6-Rheology

School of Petroleum Engineering, UNSW Open Learning - 2000

6

Rheology & Types of Flow

q Types of flowØ Laminar flowØ Turbulent flowØ Plug flow

q Rheological modelsØ TerminologyØ Newtonian modelØ Non-Newtonian modelsØ Laboratory determination of rheological properties

q Flow in pipes and annuliØ Pipe laminar flowØ Pipe turbulent flowØ Laminar flow in concentric annuliØ Turbulent flow in concentric annuliØ Flow in eccentric annuli

A proper understanding of cement slurry rheology is important to design, execute andevaluate a primary cementation. An adequate rheological characterization of cementslurries is necessary for many reasons, including

• Evaluation of slurry mixability and pumpability,• Determination of the pressure-vs-depth relationship during and after

placement,• Design of the displacement rate required to achieve optimum mud removal.

6.1 Types of FlowFluids, which are deemed to flow under steady-state conditions, may be considered todo so in one of the following regimes:

• Laminar flow or• Turbulent flow

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6.1.1 Laminar FlowWhen a fluid is defined as flowing in laminar-flow regime, the individual particles inthat fluid move forward in straight lines parallel to the pipe’s axis. The particles incontact with the pipe wall are considered stationary. The velocity within a pipe variesaccording to their proximity to the walls of the pipe, with the particles at the centre ofthe channel moving at the highest speed (Fig. 6.1).

Fig. 6.1 – Laminar flow.

6.1.2 Turbulent FlowIn turbulent flow, particles swirl within a pipe in a rolling motion that is quite distinctfrom the sliding motion of the laminar flow. With turbulent flow, the speed of flowincreases rapidly away from the wall of the pipe and becomes fairly constantthroughout the main part of the fluid (Fig. 6.2).

Fig. 6.2 – Turbulent flow.

6.1.3 Plug FlowPlug flow may be considered as an exaggerated type of laminar flow in which ashearing effect is resent near the wall, while the central part of the fluid moves as asolid plug with similar particle velocities.

When certain fluids have their flow rate changed within a pipe, they can exhibit thefollowing patterns (Fig. 6.3):

1. Plug flow.



2. Transition period between plug and laminar flow patterns (in which that partof the fluid moving in a plug-flow regime (d) is decreasing as the flow rateincreases).

3. Laminar flow.4. Transition period between laminar and turbulent flow patterns.5. Turbulent flow.

d

1 2 3 4 5

INCREASING FLOW RATE

Fig. 6.3 – Flow patterns.

6.2 Rheological ModelsRheology is concerned with the flow and deformation of materials in response toapplied stresses. It describes the relationship between the flow rate (shear rate) andpressure (shear stress) that causes movement. It enables one to determine the flowregime needed for optimum cement slurry placement and to calculate values forfriction pressure within the pipe and annulus.

6.2.1 TerminologyConsider a fluid contained between two large parallel plates of area A, which areseparated by small distance r (see Fig. 6.4). The two plates move parallel to oneanother but at differing velocities. For steady motion to be achieved, a constant forceF is required to keep the upper the moving at a constant velocity, relative to the lowerplate, (V1 – V2). The magnitude of the force F was found experimentally to be givenby:

rVV

AF )( 21 −= µ

FF

V1

V2

r

A

A

Fig. 6.4 - Fluid behaviour.



The term F/A is called the shear stress exerted on the fluid or

AF=τ (6.1)

where, dimensions of Eq. (6.1) are ML-1T-2 and units are Pascal (Pa) in the metricsystem, lbf/100ft2 in the field units.

The velocity gradient (V1-V2)/r is an expression of the shear rate:

drdv

rVV =−= 21γ& (6.2)

where dimensions of Eq. (6.2) are T-1 and units are sec-1.

Apparent viscosity is defined as:

γτµ&

==rateshear

stressshear a (6.3)

where dimensions of the equation are ML-1T-1 and units are Pa.sec in the metricsystem, centipoise in the field units.

Viscosity being considered as the internal resistance a fluid offers to the flow,stemming from the frictional force arising from one layer of fluid rubbing againstanother.

