SPE-of~Gwinf=a

SPE 15416

Rheology of Oil-Base Muds

by O,H. Houwen, Sch/urnberger Cambridge Research, and T, Geehan, SEDCO FOREX

SPE Members

Copyright 1986, Society of Petroleum Engineers

This paper was prepared for presentation al Ihe 61st Annual Technicsl Conference and Exhibition of Ihe Society of Petroleum Engineers held in NewOrleans, LA October 5-6, 19S6.

This paper waa selected for prasentafion by an SPE Program Committee fotlowing review of information confained in an abstract submitfec by (heauthor(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by theauthor(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Paperspresented at SPE meetings are subject fo publication review by Editorial Committees of fhe SOciely of Petroleum Engineers. Permission to copy isrestricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment ofwhere and by whom the paper ia presented. Write Publications Manager, SPE, P.O. Box 833S36, Richardson, TX 760S3-3S36. Telex, 7309S9 SPEDAL.

INTRODUCTION

Although a great deal has been written about the pressureand temperature behaviour of the viscosity of simple non-

ABSTRACT Newtonian fluids, and an understanding of this behaviour atThe theological behaviour of invert emulsion muds has been the molecular level is emerging, no consensus exists on howstudied at pressures up to 1000 bar and temperatures up to to deal with concentrated suspensions. This can easily be240°C. Theological parameters were calculated for the Bing- understood, considering the widely different nature of non-ham, Herschel-Bulkley and Cssson theological models. The Newtonian fluids, Invert emulsion muds are suspensions ofIierachel-Bulkley and Cesson modeb both give good fits to the solids and emulsions at the same time, and as there is no

experimental rheograms. The Cesson model is more reliable generally acceptedtheological model that can be applied tofor extrapolation purposes than the Herschel-Bulkley model. emulsions and suspensions, the engineering aspects of invertA pair of two similar exponential expressions were found to emulsion muds are not always based on very sound scientificbe able to model the pressure and temperature behaviour of principles, Thus, while it is known that at pressures and tem-the two parameters of the Casson model. The expressions, peratures encountered in the wellbore the rheology of the mudwhich are baaed on the relation for pure liquids derived the-oretically by Eyring, contain temperature dependent pressure

will be different from that measured at the surface, lack of

coefficients. The simplifications inherent in the temperaturethe ability to quantify the effects involved has perpetuated the

and pressure model are dkcussed in the light of the tempera-field practice of using theological parameters measured at at-

ture and pressure behaviour of the viscosity of common basemospheric pressure. Traditionally the mud industry has, with

oils and their constituent hydrocarbons. Field application ofa few exceptions, adhered to the uee of the Bingham and power

the model requires measurement of the rheology of the mud atlaw theological models, which have the advantage that hy-

two or more temperatures and knowledge of the pressure co-draulics calculations are available for fluids obeying these mod-

efficients relating the behaviour of the plastic viscos~ty to thatels. Hence, it is not surprising that the existing techniques for

of the yield point, or the Casson high shear visccgity to thatprediction of downhole rheology are based on these models.

of the Casson yield stress. Pressure meaauremer !S or other A number of recent publications have dealt with the problem.

information are then not required. Applications can be baaed Combs and Whitmire (1) showed that the change in the viscos-

on Caason or Bingham theological meesurements. The rela- ity of the continuous phase is the main factor in controlling the

tionships between the parameters of the Casson and Bingham change in the viscosity of the mud with pressure. Both yield

models are disussed. point and plastic viscosity seemed to be governed by this effect.McMordie et af, (2) concluded that the power law model givesthe beat mathematical description of the viscosity of an oil basemud at constant temperature and pressure, They require twosets of constants for shear rates below 200s- 1 and above thisvalue, which suggests that the choice of the power law model

References and illustrations at end of paper. is not the best one to be made. They found that the loga-rithm of the shear stress is proportional to the pressure, giving

2 RHEOLOGY OF OIL BASE MUDS SPE 15416

rise to the exponential law for the pressure dependence thatalso appeared in the appropriate API Bulletin (3). Again, thechoice of the power law model necessitated the use of differentconstants for high and low shear rat es. De Wolfe et al. (4)studied a number of less toxic oils. They report a close corre-lation of the results to the Herschel-Bulkley model. Since theirresults are presented as apparent viscosities, no further infer-ences can be made as to the manner in which the theologicalparameters of this model individually depend on temperatureand pressure. Politte (5) chose to model the invert muds asBingham fluids, which they resembled more closely than powerlaw fluids, A multi-term equation with 13 numerical constantswas presented to model the viscosity behaviour of diesel oil atpressures over 1000 psi. Politte concluded that the plastic vis-cosity could be normalized with the viscosity of the oil, Theyield point was found to be more of a problem, since it is nota true physical parameter, and much more susceptible to ex-perimental error than the plastic viscosity. It was found to bea weak function of pressure, the effect of pressure decreasingas temperature increases. Bailey et al. (6) used the Binghammodel to describe the rheology of low-toxicity oil mud, butnoted that at higher temperatures departure occurs from thismodel.

