (L/T)q Flow rate (L3/T)Q2 Sum of error squarestol Half width of search intervalu Liquid velocity (L/T)U Upper limit for the estimation of yield stressVn Liquid velocity at the drill bit nozzle (L/T)ya Distance of top of inner layer from bottom

plate (L)yb Distance of bottom of top layer from bottom

plate (L)yi Measured parameterŷi Predicted parametery Average value of the parameterw Width of the slot (L)

Fig. A1. Geometry and parameters for laminar flow in pipes.

Greek lettersγ Shear rate (T− 1)γ Geometric mean shear rate (T− 1)Δ pressure drop per unit length (M/L2T2)μ Liquid viscosity (M/LT)ξ Dimensionless parameter, ratio of yield stress

to wall shear stressρ Liquid density (M/L3)τ Shear stress (M/LT2)τ Geometric mean shear stress (M/LT2)τy Yield stress (M/LT2)τy0 Initial estimate of yield stress (M/LT2)τo Integer constant

Subscriptsmin minimum valuemax maximum valueo initial valuew walla apparent valuea, bit at the bit

Appendix A. Flow of Herschel–Bulkley drillingfluids in pipes and concentric annuli

A.1. Flow in pipes

The geometry for pipe flow together with theappropriate parameters is shown in Fig. A1. There is acentral core of the fluid which moves as a rigid plug ifthe shear stress levels are smaller than the yield stress ofthe fluid. Letting Δ=dp/dL, with dp/dL the pressuredrop per unit length, and for values of the shear stress τgreater than the yield stress of the fluid τy, τ≥τy,balance of forces gives,

dðsrÞdr

¼ rdpdL

¼ rD ðA1Þ

Eq. (A1) upon integration gives,

s ¼ r2Dþ C1

rðA2Þ

The shear stress is finite at r=0, hence, C1=0. Theradius at which there is an unsheared portion of the fluid,rp, is given by

sy ¼ rp2D ðA3Þ

and the wall shear stress, τw, is given by,

sw ¼ R2D ðA4Þ

Fig. A2. Representation of the concentric annulus as a slot.


where R is the pipe radius. The flow curve of theHerschel–Bulkley fluid is given by,

s ¼ sy þ K −dudr

� �n

ðA5Þ

where K, n are the fluid consistency and flow behaviorindices respectively. Hence,

sy þ K −dudr

� �n

¼ r2D ðA6Þ

The solution of Eq. (A6), utilizing the boundarycondition that the fluid velocity u is zero at the pipewall, u=0 at r=R, is finally given by,

u¼ 1D=ð2KÞ

1mþ 1

D2K

R−syK

� �mþ1

−D2K

r−syK

� �mþ1( )

;

rp ¼ syD=2

VrVR

ðA7aÞ

and

u ¼ up ¼ ðD=2KÞmmþ 1

R−syD=2

� �mþ1

; 0VrVrp ðA7bÞ

with m=1/n.The flow rate q can be derived as,

q ¼ 2kZ R

0

urdr

¼ knK1=n

ðDR=2−syÞ1=nþ1

ðD=2Þ3

� ðDR=2−syÞ21þ 3n

þ 2syðDR=2−syÞ1þ 2n

þ s2y1þ n

" #

ðA8Þ

Eq. (A8) relates pressure drop (Δ=dp/dL) withflow rate (q) for flow of Herschel–Bulkley drillingfluids in a pipe of radius R for laminar flow. If thepressure drop is known, the flow rate can be directlycomputed. On the other hand, if the flow rate isknown, Eq. (A8) can be solved by trial-and-error forthe estimation of pressure drop. The velocity profilescan be computed readily from Eqs. (A7a) and (A7b)for both cases.

A.2. Flow in concentric annuli

For flow in a concentric annulus, the annulus can beapproximated by a slot of gap h=R2−R1, as representedin Fig. A2. The solution that will be derived follows theprocedure for flow of Bingham plastic fluids in a slot,given by Bourgoyne et al. (1991).

There is a central core of the fluid which moves as arigid plug if the shear stress levels are smaller than theyield stress of the fluid. Integration of the force balanceperformed on a fluid element gives

s ¼ yDþ s0 ðA9Þ

with τ0 an integration constant to be determined. Let ya,yb the distances of the lower sheared and upper shearedsurfaces from the bottom plate respectively. At the innerlayer of the plug, ya, the shear stress τa must equal(−τy). It follows then from (A9),

sa ¼ −sy ¼ s0 þ yaD ðA10Þ

which gives,

ya ¼ −sy þ s0

DðA11Þ

Similarly, for the outer plug region one obtains forthe shear stress at yb,

sb ¼ sy ¼ s0 þ ybD ðA12Þ

In the fluid layer enclosed by the inner layer of theplug and the bottom plate, the shear stress is,

s ¼ −sy−Kdudy

� �n

¼ s0 þ yD ðA13Þ


The solution to differential Eq. (A13), utilizingboundary condition u=0 at y=0 is finally given by,

u ¼ −Kðmþ 1ÞD − −

sy þ s0K

� �mþ1

þ −sy þ s0

K−DKy

� �mþ1( )