Thixotropy - A fluid is said to be thixotropic if it is thinned when mixed or shaken,gels when it is allowed to stand for a short period and then becomes thin again whensubjected to shear. The shear rate/shear stress relationship for a thixotropic fluid isillustrated in Figs. 6.5 and 6.6.

Fig. 6.5 - Thixotropy.



Fig. 6.6 - Thixotropic behaviour.

Rheological models are a mathematical expression for the shear stress or theviscosity as a function of the shear rate. The rheological models generally used bydrilling engineers to approximate fluid behaviour are:

1. Newtonian model,2. Bingham plastic model and3. Power-law model.

6.2.2 Newtonian ModelIn this model, the shear stress is directly proportional to the rate of shear, therefore,the viscosity (µ) is a constant.

−==

drdvµγµτ & (6.4)



The rheogram (shear rate vs. shear stress curve) of the fluid is a straight line of slopeµ passing through the origin (Fig. 6.7).

Bingham Plastic

Shear Rate

Power Law

Herschel-Bulkley

She

ar S

tres

s

Newtonian

Fig. 6.7 - Examples of flow curves used in the petroleum industry.

Typical Newtonian fluids used in cementing operations are water, some chemicalwashes, gasoline and light oil.

6.2.3 Non-Newtonian ModelsThe term non-Newtonian covers any fluid that cannot be classified as Newtonian.These include drilling fluids, cement slurries, heavy oils, etc.

Generally, their viscosity is a function of the shear rate and also of the shear history.A distinction is usually made between shear-thinning fluids for which the viscositydecreases with the rate of shear, and shear-thickening fluids for the reverse is true.Generally speaking, cement slurries fall in the first category.

The two mathematical models commonly used to describe the rheological propertiesof cement slurries and drilling fluids are the power-law model and the Binghamplastic model. Both models only apply where laminar flow regime is prevalent.

q POWER-LAW MODELThe equation for the power-law model can be written as (Fig. 6.8):

nk γτ & ×= (6.5)

where n = flow behaviour index or power-law index, dimensionless k = consistency index, lbf.sn/ft2



The power-law fluids start to flow, just like Newtonian fluids, as soon as initialpressure is applied, but do not exhibit the characteristic Newtonian proportionalitybetween shear stress and shear rate (see Fig. 6.7).

LAMINAR FLOWTURBULENT

FLOW

LOG SCALE

SHEAR RATE (dv/dr)

SH

EA

R S

TRE

SS

(τ)

LOG SCALEK

'

1

n'

Fig. 6.8 – Power-law model.

The power-law index (n) quantifies the degree of non-Newtonian behaviour of thefluid. For shear-thinning fluids n < 1 and the fluids behave as actual pseudoplasticfluids. As n à 1, the fluid behaviour increasingly approximates to the status of aNewtonian fluid. Dilatant fluids are those defined by n > 1 and they arecommonly called shear-thickening fluids.

q BINGHAM PLASTIC MODEL It is represented by the equation

γµττ &×+= py (6.6)

where τy = yield stress, lbf/100 ft2

µp = plastic viscosity, cp

This is the simplest model describing the behaviour of a special kind of fluidwhich does not flow unless submitted to a minimum stress, i.e., yield stress τy – aphenomenon which is very common in concentrated suspension such as cementslurries. Above the yield stress, the Bingham plastic model assumes that the shearstress is linearly related to the shear rate (see Fig. 6.7). In this case, thecorresponding apparent viscosity decreases from infinity at zero shear rate to theplastic viscosity (µp) at infinite shear rate (Fig. 6.9).



LAMINAR FLOWTURBULENT

FLOW

SHEAR RATE (dv/dr)

SH

EA

R S

RE

SS

(τ)

1

PLASTIC VISCOSITY

APPARENTVISCOSITYBINGHAM

YIELDτy

TRUEYIELDVALUE

PLUGFLOW

µp

µa

Fig. 6.9 - Bingham plastic model.

