INTRODUCTIONSuccess of a drilling, workover or completion fluid canbe directly attributed to its viscosity at the shear rate ofinterest. In practice, fluids are subjected to a wide rangeof shear rates; therefore, a thorough understanding offluid rheology and the impact of shear is necessary tooptimize fluid design. Ideally, high viscosity is desirableonly under low shear rate conditions, and should de-crease as flow rate increases. Among the viscosifiersused, biopolymers exhibit unique rheological propertiesincluding increased low shear rate viscosity and anextended power law region. These differences stressthe importance of understanding the rheological behav-ior of fluids in order to maximize performance andevaluate the relative merits of each fluid.

DEFINITION OF TERMSMost drilling, workover and completion operationsinvolve the movement of fluids. The study of flow anddeformation of matter, including liquids and solids, iscalled rheology. To better understand the impact ofrheological properties, Figure 1 can be used to provide a definition of some basic rheological terms.

1 sq. meter

1 sq. meter

liquid

1 meter/sec1 Newton (force)

1 meter

Figure 1. Diagram for Rheological Definitions

This illustration depicts the forces acting on a liquidbetween two 1 square meter plates, which are separatedby 1 meter. The bottom plate is stationary and the topplate is moved at a rate of 1 meter per second. Theamount of force required to do this is measured inNewtons. Shear Stress () is defined as the forcerequired to move a given area of the fluid. In this caseone Newton is required for each square meter of area.

RHEOLOGYTECHNICAL BULLETIN

The units of shear stress are Newtons per square meter,also known as Pascals. Alternative units for shearstress are dynes per square centimeter and poundsforce per square inch. Shear Rate ( . ) is defined as therate of movement of the fluid between the plates. It isdetermined by dividing the velocity difference betweenthe plates by the distance between them. This can alsobe called the velocity gradient. In this case, it is onemeter per second per meter of fluid and is thus mea-sured in reciprocal seconds (sec-1). Viscosity () isdefined as the ratio of shear stress over shear rate.Consequently, the units are Newton seconds per squaremeter or Pascal seconds. Another common unit ofviscosity is the Poise (dyne.second/centimeter2). OnecentiPoise is equal to one milliPascal second (1 cP = 1mPas). Stated another way, 1000 cP is equal to 1 Pas.NOTE: in oilfield literature, () is most often substitutedwith () to designate viscosity.

FUNCTIONS OF RHEOLOGYThe rheological characteristics of a fluid are important inevaluating the ability of a fluid to perform a specificfunction. Oil field fluids are often used to transportsolids, suspend solids, reduce pressure drop, or controlfluid loss. Examples of how flow properties affect thesefunctions include Stokes law, which states that thesettling velocity of a particle depends directly on theviscosity of the fluid. In order to transport a solid verti-cally, the upward velocity of the particle must be greaterthan the settling velocity of the particle. To suspend asolid, the shear stress exerted by the particle must beless than the shear stress provided by the fluid. Certainhigh molecular weight polymers are used to reducefriction pressure losses of a base fluid and improve flowrates. Darcys law states that fluid loss is inverselyproportional to the viscosity, so higher viscositiesdecrease fluid loss to the formation.

Rheological characteristics of a fluid can be radicallymodified through the addition of a polymer. By choosingthe proper polymer and polymer concentration, flowproperties can be optimized for a given application toachieve maximum results.

1

GENERALIZED FLOW CURVESFigure 2 illustrates the viscosity versus shear rate curvefor water, a simple Newtonian fluid. For a Newtonianfluid, viscosity is independent of shear rate.

log (viscosity)

log (shear rate)Figure 2. Viscosity of a Simple Newtonian Fluid.

When a water-soluble polymer is added to water, thecurve changes dramatically as indicated in Figure 3.

log (viscosity),

log (shear rate),

Lower Newtonian Region

Upper Newtonian Region

.

Figure 3. Viscosity of a Non-Newtonian Fluid.

Of significance, there is usually a region at both low andhigh shear rates where viscosity is independent ornearly independent of shear rate, and a section inbetween that exhibits strong shear rate dependence.For reasons that will become apparent later, this middleregion is usually referred to as the power law region. Insome fluids, it can be difficult to reach shear rates thatare high or low enough to observe the upper and lowerNewtonian regions.

Fluids are generally classified as Newtonian (shear rateindependent) or non-Newtonian (shear rate dependent).Newtonian fluids follow a simple relationship betweenshear stress and shear rate (Equation 1). Their viscosityis a constant as described by the slope of the line on alinear plot of shear stress versus shear rate (Figure 2).Air, water and light hydrocarbon oils are examples ofNewtonian fluids.

= . ................................................................ (1)

The viscosity of Non-Newtonian Fluids at a specificshear rate can also be defined by Equation 1; however,for these fluids, viscosity will vary dependent on theshear rate.