;

0VyVya ðA14Þ

with m=1/n. In terms of the parameter ya this equationbecomes

u ¼ −ðD=KÞmðmþ 1Þ −ðyaÞmþ1 þ ðya−yÞmþ1

n o; 0VyVya

ðA15Þ

The plug velocity, up, is the velocity at y=ya, givenby,

up ¼ ymþ1a

ðmþ 1ÞDK

� �m

; yaVyVyb ðA16Þ

For the fluid region enclosing the plug and the upperplate, the shear stress is

s ¼ sy þ K −dudy

� �n

¼ s0 þ yD ðA17Þ

and following similar approach as above, utilizing theboundary condition u=0 at y=h, the velocity is givenby,

u ¼ 1DKmðmþ 1Þ

�ðs0−sy þ hDÞmþ1

−ðs0−sy þ yDÞmþ1�; ybVyVh ðA18Þ

In terms of yb, Eq. (A18) becomes,

u ¼ DK

� �m 1ðmþ 1Þ

�ðh−ybÞmþ1−ðy−ybÞmþ1

�; ybVyVh

ðA19Þ

The velocity of the plug is given when y=yb, hence,

up ¼ DK

� �mðh−ybÞmþ1

ðmþ 1Þ ; yaVyVyb ðA20Þ

The plug velocity, given by Eqs. (A16) and (A20) isthe same, hence,

ymþ1a ¼ ðh−ybÞmþ1 ðA21Þ

Taking the (m+1)th root and keeping only thepositive value, it follows that,

ya ¼ h−yb Z ya þ yb ¼ h ðA22Þ

Substituting the values of ya, yb, given above, theunknown τ0 can now be determined as,

s0 ¼ −h2D ¼ −

h2dpdL

ðA23Þ

Furthermore, the following holds,

sw ¼ −s0 ¼ h2dpdL

ðA24Þ

The summary of the equations governing flow ofHerschel–Bulkley drilling fluids in a slot is given below.

There is fluid flow only if,

D ¼ dpdL

>syh=2

ðA25Þ

When there is flow, then,

s0 ¼ −h2D ¼ −

h2dpdL

¼ −sw ðA26Þ

ya ¼ h2−syD

ðA27Þ

yb ¼ h2þ sy

DðA28Þ

The velocity profile is given by,

u ¼ −ðD=KÞmðmþ 1Þ −ðyaÞmþ1 þ ðya−yÞmþ1

n o0VyVya

ðA29aÞ

u ¼ ymþ1a

ðmþ 1ÞDK

� �m

yaVyVyb ðA29bÞ

u ¼ DK

� �m 1ðmþ 1Þ

�ðh−ybÞmþ1−ðy−ybÞmþ1

�; ybVyVh

ðA29cÞwith m=1/n.


The flow rate per unit width of the slot, w, is givenby,

qw¼Z h

0udy ¼

Z ya

0udyþ up

Z yb

ya

dyþZ h

yb

udy

which can be written as,

qw¼ uy

��h

0

−Z h

0ydudy

dy ¼ 0 −Z h

0y

dudy

� �dy

¼ −Z ya

0y

dudy

� �dy −

Z h

yb

ydudy

� �dy ¼ −I1−I2

With some mathematical manipulation, it can beshown that,

I1 ¼ DK

� �m ymþ2a

ðmþ 1Þðmþ 2Þ

and

I2 ¼ −DK

� �m ðh−ybÞmþ2

mþ 2þ ybðh−ybÞmþ1

mþ 1

" #

So finally,

qw¼ D

K

� �m 1ðmþ 1Þðmþ 2Þ

� f−ymþ2a þ ðmþ 1Þðh−ybÞmþ2

þ ðmþ 2ÞðybÞðh−ybÞmþ1g ðA30Þ

The flow rate can be expressed in terms of parametersτy, Δ by replacing ya, yb and noting that τ0=−hΔ/2, sothat,

q ¼ DK

� �m wðmþ 1Þðmþ 2Þ� �

��− −

sy þ s0D

� �mþ2

þðmþ 1Þ h−sy−s0D

� mþ2

þðmþ 2Þ sy−s0D

� h−

sy−s0D

� mþ1

ðA31Þ

If now ξ is defined as,

n ¼ −sys0

¼ 2syhD

¼ sysw

ðA32Þ

and noting that for partial or full flow it must be true thatξ<1, the flow rate is,

q ¼ DK

� �m2wðh=2Þmþ2ð1−nÞmþ1

ðmþ 1Þðmþ 2Þ ½nþ ðmþ 1Þ�

ðA33Þ

An equivalent expression has been derived byGrinchik and Kim (1974) as referenced by Fordham etal. (1991), where though, only the final result was given.The flow rate also can be expressed in terms of dp/dLand τy as,

q¼2wðh=2Þ2þ1=n 1

KdpdL

� 1=nð1=nþ 1Þð1=nþ 2Þ 1−

syðh=2Þðdp=dLÞ

� �1þ1=n

� syðh=2Þðdp=dLÞ þ

1nþ 1

� �ðA34Þ

The final result for the flow rate in terms of annulusgeometry parameters, noting that, wh=π(R2