6.2.4 Laboratory Determination of Rheological PropertiesThe rotational viscometer is used to measure the level of shear stress in the presenceof several pre-selected fixed shear rates. The dial readings and rotational speeds areconverted to shear stress in lbf/ft2 and shear rate in sec-1, respectively.

By plotting, or curve fitting by computer, measured shear stresses against shear rates,one can accurately determine the most appropriate rheological model (Newtonian,power-law or Bingham plastic) for any given case.

Once the appropriate model is determined, the parameters of the fluid (µp and τy forBingham plastic, or n and k for power-law) can be calculated.

The following calculations were developed for field application only. In consequence,they should only be considered relevant where field viscometers are used (Fig. 6.10).

TORSION SPRING(DEFLECTIONRELATED TO

SHEAR STRESS)

BOB(STATIONARY)

ROTOR(VARIABLE RPM)

SLURRY CUP

TEST FLUID(CEMENT SLURRY)

Fig. 6.10 - Determination of rheological properties.



For a power-law fluid:

300

600log32.3n

n

FF

n = (6.7)

nnFf

ftk(5.11) 100

068.1)/(lb.s 3002n' ××= (6.8)

where Fn600 = reading at 600 rpmFn300 = reading at 300 rpmf = spring factor

API measurement of apparent viscosity using an R1-B1 rotor and bob combinationand a spring with a spring factor f = 1, is given by:

rpm6.1)(lbf.s/ft )(2

×== rpmn

a

F

γτµ (6.9)

where Fn(rpm) = reading at a given rpm

For a Bingham plastic fluid:

)( (cp) 300600 nnp FFf −=µ (6.10)

)2(068.1)(lbf/100ft 6003002

nnp FFf −=τ (6.11)

6.3 Flow in Pipes and AnnuliThe drilling engineer frequently deals with the flow of drilling fluids and cementslurries down the circular bore of the drillstring and up the circular annular spacebetween the drillstring and the casing or open hole (Fig. 6.11). If the pump rate is lowenough for the flow to be laminar, the Newtonian, Bingham plastic or power-lawmodel can be employed to develop the mathematical relation between flow rate andfrictional pressure drop.

RELATIVEFLOW RATE

HIGH

MODERATE

LOW

FLOW REGIME

TURBULENT

LAMINAR

PLUG

Fig. 6.11 - Annular velocity profiles for a non-Newtonian fluid.



6.3.1 Pipe Laminar FlowFor steady state flow through a pipe, flow equations can be derived with the followingassumptions (Fig. 6.12).

1. The velocity of the fluid contacting the wall is zero (i.e., there is noslippage).

2. All particles situated within a cylindrical shell of radius r and thickness drare travelling at a constant velocity (v = v(r)) and parallel to the axis of thepipe.

3. The shear rate at any point in the flow is a function only of the shear stressat that point.

rr

τ

τ

L FLOW

P2P1

R

O τdd

Fig. 6.12 - Laminar flow.

The equations for the velocity profile and for the volume flux for laminar flow inpipes are summarized in Appendix A.

It is to be noticed that the velocity profiles for power-law fluids depend only on thepower-law index (n). The lower the n the flatter the velocity profile, whatever theflow rate or pipe diameter, provided the flow regime remains laminar (Fig. 6.13). ForBingham plastic fluids, part of the velocity profile is flat around the pipe axis whilethe rest is a parabola. The normalized velocity profiles are flow-rate dependent. Giventhe pipe diameter or the annular gap, the smaller the average velocity and the plasticviscosity-to-yield stress ratio (µp/τy), the flatter the velocity profile laminar (Fig.6.14). Notice that the dimensionless shear stress ψ (= τy/τw) also represents thefraction of the pipe diameter where the profile is totally flat. This is why thisparameter is sometimes called the plug-to-pipe ratio. The other dimensionlessparameter is the dimensionless shear rate or )/( ypNw τµγξ ×= , where Nwγ is theNewtonian shear rate at the wall which is:

RV

Nw4=γ& (for pipes)

)(6

ioNw RR

V−

=γ& (for narrow concentric annuli)



Nwγ& represents a lower limit for the shear rate at the wall for non-Newtonian fluids,provided they are shear-thinning (which is the case of most cement slurries).