RHEOLOGICAL MODELSSeveral mathematical models have been developed todescribe the shear stress/shear rate relationship of non-Newtonian fluids. These models are used to character-ize flow properties in an effort to determine the ability ofa fluid to perform specific functions. Misapplication ofrheological data can result in an over-simplification orexaggeration of fluid features, accompanied by thefailure to perform a specific task. In order to optimizefluid performance, an in-depth discussion of dataacquisition, rheological models and their inherentlimitations is necessary.

Rheological evaluation of oil field fluids is generallyaccomplished using concentric cylinder viscometers.Typically, these instruments provide a limited number ofshear rates ranging from 5.1 to 1022 sec-1. Data gener-ated with these instruments are analyzed using empiricalmodels developed to describe the flow of non-Newtonianfluids. The most frequently applied models are theBingham Plastic and Power Law models. While theBingham model may have sufficed during the evolutionof clay based drilling fluids, it is deficient when describ-ing the overall rheological profile of polymer basedsystems. Furthermore, the presence of an upper andlower Newtonian region, coupled with a region of powerlaw behavior (Figure 3), make the interpretation andapplication of rheological data a challenging task. Thisis further complicated by the fact that the onset ofNewtonian behavior varies, depending on the viscosifieror fluid system under consideration (Figure 4).

Visc

osity

, cP

107

103

0.5% = 1.75 lbs/bbl= 42 lbs/1000 gal.

0.5% Xanthan

1.0% HEC

1.0% Guar

Shear Rate, sec-1

106

105

104

102

101

100

10-5 10-4 10-3 10-210-1 100 101 102 103 104 105 106

Figure 4. Viscosity Profiles of Xanthan, HECand Guar, 75F, 2% KCl

POWER LAW MODELOne of the most widely used models for describing thebehavior of oil field fluids is the power law model (Bird,Armstrong et al. 1977). This model (Equation 2) is validfor the linear, i.e., center section, of the curve shown inFigure 3. In the power law model, the viscosity termfrom the Newtonian model is replaced with a constant,

2

K, termed the consistency index, which serves as aviscosity index of the system. The consistency indexhas the unusual set of units, Force-secn/Area. Inaddition, the shear rate term is raised to the nth power,thus the term power law. The factor, n, is called the flowbehavior index which indicates the tendency of the fluidto shear thin. As the value of the flow behavior indexdeviates from one, the fluid becomes increasingly non-Newtonian. In oil field (especially fracturing) literature, nand K are usually expressed as n' and K''. The expo-nent n is equal to n' and K' is equal to K times a geom-etry correction factor. Because the equations discussedhere are given as general examples, n and K will beused. All polymer solutions exhibit power law behaviorover part of the shear rate range.

= n. ................................................... (2)

This equation can also be expressed in terms of appar-ent viscosity, (

a).

a = n-1.

.............................................. (3)

When the log of the shear stress is plotted against thelog of the shear rate (Figure 5), a straight line with aslope equal to n and an intercept (at a shear rate of one)equal to log K results. A plot of the log of the viscosityversus log of the shear rate also results in a straight line.

0.1 10 100 1000

10

1

Shear Rate, sec -1

Shea

r Stre

ss, d

ynes

/cm

2

Viscosity, cP

Shear StressViscosity

100

100

1

10000

1000

10

1

Figure 5. Idealized Power Law Fluid.

Most polymer solutions are pseudoplastic. In this case,increased shear rate causes a progressive decrease inviscosity. This is due to alignment of the polymer chainsalong the flow lines. For pseudoplastic fluids, the valueof the flow behavior index, n, ranges from about 0.1 to

CASSON MODELThe Casson model (Equation 5) has been used todescribe the behavior of drilling fluids (Lauzon and Reid1979). In general, it has limited application because ofthe difficulty in using it as a model to predict pressurelosses.

1/2 o1/2 = 1/2 1/2 .

........................... (5)

Plots of 1/2 versus 1/2 yield an intercept of

1/2 and a

slope of 1/2 (Figure 9).

1

10

100

1000

1 10 100 10002 3 4 5 6 2 3 4 5 6 2 3 4 5 6

1

10

20

30

40

500 5 10 15 20 25 30

(Shear Rate)1/2

(Shear Stress) 1/2

(left and bottom axes)

Square Root plot (right and top axes)

Shear Rate, sec -1

Shea

r Stre

ss, d

ynes

/cm

2

Figure 9. Idealized Casson Model Behavior.

HERSCHEL-BULKLEY MODELFluids that exhibit a yield point and viscosity that isstress or strain rate dependent cannot be adequatelydescribed by the Bingham Plastic model. The Herschel-Bulkley model (Figure 10) corrects this deficiency byreplacing the plastic viscosity term in the Bingham modelwith a power law expression (Equation 6).