OBJECTIVESWe decided to reinvestigate the problem. Our first objectivewas to find a pressure and temperature dependent model witha wide app!icabilit y. The second objective was to give prefer-ence to models which would offer some physical interpretationof the theological phenomena. This would give hope that ulti-mately a model can be constructed that relates the theologicalparameters to mud composition and changes thereof (our thirdobjective). It was apparent from previous work that the mostelusive part of the prediction of the temperature and pressurebehaviour of the viscosity is at the low shear rate end. In orderto study this problem under the best conditions we preparedmuds with appreciable yield points, so that we were able toanal yse numerical values well above the limits set by experi-mental precision.

EXPERIMENTALWe carried out the investigation with a Haake 1000 bar rheome-ter, essentially a modern version of the instrument describedas the BHC Viscometer (7). The nominal shear rate range iso-1200 s-l, continuously variable. The bob of this rheometer ismagnetically coupled to the drive and torque measuring unit,

which can measure shear stresses up to about 60 Pa. In ourexperience an accuracy of about 1 Pain the shear stress read-ings can be achieved with the system. The shear stress vs.time profile of the rheometer was controlled by a home madesystem, incorporating a micro computer, which also continu-ously recorded the shear stress readings. Rheograms thus col-lected were reformatted and stored in a main frame databasesystem, for later recall and plotting. We also used a moreprecise thermostated atmospheric Haake RV1OO system with ashear rate range to about 2700 s–l. Muds were prepared inthe laboratory, using commercially available products for for-mulation of a currently popular less toxic invert mud systemused in the North Sea, and also used in the study on solids con-trol equipment published from our laboratories (8). Viscosities

were changed by addition of barite or organophilic clay. Greatcare wrw taken to use only reproducible results, After viscome-ter runs at elevated temperatures rheograms were re-recorded,to see if any irreversible chemical change had occurred. Gen-erally this started to happen at temperatures above 140°C.As we are then dealing with a problem of a different nature,the irreversible effect of temperature on mud systems whichcan only be alleviated by chemical treatment, we decided toexclude these measurements from our data set.

THEOLOGICAL MODELS

An obvious strategy is to decide upol: a theological model andto investigate the pressure and temperature variation of theparameters involved in the model. In order for this strategy towork, it is necessary that this model is applicable with the samedegree of accuracy at all pressures and temperatures that arerelevant to the rig operations. When the rheograms obtainedwith the HTHP rheometer were examined, it became imnledi-ately apparent that they all show curvature. It would thus bea simplification to analyse the rheology of the muds in termsof the Bingham model. The power law model leads to curvedrheogranl& but is not applicable either, as can be seen in Fig-ure 1. For a power law fluid a plot of log p vs. log ~ shouldgive a straight line, since

~ = l+, (1)

hence,

log; =logp=log k+(?t–l)log+. (2)

Figure 1 also shows a remarkable similarity of the curves de-termined at temperatures between 25°C and 110°C, indicat-ing some invariant feature of the rheology as temperature is

changed. The same observation was also made for curves de-termined at constant temperature and increasing pressure.The modified power law model of Herschel and Bulkley hasbeen advocated (9) as a more realistic description of muds,since it contains a yield stress term:

T = To -i- k~n. (3)

In order to make some judgement of the closeness of fit of theexperimental data with theological models, a set of 20 shearstresses was extracted from each complete rheogram at pre-selected ~ values, sampling relatively more points from theinitial curved portion of the rheogram than at the high shearrate end. The differences of these experimental shear stressvalues with the predictions based on the model were calculated.Two numerical techniques were used in order to calculate theHerschel-Bulkley parameters. In the first a variable trial valueof TO was subtracted from all 7 values, and a least squaresregression performed on the logarithms of the values of (T – TO)

and ~. Maximisation of the correlation coefficient gave the finalvalue of TO. In method 2 the function of TO,k, and n:

(4)

SPF 15AIA O.H. HOUWEN ANO T, ~FFHAN 3

where i runs over the zo data points extracted from the ex-perimental rheogramwaa minimized, by considering that atthe optimum values of TO,k, and n the partial derivatives ofE with respect to r., k, and n will be zero. In essence thismethod does a least squares regression on the 4, r data pointsdirectly without having to resort to taking logarithms. A con-sequence of this difference and the way in which we have spacedout our sampling points is that, if the Herschel-Bulkley modelis not the ideal representation of the datapoints, method 2 isexpected to give a closer fit than method 1 at the high shearrate end, at the expense of the fit at the low shear rate end.