Fig. 6.13 - Normalized velocity and shear rate profiles for a power law fluid flowing in pipe (n = Power Law Index)1.

Fig. 6.14 - Normalized velocity and shear rate profiles for a Bingham plastic fluid flowing in a pipe (? = Dimensionless shear stress, ?= Dimensionless shear rate)1.



6.3.2 Pipe Turbulent Flow

q NEWTONIAN FLUIDSRegardless of the type of fluid, once a critical flow rate in a given pipe isexceeded, streamlines are no longer parallel to the main direction of flow. Fluidparticles become subject to random fluctuations in velocity both in amplitude anddirection (see Fig. 6.15). Such flow instability starts for a given value of adimensionless parameter, the Reynolds number (Re) which, for Newtonianfluids, is defined by:

µρVD=Re (6.12)

where ρ = fluid densityV = average velocity (= flow rate / cross-sectional area = Q / A)D = diameterµ = fluid viscosity

Fig. 6.15 - Laminar and turbulent flow patterns in a circular pipe: (a) Laminar flow,(b) transition between laminar and turbulent flow, and (c) turbulent flow2.

With the Reynolds number being dimensionless, any consistent units may be usedto obtain the same numerical values.

The units used in the oil industry are as follows:

ρ is in lb/galV is in ft/sD is in inchesµ is in centipoise

Using these field units, Reynolds number is given by:

µρVD928Re = (6.13)

Departure from laminar flow occurs as the Reynolds number increases beyond avalue of 2100. A transition regime, which is not very well defined, exists up to Re= 3000. Above this value, flow becomes turbulent. The resistance to flow at thepipe wall is then expressed as:



CfrAfr

+= ]log[Re1 (6.14)

where fr, the Fanning friction factor, is defined by:

2

2V

fr w

ρτ= (6.15)

where τw is shear stress at the wall.

In Eq. (6.14), parameters A and C depend on the roughness of the pipe. Forturbulent flow in smooth pipes, A = 4.0 and C = -0.4.

With these definitions, it should be noticed that, in laminar flow

Re16=fr (6.16)

In the transition regime, the friction factor-Re relationship is not uniquely defined,but for most engineering applications, a linear interpolation is made on a log-logscale between the laminar value of fr at a Re of 2100 and its value at a Re of 3000(Fig. 6.16).

Fig. 6.16 - Relationship between Fanning friction factor and the generalised Reynoldsnumber. Note that, for a given Reynolds number, fr is strongly dependent onthe value of n'. (Dodge and Metzner, 1959)



q NON-NEWTONIAN FLUIDSSimilar equations have been developed for non-Newtonian fluids. The mainproblem here is to determine which viscosity should be used in the expression forthe Reynolds number, because it is shear-rate dependent.

For Bingham plastic fluid, the simplest method consists of assuming that onceturbulent flow is reached, the fluid behaves like a Newtonian fluid with a viscosityequal to its plastic viscosity. This indicates that the relevant Reynolds number inturbulent flow is:

pBG

VDµ

ρ=Re (6.17)

Equation (6.14) is then used to calculate friction pressures for a given flow rate.This assumption has been established empirically for smooth pipes by severalauthors working with different types of fluids. Unfortunately, it does not seem tohole true for all cement slurries.

Dodge and Metzner (1959) proposed a more general approach, which is oftenpreferable. They proposed to generalize Eq. (6.14) to describe the turbulent flowof nonelastic non-Newtonian fluids in smooth pipes:

'2/'1' ]log[Re1n

nMRn CfrA

fr+= − (6.18)

where 'nA and '

nC are a function of n’ only. The generalized Reynolds number,ReMR, is defined by Metzner and Reed (1955) as

'8Re 1'

''2

kDV

n

nn

MR −

−

= ρ(6.19)

For power-law fluids,

n’ = n (6.20)

and

kn

nk

n

+=

413

' (6.21)