= o + n .

............................................ (6)

This model is useful for describing a wide range of fluidsused in oil field applications. The model is reduced tothe power law if there is no yield point and to theBingham model if n is equal to one. The primary limita-tion of this and all other models that cannot be easilylinearized is curve fitting to evaluate model parameters.Computers and good non-linear curve fitting techniqueshave overcome this limitation. Recent publications(Clark, Sundaram et al. 1991; and Hemphill, Campos etal. 1993) have shown the usefulness of the threeparameter Herschel-Bulkley model for drilling fluids.

These examples illustrate the need to apply this modelwithin the power law region of a particular fluid to assureaccurate predictions of viscosity. The power law modelcannot be used to predict viscosities at shear rates inthe upper and lower Newtonian regions.

BINGHAM PLASTIC MODELThe other two parameter model which has been widelyused in drilling fluid applications, is the Bingham Plasticmodel (Equation 4). Fluids that exhibit Bingham Plasticbehavior are characterized by a yield point (0) and aplastic viscosity () that is independent of the shearrate (Figure 8).

= o + p .

......................................... (4)

100

80

60

40

20

00 200 400 600 800

Shear Rate, sec -1

Shea

r Stre

ss, d

ynes

/cm

2

Slope = plastic viscosity, p

Figure 8. Idealized Bingham Plastic Fluid.

The presence of a yield stress means that a certaincritical shear stress must be exceeded before flow canbegin. If the fluid exhibits Newtonian flow after the yieldvalue is exceeded, it is called a Bingham Plastic fluid.While some fluids exhibit a yield point, most oil fieldfluids show shear rate dependence after flow is initiated.This model is interesting but not very useful in describ-ing the behavior of polymer based fluids. In the oil field,the Bingham Plastic model has been used to describethe flow behavior of some clay based drilling muds and afew cement slurries. An extension of the BinghamPlastic model to include shear rate dependence is theHerschel-Bulkley model, described later in the text.

There are several rheological models that involve theuse of three or more adjustable parameters. It isnecessary to include a third parameter to describe theflow of the fluids in the upper or lower Newtonian regionas well as the power law region.

4

1000

800

600

400

200

10008006004002000

Shear Rate, sec -1

Shea

r Stre

ss, d

ynes

/cm2

Figure 10. Idealized Herschel-Bulkley Fluid.

ELLIS MODELIn all previously discussed models, viscosity or shearstress has been expressed as a function of shear rate.The Ellis model (Equation 7) expresses viscosity as afunction of shear stress (Figure 11).

o

= 1 +

1/2

1 .......................... (7)

In this equation, is equal to 1/n and 1/2 is the shearstress at equal to 0/2. Plotting log 0/( - 1) versuslog /1/2) yields a straight line with a slope of ( - 1).

10

100

1000

10,000

102 3 4 5 6 7 8 9

1002 3 4 5 6 7 8 9

1000

Visc

osity

, cP

Shear Rate, sec -1

Figure 11. Idealized Ellis Model Fluid.

There are many examples in literature where the threeparameter Ellis model is used to describe the behaviorof fluids that exhibit a lower Newtonian region followedby a power law region. While viscosity can be ex-pressed as a function of shear stress, it is not commonlydone in oil field applications, since most data is obtainedfrom instruments that use shear stress as the dependentvariable. Controlled stress instruments are finding wideruse, but these instruments normally require a com-pressed air flow and do not work well outside a labora-tory. Plots of log (viscosity) versus log (shear stress),Figure 12, indicate the sensitivity of viscosity to changesin shear stress.

1

10

1000

0.1 1 10 100Shear Stress, dynes/cm2

100

10000

0.1

Vi

scos

ity,P

Figure 12. Viscosity as a Function of ShearStress for Xanthan, 1.75 lb/bbl, 2% KCl

SISKO MODELThe Sisko model (Equation 8) is another three param-eter model, which is useful in describing flow in thepower law and upper Newtonian regions.

=

+

n-1 . ................................. (8)

The Sisko model (Figure 13) is best suited to describethe flow behavior of fluids in the high shear rate regions(1000-10,000 sec-1). This model may be useful inevaluating fluid behavior at high flow rates to determinethe impact of viscosity on pressure drops.

1

2

468

10

2

68

100

1 10 100 1000 10000

Visc

osity

, cP

2

4

Shear Rate, sec -1

Figure 13. Idealized Sisko Model Fluid.

CARREAU MODELThis model (Equation 9) is an example of a four param-eter model that should describe the flow behavior overthe entire range of shear rates.

.