It was possible to fit our experimental rheograms to this modelquite well over the whole range of shear rates (O -1200 s-l),Table 1 shows an example of the results obtained by both meth-ods. A similar closeness of fit was observed for all our HTHPrheograms. It is indeed clear that the predicted discrepancybetween methods 1 and 2 at high shear rates occurs, method2 giving an overall better fit except at shear rates below about15 s-l.

The muds were also run in the atmospheric rheometer. A fitwas made to the 0- 12005-1 portion of these rheograms. Withthe fitted parameters an extrapolation was made fro,m 1200-2700 s-1 (Figure 2). It can be seen that at the highest shearrates the experimental rheograms depart from the Herschel-Bulkley model, which consistently gives too low values in thisrange, regardless of whether the Herschel-Bulkley parameterswere obtained by method 1 or 2. This is a consequence of thenature of the model which predicts an ever decreasing viscosityWith increasing shear rate. However, the behaviour of the log pm, log ~ curves (Figure 1) suggests a leveling off of the viscosityto some constant Newtonian plateau value, as is generally ob-served for suspensions (10). Thus, while the Herschel-Bulkleymodel is capable of describing the rheology of the muds in the 1-1200 s–l range quite adequately, it cannot be used with confi-dence for extrapolation. Another objection is the fact that thepower law index n haa no clear physical interpretation otherthan being a convenient curve fitting parameter, which wouldstand in the way of reaching our third objective of relatingrheology to mud composition.

As an alternative to tlie Herschel-Bulkley model, we decidedto investigate the Casson model, which has been claimed byseveral authors to give a good represen~ation of the rheologyDf water based muds (11-15). This is a two-parameter model

and can be written as (16)

The constants k. and kl can conveniently be evaluated fromthe experimental data by least squares linearization of thesquare roots of the r and i values. The closeness of fit ofthe calculated shear stress values was analysed in the sameway as outlined before and a similar analysis as performed forthe Herschel-Bulkley model was made. Within the experimen-tal errors of the HTHP rheometer, the Caeson and Herechel-Bulkley models are seen in Table 1 to give about equally satis-factory results. The reliability of extrapolations based on Cae-

----- . . . .

son constants determined in the shear rate interval 1-1200 s-1was again judged by comparison with rheograms determinedbetween 1 and 2700 s-l (Figure 2). The Casson rheogramremains closer to the experimental curve than the Herschel-Bulkley rheogram, In fact, if one would compare extrapolatedviscosities at much higher shear rates, those calculated fromthe Herschel-Bulkley parameters are significantly lower thanthose calculated from the Casson parameters, To put this inperspective, we will summarize here the physical interpretationof the two Casson constants.Squaring both sides of equation 4 gives

T = koz + 2kok1-j* + klzi. (6)

Equation 6 shows that for ~ approaching zero, the shear stressbecomes equal to k.’. Hence, k.’ is identified = a yield stress.The ko2 parameter, which we will call the Casson yield stressTY, plays much the same role in the Caason model as the fieldPoint (YP) and r. play in the Bingham and Herschel-Bulkleymodel, respectively, and like these should be reported in Pa(N/m2) or lb/100sqft. It should also be noted that, as Table1 shows, the experimentally determined values of the Cassonyield stress are very close to the Herschel-Bulkley yield stresses.Division of both sides of equation 6 by ~ gives

p = ko2~-1 +2kok1~-* + klz. (7)

As ~ goes to infinity, only the last term in this equation re-mains, Hence, kl 2 is the viscosity of the fluid at infinitelylarge shear rate, and its interpretation as such makes it sim-ilar to the Bingham plastic viscosity (~). If r is measuredin Pa, and ~ in s-1, then kl 2 has units of Pas. To make its

relationship to the Bingham ~ more apparent, we have mul-tiplied klz by 1000 to give it units of mPas or cp, and call thisquantity the Caeson high shear viscosity (PC), As far ea mudsare concerned, the Casson yield stress is always smaller thanthe ~, and the Casson high shear viscosity is always smallerthan the ~. See also the Appendix for further comments onthe Casson model.