For the non-Newtonian fluids Dodge and Metzner (1959) tested, with n’ valuesfrom 0.36 to 1.0, and ReMR values from 2900 to 35,000, they empirically foundthat, for smooth pipes

75.0'

)'(0.4

nAn = and 2.1

'

)'(4.0

nCn

−=



6.3.3 Laminar Flow in Concentric AnnuliFlow equations in narrow concentric annuli can be approximated by using those withapply to fluids in laminar flow through a narrow slot. Equations describing the flow innarrow concentric annuli are given in Appendix A. Qualitatively speaking, the resultsare the same as for pipe flow. Examples of velocity profiles for power-law andBingham plastic fluids are given in Figs. 6.17 and 6.18, respectively.

Fig. 6.17 - Normalized velocity and shear rate profiles for a power law fluid flowing in a slot or narrow annulus (? = Dimensionless shear stress, ?= Dimensionless shear rate)1.



Fig. 6.18 - Normalized velocity and shear rate profiles for a Bingham plastic fluid flowing in a slot or narrow annulus. (? = Dimensionless shear stress, ?= Dimensionless shear rate)1.

For large concentric annuli, flow equations are implicit and they can only be solvednumerically. Since the narrow gap equations are much simpler to solve, the questionthat needs to be addressed is “What are the errors associated with thisapproximation?” This really depends on the application. If one is trying to determinethe flow rate corresponding to a given friction pressure this approximation is not veryaccurate, especially for large gap sizes, as shown in Fig. 6.19 for different power-lawindices. Similar errors are obtained with Bingham plastic fluids.



0.2 0.4 0.6 10.801

1.05

1.10

1.15

1.20

1.25

1.30

Annulus Diameter Ratio (Di / D0 )

Q (A

ctua

l) / Q

(Sho

t) n = 1.0n = 0.5n = 0.2

Fig. 6.19 - Comparison of flow rates at the same friction pressures.

On the other hand, when trying to do the reverse calculation (i.e., determine thefriction pressure corresponding to a given flow rate), even for an annulus diameterratio as low as 0.3 the corresponding error is less than 2.5% for both rheologicalmodels. This is likely to be true for any generalized non-Newtonian model, providedthat the fluid is shear thinning. Therefore, it is reasonable to conclude that the narrowgap approximation is a good engineering approximation to determine laminar frictionpressure of cement slurries in annuli because:

• In most circumstances, annuli are relatively narrow during cementingoperations,

• For the diameter in question, this approximation provides an upper limit forthe friction pressures and

• In practice, friction pressures are often negligible for large-diameter ratios.

6.3.4 Turbulent Flow in Concentric Annuli

q NEWTONIAN FLUIDSFlow pattern theories were originally developed for use in pipe flow. To applypipe-flow theories to annular flow, correction factors must be introduced. In otherwords, we have to determine an equivalent diameter (De) of a circular pipe inwhich the characteristics of pressure loss versus velocity actually duplicate thoseof an annular system.

The oil industry usually adopts the simplest form (Do – Di), which in factcorresponds to the hydraulic diameter of the annulus. Therefore, the Reynoldsnumber expression for a Newtonian fluid become:

µρ )(

Re io DDV −= (6.22)



When the definition of the friction factor remains the same (Eq. (6.15)), thelaminar flow equation for a Newtonian flowing in a narrow concentric annulus isgiven by:

Re24=fr (6.23)

q NON-NEWTONIAN FLUIDSFor this expression to remain valid for non-Newtonian fluids, following Metznerand Reed (1955), one can define the generalized Reynolds number as

'12)(

Re 1'

''2

kDDV

n

nio

n

AN −

− −= ρ(6.24)

for power-law fluids

nn ='

and

kn

nk

n

+=

312

'

The definition of the Reynolds number is quite arbitrary and, therefore, it is notobvious that Eqs. (6.14) and (6.18) can be used to calculate turbulent frictionpressures in annuli. For Newtonian fluids, it seems that turbulent friction factorslie between the curve defined by Eq. (6.14) for low-diameter ratios (Di/Do) andthe curve corresponding to