= [ 1 + (

) ] 2

(n-1)/2............ (9)

The parameter, , is a time constant, calculated from thepoint on the viscosity versus shear rate curve where

5

the flow changes from the lower Newtonian region to thePower Law region. The other parameters have beenpreviously defined. This model has received a lot ofattention in rheological literature; however, it has notbeen extensively used in describing oil field fluids due todifficulties in obtaining data to define upper and lowerNewtonian regions. Figure 14 is an idealized fluid with alower and upper Newtonian region. This type of fluidcan be modeled using the Carreau, Cross or Metermodels.

12

46

102

46

1002

4

1 10 100 1000 10000

Visc

osity

, cP

Shear Rate, sec -1

Figure 14. Idealized Non-Newtonian Fluid.

CROSS MODELThe Cross model (Equation 10) is similar to the Carreaumodel in that it is a four parameter model that covers thewhole shear rate range (Figure 14).

.= [ 1 + (

) ]1

m

In this equation, is a time constant and m is a dimen-sionless exponent.

METER MODELThe Meter equation (11) is a four parameter equationthat expresses viscosity as a function of shear stress.Because it contains terms for the upper and lowerNewtonian viscosities and terms that describe the powerlaw region, this model can be used over a wide range ofshear rates.

= 1 +

1/2

(1) 1

......... (11)

In general, the four parameter models are difficult toapply, because there is seldom enough data to allowgood model fitting.

TIME DEPENDENT FLUIDSDue to physical and/or chemical associations betweenparticles in a fluid, time dependent behavior can beexhibited in response to an applied stress or strain.Viscosity of the fluid will increase or decrease with timewhile shearing at a constant rate. There are two pos-sible time-dependent effects as shown in Figure 15.Time dependence can be manifested in viscositymeasurements as hysteresis in flow curves (Figure 16).

Rheopectic

Time Independent

Thixotropic

Time

Visc

osity

Figure 15. Time Dependent Effects.

Rheopectic

Shear Rate

Shea

r Stre

ssThixotropic

Shear Rate

Shea

r Stre

ss

Figure 16. Manifestation of Time Dependencyin Rheological Measurements.

MEASUREMENTSThe viscosity of a thixotropic fluid decreases with time.Many polysaccharides such as guar gum, carboxym-ethyl cellulose and starch are thixotropic to a certaindegree. At concentrations used in most field applica-tions, biogum solutions do not show significant thixotro-pic behavior. In some cases, fluids can display shearthickening or rheopectic behavior. This occurs in solidsladen fluids containing high concentrations of bentonite(>20 lbs/bbl) or drilled solids. This behavior is alsoknown as anti-thixotropic or negative-thixotropic behav-ior. Thixotropic or rheopectic behavior is often a mani-festation of a structure build-up or breakdown, as aresult of shear.

MEASUREMENT METHODSMeasurement of fluid rheology presents a number ofinteresting challenges. Flow must be steady anduniform (laminar flow) in a well defined geometry.Numerous viscometers are available for measuring therheological properties of a fluid. The most basic viscom-eters provide shear stress versus shear rate data.

.......(10)

6

In the oil industry, rotational and tube type viscometersare the most popular. Tube or capillary viscometers areeasy to set up and run, but the fluid has a relatively shortresidence time in the measurement section and largevolumes of fluid are required. Rotational viscometers,on the other hand, use small volumes of fluid and offerthe option of long residence times. Measurements atelevated temperatures (above 200F) are difficultbecause the viscometer must be pressurized. Histori-cally, the Marsh Funnel has been used to provide aquick and easy measurement in the field. However, it isnot a viscometer, rather it only gives relative measure-ments, which cannot be converted to shear rate/shearstress data.

TUBE OR CAPILLARY VISCOMETERSDesign of the tube or capillary viscometer is based onthe measurement of flow rate and pressure drop througha small diameter tube. Although the principles of thecapillary viscometer were employed as early as 3000B.C., it was not until 1839 that Hagen reported hisexperimental data. This later resulted in the Hagen-Poiseuille equation (12) for laminar flow that is the basicequation for a capillary viscometer:

=pDc4

128 QLc................................. (12)

where is the pressure drop over a length Lc of the

capillary of diameter Dc and Q is the flow rate through

the capillary (Figure 17).

DcLc

p * Dc4 * Lc

= =32 * Q

* Dc3.

Figure 17. Diagram of Capillary ViscometerWith Calculations.

The capillary viscometer usually consists of a reservoir,a capillary, and methods to control and measure flowrate, pressure drop, and temperature. Although glasscapillary viscometers are sometimes used in labs, metaltubes are typically selected for field conditions. TheCombs capillary viscometer (Figure 18) is a commer-cially available oil field version.

ReservoirThermocouple

A pipe viscometer is a variation of the capillary viscom-eter in which larger diameter tubes or pipes are utilized.Although larger diameter pipes are used, the same basicequation is applicable. Rheological data is usually takenin laminar flow; however, data can be obtained underturbulent flow conditions to obtain friction pressureinformation for design purposes.