HTHP RHEOMETRY RESULTSThe Bingham ~ and ~ were calculated from the shearstresses measured at shear rate values of 500 and 1000 see-1,hence at practically the same values as are used in field prac-tice. Casson and Herschel-Bulkley (by methods 1 and 2) pa-rameters were determined from computer fits to the 20 selecteddata points in the manner described before. Examples of plotsof theological parameters as a function of pressure at constanttemperature are shown in Figures 3 and 4. The same generaltrends as observed by previous workers were observed for the~. Thus, the ~ dropped with rising temperature rmd in-creased with rising pressure. The hypothesis was tested thatthe Arrhenius equation

p= Aexp$, (8)

where T is the absolute temperature (K), and the simple rela-tionship

p = A exp CP, (9)

h RHEOLOGY OF OIL BASE MUDS <PF 75[,IL

ihat had previously been advanced for the pressure behaviour~f the viscosity of muds (2,3), and in which C is a temperaturendependent constant, would apply tosome of the theological>arameterso When for this purpose plots of the logarithms>f the theological parameters were studied, it appeared that;he slopes B of the lines relating log ~ to l/T at constant—?ressure were dependent on pressure. Also, slopes of log PVUS.P at constant temperature were dependent on temperature.I’hisbehaviour suggests thateq. 8andeq, 9can recombinedn a law of the type

(lo)w = Aexp[~ + C(T) P],

tvhere the pressure coefficient C of eq. 9 is now a function oftemperature, yet to be specified.Previous workers have shown that the ~ of invert muds isproportional to the viscosity of the base oil at elevated tem-perature and pressure (1,5). We therefore turned to viscosity

iata for the base oils. An extensive set of measurements ondiesel oil is embodied in Figure z of Politte’s paper (5); similarbut less detailed figures showing the viscosities of naphthenicand solvent based oils can be found in the paper by De Wolfeet al. (4). We read the experimentally determined viscositiesfrom these graphs, and replotted their logarithms as functionsDf pressure. It became clear that (~ log p/dP)~ had a similardependence on T as was displayed by the ~’s of our muds,k .g-ure 5, where we have plotted (~ log p/8P)zI as a functionof (1/2’), shows a least squares fit representing the trend dis-played by the oils in this way, including the base oil used forour work. Because of the experimental uncertainty of some ofthe data, there is still room for the question as to whether thisline should go through the origin of the plot, in which case theterm C(T) in equation 10 bec~,,:es C/T. We have looked atthis question in the light of what ;s known about the bahaviourof pure hydrocarbons, and the experimental data available formuds.

The viscosities of some hydrocarbons of interest have been re-ported at pressures up to several thousands of bars, far ex-ceeding the range over which the available data on invert mudbase oils extend, and exceeding the range of wellbore pres-wres. At that scale of measurement log p us. P isothermsgenerally are concave towards the P axis. (But as noted byseveral authors, can also become convex at low temperatures;e.g. Hogenboom et al. (17) show this to occur at pressures ofabout 2000 to 3000 bar for the 16° C and 38°C isotherms ofci-decalin.) This is of no practical concern to our problem,however; below 1000 bar the isotherms can be represented bystraight lines without much harm to the accuracy of the rheo-Iogical predictions. In this way we have calculated the slopes ofthe (~ log g/i3P)T isotherms from published data, generally inthe 200-1000 bar range. Results are shown in Figure 5 for thestraight chain alkanea *octane (22) and whexadecane (16),the naphthenes cyclooctane (22) and cisdecalin (17), and thearomatics butylbenzene (23) and octylbenzene (23), the lastthree compounds all having 10 carbon atoms. The regressionlines for the base oils and the n-alkanes have negative inter-cepts on the (~ log p/aP)T axis; most of the naphthenes andaromatics have positive intercepts.

We analyzed the mud data in a similar way, taking as the vis-cosity parameters both the _ and Casson high shear viscosi-ties. The comparison between these two parameters was madebecause both should reflect the effect of the continuous phase

on the high shear rate viscosity, where inter-particle effects be-come small, The PV data were taken from our own work, inaddition to the data published by Politte (5) and Bailey (6).Casson high shear viscosities were deduced from the literaturedata by means of the algorithm discussed in the Appendix.We found that a reasonable fit to the data was achieved if wetook (a log ~/aP)T and (a log pc/~P)~ to be directly pro-portional to l/T, with proportionality constants CB and CC,respectively,Analogous behaviour to the ~ and Casson high shear vis-cosity wss displayed by the YP and Casson yield stress, Theslopes of the lines relating the logarithms of these quantities toP at constant temperature were found to be smaller than thecorresponding slopes of the ~ and Casson high shear viscos-ity lines. We found by doing a similar linear regression throughthe (8 log YP/aP)T or (~ log ry /ap)T us. l/T plots that thepressure dependence of YP and rY can be represented by twoequations:

(B’~ = A’ exp

YBCBPy+-y-

)(11)

and

(

B/l YCCCPTy = A“ exp ~+—

T )(12)

where YB and YC are multipliers smaller than unity modify-ing the same pressure coefficients CB and Cc that were foundapplicable to _ and Casson high shear viscosity.We did not attempt to fit the Herschel-Bulkley parameters toany P, T model. As seen in Figure 4 the three Herschel-Bulkleyparameters show less regular behaviour than those belonging tothe two-parameter theological models. No doubt this situationcould be improved if we would impose the condition that nremain constant over the range of pressure and temperature.The objection against n as a curve fitting parameter remainshowever.

DISCUSSION OF THE P,T MODELWe can rewrite the expressions found for the temperature andpressure dependence of the theological parameters in the fol-lowing generalized form:

VIS(P, T) = A exp(

YvaP;+=

)(13)

In this equation VIS stands for ~, Casson high shear viscos-ity, ~, or Ty. A has the meaning defined before in eq. 8-10.El.2 replaces B in eqs. 8 and 10. Va/R takes the value of CBor . .*c. Y is equal to unity if ~ stands for ~ or Cassonhigh shear viscosity, and is equal to YB or YC if VIS standsfor ~ or TY, respectively.The values of A will be different for ~ (or PC) and ~ (orry ). We also found that the values of E used in the forms of

SPE 15416 O.H. HOUWEN AND T. GEEHAN 5

eq. 13 describing ~ (~C) ,YP (~Y), or the base oil viscosity An alternative method is to replace the Casson parameters byare not necessarily equal. Bingham ~ and ~ in steps 2 and 3. Step 5 gives then theIf R is the gas constant, then E has the dimension of energy Bingham rheology at the desired temperature and pressure. Ifand V@of volume, We have thus obtained a modified form of a rheogram is to be produced, then, before applying equationthe equation derived theoretically by Eyring (18) 6, the Caason kl and kO are calculated from ~ and ~ by

the method given in the Appendix.

‘f ‘xp(AE”~~+)9 C!ssrjonrheologicalmodel. The fhstusesCaasonconstantsP = #27rmkT)l/2 ~/3 (14) Thus, we have two ways of drawing rheograms based on the

where AEvia is the activation energy for flow and V~ is the throughout, that are directly obtained from measurements at

molar volume. We have used the constant A instead of the surface conditions. The second uses Bingham constants in the

preexponential factor derived by Eyring, on the grounds that initial measurements and the P,T prediction, and then converts

this seemed experimental y justified. Indsed, if for example we these into Casson constants. The question has to be asked if

define a reduced ~ these two routes yield different results, given the rather com-plicated quadratic equations underlying the conversion. We

-()

–YvaPPv,ed = Pvexp ~ , (17)

have applied the two methods using pairs of A and E/R val-ues that were obtained separately for the Bingham constants(determined 500 and 1000 see-l and for the Caason constants

then plots of log ~r.d vs. l/T (Figure 6), using the appropri- (determined by least squares fit to 20 selected points on the

ate values for Y and Va, are straight within experimental error, input rheograms). Results are shown in Figures 9 and 10. The

and introduction of a more rigorous temperature dependence differences are very small and of the order of the experimen-

seems unnecessary. Our numerical choice for Va/R leads to tal error as can be seen by comparison with the experimental

values for the factor n in Eyring’s relation (eq, 14) which are rheograms presented in Figures 7 and 8.in agreement with n values published for hydrocarbons (18).A different issue raised by our treatment of YP and TY which CONCLUSIONS

we implicitly consider to be a f~nction of the base oil viscosity, We have found a simple model for the description of the rheol-

is the precise nature of this function. Through the introduc- ogy of invert emulsion muds under downhole conditions, The

tion of our factors YB and Yc, we make clear that ~ and model needs as input parameters four constants which are spe

TY vary less with pressure than PV and the Casson high shear cific for each mud. They are two activation energies and two

viscosity. The details of the mechanism by which the viscosity preexponential factors. These can easily be obtained from Ar-

Df the base oil governs ~ and rY warrants much more inves- rhenius plots of two theological parameters, measured at the

tigation. Firth and Hunter (19) and Van de Ven and Hunter rig site with a conventional rheometer at two or more different