CfrAfr

+×= ]Re)3/2log[(1 (6.25)

for high-diameter ratios (i.e., for narrow annuli). Therefore, for the sake ofsimplicity, the narrow gap approximation (Eq. (6.25)) can be used for all diameterratios because, as in the case of laminar flow, it gives an upper limit for thefriction factor whatever the diameter ratio is. For non-Newtonian fluids, it appearsreasonable to follow the same approach and to replace Eq. (6.18) by

'2/'1' ]Re)3/2log[(1n

nANn CfrA

fr+×= − (6.26)



6.3.5 Flow in Eccentric AnnuliPipe eccentricity plays a predominant role in mud-circulation and mud-displacementprocesses. The effect of eccentricity on velocity profiles and pressure gradients ofnon-Newtonian fluids in annuli has been studied by many researchers. Since there isno simple analytical solution to such a difficult problem, especially for fluidsexhibiting a yield stress, several simplified approaches have been adopted. In thischapter, a simple model (i.e., basic slot model) is chosen to present the qualitativeeffect of casing eccentricity on circulation efficiency.

Fig. 6.20 - Profile of the slot equivalent to the eccentric annulus. (After Iyoho and Azar, 1981)

In this model, the eccentric annular geometry is considered being equivalent to aseries of independent rectangular slots of varying heights (Fig. 6.20). For a fixedpressure gradient, the contribution from each slot to the flow rate is determined usingequations given in Appendix A. The reverse problem of calculating the frictionpressure knowing the flow rate is then solved numerically. Thus, this model is basedon a narrow annulus approximation where the annular gap varies slowly withazimuthal position; therefore, results will be presented only for a high-diameter ratio(i.e., Di/Do – 0.8).



Fig. 6.21 - Typical example of velocity profile on the narrow andwide sides of eccentric annuli for model fluids1.

The major effect of eccentricity is to distort the velocity distribution around theannulus, the flow favoring the widest part of the annulus to the narrowest part (Fig.6.21). Hence, in an eccentric annulus, the flow regime can vary azimuthally fromwholly laminar to wholly turbulent as illustrated in Fig. 6.22.

Note that eccentricity (ε) is defined as the distance between the axis of the cylindersdivided by the average annular gap (Fig. 6.20). However, in the oil industry, the pipestandoff (STO) is more commonly used, which is defined as STO = (1 - ε) x 100.



Fig. 6.22 - Combined turbulent and laminar flow in eccentric annulus.

REFERENCES

1. “Well Cementing” edited by E.B. Nelson. Published by Elsevier (1990).

2. “Applied Drilling Engineering” by A.T. Bourgoyne Jr., et al. SPE textbook series, Vol. 2(1991).

3. Dodge, D.W. and Metzner, A.B.: “Turbulent Flow of non-Newtonian Systems”, AIChEJ.(June 1959) 5, No. 2, 189-204.

4. Metzner, A.B. and Reed, J.C.: “Flow of non-Newtonian Fluids – Correlations of theLaminar and Turbulent Flow Regions”, AIChEJ. (1955) 1, 434-440.

5. Iyoho, A.W. and Azar, J.J.: “An accurate Slot Flow Model for non-Newtonian FlowThrough Eccentric Annuli”, SPEJ (Oct. 1981) 565-572.



REVIEW QUESTIONS

1. List three reasons why an adequate rheological characterization of cement slurries isnecessary?

2. Distinguish between plug flow, laminar flow and turbulent flow.

3. Define the following terms: rheology, rheological models and thixotropy.

4. Draw the rheograms for Newtonian, power-law and Bingham plastic models.

5. Write down the equation for the power-law model. Explain the major difference betweena power-law fluid and a Newtonian fluid.

6. Define yield stress.

7. Sketch velocity profiles (i.e., plug, laminar and turbulent flows) for a non-Newtonianfluid in an annulus.

8. Assuming the flow regime remains laminar, what effect does lowering of ‘n’ of a power-law fluid has on its velocity profile?

9. Define Reynolds number and what is the critical Reynolds number where flow becomesturbulent?

10. What is the major effect of eccentricity on the velocity profile?

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6-Rheology