ROTATIONAL VISCOMETERSIn rotational viscometry, torque is applied to a fluid bythe rotating element of the viscometer. This torque,which is proportional to the shear stress of the fluid, istransmitted to a dial. Torque measurements can beconverted to shear stress units and the rotation rateconverted to shear rate. The two can then be used tocalculate viscosity.

Concentric Cylinder Viscometers. In 1890 Couettedescribed what was the first practical rotational viscom-eter. It was a concentric cylinder design that had arotating outer cup and an inner cylinder on which thetorque was measured. The Fann viscometers are basedon this design (Figure 19). This type of viscometer isprobably the most popular in the oil industry.

DcDb

Hb

2T

=

=15

Db Hb 2

rpm

Dc2

Dc2 - Db2.

T= torque

Figure 19. Diagram of a Couette ViscometerWith Calculations.

The Fann or similar viscometers as used in the oilindustry offer either six or twelve rotational rates.Values of the rotational rate chosen for these instru-ments tend to emphasize shear rates above 100 sec-1.Because of the mechanical configuration of the viscom-eter, the low shear rate measurements have limitedaccuracy. Recent studies have shown that low shearrate data is useful for understanding particle transport(Bloodworth, Keely et al., 1992). Applications of thistechnology include the use of a Brookfield viscometer,which is capable of measurements at shear rates as lowas 0.06 sec-1.

Cone and Plate and Parallel Plate Viscometers.Laboratory measurements on water-soluble polymersare often made with cone and plate or parallel plateviscometers. There are advantages to each of thesegeometries, but both are difficult to use at elevatedFigure 18. Capillary Viscometer Schematic.

7

temperatures with water-base fluids. They are used toprovide high quality data at or near ambient temperatures.

Cone and plate viscometers (Figure 20) are equippedwith small angle cones (

The most common way of quantifying the viscoelasticproperties of fluids is by measurement of their elasticmodulus (G') and viscous modulus (G''). G' is alsocalled the storage modulus (since elastic energy isstored), while G'' is often referred to as the loss modulus(since viscous energy is lost).

It is convenient to think of G' and G'' as vector quanti-ties located 90 apart. Vector quantities have a magni-tude (which is indicated by the length of the vector) aswell as a direction. Figure 21 illustrates these values fortwo fluids.

Elastic Liquid Viscous Liquid

Viscous Modulus G" Viscous Modulus G"

Elas

tic M

odul

us G

'

Elas

tic M

odul

us G

'

Figure 21. Modulus Values For Two Fluids

Both fluids would exhibit the same constant shearviscosity since the resultant vector of the two moduli isthe same magnitude. However, viscoelastic measure-ment reveals that from a rheological standpoint, they arequite different. In the elastic fluid, viscosity is primarilydue to its elastic modulus. On the other hand, theviscous fluid derives most of its viscosity from its viscousmodulus. The elastic fluid would provide improvedsuspension of insoluble particles or a more stableemulsion. These differences can only be revealedthrough viscoelastic measurements.

Since viscoelasticity cannot be measured in the steady,uniform flow field found in viscometers, oscillatorymethods of measurement must be used. The rheometerapplies a sinusoidally varying strain (deformation) to thesample. An appropriate transducer measures theresulting stress (force). In samples that are highlyelastic, the stress and strain sine waves are in phase. If the material is perfectly viscous (such as a an oil) thestress and strain will be 90 out of phase (Figure 22). All materials have a phase angle somewhere between 0 and 90 degrees. Instruments commonly used for vis-coelastic measurement include the Rheometrics,Weissenberg and some Haake and Contraves models.

Stress (solid line) and strain (dashed line) are 90 out of phase

Stress (solid line) and strain (dashed line) are in phase

Perfectly Viscous

TIME

Perfectly Elastic

TIME

Figure 22. Viscous and Elastic Responses toan Applied Strain.

MEASUREMENT METHODSThe technique described above for measuring vis-coelastic properties can be used a number of ways.These are commonly called a frequency sweep, astrain sweep and a dynamic recovery test.

As the name implies, the frequency sweep varies theperiod or frequency of the sinusoidal oscillation as thetest progresses. The amplitude or strain level is heldconstant. In many cases the frequency is expressed inradians/second (rad/s) to make comparison with steadyshear data more straightforward. Because of the Cox-Merz rule, frequency in rad/s is equivalent to shear ratein sec-1. In this way, the complex viscosity (*) can becompared to the shear viscosity () for materials thatare not highly associated. Most often, a frequencysweep is run from 0.1 to 100 rad/s with a strain level inthe linear viscoelastic region (to be explained in thenext section). Fluids that are liquid-like have a G'' thatis higher than G' at low to moderate frequency values.As a fluid becomes more gel-like and structured, G'increases faster than G''; therefore, G' predominates.In addition, these fluids tend to have G' curves that arerelatively flat (independent of frequency).