(2o) studied the rheology of mineral suspensions; it is interest- temperatures. Also required are a knowledge of the temperi-

ng to note that they found a linear relation for the viscosity ture dependent pressure coefficients, expressed in this paper as

Df the continuous phase with the Bingham fi. It should also the product of the constants Y and C, which are specific for a

be mentioned that our finding is in qualitative agreement with mud Q’pe”the work done by Combs and Whitmire (1) and by Politte (5). The pressure behaviour of the muds was found to obey an expo-

nential law. The limitations of this conclusion within the realmAPPLICATIONS OF THE P,T MODEL of pressure dependencies of related hydrocarbons has been dis-We can use equation 13 for predictions of downhole rheology. cussed, Because base oils are complex mixtures of these classesAccording to our model it would be sufficient to determine the of hydrocarbons, the pressure behaviour of flifferent oils showsArrhenius constants A and E/Rat atmospheric pressure. This little variation. It is interesting to note thee Politte (5), using arequires the following steps. different line of reasoning, also tentatively concluded that two

1. Measure the rheology of the mud with a standard oil field of the oils studied by De Wolfe (4), paraffinic oil and solvent oil,rheometer, at least at two different temperatures. could be modelled by the equation developed by him for diesel

2. Determine the Casson high shear viscosity PC and Cas- oil. Our theory is sufficiently general to allow for radicallyson yield stress ry by one of the methods given in the different oil mixtures in a straightforward manner. The acti-Appendix, vation volumes would then simply be replaced by other values,

3. Plot In WC and In ry as functions of I/T. and even a constant factor (not dependent on temperature)4. The slopes of these lines give the values of E/R; the inter- can be added to the exponential term. This would correspond

cepts give the values of A. Two sets of these parameters to abandoning the decision to take the functions (tl log /.t/8P)Twill result, to be directly proportional to l/T and to accept that plots of

5, If Cc and YC are given, equation 13 will give the Casson the type presented in Figure 5 have intercepts.

high shear viscosity and rY at the desired combinations of The non-Newtonian character of invert emulsion muds can betemperature and pressure, described by the two parameter Casson model, which by others

6. If a rheogram is to be produced, convert Casson high shear haa been shown to be applicable to water based muds as well.viscosity and TY into kl and ko and use equation 6. The two Caason parameters can conveniently be presented as

Of course, these steps can be facilitated by the use of a pro- the viscosity at infinitely high shear rate, and the apparent

grammable hand-held calculator or a computer. An example physical yield stress, which play roles analogously to the ~of computer drawn rheograrns and experimentally determined and ~ of the Bingham model.rheograma can be seen in Figures 7 and 8. We have shown in the Appendix that the Caason and Bhgham

6 RHEOLOGY OF OIL BASE MUEJS SPE 15L16

parameters are linked by a set of quadratic equations. Hencethe tworheological parameters required for application of theP,T model can be taken from either of the two models. We havealso shown that modelling of the rheology by the Herschel-Bulkley model gives shear stresses which are quite close tothose calculated from the Casson model.

The advantage of the Caason model over the Bingham model isthat it reproduces the curvature of the rheograme at shear ratesabove about 1 s-1, It can therefore serve as a mathematicalbasis for advanced hydraulic programs which are based on asclose an approximation of the real non-Newtonian character ofthe fluid as possible.

NOMENCLATUREA, A’,At’ = Constants in Arrhenius equationB, B’,B” = Constants in Arrhenius equationc;ccE

+kkOkl

PPcn

PPvRTrToryYBYcw

= Pressure coefficient in eq. 11, K-bar-1= Pressure coefficient in eq. 12, K- bar-1= Least squares sum= Shear rate, s– 1= Consistency index, Pa.sn. Casson intercept, Pa1i2= Caason slope, (Pas) 1/2= Newtonian viscosity, Pa,s= Casson high shear viscosity, mPa.s= Power law index= Mud pressure, bar= Bingham plastic viscosity mPa.s= Gas constant= Absolute temperature, K= Shear stress, Pa= Herschel-Bulkley yield stress, Pa= Cssson yield stress, Pa= Multiplier in eq. 11= Multiplier in eq. 12= Bingham yield point, Pa

ACKNOWLEDGEMENTS

The authors thank their managements for permission to pub-lish this paper. They further thank I. Bratchie of SCR formaking a computer program available to calculate Herechel-Bulkley constants by Method 2, and I. Chalmers, N. Alderman,and H. Ladva for technical assistance.