2−R12), and

h=R2−R1, is,

q ¼kðR2

2−R21ÞðR2−R1Þ1þ1=n 1

KdpdL

� 1=n21=nð1=nþ 1Þð2=nþ 4Þ

264

375

� 1−sy

½ðR2−R1Þ=2�ðdp=dLÞ� �1þ1=n

�sy

½ðR2−R1Þ�ðdp=dLÞ þ1n þ 1

21=nð1=nþ 1Þð2=nþ 4Þ

" #ðA35Þ

Eq. (A35) provides the relationship between flowrate and pressure drop for Herschel–Bulkley drillingfluids flowing in laminar flow in concentric annuli,modeled as a slot. The flow rate can be directlydetermined given the pressure drop but trial-and-errorsolution is required if the pressure drop is to bedetermined for a given flow rate.

Sample number S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12

Sample source A hydrated25 °C

A aged25 °C

A+L1, 3.0%aged 25 °C

A+L2 3.0%aged 25 °C

A+L4 3.0%aged 25 °C

A+L5 3.0%aged 25 °C

A+L6 3.0%aged 25 °C

A+L7 0.5%hydrated 25 °C




A+L3 3.0%aged 25 °C

Shear rate Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

Shearstress

(1/s) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa)

1021.38 28.50 48.42 21.92 6.92 10.25 20.75 18.17 11.75 12.83 4.50 9.33 12.17851.15 27.75 46.83 22.75 6.17 8.92 19.25 16.67 11.08 9.75 4.17 8.58 11.00680.92 26.83 44.58 26.08 5.58 7.83 17.92 14.75 9.75 8.58 3.58 7.92 9.83510.69 25.00 41.58 22.42 4.75 5.92 17.08 12.75 8.58 8.00 3.08 6.83 8.33340.46 22.58 37.83 18.58 3.67 4.75 15.17 9.92 6.58 7.33 2.42 5.58 6.83170.23 21.00 35.67 13.58 2.17 3.42 12.33 6.50 4.25 5.58 1.75 3.42 4.00136.18 21.00 33.25 12.00 1.83 3.08 11.67 6.17 4.08 5.42 1.50 2.92 3.42102.14 20.08 30.83 10.50 1.33 2.92 10.75 5.75 3.75 5.08 1.42 2.42 2.7551.07 16.67 27.42 6.92 0.58 2.00 9.00 4.00 2.58 4.00 1.08 1.50 1.5834.05 17.50 27.83 7.42 0.50 1.67 8.83 4.17 2.42 4.17 1.00 1.33 1.4217.02 16.00 21.90 6.67 0.08 1.25 8.08 3.33 2.08 3.67 0.67 1.00 1.1710.21 14.67 23.75 6.25 0.00 1.00 8.17 3.25 1.83 3.67 0.58 0.58 0.925.11 13.20 19.90 5.10 0.00 0.50 7.70 2.80 1.60 3.25 0.50 0.50 0.10

Appendix B. Rheological data

Table B1. Rheological data for bentonite–water suspensions and bentonite–lignite–water suspensions (A: bentonite suspension, L: lignite) (from Kelessidiset al., 2005).

222V.C

.Kelessidis

etal.

/Journal

ofPetroleum

Scienceand

Engineering

53(2006)

203–224


Table B2: Rheological data of drilling fluids (from Merlo et al., 1995)

Samplenumber

S13 S14 S15 S16 S17 S18

Sample source Mud 20 °C 0.1 MPa Mud 30 °C 0.1 MPa Mud 45 °C 0.2 MPa Mud 85 °C 0.5 MPa Mud 100 °C 1 MPa Mud 20 °C 2008 m

Shear rate Shear stress Shear stress Shear stress Shear stress Shear stress Shear stress

(1/s) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa)

1021.32 26.6 22.0 20.4 16.4 14.3 27.1510.66 19.4 16.9 15.8 12.3 10.7 17.4340.44 16.9 14.8 13.8 10.2 9.2 14.8170.22 13.3 11.8 11.2 8.2 7.2 10.210.21 5.6 4.6 4.1 2.6 2.0 4.15.11 3.6 3.1 2.6 2.0 1.5 3.1

Table B3: Rheological data of drilling fluids (from Blick, 1992, as reported by Al-Zahrani, 1997)

Sample number S19 S20 S21

Sample source 10% bentonite 12% bentonite 28% bentonite

Shear rate Shear stress Shear stress Shear stress

(1/s) (Pa) (Pa) (Pa)

1020.80 23.46 46.44 78.52765.60 21.07 42.61 72.78510.40 17.72 37.35 63.20340.27 14.84 32.56 55.06238.19 12.93 28.25 47.40153.12 10.05 23.46 40.22119.09 8.62 21.55 36.3985.07 8.14 18.67 32.5651.04 5.75 14.84 25.8617.01 4.31 10.53 17.72

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