In the strain sweep test, frequency is held constantwhile the amplitude of the oscillation (and thus thestrain) is increased in some progression. The modulusvalues will remain relatively constant with respect tostrain until a critical strain value is exceeded. Thisregion, where modulus is constant, is referred to as thelinear viscoelastic region. With these relatively small

9

0.5% Xanthan

1.0% HEC

1.0% Guar

1000

100

10

10.1 1 10 100

Frequency, rad/sec

Elas

tic M

odul

us G

' (dy

nes/c

m2)

0.5% Xanthan

1.0% HEC

1.0% Guar

1000

100

10

10.1 1 10 100

Visc

ous

Mod

ulus

G" (dy

nes/c

m2)

Frequency, rad/sec

Figure 23. Viscoelastic Measurements forXanthan, HEC and Guar in 3% KCl, 75F

EXTENSIONAL RHEOLOGYExtensional viscosity is a relatively new concept formany people; however, these types of flow are importantin polymer processing and fiber spinning. Flow does notneed to be complex to have extensional components.Pouring water from a container involves an extensionalflow with gravity acting as the extensional force. Flowinto expansions or contractions are other examples ofextensional flow. The extensional component of thistype of flow is one of the reasons why non-Newtonianfluids show higher pressure drops in flow throughexpansions and contractions than is observed forNewtonian fluids. The idea of stretching a liquid solutionmay seem unlikely and of little relevance; however, suchmeasurements can be very revealing for many poly-mers. Diverse processes such as flow through porousmedia and drag reduction involve highly extensionalflow.

strain values the associations between components inthe system are not significantly disrupted. This is whatleads to nearly constant modulus values. Systems thatare highly loaded with suspended particles have a muchsmaller linear viscoelastic region than do solutions ofpure polymer. In the same way, polymer solutions withrobust intermolecular associations (like biopolymers)have a longer viscoelastic region than do more cross-linked fluids. If the linear viscoelastic region is notknown, the strain sweep is run before the frequencysweep so that the latter can be run in the linear strainregion.

The final common type of viscoelastic test is the dy-namic recovery test. For this test, the sample issheared at steady shear for some time to fully breakdown all molecular associations. Following this, therotation is immediately stopped and a dynamic oscilla-tion at some low strain level is begun. Points of associa-tion within the sample begin to reform and the elasticmodulus (G') begins to increase. If the associationsrequire extensive time to rebuild (as is the case withmany particulate thickeners such as clays) the rate ofincrease of G' will be slow. On the other hand, biopoly-mer formulated systems will frequently show negligibleloss of modulus from shear. Any loss measured isquickly rebuilt.

While it is undoubtedly a sophisticated technique,viscoelastic measurement can add a great deal to theunderstanding and differentiation of polymer systems. Itprovides a useful addition to the traditional, steady shearmeasurements.

TYPICAL CURVESFigure 23 shows viscoelastic results for the samepolymers illustrated in Figure 4 previously discussed.The xanthan exhibits a higher degree of elasticity at thelower frequencies than the HEC or guar. In addition, thexanthan G' curve is much flatter than either the HEC orguar G' curves which indicates gel-like behavior in thexanthan solution. For xanthan, the G' curve is greaterthan the G'' curve giving further evidence for gel-likebehavior which indicates good suspension properties.

10

MEASUREMENT METHODSExtensional flow devices for liquids can be divided intotwo broad categories. In the filament stretching device,a thin filament of the liquid is stretched, usually bypulling a vacuum. The flow rate and change of crosssectional area of the filament is used to determine theextensional strain rate. A force measuring device, suchas a fiber spinning instrument, is used to determine thestress applied to the sample. A second class of instru-ments capable of providing extensional flow fields isrepresented by the stagnation point devices. Theseinclude the 4 roll mill, cross slot flow and opposing jetdevices. With these, an area of intense extensional flowcan be created without the need for a spinnable sample.That is, extensional measurements are possible formaterials not able to sustain a continuous filament. Inaddition, stagnation point devices are capable of higherstrain rates than are fiber spinning instruments. Stagna-tion point devices have a long history of use for opticalrheological measurements. The area near the stagna-tion point shows birefringence due to chain alignmentabove a critical strain for most polymers.

A schematic of the opposing jets device is shown inFigure 24. The sample is held in a jacketed beaker witha temperature probe and two jets immersed in the fluid.The left arm and jet are mounted on a force rebalancetransducer. The right arm and jet are mounted on amoveable mechanical stage to control jet spacing andalignment. Fluid is drawn into the two jets simulta-neously by dual syringe pumps. Flow rate is controlledby a computer driven stepper motor. When desired,temperature control is achieved with a circulating waterbath and jackets around the beaker and syringes. Thisdevice has the advantage of producing an extensionalflow field in a relatively small volume of liquid. It isreadily adaptable to the water-soluble polymer solutionsused in the oil field.