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Lauzon, R. V., and Reid, K. I. G., “New theological modeloffers field alternative~, Oil and Gas Journal (May 211979) 52.Lauzon, R. V., and Short, J. S., “The colloidal interac-tion of ferrochrome Iignosulfonate with montmorillonite indrilling fluid applications”, SPE 8225, (Laa Vegas, Sept.23-26, 1979).Zhongying, W. and Songran, T., “Casson theologicalmodel in drilling fluid mechanics”, SPE 10564 (Beijing,March 21-23, 1982).

Zhongying, W. and Songran, T., ‘The flow effect of mudin annulus and the selection of theological parameters”,SPE 14871 (Beijing, March 17-20, 1986).

Wanneng, S., Jianping, C., and Zhenxue, L., “Compar-ison of theological models in iiigh shear rate range andexperimental relationship between penetration rate andhigh shear viscosities”, SPE 14858 (Beijing, March 17-20,1986) .

Caason, N., ‘A flow equation for pigment-oil suspensionsof the printing ink typen, in Rhwlogy of disperse systems,Mill, C. C., cd., (Pergamon, London, 1959) 84.

Hogenboom, D. L., Webb, W., Dixon, J. A., “Viscosity ofseveral liquid hydrocarbons as a function of temperature,pressure, and free volume”, J. G’hem. Phys. (1967) 46,2586.

Ghisstone, S., Laidler, K. J., and Eyring, H., The theory ofrate processes, (McGraw-Hill, New York, 1941) ChapterIx.Firth, B.A. and Hunter, R.J., “Flow properties of coagu-lated colloidal suspensions III. The elastic floe model”, J.Colloid Interface Sci, (1976) 57,266,

Van de Ven, T. G.M., and Hunter, R. J., “The energy dis-sipation in sheared coagulated SOIS”, Rheol. Acts (1977)16,534.Speers, A., “Computer aids analysis of drilling fluids”, Oiland Gas Jcurnaf (Nov. 19 1984) 118.

Gouel, P., “Viscosity of alkanes (C13to Cle), cycha and

alkyl-benzenes”, Bull. Cent. Rech. Ezplor.-Prod. Elf-Aquitaine (1978) 2, 439.

Ducoulombier, D., Zhou, H., Boned, C., Peyrehiase, J.,Saint-Guirona, H., and Xans, P., “Pressure (1-1000 bars)

and temperature (20-100 “C) dependence of the viscosityof liquid hydrocarbons”, J. Chem. Phys. (1986) 90, 1692.

5PE 15416 O.tl. HOUWEN AND T. GFFHAN .----- . . . . I

APPENDIX

CALCULATIONOF THE CASSON CONSTANTS

It shouldbe noted that we have adopted a different nomen-clature from the one used by Lauzon and Reid (11) and insubsequent papers (13,14,15,21). Our definition ofkl and kois as given originally by Caason (16). Our use of Casson highshear viscosity and TY is designed to remain numerically andconceptually close to the framework of the Bingham model.

If a conventional six-speed field rheometer is used, then themost accurate way is to plot the square roots of the shear stressvalues against the square roots of the shear rates. The interceptof the line through the data points is k. and kl is the slope.This can best be done by a programmable calculator with aleast squares subroutine. Alternatively, but employing onlytwo of the data points and therefore less accurately, Lauzonand Reid’s formulae using the 100 and 600 rpm readings maybe used (11).

If a two-speed rheometer is used of the conventional type, thenthe Caason constants can be calculated from the 600 and 300

rpm dial readings by using

andk. = 2.44068:~; – 1.72588:{; (A -1)

where kl is in (Prvs) 112 and k. is in Pa]/z. Of course, if only~ and ~ are given, a simple conversion gives &jOO and 0600,from which the Casson constants are then calculated.[f we calculate the Bingham parameters in the usual way from”300 and 600 rpm readings, then the following interesting rela-tions hold:

~ = kl(kl + 0.03665kO),——YP = kO(kO+ 26.484kl), (A -2)

where YP is in Pa, It can be shown that k. and kl are——quadratic functions of the PV/YP ratio. For example,

(A -3)

where

( )$

~ _ 0,0183 0.000336 0.5147

– W/v-13.2+ 175.4+ ~+—— .

(Pv/YP) pv/yp(A -4)

SPE 15416

TABLE ]

RNEoLOGICALMODELSANo EXPBNIMENTALitNEOWAltS

Sxperi8ent CASSON EBRSCEEL-BUWL8Y(1) BSRSCSIEL-BULKLSY(2)

sac’-l’ Pa Pa cliff. Pa cliff. Pa cliff.