The equations required to calculate extensional stress() and extensional strain rate (. ), as well as exten-sional viscosity ( ), are very simple for the opposing jetdevice where Fj is the force at the jet, A is the crosssectional area of the jet, Q is the volumetric flow rate, Gis the gap between the jets and Dj is the diameter of thejets:

=FjA

=8Q

G (Dj) =

2

Force RebalanceTransducer Movable

Stage

SampleUnderTest

ExtensionalControl Volume

Syringe PumpsOpposing Jets

Figure 24. Schematic of the Opposing Jets Device.

Effective strain rate may be varied by two methods:altering the flow rate or by changing the jet area. Largerjets and/or lower flow rates reduce the effective strainrate,

.

.

The Rheometrics Fluids Extensional Rheometer (RFX)measures extensional viscosity by means of an oppos-ing jet design. Strain rates from 10-1 to 104 may beachieved for materials of sufficient viscosity, as illus-trated in Figure 25.

1

10

100

1000

0.1 1 10 100 1000 10000Strain Rate, sec-1

Exte

nsio

nal V

isco

sity

,P HEC, 3.0 lb/bbl

XANTHAN, 1.5 lb/bbl

Figure 25. Extensional Viscosity of Xanthanand HEC, 3% KCl, 75F

The Trouton ratio is the extensional viscosity over theshear viscosity, / and is used as a reference forextensional viscosity measurement. It is an excellentway to check the validity of a technique, since theextensional viscosity should be three times the shearviscosity for simple Newtonian materials.

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1

10

100

1000

0.1 1 10 100 1000 10000Strain Rate, sec -1

TTro

uton

Rat

io

HEC, 3.0 lb/bbl

XANTHAN, 1.5 lb/bbl

Figure 26. Trouton Ratio of Xanthan and HEC,3% KCl, 75F

Trouton ratios are frequently a good indicator of molecu-lar conformation. When the extensional viscosity in-creases relative to the shear viscosity, the Trouton ratioincreases. This increase is indicative of an increase inhydrodynamic size due to molecular stretching. If theTrouton ratio is constant and low (theoretical minimum is3) then the time scale (inverse of the strain rate) is toolong to influence the molecules shape. If the Troutonratio is constant and high, it is an indicator that themolecule is fully stretched. The absolute magnitude ofthe Trouton ratio in the upper plateau (if measurable) isan indicator of how much the molecule can be stretchedout. A large difference between the ratio at low and highstrain rates indicates that the molecule is very flexibleand can be extended to a high degree. A large moleculewill show a greater Trouton ratio than a small one.Figure 26 provides a comparison of Trouton ratios forxanthan (1.5 lb/bbl) and HEC (3.0 lb/bbl). Since thexanthan molecule is stiffer and easier to stretch, Troutonratios for the xanthan increase at lower strain ratesrelative to the HEC. The increased Trouton ratiosobserved for xanthan are also indicative of the highdegree of inter-molecular associations, typical for thisconcentration of xanthan.

SUMMARYThe ability of a fluid to perform a specific function isdependent on its rheological properties. Establishingcorrelations between rheological parameters and fluidfunctionality is the first step toward fluid design andoptimization for any application.

Mathematical equations describing the rheologicalproperties of fluids have been used to predict flowbehavior under a variety of conditions. When modelingfluid flow, extreme care should be taken to validate themodel for each fluid and to use the model only forconditions under which it has been shown to be valid.There is no universal model for fluids. All the modelshave limitations; for example, the power law modelcannot be used to accurately predict low shear ratebehavior of fluids having a lower Newtonian region.

Measurement of rheological properties is a difficultprocess which is complicated by the presence of solids,viscoelasticity, and the need to measure rheologicalproperties over a wide range of shear rates. Currentinstrumentation, usually a concentric cylinder viscom-eter, does not adequately describe the rheologicalbehavior of fluids in the low shear rate range thatcharacterizes hole cleaning and suspension propertiesof the fluid. For example, traditional measurementsderived from the Fann model 35 viscometer includingyield point and viscosity at 3 rpm (5.1 sec-1) do not fullydescribe or predict performance of polymer based fluids.This has mandated the adoption of more sophisticatedinstruments. Even the use of the Brookfield LV seriesviscometers is an improvement because they allow datato be acquired at much lower shear rates. Theseinstruments provide a means of evaluating fluid behaviorat low shear rates and verifying the accuracy of extrapo-lations made from high shear rate measurements. Untilthese and other instruments, which provide data over awider shear rate range, are adopted, there will continueto be limited understanding of fluid functionality.

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REFERENCESAstaria, G (1990). The Engineering Reality of the YieldStress. J. Rheol. 34(2): 275-277.

Barnes, H. A. and K. Walters (1985). The Yield StressMyth. Rheol. Acta 24: 323-325.

Bird, R. B., R. C. Armstrong, et al. (1977). Dynamics ofPolymeric Liquids John Wiley & Sons, New York.

Bloodworth, B. R., G. J. Keely, et al. (1992). MudRelaxation Measurements Help Predict Hole CleaningAbility. OGJ. June 1, 1992, 73-77.

Clark, P.E., L. Sundaram, et al. (1991). Yield Points inOil Field Cements. Production Operations Symposium,Oklahoma City, OK, Society of Petroleum Engineers.

Durst, F., R. Haas, et al. (1981). Flows of DiluteHydrolyzed Polyacrylamide Solutions in Porous Mediaunder Various Solvent Conditions. J. Appl. Poly. Sci.26: 3125-3149.

Hartnett, J. P. and R. Y. Z. Hu (1989). The Yield Stress- An Engineering Reality. J. Rheol. 33(4): 671-679.

Hemphill, T. W. Campos, et al. (1993). Yield-Power LawModel More Accurately Predicts Mud Rheology. OGJ.August 23, 1993, 45-50.

Lauzon, R. V. and K. I. G.. Reid (1979). New Rheologi-cal Model Offers Field Alternative. OGJ.May 21,1979. 51

SymbolsA - surface aread - diameter or thicknessd

c - diameter of a tube or capillary

f - friction factorF - forceG - modulus of elasticityG' - elastic storage modulusG'' - viscous loss modulusK, K' - consistency indexL

c - length of a tube or capillary

n, n' - flow behavior indexNRe - Reynolds number - pressure dropQ - flow rates - displacementv - velocity

Greek Symbols - exponent usually equal to 1/n - shear rate

- shear strain - extensional strain - shear viscosity

a - apparent viscosity

- limiting high shear rate viscosity, upper Newtonianregion

- extensional viscosity* - complex viscosity - Newtonian viscosity - fluid density

s - solid density

- extensional stress - shear stress0 - yield stress1/2 - stress at which the initial viscosity drops by half - porosity

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Apparent Viscosity (a) - Fluids that exhibit a non-linear

relationship between shear stress and shear rate willexhibit a spectrum of viscosities depending upon theshear rate. To describe the behavior of these fluids, anapparent Newtonian viscosity has been defined as theslope of the line that connects the origin with a point onthe curve at the shear rate of interest. Normally, thisvalue is reported along with the corresponding shearrate.

Non-Newtonian Fluid - fluids that exhibit a non-linearrelationship between stress and strain are termed non-Newtonian fluids. There are a number of different non-Newtonian fluids. The basic types of non-Newtonianresponse curves are shown below:

Ellis

Shear RateSh

ear S

tress

Bingham

Pseudoplastic

Newtonian Dilatant

Laminar Flow - is a smooth flow that is devoid of flowdisturbances such as eddies.

Turbulent Flow - is a chaotic flow characterized byintense mixing caused by eddies.

Reynolds Number (NRe) - a dimensionless number thatis the ratio of the inertial to the viscous forces,

NRe = vd

where v is velocity, is the fluid density, d is a diameterand is the viscosity.

Velocity Profile - when a fluid flows in a confinedregion, the velocity of the fluid will vary with position.The shape of the velocity profile is a function of non-Newtonian nature of the fluid, and flattens as the powerlaw n value decreases.

APPENDIXDefinition of Terms

Rheology - The science of the deformation and flow ofmatter.

Rheological properties - Rheological properties offluids are usually derived from measurements usinginstruments with carefully defined flow geometries.

Normal Stress - A stress that acts perpendicular to a face.

Shearing Stress - A stress that acts tangentially to a face.

Shear Stress () - The shear stress is defined as the forceper unit area.

Shear Stress () = shear forcearea faction

A

F=

Shear Strain () - The shear strain is defined as thedisplacement divided by the height of the fluid element.

Shear Strain = shear displacement

height of fluid elements

h=

Shear Rate ( ) - The shear rate is also called the shearstrain rate. It is the time rate of change of the shearstrain.

Shear Rate () = d (s/d)dt

ddt

=.

Viscosity ( or ) - The resistance to flow. ForNewtonian fluids when shear stress is plotted againstshear rate, a straight line results. The mathematicalrelationship that describes this line is = . The slopeof the line is the constant of proportionality betweenshear stress and shear rate, and is known as theviscosity.

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Kelco Oil Field GroupAberdeen

Scotland, UK

Tel: (713) 895-7575 (800) 331-3677

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