50015:00

5 93 6UD O 08 5 96 0 037:33 7:30 - :G 03 7:18 -0:15

30.00 8.60 8.68 -0.09 8.58 0,0145.00 9.88 9.83 0.O6 9.78 0.1060.00 10.79 10.85 -0.05 10.06 -0.0675.00 11.81 11.79 0.02 11.85 -0.0590.00

150. W200.00250.00300.00350.00400.00450.00500.00600.00700.00800. 0+3900.00

1000.00

12.9216,0917.9820.4222.6124.7626.7828.6830! 5434.2137.8941.4844.8448.34

12.6715.86lB.2520.5022,6524.7126.7228.6730.5834.3037.9141.4344.8B48.27

0.250.23

-0.28-0.08-0.04

0.050.070.01

-0.05-0.10-0.02

0,05-0.04

0.07

12.7916.14

0.13-0.05

18.6220.9123.0625.1127,0628.9430,7634.2437.5440.7043.7446.68

-0,64-0.48-0.45-0.35-0.28-0.26-0.23-0.03

0.350,781.101.67

6 427:41

049:0:08

8.63 -0.039.70 0.19

10.68 0,1111.61 0.2012.49 0.4315.72 0.3718.18 -0.2020.48 -0.0622.68 -0.0724.79 -0.0326.83 -0.0428.80 -0.1230.73 -0.1934.44 -0.2338.01 -0!1241.45 0.0344.79 0.0548.04 0.30

!i.,!:

,[:,I

,:.

. .,:

I I 1

10 100 1000

Shear Rate, s“’

Fig. l-Vlecoslty VS. shear rate at 800 bar and (from top WIbottom) 25, 50, 80, and 11O°C.

●✎

W 15416

0 I

0 250 500 750 1000 1260 1500 1750 200022502500

Shear Rate, s-’Fig. 2—Comparlscmof theological models with experimental rheogram.

CASSON MODELHigh Shear Viscosity

BINGHAM MODELPlastic Viscosity

● 25*C

● 50”CO 80”Co !.10”C

——

b

●

●■

m■

0

00D ❑

0

●

■

0

n

I 1 1 1,,, ,0 2004006008001000

Pressure (Bar)

●

b

●

■

●■

m■

00

00 c1•1 0 0

I I I I

20040060080010

Pressure (Bar)10

Fig. 3–Prnsw9 ●d temperature Lmhsviorof Casson ●nd Rlnghsm parmnetemt

HERSCHEL - B(JLKLEY MODELConsistency index

o00v-00 ● 25°Cm a 50°co0m O 80”C

o ❑ llo”co*

‘q :

2:gg

y: .00mo

‘Lo.04 0

00 9

0

●

o■D

I

●

c1o

I

0 200400600 8001(

Pressure (Bar)D

Power Law index

m

0:

w

0:

Ic

?y .]o–

w

o–.

q :0 .

0■

■

●c10 ❑

8

I I 1

m

0●

n

0 2004006006001

Pressure (Bar)o

Yieid Stress

●

m

El

I

●

✘

0

0

● ☛

m

=

o0

OQ

I I

F4f, 4—Pre8sum and temperature bhsvior of Hemchel.hlkley pammetcm.

2004006008001000

Pressure (Bar)

(y~ octylbenzene

o [ i 1 I I 1 I I I I I

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 :

1000/T (K)F~. S–lm-re dqwxlmccof pressure coeftlclcat for Imu 0118●nd hydrocarbons.

.5

104–

:-Co

g

&_

m

5

q -0

2.6 2.7 2.6 2.9 3 3.1 3.2 3.3 3.4 3.5

1000/T (K)F@.6-FMUCNIPVat1, 200, 400, WO, BOO,●d 1,000 ban four tmnpemlur= at MCh Pr~ura.

0 100 200 300 400 500 800 700 800 900 10001100

Shear Rate, s-’Fig. 7-PrediCted and experimental dWOgMM$ at 400 bar ●nd (frOM top to bottom) 25, 50, 80, and 11OQC.

tio js:u)

5

%o :

00 100 200 300 400 500 800 700 800 900 10001100

Shear Rate, S-’

Fig. O-PmdktOd and expwlmental ttDOWramsatM-C and (from bottom to top) 200,400,400,800, and 1,000hr.

Shear R@e, s-’Fig. 9-Comparlbon of modellng vla Elngham ●nd Cas80n pmmetem, conditions of Fig. 7.

$~ 15416

100 200 300 400 500 600 700 800 000 10001100

00 100 200 300 400 500 600 700 800 900 1000 1“

Shear Rate, S-l

F@ 10-ComparisC+I of mod.llng via Bingham ●nd CmaOn pamm.lem at E(I”G 200 ●nd 1,000 Imr.